Samm Laboratoire

Ea

Thoi-Nhan Ngo

Joël Blot

M Me

Sabine Pickenhain

Rapporteur M Vladimir Veliov

M Georges Zaccour

M Joël Blot

Contrôle optimal en temps discret et en horizon infini

Keywords: Contrôle Optimal, temps discret, horizon infini, principe de Pontryagin, système dynamique

où y ∞ est donné. En utilisant la structure d'espace affine de Banach de l'ensemble des suites convergentes vers y ∞ , et la structure d'espace vectoriel de Banach de l'ensemble des suites bornées, nous traduisons ce problème en un problème d'optimisation statique dans des espaces de Banach. Après avoir établi des résultats originaux sur les opérateurs de Nemytskii sur les espaces de suites et après avoir adapté à notre problème un théorème d'existence de multiplicateurs (au sens de Fritz John et au sens de Karush-Kuhn-Tucker), nous établissons un nouveau principe de Pontryagin faible pour notre problème.

Dans le chapitre 3, nous établissons un principe de Pontryagin fort pour les problèmes considérés au chapitre 2 en utilisant un résultat de Ioffe-Tihomirov. Nous établissons aussi un théorème de conditions suffisantes qui est nouveau, sous des conditions adaptées de concavité.

Le chapitre 4 est consacré aux problèmes de Contrôle Optimal, en temps discret et en horizon infini, generaux avec plusieurs critères différents, sans condition de borne ou de comportement asymptotique sur la variable d'état et la variable de contrôle. La méthode utilisée est celle de la réduction à l'horizon fini, initiée par J. Blot et H. Chebbi en 2000. Les problèmes considérés sont gouvernés par des équations aux différences ou des inéquations aux différences. Un nouveau principe de Pontryagin faible est établi en utilisant un résultat récent de J. Blot sur les multiplicateurs à la Fritz John.

Le chapitre 5 est consacré aux problèmes multicritères de Contrôle Optimal en temps discret et en horizon infini. De nouveaux principes de Pontryagin faibles et forts sont établis, là-aussi en utilisant des résultats récents d'optimisation, sous des hypothèses plus faibles que celles des résultats existants. En corollaires de nouveaux résultats sur les problèmes multicritères, on obtient de nouveaux résultats sur les problèmes avec un seul critère.

Acknowledgement

Realizing PhD at Laboratory SAMM, University of Paris 1 Pantheon-Sorbonne has been one of the most memorable time in my life. I would like to thank all people who contributed in various ways to this dissertation.

First and foremost, I am extremely grateful to my supervisor Professor Joël Blot. It has been a great opportunity for me to be inspired and worked under the supervision of the main contributor on the subject of my thesis. He spent a lot of time taking care of my thesis progress, encouraging me, sharing with me all of his ideas on the subject, and yet not forgetting to give me the freedom to explore on my own. I still remember the first time when I met Professor Blot. He gave me a warm welcome and also 7 of his recent publications on my subject. He always listens patiently to all of my questions and explains many technical details on the subject until I could get the point. Without his help, it would be impossible for me to overcome many difficulties in my research. I have learned so many things from him and his results. I would never forget the day in the spring of 2014 when he spent the whole afternoon to explain all the things that I did not understand about his great book on my subject, which I always have by my side. I am thankful to Professor Blot for introducing me to work with Professor Hayek who is the co-author of that book. I am also deeply grateful to him for spending a lot of time for the preparation of my dissertation's defense. He did a lot more than the things a supervisor does and I appreciate him very much. During three years of my PhD, I am indebted to him for all the things he did for me, for his lessons not only in mathematics but also in philosophy, in writing style and in daily life. Those will go with me for the rest of my life.

My sincerest thanks go to Professor Naila Hayek. She is technically not my supervisor but during more than 8 months working with her, she has been truly an advisor to me, who not only took care a lot of the last chapter of my dissertation but also gave me many great advises for my work, my career and also for the life in Paris. I always admire her productivity of working because although she has so many works to do, she has always scheduled her valuable time for me and given me many treasure discussions which help me so much to write the last part of my thesis. I thank her so much for always encourage me in PhD work and show the great care of my family.

Next, I am truly grateful to Professors Vladimir Veliov and Sabine Pickenhain, the two reviewers of this dissertation. It is my great honor to have my thesis evaluated by these admirable experts in Optimal Control Theory.

I would like to express my gratitude to my dissertation committee, Professors Joël Blot, Naila Hayek, Vladimir Veliov, Sabine Pickenhain, Georges Zaccour, Marie Cottrell, Jean-Bernard Baillon and Bruno Nazaret. I am very honored by their presence and appraisement.

I would like to give my special thanks to Professor Le Van Cuong. Without his introduction to Professor Blot, I could not have been here in Paris to realize a PhD. In the last 3 years, he has always cared for my daily life and my progress in PhD work. For me, he is like a special teacher and in some aspects, like a father. Besides my supervisor, he has always been the first one who shares the happiness with me when I have a published paper. His apartment is the place where our group of Vietnamese scientists (mainly economists) usually gathers to share the things in life, to discuss the scientific problems, to help each other and sometimes just to celebrate a festival or a holiday. Those are the unforgettable memories with me.

I am grateful to VIED and Campus France, who provide me the financial support during the last 3 years.

Many thanks to all the members of SAMM for providing a friendly and academic atmosphere to work in.

I would like to thank Professor Jean-Marc Bardet firstly for his invitation letter that helped me go joining SAMM as a PhD student and for his great work as a director that I was benefited a lot during my time in SAMM. I always feel in debt to him for his care of my working condition and my French language.

I am grateful to Professor Marie Cottrell for always being nice to me, especially during the ICOR 2016 in Cuba. Marie, you are the best Havana's tour guide. I will miss your handmade gastronomies at each SAMM's mid-year festival.

I would like to say thank you to my "sister" Diem, who is the secretary of SAMM, for her administrative assistance and her care of my work, my family. It is my luck to begin my PhD the same time with her job in SAMM.

Thank you, Dhaou, Julien and Anthikos, the ones who helped me when I first come to SAMM, especially Dhaou, who shared many experiences in PhD life with me. Thank you, Cynthia, Tsirizo "Zizou", Ibra, Bienvenue, Alexis, Fania, Jihed, Marco, Arnaud, Rawya, Clara and Latif for always carefully listening and talking to me, especially Cynthia, Tsirizo and Ibra who have listened "a lot" and always been patient with me. Without you guys, my PhD time would be very lonely.

Thank you, Professor Dominique Haughton, (Cam on Co Dominique rat nhieu) for bringing me many surprises, for speaking Vietnamese with me and inviting me to have Pho at my first day at SAMM.

Thank you, Professor Jean-Bernard Baillon, for giving me many interesting pieces of advice and for your great sense of humor.

Many thanks to French and Vietnamese friends and relatives for your help, your encouragement. You have made my PhD life more colorful and interesting.

I have been very happy to meet and get acquainted with many nice people in the last three years. They helped me to make a lot of beautiful memories to treasure forever. I am grateful to my "brother and sister", Ngoc Sang and Hoang Quynh, who provided me a place to live in my first 2 weeks in France. Moreover, they always made me have a feeling of family. For the same reason, I sincerely thank Cao Tung and Thu Trang and their son, Tuntin, especially Tung for his great help for my daily life in Paris. Thank you very much, Ha-Huy Thai, my "big brother", for always being interested in my subject and providing me many great materials. Thank you, my "brother", Thanh Tam for your great enthusiasm and care. Thank you, Tran Nhat Thien and Tran Viet Cuong. Thank you, Bui Quang Vu. Thank you, "boss" Nguyen Duc Khuong for helping me in some administrative procedures and sharing many great talks. Thank you, Nicolas, my French friend since 2012. I will remember your great French repas and your interesting talk about traveling in Asian countries. I also thank my best friends and colleagues in Vietnam for their encouragement: Huu Tho, Nhat Trung, Vinh, Quang Tam, Thanh Tam, Truc, Khanh, Khiem, Hoa Cuong, Thanh Tuan, Thai Hoa, Minh Tan, Khanh Linh, Hai Phuong, Thay Hung, Thay Xung.

My special thanks go to my cousin Quang Binh and his parents, my uncles Cheo-Thu, who always care about my daily life and my progress. I would like to thank my friends from Residence Universitaire Francis de Croisset, where I have been living during last 3 years. Thank you, Thuy, Van, Nghi, Hai Minh, Ly Hung, Hien and Hoa for all the nice small parties and nice time we had together. Thank you, "brother" Tran Quang Hoa for sharing his room with me in my last 3 months. Thank you, Marcelle, for always being nice to me.

The family is the greatest motivation. Last but not least, from the bottom of my heart, I want to thank my whole family, firstly, my parents for their limitless love, their constant encouragement to me and their care of my wife and my daughter. I am deeply indebted to my wife who spent 3 years raising my little child without me. Finally, thank you, my little girl Susu, for bringing me so many smiles and happiness that motivate me to go to the end of this way to come back to you. This thesis is dedicated to you, my family.

Résumé Résumé

Une première théorie consacrée à l'optimisation des problèmes dynamiques est le Calcul des Variations. Les mathématiciens qui ont créé cette théorie sont: Jean BERNOULLI (1667-1748), Leonhard EULER (1707-1793), Joseph Louis LAGRANGE (1736-1813), Adrien LEGENDRE (1752-1833), Carl JACOBI (1804-1851), William HAMILTON (1805-1865), Karl WEIERSTRASS (1815-1897), Adolph MAYER (1819-1904) et Oscar BOLZA . Dans les problèmes de Calcul des Variations, l'observateur n'intervient pas sur le problème. Aujourd'hui, le Calcul des Variations est encore un champ de recherche très actif.

Pour répondre à des questions technologiques, issues de diverses industries, une seconde théorie, consacrée à l'optimisation des problèmes dynamiques, naît au milieu du vingtième siècle: la théorie du Contrôle Optimal. Dans cette théorie, l'observateur agit sur le problème. Outre la "variable d'état" qui décrit le comportement du système dynamique, il y a une "variable de contrôle" qui est pilotée par l'observateur.

Historiquement, il y a deux grands points de vue en théorie du Contrôle Optimal: le point de vue de Lev Pontryagin (Principe du Maximum) et le point de vue de Richard Bellman (Programmation Dynamique).

Le premier cadre qui fut utilisé en Contrôle Optimal est celui du temps continu en horizon fini. Plus tard le cadre du temps discret fut aussi étudié. Le développement de l'utilisation des ordinateurs pour faire des calculs approchés ou des simulations constitue une motivation supplémentaire pour étudier le cadre du temps discret en Contrôle Optimal.

À propos de l'horizon infini en Calcul des Variations et en Contrôle Optimal, des motivations importantes et historiques viennent de la théorie économique: les travaux de F. P. [START_REF] Ramsey | A Mathematical Theory of Saving[END_REF], de H. [START_REF] Hotelling | The Economics of Exhaustive Resources[END_REF], de C. C. von Weizsacker (1965). Plus tard, des modèles de développement durable, par exemple la gestion optimale des forêts et des pêcheries, ont été traités par la théorie du Contrôle Optimal en horizon infini.

Cette thèse contient des contributions originales à la théorie du Contrôle Optimal en temps discret et en horizon infini du point de vue de Pontryagin. Le point de vue de Pontryagin fournit des conditions nécessaires d'optimalité. De telles conditions ont du sens dans les problèmes considérés; ce sont des lois de comportement. De plus, l'utilisation de telles conditions nécessaires d'optimalité peut permettre d'améliorer la modélisation du phénomène étudié, par exemple en montrant que les seuls candidats possibles à l'optimalité sont inadaptés au problème. Dans certain cas, il est possible d'établir des théorèmes de condition suffisante dans le point de vue de Pontryagin. Nous décrivons maintenant le contenu de cette thèse. Dans le chapitre 1, nous rappelons des résultats sur les espaces de suites à valeur dans R k . Nous rappelons aussi des résultats de Calcul Différentiel: sur la dérivabilité directionnelle, sur la Gâteaux différentiabilité, sur la Fréchet-différentiabilité et sur la stricte différentiabilité.

Dans le chapitre 2, nous étudions le problème Maximiser K(y, u) := � +∞ t=0 β t ψ(y t , u t ) quand y = (y t ) t∈N ∈ (R n ) N , u = (u t ) t∈N ∈ U N y 0 = η, lim t→+∞ y t = y ∞ , u est bornée, ∀t ∈ N, y t+1 = g(y t , u t ),

            

Introduction

A first theory which is devoted to the optimization of dynamical problems is the theory of Calculus of Variations, a field of mathematical analysis that deals with maximizing or minimizing functionals. The first mathematicians who gave contribution to the theory of Calculus of Variations are Johann Bernoulli (1667-1748), Leonhard Euler (1707-1793), Joseph-Louis Lagrange (1736-1813), Andrien Legendre (1752-1833), Carl Jacobi (1804-1851), William Hamilton (1805-1865), Karl Weierstrass (1815-1897), Adolph Mayer (1839-1907), and Oskar Bolza . Today it is ever an important field of research. In theory of Calculus of Variations, the researcher does not take action on the considered problem but to play the role of an observer. He only acknowledges the behaviour of system and understands it through the observations.

Along with the development of technology, there was a need to answer the technological problem of finding a control law for a given system such that a certain optimality criterion is achieved. Then a second theory devoted to the optimization of dynamical problems was born in the middle of twentieth century: the Optimal Control Theory, a mathematical optimization method for deriving control policies. Beside the so-called "state variable" which represents the behaviour of the dynamical system, there is also a so-called "control variable", which is chosen by the researcher over the time. Now not only that the researcher observes, but he also takes action on the dynamical system over the time and plays the role of a controller. By this theory, studying the optimization of a dynamical problems becomes more interactive.

Historically, there exist two great methods in Optimal Control Theory, which are due to the work of Lev Pontryagin (1908Pontryagin ( -1988) ) and Richard Bellman in the 1950s: The minimum (maximum) principle of Pontryagin and the dynamic programming of Bellman.

Optimal Control Theory has found applications in many different fields of science, including aerospace, process control, robotics, bioengineering, economics, finance, and management science, and it continues to be an active research area within control theory. The first framework which was used in Optimal Control Theory is the continuous-time framework. Later, the discrete-time framework was also studied. The reason is that while theory of differential equations in continuous-time models is not well known by all the scientists, except for mathematicians and physicists, the equations of a discretetime dynamical system do not require sophisticated mathematical tools. Thus, discretetime models can simplify the communication between mathematicians and the researchers of other scientific fields. Besides, studying the same phenomenon using both discretetime model and continuous-time model can lead to a comparison between their respective results and can provide interesting consequences. Moreover, the development of the use of electronic computers to calculate approximations or to realize simulations of the optimal solutions is an additional motivation to study the discrete-time framework in Optimal Control Theory. In fact, contemporary control theory is now primarily concerned with discrete-time systems and their solutions.

Introduction

A historical motivation for infinite-horizon variational problems and infinite-horizon optimal control problems is found in the macroeconomic optimal growth theory in the works of Ramsey [START_REF] Ramsey | A Mathematical Theory of Saving[END_REF], Hotelling [START_REF] Hotelling | The Economics of Exhaustive Resources[END_REF], von Weizsäcker [START_REF] Weizsäcker | Existence of Optimal Programs of Accumulation for an Infinite Time Horizon[END_REF] and Brock [START_REF] Brock | On Existence of Weakly Maximal Programmes in a Multi-sector Economy[END_REF]. In such a theory, an agent represents itself and all its progeny, and the infinite horizon avoids to deal with the problems of the end of the world. Another important field of knowledge which uses the infinite-horizon optimal control is the management of natural resources as forests and fisheries, which are introduced in [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF]. More generally, the study of some aspect of sustainable development naturally leads to a framework where a final time does not exist.

Those are the reasons for the author to choose infinite-horizon discrete-time optimal control problems to be the studied object of the thesis.

In finite-horizon continuous-time optimal control theory, there exists two main historical approaches: Pontryagin's approach and Bellman's approach. In infinite-horizon discrete-time optimal control problems, the dynamic programming of Bellman is currently used. In this thesis, we follow the other approach: the viewpoint of Pontryagin.

Pontryagin's viewpoint provides necessary conditions of optimality which are principles that the optimal solutions ought to satisfy, and these principles possess a meaning in the considered phenomenon. Moreover, the role of necessary conditions of optimality is to narrow the set of all processes which are candidates to be solutions of the problem, and this can also improve the modeling. In some cases, it is also possible to formulate sufficient conditions of optimality in the spirit of the conditions initiated by Seierstad and Sydsaeter for the continuous-time problems. During the process of establishing Pontryagin principles for discrete-time optimal control problems in infinite horizon, in some cases, we can also establish those for such problems in finite horizon.

The structure of this thesis is as follows. In Chapter 1, firstly, we recall preliminary basis of sequence spaces. In this part, we introduce some classical sequence spaces in (R k ) N . Then we study their norms, their dualities and their completeness. In second part, we recall some basic results on differential calculus in normed linear space, particularly, the various types of differential in normed linear space, their properties, the Mean Value Theorem and the differential in product space. Those are the fundamental mathematical tools that we use throughout the thesis.

Chapters 2-5 are the main chapters of this thesis which contain new results on Pontryagin principles for infinite-horizon discrete-time optimal control problems. There are several ways to establish Pontryagin principles for our considered problems. The first method is to translate the original problem into an optimization problem that defined in Banach spaces; then use an appropriate multiplier rule in Banach spaces to obtain Pontryagin princiles. This method is direct and it requires the considered problems to have the capability of being translated into an optimization problem in Banach spaces. It is used in Chapter 2 and Chapter 3. The second method, which is first proposed by Blot and Chebbi in 2000, is to reduce the infinite-horizon problems into families of finite-horizon problems; then use an appropriate multiplier rule to obtain Pontryagin principle for the finite-horizon problem; and finally, extend that result to the infinite-horizon case by using some additional assumptions. Chapter 4 and Chapter 5 follow the second method.

In Chapter 2, we study the following problem:

            
Maximize K(y, u) := now the problem contains an asymptotic constraint at infinity on the state variable. We will use an approach of functional analytic for this problem after translating it into the form of an optimization problem in Banach (sequence) spaces. Then a weak Pontyagin principle is established for this problem by using a classical multiplier rule in Banach spaces. Some new properties of Nemytskii operators are also studied in this chapter.

In Chapter 3, we establish a strong Pontryagin principle and a sufficient condition for the considered problems in Chapter 2. To obtain the strong principle, we use the non linear functional analytic approach as in Chapter 2 and apply a multiplier rule of Ioffe and Tihomirov in which a convexity condition is necessary. Sufficient condition is obtained by using the weak Pontryagin principle's conclusions as assumptions and an assumption of concavity of the Hamiltonian.

In Chapter 4, we study the infinite-horizon discrete-time single-objective optimal control problem, which is more general than the problem considered in Chapter 2 and Chapter 3. For such problems, we consider the dynamical system with difference equation and with difference inequation and in the presence of the constraints on optimal control at each period of time. There exists several results on Pontryagin principles for such problems which are established in Blot and Hayek [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF]. However, some require the Lipschitzian conditions to use Clarke's calculus, while others require the smoothness or at least, the Fréchet differentiability and the continuity on a neighborhood of the optimal solution of the functions, which are present in the problem. The aim of this chapter is to establish Pontryagin maximum principle under the weak form for such problems using lighter assumptions than the usual ones by applying recent result of Blot on the multiplier rule in [START_REF] Blot | On the Multiplier Rules[END_REF].

In Chapter 5, necessary conditions of Pareto optimality under the form of Pontryagin principles for finite-horizon and infinite-horizon multiobjective optimal control problems in discrete-time framework are studied. The considered problems in this chapter are similar to the ones in Chapter 4 but with multicriteria objective function. The aim of this chapter is to establish weak and strong maximum principles of Pontryagin for problems in the presence of constraints and under assumptions which are weaker than the usual ones. In this way, this chapter generalizes existing results for single-objective optimal control problems and for multiobjective optimal control problems with or without constraints. To establish weak principles of Pontryagin, we provide new multiplier rules for static multiobjective optimization problems, which are in the spirit of the multiplier rule for static single-objective optimization problems of Blot in [START_REF] Blot | On the Multiplier Rules[END_REF]. The strong principles of Pontryagin are established relied on the multiplier rule of Khanh and Nuong in [START_REF] Khanh | On Necessary Optimality Conditions in Vector Optimization Problems[END_REF]. Sufficient conditions of optimality for the considered problems are also provided in the end of this chapter by using the weighting method.

Chapter 1

Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces

Sequence Spaces

In this section we provide elements on sequence spaces, essentially on the space � ∞ (N, R k ) of the bounded sequences in R k and the space c 0 (N, R k ) of bounded sequences in R k which converge to 0. Firstly, we define our notation and we recall some basic facts. Then we provide some basic analyses on sequence spaces in R k and recall the results on the dual spaces of c 0 (N, R k ) and � ∞ (N, R k ) which are useful in the establishment of Pontryagin principles for optimal control problems with processes from these spaces. Finally we prove the completeness of the classical sequence spaces in R k .

The main references that we use on the sequence spaces are Chapter 16 in Aliprantis and Border [START_REF] Aliprantis | Infinite-dimensional Analysis[END_REF], Section 31 in Köthe [START_REF] Köthe | Topological Vector Spaces[END_REF] and Appendix A in Blot and Hayek [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF]. -R is the set of all real numbers.

Notation and Recall

-When k ∈ N * , R k is the space of all k-dimensional real vectors. If v ∈ R k then v = (v 1 , v 2 , . . . , v k ) where v i ∈ R, i ∈ {1, . . . , k}. The canonical basis of R k is denoted by (e j ) 1≤j≤k where e i j := δ i j for all i, j ∈ {1, . . . , k}. (δ i j is the Kronecker symbol).

-R k * denotes the dual space of R k which is the space of all linear functionals from R k into R.

-Let E be a set. Then E N denotes the set of all sequences in E. For E N , x = (x t ) t∈N denotes its element. Here, x t ∈ E for each t ∈ N. -Let M be a finite set. |M | denotes the number of elements of M .

Definition 1.2. (Norm and normed space)

Given a vector space E over the field R. A norm on E is a map �•� : E → R that satisfies the following conditions: The pair (E, �•�) is then called a normed space.

Definition 1.3. (Banach space)

A Banach space is a vector space E over the field R which is equipped with a norm �•� and which is complete with respect to that norm. That is to say, for every Cauchy sequence (x n ) n∈N in E, there exists an element x in E such that When E = R k , (E, |•|) is a finite-dimensional normed real vector space, we consider E N , the set of all sequences in E. This set can be turned into a vector space by defining vector addition as follows (x t ) t∈N + (y t ) t∈N := (x t + y t ) t∈N for all x, y ∈ E N, and the scalar multiplication as follows α(x t ) t∈N := (αx t ) t∈N for all x ∈ E N .

There are some basic sequence spaces which are subspaces of E N and are defined as follows: -Space c 00 (N, E) : c 00 (N, E) := {x ∈ E N : x t = 0 except for finitely many indexes t}. It is a subspace of � p (N, E) for all p ≥ 1.

By direct verification, it is evident that �•� ∞ satisfies all conditions (N1), (N2), (N3) of the definition of norm. Hence, � ∞ (N, E), c(N, E) and c 0 (N, E) are normed spaces. We will show that �•� p also satisfies the definition of a norm in next subsection. The completeness of space � p (N, E), � ∞ (N, E), c(N, E) and c 0 (N, E) will be proven in the end of this section.

For the above-mentioned sequence spaces, we have the following theorem:

Theorem 1.6. For all p, q ∈ [1, +∞) such that p ≤ q, the following inclusions hold:

c 00 (N, E) ⊂ � p (N, E) ⊂ � q (N, E) ⊂ c 0 (N, E) ⊂ c(N, E) ⊂ � ∞ (N, E).
Proof. We will prove the above-mentioned inclusions by the inverse order.

-Prove c(N, E) ⊂ � ∞ (N, E)

: let x = (x t ) t∈N ∈ c(N, E) then (x t ) t∈N is convergent. A convergent sequence is clearly bounded. So x ∈ � ∞ (N, E). -Prove c 0 (N, E) ⊂ c(N, E) : it is clear from the definition of these two spaces. -Prove � q (N, E) ⊂ c 0 (N, E) with q ≥ 1 : let x = (x t ) t∈N ∈ � q (N, E) then � t∈N |x t | q < +∞.
From this we deduce that lim t→+∞ |x t | q = 0 and hence lim t→+∞ x t = 0. And so,

x ∈ c 0 (N, E). -Prove � p (N, E) ⊂ � q (N, E) for all p, q ∈ [1, +∞) such that p ≤ q: Let x = (x t ) t∈N ∈ � p (N, E).
From the above-mentioned inclusion, we know that lim t→+∞ x t = 0. Then there exists T ∈ N * big enough such that when t > T we have |x t | < 1. Now we consider the series

� t∈N |x t | q = T � t=0 |x t | q + ∞ � t=T +1 |x t | q ≤ T � t=0 |x t | q + ∞ � t=T +1 |x t | p ≤ T � t=0 |x t | q + � t∈N |x t | p .
It is obvious that

� T t=0 |x t | q < +∞ as it is a finite sum of positive numbers. Besides, � t∈N |x t | p < +∞ since x = (x t ) t∈N ∈ � p (N, E). Hence, � t∈N |x t | q < +∞ which means that x ∈ � q (N, E). -Prove c 00 (N, E) ⊂ � p (N, E) : Let x = (x t ) t∈N ∈ c 00 (N, E). We denote N Z = {t ∈ N : x t � = 0}.
From the definition of space c 00 (N, E) we know that |N Z| < +∞. Now, consider the series

� t∈N |x t | p = � t∈N Z |x t | p < +∞.
Then x ∈ � p (N, E).

Some Basic Analyses on Sequence Spaces

In this subsection, we will provide some basic analyses on sequence spaces. For simplicity, we set by default E = R k where k ∈ N * . A map f : R m → R is called convex if for all x, y ∈ R m and for all α ∈ [0, 1] we have:

f (x + α(y -x)) ≤ f (x) + α(f (y) -f (x)).
For the convexity of real function we have the following theorem: Proposition 1.10. (Hölder inequality) Let p, q ∈ (1, ∞) with 1 p + 1 q = 1. Then for x ∈ � p (N, E) and y ∈ � q (N, E * ) one has

� t∈N |�y t , x t �| ≤ �x� p �y� q .
Proof. For a, b > 0, set A := p ln a, B := q ln b. The exponential function e x is convex, thus:

e A p + B q = e B+ 1 p (A-B) ≤ e B + 1
p (e A -e B ) = 1 p e A + 1 q e B .
Then after substituting A = p ln a and B = q ln b, we have

ab ≤ 1 p a p + 1 q b q .
It is obvious that when x = 0 or y = 0, the Hölder inequality holds. Now, for x = (x t ) t∈N ∈ � p (N, E) and y = (y t ) t∈N ∈ � q (N, E * ) such that �x� p = 1 = �y� q , from the above-mentioned result one has

∀t ∈ N, |�y t , x t �| ≤ |x t | |y t | * ≤ 1 p |x t | p + 1 q |y t | q * ⇒ � t∈N |�y t , x t �| ≤ 1 p � t∈N |x t | p + 1 q � t∈N |y t | q * = 1 p + 1 q = 1.
For x ∈ � p (N, E), x � = 0 and y ∈ � q (N, E * ), y � = 0, using the above-mentioned result for x � := x �x� p and y � := y �y� q we come to the following inequality:

� t∈N � � � y � t , x � t �� � ≤ 1 ⇔ � t∈N � � � � � � y t �y� q , x t �x� p �� � � � � ≤ 1 ⇔ 1 �x� p �y� q � t∈N |�y t , x t �| ≤ 1.
Multiply both sides of the last inequality with �x� p �y� q we obtain the result of this proposition.

Sequence Spaces

Lemma 1.11. (Supremum formula)

Let p, q ∈ (1, +∞) with 1 p + 1 q = 1. Then for all x ∈ E N we have

�x� p = sup � | � t∈N �y t , x t �| : y ∈ c 00 (N, E * ), �y� q ≤ 1 � , whereas, equality is meant in [0, +∞].
Proof. Let x ∈ E N and let p, q ∈ (1, +∞) with 1 p + 1 q = 1. We denote

C = sup � | � t∈N �y t , x t �| : y ∈ c 00 (N, E * ), �y� q ≤ 1 � .
-Prove that �x� p ≥ C : If �x� p = +∞ then �x� p ≥ C is true. Now we assume that �x� p < +∞. Following Hölder inequality:

| � t∈N �y t , x t �| ≤ � t∈N |�y t , x t �| ≤ �x� p for all y ∈ � q (N, E * ) (including y ∈ c 00 (N, E * ) ) with �y� q ≤ 1. And so, C ≤ �x� p . -Prove that �x� p ≤ C :
It is obviously that when �x� p = 0 we have C = 0. So the inequality is satisfied for this case. If

�x� p � = 0 then x � = 0. Choose λ = (λ t ) t∈N ∈ (E * ) N such that |λ t | * = 1 and �λ t , x t � = |x t | for all t ∈ N.
For sufficient big N ∈ N, we know that

A := � � N t=0 |x t | p
� -1/q exists and A ≥ 0. Define y = (y t ) t∈N ∈ c 00 (N, E * ) as follows:

y t := A |x t |
p q λ t for 0 ≤ t ≤ N and y t := 0 for t > N . Then we have:

�y� q = � � N t=0 � � �Aλ t |x t | p q � � � q * � 1 q = A � � N t=0 |x t | p |λ t | q * � 1 q |λt| * =1 = A � � N t=0 |x t | p � 1 q = 1.
Now with this y ∈ c 00 (N, E * ) we have

C ≥ � � � � N t=0 �y t , x t � � � � = � � � � N t=0 � A |x t | p q λ t , x t �� � � = A � � � � N t=0 |x t | p q �λ t , x t � � � � �λt,xt�=|xt| = A � N t=0 |x t | 1+ p q = A � N t=0 |x t | p = A � N t=0 |x t | p = � � N t=0 |x t | p � 1-1 q = � � N t=0 |x t | p � 1 p .
The last inequality:

C ≥ � � N t=0 |x t | p � 1
p is satisfied for all N ∈ N, hence C ≥ �x� p . From the above-mentioned arguments, we have �x� p = C. Proposition 1.12. For 1 ≤ p < +∞, � p (N, E) is a normed sequence space.

Proof. We will prove this statement for the case p = 1 and for the case p ∈ (1, +∞) individually.

-Case p = 1:

We have � Obviously, �•� p satisfies properties (N1) and (N2). Now let q = p p-1 , thus 1 p + 1 q = 1. Then, for any x, y ∈ � p (N, E) and for all z ∈ E N (including all z ∈ c 00 (N, E * )) such that �z� q ≤ 1 the following triangle inequality takes place:

∀N ∈ N, � � � � N t=0 �z t , x t + y t � � � � ≤ � � � � N t=0 �z t , x t � � � � + � � � � N t=0 �z t , y t � � � � .
Let N → +∞ and using supremum formula, we have:

| � t∈N �z t , x t + y t �| ≤ | � t∈N �z t , x t �| + | � t∈N �z t , y t �| ≤ �x� p + �y� p . ⇒ | � t∈N �z t , x t + y t �| ≤ �x� p + �y� p
Take supremum both sides of the last inequality and using the supremum formula we obtain the property (N2) of �•� p :

�x + y� p ≤ �x� p + �y� p . Hence � p (N, E) is a normed space. Lemma 1.13. (Convergence of norm) Let x ∈ � p (N, E) where p ∈ [1, +∞) . Then �x� p p→+∞ -→ �x� ∞ . Proof. Since |x k | ≤   � t∈N |x t | p   1 p
for all k ∈ N and p ≥ 1. So we have �x� ∞ ≤ �x� p . Thus, in particular

�x� ∞ ≤ lim inf p→+∞ �x� p .
On the other hand, we know that

�x� p =   � t∈N |x t | p-q • |x t | q   1 p ≤ �x� p-q p ∞ •   � t∈N |x t | q   1 p = �x� 1-q p ∞ • �x� q p q ,
for all q < p where we used |x t | ≤ �x� ∞ for all t ∈ N. Therefore, we arrive at lim sup

p→+∞ �x� p ≤ lim sup p→+∞ � �x� 1-q p ∞ • �x� q p q � = �x� ∞ • 1.
We conclude lim sup

p→+∞ �x� p ≤ �x� ∞ ≤ lim inf p→+∞ �x� p .
This shows that lim p→+∞ �x� p exists and equals �x� ∞ .

Corollary 1.14. (Extended supremum formula)

The supremum formula is also true for p = ∞ and p = 1:

�x� ∞ = sup � | � t∈N �y t , x t �| : y ∈ c 00 (N, E * ), �y� 1 ≤ 1 � and �x� 1 = sup � | � t∈N �y t , x t �| : y ∈ c 00 (N, E * ), �y� ∞ ≤ 1 � .
Proof. In supremum formula we let p → +∞ with q = p p-1 and q → +∞ with p = q q-1 , respectively. When P and Q are sequence spaces, Q is called the dual space of P if (D1) For each y ∈ Q the series � t∈N �y t , x t � =: y(x) is convergent for all x ∈ P and defines an element y(.) in P * , with �y(.)� P * = �y� Q ;

(D2) For each η(.) ∈ P * , an element y ∈ Q exists with y(.) = η(.).

and we can write P * = Q.

So the notation P * = Q means that the map y � -→ y(.) from Q into P * is an isometric isomorphism between Q and P * .

We also recall that norm of a bounded linear operator is defined as follows.

�y(.)� P * = sup{|y(x)| : x ∈ P, �x� P ≤ 1}.

Proposition 1.16. Let p, q ∈ (1, +∞) such that 1 p + 1 q = 1, then � p (N, E) * = � q (N, E * ). Furthermore, c 0 (N, E) * = � 1 (N, E * ) and � 1 (N, E) * = � ∞ (N, E * ).
Proof. Let p, q ∈ (1, +∞) and 1 p + 1 q = 1. For simplicity, in this proof we denote the norm on dual space by �•� * .

-Prove � p (N, E) * = � q (N, E * ): -(D1) For each y ∈ � q (N, E * ) the series � t∈N �y t , x t � =: y(x) is absolutely convergent (from Hölder inequality) for all x ∈ � p (N, E) and hence it defines an element y(.) in � p (N, E) * . Using Hölder inequality we have:

|y(x)| = | � t∈N �y t , x t �| ≤ �x� p �y� q .
Therefore, from the definition of norm of linear operator, we obtain �y(.)� * ≤ �y� q . Now, using supremum formula with interchange x and y, p and q, we have:

�y� q = sup � | � t∈N �x t , y t �| : x ∈ c 00 (N, (E * ) * ), �x� p ≤ 1 � E=R k = sup � | � t∈N �x t , y t �| : x ∈ c 00 (N, E), �x� p ≤ 1 � ≤ sup � | � t∈N �x t , y t �| : x ∈ � p (N, E), �x� p ≤ 1 � = sup � |y(x)| : x ∈ � p (N, E), �x� p ≤ 1 � = �y(.)� * .
Then, we have proven �y(.)

� * = �y� q . -(D2) Let η(.) ∈ � p (N, E) * . If y t : E → R defined by y t (x t ) =
η(0, 0, . . . , x t , 0, 0, . . . ) for all t ∈ N then y t ∈ E * and y = (y t ) t∈N ∈ (E * ) N . Now, for all x ∈ � p (N, E) we have

η(x) = η(x 1 , x 2 , . . . ) = � t∈N η(0, 0, . . . , x t , 0, 0, . . . ) = � t∈N y t (x t ) = � t∈N �y t , x t � .
From here, using supremum formula as before we obtain �y� q ≤ �η(.)� * < +∞ hence y ∈ � q (N, E * ). Therefore, if from y we define y(.) ∈ � p (N, E) * by setting y(x) := � t∈N �y t , x t � then η(.) = y(.). And so (D2) is satisfied. From the above-mentioned arguments, we can conclude that � p (N, E) * = � q (N, E).

-Prove c 0 (N, E) * = � 1 (N, E * ):

N →+∞ � N t=0 |�y t , x t �| ≤ lim N →+∞ � N t=0 |y t | * |x t | ≤ sup t∈N |x t | lim N →+∞ � N t=0 |y t | * = �x� ∞ . � t∈N |y t | * = �x� ∞ .�y� 1 < +∞.
And so, for each y ∈ � 1 (N, E * ) the series � t∈N �y t , x t � =: y(x) is absolutely convergent for all x ∈ c 0 (N, E) and hence, we can define an element y(.) ∈ c 0 (N, E) * . Moreover, from the previous inequality we have �y(.)� * ≤ �y� 1 . Now, using the extended supremum formula

�y� 1 = sup {| � t∈N �y t , x t �| : x ∈ c 00 (N, E), �x� ∞ ≤ 1} ≤ sup {| � t∈N �y t , x t �| : x ∈ c 0 (N, E), �x� ∞ ≤ 1} = sup {|y(x)| : x ∈ c 0 (N, R), �x� ∞ ≤ 1} = �y(.)� * .
Then, we have proven �y(.)� * = �y� 1 . And so condition (D1) is satisfied.

-(D2) Let η(.) ∈ c 0 (N, E) * . If y t : E → R defined by y t (x t ) =
η(0, 0, . . . , x t , 0, 0, . . . ) for all t ∈ N then y t ∈ E * and y = (y t ) t∈N ∈ (E * ) N . Now, for all x ∈ c 0 (N, E) we have

η(x) = η(x 1 , x 2 , . . . ) = � t∈N η(0, 0, . . . , x t , 0, 0, . . . ) = � t∈N y t (x t ) = � t∈N �y t , x t � .
From here, using supremum formula as before we obtain �y� 1 ≤ �η(.)� * < +∞ hence y ∈ � 1 (N, E * ). Therefore, if from y we define y(.) ∈ � 1 (N, E) * by setting y(x) := � t∈N �y t , x t � then η(.) = y(.). And so (D2) is satisfied. From the above-mentioned arguments, we can conclude that c 0 (N,

E) * = � 1 (N, E * ). -Prove � 1 (N, E) * = � ∞ (N, E * ): -(D1) For each y ∈ � ∞ (N, E * ), we consider series � t∈N �y t , x t � where x = (x t ) t∈N ∈ � 1 (N, E) arbitrary. We have | � t∈N �y t , x t �| ≤ � t∈N |�y t , x t �| = lim N →+∞ � N t=0 |�y t , x t �| ≤ lim N →+∞ � N t=0 |y t | * |x t | ≤ sup k∈N |y k | * lim N →+∞ � N t=0 |x t | = �y� ∞ . � t∈N |x t | = �y� ∞ �x� 1 < +∞.
And so, for each y ∈ � ∞ (N, E * ) the series � t∈N �y t , x t � =: y(x) is absolutely convergent for all x ∈ � 1 (N, E) and hence, we can define an element y(.) ∈ c 0 (N, E) * . Moreover, from the previous inequality we have �y(.)� * ≤ �y� ∞ . Now, using the extended supremum formula, we have

�y� ∞ = sup {| � t∈N �y t , x t �| : x ∈ c 00 (N, E), �x� 1 ≤ 1} ≤ sup � | � t∈N �y t , x t �| : x ∈ � 1 (N, E), �x� 1 ≤ 1 � = sup � |y(x)| : x ∈ � 1 (N, E), �x� 1 ≤ 1 � = �y(.)� * .

Sequence Spaces

Then, we have proven �y(.)� * = �y� 1 . And so condition (D1) is satisfied.

-(D2) Let η(.) ∈ � 1 (N, E) * . If y t : E → R defined by y t (x t ) =
η(0, 0, . . . , x t , 0, 0, . . . ) for all t ∈ N then y t ∈ E * and hence, y = (y t ) t∈N ∈ (E * ) N . Now, for all x ∈ � 1 (N, E) we have

η(x) = η(x 0 , x 1 , . . . ) = � t∈N η(0, 0, . . . , x t , 0, 0, . . . ) = � t∈N y t (x t ) = � t∈N �y t , x t � .
From here, using supremum formula as before we obtain �y� ∞ ≤ �η(.)� * < +∞ hence y ∈ � 1 (N, E * ). Therefore, if from y we define y(.) ∈ � 1 (N, E) * by setting y(x) := � t∈N �y t , x t � then η(.) = y(.). And so (D2) is satisfied. From the above-mentioned arguments, we can conclude that

� 1 (N, E) * = � ∞ (N, E * ).
The dual space of � ∞ (N, R k ):

In [START_REF] Aliprantis | Infinite-dimensional Analysis[END_REF] the following space is defined Definition 1.17.

� 1 d (N, R) is the set of all linear functionals θ ∈ � ∞ (N, R) * such that there exists ξ ∈ R satisfying �θ, x� = ξ lim t→+∞ x t for all x ∈ c(N, R). Its elements are called the singular functionals of � ∞ (N, R) * .
In [START_REF] Aliprantis | Infinite-dimensional Analysis[END_REF] the following result is established.

Theorem 1.18. � ∞ (N, R) * = � 1 (N, R) ⊕ � 1 d (N, R
). The meaning of this equality is the following: for all Λ ∈ � ∞ (N, R) * there exists a unique (q, θ)

∈ � 1 (N, R) × � 1 d (N, R) such that �Λ, x� = � x, q � + �θ, x� for all x ∈ � ∞ (N, R).
Now we extend this space and the previous description to sequences in R k .

Definition 1.19. � 1 d (N, R k ) is the set of all linear functionals θ ∈ � ∞ (N, R k * ) such that there exists ζ ∈ R k * satisfying �θ, x� = �ζ, lim t→+∞ x t � for all x ∈ c(N, R k ). Its elements are call the singular functionals of � ∞ (N, R k ) * . Proposition 1.20. � ∞ (N, R k ) * = � 1 (N, R k * ) ⊕ � 1 d (N, R k ). Proof. Let Λ ∈ � ∞ (N, R k ) * . When x ∈ � ∞ (N, R k ), we can identify it with (x 1 , x 2 , . . . , x k ) ∈ (� ∞ (N, R)) k . And we can write �Λ, x� = � k i=1 � Λ i , x i � where � Λ i , x i � = � Λ, (0, . . . , 0, x i , 0, . . . , 0) � .
Note that Λ i ∈ � ∞ (N, R) * and then, using Theorem 1.18, we know that there exist

q i ∈ � 1 (N, R) and θ i ∈ � 1 d (N, R) such that �Λ i , r� = � r, q i � + � θ i , r � for all r ∈ � ∞ (N, R).

Denoting by (e *

i ) 1≤i≤k the dual basis of the canonical basis of R k , we set q t :=

� k i=1 q i t e * i . Since |q t | * = � k i=1 � � q i t � � we obtain that q = (q t ) t∈N ∈ � 1 (N, R k * ). We set �θ, x� := � k i=1 � θ i , x i � . We see that θ is a linear functional from � ∞ (N, R k ) into R. Since the projection π i : � ∞ (N, R k ) → � ∞ (N, R), π i (x) := x i , are continuous, θ = � k i=1 θ i • π i is continuous as a finite sum of compositions of continuous functions. And so we obtain θ ∈ � ∞ (N, R k ) * . When x ∈ c(N, R k ) we have x i ∈ c(N, R) and since θ i ∈ � 1 d (N, R) there exists ξ i ∈ R such that � θ i , x i � = ξ i lim t→+∞ x i t . We set ξ := � k i=1 ξ i e * i ∈ R k * ,and then we have �θ, x� = � k i=1 � θ i , x i � = � k i=1 ξ i lim t→+∞ x i t = �ξ, lim t→+∞ x t � . Hence, θ ∈ � 1 d (N, R k ).
The existence is proven.

Chapter 1. Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces

To check the uniqueness we assume that there exist

p ∈ � 1 (N, R k * ) and ρ ∈ � 1 d (N, R k ) such that � x, q � + �θ, x� = � x, p � + � ρ, x � for all x ∈ � ∞ (N, R k ). When x ∈ c 0 (N, R k ), this equality becomes � x, q � = � x, p �
, and since � 1 (N, R k * ) is the dual space of c 0 (N, R k ) we obtain q = p, from which we deduce θ = ρ.

Completeness of Sequence Spaces

In this subsection, we will show that � p (N, E) with p ∈ [1, +∞), � ∞ (N, E), c(N, E) and c 0 (N, E) are Banach spaces.

Theorem 1.21. � p (N, E) where p ∈ [1, +∞] is a Banach space.
Proof. We will prove this theorem for the cases p ∈ (1, +∞), p = 1 and p = +∞ individually.

-Case p ∈ (1, +∞): Let (x m ) m∈N = (x 0 , x 1 , . . . , x n , . . . ) be a Cauchy sequence in � p (N, E). Here, for each n ∈ N, x n = (x n t ) t∈N ∈ � p (N, E). Now, since (x m ) m∈N is a Cauchy sequence then for all ε > 0 small enough, there exists M ∈ N * such that when m, n > M we have

�x m -x n � p < ε ⇔   � t∈N |x m t -x n t | p   1/p < ε ⇔ � t∈N |x m t -x n t | p < ε p < ε.
Then for every t ∈ N we have |x m tx n t | p < ε and it means that for each t ∈ N sequence (x m t ) m∈N is a Cauchy sequence in space E = R k . Since R k is a Banach space we deduce that lim m→+∞ x m t = x t for each t ∈ N. We set x = (x t ) t∈N . For any N > 1 and n, m > M we have:

N � t=0 |x m t -x n t | p ≤ � t∈N |x m t -x n t | p < ε p < ε.
Let n → +∞ in the previous inequality, we obtain the following:

N � t=0 |x m t -x t | p < ε p < ε
This inequality holds for all N > 1. Then we can take the limit as N → +∞ :

+∞ � t=0 |x m t -x t | p ≤ ε p < ε ⇔ �x m -x� p < ε.
And so,

x m -x ∈ � p (N, E) hence x ∈ � p (N, E) since x m ∈ � p (N, E). Moreover, �x m -x� p < ε for all m > M then we deduce that lim m→+∞ x m = x ∈ � p (N, E). Hence, � p (N, E) is a Banach space. -Case p = 1 :
Let (x m ) m∈N = (x 0 , x 1 , . . . , x n , . . . ) be a Cauchy sequence in � 1 (N, E). Here, for each n ∈ N, x n = (x n t ) t∈N ∈ � 1 (N, E). Now, since (x m ) m∈N is a Cauchy sequence then for all ε > 0, there exists M ∈ N * such that when m, n > M we have

�x m -x n � 1 < ε ⇔ � t∈N |x m t -x n t | < ε.
Then for every t ∈ N we have |x m tx n t | < ε and it means that for each t ∈ N sequence (x m t ) m∈N is a Cauchy sequence in space E = R k . Since R k is a Banach space we deduce that lim m→+∞ x m t = x t for each t ∈ N. We set x = (x t ) t∈N . For any N > 1 and n, m > M we have:

N � t=0 |x m t -x n t | ≤ � t∈N |x m t -x n t | < ε.
Let n → +∞ in the previous inequality, we obtain the following:

N � t=0 |x m t -x t | p < ε.
This inequality holds for all N > 1. Then we can take the limit as N → +∞ :

+∞ � t=0 |x m t -x t | p < ε ⇔ �x m -x� 1 < ε.
And so,

x m -x ∈ � 1 (N, E) hence x ∈ � 1 (N, E) since x m ∈ � 1 (N, E). Moreover, �x m -x� 1 < ε for all m > M then we deduce that lim m→+∞ x m = x ∈ � 1 (N, E). Hence, � 1 (N, E) is a Banach space. -Case p = +∞ : Let (x m ) m∈N = (x 0 , x 1 , . . . , x n , . . . ) be a Cauchy sequence in � ∞ (N, E). Here, for each n ∈ N, x n = (x n t ) t∈N ∈ � ∞ (N, E). Now, since (x m
) m∈N is a Cauchy sequence then for all ε > 0, there exists M ∈ N * such that when m, n > M we have

�x m -x n � ∞ < ε ⇔ sup t∈N |x m t -x n t | < ε.
Then for every t ∈ N we have |x m tx n t | < ε and it means that for each t ∈ N sequence (x m t ) m∈N is a Cauchy sequence in space E = R k . Since R k is a Banach space we deduce that lim m→+∞ x m t = x t for each t ∈ N. We set x = (x t ) t∈N . For any N > 1 and n, m > M we have:

max 0≤t≤N |x m t -x n t | ≤ sup t∈N |x m t -x n t | < ε.
Let n → +∞ in the previous inequality, we obtain the following:

max 0≤t≤N |x m t -x t | < ε.
This inequality holds for all N > 1. Then we can take the limit as N → +∞ :

sup t∈N |x m t -x t | < ε ⇔ �x m -x� ∞ < ε.
And so,

x m -x ∈ � ∞ (N, E) hence x ∈ � ∞ (N, E) since x m ∈ � ∞ (N, E). Moreover, �x m -x� ∞ < ε for all m > M then we deduce that lim m→+∞ x m = x ∈ � ∞ (N, E). Hence, � ∞ (N, E) is a Banach space.
Lemma 1.22. c(N, E) and c 0 (N, E) are closed subspace of � ∞ (N, E).

Chapter 1. Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces

Proof.

-Prove c 0 (N, E) is a closed subspace of � ∞ (N, E) : Let (x m ) m∈N = (x 0 , x 1 , . . . , x n , . . . ) be a convergent sequence of elements from c 0 (N, E) and let lim m→+∞ x m = x. Since � ∞ (N, E) is a Banach space, we know that x ∈ � ∞ (N, E).Now we need to prove x ∈ c 0 (N, E). From lim m→+∞ x m = x, for all ε > 0 there exists N 1 ∈ N * such that when m > N 1 we have:

�x m -x� ∞ < ε 2 ⇔ sup t∈N |x m t -x t | < ε 2 .
From this we deduce that

|x m t -x t | < ε 2 for each t ∈ N when m > N 1 . Now, x m ∈ c 0 (N, E) for all m ∈ N, hence lim t→+∞ x m t = 0.
It means that for all ε > 0, there exists N 2 ∈ N * such that when t > N 2 we have |x m t | < ε 2 for all m ∈ N. And so, when m > N 1 and t > N 2 we have

|x t | = |x t -x m t + x m t | ≤ |x t -x m t | + |x m t | ≤ ε 2 + ε 2 = ε.
The last inequality holds for all t > N 2 , hence we can conclude that lim t→+∞ x t = 0 and x ∈ c 0 (N, E). Then we obtain

c 0 (N, E) is closed subspace of � ∞ (N, E). -Prove c(N, E) is a closed subspace of � ∞ (N, E) :
Let (x m ) m∈N = (x 0 , x 1 , . . . , x n , . . . ) be a convergent sequence of elements from c(N, E) and let lim m→+∞ x m = x. We know that x ∈ � ∞ (N, E) and we need to prove x ∈ c(N, E). From lim m→+∞ x m = x, for all ε > 0 there exists N 1 ∈ N * such that when m > N 1 we have:

�x m -x� ∞ < ε 3 ⇔ sup t∈N |x m t -x t | < ε 3 .
From this we deduce that |x m tx t | < ε 3 for each t ∈ N when m > N 1 . Now, x m ∈ c(N, E) for all m ∈ N, hence lim t→+∞ x m t exists for all m ∈ N and from this we obtain (x m t ) t∈N is a Cauchy sequence. Therefore, for all ε > 0 there exists N 2 ∈ N * such that when t, t � > N 2 we have:

|x m t -x m t � | ≤ ε 3 .
Notice that the previous inequality holds for each m ∈ N. Now consider

|x t -x t � | = |x t -x m t + x m t -x m t � + x m t � -x t � | ≤ |x t -x m t | + |x m t -x m t � | + |x m t � -x t � |
In the previous inequality, we take t, t � > N 2 and m > N 1 then:

|x t -x t � | ≤ ε 3 + ε 3 + ε 3 = ε. It means that x = (x t ) t∈N is a Cauchy sequence of element from E = R k . Since R k is complete, we deduce that x = (x t ) t∈N is convergent. Hence, x ∈ c(N, E) and therefore, c(N, E) is closed subspace of � ∞ (N, E).
Lemma 1.23. A closed subspace of a Banach space is also a Banach space. Proof. c(N, E) and c 0 (N, E) are closed subspaces of � ∞ (N, E); hence, they are also Banach spaces since � ∞ (N, E) is a Banach space.

Recall of Differential Calculus in Normed Linear Spaces

In this section, we recall the basis of differential calculus in normed linear spaces, especially on diverse kinds of differentials and their properties which will be useful for the proofs in later chapters. The main references for this section are Chapter XIII in Lang [START_REF] Lang | Real and Functional Analysis[END_REF] and Section 2.2 in Alekseev-Tihomirov-Fomin [START_REF] Alekseev | Optimal Control[END_REF].

Directional Derivative, Gâteaux and Fréchet Differentials and Strict Differentiability

Let X and Y be normed linear spaces, let U be a neighborhood of a point x in X, and let F be a mapping from U into Y . The directional derivative is usually defined as follows Definition 1.25. Let h ∈ X. If the limit lim

λ→0 + F (x + λh) -F (x) λ (1.1)
exists, it is called the directional derivative of F at the point x in the direction h and it is denoted by � DF (x; h).

Definition 1.26. Let us suppose that for any h ∈ X there exists the directional derivative � DF (x; h) and there exists a continuous linear operator Λ ∈ L(X, Y ) such that � DF (x; h) ≡ Λh. Then the operator Λ is called the Gâteaux differential of the mapping F at the point x and is denoted by D G F (x).

Thus, D G F (x) is an element of L(X, Y ) such that, given any h ∈ X, the relation

F (x + λh) = F (x) + λD G F (x) • h + λ • ρ h (λ) (1.2)
holds when λ is positive and small enough. Here ρ h : R → Y is a mapping satisfying lim λ→0 + ρ h (λ) = 0. It readily follows that the Gâteaux differential is determined uniquely since the directional derivatives are determined uniquely.

Definition 1.27. Let it be possible to represent a mapping F in a neighborhood of a point x in the form

F (x + h) = F (x) + Λ.h + α(h). �h� , (1.3)
where Λ ∈ L(X, Y ) and α : X → Y is a mapping which is defined for all sufficient small h in X and such that lim

h→0 α(h) = 0. (1.4)
Then the mapping F (•) is said to be Fréchet differentiable at the point x. The operator Λ is called the Fréchet differential of the mapping F at the point x and is denoted by DF (x).

Chapter 1. Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces

From this definition, DF (x) = Λ is a continuous linear function which belongs to L(X, Y ). Relations (1.3) and (1.4) can also be written thus:

F (x + h) = F (x) + DF (x).h + o(�h�), (1.5)
It readily follows that the Fréchet differential is determined uniquely because if Λ 1 and Λ 2 from L(X, Y ) simultaneously satisfy relation (1.5) then

�Λ 1 h -Λ 2 h� = o(�h�). It is only possible when Λ 1 = Λ 2.
Moreover, if a mapping is Fréchet differentiable at a point then it is also Gâteaux differentiable at that point and DF (x) ≡ D G F (x). This assertion is easily verified by setting h = λv in the definition of Fréchet differential. Finally, in the term of the ε, δ formalism the relations Relations (1.3) and (1.4) are stated thus: given an arbitrary ε > 0, there is δ > 0 for which the inequality

�F (x + h) -F (x) -Λ.h� ≤ ε �h� (1.6)
holds for all h such that �h� < δ. This naturally leads to a further strengthening:

Definition 1.28. A mapping F is said to be strictly differentiable at a point x if there is an operator Λ ∈ L(X, Y ) such that for any ε > 0 there is δ > 0 such that for all y and z satisfying the inequalities �y -x� < δ and �z -x� < δ the inequality

�F (y) -F (z) -Λ.(y -z)� ≤ ε �y -z� (1.7) holds.
Putting z = x and y = x + h in (1.7) we obtain (1.6), and hence a strictly differentiable mapping is Fréchet differentiable and Λ = DF (x) .

Example 1.29. A mapping A : X → Y of one linear space into another is said to be affine if there exists a linear mapping Λ : X → Y and a constant α ∈ Y such that A(x) = Λ.x + α. If X and Y are normed spaces and Λ ∈ L(X, Y ), then the mapping A is strictly differentiable at any point x, and, moreover, DA(x) = Λ. This assertion can be verified directly. Indeed, we have:

�A(y) -A(z) -Λ.(y -z)� = �Λ.y -Λ.z -Λ.(y -z)� = 0 ≤ ε �y -z�
for any ε > 0 and for all y, z ∈ X (in particular, including y, z ∈ X such that �y -x� < δ and �z -x� < δ where δ is some sufficient small positive number). Hence, A is strictly differentiable at any point x ∈ X. When α = 0 the mapping A degrades into a continuous linear mapping Λ ∈ L(X, Y ). And from the above-mentioned argument, we can deduce that a mapping Λ ∈ L(X, Y ) is also a strictly differentiable at any point x ∈ X and DΛ(x) = Λ.

For Fréchet differentiable mapping, we have the following well-known property:

Theorem 1. Proof. Let F be a mapping which is strictly differentiable at the point x. Then for any ε > 0 there exists δ > 0 such that for all x 1 and x 2 satisfying the inequalities �x 1 -x� < δ and �x 2 -x� < δ we have the inequality

�F (x 1 ) -F (x 2 ) -Λ.(x 1 -x 2 )� ≤ ε �x 1 -x 2 � .
Let z be a point in a δ 2 -neighborhood of x.We will show that F is continuous at z. We set y = z + h where h ∈ X and �h� < δ 2 . Then

�y -x� = �z + h -x� ≤ �z -x� + �h� ≤ δ 2 + δ 2 = δ.
From this we deduce that both y and z belongs to δ -neighborhood of x, hence:

�F (y) -F (z) -Λ.(y -z)� ≤ ε �y -z� ⇔ �F (z + h) -F (z) -Λ.h� ≤ ε �h� .
The last inequality shows that F is Fréchet differentiable at z and hence, continuous at that point. Since taking z is arbitrary in δ 2 -neighborhood of x, then one can conclude that F is continuous on this neighborhood of x. Definition 1.32. If F is (Gâteaux or Fréchet or strictly) differentiable at every point x of U , then we say that F is ( Gâteaux or Fréchet or strictly) differentiable on U . In that case, the differential DF (DF can be DF or D G F ) is a mapping

DF : U → L(X, Y )
from U into the space of continuous linear maps L(X, Y ), and thus to each x ∈ U , we have associated the linear map DF (x) ∈ L(X, Y ). If DF is continuous, we say that F is continuously differentiable or simply, F is of class C 1 .

Since DF maps U into the Banach space L(X, Y ), we can define inductively F to be of class C p if all the differentials D k F exist and are continuous for 1 ≤ k ≤ p.

When a mapping F is Fréchet differentiable on an open set, the following theorem shows the necessary and sufficient condition for F to be strictly differentiable: Theorem 1.33. If F is Fréchet differentiable on an open set, then F is continuously differentiable if and only if F is strictly differentiable on the same set.

The proof of this theorem can be found in [START_REF] Schechter | Handbook of Analysis and Its Foundations[END_REF], page 682.

Subgradient and subdifferential

Let X be a normed linear space, F : X → R is a functional and x ∈ X. Definition 1.34. An element ζ of X * is called a subgradient of F at x (in the sense of convex analysis) if it satisfies the following subgradient inequality:

F (y) -F (x) ≥ �ζ, y -x�, y ∈ X.
The set of all subgradients of F at x is called the subdifferential of F at x and is denoted by ∂F (x). When X = X 1 × X 2 , x = (x 1 , x 2 ), the partial subdifferential of F with respect to x 1 (respectively, x 2 ) at (x 1 , x 2 ) is denoted by

∂ 1 F (x 1 , x 2 ) (respectively, ∂ 2 F (x 1 , x 2 )).
The following properties of subdifferential are taken from [START_REF] Alekseev | Optimal Control[END_REF] and [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF]. (ii) If F is convex then

∂F (x) = {ζ ∈ X * : � DF (x; h) ≥ �ζ, h�, ∀h ∈ X}. (iii) If F is convex and it is Gâteaux differentiable at x then ∂F (x) = {D G F (x)}.
(iv) If F is convex and continuous at x then ∂F (x) is nonempty closed convex. When X = R n , then for any x ∈ R n , ∂F (x) is a non empty convex compact set.

Properties of Differentials

We recall some basic properties of differentials.

Sum: Let X, Y be normed vector spaces, and let U be open in X. Let f, g : U → Y be maps which are differentiable (in the sense of Gâteaux or Fréchet or strict differentiability) at x ∈ U . Then f + g is differentiable at x and

D(f + g)(x) = Df (x) + Dg(x).
If c is a real number, then

D(cf )(x) = cDf (x).
Product: Let X be a normed vector space. Let Y , Z and W be complete normed vector spaces, and let Y × Z → W be a continuous bilinear map. Let U be open in X and let f : U → Y , and g : U → Z be maps differentiable at x ∈ U . Then the product map f g is differentiable at x and for all h ∈ X, one has

D(f g)(x) • h = (Df (x) • h)g(x) + f (x)(Dg(x) • h).
Chain rule: Let X, Y, Z be normed vector spaces, let U be open in X and let V be open in Y , let f : U → V and g : V → Z be mappings and let h = g • f : U → Z be the composition of the mapping f and g. Let x ∈ U and ŷ = f (x) ∈ V.

Assume that f is differentiable (in Gâteaux or Fréchet's sense) at x and g is Fréchet differentiable at ŷ. Then h is differentiable (in the same sense of f ) at x and for all h ∈ X,

D(g • f )(x) • h = Dg(ŷ)(Df (x) • h). (1.8)
If f is strictly differentiable at x and g is strictly differentiable at ŷ then h is strictly differentiable at x.

Those are well-known properties of differentiable mapping in normed space. The proofs for them can be found for instance in [START_REF] Alekseev | Optimal Control[END_REF] or [START_REF] Lang | Real and Functional Analysis[END_REF].

Example 1.35. Let f : U → Y be a (Gâteaux or Fréchet or strictly) differentiable map, and let λ : Y → Z be a continuous linear map. Then for each x ∈ U , λ • f is differentiable in the same sense of f at x, and for every v ∈ U we have

D(λ • f )(x) • v = λ(Df (x) • v).
This result follows directly from Example 1.29 and chain rule.

Mean Value Theorem

Let X be normed space and let a, b ∈ X. We denote closed line segment which connects a and b as follows:

If f : U → Y is a Gâteaux differentiable function at each point x ∈ [a, b], then �f (b) -f (a)� ≤ sup c∈[a,b] �D G f (c)� �b -a� .
Proof. Let us take an arbitrary y * ∈ Y * and consider the function Φ : R → R defined by

Φ(t) := �y * , f (a + t(b -a))� .
This function possesses derivatives at each point of the closed interval [0, 1]:

Φ � (t) = lim λ→0 Φ(t + λ) -Φ(t) λ = � y * , lim λ→0 f (a + (t + λ)(b -a)) -f (a + t(b -a)) λ � = � y * , lim λ→0 f (a + t(b -a) + λ(b -a)) -f (a + t(b -a)) λ � = �y * , D G f (a + t(b -a)).(b -a)� .
Hence, function Φ is differentiable (in the ordinary sense) on the closed interval [0, 1] and therefore it is continuous on this interval. By Lagrange's formula, there exists θ ∈ (0, 1) such that

�y * , f (b) -f (a)� = Φ(1) -Φ(0) = Φ � (θ) = �y * , D G f (a + θ(b -a)) • (b -a)� .
Now we shall make use of Corollary 1 of the Hahn-Banach Theorem in [START_REF] Alekseev | Optimal Control[END_REF] (page 76), according to which, for any element y ∈ Y , there is a linear functional y * ∈ Y * such that �y * � = 1 and �y * , y� = �y�. Taking this functional y * for the element y = f (b)f (a), we obtain

�f (b) -f (a)� = �y * , f (b) -f (a)� = �y * , D G f (a + θ(b -a)).(b -a)� ≤ �y * � �D G f (a + θ(b -a))� �(b -a)� = �D G f (a + θ(b -a))� �(b -a)� ≤ sup c∈[a,b] �D G f (c)� �b -a� .
Corollary 1.37. Let all the conditions of the Mean Value Theorem be fulfilled, and let

Λ ∈ L(X, Y ). Then �f (b) -f (a) -Λ(b -a)� ≤ sup c∈[a,b] �D G f (c) -Λ� �b -a� .
Proof. The proof reduces to the application of the Mean Value Theorem to the mapping

g(x) = f (x) -Λ • x.
Chapter 1. Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces

Corollary 1.38. Let X and Y be normed spaces, let U be a neighborhood of a point x in X, and let a mapping f : U → Y be Gâteaux differentiable at each point x ∈ U . If the mapping x � → D G f (x) is continuous at the point x, then the mapping f is strictly differentiable at x (and consequently it is Fréchet differentiable at that point).

Proof. Given ε > 0, there is δ > 0 such that the relation

�x -x� < δ =⇒ �D G f (x) -D G f (x)� < ε (1.9) holds. If �x 1 -x� < δ and �x 2 -x� < δ, then for any x = x 1 + t(x 2 -x 1 ) ∈ [x 1 , x 2 ], 0 ≤ t ≤ 1, we have �x -x� = �x 1 + t(x 2 -x 1 ) -x� = �t(x 2 -x) + (1 -t)(x 1 -x)� ≤ t �x 2 -x� + (1 -t) �x 1 -x� < tδ + (1 -t)δ = δ,
and therefore, by virtue of (1.9), we have

�D G f (x) -D G f (x)� < ε. Applying Corollary 1.37 to Λ = D G f (x), we obtain �f (x 1 ) -f (x 2 ) -D G f (x) • (x 1 -x 2 )� ≤ sup x∈[x 1 ,x 2 ] �D G f (x) -D G f (x)� �x 1 -x 2 � ≤ ε �x 1 -x 2 � ,
which implies the strict differentiability of f at x. In this subsection X, Y and Z are normed spaces. We shall begin with the case of a mapping whose values belong to the product space Y × Z, i.e., F : U → Y × Z, U ⊂ X. Since a point of Y × Z is a pair (y, z), the mapping F also consists of two components: F (x) = (G(x), H(x)), where G : U → Y and H : U → Z. The corresponding definitions immediately imply: Proposition 1.39. Let X, Y and Z be normed spaces, let U be a neighborhood of a point x in X, and let G : U → Y and H : U → Z.

For the mapping F = (G, H) : U → Y × Z to be differentiable at the point x in the sense of Gâteaux or Fréchet or strict differentiability, it is necessary and sufficient that G and H possess the same property. Moreover, in this case

DF (x) = (DG(x), DH(x)),
or for all h ∈ X, we have

DF (x) • h = (DG(x) • h, DH(x) • h).
Now we turn to the case when the domain of the mapping F : U → Z belongs to the product space: U ⊂ X × Y . Definition 1.40. Let X, Y and Z be normed spaces, let U be a neighborhood of a point (x, ŷ) in X × Y , and let F : U → Z. If the mapping x � → F (x, ŷ) is (Gâteaux or Fréchet or strictly) differentiable at the point x, its differential is called the partial differantial of the mapping F with respect to x at the point (x, ŷ) and is denoted D 1 F (x, ŷ). The partial differential D 2 F (x, ŷ) with respect to y is defined in an analogous manner.

Theorem 1.41. (Theorem on the Total Differential) Let X, Y and Z be normed spaces, let U be a neighborhood of a point (x, ŷ) in X × Y and let F : U → Z be a mapping possessing the partial derivatives D 1 F (x, y) and D 2 F (x, y) in Gâteaux' sense at each point (x, y) ∈ U .

If the mappings (x, y) � → D 1 F (x, y) and (x, y) � → D 2 F (x, y) are continuous at a point (x, ŷ) ∈ U in the uniform topology of operators, then F is strictly differentiable at that point, and moreover,

DF (x, ŷ).(ξ, η) = D 1 F (x, ŷ).ξ + D 2 F (x, ŷ).η for all (ξ, η) ∈ X × Y .
Proof. Given an arbitrary ε > 0, let us choose δ > 0 small enough so that the neighborhood

V = {(x, y) : �x -x� < δ, �y -ŷ� < δ} of the point (x, ŷ) is contained in U and the inequalities �D 1 F (x, y) -D 1 F (x, ŷ)� < ε, �D 2 F (x, y) -D 2 F (x, ŷ)� < ε (1.10)
hold in that neighborhood. Now we have

∆ = F (x 1 , y 1 ) -F (x 2 , y 2 ) -D 1 F (x, ŷ)(x 1 -x 2 ) -D 2 F (x, ŷ)(y 1 -y 2 ) = (F (x 1 , y 1 ) -F (x 1 , y 2 ) -D 1 F (x, ŷ)(x 1 -x 2 )) + (F (x 1 , y 2 ) -F (x 2 , y 2 ) -D 2 F (x, ŷ)(y 1 -y 2 )).
It can readily be seen that if the points (x 1 , y 1 ), (x 2 , y 2 ) belong to V , then (x 2 , y 1 ) ∈ V , and moreover the line segments [(x 1 , y 1 ), (x 2 , y 1 )] and [(x 2 , y 1 ), (x 2 , y 2 )] are contained in V ⊂ U . Therefore, the functions x � → F (x, y 1 ) and y � → F (x 2 , y) are Gâteaux differentiable: the first of them possesses the differential D 1 F on [x 1 , x 2 ] and the other possesses the differential D 2 F on [y 1 , y 2 ]. Applying the Mean Value Theorem to these functions we obtain, by virtue of (1.10), the relations

(x 1 , y 1 ), (x 2 , y 2 ) ∈ V =⇒ �∆� ≤ sup z∈[x 1 , x 2 ] �D 1 F (z, y 1 ) -D 1 F (x, ŷ)� �x 1 -x 2 � + sup w∈[y 1 , y 2 ] �D 2 F (x 2 , w) -D 2 F (x, ŷ)� �y 1 -y 2 � ≤ ε(�x 1 -x 2 � + �y 1 -y 2 �).
And so, F is strictly differentiable at (x, ŷ).

Chapter 2

Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle

Introduction

The aim of this chapter is to introduce the infinite-horizon optimal control problem in discrete time framework with asymptotical constraint and to establish for it necessary condition of optimality in the form of weak Pontryagin principle. We consider the following optimal control problem Maximize K(y, u) :=

� +∞ t=0 β t ψ(y t , u t ) when y = (y t ) t∈N ∈ (R n ) N , u = (u t ) t∈N ∈ U N , y 0 = η, lim t→+∞ y t = y ∞ , u is bounded, ∀t ∈ N, y t+1 = g(y t , u t ),              (P m )
where U ⊂ R d is nonempty; β ∈ (0, 1); η, y ∞ ∈ R n fixed; function ψ : R n × U → R and function g : R n × U → R n , and (R n ) N (respectively U N ) denotes the set of the sequences in R n (respectively U ). In comparison with existing results on bounded processes like [START_REF] Blot | Infinite Horizon Discrete Time Control Problems for Bounded Processes[END_REF] and [START_REF] Blot | Pontryagin Principles for Bounded Discrete-time Processes[END_REF], the specificness of the present work is the presence of the asymptotical constraint on the state variable lim t→+∞ y t = y ∞ . Its meaning is that the optimal state of the problem stays near a "good" or "expected" value on the long run.

The approach to this problem is functional analytic; we translate our problem into static form of optimization in suitable Banach sequence spaces. We describe the content of this chapter as follows.

-In Section 2.2 we introduce a problem of optimal control which is equivalent to the initial problem and is defined in the following classical sequence spaces: c 0 (N, R n ) the space of all sequences in (R n ) N which converge to zero at infinity, and � ∞ (N, U ) the space of all sequences in U N which are bounded.

-In Section 2.3 we study the properties of operators and functionals on sequence spaces. A first novelty is a characterization of the operators which send

c 0 (N, R n ) × � ∞ (N, U ) into c 0 (N, R n ) (Theorem 2.

2). The other results use this characterization and existing results on

Nemytskii operators from � ∞ (N, R n ) × � ∞ (N, U ) into � ∞ (N, R m ).
Chapter 2. Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle -Section 2.4 is devoted to the solutions which converge toward zero of a linear difference equation. These results are useful to establish regularity properties of the differential of operators formalizing the nonlinear difference equation which governs the dynamical system.

-In Section 2.5 we establish a variation of a Karush-Kuhn-Tucker theorem which is useful for weak Pontryagin principles in later subsections.

-In Section 2.6 and Section 2.7 we establish weak Pontryagin principles for the considered problems.

The Supporting Problem

In this section we introduce a supporting problem which can be translated equivalently into the form of Problem (P m ). The purpose of this is that with the supporting problem, we can work in classical Banach sequence spaces. We consider the following optimal control problem:

Maximize J(x, u) :=

� +∞ t=0 β t φ(x t , u t ) when x = (x t ) t∈N ∈ c 0 (N, R n ), u = (u t ) t∈N ∈ � ∞ (N, U ), x 0 = η, ∀t ∈ N, x t+1 = f (x t , u t ).              (P s )
Here U, β and η are exactly the same like those in the main problem. Functions φ and f are defined like functions ψ and g, respectively. Notice that Problem (P m ) can be translated into the form of Problem (P s ) by using the following transformation -Let for all t ∈ N, z t := y ty ∞ . Then z = (z t ) t∈N ∈ c 0 (N, R n ) since lim t→+∞ z t = lim t→+∞ (y ty ∞ ) = 0.

-Now K(y, u) =

� +∞ t=0 β t ψ(y t , u t ) = � +∞ t=0 β t ψ(z t + y ∞ , u t ). We set N (z, u) := � +∞ t=0 β t ϕ(z t , u t ) where ϕ(z, u) := ψ(z + y ∞ , u) for all (z, u) ∈ R n × U then K(y, u) = N (z, u).
-Finally, we set �(z, u) := g(z + y ∞ , u)y ∞ for all (z, u) ∈ R n × U then for all t ∈ N, from y t+1 = g(y t , u t ) we obtain the equivalent equation z t+1 = �(z t , u t ). After these settings, Problem (P m ) becomes:

Maximize N (z, u) := � +∞ t=0 β t ϕ(z t , u t ) when z = (z t ) t∈N ∈ c 0 (N, R n ), u = (u t ) t∈N ∈ � ∞ (N, U ), z 0 = η -y ∞ , ∀t ∈ N, z t+1 = �(z t , u t ).              (P 1)
This problem has precisely the form of Problem (P s ).

Some Useful Properties of Nemytskii Operators

This section is devoted to the study of several operators between sequence spaces; notably the Nemytskii operators, and to the study of functionals and the mappings which define the criterion and the dynamical system of our maximization problems. We also establish results of continuity and Fréchet differentiability for those operators. Lemma 2.1. Let ζ be a mapping from X × V into Y where X, V and Y are normed spaces and U be a nonempty subset of V . Then from (i) For all B bounded, nonempty in U : lim x→0 (sup u∈B �ζ(x, u)�) = 0, we obtain

(ii) For all x ∈ c 0 (N, X), for all u ∈ � ∞ (N, U ) : (ζ(x t , u t )) t∈N ∈ c 0 (N, Y ) or equivalently, lim t→+∞ ζ(x t , u t ) = 0. Proof. Let x ∈ c 0 (N, X) and u ∈ � ∞ (N, U ). Take B = {u t , t ∈ N}, then B is bounded since (u t ) t ∈ � ∞ (N, U ).
Using the hypothesis (i) of this lemma, we know that for this set B, lim x→0 (sup u∈B �ζ(x, u)�) = 0. It means that with ε > 0 arbitrarily, there exists δ ε > 0 such that for all x ∈ X, sup u∈B �ζ(x, u)� < ε when �x� < δ ε which also means that for all t ∈ N, �ζ(x, u t )� < ε when �x� < δ ε . From (x t ) t ∈ c 0 (N, X) we obtain lim t→+∞ x t = 0. It means that with δ ε > 0 above, there exists T ∈ R + such that for all t ∈ N, when t > T we have �x t � < δ ε .

From these arguments, we derive that with ε > 0 arbitrarily, there exists T ∈ R + such that for all t ∈ N, when t > T we have �x t � < δ ε and hence, �ζ(x t , u t )� < ε. And so, we have proven lim t→∞ ζ(x t , u t ) = 0 and we obtain (ζ(x t , u t )) t∈N ∈ c 0 (N, Y ).

Theorem 2.2. Let X, V, Y be three normed spaces, U be a nonempty subset of V and ζ : X × U → Y be a mapping such that, for all x ∈ X, the partial mapping u � → ζ(x, u) transforms the bounded subsets of U into bounded subsets of Y. Then the assertions (i) and (ii) of Lemma 2.1 are equivalent.

Proof. Using Lemma 2.1 we have (i) =⇒ (ii). Now we prove the converse implication (ii) =⇒ (i). Let B be nonempty bounded subset of U . Let x ∈ c 0 (N, X). From the assumption on ζ, we know that, for all t ∈ N, we have sup u∈B �ζ(x t , u)� < +∞. Therefore, for all t ∈ N, there exists u t ∈ B such that

0 ≤ sup u∈B �ζ(x t , u)� ≤ �ζ(x t , u t )� + 1 t + 1 .
Note that for all t ∈ N, u t ∈ B, then u = (u t ) t∈N ∈ � ∞ (N, U ). Using (ii), we obtain lim t→+∞ �ζ(x t , u t )� = 0 and from previous inequality we obtain lim t→+∞ sup u∈B �ζ(x t , u)� = 0, and since we work in normed spaces we can use the sequential characterization of the limit (which will be proven in Remark 2.6) and assert that we obtain (i).

Remark 2.3. Assertion (ii) of Lemma 2.1 permits to define the Nemytskii operator

N ζ : c 0 (N, X) × � ∞ (N, U ) → c 0 (N, Y ), N ζ (x, u) := (ζ(x t , u t )) t∈N . Proof. Let x ∈ c 0 (N, X) and u ∈ � ∞ (N, U ). From assertion (ii) of Lemma 2.1 we have lim t→+∞ ζ(x t , u t ) = 0. It means that N ζ (x, u) = (ζ(x t , u t )) t∈N ∈ c 0 (N, Y ). Hence, N ζ is well defined.
Remark 2.4. Let B R := {v ∈ V : �v� ≤ R} when R ∈ (0, +∞). We have the following statements:

1. In the setting of Theorem 2.2, the assumption on ζ is equivalent to the following condition: ∀x ∈ X, ∀R ∈ (0, +∞), sup (iv) For all K compact in U , lim x→0 (sup u∈K �ζ(x, u)�) = 0.

Proof. We will prove the equivalent of (i) and (ii), then we will prove that (i) ⇐⇒ (iii) and finally, (i) ⇐⇒ (iv).

-Prove (i) ⇐⇒ (ii): it is a direct consequence of Remark 2.5 and Theorem 2.2.

-Prove (i) ⇐⇒ (iii) -(i) =⇒ (iii): suppose that we have (i). Let B be some bounded nonempty set in U and sequence (x t ) t ∈ c 0 (N, X). Now lim x→0 (sup u∈B �ζ(x, u)�) = 0 means that for all ε > 0, there exists δ > 0 such that sup u∈B �ζ(x, u)� < ε when �x� < δ. Now since (x t ) t ∈ c 0 (N, X) then with δ above there exists T ∈ R + such that when t > T, t ∈ N we have �x t � < δ and hence, sup u∈B �ζ(x t , u)� < ε from the above-mentioned argument. And so lim t→+∞ (sup u∈B �ζ(x t , u)�) = 0.

-(iii) =⇒ (i): Let B be a bounded nonempty set in space U . For all x ∈ X, we set S(x) := sup u∈B �ζ(x, u)�. Obviously, S(x) ≥ 0 on X. Now we can rewrite (iii) ⇒ (i) as follows

∀(x t ) t ∈ c 0 (N, X), lim t→+∞ S(x t ) = 0 ⇒ lim x→0 S(x) = 0.
We will prove this implication by contradiction. Suppose that we have (ii) and not (i). We have

not (i) ⇔ not[∀ε > 0, ∃δ ε > 0, ∀x ∈ X, �x� ≤ δ ε ⇒ S(x) ≤ ε] ⇔ [∃ε > 0, ∀δ > 0, ∃x δ ∈ X, �x δ � ≤ δ and S(x δ ) > ε].
We take δ = 1 n when n ∈ N * . Then from the meaning of not (i), the following statement holds

[∃ε > 0, ∀n ∈ N * , ∃z n := x 1 n ∈ X, �z n � ≤ 1 n and S(z n ) > ε].
Then there exists (z n ) n∈N * ∈ c 0 (N, X) such that S(z n ) > ε for all n ∈ N * . It means that lim n→+∞ S(z n ) ≥ ε > 0. Using (ii), since (z n ) n∈N * ∈ c 0 (N, X), we have lim n→+∞ S(z n ) = 0. Hence, we obtain the contradiction and so the implication (iii) ⇒ (i) is proven. -Prove (i) ⇐⇒ (iv):

-First, we prove that (i) =⇒ (iv). Let K be some compact set in U . Then K is bounded. From (i) we immediately obtain the result of (iv). -Now on the other hand, suppose that, for all K compact in U , lim x→0 (sup u∈K �ζ(x, u)�) = 0. Let B be some bounded nonempty subset of U . Then its closure B is compact and B ⊂ U since dim V < +∞ and U is closed. From (iv) we know that lim x→0 (sup u∈ B �ζ(x, u)�) = 0. Hence, we only need to prove that for all x ∈ X, sup u∈ B �ζ(x, u)� = sup u∈B �ζ(x, u)�. When x ∈ X, we set ψ(u) := �ζ(x, u)� then ψ is continuous on U since �•� and ζ(x, .) are continuous. It is known that sup ψ(B) = sup u∈B �ζ(x, u)� ≤ sup u∈ B �ζ(x, u)� = sup ψ( B). Now B is compact and then from Weierstrass theorem there exists û ∈ B such that ψ(û) = sup ψ( B). Since û ∈ B, there exists a sequence (u n ) n ∈ B N such that lim n→+∞ u n = û. Since ψ is continuous, we deduce that lim n→+∞ ψ(u n ) = ψ(û). Hence, for all ε > 0, there exists

n ∈ N such that ψ(û) ≤ ψ(u n )+ε ≤ sup ψ(B)+ε. Then for all ε > 0, sup ψ( B) ≤ sup ψ(B) + ε. Let ε → 0, we have sup ψ( B) ≤ sup ψ(B).
From the above arguments, we can conclude that sup ψ(B) = sup ψ( B) which means that for all x ∈ X,

sup u∈ B �ζ(x, u)� = sup u∈B �ζ(x, u)� . Hence, from lim x→0 (sup u∈ B �ζ(x, u)�) = 0 we have lim x→0 (sup u∈B �ζ(x, u)�) = 0. Let U be a nonempty closed subset of R d . Let us call (α.1) ζ ∈ C o (R n × U, R m ) ; (α.2) For all B bounded, nonempty in U : lim x→0 (sup u∈B �ζ(x, u)�) = 0.
Now we introduce some important facts before proving the continuity of

N ζ . Let us fix an arbitrary point (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ). Since x ∈ c 0 (N, R n ) ⊂ � ∞ (N, R n ) and u ∈ � ∞ (N, U ) we can assert that {(x t , u t ) : t ∈ N} is bounded. Moreover, since dim(R n × U ) < +∞ then K := {(x t , u t ) : t ∈ N}, the closure of {(x t , u t ) : t ∈ N}, is Chapter 2. Infinite-Horizon Optimal Control Problem in Presence of
Asymptotical Constraint and a Weak Pontryagin Principle compact and K ⊂ R n × U since U is closed. Using a property of locally compact metric spaces, we know that since dim R n × U < +∞, there exists ρ ∈ (0, +∞) such that

L := {ν = (x, u) ∈ R n × U : d(ν, K) < ρ}, the closure of {ν = (x, u) ∈ R n × U : d(ν, K) < ρ}, is also a compact set (see [26], page 65) and it is contained in R n × U since U is closed. Here d(ν, K) is the distance from a point ν to set K which is defined by d(ν, K) := inf ν � ∈K �ν -ν � � .
Remark 2.7. Let U be a nonempty closed subset of R d . Under condition (α.1), ζ is uniformly continuous from L into R m where L is defined as above.

Proof. We recall Heine -Cantor's theorem: if ζ : M → N is a continuous mapping between two metric spaces and M is compact then ζ is uniformly continuous on M (see Theorem 4.19, page 91 of [START_REF] Rudin | Principles of Mathematical Analysis[END_REF]). Under (α.1) we have ζ : L → R m is a continuous mapping. Besides, L is compact as already proven above. Hence, apply Theorem Heine -Cantor for our case, we deduce that that

ζ is uniformly continuous from L into R m . It means that ∀ε > 0, ∃δ ε > 0, ∀ν, ν � ∈ L, � � ν -ν � � � < δ ε ⇒ � � ζ(ν) -ζ(ν � ) � � < ε.
We will use these facts to prove the continuity of operator N ζ in the following lemma.

Theorem 2.8. Let U be a nonempty closed subset of R d and let ζ : R n × U → R m be a mapping that satisfies conditions (α.1) and (α.2). Then the Nemytskii operator N ζ is well defined and is continuous on

c 0 (N, R n ) × � ∞ (N, U ), i.e. N ζ ∈ C 0 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )).
Proof. It can be seen that condition (α.1) is the condition on ζ and condition (α.2) is the assertion (i) in Remark 2.6. Then using Remark 2.6 we can deduce that N ζ is well defined. Now we prove its continuity. Take (x, u)

∈ c 0 (N, R n ) × � ∞ (N, U ) arbitrarily.
Then K = {(x t , u t ) : t ∈ N} is compact and there exists ρ ∈ (0, +∞) such that L = {ν = (x, u) ∈ R n × U : d(ν, K) < ρ} is also compact. From Remark 2.7 we know that ζ is uniformly continuous on L. Then for any ε > 0, there exists

δ ε > 0 such that for all ν, ν � ∈ L, if �ν -ν � � < δ ε then �ζ(ν) -ζ(ν � )� < ε. Let (x � , u � ) ∈ c 0 (N, R n ) × � ∞ (N, U ) such that �x -x � � + �u -u � � < δ := min{δ ε , ρ}. From this we obtain for all t ∈ N, �x t -x � t � + �u t -u � t � < ρ . Hence, for all t ∈ N, if we set ν t := (x t , u t ) and ν � t := (x � t , u � t ) then from the definition of set L we see that for all t ∈ N, ν t and ν � t belong to L (because d(ν t , K) = 0 and d(ν � t , K) ≤ d(ν t , ν � t ) = �x t -x � t � + �u t -u � t � < ρ).
On the other hand, since for all t ∈ N, ν t and ν � t belong to L and �ν t -

ν � t � = �x t -x � t � + �u t -u � t � < δ ε then �ζ(ν t ) -ζ(ν � t )� < ε. This inequality is satisfied for all t ∈ N, then sup t∈N �ζ(x t , u t ) -ζ(x � t , u � t )� ≤ ε or equivalently, �N ζ (x, u) -N ζ (x � , u � )� ≤ ε. The last inequality holds for all (x � , u � ) ∈ c 0 (N, R n ) × � ∞ (N, U ) satisfying �x -x � �+�u -u � � < δ and it means that operator N ζ is continuous at (x, u). From the arbitrary choice of (x, u) in c 0 (N, R n ) × � ∞ (N, U ) we obtain N ζ ∈ C 0 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )).
Remark 2.9. Let X, V , W be real Banach spaces, and U be a nonempty subset of V . Let F : X × U → W be a mapping. We say that F is of class C 1 at (x, u) ∈ X × U when there exist an open neighborhood N x,u of (x, u) in X × V and a mapping [START_REF] Munkres | Elementary Differential Topology[END_REF], page 7).

F 1 ∈ C 1 (N x,u , W ) such that F 1| Nx,u∩X×U = F | Nx,u∩X×U . In finite dimension, it is showed that if F is of class C 1 at each point of X × U , then there exist an open subset G of V and a mapping F ∈ C 1 (X × G, W ) such that U ⊂ G and F| X×U = F (see
Remark 2.10. Recall that U is star-shaped with respect to u 0 means that, for all u ∈ U , the segment [u 0 , u] is included in U (see page 93 of [START_REF] Spivak | Calculus on Manifolds[END_REF]). It is also noticed that if U is convex then U is star-shaped with respect to all of its elements.

Remark 2.11. When F 1 , F 2 ∈ C 1 (X × G, W ) such that F 1| X×U = F 2| X×U = F,
when U is star-shaped with respect to u 0 , when u, u 0 ∈ U and x 0 ∈ X, note that, for all θ ∈ (0, 1), we have

F 1 (x 0 , (1 -θ)u 0 + θu) = F 2 (x 0 , (1 -θ)u 0 + θu) = F (x 0 , (1 -θ)u 0 + θu). Therefore we have D 2 F 1 (x 0 , u 0 )(u-u 0 ) = d dθ | θ=0 F 1 (x 0 , (1-θ)u 0 +θu) = d dθ | θ=0 F 2 (x 0 , (1-θ)u 0 +θu) = D 2 F 2 (x 0 , u 0 )(u -u 0 ), and so D 2 F (x 0 , u 0 )(u -u 0 ) does not depend of extension of F .
To treat the differentiability of Nemytskii operator N ζ , we introduce the two following conditions:

(α.3) For all x ∈ R n and for all u ∈ U, the Fréchet differential Dζ(x, u) exists and Dζ is of class C 0 on R n × U ;

(α.4) For all B bounded, nonempty in U : lim x→0 (sup u∈B �Dζ(x, u)�) = 0. Under (α.1), (α.3) and (α.4), condition (α.2) can be lightened into a weaker one which is shown in the following remark.

Remark 2.12. Let U be a nonempty closed subset of R d and let ζ : R n × U →R m be a mapping which belongs to class C 1 (R n × U, R m ). Consider the following assumptions.

(i) lim x→0 (sup u∈B �Dζ(x, u)�) = 0 for all B bounded, nonempty in U .

(ii) There exists u 0 ∈ U such that ζ(0 , u 0 ) = 0 and U is star-shaped with respect to u 0 . If (i) and (ii) hold, then we obtain lim x→0 (sup u∈B �ζ(x, u)�) = 0 for all B bounded, nonempty in U.

Proof. Let ζ be a mapping which satisfies all the assumptions of this remark. It is noticed that in this remark, assumption

ζ ∈ C 1 (R n × U, R m ) is equivalent to ζ satisfies (α.1)
and (α.3); assumption (i) is (α.4); and the conclusion is (α.2). Let B be a nonempty bounded set in U then there exists R ∈ (0, +∞) such that

� B ∪ {u 0 } � ⊂ B R where B R = {u ∈ R d : �u� ≤ R}. Using Mean Value Theorem, for all u ∈ B and for all x ∈ R n , we have �ζ(x, u)� - � � �ζ(x, u 0 ) � � � ≤ � � �ζ(x, u) -ζ(x, u 0 ) � � � ≤ sup w∈[u 0 ,u] �D 2 ζ(x, w)� � � �u -u 0 � � � ≤ 2R. sup w∈B R ∩U �Dζ(x, w)� since sup w∈[u,u 0 ] �D 2 ζ(x, w)� ≤ sup w∈[u,u 0 ] �Dζ(x, w)� ≤ sup w∈B R ∩U �Dζ(x, w)�.
Here we notice that U contains segment [u 0 , u] since U is star-shaped with respect to u 0 and hence, for all x ∈ R n , Dζ(x, .) exists at any w

∈ [u 0 , u]. Besides, B R ∩ U is nonempty bounded subset of U since it contains u 0 and B R ∩ U is a subset of B R . The last inequality is equivalent to the following �ζ(x, u)� ≤ � � �ζ(x, u 0 ) � � � + 2R. sup w∈B R ∩U �Dζ(x, w)� .
This inequality holds for all u ∈ B, hence

sup u∈B �ζ(x, u)� ≤ � � �ζ(x, u 0 ) � � � + 2R. sup w∈B R ∩U �Dζ(x, w)� . (2.1)
Using continuity of ζ and assumption (ii) of this remark, we obtain

lim x→0 ζ(x, u 0 ) = ζ(0, u 0 ) = 0,
Chapter 2. Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle and using assumption (i), since (B R ∩ U ) is a nonempty bounded subset of U , we obtain lim x→0 (sup w∈B R ∩U �Dζ(x, w)�) = 0. Hence, if we take limit when x → 0 on both sides of (2.1), we obtain

lim x→0 � sup u∈B �ζ(x, u)� � = 0.
We will call the weaker condition as follows.

(α.2 � ) There exists u 0 ∈ U such that ζ(0 , u 0 ) = 0 and U is star-shaped with respect to u 0 . From these conditions, we have following theorem on the differentiability of Nemytskii operator.

Theorem 2.13. Let U be a nonempty closed subset of R d and let ζ : R n × U →R m be a mapping which satisfies conditions (α.1), (α.2 � ), (α.3) and (α.4). Then

N ζ ∈ C 1 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )). Moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, U ), we have DN ζ (x, u).(δx, δu) = (Dζ(x t , u t ).(δx t , δu t )) t∈N = (D 1 ζ(x t , u t ).δx t + D 2 ζ(x t , u t ).δu t ) t∈N
where D 1 and D 2 denote the partial Fréchet differentiations.

Proof. Under (α.1), (α.2 � ), (α.3) and (α.4), after Remark 2.12, we can assert that ζ satisfies condition (α.2). Under conditions (α.1) and (α.2), using Theorem 2.8, we obtain

N ζ ∈ C 0 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )). Now we define a new Nemytskii operator N Dζ on c 0 (N, R n ) × � ∞ (N, U ) by setting N Dζ (x, u) := (Dζ(x t , u t )) t∈N for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ).
Under conditions (α.3) and (α.4), by realizing a similar interpretation as in the proof of Theorem 2.8, we obtain

N Dζ is of class C 0 on c 0 (N, R n ) × � ∞ (N, U ). Take (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and ε > 0 arbitrarily. Since N Dζ is continuous at (x, u), there exists δ � > 0 such that for all (x � , u � ) ∈ c 0 (N, R n ) × � ∞ (N, U ) satisfying �x � -x� + �u � -u� < δ � , the following inequality holds �N Dζ (x, u) -N Dζ (x � , u � )� < �. That means for all t ∈ N, �Dζ(x t , u t ) -Dζ(x � t , u � t )� < �.

Now we consider the following expression

G t = � � ζ(x � t , u � t ) -ζ(x t , u t ) -Dζ(x t , u t ).(x � t -x t , u � t -u t ) � � .
For all t ∈ N, using Corollary 1.37 in Chapter 1, we know that

G t ≤ sup (zt,wt)∈[(xt,ut),(x � t ,u � t )] �Dζ(z t , w t ) -Dζ(x t , u t )� �(δx t , δu t )� ,
where

δx t := x � t -x t ∈ R n ; δu t := u � t -u t ∈ R d . For all (z t , w t ) ∈ [(x t , u t ), (x � t , u � t )], we have �z t -x t � + �w t -u t � ≤ �x t -x � t � + �u t -u � t � < δ, hence �Dζ(z t , w t ) -Dζ(x t , u t )� < ε. Therefore, we obtain sup (zt,wt)∈[(xt,ut),(x � t ,u � t )] �Dζ(z t , w t ) -Dζ(x t , u t )� < ε.
Thus, we have

∀t ∈ N, G t ≤ ε �(δx t , δu t )� = ε( � � x t -x � t � � + � � u t -u � t � � ),
which can be rewritten as follows

sup t∈N G t ≤ ε( � � x � -x � � + � � u � -u � � ) = ε.(�δx� + �δu�) ⇐⇒ � � � � ζ(x � t , u � t ) � t∈N -(ζ(x t , u t )) t∈N -(Dζ(x t , u t ).(δx t , δu t )) t∈N � � � ≤ ε.(�δx� + �δu�) ⇐⇒ � � N ζ (x � , u � ) -N ζ (x, u) -N Dζ (x, u).(δx, δu) � � ≤ ε(�δx� + �δu�). The last inequality means that N ζ is Fréchet differentiable at the point (x, u). Since the choice of (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) is arbitrary, N ζ is Fréchet differentiable on c 0 (N, R n ) × � ∞ (N, U ) and moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ) we have DN ζ (x, u).(δx, δu) = N Dζ (x, u).(δx, δu) = (Dζ(x t , u t ).(δx t , δu t )) t∈N = (D 1 ζ(x t , u t ).δx t + D 2 ζ(x t , u t ).δu t ) t∈N . It is noticed that DN ζ (x, u) and N Dζ (x, u) are equal so DN ζ (x, u) is continuous on c 0 (N, R n ) × � ∞ (N, U ) from the continuity of N Dζ (x, u). Hence, N ζ ∈ C 1 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )).
Finally, we introduce a theorem which relates to the functionals in the criterion of Problem (P s ).

Theorem 2.14. Let U be a nonempty closed subset of R d and let φ :

R n × U → R be a mapping. If φ ∈ C 1 (R n × U, R) then the following assertions hold (i) N φ : c 0 (N, R n ) × � ∞ (N, U ) → � ∞ (N, R) is well defined where N φ (x, u) := (φ(x t , u t )) t∈N . (ii) N φ is continuous on c 0 (N, R n ) × � ∞ (N, U ). (iii) For all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), DN φ (x, u) exists and DN φ is continuous on c 0 (N, R n ) × � ∞ (N, U ). Moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ) we have DN φ (x, u).(δx, δu) = (D 1 φ(x t , u t ).δx t + D 2 φ(x t , u t ).δu t ) t∈N . Proof. (i) Let (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ). Let V = {ξ t = (x t , u t ) : t ∈ N}. Then V ⊂ R n × U and V is bounded since sup t∈N �ξ t � ∞ = sup t∈N �(x t , u t )� ∞ ≤ sup t∈N �x t � ∞ + sup t∈N �u t � ∞ = �x� + �u� < +∞. Since φ is continuous on V , φ(V ) is bounded in R, i.e. sup t∈N |φ(ξ t )| = sup t∈N |φ(x t , u t )| < +∞ =⇒ (φ(x t , u t )) t∈N ∈ � ∞ (N, R).
Hence, N φ is well defined and (i) holds.

(ii) After (i) we know that N φ : c 0 (N, R n ) × � ∞ (N, U ) → � ∞ (N, R
) is well defined. By a similar proceeding like in the proof of Theorem 2.8 we obtain the continuity of

N φ on c 0 (N, R n ) × � ∞ (N, U ). (iii) We know that φ ∈ C 1 (R n × U, R) then Dφ is of class C 0 on R n × U . We define new Nemytskii operator DΦ : c 0 (N, R n ) × � ∞ (N, U ) → � ∞ (N, L(R n × R d , R)),
where DΦ(x, u) := (Dφ(x t , u t )) t∈N . Proceeding like in (i) and (ii), we obtain DΦ is well defined and is continuous on c 0 (N, R n ) × � ∞ (N, U ). Then using a similar interpretation like the proof of Theorem 2.13, we obtain assertion (iii), in which DN φ coincides with DΦ. After (i), (ii) and (iii) we can assert that

N φ ∈ C 1 (c 0 (N, R n ) × � ∞ (N, U ); � ∞ (N, R)).

Linear Difference Equations

We will establish a result on the existence of a solution of a nonhomogeneous linear equation which belongs to c 0 (N, R n ) when the second member belongs to c 0 (N, R n ). L(R n , R n ) denotes the space of linear mappings from R n into R n , and

�A� L := sup{�Ax� : x ∈ R n , �x� ≤ 1} is the norm on L(R n , R n ). Proposition 2.15. Let (A t ) t∈N be a sequence in L(R n , R n ), z ∈ c 0 (N, R n ) and w ∈ R n .
We consider the following Cauchy problem

� t+1 = A t � t + z t � 0 = w. � (DE)
We assume that sup t∈N �A t � L < 1. Then the solution of (DE) belongs to c 0 (N, R n ).

Proof. We denotes by � = (� t ) t∈N the solution of (DE). We need to prove that � exists and

� ∈ c 0 (N, R n ).
Obviously, for any z = (z t ) t∈N ∈ c 0 (N, R n ) this system of difference equations has an unique solution which can be generalized by induction as follows

� 0 = w; � 1 = A 0 � 0 + z 0 = A 0 w + z 0 ; � 2 = A 1 � 1 + z 1 = A 1 A 0 w + A 1 z 0 + z 1 ; � 3 = A 2 � 2 + z 2 = A 2 A 1 A 0 w + A 2 A 1 z 0 + A 2 z 1 + z 2 ;
. . .

� t+1 = ( � t i=0 A i )w + ( � t i=1 A i )z 0 + ( � t i=2 A i )z 1 + • • • + A t z t-1 + z t . Now we prove that � = (� t ) t∈N ∈ � ∞ (N, R n ).
From the last equality, we have:

�� t+1 � ≤ ( � t i=0 �A i � L ) �w� + ( � t i=1 �A i � L ) �z 0 � +( � t i=2 �A i � L ) �z 1 � + • • • + �A t � L �z t-1 � + �z t � . Let r = max{�z� ∞ , �w�} < +∞ and M = sup t∈N �A t � < 1. Then we have �� t+1 � ≤ r � ( � t i=0 �A i � L ) + � t i=1 �A i � L + � t i=2 �A i � L + • • • + �A t � L + 1 � < r(1 + M + M 2 + • • • + M t+1 ) ⇒ sup t∈N �� t+1 � ≤ r � +∞ i=0 M i = r 1 1-M < +∞. And so, (� t ) t∈N ∈ � ∞ (N, R n ).
Finally, we prove that � = (� t ) t∈N ∈ c 0 (N, R n ). We have:

∀t ∈ N, � t+1 = A t � t + z t ⇒ ∀t ∈ N, �� t+1 � ≤ �A t � �� t � + �z t � ≤ M �� t � + �z t � . Since (� t ) t∈N ∈ � ∞ (N, R n ), lim sup t→+∞ �� t �
exists and it is finite. Take lim sup t→+∞ of both sides of last inequation and notice that lim sup t→+∞ �� t+1 � = lim sup t→+∞ �� t � and lim sup t→+∞ �z t � = 0 (since (z t ) t ∈ c 0 (N, R n )) then we obtain the following inequality lim sup

t→+∞ �� t � ≤ M lim sup t→+∞ �� t � ⇔ (1 -M ) lim sup t→+∞ �� t � ≤ 0.
From this we obtain lim sup t→+∞ �� t � = 0 because M < 1. Now since lim sup t→+∞ �� t � = 0 we obtain lim t→+∞ �� t � = 0. Therefore,

(� t ) t∈N ∈ c 0 (N, R n ). Corollary 2.16. Let (B t ) t∈N be a sequence in L(R n , R n ), d ∈ c 0 (N, R n
) and e ∈ R n . We consider the following Cauchy problem

k t+1 = B t k t + d t , k 0 = e. � (DE1)
We assume that there exists t * ∈ N such that sup t≥t * �B t � L < 1. Then the solution of (DE1) belongs to c 0 (N, R n ).

Proof. For all t ∈ N, we set

A t := B t+t * ∈ L(R n , R n ) and z t := d t+t * . Then we have sup t∈N �A t � L < 1 and z ∈ c 0 (N, R n ).
We denote by k the solution of (DE1). We set

� t := k t+t * for all t ∈ N. Then we have � t+1 = k t+t * +1 = B t+t * k t+t * + d t+t * = A t � t + z t for all t ∈ N and � 0 = k t * ∈ R n . Using Proposition 2.
15 we obtain lim t→+∞ � t = 0, i.e. lim t→+∞ k t+t * = 0 which implies lim t→+∞ k t = 0.

Static Optimization -a Result in Abstract Banach Spaces

In this section we establish a result in the form of Karush-Kuhn-Tucker theorem in abstract Banach spaces. It will be useful for the proof of weak Pontryagin principles in next sections.

Lemma 2.17. Let X , V, W be real Banach spaces, and U be a nonempty subset of V.

Let J ∈ C 1 (X × U, R) and Γ ∈ C 1 (X × U, W). Let (x, û) be a solution of the following optimization problem � Maximize J (x, u) when x ∈ X , u ∈ U, Γ(x, u) = 0.
We assume that D 1 Γ(x, û) is invertible and that U is star-shaped with respect to û. Then there exists M ∈ W * which satisfies the following conditions.

(i) D 1 J (x, û) + M • D 1 Γ(x, û) = 0. (ii) ∀u ∈ U, �D 2 J (x, û) + M • D 2 Γ(x, û), u -û� ≤ 0.
Proof. Let U 1 be an open subset of V such that U ⊂ U 1 and such that there exists

Γ 1 ∈ C 1 (X × U 1 , W) such that Γ 1| X ×U = Γ. Since D 1 Γ 1 (x, û) = D 1 Γ(x, û
) is invertible, we can use Implicit Function Theorem and assert that there exist where B(u) = J (π(u), u). Since B is differentiable (as a composition of differentiable mappings) and N û ∩ U is also star-shaped with respect to û, a necessary condition of optimality for the last problem is

N x an open neighborhood of x in X , N û an open convex neighborhood of û in U 1 , and a mapping π ∈ C 1 (N û, N x) such that {(x, u) ∈ N x × N û : Γ 1 (x, u) = 0} = {(π(u), u) : u ∈ N û}. Differentiating Γ 1 (π(u), u) = 0 at û we obtain D 1 Γ 1 (x, û) • Dπ(û) + D 2 Γ 1 (x, û) = 0 which implies Dπ(û) = -(D 1 Γ(x, û)) -1 • D 2 Γ(x, û). ( 2
∀u ∈ N û ∩ U, �DB(û), u -û� ≤ 0, (2.3) since 0 ≥ lim θ→0+ 1 θ (B(û + θ(u -û)) -B(û)) = �DB(û), u -û�. When u ∈ U, there exists θ u ∈ (0, 1) such that (1 -θ u )û + θ u u ∈ N û ∩ U. Using (2.3) we obtain θ u • �DB(û), u -û�) = �DB(û), θ u (u -û)� = �DB(û), [(1 -θ u )û + θ u u] -û� ≤ 0,
and so we obtain ∀u ∈ U, �DB(û), u -û� ≤ 0.

(2.4)

Using the chain rule we obtain

DB(û) = D 1 J (x, û) • Dπ(û) + D 2 J (x, û). (2.5)
We define

M := -D 1 J (x, û) • (D 1 Γ(x, û)) -1 ∈ W * . (2.6) From (2.6) we obtain D 1 J (x, û) + M • D 1 Γ(x, û) = 0. (2.7)
Using (2.5) and (2.2) we have

DB(û) = -D 1 J (x, û) • (D 1 Γ(x, û)) -1 • D 2 Γ(x, û) + D 2 J (x, û) = M • D 2 Γ(x, û) + D 2 J (x, û)
and therefore, from (2.4) we obtain

∀u ∈ U, �D 2 J (x, û) + M • D 2 Γ(x, û), u -û� ≤ 0. (2.8)
Remark 2.18. There exist several results like this one in the books [START_REF] Colonius | Optimal Periodic Control[END_REF] and [START_REF] Werner | Optimization Theory and Applications[END_REF] which use the convexity of U . In the necessary conditions of optimality we prefer to avoid the convexity of the sets; it is why we have established this lemma.

Weak Pontryagin Principle for Problem (Ps)

We start by a translation of Problem (Ps) into a simpler abstract optimization problem in Banach spaces. Let T :

c 0 (N, R n ) × � ∞ (N, U ) → c 0 (N, R n ) × R n where T (x, u) = (F (x, u), h(x, u)) . Here F (x, u) := (f (x t , u t ) -x t+1 ) t∈N and h(x, u) := x 0 -η.
Then we can translate Problem (P s ) into the following problem.

(P 2)      Maximize J(x, u) when x ∈ c 0 (N, R n ), u ∈ � ∞ (N, U ) T (x, u) = 0.
We consider the following list of assumptions:

(A1) U is a nonempty closed subset of R d . (A2) φ ∈ C 1 (R n × U, R) and f ∈ C 1 (R n × U, R n ).
(A3) There exists u 0 ∈ U such that f (0, u 0 ) = 0 and U is star-shaped with respect to u 0 .

(A4) lim x→0 (sup u∈B �Df (x, u)� L ) = 0 for all nonempty bounded subset B ⊂ U .

Lemma 2.19. Under condition (A2), the functional J is of class

C 1 on c 0 (N, R n ) × � ∞ (N, U
) and moreover, the following formula holds for all (x, u), (δx, δu)

∈ c 0 (N, R n ) × � ∞ (N, U ) DJ(x, u).(δx, δu) = +∞ � t=0 β t D 1 φ(x t , u t )δx t + +∞ � t=0 β t D 2 φ(x t , u t )δu t .
Proof. Under condition (A2), the Nemytskii operator

N φ : c 0 (N, R n ) × � ∞ (N, U ) → � ∞ (N, R) defined by N φ (x, u) := (φ(x t , u t )) t∈N is of class C 1 on c 0 (N, R n ) × � ∞ (N, U ) using Theorem 2.14. Moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ), we have DN φ (x, u).(δx, δu) = (D 1 φ(x t , u t )δx t ) t∈N + (D 2 φ(x t , u t )δu t ) t∈N .
Consider the functional S :

� ∞ (N, R) → R defined by S(r) := � +∞ t=0 β t r t . It is easy to verify that S(α 1 r 1 + α 2 r 2 ) = α 1 S(r 1 ) + α 2 S(r 2 ) for all α 1 , α 2 ∈ R and for all r 1 , r 2 ∈ � ∞ (N, R). Hence, S is linear. Besides, |S(r)| ≤ ( � +∞ t=0 β t ) �r� ∞ = 1 1-β �r� ∞ < +∞ so S is bounded and consequently, continuous. From Example 1.29 in Chapter 2, we obtain S is of class C 1 and moreover, for all r, δr ∈ � ∞ (N, R), we have DS(r).δr = S(δr) = � +∞ t=0 β t δr t . We see that J = S • N φ is of class C 1 as a composition of two C 1 - mappings. Using the chain rule we obtain for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ) we have DJ(x, u).(δx, δu) = DS(N φ (x, u)) • DN φ (x, u).(δx, δu) = S(DN φ (x, u).(δx, δu)) = +∞ � t=0 β t D 1 φ(x t , u t )δx t + +∞ � t=0 β t D 2 φ(x t , u t )δu t . Lemma 2.20. Under conditions (A1-A4), the Nemytskii operator N f : c 0 (N, R n )× � ∞ (N, U ) → c 0 (N, R n ) defined by N f (x, u) := (f (x t , u t )) t∈N and the operator F defined before (F (x, u) := (f (x t , u t ) -x t+1 ) t∈N ) are of class C 1 on c 0 (N, R n ) × � ∞ (N, U ) and moreover, the following formulas hold for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ): 1. DN f (x, u).(δx, δu) = (D 1 f (x t , u t )δx t ) t + (D 2 f (x t , u t )δu t ) t ; 2. DF (x, u).(δx, δu) = (D 1 f (x t , u t )δx t -δx t+1 ) t + (D 2 f (x t , u t )δu t ) t .
Proof. Under conditions (A2-A4), we can assert that the mapping f satisfies all the conditions (α.1), (α.2 � ), (α.3) and (α.4) (by replacing ζ by f ). Then using condition (A1) and Theorem 2.13, we know that

N f ∈ C 1 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R n ). Moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ), we have DN f (x, u).(δx, δu) = (D 1 f (x t , u t )δx t ) t∈N + (D 2 f (x t , u t )δu t ) t∈N .
So conclusion 1 is proven. Now we consider operator A(x, u) We see that

:= (-x t+1 ) t∈N from c 0 (N, R n ) × � ∞ (N, U ) into c 0 (N, R n ). Obviously, A is linear and from �A(x, u)� ≤ �x� ∞ ≤ �x� ∞ + �u� ∞ < +∞, we obtain A is bounded and hence, continuous. So A is of class C 1 and for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ) we have DA(x, u).(δx, δu) = A(δx, δu) = (-δx t+1 ) t∈N .
F = N f + A then F is of class C 1 as a sum of two C 1 -mappings. Moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ) and for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ) we have DF (x, u).(δx, δu) = DN f (x, u).(δx, δu) + DA(x, u).(δx, δu) = (D 1 f (x t , u t )δx t ) t + (D 2 f (x t , u t )δu t ) t + (-δx t+1 ) t = (D 1 f (x t , u t )δx t -δx t+1 ) t + (D 2 f (x t , u t )δu t ) t .
Hence, conclusion 2 is proven.

Under conditions (A1-A4), using Lemma 2.19 and Lemma 2.20 we obtain J and

F are continuously differentiable on c 0 (N, R n ) × � ∞ (N, U ). Now we consider T (x, u) = (F (x, u), h(x, u)) where F (x, u) = (f (x t , u t ) -x t+1
) t∈N and h(x, u) = x 0η as they were defined before. The operator h is obviously an affine continuous mapping since we can consider h as a sum of linear continuous mapping Pr 0 (projection with respect to the 0-coordinate of x) and a constant -η. Here Pr 0 :

c 0 (N, R n ) × � ∞ (N, U ) → R n defined by Pr 0 (x, u) := x 0 . We know that mapping Pr 0 is of class C 1 on c 0 (N, R n ) × � ∞ (N, U ) and for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ), we have D Pr 0 (x, u).(δx, δu) = Pr 0 (δx, δu) = δx 0 . In addition, a constant -η is continuously differentiable on c 0 (N, R n ) × � ∞ (N, U ) and its differential is null function. Therefore, h is of class C 1 on c 0 (N, R n ) × � ∞ (N, U ) and for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), for all (δx, δu) ∈ c 0 (N, R n ) × � ∞ (N, R d ) we have Dh(x, u).(δx, δu) = (D Pr 0 +0)(x, u) • (δx, δu) = Pr 0 (δx, δu) = δx 0 .
More specifically, Dh(x, u).(δx, δu) = D 1 h(x, u).δx + D 2 h(x, u).δu and from the abovementioned arguments, we obtain D 1 h(x, u).δx = δx 0 and D 2 h(x, u).δu = 0. From this assertion, we obtain T is of class

C 1 on c 0 (N, R n ) × � ∞ (N, U ) since each of its element mappings F and h are of class C 1 on c 0 (N, R n ) × � ∞ (N, U ). Lemma 2.21. We assume assumptions (A1-A4) fulfilled. Let (x, û) ∈ c 0 (N, R n ) × � ∞ (N, U
) be a solution of (P2) and assume that for all t ∈ N, U is star-shaped with respect to ût . Then there exists (q, µ) ∈ � 1 (N, R n * ) × R n * which satisfies the two following conditions.

(i) D 1 J(x, û) + (q, µ) • D 1 T (x, û) = 0. (ii) For all u ∈ � ∞ (N, U ), � D 2 J(x, û) + (q, µ) • D 2 T (x, û), u - û� ≤ 0.
Proof. Under conditions (A1-A4), after Lemma 2.19 and Lemma 2.20 we know that the mappings J and T are of class

C 1 on c 0 (N, R n ) × � ∞ (N, U ).
Hence, the partial Fréchet differential with respect to x of mapping T exists at the optimal solution (x, û). We will prove that

D 1 T (x, û) is invertible. Let d = (d t ) t∈N ∈ c 0 (N, R n
) and e ∈ R n arbitrarily. We need to show that there exists an unique k

= (k t ) t∈N ∈ c 0 (N, R n ) such that D 1 T (x, û).k = (-d, e). Since D 1 T (x, û).k = (D 1 F (x, û).k, D 1 h(x, û).k),
2.6. Weak Pontryagin Principle for Problem (Ps)

49
the problem is equivalent to the following one

� D 1 F (x, û).k = -d, D 1 h(x, û).k = e ⇔ � D 1 F (x, û).k = -d, k 0 = e.
Now using Lemma 2.20, we know that D 1 F (x, û).k = (D 1 f (x t , ût )k tk t+1 ) t∈N then system above can be translated into the following problem of solutions in space c 0 (N, R n ) of difference equations

� k t+1 = B t k t + d t , t ∈ N, k 0 = e,
where

B t = D 1 f (x t , ût ) ∈ L(R n , R n ).
We will prove that sequence (B t ) t∈N satisfies the property mentioned in Corollary 2.16. Set B := {û t : t ∈ N} then B is nonempty bounded set in U since û ∈ � ∞ (N, U ). For all t ∈ N, we have

0 ≤ �B t � L = �D 1 f (x t , ût )� L ≤ �Df (x t , ût )� L ≤ sup u∈ B �Df (x t , u)�
and therefore, using (A4) and Remark 2.6 we obtain lim t→+∞ �B t � L = 0. Hence, there exists t * ∈ N such that when sup t≥t * �B t � L < 1. Finally, using Corollary 2.16 we obtain the unique solution k = (k t ) t∈N of the above system belongs to c 0 (N, R n ).

And so, we have proven that for any

d = (d t ) t∈N ∈ c 0 (N, R n ) and e ∈ R n the equation D 1 T (x, û).k = (-d, e
) always has an unique solution (k t ) t∈N that belongs to c 0 (N, R n ). It means that operator D 1 T (x, û) is bijective, hence D 1 T (x, û) is invertible. Now take u in � ∞ (N, U ) and take α ∈ (0, 1) arbitrarily. From the boudedness of û and u we obtain αû + (1α)u is bounded. Moreover, for all t ∈ N, since U is star-shaped with respect to ût we have αû t +(1-α)u t ∈ U . Hence, we deduce that û+(1-α)u ∈ � ∞ (N, U ). It means that � ∞ (N, U ) is star-shaped with respect to û.

Recall that in Chapter 1, we have proven that � 1 (N, R n * ) can be assimilated to the dual topological space of c 0 (N, R n ), i.e. an element of � 1 (N, R n * ) can be considered as a continuous linear functional on c 0 (N, R n ). Now all the assumptions of Lemma 2.17 are fulfilled and we can use Lemma 2.17 and assert that there exists Lagrange multiplier (q, µ)

∈ (c 0 (N, R n ) × R n ) * = � 1 (N, R n * ) ×R n *
which satisfies the announced conclusions.

Theorem 2.22. We assume assumptions (A1-A4) fulfilled. Let (x, û) ∈ c 0 (N, R n ) × � ∞ (N, U ) be a solution of Problem (P s ) and assume that for all t ∈ N, U is star-shaped with respect to ût . Then there exists p

∈ � 1 (N * , R n * ) such that (AE1) p t = p t+1 • D 1 f (x t , ût ) + β t .D 1 φ(x t , ût ) for all t ∈ N * ; (WM1) � p t+1 • D 2 f (x t , ût ) + β t .D 2 φ(x t , ût ), u -ût � ≤ 0 for all t ∈ N, for all u ∈ U. Proof. Since (x, û) ∈ c 0 (N, R n )×� ∞ (N, U ) is a solution of Problem (P s ), it is also a solution of Problem (P2). Then Lemma 2.21 provides (q, µ) ∈ � 1 (N, R n * ) ×R n * such that D 1 J(x, û) + (q, µ) • D 1 T (x, û) = 0, � D 2 J(x, û) + (q, µ) • D 2 T (x, û), u - û� ≤ 0 � (2.9)
for all u ∈ � ∞ (N, U ). In (2.9), the equation can be rewritten equivalently as follows Then for all δx ∈ c 0 (N, R n ) such that δx 0 = 0 we have

D 1 J(x, û) + q • D 1 F (x, û) + µ • D 1 h(x, û) = 0.
� D 1 J(x, û) + q • D 1 F (x, û) + µ • D 1 h(x, û) � .δx = 0. (2.10)
Using results of Lemma 2.19 and Lemma 2.20, we know that

D 1 J(x, û).δx = +∞ � t=0 β t D 1 φ(x t , ût )δx t , D 1 F (x, û).δx = (D 1 f (x t , u t )δx t -δx t+1 ) t∈N and D 1 h(x, û).δx = δx 0 = 0. Hence, equation (2.10) becomes +∞ � t=0 β t D 1 φ(x t , ût )δx t + +∞ � t=0 �q t , D 1 f (x t , ût )δx t -δx t+1 � = 0 ⇐⇒ +∞ � t=1 β t D 1 φ(x t , ût )δx t + +∞ � t=1 �q t , D 1 f (x t , ût )δx t -δx t+1 � -�q 0 , δx 1 � = 0. (2.11)
We fix t ∈ N * arbitrarily. Take δx = (δx s ) s∈N * such that when s � = t, δx s = 0 and δx t varies in R n . Then the last equation becomes

β t D 1 φ(x t , ût )δx t + (q t • D 1 f (x t , ût )) • δx t -q t-1 • δx t = 0 ⇔ � β t D 1 φ(x t , ût ) + q t • D 1 f (x t , ût ) -q t-1 , δx t � = 0.
This equality is verified for all δx t ∈ R n and for all t ∈ N * . And so we have

∀t ∈ N * , q t-1 = q t • D 1 f (x t , ût ) + β t D 1 φ(x t , ût ).
(2.12)

Now in (2.9), the inequation can be rewritten equivalently as follows

� D 2 J(x, û) + q • D 2 F (x, û) + µ • D 2 h(x, û), u - û� ≤ 0 for all u ∈ � ∞ (N, U ).
Using results of Lemma 2.19 and Lemma 2.20, we know that D 2 J(x, û).(u -

û) = +∞ � t=0 β t D 2 φ(x t , ût )(u t -ût ), D 2 F (x, û).(u -û) = (D 2 f (x t , u t ).(u t -ût )) t∈N and D 2 h(x, û).(u -û) = 0. Hence, the last inequation becomes +∞ � t=0 β t D 2 φ(x t , ût )(u t -ût ) + +∞ � t=0 �q t • D 2 f (x t , ût ), (u t -ût )� ≤ 0 for all (u t ) t∈N ∈ � ∞ (N, U ).
(2.13) Fix t ∈ N arbitrarily. Take u = (u s ) s∈N such that when s � = t, u s = ûs and u t varies in U . Then from equation (2.12) we have

β t D 2 φ(x t , ût )(u t -ût ) + �q t • D 2 f (x t , ût ), (u t -ût )� ≤ 0 ⇐⇒ � q t • D 2 f (x t , ût ) + β t .D 2 φ(x t , ût ), u t -ût � ≤ 0.
This inequation is verified for all u t ∈ U and for all t ∈ N. Hence, we have

∀t ∈ N, ∀u ∈ U, � q t • D 2 f (x t , ût ) + β t .D 2 φ(x t , ût ), u t -ût � ≤ 0. (2.14)
In (2.12) and (2.14), by setting p t+1 := q t for all t ∈ N, we obtain p = (p t+1 ) t∈N ∈ � 1 (N * , R n * ) such that the following statements hold

(i) p t = p t+1 • D 1 f (x t , ût ) + β t .D 1 φ(x t , ût ) for all t ∈ N * . (ii) � p t+1 • D 2 f (x t , ût ) + β t .D 2 φ(x t , ût ), u -ût � ≤ 0 for all t ∈ N, for all u ∈ U.
These statements are conclusions (AE1) and (WM1). The proof is complete. Maximize K(y, u) :=

� +∞ t=0 β t ψ(y t , u t ) when y = (y t ) t∈N ∈ (R n ) N , u = (u t ) t∈N ∈ U N y 0 = η, lim t→+∞ y t = y ∞ , u is bounded, ∀t ∈ N, y t+1 = g(y t , u t ).              (P m ) By setting z t := y t -y ∞ for all t ∈ N, N (z, u) := � +∞ t=0 β t ϕ(z t , u t ) where ϕ(z, u) := ψ(z + y ∞ , u) for all (z, u) ∈ R n × U and �(z, u) := g(z + y ∞ , u) -y ∞ for all (z, u) ∈ R n × U ,
we can translate Problem (P m ) into Problem (P 1) which has the form of Problem (P s )

Maximize N (z, u) :=

� +∞ t=0 β t ϕ(z t , u t ) when z = (z t ) t∈N ∈ c 0 (N, R n ), u = (u t ) t∈N ∈ � ∞ (N, U ), z 0 = η -y ∞ , ∀t ∈ N, z t+1 = �(z t , u t ).              (P 1)
Let (ẑ, û) be a solution of Problem (P 1) and assume that for all t ∈ N, U is star-shaped with respect to ût . Then by the inverse transformation, (ŷ, û) is a solution of Problem (P m ). Apply Theorem 2.22 to Problem (P 1) we know that if in Problem (P 1) the following conditions

(1) U is a nonempty closed subset of R d .
(

) ϕ ∈ C 1 (R n × U, R), � ∈ C 1 (R n × U, R n ), 2
(3) There exists u 0 ∈ U such that � (0, u 0 ) = 0 and U is star-shaped with respect to u 0 , (4) lim x→0 (sup u∈B �D�(z, u)�) = 0 for all nonempty bounded subset B ⊂ U are fulfilled then there exists (p t+1

) t∈N ∈ � 1 (N * , R n * ) such that (i) p t = p t+1 • D 1 �(ẑ t , ût ) + β t .D 1 ϕ(ẑ t , ût ) for all t ∈ N * . (ii) � p t+1 • D 2 �(ẑ t , ût ) + β t .D 2 ϕ(ẑ t , ût ), u -ût � ≤ 0 for all t ∈ N, for all u ∈ U.
Now we will study the statement of weak Pontryagin principle for our main problem based on that for Problem (P 1).

-Study the assumptions -The assumption on U and assumption (1) remain unchanged.

-Because of the definitions of functions ψ, g, ϕ and �, it is evident that the assumption (2) above is equivalent to following condition.

(

) ⇐⇒ ψ ∈ C 1 (R n × U, R) and g ∈ C 1 (R n × U, R n ). 2
-From assumption (3) above, we make the equivalent changes as follows:

�(0, u 0 ) = 0 ⇐⇒ g(0 + y ∞ , u 0 ) -y ∞ = 0 ⇐⇒ g(y ∞ , u 0 ) = y ∞ .
And finally, we translate assumption (4) as follows.

lim z→0 (sup u∈B �D�(z, u)�) = 0 ⇔ lim z→0 (sup u∈B �D(g(z + y ∞ , u) -y ∞ )�) = 0 y:=z+y∞ ⇔ lim y→y∞ (sup u∈B �D(g(y, u) -y ∞ )�) = 0 ⇔ lim y→y∞ (sup u∈B �Dg(y, u)�) = 0.
Chapter

(i) p t = p t+1 • D 1 g(ŷ t , ût ) + β t .D 1 ψ(ŷ t , ût ) for all t ∈ N * , (ii) � p t+1 • D 2 g(ŷ t , ût ) + β t .D 2 ψ(ŷ t , ût ), u -ût � ≤ 0 for all t ∈ N, for all u ∈ U.
Here we notice that

D 1 g(ŷ t , ût ) = D 1 �(ẑ t , ût ); D 1 ψ(ŷ t , ût ) = D 1 ϕ(ẑ t , ût ); D 2 g(ŷ t , ût ) = D 2 �(ẑ t , ût ) and D 2 ψ(ŷ t , ût ) = D 2 ϕ(ẑ t , ût ).
The reasoning is like what we did when studying assumption (4). Finally, we can state the weak Pontryagin principle for main problem.

Theorem 2.24. Assume that the following conditions are fulfilled

(B1) U is a nonempty closed subset of R d . (B2) ψ ∈ C 1 (R n × U, R), g ∈ C 1 (R n × U, R n ).
(B3) There exists u 0 ∈ U such that g(y ∞ , u 0 ) = y ∞ and U is star-shaped with respect to u 0 .

(B4) lim y→y∞ (sup u∈B �Dg(y, u)�) = 0.

Let (ŷ, û) in c 0 (N, R n ) × � ∞ (N, U
) be a solution of Problem (P m ) and assume that for all t ∈ N, U is star-shaped with respect to ût . Then there exists

(p t+1 ) t∈N ∈ � 1 (N * , R n * ) such that (AE) p t = p t+1 • D 1 g(ŷ t , ût ) + β t .D 1 ψ(ŷ t , ût ) for all t ∈ N * . (WM) � p t+1 • D 2 g(ŷ t , ût ) + β t .D 2 ψ(ŷ t , ût ), u -ût � ≤ 0 for all t ∈ N, for all u ∈ U.
Chapter 3

Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality

Introduction

This chapter is devoted to establish strong Pontryagin principles and sufficient conditions of optimality for the problems introduced in Chapter 2. The content of this chapter is as follows:

-In Section 3.2, we recall the main problem and the supporting problem which were introduced in Chapter 2.

-In Section 3.3, by fixing the sequence of control variable, we study the properties of Nemytskii operator from c 0 (N, R n ) into c 0 (N, R m ) and of Nemystskii operator from

c 0 (N, R n ) into � ∞ (N, R).
-In Section 3.4, we recall a result on static optimization in Banach spaces which is useful for establishing strong Pontryagin principles.

-In Section 3.5 and 3.6, we establish strong Pontryagin principles for the supporting problem and the main problem.

-In Section 3.7 and 3.8, we establish results on sufficient condition of optimality for them.

Recall of the Main Problem and the Supporting Problem

Let U be nonempty subset of R d . We consider the same problem like in Chapter 2.

Maximize K(y, u) :=

� +∞ t=0 β t ψ(y t , u t ) when y = (y t ) t∈N ∈ (R n ) N , u = (u t ) t∈N ∈ U N y 0 = η, lim t→+∞ y t = y ∞ , u is bounded, ∀t ∈ N, y t+1 = g(y t , u t ).              (P m )
Chapter 3. Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality Here functional K, functions ψ, g, real number β and elements η, y ∞ were already defined in Chapter 2. For this problem, we have introduced its supporting problem in previous chapter as follows Maximize J(x, u) :=

� +∞ t=0 β t φ(x t , u t ) when x = (x t ) t∈N ∈ c 0 (N, R n ), u = (u t ) t∈N ∈ � ∞ (N, U ), x 0 = η, ∀t ∈ N, x t+1 = f (x t , u t ).              (P s )
Functional J, functions φ, f , real number β and element η were defined in Chapter 2. We knew that Problem (P m ) can be translated into the form of Problem (P s ) by the following transformation:

-For all t ∈ N,

z t := y t -y ∞ . Then z = (z t ) t∈N ∈ c 0 (N, R n ). -Set N (z, u) := � +∞ t=0 β t ϕ(z t , u t ) where ϕ(z, u) := ψ(z +y ∞ , u) for all (z, u) ∈ R n ×U then K(y, u) = N (z, u). -Set �(z, u) := g(z + y ∞ , u) -y ∞ for all (z, u) ∈ R n × U then from y t+1 = g(y t , u t ) we get the equivalent equation z t+1 = �(z t , u t ).

Now the main problem becomes the following one

Maximize N (z, u) :=

� +∞ t=0 β t ϕ(z t , u t ) when z = (z t ) t∈N ∈ c 0 (N, R n ), u = (u t ) t∈N ∈ � ∞ (N, U ), z 0 = η -y ∞ , ∀t ∈ N, z t+1 = �(z t , u t ),              (P 1)
which has the form of supporting problem.

Some Useful Properties of Nemytskii Operators

Let U be nonempty subset of R d . We recall the conditions (α.1) and (α.2) in Chapter 2. For the sake of consistence, in this chapter, we will call those conditions by (β.1) and (β.2).

(β.1) ζ ∈ C o (R n × U, R m ) . (β.2) For all B bounded, nonempty in U , lim x→0 (sup u∈B �ζ(x, u)�) = 0.
Now to deal with partial Fréchet differential of Nemytskii operator, we introduce the following conditions. Proof. Let B be a nonempty bounded subset of U . We fix R ∈ (0, +∞). For all x ∈ R n such that �x� ≤ R, using (β.3) and the Mean Value Theorem we have

(β.3) For all (x, u) ∈ R n × U, D 1 ζ(x, u) exists and, for all u ∈ U , D 1 ζ(., u) ∈ C 0 (R n , L(R n , R m )).
�ζ(x, u)� -�ζ(0, u)� ≤ �ζ(x, u) -ζ(0, u)� ≤ sup z∈[0,x] �D 1 ζ(z, u)� �x� ≤ �x� . sup �z�≤R �D 1 ζ(z, u)� .
Then using (β.6) we have

�ζ(x, u)� ≤ �ζ(0, u)� + �x� . sup �z�≤R �D 1 ζ(z, u)� ≤ �x� . sup �z�≤R �D 1 ζ(z, u)� .
Take supremum when u ∈ B on both sides of the last inequality, we obtain

sup u∈B �ζ(x, u)� ≤ �x� . sup u∈B sup �z�≤R �D 1 ζ(z, u)� . (3.1) Let V = {z ∈ R n : �z� ≤ R} × B. Then V is nonempty bounded subset of R n × U . Using (β.5) we obtain sup u∈B sup �z�≤R �D 1 ζ(z, u)� = sup (z,u)∈V �D 1 ζ(z, u)� < +∞.
Finally, take limit when x → 0 on both sides of (3.1) we have

lim x→0 sup u∈B �ζ(x, u)� ≤ lim x→0 � �x� . sup u∈B sup �z�≤R �D 1 ζ(z, u)� � = sup u∈B sup �z�≤R �D 1 ζ(z, u)� lim x→0 �x� = 0.
The final expression indicates that condition (β.2) holds.

Theorem 3.2. Let U be nonempty subset of R d and let ζ : R n × U → R m be a mapping. Under conditions (β.1), (β.3β.6), the following assertions hold.

1. N ζ ∈ C 0 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )). 2. For all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), D 1 N ζ (x, u) exists. 3. For all u ∈ � ∞ (N, U ), the partial differential D 1 N ζ (., u) belongs to C 0 (c 0 (N, R n ); L(c 0 (N, R n ), c 0 (N, R m ))). Moreover, for all x ∈ c 0 (N, R n ) and for all δx ∈ c 0 (N, R n ) we have D 1 N ζ (x, u).δx = (D 1 ζ(x t , u t ).δx t ) t∈N .
Proof. Under conditions (β.1), (β.3), (β.5) and (β.6), using Remark 3.1 we obtain ζ satisfies property (β.2). Then under (β.1) and (β.2), using Theorem 2.8 we deduce that

N ζ ∈ C 0 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R m )). Let D 1 N : c 0 (N, R n ) × � ∞ (N, U ) → c 0 (N, L(R n , R m )) defined by D 1 N (x, u) := (D 1 ζ(x t , u t )) t∈N . Under (β.4) and (β.5), after Theorem 2.2, D 1 N is well defined. Now we fix u ∈ � ∞ (N, U ). We take x = (x t ) t∈N ∈ c 0 (N, R n ) arbitrarily. Let K = {(x t ) : t ∈ N} -the closure of the bounded set {(x t ) : t ∈ N} then K is compact since {(x t ) : t ∈ N} is a bounded subset of R n . Let L = {x ∈ R n : d(x, K) < ρ}.
By the analogous argument as in Chapter 2, we obtain that L is also compact. Under (β.3), by fixing u and proceeding as in the proof of Theorem 2.8 we deduce that for all t ∈ N, D 1 ζ(., u t ) is uniformly continuous on L and hence, D 1 N (., u) is of class C 0 on c 0 (N, R n ). That means for all ε > 0, there exists δ ε ∈ (0, ρ) such that

1 N (x, u) -D 1 N (x � , u)� < ε for all x � ∈ c 0 (N, R n ) satisfying �x -x � � < δ ε . Then we deduce that, for all t ∈ N, �D 1 ζ(x t , u t ) -D 1 ζ(x � t , u t )� < ε whenever �x t -x � t � < δ ε .

Let us consider the following expression

G t = � � ζ(x � t , u t ) -ζ(x t , u t ) -D 1 ζ(x t , u t ).(x � t -x t ) � � where t ∈ N.
For all t ∈ N, using Corollary 1.37 in Chapter 1, we have

G t ≤ sup zt ∈[xt,x � t ] �D 1 ζ(z t , u t ) -D 1 ζ(x t , u t )� �δx t � ,
where

δx t := x � t -x t . Now for all t ∈ N, for all z t ∈ [x t , x � t ] , we have�z t -x t � ≤ �x t -x � t � < δ ε and hence �D 1 ζ(z t , u t ) -D 1 ζ(x t , u t )� < ε, which implies that sup zt ∈[xt,x � t ] �D 1 ζ(z t , u t ) -D 1 ζ(x t , u t )� ≤ ε. So we have ∀t ∈ N, G t ≤ ε �δx t � .
Hence,

sup t∈N G t ≤ ε. �δx� where δx = (δx t ) t∈N ⇐⇒ � � � � ζ(x � t , u t ) � t∈N -(ζ(x t , u t )) t∈N -(D 1 ζ(x t , u t )) t∈N .(δx t ) t∈N � � � ≤ ε �δx� ⇐⇒ � � N ζ (x � , u) -N ζ (x, u) -D 1 N (x, u).δx � � ≤ ε �δx� . The last inequality means that N ζ is partial Fréchet differentiable with respect to x at point (x, u). Since the choice of x ∈ c 0 (N, R n ) is arbitrary, N ζ is partially Fréchet differentiable with respect to x on c 0 (N, R n ). Moreover, for all x ∈ c 0 (N, R n ) and for all δx ∈ c 0 (N, R n ) we have D 1 N ζ (x, u).δx = D 1 N (x, u).δx = (D 1 ζ(x t , u t ).δx t ) t∈N .
Therefore, D 1 N ζ (., u), and D 1 N (., u) are equal. And so,

D 1 N ζ (., u) is continuous on c 0 (N, R n ) from the continuity of D 1 N (., u). Hence, for all u ∈ � ∞ (N, U ), D 1 N ζ (., u) ∈ C 0 (c 0 (N, R n ); L(c 0 (N, R n ), c 0 (N, R m ))).
Theorem 3.3. Let U be nonempty subset of R d and let φ : R n × U → R be a mapping which satisfies conditions (β.1) and (β.3) with m = 1. Then the following assertions hold

(i) N φ : c 0 (N, R n ) × � ∞ (N, U ) → � ∞ (N, R) is well defined. (ii) N φ is continuous on c 0 (N, R n ) × � ∞ (N, U ). (iii) For all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), D 1 N φ (x, u) exists and for all u ∈ � ∞ (N, U ), D 1 N φ (., u) is continuous on c 0 (N, R n ). Moreover, for all x ∈ c 0 (N, R n )
and for all δx ∈ c 0 (N, R n ) we have

D 1 N φ (x, u).(δx, δu) = (D 1 φ(x t , u t ).δx t ) t∈N .
Proof. The assertions (i) and (ii) are directly derived from (i) and (ii) of Theorem 2.14. Now we prove the assertion (iii). Under (β.3) we know that for all (x, u) ∈ R n × U, D 1 φ(x, u) exists and, for all

u ∈ U , D 1 φ(., u) ∈ C 0 (R n , L(R n , R)). We fix u ∈ � ∞ (N, U ) and we set D 1 Φ(., u) : c 0 (N, R n ) → � ∞ (N, L(R n , R)) where D 1 Φ(x, u) := (D 1 φ(x t , u t )) t∈N . Let x ∈ c 0 (N, R n ) arbitrarily and let V = {x t : t ∈ N}. Then V ⊂ R n and V is bounded since sup t∈N �x t � ∞ = �x� < +∞. Since for all t ∈ N, D 1 φ(., u t ) is continuous on V then for all t ∈ N, the image of V under the mapping D 1 φ(., u t ) is bounded in L(R n , R). Then sup t∈N �D 1 φ(x t , u t )� < +∞ =⇒ D 1 Φ(x, u) ∈ � ∞ (N, L(R n , R)). Hence, D 1 Φ(., u) : c 0 (N, R n ) → � ∞ (N, L(R n , R)
) is well-defined. By an analogous argument 3.4. Static Optimization -a Result in Abstract Banach Spaces 57 like that in Theorem 3.2, we obtain D 1 Φ(., u) is of class C 0 on c 0 (N, R n ) and assertion (iii) holds with D 1 N φ (., u) coincides with D 1 Φ(., u). Moreover, for all x ∈ c 0 (N, R n ) and for all δx ∈ c 0 (N, R n ) we have

D 1 N φ (x, u).(δx, δu) = (D 1 φ(x t , u t ).δx t ) t∈N .

Static Optimization -a Result in Abstract Banach Spaces

In this section we recall a result issued from the book of Ioffe and Tihomirov [START_REF] Ioffe | Theory of Extremal Problems[END_REF]. As a corollary of the extremal principle in mixed problems (Theorem 3, page 71 in [START_REF] Ioffe | Theory of Extremal Problems[END_REF]), we obtain the following lemma. Lemma 3.4. Let X , V, W be real Banach spaces and U be a nonempty set in V. Let J : X × U → R -a functional and Γ : X × U → W -a mapping. Let (x, û) ∈ X × U be a solution of the following optimization problem

Maximize J (x, u) when (x, u) ∈ X × U, Γ(x, u) = 0. � (P )
We assume that the following conditions are fulfilled.

(a) For all u ∈ U, [x → Γ(x, u)] and [x → J (x, u)] are of class C 1 at x;
(b) There exists a neighborhood N of x in X such that for all x ∈ N , for all u � , u �� ∈ U and for all θ ∈ [0, 1] there exists u ∈ U satisfying

� Γ(x, u) = (1 -θ)Γ(x, u � ) + θΓ(x, u �� ), J (x, u) ≥ (1 -θ)J(x, u � ) + θJ(x, u �� ). (c) The codimension of D 1 Γ(x, û) is finite. (d) The set {D 1 Γ(x, û).x + Γ(x, u) : x ∈ X , u ∈ U} contains a neighborhood of the origin of W.
Then there exists M ∈ W * such that the following assertions hold.

(i) D 1 J (x, û) + M • D 1 Γ(x, û) = 0. (ii) For all u ∈ U, J (x, û) + �M, Γ(x, û)� ≥ J (x, u) + �M, Γ(x, u)� .

Strong Pontryagin Principle for Problem (Ps)

First we introduce the Hamiltonian of Pontryagin which is defined, for all t ∈ N, as follows

H t : R n × U × R n * → R, H t (x, u, p) := β t φ(x, u) + �p, f (x, u)� . ( 3
(C1) U is a nonempty compact subset of R d . (C2) φ ∈ C 0 (R n × U, R) and f ∈ C 0 (R n × U, R n ). (C3) For all u ∈ U, f (0, u) = 0. (C4) For all u ∈ U, D 1 φ(x, u) and D 1 f (x, u) exist for all x ∈ R n , and D 1 φ(., u) ∈ C 0 (R n , R n * ), and D 1 f (., u) ∈ C 0 (R n , L(R n , R n )). (C5) D 1 f transforms the nonempty bounded subsets of R n × U into bounded subsets of L(R n , R n ).
(C6) For all B bounded, nonempty set in U we have

lim x→0 (sup u∈B �D 1 f (x, u)�) = 0.
(C7) For all t ∈ N, for all x t ∈ R n , for all u � t , u �� t ∈ U and for all θ ∈ (0, 1) there exists

u t ∈ U such that � φ(x t , u t ) ≥ (1 -θ)φ(x t , u � t ) + θφ(x t , u �� t ), f (x t , u t ) = (1 -θ)f (x t , u � t ) + θf (x t , u �� t ).
Using these conditions, we will prove the smoothness properties of the criterion J and the Nemytskii operator F in the following lemmas.

∈ � ∞ (N, U ) the functional x � → J(x, u) is of class C 1 on c 0 (N, R n ) and moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U
) and for all δx ∈ c 0 (N, R n ) the following formula holds

D 1 J(x, u).δx = +∞ � t=0 β t D 1 φ(x t , u t )δx t .
Proof. Under the condition (C1), U is closed. Under conditions (C2) and (C4), the Nemytskii operator

N φ : c 0 (N, R n ) × � ∞ (N, U ) → � ∞ (N, R) defined by N φ (x, u) := (φ(x t , u t )) t∈N is of class C 0 on c 0 (N, R n ) × � ∞ (
N, U ) using assertions (i) and (ii) of Theorem 3.3. Besides, from assertion (iii) of Theorem 3.3, we know that for all (x, u)

∈ c 0 (N, R n ) × � ∞ (N, U ), D 1 N φ (x, u) exists; for all u ∈ � ∞ (N, U ), D 1 N φ (., u) is continuous on c 0 (N, R n )
and moreover, for all x ∈ c 0 (N, R n ) and for all δx ∈ c 0 (N, R n ) we have

D 1 N ζ (x, u).(δx, δu) = (D 1 ζ(x t , u t ).δx t ) t∈N . Consider the functional S : � ∞ (N, R) → R defined by S(r) := � +∞ t=0 β t r t . In Chapter 2, we have proven that S is of class C 1 and for all r, δr ∈ � ∞ (N, R), we have DS(r).δr = S(δr) = � +∞ t=0 β t δr t . We knew that J = S • N φ . Since S is of class C 1 and for all u ∈ � ∞ (N, U ), N φ (., u) is of class C 1 then for all u ∈ � ∞ (N, U ), [ x � → J(x, u)] is of class C 1 on c 0 (N, R n ) as a composition of two C 1 -mappings.
Using the chain rule we obtain that for all u ∈ � ∞ (N, U ), for all x, δx ∈ c 0 (N, R n ) we have

D 1 J(x, u).δx = DS(N φ (x, u)) • D 1 N φ (x, u).δx = S(D 1 N φ (x, u)).δx = +∞ � t=0 β t D 1 φ(x t , u t )δx t .
∈ � ∞ (N, U ) the mapping [x � → F (x, u)] are of class C 1 on c 0 (N, R n ) and moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U
) and for all δx ∈ c 0 (N, R n ) the following formula holds

D 1 F (x, u).δx = (D 1 f (x t , u t )δx t -δx t+1 ) t∈N. (3.3)
Proof. Under conditions (C1-C6), we can assert that U is closed and that the mapping f satisfies all the conditions (β.1), (β.3β.6) (by replacing ζ by f ). Using Theorem 3.2, we obtain

N f ∈ C 0 (c 0 (N, R n ) × � ∞ (N, U ); c 0 (N, R n )); for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), D 1 N f (x, u) exists; and for all u ∈ � ∞ (N, U ), D 1 N f (., u) ∈ C 0 (c 0 (N, R n ); L(c 0 (N, R n ), c 0 (N, R n ))). Moreover, for all x ∈ c 0 (N, R n ) and for all δx ∈ c 0 (N, R n ) we have D 1 N f (x, u).δx = (D 1 f (x t , u t ).δx t ) t∈N . Now we set A(x, u) := (-x t+1 ) t . As we knew before, A is a bounded linear operator from c 0 (N, R n ) × � ∞ (N, U ) into c 0 (N, R n ); operator A is Fréchet differentiable and its partial differential with respect to x is D 1 A(x, u) • δx = A(δx, u) = (-δx t+1 ) t∈N . Obviously, for all u ∈ � ∞ (N, U ), D 1 A(., u) is continuous on c 0 (N, R n ). In Chapter 2, we knew that F (x, u) = N f (x, u) + A(x, u) is a mapping from c 0 (N, R n ) × � ∞ (N, U ) into c 0 (N, R n ).
Using this fact we can easily obtain these following results when we fix u ∈ � ∞ (N, U ).

-

F (., u) is a mapping from c 0 (N, R n ) into itself. -x � → F (x, u) is of class C 0 on c 0 (N, R n ) because F = N f + A is of class C 0 on c 0 (N, R n ) × � ∞ (N, U ) as a sum of two C 0 -mappings on c 0 (N, R n ) × � ∞ (N, U ). -x � → F (x, u) is Fréchet differentiable on c 0 (N, R n ). This is a consequence of the existence of D 1 N f and D 1 A on c 0 (N, R n ) × � ∞ (N, U ). Moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U
) and for all δx ∈ c 0 (N, R n ) we have

D 1 F (x, u).δx = D 1 N f (x, u).δx + D 1 A(x, u).δx = (D 1 f (x t , u t )δx t -δx t+1 ) t∈N . -x � → D 1 F (x, u) is continuous on c 0 (N, R n ) as a sum of two continuous mappings on c 0 (N, R n ) (D 1 F (., u), = D 1 N f (., u) + D 1 A(., u)).
And so, for all u ∈ � ∞ (N, U ) the mapping x � → F (x, u) are of class C 1 on c 0 (N, R n ) and formula (3.3) holds. Now for T (x, u) = (F (x, u), h(x, u)), from above-mentioned argument we already know that for all u ∈ � ∞ (N, U ) the mapping

x � → F (x, u) is of class C 1 on c 0 (N, R n ). In Chapter 2, we have proven that h is of class C 1 on c 0 (N, R n ) × � ∞ (N, U ). Hence, it is trivial that for all u ∈ � ∞ (N, U ), the mapping x � → h(x, u) is of class C 1 on c 0 (N, R n ) and moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), for all δx ∈ c 0 (N, R n ) we have D 1 h(x, u) • δx = δx 0 . Then for all u ∈ � ∞ (N, U ), the mapping x � → T (x, u) is of class C 1 on c 0 (N, R n ) and moreover, for all (x, u) ∈ c 0 (N, R n ) × � ∞ (N, U ), for all δx ∈ c 0 (N, R n ), the following formula holds. D 1 T (x, u).δx = (D 1 F (x, u).δx, D 1 h(x, u).δx) = ((D 1 f (x t , u t )δx t -δx t+1 ) t∈N , δx 0 ).
In Chapter 2, we have translated Problem (P s ) into the following form

(P 2)      Maximize J(x, u) when x ∈ c 0 (N, R n ), u ∈ � ∞ (N, U ) T (x, u) = 0, Chapter
: c 0 (N, R n )×� ∞ (N, U ) → c 0 (N, R n )×R n and T (x, u) = (F (x, u), h(x, u)) . Recall that F (x, u) := (f (x t , u t ) -x t+1
) t∈N and h(x, u) := x 0η. We will make use of Lemma 3.4 and these above-mentioned lemmas to prove the following result. Lemma 3.7. Under assumptions (C1-C7), let (x, û) ∈ c 0 (N, R n ) × � ∞ (N, U ) be a solution of Problem (P2). Then there exists (q, µ) ∈ � 1 (N, R n * ) × R n * which satisfies the following properties.

1. D 1 J(x, û) + (q, µ) • D 1 T (x, û) = 0. 2. J(x, û) + � (q, µ), T (x, û) � = max u∈� ∞ (N,U ) � J(x, u) + � (q, µ), T (x, u) �� .
Proof. We want to apply Lemma 3.4 with J = J and Γ = T . Under assumptions (C1-C6), after Lemma 3.5 and Lemma 3.6 we know that for all u ∈ � ∞ (N, U ), the mappings x � → T (x, u) and x � → J(x, u) are of class C 1 on c 0 (N, R n ). Hence, assumption (a) of Lemma 3.4 is fulfilled.

Let x = (x t ) t∈N ∈ c 0 (N, R n ), u � = (u � t ) t∈N ∈ � ∞ (N, U ), u �� = (u �� t ) t∈N ∈ � ∞ (N, U
) and θ ∈ [0, 1] arbitrarily. Using condition (C7), we know that for each t ∈ N, there exists an element u t ∈ U such that

φ(x t , u t ) ≥ (1 -θ)φ(x t , u � t ) + θφ(x t , u �� t ), (3.4)
and

f (x t , u t ) = (1 -θ)f (x t , u � t ) + θf (x t , u �� t ). (3.5)
We set u = (u t ) t∈N then u ∈ � ∞ (N, U ) since U is compact (condition (C1)). From equation (3.4), for any β ∈ [0, 1] we have:

β t φ(x t , u t ) ≥ (1 -θ)β t φ(x t , u � t ) + θβ t φ(x t , u �� t ), (3.6)
Take sum both sides of equation (3.6) for all t in N we have:

� t∈N β t φ(x t , u t ) ≥ (1 -θ) � t∈N β t φ(x t , u � t ) + θ � t∈N β t φ(x t , u �� t ),
or equivalently, J(x, u) ≥ (1θ)J(x, u � ) + θJ(x, u �� ). Now, equation (3.5) holds for all t ∈ N, hence

(f (x t , u t )) t∈N = (1 -θ)(f (x t , u � t )) t∈N + θ(f (x t , u �� t )) t∈N,
or equivalently,

F (x, u) = (1 -θ)F (x, u � ) + θF (x, u �� ).
Besides, by the virtue of the definition of h, we always have

h(x, u) = (1 -θ)h(x, u � ) + θh(x, u �� ).
From the last two equations, we obtain

T (x, u) = (1 -θ)T (x, u � ) + θT (x, u �� ).
And so assumption (b) of Lemma 3.4 is fulfilled.

To verify assumption (c) of Lemma 3.4, we see that under conditions (C1-C6), after Lemma 3.6 we know that the for all u ∈ � ∞ (N, U ), the mapping x � → T (x, u) is of class C 1 on c 0 (N, R n ). Hence, the partial Fréchet differential with respect to x of mapping T exists at the optimal solution (x, û). We set B := {û t : t ∈ N} then B is nonempty bounded set in U since û ∈ � ∞ (N, U ). For all t ∈ N, we have

0 ≤ �D 1 f (x t , ût )� L ≤ sup u∈ B �D 1 f (x t , u)� L ,
and therefore, using (C6) and Remark 2.6 we obtain lim t→+∞ �D 1 f (x t , ût )� L = 0. Hence, there exists t * ∈ N such that sup t≥t * �D 1 f (x t , ût )� L < 1. Then proceeding as in Corollary 2.16, and using Lemma 2.21, we deduce that Im D 1 T (x, û) = c 0 (N, R n ) × R n . That means the codimension of D 1 T (x, û) is finite. Hence, assumption (c) of Lemma 3.4 is fulfilled.

Finally, after assumption (c), Now we can use Lemma 3.4 and we obtain (q, µ)

Im D 1 T (x, û) = c 0 (N, R n ) × R n then it is clear that the set {D 1 T (x, û).x + T (x, u) : x ∈ c 0 (N, R n ), u ∈ � ∞ (N, U )} = Im D 1 T (x, û) + T ({x} × � ∞ (N, U )) contains Im D 1 T (x,
∈ (c 0 (N, R n ) × R n ) * = � 1 (N, R n * ) × R n *
that satisfied the announced conclusions.

We will prove the following theorem which is called strong Pontryagin principle for Problem (P s ).

Theorem 3.8. Assume that conditions (C1)-(C7) are fulfilled. Let (x, û) ∈ c 0 (N, R n ) × � ∞ (N, U ) be a solution of Problem (P s ). Then there exists (p t ) t∈N * ∈ � 1 (N * , R n * ) such that the following statements hold:

(AE1) p t = p t+1 • D 1 f (x t , ût ) + β t .D 1 φ(x t , ût ) for all t ∈ N * . (MP1) β t φ(x t , ût ) + p t+1 , f (x t , ût ) = max u∈U (β t φ(x t , u) + �p t+1 , f (x t , u)�) for all t ∈ N. Proof. Let (x, û) ∈ c 0 (N, R n ) × � ∞ (N, U
) be a solution of Problem (P s ) then (x, û) is also a solution of Problem (P2). Under conditions (C1-C7), after Lemma 3.7 we obtain the existence of (q, µ)

∈ (c 0 (N, R n ) × R n ) * = � 1 (N, R n * ) × R n * such that D 1 J(x, û) + (q, µ) • D 1 T (x, û) = 0, J(x, û) + � (q, µ), T (x, û) � = max u∈� ∞ (N,U ) � J(x, u) + � (q, µ), T (x, u) �� . � (3.7)
Then from first equality of (3.7), proceeding like in the proof of Theorem 2.22 in Chapter 2, we obtain (AE1). We rewrite the second equality of (3.7) as follows

J(x, û) + � q, F (x, û) � + �µ, h(x, û)� = max u∈� ∞ (N,U ) (J(x, u) + � q, F (x, u) � + �µ, h(x, u)�). (3.8) Since for all u ∈ � ∞ (N, U ), h(x, u) = x0 -η = h(x, û) then we have (3.8) ⇔ J(x, û) + � q, F (x, û) � = max u∈� ∞ (N,U ) (J(x, u) + � q, F (x, u) � ).
That means for all u ∈ � ∞ (N, U ),

+∞ � t=0 β t φ(x t , ût ) + � +∞ t=0 �q t , f (x t , ût ) -xt+1 � ≥ +∞ � t=0 β t φ(x t , u t ) + � +∞ t=0 �q t , f (x t , u t ) -xt+1 � ⇔ +∞ � t=0 β t φ(x t , ût ) + � +∞ t=0 �q t , f (x t , ût )� ≥ +∞ � t=0 β t φ(x t , u t ) + � +∞ t=0 �q t , f (x t , u t )� .
Chapter 3. Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality Now we arbitrarily fix t ∈ N. Take u = (u s ) s∈N ∈ � ∞ (N, U ) such that when s � = t, u s = ûs and u t varies in U . Then from the last inequality we have

∀u t ∈ U, β t φ(x t , ût ) + �p t , f (x t , ût )� ≥ β t φ(x t , u t ) + �q t , f (x t , u t )� ⇔ β t φ(x t , ût ) + �q t , f (x t , ût )� = max ut∈U � β t φ(x t , u t ) + �q t , f (x t , u t )� � .
This expression is satisfied for all t ∈ N. So we have

∀t ∈ N, β t φ(x t , ût ) + �q t , f (x t , ût )� = max u∈U � β t φ(x t , u) + �q t , f (x t , u)� � .
(3.9)

From (3.9), by setting p t+1 := q t for all t ∈ N, we obtain p = (p t+1 ) t∈N ∈ � 1 (N * , R n * ) such that (SM1) holds. The proof is complete.

Remark 3.9. In Theorem 3.8, (SM1) means Strong Maximum principle.

Strong Pontryagin Principle for Main Problem

In previous section, we already proven the strong Pontryagin principle for supporting problem. Now we study that for main problem. Notice that main problem can be translated into a problem in the form of supporting problem and we named it by (P 1) as follows

Maximize N (z, u) := � +∞ t=0 β t ϕ(z t , u t ) when z = (z t ) t∈N ∈ c 0 (N, R n ), u = (u t ) t∈N ∈ � ∞ (N, U ), z 0 = η -y ∞ , ∀t ∈ N, z t+1 = �(z t , u t ).              (P 1)
Apply Theorem 3.8 to Problem (P 1), we obtain the following statement. Let (ẑ, û) ∈ c 0 (N, R n ) × � ∞ (N, U ) be a solution of Problem (P m ). Assume that these following conditions hold.

(1) U is a nonempty compact subset of R d .

(

) ϕ ∈ C 0 (R n × U, R) and � ∈ C 0 (R n × U, R n ). 2
(3) For all u ∈ U, �(0, u) = 0.

(4) For all u ∈ U, D 1 ϕ(z, u) and D 1 �(z, u) exist for all z ∈ R n , and

D 1 ϕ(., u) ∈ C 0 (R n , R n * ), and D 1 �(., u) ∈ C 0 (R n , L(R n , R n )).
( (7) For all t ∈ N, for all z t ∈ R n , for all u � t , u �� t ∈ U and for all θ ∈ [0, 1] there exists

) D 1 � transforms the nonempty bounded subsets of R n × U into bounded subsets of L(R n , R n ). 5
u t ∈ U such that � ϕ(z t , u t ) ≥ (1 -θ)ϕ(z t , u � t ) + θϕ(z t , u �� t ), �(z t , u t ) = (1 -θ)�(z t , u � t ) + θl(z t , u �� t ).
Then there exists (p t ) t∈N * ∈ � 1 (N * , R n * ) such that

(i) For all t ∈ N * , p t = p t+1 • D 1 �(ẑ t , ût ) + β t .D 1 ϕ(ẑ t ,

63

(ii) For all t ∈ N, for all u ∈ U ,

β t ϕ(ẑ t , ût ) + �p t+1 , �(ẑ t , ût )� ≥ β t ϕ(ẑ t , u) + �p t+1 , �(ẑ t , u)� .
We study the statement of strong Pontryagin principle for main problem based on the statement above. We know that if (ẑ, û) is a solution of Problem (P 1) then (ŷ, û) is a solution of Problem (P m ) where ŷt = ẑt + y ∞ , for all t ∈ N.

Study the assumptions: similar to what we did in Chapter 2, the assumptions (1-7) for Problem (P 1) can be rewritten equivalently for Problem (P m ) as follows.

(D1) U is a nonempty compact subset of R d . (D2) ψ ∈ C 0 (R n × U, R) and g ∈ C 0 (R n × U, R n ). (D3) For all u ∈ U, g(y ∞ , u) = 0.
(D4) For all u ∈ U, D 1 ψ(y, u) and D 1 g(y, u) exist for all y ∈ R n , and For assumption [START_REF] Blot | Infinite-Horizon Discrete-Time Pontryagin Principles via Results of Michel[END_REF], using the definition of mappings ϕ and � (as we know before,

D 1 ψ(., u) ∈ C 0 (R n , R n * ), and D 1 g(., u) ∈ C 0 (R n , L(R n , R n )). (D5) D 1 g transforms the nonempty bounded subsets of R n × U into bounded subsets of L(R n , R n ). (
ϕ(z, u) = ψ(z + y ∞ , u); �(z, u) = g(z + y ∞ , u) -y ∞ for all (z, u) ∈ R n × U
) and recall that z = yy ∞ for all z ∈ R n , we have the following equivalent transformations

ϕ(z t , u t ) ≥ (1 -θ)ϕ(z t , u � t ) + θϕ(z t , u �� t ) ⇔ ψ(z t + y ∞ , u t ) ≥ (1 -θ)ψ(z t + y ∞ , u � t ) + θψ(z t + y ∞ , u �� t ) ⇔ ψ(y t , u t ) ≥ (1 -θ)ψ(y t , u � t ) + θψ(y t , u �� t ), and
�(z t , u t ) = (1 -θ)�(z t , u � t ) + θh(z t , u �� t ) ⇔ g(z t + y ∞ , u t ) -y ∞ = (1 -θ) (g(z t + y ∞ , u � t ) -y ∞ ) + θ (g(z + y ∞ , u �� t ) -y ∞ ) ⇔ g(z t + y ∞ , u t ) = (1 -θ)g(z t + y ∞ , u � t ) + θg(z + y ∞ , u �� t ) ⇔ g(y t , u t ) = (1 -θ)g(y t , u � t ) + θg(y t , u �� t ).
Then we obtain assumption (D7) as follows (D7) For all t ∈ N, for all y t ∈ R n , for all u � t , u �� t ∈ U and for all θ ∈ [0, 1] there exists

u t ∈ U such that � ψ(y t , u t ) ≥ (1 -θ)ψ(y t , u � t ) + θψ(y t , u �� t ), g(y t , u t ) = (1 -θ)g(y t , u � t ) + θg(y t , u �� t ).
Study the conclusions: By the same reasoning as in Chapter 2, conclusion (i) can be rewritten as follows.

(i) For all t ∈ N * , p t = p t+1 • D 1 g(ŷ t , ût ) + β t .D 1 ψ(ŷ t , ût ). For conclusion (ii), we make the following equivalent transformations:

β t ϕ(ẑ t , ût ) + �p t+1 , �(ẑ t , ût )� ≥ β t ϕ(ẑ t , u) + �p t+1 , �(ẑ t , u)� ⇔ β t ψ(ŷ t , ût ) + �p t+1 , g(ŷ t , ût ) -y ∞ � ≥ β t ψ(ŷ t , u) + �p t+1 , g(ŷ t , u) -y ∞ � ⇔ β t ψ(ŷ t , ût ) + �p t+1 , g(ŷ t , ût )� ≥ β t ψ(ŷ t , u) + �p t+1 , g(ŷ t , u)� .
Then we can rewrite (ii) as follows (ii) For all t ∈ N, for all u ∈ U : β t ψ(ŷ t , ût )+�p t+1 , g(ŷ t , ût )� ≥ β t ψ(ŷ t , u)+�p t+1 , g(ŷ t , u)� .

Finally, we state the strong Pontryagin principle for Problem (P m ):

Theorem 3.10. Let (ŷ, û) ∈ c 0 (N, R n )×� ∞ (N, U
) be a solution of Problem (P m ). Assume that the assumptions (D1-D7) are fulfilled then there exists (p t+1 ) t∈N ∈ � 1 (N * , R n * ) such that the following assertions hold.

(AE) p t = p t+1 • D 1 g(ŷ t , ût ) + β t .D 1 ψ(ŷ t , ût ) for all t ∈ N * .

(SM) β t ψ(ŷ t , ût ) + �p t+1 , g(ŷ t , ût )� = max u∈U (β t ψ(ŷ t , u) + �p t+1 , g(ŷ t , u)�) for all t ∈ N.

Sufficient Condition for Problem (P1)

In this section we establish a result of sufficient condition of optimality which uses the adjoint equation, the weak maximum principle and the concavity of the Hamiltonian with respect to the state variable and the control variable.

Theorem 3.11. Let U be a nonempty convex subset of R d , β ∈ (0, 1), η, y ∞ ∈ R n and two mappings ϕ : R n × U → R and � : R n × U → R n . Let (ẑ, û) ∈ c 0 (N, R n ) × � ∞ (N, U ) and p ∈ � 1 (N * , R n * ).
Assume that the following conditions hold.

(i) ẑt+1 = �(ẑ t , ût ) for all t ∈ N, and ẑ0 = ηy ∞ .

(ii) ϕ ∈ C 1 (R n × U, R) and � ∈ C 1 (R n × U, R n ). (iii) ϕ transforms bounded subsets of R n × U into bounded subsets of R. (iv) p t = p t+1 • D 1 �(ẑ t , ût ) + β t D 1 ϕ(ẑ t , ût ) for all t ∈ N * . (v) �p t+1 • D 2 �(ẑ t , ût ) + D 2 ϕ(ẑ t , ût ), u -ût � ≤ 0 for all u ∈ U , for all t ∈ N. (vi) The function [(z, u) � → �p t+1 , �(z, u)� + β t ϕ(z, u))] is concave on R n × U for all t ∈ N.
Then (ẑ, û) is a solution of (P1).

Proof. Let (z, u) be an admissible process for (P1), i.e. z ∈ c 0 (N, R n ), u ∈ � ∞ (N, U ), z t+1 = �(z t , u t ) for all t ∈ N, and z 0 = ηy ∞ . From (iii), since {ϕ(z t , u t ) : t ∈ N} is bounded, N (z, u) = � +∞ t=0 β t ϕ(z t , u t ) exists in R. From (ii) and (iv) and from the definition of Hamiltonian of Pontryagin (3.2) we obtain

D 1 H t (ẑ t , ût , p t+1 ) = p t .
(3.10)

From (vi) we obtain, for all t ∈ N, For all t ∈ N we have

H t (ẑ t , ût , p t+1 ) -H t (z t , u t , p t+1 ) -�D 1 H t (ẑ t , ût , p t+1 ), ẑt -z t � -�D 2 H t (ẑ t , ût , p t+1 ), ût -u t � ≥ 0. ( 3
β t ϕ(ẑ t , ût ) -β t ϕ(z t , u t ) = H t (ẑ t , ût , p t+1 ) -�p t+1 , �(ẑ t , ût )� -H t (z t , u t , p t+1 ) + �p t+1 , �(z t , u t ) = H t (ẑ t , ût , p t+1 ) -H t (z t , u t , p t+1 )
-�p t+1 , ẑt+1z t+1 �.

Sufficient Condition for Main Problem

65

Then, using (3.10) and (3.12) we obtain

β t ϕ(ẑ t , ût ) -β t ϕ(z t , u t ) ≥ H t (ẑ t , ût , p t+1 ) -H t (z t , u t , p t+1 ) -�D 2 H t (ẑ t , ût , p t+1 ), ût -u t � -�D 1 H t+1 (ẑ t+1 , ût+1 , p t+2 ), ẑt+1 -z t+1 �,
which implies

β t ϕ(ẑ t , ût ) -β t ϕ(z t , u t ) ≥ [H t (ẑ t , ût , p t+1 ) -H t (z t , u t , p t+1 ) -�D 1 H t (ẑ t , ût , p t+1 ), ẑt -z t � -�D 2 H t (ẑ t , ût , p t+1 ), ût -u t �] +[�D 1 H t (ẑ t , ût , p t+1 ), ẑt -z t � -�D 1 H t+1 (ẑ t+1 , ût+1 , p t+2 ), ẑt+1 -z t+1 �],
and using (3.11) we obtain

β t ϕ(ẑ t , ût ) -β t ϕ(z t , u t ) ≥ [�D 1 H t (ẑ t , ût , p t+1 ), ẑt -z t � -�D 1 H t+1 (ẑ t+1 , ût+1 , p t+2 ), ẑt+1 -z t+1 �].
Therefore, using (3.10), we obtain, for all T ∈ N * ,

� T t=0 β t ϕ(ẑ t , ût ) - � T t=0 β t ϕ(z t , u t ) ≥ �D 1 H 0 (η -y ∞ , û0 , p 1 ), η -y ∞ -(η -y ∞ )� -�p T +1 , ẑT +1 -z T +1 � =⇒ T � t=0 β t ϕ(ẑ t , ût ) - T � t=0 β t ϕ(z t , u t ) ≥ -�p T +1 , ẑT +1 -z T +1 � . (3.13)
Since p ∈ � 1 (N * , R n * ), we have lim T →+∞ p T +1 = 0, and since ẑ, z ∈ c 0 (N, R n ) we have lim T →+∞ (ẑ T +1z T +1 ) = 0 which implies lim T →+∞ (-�p T +1 , ẑT +1z T +1 �) = 0, and then, from (3.13), doing T → +∞ we obtain N (ẑ, û) -N (z, u) ≥ 0. And so we have proven that (ẑ, û) is a solution of (P1).

Remark 3.12. The structure of the previous proof is inspired by the proof of Theorem 5.1 in [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF]. Note that our assumption (iii) permits to avoid to assume that U is compact. Moreover, note that we can replace the assumption (iii) by the condition: U is closed.

Remark 3.13. Note that under our assumptions, the process (ẑ, û) is also solution of the following problem

     Maximize � +∞ t=0 β t ϕ(x t , u t ) when z ∈ � ∞ (N, R n ), u ∈ � ∞ (N, U ), and ∀t ∈ N, z t+1 = �(z t , u t ), z 0 = η -y ∞ ,
since, in the previous proof, when we obtain (3.13), having ẑ and z bounded is sufficient to obtain lim T →+∞ (-�p T +1 , ẑT +1z T +1 �) = 0 and consequently to have the optimality of (ẑ, û) for the last problem.

Sufficient Condition for Main Problem

This section is devoted to the translation of the result of sufficient condition of optimality for (P 1) into an analogous result for (P m ). When y ∞ ∈ R n , we denote by c y∞ (N, R n ), the set of the sequences y in R n such that lim t→+∞ y t = y ∞ . It is a complete affine subset of � ∞ (N, R n ).

× U → R n . Let (ŷ, û) ∈ c y∞ (N, R n ) × � ∞ (N, U ) and p ∈ � 1 (N * , R n *
) which satisfy the following conditions.

(i) For all t ∈ N, ŷt+1 = g(ŷ t , ût ), and ŷ0 = η.

(ii) ψ ∈ C 1 (R n × U, R) and g ∈ C 1 (R n × U, R n ).
(iii) ψ transforms bounded subsets of R n × U into bounded subsets of R.

(iv)

p t = p t+1 • D 1 g(ŷ t , ût ) + β t D 1 ψ(ŷ t , ût ) for all t ∈ N * . (v) �p t+1 • D 2 g(ŷ t , ût ) + β t D 2 ψ(ŷ t , ût ), u -û� ≤ 0 for all u ∈ U , for all t ∈ N. (vi) The function [(y, u) � → �p t+1 , g(y, u)� + β t ψ(y, u)] is concave on R n × U for all t ∈ N.
Then (ŷ, û) is a solution of (P m ).

Proof. We set ẑt = ŷt -y ∞ for all t ∈ N. We see that (ẑ, û) ∈ c 0 (N, R n )×� ∞ (N, U ) satisfies all the assumptions of Theorem 3.11. And so (ẑ, û) is a solution of (P 1) which implies that (ŷ, û) is a solution of (P m ).

Chapter 4

Lightenings of Assumptions for Pontryagin Principles in Finite Horizon and Discrete Time

Introduction

In this chapter, we will establish maximum principles of Pontryagin under assumptions which are weaker than those of existing results for the infinite horizon optimal control problems in discrete time framework. The considered problems is stated as follows.

The discrete time is denoted by the letter t ∈ N. For all t ∈ N, X t is a nonempty open subset of R n , U t is a nonempty subset of R d , and f t : X t × U t → X t+1 is a mapping where n and d are fixed positive integers. The usual order of R n is x = (x 1 , x 2 , . . . , x n ) ≤ (y 1 , y 2 , . . . , y n ) = y defined by x i ≤ y i for all i ∈ {1, . . . , n}. And x < y means that x ≤ y and x � = y. To abridge the writing we use the notation x := (x t ) t∈N ∈ � t∈N X t and also u := (u t ) t∈N ∈ � t∈N U t where � t∈N X t and � t∈N U t are the Cartesian products. We work with two families of controlled dynamical systems: difference equations and difference inequations. They are

(De) x t+1 = f t (x t , u t )
and

(Di) x t+1 ≤ f t (x t , u t ).
The variable x t is called the state variable and the variable u t is called the control variable. When we fix an initial state η ∈ X 0 , we denote by Adm e the set of all processes (x, u) ∈ � t∈N X t × � t∈N U t which satisfy (De) at each time t ∈ N and such that x 0 = η. These processes are called the admissible for (De) and η. The letter e as lower index means equation. Similarly, we denote by Adm i the set of all processes (x, u) ∈ � t∈N X t × � t∈N U t which satisfy (Di) at each time t ∈ N and such that x 0 = η. The letter i as lower index means inequation.

For all t ∈ N, we consider a function φ t : X t × U t → R. When k ∈ {i, e}, we define Dom k as the set of the (x, u) ∈ Adm k such that the series

� +∞ t=0 φ t (x t , u t ) is convergent in R.
We define the functional J : Dom k → R by setting J(x, u) := � +∞ t=0 φ t (x t , u t ). When k ∈ {i, e}, we consider the following list of problems.

(P 1 k ) Maximize J(x, u) when (x, u) ∈ Dom k .
Chapter 4. Lightenings of Assumptions for Pontryagin Principles in Finite Horizon and Discrete Time

(P 2 k ) Find (x, û) ∈ Adm k such that, for all (x, u) ∈ Adm k lim sup h→+∞ ( h � t=0 φ t (x t , ût ) - h � t=0 φ t (x t , u t )) ≥ 0. (P 3 k ) Find (x, û) ∈ Adm k such that, for all (x, u) ∈ Adm k lim inf h→+∞ ( h � t=0 φ t (x t , ût ) - h � t=0 φ t (x t , u t )) ≥ 0.
Now we describe the content of this chapter.

-In Section 4.2, we recall the method of reduction to finite horizon (Theorem 4.1).

-In Section 4.3, firstly, we recall the Multiplier Rule of Halkin. Then we introduce the New Multiplier Rules of Blot for maximization static problems with only inequality constraints and with both equality and inequality constraints. After that, we apply the New Multiplier Rules to obtain weak Pontryagin principles for the reduced problems in Section 4.2.

-In Section 4.4, at first, we establish weak maximum principles where the values of the optimal control belong to the interior of the sets of controls for systems which governed by difference inequations (Theorem 4.14 and Theorem 4.16). These results are new and only use the Gâteaux differentiability of the criterion, of the vector field and of the inequality constraints. Neither continuity on a neighborhood of optimal solution nor Fréchet differentiability is necessary. In the end of this section, we state a a similar result for problem in which the system is governed by difference equations.

-In Section 4.5, firstly, we establish a weak maximum principle when the sets of controls are defined by inequalities (Theorem 4.21) and when the system is governed by a difference inequations. This result also only uses the Gâteaux differentiability of the criterion, of the vector field and of the inequality constraints and a condition of separation of the origin and of the convex hull of the Gâteaux differentials of the inequelities constraints in the spirit of the Mangarasian-Fromowitz condition. Secondly, we establish a weak maximum principle when the sets of controls are defined by equalities and inequalities (Theorem 4.25) when the system is governed by a difference inequation. Such a case is treated in [START_REF] Blot | Infinite-Horizon Discrete-Time Pontryagin Principles via Results of Michel[END_REF] (Theorem 3.1 and Theorem 3.2). In comparison with the result of [START_REF] Blot | Infinite-Horizon Discrete-Time Pontryagin Principles via Results of Michel[END_REF], the improvements are the following ones: we avoid a condition of continuity for the saturated inequality constraints and for the vector field, we avoid a condition of linear independence of all the differentials of the constraints. A similar result is Theorem 4.26 for which the system is governed by a difference equation.

Reduction to Finite Horizon

The general principle is the following one: when a process is optimal on N (until infinity), then for all T ∈ N * its restriction to [0, T ] ∩ N is optimal by fixing the final condition at T . Theorem 4.1. The two assertions hold.

(a) Let (x, û) be a solution of (P j e ) and let T ∈ N * when j ∈ {1, 2, 3}. Then the restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is an optimal solution of the following finite-horizon problem:

Maximize F T ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) := � T t=0 φ t (x t , u t ) when ∀t ∈ {0, . . . , T + 1}, x t ∈ X t ∀t ∈ {0, . . . , T }, u t ∈ U t ∀t ∈ {0, . . . , T }, x t+1 = f t (x t , u t ) x 0 = η, x T +1 = xT +1 .              (F T e )
(b) Let (x, û) be a solution of (P j i ) let T ∈ N * when j ∈ {1, 2, 3}. Then the restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is an optimal solution of the following finite-horizon problem:

Maximize F T ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) := � T t=0 φ t (x t , u t ) when ∀t ∈ {0, . . . , T + 1}, x t ∈ X t ∀t ∈ {0, . . . , T }, u t ∈ U t ∀t ∈ {0, . . . , T }, x t+1 ≤ f t (x t , u t ) x 0 = η, x T +1 = xT +1 .              (F T i )
Proof. For the (De) case: We will prove for each case of (P j e ) where j ∈ {1, 2, 3}. -For (P 1 e ): Let (x, û) be a solution of (P 1 e ). We proceed by contradiction. Assume that ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is not optimal for (F T e ).

Then there exists ((z 0 , . . . , z T +1 ), (w 0 , . . . , w T )) which is admissible for (F T e ) such that

T � t=0 φ t (z t , w t ) > T � t=0 φ t (x t , ût ).
When t > T + 1, we set z t := xt and when t ≥ T + 1, we set w t := ût . From the admissibility and this setting, it is obvious that z ∈ � t∈N X t and w ∈ � t∈N U t . Also, z t+1 = xt+1 = f t (x t , ût ) = f t (z t , w t ) when t ≥ T + 1. It implies that (z, w) belongs to Adm e , the admissible set of (P

> T � t=0 φ t (x t , ût ) + +∞ � t=T +1 φ t (x t , ût ) = +∞ � t=0 φ t (x t , ût )
This is contradiction since (x, û) is the optimal solution for Problem (P 1 e ). Hence, ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) must be optimal for (F T e ). -For (P 3 e ): Let (x, û) be a solution of (P 3 e ). By an analogous proceed like in the previous case, we obtain (z, w) ∈ Adm e such that

T � t=0 φ t (z t , w t ) > T � t=0 φ t (x t , ût ). Then we have when h ≥ T, lim inf h→+∞ ( h � t=0 φ t (z t , w t ) - h � t=0 φ t (x t , ût )) = T � t=0 φ t (z t , w t ) - T � t=0 φ t (x t , ût ) > 0,
which is a contradiction.

-For (P 2 e ): Let (x, û) be a solution of (P 2 e ). It is clear that it is also a solution of (P 3 e ) and hence, its restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is an optimal solution for (F T e ). For the (Di) case: the proof is completely similar to the (DE) case. We consider two nonnegative integer numbers n I and n E , a nonempty open subset Ω in R n , and functions g 0 , g 1 , . . . , g n I , h 1 , h 2 , . . . , h n E from Ω into R. With these elements we formulate the following maximization problem:

Maximize g 0 (z) when ∀α ∈ {1, 2, . . . , n I }, g α (z) ≥ 0 ∀β ∈ {1, 2, . . . , n E }, h β (z) = 0.      (M )
The conditions g α (z) ≥ 0 are called the inequality constraints, the conditions h β (z) = 0 are called the equality constraints, and g 0 (z) is called the criterion. A point z ∈ Ω which satisfies all the inequality constraints and all the equality constraints is called admissible for (M ).

Definition 4.2. The function

L : Ω × R n I × R n E → R defined by L(z, λ 1 , . . . , λ n I , µ 1 , . . . , µ n E ) := g 0 (z) + n I � α=1 λ α g α (z) + n E � β=1 µ β h β (z)
is called the Lagrangian of (M ).

The function

G : Ω × R × R n I × R n E → R defined by G(z, λ 0 , λ 1 , . . . , λ n I , µ 1 , . . . , µ n E ) := λ 0 g 0 (z) + n I � α=1 λ α g α (z) + n E � β=1 µ β h β (z) = n I � α=0 λ α g α (z) + n E � β=1 µ β h β (z)
is called the generalized Lagrangian of (M ). Note that the difference between G and L is the presence of a scalar λ 0 associated to the criterion.

The following theorem is established in the paper of Halkin [START_REF] Halkin | Implicit Functions and Optimization Problems without Continuous Differentiability of the Data[END_REF]. In [START_REF] Michel | Some Clarifications on the Transversality Condition[END_REF], Michel provides a proof which is different from this one of Halkin.

Theorem 4.3. Let z * be a solution of (M ).

We assume that the functions g 0 , g 1 , . . . , g nI and h 1 , h 2 , . . . , h nE are continuous on a neighborhood of z * and that they are Fréchet differentiable at z * .

Then there exist real numbers λ 0 , . . . , λ n I , µ 1 , . . . , µ n E which satisfy the following conditions:

(a) λ 0 , λ 1 , . . . , λ n I , µ 1 , . . . , µ n E are not simultaneously equal to zero.

(b) For all ∀α ∈ {0, 1, . . . , n I }, λ α ≥ 0.

(c) For all ∀α ∈ {1, . . . , n I }, λ α g α (z * ) = 0.

(d) D 1 G(z * , λ 0 , λ 1 , . . . , λ n I , µ 1 , . . . , µ n E ) = 0 where D 1 denotes the partial differential with respect to the first variable z.

The real numbers of the conclusion of the theorem are called the multipliers associated to z * . λ 0 is called the multiplier associated to the criterion; when α ∈ {1, . . . , n I }, λ α is called the multiplier associated to the inequality constraint g α (z) ≥ 0; and when β ∈ {1, 2, . . . , n E }, µ β is called the multiplier associated to the equality constraint h β (z) = 0. About the conclusion (a), it is easy to see that when all the multipliers are zero then all the conclusions hold even if z * is not a solution of the problem. The conclusion (c) is called the slackness condition; it means that when g α (z * ) > 0 then the associated multiplier is zero and consequently we can delete it. Conclusion (d) can be translated as follows.

n I � α=0 λ α Dg α (z * ) + n E � β=1 µ β Dh β (z * ) = 0
Note that when (λ 0 , λ 1 , . . . , λ n I , µ 1 , . . . , µ n E ) satisfies (a-d), for all real number r > 0, the new list (rλ 0 , rλ 1 , . . . , rλ n I , rµ 1 , . . . , rµ n E ) also satisfies (a-d) (this is a property of cones). Consequently it is possible to normalize a list (λ 0 , λ 1 , . . . , λ n I , µ 1 , . . . , µ n E ) which satisfies (a-d): choosing a norm �.� on R × R n I × R n E , we can choose a suitable list such that �λ 0 , . . . , λ n I , µ 1 , . . . , µ n E � = 1. Also note that the set of all lists (λ 0 , . . . , λ n I , µ 1 , . . . , µ n E ) which satisfy (a-d) is a convex subset of R 1+n I +n E .

New Multiplier Rules

Let Ω be a nonempty open subset of R n , let f i : Ω → R (when i ∈ {0, . . . , m}) be functions, let g i : Ω → R (when i ∈ {0, . . . , p}) and h i : Ω → R (when i ∈ {1, . . . , q}) be functions. With these elements, we consider the two following problems: Maximize f 0 (x) when x ∈ Ω and when ∀i ∈ {1, . . . , m}, f i (x) ≥ 0,

     (I)
and Maximize g 0 (x) when x ∈ Ω when ∀i ∈ {1, . . . , p}, g i (x) ≥ 0 and when ∀j ∈ {1, . . . , q}, h j (x) = 0.

         (M)
For Problem (I), the conditions f i (x) ≥ 0 are called the inequality constraints and f 0 (x) is called the criterion. A point x ∈ Ω which satisfies all the inequality constraints is called admissible for (I).

Similarly, for Problem (M), the conditions g i (x) ≥ 0 are called the inequality constraints, the conditions h j (x) = 0 are called the equality constraints, and g 0 (x) is called the criterion. A point x ∈ Ω which satisfies all the inequality constraints and all the equality constraints is called admissible for (M).

Recently, Blot has improved the Multiplier Rule of Halkin by the following theorems. Their statements and proofs are taken from [START_REF] Blot | On the Multiplier Rules[END_REF].

Theorem 4.4. Let x be a solution of (I). We assume that the following assumptions are fulfilled.

(i) For all i ∈ {0, . . . , m}, f i is Gâteaux differentiable at x.

(ii) For all i ∈ {1, . . . , m}, f i is lower semicontinuous at x when f i (x) > 0.

Then there exist λ 0 , . . . , λ m ∈ R + such that the following conditions hold.

(a) (λ 0 , . . . , λ m ) � = (0, . . . , 0).

(b) For all i ∈ {1, . . . , m}, λ i f i (x) = 0. (c) � 0≤i≤m λ i D G f i (x) = 0.
If, in addition, we assume that the following assumption is fulfilled, (iii) There exists w ∈ R n such that, for all i ∈ {1, . . . , m}, D G f i (x).w > 0 when f i (x) = 0, Finite Horizon and Discrete Time then we can take (d) λ 0 = 1. Theorem 4.5. (New Multiplier Rule) Let x be a solution of (M). We assume that the following assumptions are fulfilled.

(i) g 0 is Fréchet differentiable at x.

(ii) For all i ∈ {1, . . . , p}, g i is Fréchet differentiable at x when g i (x) = 0.

(iii) For all i ∈ {1, . . . , p}, g i is Gâteaux differentiable at x and lower semicontinuous at

x when g i (x) > 0.

(iv) For all i ∈ {1, . . . , q}, h i is continuous on a neighborhood of x and Fréchet differentiable at x.

Then there exist λ 0 , . . . , λ p ∈ R + and µ 1 , . . . , µ q ∈ R such that the following conditions are satisfied.

(a) (λ 0 , . . . , λ p , µ 1 , . . . , µ q ) � = (0, . . . , 0).

(b)

For all i ∈ {1, . . . , p}, λ i g i (x) = 0.

(c) λ 0 D G φ(x) + � 1≤i≤p λ i D G g i (x) + � 1≤i≤q µ i D G h i (x) = 0.
Moreover, under the additional assumption (v) Dh 1 (x), . . . , Dh q (x) are linearly independent, we can take (d) (λ 0 , . . . , λ p ) � = (0, . . . , 0).

Furthermore, under (v) and under the additional assumption

(vi) There exists w ∈ � 1≤j≤q KerDh j (x) such that, for all i ∈ {1, . . . , p}, Dg i (x).w > 0 when g i (x) = 0, we can take (e) λ 0 = 1.

In comparison with the Halkin's Multiplier Rule, for Problem (I), the assumptions of local continuity on a neighborhood of x of the f i , i ∈ {0, . . . , m} have been deleted and their Fréchet differentiability has been replaced by their Gâteaux differentiability, and for Problem (M), the assumptions of local continuity on g 0 and on the g i , i ∈ {1, . . . , p} have been deleted. In comparison with the result of [START_REF] Pourciau | Modern Multiplier Rules[END_REF] for Problem (I), the Fréchet differentiability of the f i has been replaced by their Gâteaux differentiability. Note that the Gâteaux differentiability of a mapping at a point does not imply the continuity of this mapping at this point.

Proof of Theorem 4.4

Some fundamental tools

Before proving Theorem 4.4, we recall the following well-known results. (i) For all x ∈ R n , (∀i ∈ {1, . . . , m}, ϕ i .x ≥ 0) =⇒ (a.x ≥ 0).

(ii) There exists λ 1 , . . . ,

λ m ∈ R + such that a = � 1≤i≤m λ i ϕ i .
A complete proof of this result is given in [START_REF] Van Tiel | Convex Analysis[END_REF] (Chapter 4, Sections 4. 14 -4.19) and in [START_REF] Karmanov | Programmation Mathématique. Mir[END_REF] (Chapter 2, Sections 2.5, 2.6). This result is presented in many books (for example, in [START_REF] Berge | Topological Spaces[END_REF] page 164 and in [START_REF] Truchon | Theorie de Optimisation Statique et Differentiable[END_REF]

page 176 ).

A second fundamental tool that we recall is the Implicit Function Theorem of Halkin for the Fréchet differentiable mappings which are not necessarily continuously Fréchet differentiable.

Theorem 4.7. Let X, Y, Z be three real finite-dimensional normed vector spaces. Let A ⊂ X × Y be a nonempty open subset, let f : A → Z be a mapping, and let (x, ȳ) ∈ A. We assume that the following conditions are fulfilled.

(i) f (x, ȳ) = 0.
(ii) f is continuous on a neighborhood of (x, ȳ).

(iii) f is Fréchet differentiable at (x, ȳ) and the partial Fréchet differential D 2 f (x, ȳ) is bijective.

Then there exist a neighborhood U of x in X, a neighborhood V of ȳ in Y such that U × V ⊂ A, and a mapping ψ : U → V which satisfy the following conditions.

(a) ψ(x) = y. (b) For all x ∈ U, f (x, ψ(x)) = 0. (c) ψ is Fréchet differentiable at x and Dψ(x) = -D 2 f (x, ȳ) -1 • D 1 f (x, ȳ).
This result is proven in [START_REF] Halkin | Implicit Functions and Optimization Problems without Continuous Differentiability of the Data[END_REF]. Its proof uses the Fixed Point Theorem of Brouwer. The assumptions of this theorem are well explained by the electronic paper of Border [START_REF] Border | Notes on the Implicit Function Theorem[END_REF]. Halkin does not use an open subset A. His function is defined on X × Y but it is easy to adapt his result. Since ψ is Fréchet differentiable at x, ψ is continuous at x and then we can consider a neighborhood V of ȳ and a neighborhood U of x such that ψ(U ) ⊂ V and such that U × V ⊂ A. Now we will prove Theorem 4.4. Let x be a solution of Problem (I). Doing a change of index, we can assume that I := {1, . . . , e} = {i ∈ {1, . . . , m} : f i (x) = 0}. If f i (x) > 0 for all i ∈ {1, . . . , m} then I = ∅ (or equivalently, e = 0). Using the lower semicontinuity of (ii), there exists an open neighborhood Θ of x on which f i (x) > 0 for all i ∈ {1, . . . , m}. Hence, x maximizes f 0 (without constraints) on open set Θ. Then using (i) and the definition of Gâteaux differential, we obtain D G f 0 (x) = 0, and we conclude by taking λ 0 := 1 and λ i := 0 for all i ∈ {1, . . . , m}. And so, for sequel of the proof we assume that 1 ≤ e ≤ m.

Proof of (a), (b), (c)

Ever using (ii), when e < m we can assert that there exists an open neighborhood Ω 1 ⊂ Ω of x such that, for all x ∈ Ω 1 and for all i ∈ {e + 1, . . . , m}, f i (x) > 0. When e = m we simply take Ω 1 := Ω. Then for all case of e, x is a solution of the following problem.

Maximize f 0 (x) when x ∈ Ω 1 and when ∀i ∈ {1, . . . , e}, f i (x) ≥ 0.

     (P)
For all k ∈ {0, . . . , e}, we introduce the set

A k := {v ∈ R n : ∀i ∈ {k, . . . , e}, D G f i (x).v > 0} (4.

1) Finite Horizon and Discrete Time

We will prove that A 0 = ∅. To realize that, we proceed by contradiction; we assume that A 0 � = ∅, and so there exists w ∈ R n such that D G f i (x).w > 0 for all i ∈ {0, . . . , e}.

Since Ω 1 is open, there exists θ * ∈ (0, +∞) such that x+ θw ∈ Ω 1 for all θ ∈ [0, θ * ]. After (i), for all i ∈ {0, . . . , e}, the function σ i : [0, θ * ] → R, defined by σ i (θ) := f i (x+ θw), is differentiable at 0, and its derivative is σ � i (0) = D G f i (x).w as follows

σ � i (0) = lim t→0 σ i (t) -σ i (0) t = lim t→0 f i (x + tw) -f i (x) t = � Df (x, w) = D G f i (x).w
The differentiability of at 0 implies the existence of a function ρ i : [0, θ * ] → R such that lim θ→0 ρ i (θ) = 0 and such that σ i (θ) = σ i (0) + σ � i (0)θ + ρ i (θ)θ for all θ ∈ [0, θ * ] and for all i ∈ {0, . . . , e}. Translating this last equality we obtain for all θ ∈ [0, θ * ] and for all i ∈ {0, . . . , e},

f i (x + θw) = f i (x) + θ(D G f i (x).w + ρ i (θ)).
Since D G f i (x).w > 0 and since lim θ→0 ρ i (θ) = 0, we obtain the existence of θ i ∈ (0, θ) such that D G f i (x).w + ρ i (θ) > 0 for all θ ∈ (0, θ i ]. Setting θ := min{θ i : i ∈ {0, . . . , e}} we obtain that f i (x + θw) < f i (x) for all θ ∈ (0, θ] and for all i ∈ {0, . . . , e}. Then using i ∈ {1, . . . , e}, this last relation ensures that x + θw is admissible for (P) when θ ∈ (0, θ], and using this last relation when i = 0 we obtain f 0 (x + θw) > f 0 (x) when θ ∈ (0, θ], that is impossible since x is a solution of (P). And so the reasoning by contradiction is complete, and we have proven

A 0 = ∅. ( 4

.2)

When A e = ∅ there is not any v ∈ R n such that D G f e (x).v > 0. This implies that D G f e (x) = 0 since it belongs to R n * . Then taking λ e := 1 and λ i := 0 when i ∈ {0, • • • , }\{e}, we obtain the conclusions (a), (b), (c). And so we have proven

A e = ∅ =⇒ ((a), (b), (c) hold). ( 4.3)
Now we assume that A e � = ∅. Since we have A 0 = ∅ after (4.2) and A i ⊂ A i+1 we can define

k := min{i ∈ {1, . . . , e} : A i � = ∅}. (4.4) Note that A k � = ∅ and that A k-1 = ∅. We consider the following problem Maximize D G f k-1 (x).v when v ∈ R n and when ∀i ∈ {k, . . . , e}, D G f i (x).v ≥ 0.      (Q)
We want to prove that 0 is a solution of (Q). To do that, we proceed by contradiction. We assume that there exists y ∈ R n such that (∀i ∈ {k, . . . , e}, D G f i (x).y ≥ 0) and

D G f k-1 (x).y > 0 = D G f k-1 (x).0. Since A k � = ∅, there exists z ∈ R n such that D G f i (x).z > 0 when i ∈ {k, . . . , e}. We can not have D G f k-1 (x).z > 0 since A k-1 = ∅. Therefore we have D G f k-1 (x).z ≤ 0. If D G f k-1 (x).z < 0 we can choose ε > 0 such that D G f k-1 (x).y + εD G f k-1 (x).z > 0 (i.e 0 < ε < -D G f k-1 (x).y D G f k-1 (x).z ). If D G f k-1 (x)
.z = 0 we arbitrarily choose ε ∈ (0, +∞) and we also have D G f k-1 (x).y + εD G f k-1 (x).z > 0. We set u ε := y + εz, and we note that

D G f k-1 (x).u ε = D G f k-1 (x).y + εD G f k-1 (x).z > 0.
Furthermore, when i ∈ {k, . . . , e}, we have D G f i (x).u ε = D G f i (x).y + εD G f i (x).z > 0 since the three terms are positive. Therefore we have u ε ∈ A k-1 that is impossible since A k-1 = ∅ . And so the reasoning by contradiction is complete, and we have proven

A e � = ∅ =⇒ (0 solves (Q)). (4.5)
Since 0 solves (Q), we have, for all v ∈ R n ,

(∀i ∈ {k, . . . , e}, D G f i (x).v ≥ 0) =⇒ (D G f k-1 (x).v ≤ 0) ⇐⇒ (∀i ∈ {k, . . . , e}, D G f i (x).v ≥ 0) =⇒ (-D G f k-1 (x).v ≥ 0) .
Then we use Theorem 4.6 that ensures the existence of α k , . . . ,

α e ∈ R + such that α k D G f k (x) + • • • + α e D G f e (x) = -D G f k-1 (x) or equivalently, D G f k-1 (x) + α k D G f k (x) + • • • + α e D G f e (x) = 0. We set λ i :=          0 if i < k -1, 1 if i = k -1, α i if i ∈ {k, . . . , e}, 0 if i ∈ {e + 1, . . . ,

Proof of (d)

The assumption (iii) means that A 1 � = ∅ and by (4.2) we know that A 0 = ∅. Proceeding like in the proof of (4.5) we can prove that 0 is a solution of the following problem Maximize D G f 0 (x).v when v ∈ R n and when ∀i ∈ {1, . . . , e}, D G f i (x).v ≥ 0.

    

Then using Theorem 4.6, there exist α 1 , . . . , α e ∈ R + such that

D G f 0 (x) + α 1 D G f 1 (x) + • • • + α e D G f e (x) = 0.
Then we conclude by setting

λ i :=      1 if i = 0, α i if i ∈ {1, . . . , e}, 0 if i ∈ {e + 1, . . . , m}.
And so the proof of Theorem is complete.

Remark 4.8. The use of the sets A k comes from the book of Alekseev-Tihomirov-Fomin [START_REF] Alekseev | Optimal Control[END_REF], and the proof of formula (4.6) is similar to their proof (p. 247-248). The use of the set A 0 is yet done in [START_REF] Halkin | Implicit Functions and Optimization Problems without Continuous Differentiability of the Data[END_REF].

Proof of Theorem 4.5

First step -a simple case For a simple case, if Dh 1 (x), . . . , Dh q (x) are linearly dependent, there exist µ 1 , . . . , µ q such that (µ 1 , . . . , µ q ) � = (0, . . . , 0) and such that � 1≤i≤q µ i Dh i (x) = 0. Then it suffices to take λ i = 0 for all i ∈ {0, . . . , p} to obtain the conclusions (a), (b), (c). Now in the remaining of the proof we assume that the assumption (v) is fulfilled.

Chapter 4. Lightenings of Assumptions for Pontryagin Principles in Finite Horizon and Discrete Time

Second step -deleting the non satured inequality constraints

We will delete the non satured inequality constraints. Doing a change of index, we can assume that {1, . . . , e} = {i ∈ {1, . . . , p} : g i (x) = 0}. Using the lower semicontinuity at x of the g i when i ∈ {e + 1, . . . , p}, we can say that there exists an open neighborhood Ω 1 of x in Ω such that g i (x) > 0} when x ∈ Ω 1 and when i ∈ {e + 1, . . . , p}. And so x is a solution of the following problem Maximize g 0 (x) when x ∈ Ω 1 when ∀i ∈ {1, . . . , e}, g i (x) ≥ 0 and when ∀i ∈ {1, . . . , q}, h i (x) = 0.

         (M 1 )

Third step -deleting the equality constraints

We consider the mapping h : Ω 1 → R q defined by h(x) := (h 1 (x), . . . , h q (x)). Under (iv) and (v), h is continuous on a neighborhood of x, and it is Fréchet differentiable at x with Dh(x) onto.

We set E 1 :=KerDh(x) and we take a vector subspace

E 2 of R n such that E 1 ⊕E 2 = R n . Since Dh(x) ∈ L(R n , R q )
and it is onto hence, dim E 1 = nq and dim E 2 = q. And we can do the assimilation

R n = E 1 × E 2 . We set (x 1 , x2 ) := x ∈ E 1 × E 2 .
Then the partial differential D 2 h(x) is an isomorphism from E 2 onto R q . Now we can use Theorem 4.7 and assert that there exist a neighborhood U 1 of x1 in E 1 , a neighborhood U 2 of x2 in E 2 , and a mapping ψ :

U 1 → U 2 such that ψ(x 1 ) = x2 , h(x 1 , ψ(x 1 )) = 0 for all x 1 ∈ U 1 , and such that ψ is Fréchet differentiable at x1 with Dψ(x 1 ) = -(D 2 h(x) ) -1 • D 1 h(x) = 0 since D 1 h(x) = Dh(x)| E 1 = 0.
We define f i : U 1 → R by setting f i (x 1 ) := g i (x 1 , ψ(x 1 )) for all i ∈ {0, . . . , e}. Since x is a solution of (M 1 ), x1 is a solution of the following problem without equality constraints Maximize f 0 (x 1 ) when x 1 ∈ U 1 when ∀i ∈ {1, . . . , e}, f i (x 1 ) ≥ 0.

     (R)
Fourth step: using Theorem 4.4

Since ψ is Fréchet differentiable at x1 , the mapping (x 1 � → (x 1 , ψ(x 1 )) is Fréchet differentiable at x1 , and using (i) and (ii), we obtain that f i is Fréchet differentiable (and therefore Gâteaux differentiable) at x1 , for all i ∈ {0, . . . , e}. Note that f i (x 1 ) = 0 for all i ∈ {1, . . . , e}. Consequently we can use Theorem 4.4 on (R) that permits us to ensure the existence of λ 0 , λ 1 , . . . , λ e ∈ R + such that (λ 0 , λ 1 , . . . , λ e ) � = (0, 0, . . . , 0), (4.7)

∀i ∈ {1, . . . , e}, λ i f i (x 1 ) = 0, (4.8)

� 0≤i≤e λ i D G f i (x 1 ) = 0. (4.9)
The proof of (a), (b), (c)

We have for all i ∈ {0, . . . , e}, D

G f i (x 1 ) = Df i (x 1 ) = Dg i (x 1 , ψ(x 1 )) = D 1 g i (x 1 , ψ(x 1 )) + D 2 g i (x 1 , ψ(x 1 )) • D 1 ψ(x 1 ) = D 1 g i (x) since D 1 ψ(x 1 ) = 0. The formula (4.9) implies � 0≤i≤e λ i D 1 g i (x) = 0.
(4.10)

We will find operator M ∈ R q * such that

� 0≤i≤e λ i D 2 g i (x) + M • D 2 h(x) = 0. Since D 2 h(x) is invertible, M is easily found as follows M =   - � 0≤i≤e λ i D 2 g i (x)   • (D 2 h(x)) -1 . (4.11)
Denoting by µ 1 , . . . , µ q the coordinates of M in the canonical basis of R q * , we obtain

� 0≤i≤e λ i D 2 g i (x) + � 1≤j≤q µ j D 2 h j (x) = 0. (4.12) Since E 1 :=KerDh(x) = � 1≤j≤q KerDh j (x), we have D 1 h j (x) = Dh j (x)| E 1 =
0 for all j ∈ {1, . . . , q}, from (4.10) we obtain

� 0≤i≤e λ i D 1 g i (x) + � 1≤j≤q µ j D 1 h j (x) = 0. (4.13)
From (4.12) and (4.13) we obtain

� 0≤i≤e λ i Dg i (x) + � 1≤j≤q µ j Dh j (x) = 0. (4.14)
We set λ i := 0 when i ∈ {e + 1, . . . , p}, and so (4.14) implies (c). With (4.7) we obtain (a) and with (4.8) we obtain (b). And so the proof of (a), (b), (c) is complete.

The proof of (d)

The relation (4.7) provides the conclusion (d).

The proof of (e)

When i ∈ {1, . . . , e}, we have yet seen that Df i (x 1 ) = D 1 g i (x) = Dg i (x)| E 1 . And so the translation of the assumption (vi) gives ∃w ∈ E 1 such that ∀i ∈ {1, . . . , e}, Df i (x 1 ).w > 0.

That permits us to use the last assertion of Theorem 4.4 on (R) to ensure that we can choose λ 0 = 1.

Then the proof of Theorem 4.5 is complete. Finite Horizon and Discrete Time Remark 4.9. We see that in this proof, the assumption of Fréchet differentiability of the h j is used in order to apply the Implicit Function Theorem of Halkin. The assumption of Fréchet differentiability of g 0 and of the g i for which the associated constraint is satured is used to obtain the differentiability when we compose them with h j (to obtain the differentiability of the f j ). The Hadamard differentiability is sufficient to do that, but in finite-dimensional spaces, the Hadamard differentiability coincides with the Fréchet differentiability ( [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF], page 266).

Weak Pontryagin Principles for The Reduced Problems

We will apply New Multiplier Rule for the reduced to finite-horizon problems (F T e ) and (F T i ). To do it, we will translate these problems into static optimization problems. Note that, in the reduced problems, x 0 and x T are fixed and so they are not unknown variables. Like in [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF], page 11, we will assume that for all t ∈ N, X t is open and the sets of admissible controls U t , with t ∈ N are defined by equalities and inequalities

U t = � m i � i=1 {u ∈ R d : g i t (u) ≥ 0 � ∩ � me � k=1 {u ∈ R d : e k t (u) = 0 � , (4.15)
where for all i ∈ {1, . . . , m i }, for all k ∈ {1, . . . , m e }, for all t ∈ N, g i t : R d → R and e k t : R d → R. We also assume that U t � = ∅ for all t ∈ N.

For Problem (F T e ) We arbitrarily fix T ∈ N * . Let η and xT +1 be given. We rewrite Problem (F T e ) under model of (M).

Let Ω = T � t=1 X t × � R d � T
, then Ω is open as a finite product topology of open sets and the element z from Ω has the form z = (x 1 , x 2 , . . . , x T , u 0 , u 1 , . . . , u T ). We set: g0 (x 1 , . . . , x T , u 0 , . . . , u T ) := φ 0 (η, u 0 ) + � T t=1 φ t (x t , u t ); and for all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, we set gi t (x 1 , . . . , x T , u 0 , . . . , u T ) := g i t (u t ); and for all k ∈ {1, . . . , m e }, for all t ∈ {0, . . . , T }, we set ẽk t (x 1 , . . . , x T , u 0 , . . . , u T ) := e k t (u t ); and for all α ∈ {1, . . . , n},

ψ α 0 (x 1 , . . . , x T , u 0 , . . . , u T ) := f α 0 (η, u 0 ) -x α 1 ;
and for all α ∈ {1, . . . , n}, for all t ∈ {1, . . . , T -1}

ψ α t (x 1 , . . . , x T , u 0 , . . . , u T ) := f α t (x t , u t ) -xα t+1 ;
and for all α ∈ {1, . . . , n},

ψ α T (x 1 , . . . , x T , u 0 , . . . , u T ) := f α T (x T , u T ) -xα T +1 .
So we have translated Problem (F T e ) into this form:

Maximize g0 (x 1 , . . . , x T , u 0 , . . . , u T ); when ∀t ∈ {1, . . . , T }, x t ∈ X t , ∀t ∈ {0, . . . , T }, u t ∈ R d , ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, gi t (x 1 , . . . , x T , u 0 , . . . , u T ) ≥ 0, ∀k ∈ {1, . . . , m e }, ∀t ∈ {0, . . . , T }, ẽk t (x 1 , . . . , x T , u 0 , . . . , u T ) = 0, ∀α ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ α t (x 1 , . . . , x T , u 0 , . . . , u T ) = 0.

                 (R T e ) This is equivalent to:             
Maximize g0 (z); when z ∈ Ω; ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, gi t (z) ≥ 0, ∀k ∈ {1, . . . , m e }, ∀t ∈ {0, . . . , T }, ẽk t (z) = 0, ∀α ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ α t (z) = 0. Now we see that the Problem (R T e ) has the form of Problem (M). Let ẑ = (x 1 , . . . , xT , û0 , . . . , ûT ) be a solution of the problem above. If all the above-mentioned functions (functions g0 , gi t , ẽk t and ψ α t ) satisfy all the assumptions of the New Multiplier Rule then we can apply Theorem 4.5 to obtain the multiplier λ T o ∈ R + for the criterion function g0 (z), multipliers λ T i,t ∈ R + for inequality constraints gi t (z) ≥ 0, multipliers µ T k,t ∈ R for equality constraints ẽi t (z) = 0 and multipliers p T t+1,α ∈ R for the equality constraints ψ α t (z) = 0 where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i }, α ∈ {1, . . . , n} and k ∈ {1, . . . , m e } such that all the conclusions of Theorem 4.5 are satisfied.

From conclusion (a) of Theorem 4.5, we know that the multipliers are not simultaneously equal to zero.

From conclusion (b) of Theorem 4.5, we have ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, λ T i,t gi t (ẑ) = 0 i.e.

∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, λ T i,t g i t (û t ) = 0.

We set p T t+1 :

= n � α=1 p T t+1,α e * α ∈ R n *
where (e * α ) 1≤α≤n is the dual basis of the canonical basis of R n . Then the generalized Lagrangian of this problem is:

L(z, λ T o , λ T 1,0 , . . . , λ T m i ,T , µ T 1,0 , . . . , µ T me,T , p T 1,1 , . . . , p T T +1,n ) = λ T o .g 0 (z) + T � t=0 m i � i=1 λ T i,t gi t (z) + T � t=0 me � k=1 µ T k,t ẽi t (z) + T � t=0 n � α=1 p T t+1,α ψ α t (z) = λ T o (φ 0 (η, u 0 ) + � T t=1 φ t (x t , u t )) + T � t=0 m i � i=1 λ T i,t g i t (u t ) + T � t=0 me � k=1 µ T k,t e i t (u t ) + T � t=0 � p T t+1 , f t (x t , u t ) -x t+1 � .
As preliminary calculations, since all the functions in L are Gâteaux differentiable, the differential of the generalized Lagrangian with respect to x t (in the Gâteaux' sense) is

(z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = D G,xt L(z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = λ T o .D G,1 φ t (x t , u t ) + p T t+1 • D G,1 f t (x t , u t ) -p T t , for all t ∈ {1, . . . , T }.
and the differential of the generalized Lagrangian with respect to u t is

∂L ∂ut (z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = D G,ut L(z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = λ T o .D G,2 φ t (x t , u t ) + p T t+1 • D G,2 f t (x t , u t ) + m i � i=1 λ T i,t D G g i t (u t ) + me � k=1 µ T k,t D G e i t (u t ), for all t ∈ {0, . . . , T }.
From the conclusion (c) of Theorem of New Multiplier Rule, the Gâteaux differential of L with respect to z vanishes at ẑ. It means that

� ∀t ∈ {1, . . . , T }, D G,xt L(ẑ, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = 0, ∀t ∈ {0, . . . , T }, D G,ut L(ẑ, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = 0 ⇐⇒ ∀t ∈ {1, . . . , T }, p T t = λ T o .D G,1 φ t (x t , ût ) + p T t+1 • D G,1t f t (x t , ût ), ∀t ∈ {0, . . . , T }, λ T o .D G,2 φ t (x t , ût ) + p T t+1 • D G,2 f t (x t , ût ) + m i � i=1 λ T i,t D G g i t (û t ) + me � k=1 µ T k,t D G e i t (û t ) = 0        (4.16)
And so, we can state the following result which is called weak Pontryagin principle for the finite-horizon (reduced) problem (F T e ).

Proposition 4.10. Let (x, û) be a solution of (P j e ) where j ∈ {1, 2, 3}. We assume that X t are open for all t ∈ N and U t is defined by (4.15). We also assume that the following assumptions are fulfilled:

(i) ∀t ∈ N, φ t is Fréchet differentiable at (x t , ût ).
(ii) For all i ∈ {1, . . . , m i }, for all t ∈ N, g i t is Fréchet differentiable at ût when g i t (û t ) = 0. (iii) For all i ∈ {1, . . . , m i }, for all t ∈ N, g i t is Gâteaux differentiable at ût and lower semicontinuous at ût when g i t (û t ) > 0. (iv) For all k ∈ {1, . . . , m e }, for all t ∈ N, e i t is continuous on a neighborhood of ût and Fréchet differentiable at ût . For all t ∈ N, f t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

Then for all T ∈ N * there exist λ T o , λ T i,t ∈ R + , µ T k,t ∈ R and p T t+1 ∈ R n * where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i } and k ∈ {1, . . . , m e } such that the following conditions are satisfied.

(a) The multipliers are not simultaneously equal to zero.

(b) ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, λ T i,t g i t (û t ) = 0. (c) ∀t ∈ {1, . . . , T }, p T t = λ T o .D 1 φ t (x t , ût ) + p T t+1 • D 1 f t (x t , ût ). (d) ∀t ∈ {0, . . . , T }, λ T o .D 2 φ t (x t , ût ) + p T t+1 • D 2 f t (x t , ût ) + m i � i=1 λ T i,t D G g i t (û t ) + me � k=1 µ T k,t Dh i t (û t ) = 0.
Proof. Let (x, û) be a solution of (P j e ) where j ∈ {1, 2, 3}. Using Theorem 4.1, the restriction ẑ = ((x 0 , . . . , xT ), (û 0 , . . . , ûT -1 )) is an optimal solution of Problem (F T e ), and then it is a solution of Problem (R T e ). Under (i), (ii), (iii) and (iv) we obtain the following assertions in turn:

-Function g0 is Fréchet differentiable at ẑ as a sum of T Fréchet differentiable functions.

-For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, functions gi t is Fréchet differentiable at ẑ when gi t (ẑ) = 0. -For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, functions gi t is Gâteaux differentiable at ẑ and lower semicontinuous at ẑ when gi t (ẑ) > 0. -For all k ∈ {1, . . . , m e }, for all t ∈ {0, . . . , T }, ẽi t is continuous on a neighborhood of ẑ and Fréchet differentiable at ẑ. For all α ∈ {1, . . . , n}, for all t ∈ {0, . . . , T }, ψ α t is continuous on a neighborhood of ẑ and Fréchet differentiable at ẑ. In the special case when for all t ∈ N, U t is an arbitrary subset of R d and ût belongs the interior of U t , Problem (R T e ) now contains only phase constraints an can be rewritten as follows.

Maximize g0 (z); when z ∈ Ω; ∀α ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ α t (z) = 0.

     (int T e )
Then applying Theorem of New Multiplier Rule, we obtain the following simpler statement.

Corollary 4.11. Let (x, û) be a solution of (P j e ) where j ∈ {1, 2, 3}. We assume that X t is open and ût ∈ int(U t ) for all t ∈ N . We also assume that the following assumptions are fulfilled:

(i) ∀t ∈ N, φ t is Fréchet differentiable at (x t , ût ).
(ii) For all t ∈ N, f t are continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

Then for all T ∈ N * there exist λ T o ∈ R + and p T t+1 ∈ R n * where t ∈ {0, . . . , T } such that the following conditions are satisfied.

(a) λ T o , p T t+1 , t ∈ {0, . . . , T } are not all zeros. (b) ∀t ∈ {1, . . . , T }, p T t = λ T o .D 1 φ t (x t , ût ) + p T t+1 • D 1 f t (x t , ût ). (c) ∀t ∈ {0, . . . , T }, 0 = λ T o .D 2 φ t (x t , ût ) + p T t+1 • D 2 f t (x t , ût ).
For Problem (F T i ): the way we treat it is almost similar to what we did with the (De) case. The different point is just the following: In (R T e ) the equal sign in ψ α t (x 1 , . . . , x T , u 0 , . . . , u T ) = 0 will be replaced by the greater or equal sign (≥) because here the phase constraint has the form x t+1 ≤ f t (x t , u t ). And then, we obtain following problem Finite Horizon and Discrete Time Maximize g0 (x 1 , . . . , x T , u 0 , . . . , u T ); when ∀t ∈ {1, . . . , T }, x t ∈ X t , ∀t ∈ {0, . . . , T }, u t ∈ R d , ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, gi t (x 1 , . . . , x T , u 0 , . . . , u T ) ≥ 0, ∀k ∈ {1, . . . , m e }, ∀t ∈ {0, . . . , T }, ẽk t (x 1 , . . . , x T , u 0 , . . . , u T ) = 0, ∀α ∈ {1, . . . , n}∀t ∈ {0, . . . , T }, ψ α t (x 1 , . . . , x T , u 0 , . . . , u T ) ≥ 0,

                
(R T i ) Apply the Theorem 4.5 for this problem with a notice that now the inequality constraints group includes functions gi t and ψ α t and the equality constraints group includes only functions e k t , we receive the weak Pontryagin principle like before but with a slight difference for the inequalities ψ α t (z) ≥ 0. The whole new proposition is presented here: Proposition 4.12. Let (x, û) be a solution of (P j i ) when j ∈ {1, 2, 3}. We assume that X t are open for all t ∈ N and U t is defined by (4.15). We also assume that the following assumptions are fulfilled:

(i) ∀t ∈ N, φ t is Fréchet differentiable at (x t , ût ).
(ii) For all i ∈ {1, . . . , m i }, for all t ∈ N, g i t is Fréchet differentiable at ût when g i t (û t ) = 0. For all α ∈ {1, . . . , n}, for all t ∈ N, f α t is Fréchet differentiable at (x t , ût ) when f α t (x t , ût ) = xα t+1 . (iii) For all i ∈ {1, . . . , m i }, for all t ∈ N, g i t is Gâteaux differentiable at ût and lower semicontinuous at ût when g i t (û t ) > 0. For all α ∈ {1, . . . , n}, for all t ∈ N, f α t is Gâteaux differentiable and and lower semicontinuous at (x t , ût ) when f α t (x t , ût ) = xα t+1 . (iv) For all k ∈ {1, . . . , m e }, for all t ∈ N, e i t is continuous on a neighborhood of ût and Fréchet differentiable at ût . Then for all T ∈ N * there exist λ T o , λ T i,t ∈ R + , µ T k,t ∈ R and p T t+1 ∈ R n * + where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i } and k ∈ {1, . . . , m e } such that the following conditions are satisfied.

(a) The multipliers are not simultaneously equal to zero. (b) For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, λ T i,t g i t (û t ) = 0. For all t ∈ {0, . . . , T },

� p T t+1 , f t (x t , ût ) -xt+1 � = 0. (c) ∀t ∈ {1, . . . , T }, p T t = λ T o .D 1 φ t (x t , ût ) + p T t+1 • D 1 f t (x t , ût ). (d) ∀t ∈ {0, . . . , T }, λ T o .D 2 φ t (x t , ût ) + p T t+1 • D 2 f t (x t , ût ) + m i � i=1 λ T i,t D G g i t (û t ) + me � k=1 µ T k,t De i t (û t ) = 0.
In the special case when for all t ∈ N, U t is an arbitrary subset of R d and ût belongs the interior of U t , Problem (R T i ) now contains only phase constraints an can be rewritten as follows.

     Maximize g0 (z); when z ∈ Ω; ∀α ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ α t (z) ≥ 0. (int T i )
This has the form of Problem (I). Then we obtain the following corollary after applying Theorem 4.4.

Weak Pontryagin Principles for Infinite-Horizon Problems with Interior Optimal Controls 83

Corollary 4.13. Let (x, û) be a solution of (P j i ) when j ∈ {1, 2, 3}. We assume that for all t ∈ N, X t are open and ût ∈ int(U t ). We also assume that the following assumptions are fulfilled:

(i) ∀t ∈ N, φ t , f t is Gâteaux differentiable at (x t , ût ).
(ii) For all α ∈ {1, . . . , n}, for all t ∈ N, f α t is lower semicontinuous at (x t , ût ) when f α t (x t , ût ) > xα t+1 . Then for all T ∈ N * there exist λ T o ∈ R + and p T t+1 ∈ R n * + where t ∈ {0, . . . , T } such that the following conditions are satisfied.

(a) λ T o , p T t+1 , t ∈ {0, . . . , T } are not all zeros. (b) For all t ∈ {0, . . . , T },

� p T t+1 , f t (x t , ût ) -xt+1 � = 0. (c) ∀t ∈ {1, . . . , T }, p T t = λ T o .D 1 φ t (x t , ût ) + p T t+1 • D 1 f t (x t , ût ). (d) ∀t ∈ {0, . . . , T }, 0 = λ T o .D 2 φ t (x t , ût ) + p T t+1 • D 2 f t (x t , ût ).

Weak Pontryagin Principles for Infinite-Horizon Problems with Interior Optimal Controls

In this section we consider the case where values of the optimal control sequence belong to the topological interior of the set U t of the considered controls at each time t, and where the system is governed by the difference inequation (Di).

Theorem 4.14. Let (x, û) be a solution of (P j i ) when j ∈ {1, 2, 3}. We assume that the following assumptions are fulfilled.

(i) For all t ∈ N, ût ∈ intU t .

(ii) For all t ∈ N, φ t and f t are Gâteaux differentiable at (x t , ût ).

(iii) For all t ∈ N, for all α ∈ {1, . . . , n}, f α t is lower semicontinuous at (x t , ût ) when

f α t (x t , ût ) > xα t+1 . (iv) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible.
Then there exist λ 0 ∈ R and (p t ) t∈N * ∈ (R n * ) N * which satisfy the following properties.

(NN) (λ 0 , p 1 ) � = (0, 0).

(Si) λ 0 ≥ 0 and, for all t ∈ N * , p t ≥ 0.

(S�) For all t ∈ N, for all α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût ) -xα t+1 ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D G,1 φ t (x t , ût ). (WM) For all t ∈ N, p t+1 • D G,2 f t (x t , ût ) + λ 0 D G,2 φ t (x t , ût ) = 0.
Proof. Our assumptions (i, ii, iii) imply that the assumptions of Corollary 4.13 in previous section are fulfilled and so we know that, for all T ∈ N * , there exists (λ T 0 , p

= p T t • [D 1 f t (x t , ût )] -1 -λ T o .D 1 φ t (x t , ût ) • [D 1 f t (x t , ût )] -1 . (4.22)
From this last equation we easily see that (λ T 0 , p T 1 ) = (0, 0) =⇒ (λ T 0 , p T 1 , . . . , p T T +1 ) = (0, 0, . . . , 0) and then from (4.17) we have a contradiction. Therefore we can assert that (λ T 0 , p T 1 ) � = (0, 0). (4.23)

Since the set of the lists of multipliers of Problem (int T i ) is a cone, we can normalize the multipliers by setting

� � �(λ T 0 , p T 1 ) � � � = � � �λ T 0 � � � + � � �p T 1 � � � = 1. (4.24)
Since the values of the sequence (λ T 0 , p T 1 ) T ∈N * belong to the unit sphere of R × R n * which is compact, using the Bolzano-Weierstrass theorem we can say that there exist an increasing function ϕ :

N * → N * and (λ 0 , p 1 ) ∈ R × R n * such that |λ 0 | + �p 1 � = 1, lim T →+∞ λ ϕ(T ) 0 = λ 0 and lim T →+∞ p ϕ(T ) 1 = p 1 . Note that p ϕ(T ) 2 = (p ϕ(T ) 1 -λ ϕ(T ) 0 D G,1 φ 1 (x 1 , û1 )) • D G,1 f 1 (x 1 , û1 ) -1 for all T ≥ t -1, which implies that p 2 = lim T →+∞ p ϕ(T ) 2 = (p 1 -λ 0 .D G,1 φ 1 (x 1 , û1 )) • D G,1 f 1 (x 1 , û1 ) -1
Proceeding recursively we define, for all t ∈ N * ,

p t+1 := lim T →+∞ p ϕ(T ) t+1 = lim T →+∞ (p ϕ(T ) t -λ ϕ(T ) 0 D G,1 φ t (x t , ût )) • D G,1 f t (x t , ût ) -1 = (p t -λ 0 D G,1 φ t (x t , ût )) • D G,1 f t (x t , ût ) -1 .
And so we have built λ 0 ∈ R and a sequence (p t ) t∈N * ∈ (R n * ) N * which satisfies (AE). We have yet seen that (NN) is satisfied. From (4.18) we obtain (Si). From (4.19) we obtain (S�). From (4.21) we obtain (WM). Notation 4.15. Note that, for all t ∈ N * , D 1 f t (x t , ût ) belongs to L(R n , R n ), the space of all continuous linear operators from R n into R n . It can be represented by a n × n Jacobian matrix. We will denote the element at position (α, β) of this matrix by ∂f α t (x t , ût ) ∂x β t where α, β ∈ {1, . . . , n}.

In the next theorem, we will replace the assumption of invertibility (assumption (iv)) in previous theorem by another one, which is called positivity assumption. The theorem is stated as follows.

Theorem 4.16. Let (x, û) be a solution of (P j i ) when j ∈ {1, 2, 3}. We assume that the following assumptions (i,ii,iii) of Theorem 4.14 are fulfilled. Moreover we assume that the following assumption is fulfilled.

(v) For all t ∈ N * , for all α, β ∈ {1, . . . , n},

∂f α t (x t , ût ) ∂x β t ≥ 0 and ∂f α t (x t , ût ) ∂x α t > 0.
Then the conclusions of Theorem 4.14 hold.

Before proving this theorem, we recall the following useful lemma which related to condition (v).

π • D 1 f t (x t , ût ).y = π(D 1 f t (x t , ût ).y) = π(z + ρ t .y) where z ∈ R n + = π(z) + π(ρ t .y) ≥ π(ρ t .y) = ρ t .π(y).
We have the change above because π is linear. The last inequality means

π • D 1 f t (x t , ût ) ≥ ρ t .π.
Now we move to the proof of Theorem 4.16.

Proof. Proceeding as in the proof of Theorem 4.14we obtain for all T ∈ N * , there exists

(λ T 0 , p T 1 , . . . , p T T +1 ) ∈ R × (R n * + ) T +1
which satisfies the conditions (4.17-4.21). We have

∀t ∈ {1, . . . , T }, p T t = λ T o .D G,1 φ t (x t , ût ) + p T t+1 • D G,1 f t (x t , ût ) ⇒ ∀t ∈ {1, . . . , T }, p T t+1 • D G,1 f t (x t , ût ) = p T t -λ T o .D G,1 φ t (x t , ût ). Since p T
t+1 ≥ 0, using Lemma 4.17 we have:

∀t ∈ {1, . . . , T }, p T t+1 • D G,1 f t (x t , ût ) ≥ ρ t .p T t+1
where ρ t := min

1≤α≤n ∂f α t (x t , ût ) ∂x α t . Then ∀t ∈ {1, . . . , T }, 0 ≤ ρ t .p T t+1 ≤ p T t -λ T o .D G,1 φ t (x t , ût ) ⇒ ρ t � � �p T t+1 � � � = � � �ρ t .p T t+1 � � � ≤ � � �p T t � � � + λ T o . �D G,1 φ t (x t , ût )� , (since λ T o ≥ 0) ⇒ ∀t ∈ {1, . . . , T }, � � �p T t+1 � � � ≤ 1 ρ t �� � �p T t � � � + λ T o . �D G,1 φ t (x t , ût )� � ,
which implies that if (λ T 0 , p T 1 ) = (0, 0) then (λ T 0 , p T 1 , . . . , p T T +1 ) = (0, 0, . . . , 0) which is a contradiction with condition (4.17). Then condition (4.23) holds and by normalizing we obtain (4.24). Then 0 ≤ λ T o ≤ 1 and the following relation holds for all t ∈ N * and for all T ≥ t -1.

� � �p T t+1 � � � ≤ 1 ρ t �� � �p T t � � � + �D G,1 φ t (x t , ût )� � (4.25)
Now we want to prove the following assertion: We proceed by induction. When t = 1, from (4.24), we know that

∀t ∈ N * , ∃ζ t ∈ (0, +∞), ∀T ≥ t -1, � � �p T t � � � ≤ ζ t . ( 4
� � �p T 1 � � � ≤ 1.
And so it suffices to take ζ 1 := 1. Assume that (4.26) holds for t, then for t + 1, from (4.25) we obtain

� � �p T t+1 � � � ≤ 1 ρ t (ζ t + �D G,1 φ t (x t , ût )�) =: ζ t+1 .
Hence, assertion (4.26) is proven. Using (4.26) and the diagonal process of Cantor as it is formulated in [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF] (Theorem A.1, p.94), we can assert that there exist an increasing function ϕ : N * → N * , a real number λ o ≥ 0 and a sequence (p t ) t∈N * ∈ (R n * + ) N * such that lim T →+∞ λ ϕ(T ) 0

= λ 0 and lim T →+∞ p ϕ(T ) t = p t for all t ∈ N * . Now we conclude as in the proof of Theorem 4.14.

Weak Pontryagin Principles for Infinite-Horizon Problems with Constrained Controls

In this section we first consider the case where the sets of controls are defined by inequalities for each t ∈ N.

U t = � 1≤k≤m {u ∈ R d : g k t (u) ≥ 0} (4.27)
where g k t : R d → R and we assume that U t � = ∅ for all t ∈ N. In this subsection, we use the following notations for linear span and convex hull of a finite set of vectors in real normed vector space. Notation 4.18. Let S be a finite subset of a normed vector space X. We denote the linear span of S by span(S) and the convex hull of S by conv(S). Lemma 4. [START_REF] Border | Notes on the Implicit Function Theorem[END_REF]. Let E be a finite-dimensional real normed vector space and I be a nonempty finite set. Let (ϕ i ) i∈I ∈ (E * ) I . The three following assertions are equivalent.

(i) 0 / ∈ conv{ϕ i : i ∈ I}.
(ii) For all (λ i ) i∈I ∈ (R + ) I , � i∈I λ i ϕ i = 0 =⇒ λ i = 0 for all i ∈ I. (iii) There exists w ∈ E such that, �ϕ i , w� > 0 for all i ∈ I.

Proof. Firstly, we prove that non(ii) implies non(i). From non(ii) we deduce that there exists (λ i ) i∈I ∈ (R + ) I such that (λ i ) i∈I � = 0 and

� i∈I ( λ i � j∈I λ j )ϕ i = 0 which implies non(i).
Secondly, we prove that non(i) implies non(ii). From non(i) there exists (α i ) i∈I ∈ (R + ) I such that � i∈I α i = 1 and 0 = � i∈I α i ϕ i , and since (α i ) i∈I is non zero, non(ii) is fulfilled. And so we have proven that non(i) and non(ii) are equivalent.

To prove that (i) implies (iii), note that 0 / ∈ conv{ϕ i : i ∈ I} =: K, and K is a nonempty convex compact set. Using the theorem of separation of Hahn-Banach, we can assert that there exist ξ ∈ R n * * and a ∈ (0, +∞) such that �ξ, ϕ� ≥ a for all ϕ ∈ K, and �ξ, 0� = 0 < a. Since R n is reflexive, there exists w ∈ R n such �ξ, ϕ� = �ϕ, w� for all ϕ ∈ R n * . Therefore for all i ∈ I, we have �ϕ i , w� ≥ a > 0 that is (iii).

To prove that (iii) implies (i) we set γ := min i∈I �ϕ i , w� > 0. When ϕ ∈ conv{ϕ i : i ∈ I}, there exists (α i ) i∈I ∈ (R + ) I such that

� i∈I α i = 1 and ϕ = � i∈I α i ϕ i . Then we have �ϕ, w� = � i∈I α i �ϕ i , w� ≥ � i∈I α i γ = γ >
0 which implies ϕ � = 0, and so (i) is satisfied.

Weak Pontryagin Principles for Infinite-Horizon Problems with Constrained Controls 87

Lemma 4.20. Let E be a finite-dimensional real normed vector space and I be a nonempty finite set. Let (ϕ i ) i∈I ∈ (E * ) I such that 0 / ∈ conv{ϕ i : i ∈ I}. For all i ∈ I, let (r h i ) h∈N * ∈ R N * + . We assume that the sequence (ψ h ) h∈N * := (

� i∈I r h i ϕ i ) h∈N * is bounded in E * .
Then there exists an increasing function ρ : N * → N * such that, for all i ∈ I, the sequence (r

ρ(h) i ) h∈N * is convergent in R + .
Proof. Firstly, we prove that lim inf h→+∞ � i∈I r h i < +∞. We proceed by contradiction: we assume that lim inf h→+∞ � i∈I r h i = +∞. Therefore we have lim h→+∞ � i∈I r h i = +∞. We set s h i :=

r h i � j∈I r h j ∈ R + . We have � i∈I s h i = 1 and so � i∈I s h i ϕ i ∈ conv{ϕ i : i ∈ I}. Note that � � i∈I s h i ϕ i � = 1 � j∈I r h j �ψ h � converges to 0 when h → +∞ since (ψ h ) h∈N *
is bounded. Thus, we have lim h→+∞ � i∈I s h i ϕ i = 0 which implies that 0 ∈ conv{ϕ i : i ∈ I} and that is a contradiction with one assumption. And so we have proven that s := lim inf h→+∞ � i∈I r h i < +∞. Now we can assert that there exists an increasing function τ :

N * → N * such that lim h→+∞ � i∈I r τ (h) i = s. Therefore there exists M ∈ R + such that 0 ≤ � i∈I r τ (h) i ≤ M for all h ∈ N * . Since for all i ∈ I, we have 0 ≤ r τ (h) i ≤ � j∈I r τ (h) j ≤ M , i.e. the sequence (r τ (h) i
) h∈N * is bounded in R + . Using several times the Bolzano-Weierstrass theorem we can assert that there exist an increasing function τ 1 :

N * → N * and r * i ∈ R + for all i ∈ I, such that lim h→+∞ r τ •τ 1 (h) i = r * i . It suffices to take ρ := τ • τ 1 .
Theorem 4.21. Let (x, û) be a solution of (P j i ) where j ∈ {1, 2, 3} and where the sets U t are defined by (4.27). We assume that the following assumptions are fulfilled.

(i) For all t ∈ N, φ t and f t are Gâteaux differentiable at (x t , ût ).

(ii) For all t ∈ N, for all k ∈ {1, . . . , m}, g k t is Gâteaux differentiable at ût . (iii) For all t ∈ N, for all α ∈ {1, . . . , n}, f α t is lower semicontinuous at (x t , ût ) when f α t (x t , ût ) > xα t+1 . (iv) For all t ∈ N, for all k ∈ {1, . . . , m}, g k t is lower semicontinuous at ût when

g k t (û t ) > 0. (v) For all t ∈ N, 0 / ∈ conv{D G g k t (û t ) : k ∈ I s t } where I s t := {k ∈ {1, . . . , m} : g k t (û t ) = 0}. (vi) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible.
(vii) For all t ∈ N * , for all α, β ∈ {1, . . . , n},

∂f α t (xt,ût) ∂x β
≥ 0 and for all α ∈ {1, . . . , n},

∂f α t (xt,ût) ∂x α > 0.
Then, under (i-vi) or under (i-v) and (vii), there exist

λ 0 ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (λ 1 t ) t∈N ∈ R N , .
. . , and (λ m t ) t∈N ∈ R N which satisfy the following conditions.

(NN) (λ 0 , p 1 ) � = (0, 0).

(Si) λ 0 ≥ 0, , p t ≥ 0 for all t ∈ N * , and λ k t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m}. (S�) For all t ∈ N, for all α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût ) -xα t+1 ) = 0, and for all k ∈ {1, . . . , m},

λ k t g k t (û t ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D G,1 φ t (x t , ût ). (WM) For all t ∈ N, p t+1 • D G,2 f t (x t , ût ) + λ 0 D G,2 φ t (x t , ût ) + � m k=1 λ k t D G g k t (û) = 0.
Finite Horizon and Discrete Time Proof. From Theorem 4.1 (b) we know that, for all T ∈ N * , (x 0 , . . . , xT +1 , û0 , . . . , ûT ) is a solution of the following finite-horizon problem.

             Maximize J(x 0 , . . . , x T +1 , u 0 , . . . , u T ) = � T t=0 φ t (x t , u t ) when ∀t ∈ {0, . . . , T }, f t (x t , u t ) -x t+1 ≥ 0 ∀t ∈ {0, . . . , T + 1}, x t ∈ X t x 0 = η, x T +1 = xT +1 ∀t ∈ {0, .
. . , T }, ∀k ∈ {1, . . . , m}, g k t (u t ) ≥ 0. From Theorem 4.4 we can assert that there exists

(λ T 0 , p T 1 , . . . , p T T +1 , λ 1,T 1 , . . . , λ m,T T ) ∈ R × (R n * ) T + 1 × R mT which
p T t = p T t+1 • D G,1 f t (x t , ût ) + λ T 0 D G,1 φ t (x t , ût ). � (4.31) ∀t ∈ {0, . . . , T }, p T t+1 • D G,2 f t (x t , ût ) + λ T 0 D G,2 φ t (x t , ût ) + � m k=1 λ k,T t D G g k t (û t ) = 0. � (4.32)
Using (4.31) under (vi) or (vii) and working as in the proof of Theorem 4.14 or Theorem 4.16, we obtain (λ T 0 , p T 1 ) = (0, 0) =⇒ (λ T 0 , p T 1 , . . . , p T T +1 ) = (0, 0, . . . , 0). Proceeding by contradiction, assuming that (λ T 0 , p T 1 ) = (0, 0), from the previous implication and (4.32) we obtain

� m k=1 λ k,T
t D G g k t (û t ) = 0, and then using Lemma 4.19 we obtain the λ k,T t = 0. Therefore, we obtain a contradiction with (4.28). And so we have proven that (λ T 0 , p T 1 ) � = (0, 0). Under (vi), proceeding as in the proof of Theorem 4.14 and under (vii), proceeding as in the proof of Theorem 4.16 we obtain the existence of an increasing function ρ : N * → N * and of λ 0 ∈ R + and of (p

t ) t∈N * ∈ (R n * + ) N * such that λ 0 = lim T →+∞ λ ρ(T ) 0 , p t = lim T →+∞ p ρ(T ) t
, (λ 0 , p 1 ) � = (0, 0), and p α t • (f α t (x t , ût ) -xα t+1 ) = 0 for all t ∈ N * and for all α ∈ {1, . . . , n}.

We fix t ∈ N and we consider, for all T ∈ N * ,

ϕ T := � k∈I s t λ k,ρ(T ) t D G g k t (û t ) = � m k=1 λ k,ρ(T ) t D G g k t (û t ) = -(p ρ(T ) t+1 • D G,2 f t (x t , ût ) + λ ρ(T ) 0 D G,2 φ t (x t , ût )).
Therefore, we have lim

T →+∞ ϕ T = -(p t+1 • D G,2 f t (x t , ût ) + λ 0 D G,2 φ t (x t , ût ))
, and consequently the sequence (ϕ T ) T ∈N * is bounded in R n * . Using Lemma 4.20 we can assert that there exist an increasing function ρ 1 : N * → N * and λ 1 t , . . . ,

λ m t ∈ R + such that lim T →+∞ λ k,ρ•ρ 1 (T ) t = λ k t ∈ R + .
And then the assertions (NN), (Si), (S�), (AE) and (WM) are satisfied. Now we consider the case where the sets of controls are defined by equalities and inequalities for each t ∈ N as in (4.15), The following lemmas are useful for the proofs of the weak Pontryagin principles for problems with constraints on control sets defined as above.

U t =   � 1≤k≤m i {u ∈ R d : g k t (u) ≥ 0}   ∩   � 1≤k≤me {u ∈ R d : e k t (u) = 0}   . ( 4
Lemma 4.22. Let E be a real finite-dimensional normed vector space; let J and K be two nonempty finite sets, and let (ψ j ) j∈J and (ϕ k ) k∈K be two families of elements of the dual E * . Then the two following assertions are equivalent.

(i) span{ψ j : j ∈ J} ∩ conv{ϕ k : k ∈ K} = ∅.
(ii) There exists w ∈ E such that �ψ j , w� = 0 for all j ∈ J and �ϕ k , w� > 0 for all k ∈ K.

Proof. We set S := span{ψ j : j ∈ J} and C := conv{ϕ k : k ∈ K}.

[i =⇒ ii] Under (i) using the theorem of separation of Hahn-Banach, there exist ξ ∈ E * * and a ∈ (0, +∞) such that �ξ, ψ� ≤ a for all ψ ∈ S, and �ξ, ϕ� > a for all ϕ ∈ C. When ψ ∈ S is non zero, we have |�ξ, ψ�| ≤ a since -ψ ∈ S, and therefore, for all λ ∈ R, we have |λ| • |�ξ, ψ�| ≤ a which impossible if |�ξ, ψ�| � = 0, therefore we have �ξ, ψ� = 0 for all ψ ∈ S.

Since E * * is isomorphic to E there exists w ∈ E such �ξ, χ� = �χ, w� for all χ ∈ E * , and then we obtain (ii).

[ii =⇒ i] Under (ii) we define a := min k∈K �ϕ k , w� > 0. When ϕ ∈ C there exists (θ k ) k∈K ∈ R K + such that

� k∈K θ k = 1 and � k∈K θ k ϕ k = ϕ. Then �ϕ, w� = � k∈K θ k �ϕ k , w� ≥ � k∈K θ k • a > 0. When ψ ∈ S there exists (ζ j ) j∈J ∈ R J such that � j∈J ζ j ψ j = ψ.
Therefore we have �ψ, w� = � j∈J ζ j �ψ j , w� = 0. We have proven that �ψ, w� = 0 for all ψ ∈ S and �ϕ, w� > 0 for all ϕ ∈ C, which implies (i).

Lemma 4.23. In the framework of Lemma 4.22, under condition (i) of Lemma 4.22, when (µ j ) j∈J ∈ R J and (λ k ) k∈K ∈ R K + , we have

� j∈J µ j ψ j + � k∈K λ k ϕ k = 0 =⇒ (∀k ∈ K, λ k = 0).
Proof. We proceed by contraposition, we assume that there exists k ∈ K such that λ k � = 0. Then λ := � k∈K λ k > 0 and so

� k∈K λ k λ ϕ k ∈ conv{ϕ k : k ∈ K} and � k∈K λ k λ ϕ k = - � j∈J µ j
λ ψ j ∈ span{ψ j : j ∈ J} which provides a contradiction with condition (i). Lemma 4.24. Let E be a real finite-dimensional normed vector space; let J and K be two nonempty finite sets, and let (ψ j ) j∈J and (ϕ k ) k∈K be two families of elements of the dual E * . We assume that the following assumptions are fulfilled.

(a) The family (ψ j ) j∈J is linearly independent.

(b) span{ψ j : j ∈ J} ∩ conv{ϕ k : k ∈ K} = ∅. Let (µ h j ) j∈J ∈ R J and (λ h k ) k∈K ∈ R K + for all h ∈ N * such that the sequence (χ h ) h∈N * := ( � j∈J µ h j ψ j + � k∈K λ h k ϕ k ) h∈N * is bounded in E * .
Then there exists an increasing function ρ : N * → N * such that the sequences (µ ρ(h) j

) h∈N * are convergent in R for all j ∈ J and the sequences and (λ

ρ(h) k ) h∈N * are convergent in R + for all k ∈ K.
Proof. We set S := span{ψ j : j ∈ J} and C := conv{ϕ k : k ∈ K}. Firstly, we prove that lim inf h→+∞ � k∈K λ h k < +∞. We proceed by contradiction, we assume that lim inf h→+∞ � k∈K λ h k = +∞. Therefore we have s := lim h→+∞

� k∈K λ h k = +∞. We set π h k := λ h k � k � ∈K λ h k � ∈ R + . We have � k∈K π h k = 1
, and therefore

� k∈K π h k ϕ k ∈ C. Note that � � j∈J µ h j � k � ∈K λ h k � ψ j + � k∈K π h k ϕ k � = 1 � k � ∈K λ h k � �χ h � → 0
Chapter 4. Lightenings of Assumptions for Pontryagin Principles in Finite Horizon and Discrete Time

when h → +∞, therefore lim h→+∞   � j∈J µ h j � k � ∈K λ h k � ψ j + � k∈K π h k ϕ k   = 0.
Since C is compact there exists an increasing function τ :

N * → N * and ϕ * ∈ C such that lim h→+∞ � k∈K π τ (h) k ϕ k = ϕ * . Consequently lim h→+∞ � j∈J -µ τ (h) j � k � ∈K λ τ (h) k � ψ j = ϕ * . Since a
finite-dimensional normed vector space is complete, S is closed in E * , and consequently we have ϕ * ∈ S, and then ϕ * ∈ S ∩ C which is a contradiction with assumption (b). And so we have proven that lim inf h→+∞ � k∈K λ h k < +∞. Now, the previous result implies that there exists an increasing function r :

N * → N * such that lim h→+∞ � k∈K λ r(h) k = lim inf h→+∞ � k∈K λ h k .
Thus, the sequence (

� k∈K λ r(h) k ) h∈N * ) h∈N * is bounded in R + . Since 0 ≤ λ r(h) k ≤ � k∈K λ r(h) k , we obtain (λ r(h) k ) h∈N * is bounded in R + for all k ∈ K. Therefore ( � k∈K λ r(h) k ϕ k ) h∈N * is bounded in E * . Consequently, ( � j∈J µ r(h) j ψ j ) h∈N * = (χ r(h) - � k∈K λ r(h) k ϕ k ) h∈N * is
bounded as a difference of two bounded sequences. Under assumption (a) we can use Lemma 5.5 in [START_REF] Blot | Infinite-Horizon Discrete-Time Pontryagin Principles via Results of Michel[END_REF] and assert that there exists an increasing function r 1 :

N * → N * such (µ r•r 1 (h) j
) h∈N * is convergent in R for all j ∈ J. Using |K| times the Bolzano-Weierstrass theorem, there exists an increasing function r 2 :

N * → N * such that λ r•r 1 •r 2 (h) k h∈N * is convergent in R + .
Taking ρ := r • r 1 • r 2 we have proven the lemma. Theorem 4.25. Let (x, û) be a solution of (P j i ) where j ∈ {1, 2, 3} and where the sets U t are defined in (4.33). We assume that the following assumptions are fulfilled for all t ∈ N.

(i) φ t is Fréchet differentiable at (x t , ût ).

(ii) For all α ∈ {1, . . . , n}, f α t is Fréchet differentiable at (x t , ût ) when f α t (x t , ût ) = x α t+1 . (iii) For all α ∈ {1, . . . , n}, f α t is lower semicontinuous and Gâteaux differentiable at (x t , ût ) when f α t (x t , ût ) > x α t+1 . (iv) For all k ∈ {1, . . . , m i }, g k t is Fréchet differentiable at ût when g k t (û t ) = 0. (v) For all k ∈ {1, . . . , m i }, g k t is lower semicontinuous and Gâteaux differentiable at ût when g k t (û t ) > 0. (vi) For all j ∈ {1, . . . , m e }, e j t is continuous on a neighborhood of ût and Fréchet differentiable at ût .

(vii) span{De j t (û t ) : j ∈ {1, . . . , m e }} ∩ conv{D G g k t (û t ) : k ∈ I s t } = ∅, where I s t := {k ∈ {1, . . . , m i } : g k t (û t ) = 0}. (viii) De 1 t (û t ), . . . , De me (û t ) are linearly independent. (ix) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible. (x) For all t ∈ N * , for all α, β ∈ {1, . . . , n}, ∂f α t (xt,ût) ∂x β
≥ 0 and for all α ∈ {1, . . . , n},

∂f α t (xt,ût) ∂x α > 0.
Then under (i-ix) or under (i-viii) and (x) there exist λ 0 ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ 1,t ) t∈N ∈ R N , . . . , (µ me,t ) t∈N ∈ R N , (λ 1,t ) t∈N ∈ R N , . . . , (λ m i ,t ) t∈N ∈ R N which satisfy the following conditions.

(NN) (λ 0 , p 1 ) � = (0, 0).

(Si) λ 0 ≥ 0, p t ≥ 0 for all t ∈ N * , λ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }. (S�) For all t ∈ N, for all α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût )x α t+1 ) = 0, and for all k ∈ {1, . . . ,

m i }, λ k,t • g k t (û t ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D 1 φ t (x t , ût ).
(WM) For all t ∈ N,

p t+1 • D G,2 f t (x t , ût ) + λ 0 D 2 φ t (x t , ût ) + m i � k=1 λ k,t D G g k t (û t ) + me � j=1 µ j,t De j (û t ) = 0.
Proof. Under assumptions (i-vi), using Proposition 4.12 we obtain the existence of real numbers λ T 0 , p T t,α (for t ∈ {1, . . . , T + 1} and α ∈ {1, . . . , n}), µ T t,j (for t ∈ {0, . . . , T } and j ∈ {1, . . . , m e }), λ T t,k (for t ∈ {0, . . . , T } and k ∈ {1, . . . , m i }) which satisfy the following conditions:

(λ T 0 , p T 1,1 , . . . , p T T +1,n , µ T 1,0 , . . . , µ T me,T , λ T 1,0 , . . . , λ T m i ,T ) � = 0 (4.34) λ T 0 ≥ 0, (∀t ∈ {1, . . . , T + 1}, ∀α ∈ {1, . . . , n}, p T t,α ≥ 0) (∀t ∈ {0, . . . , T }, ∀k ∈ {1, . . . , m i }, λ T t,j ≥ 0) � (4.35) ∀t ∈ {0, . . . , T }, ∀α ∈ {1, . . . , n}, p T t+1,α .(f α t (x t , ût ) -xα t ) = 0 (4.36) ∀t ∈ {0, . . . , T }, ∀k ∈ {1, . . . , m i }, λ T t,k .g k t (û t ) = 0 (4.37) ∀t ∈ {1, . . . , T }, λ T 0 D 1 φ t (x t , ût ) + p T t+1 • D G,1 f t (x t , ût ) = p T t . (4.38)
λ T 0 D 2 φ t (x t , ût ) + p T t+1 • D G,2 f t (x t , ût ) + � me j=1 µ T t,j De j t (û t ) + � m i k=1 λ T t,k D G g k t (û t ) = 0. � (4.39)
Using (ix) and working as in the proof of Theorem 4.14 or using (x) and working as in the proof of Theorem 4.16 , from (4.38) we obtain the following condition.

(λ T 0 , p T 1 ) = (0, 0) =⇒ (λ T 0 , p T 1 , . . . , p T T +1 ) = (0, 0, . . . , 0). (

If (λ T 0 , p T 1 ) = (0, 0), using (4.39), (

� j=1 µ T t,j De j t (û t ) + m i � k=1 λ T t,k D G g k t (û t ) = 0, 4.40) implies me
and using (4.37) we obtain λ T t,k = 0 if k / ∈ I s t , and so we obtain the following relation

� me j=1 µ T t,j De j t (û t ) + � k∈I s t λ T t,k D G g k t (û t ) = 0.
Then using (vii) and Lemma 4.23 we obtain λ T t,k = 0 for all k ∈ I s t , and consequently we have λ T t,k = 0 for all k ∈ {1, . . . , m i }. Therefore we have � me j=1 µ T t,j De j t (û t ) = 0. Using (viii) we obtain µ T t,j = 0 for all j ∈ {1, . . . , m e }. And so we have proven that (λ T 0 , p T 1 ) = (0, 0) implies (λ T 0 , p T 1,1 , . . . , p T T +1,n , µ T 1,0 , . . . , µ T me,T , λ T 1,0 , . . . , λ T m i ,T ) = (0, . . . , 0) which is a contradiction with (4.34). And so we have proven the following condition.

(λ T 0 , p T 1 ) � = (0, 0). ( 4.41)
From (4.41) under (ix) proceeding as in the proof of Theorem 4.14 or, under (x) proceeding as in the proof of Theorem 4.16 we obtain the existence of an increasing function r :

N * → N * , of λ 0 ∈ R and of (p t ) t∈N * ∈ (R n * ) N * such that lim T →+∞ λ r(T ) 0 = λ 0 , (∀t ∈ N * , lim T →+∞ p r(T ) t = p t ), (λ 0 , p 1 ) � = (0, 0). (4.42)
then to translate the finite-horizon multiobjective optimal control problems into static multiobjective optimization problems, then use an appropriate multiplier rule for such problems. However, many existing multiplier rules require the smoothness or at least, the Fréchet differentiability at the optimal solution and the continuity on a neighborhood of the optimal solution of the functions issued in the problem. To establish weak maximum principles, we provide new multiplier rules for static multiobjective optimization problems, which are built in the spirit of the single-objective static optimization rules that Blot presents in [START_REF] Blot | On the Multiplier Rules[END_REF]. In these rules, in some places, instead of the usual Fréchet differentiability at the optimal solution we use Gâteaux differentiability at it and we replace the continuity on a neighborhood of the optimal solution by lower semicontinuity at the optimal solution.

To establish strong maximum principles, we rely on a multiplier rule of Khanh and Nuong [START_REF] Khanh | On Necessary Optimality Conditions in Vector Optimization Problems[END_REF]. So in some places, instead of the usual C 1 differentiability with respect to the optimal state variable, we use the directional differentiability with respect to the optimal state variable with a concavity property, and the continuity at the optimal state variable. Since we study multiobjective optimal control problems in the presence of constraints, this chapter generalizes to the multiobjective case some results of Blot [START_REF] Blot | A Pontryagin Principle for Infinite-horizon Problems under Constraints[END_REF] and Blot and Hayek [START_REF] Blot | Infinite-Horizon Pontryagin Principles with Constraints[END_REF][START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF], where infinite-horizon single-objective optimal control problems under constraints were studied. It also generalizes some results of strong Pontryagin principles and results of weak Pontryagin principles when replacing weak Pareto optimality by Pareto optimality from Hayek [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF], where infinite-horizon multiobjective optimal control problems in the discrete time framework are studied. In this chapter, we provide weaker smoothness assumptions and moreover, we study problems under constraints. Sufficient conditions of optimality for considered problems are also studied in this chapter, in which, similar to the one in Chapter 3, concavity assumption on the Hamiltonian is required. The structure of this chapter is as follows: After the introduction, the infinite-horizon multiobjective optimal control problems are presented in Section 5.2 and Theorem of Reduction to finite horizon is provided in Section 5.3. In Section 5.4, New Multiplier Rules for multiobjective static optimization problems are established. Then, in Section 5.5, weak and strong Pontryagin principles for the multiobjective optimal control problems in the finite-horizon setting are given where the weak ones rely on the New Multiplier Rules for static multiobjective optimization problems. In Section 5.6, weak and strong Pontryagin principles for multiobjective optimal control problems in infinite horizon are provided. Moreover, using weak principles, by adding additional condition, a transversality condition is achieved. Finally, in Section 5.7, we provide sufficient conditions of optimality.

The Multiobjective Optimal Control Problems

In previous chapter, we have established weak Pontryagin principles for the singleobjective optimal control problems (P j k ) where k ∈ {i, e} and j ∈ {1, 2, 3}. It is worth to recall that in these problems, there are two families of controlled dynamical systems that governed by difference equations (De) and by difference inequations (Di).

In this chapter, we will study Pontryagin principles and sufficient condition of optimality for the very same problems but with multiobjective criterion. All the settings remain the same like those of single-objective problems, except for the criterion and their domain. Recall that when k ∈ {i, e}, Adm k is the set of all processes (x, u) ∈ � t∈N X t × � t∈N U t which satisfy (Dk) at each time t ∈ N and such that x 0 = η. These processes are called admissible for (Dk) and η. For all t ∈ N, for all j ∈ {1, . . . , �}, we consider a functional φ j t : X t × U t → R. For each j ∈ {1, . . . , �}, we set J j (x, u) := � +∞ t=0 φ j t (x t , u t ) and we also denote by Dom k (J j ) as the set of the (x, u) ∈ Adm k such that the series

� +∞ t=0 φ j t (x t , u t ) is convergent in R.
The optimality criterion that we consider here is defined by using the vector-function (multiobjective) J(x, u) := (J 1 , . . . , J � ). The order for criterion is the usual order in R � . Now, we introduce the domain for the multiobjective optimal control problems with criterion J, denoted by DOM k (J) :=

� � � j=1 Dom k J j �
where k ∈ {e, i}. We define the following multiobjective problem when k ∈ {e, i} :

(PM 1 k ) Maximize J(x, u) when (x, u) ∈ DOM k (J).

Definition 5.1. A process (x, û) ∈ DOM k (J) is called a Pareto optimal solution of Problem (P M 1 k ), if there does not exist a process (x, u) ∈ DOM k (J) such that for all j ∈ {1, . . . , �}, J j (x, u) ≥ J j (x, û) and for some i ∈ {1, . . . , �}, J i (x, u) > J i (x, û).

A process (x, û) ∈ DOM k (J) is called a weak Pareto optimal solution of Problem (P M 1 k ), if there does not exist a process (x, u) ∈ DOM k (J) such that for all j ∈ {1, . . . , �}, J j (x, u) > J j (x, û).

It is obvious that a Pareto optimal solution of Problem (P M 1 k ) is a weak Pareto optimal solution of Problem (P M 1 k ). Consider now the following problems for the cases where the infinite series does not necessarily converge:

(PM 2 k ) Find (x, û) ∈ Adm k such that, there does not exist a process (x, u) ∈ Adm k such that for all j ∈ {1, . . . , �}, limsup

h→+∞ ( � h t=0 φ j t (x t , u t ) - � h t=0 φ j t (x t , ût )) ≥ 0 and for some i ∈ {1, . . . , �}, limsup h→+∞ ( � h t=0 φ i t (x t , u t ) - � h t=0 φ i t (x t , ût )) > 0. (PM 2 � k ) Find (x, û) ∈ Adm k such that, there does not exist a process (x, u) ∈ Adm k such that for all j ∈ {1, . . . , �}, limsup h→+∞ ( � h t=0 φ j t (x t , u t ) - � h t=0 φ j t (x t , ût )) > 0.
(PM 3 k ) Find (x, û) ∈ Adm k such that, there does not exist a process (x, u) ∈ Adm k such that for all j ∈ {1, . . . , �}, liminf h→+∞ ( � h t=0 φ j t (x t , u t ) -� h t=0 φ j t (x t , ût )) ≥ 0 and for some i ∈ {1, . . . , �}, liminf h→+∞ (

� h t=0 φ i t (x t , u t ) - � h t=0 φ i t (x t , ût )) > 0. (PM 3 � k ) Find (x, û)
∈ Adm k such that, there does not exist a process (x, u) ∈ Adm k such that for all j ∈ {1, . . . , �}, liminf

h→+∞ ( � h t=0 φ j t (x t , u t ) - � h t=0 φ j t (x t , ût )) > 0.
Since for all j ∈ {1, . . . , �}, limsup

h→+∞ ( � h t=0 φ j t (x t , u t ) - � h t=0 φ j t (x t , ût )) ≥ liminf h→+∞ ( � h t=0 φ j t (x t , u t ) - � h t=0 φ j t (x t , ût )), then a solution of Problem (P M 3 k ) is also a solution of Problem (P M 2 k ).
Besides, it is obvious that a solution of Problem (P M j � k ) is also a solution of Problem (P M j k ) when j ∈ {2, 3}.

Reduction to Finite Horizon

Let T be a fixed number in N * . We set J T j ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) := T � t=0 φ j t (x t , u t ) and J T := (J T 1 , . . . , J T � ). Consider the following reduced problems when k ∈ {e, i} Maximize J T ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) when ∀t ∈ {0, . . . , T + 1},

x t ∈ X t ∀t ∈ {0, . . . , T }, u t ∈ U t ∀t ∈ {0, . . . , T }, (D k ) holds x 0 = η, x T +1 = xT +1 .              (F M T k )
Chapter 5. Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems Definition 5.2. ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is called a Pareto optimal solution of Problem (F M T k ) where k ∈ {e, i}, if there does not exist any ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) admissible for Problem (F M T k ) such that for all j ∈ {1, . . . , �}, J T j ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) ≥ J T j ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) and for some i ∈ {1, . . . , �}, J T i ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) > J T i ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )).

((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is called a weak Pareto optimal solution of Problem (F M T k ) where k ∈ {e, i}, if there does not exist any ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) admissible for Problem (F M T k ) such that for all j ∈ {1, . . . , �}, J T j ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) > J T j ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )).

Here admissibility means that all the constraints, including the dynamical system, the initial and final conditions, are satisfied. Then we have the following theorem.

Theorem 5.3. The two following assertions hold.

(i) Let (x, û) be a Pareto optimal solution of Problem

(P M 1 k ) (respectively, solution of (P M 2 k ), (P M 3 k ))
where k ∈ {e, i} and let T ∈ N * . Then the restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is a Pareto optimal solution of the finite-horizon problem (F M T k ).

(ii) Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 k ) (respectively, solution of (P M 2 � k ), (P

M 3 � k ))
where k ∈ {e, i} and let T ∈ N * . Then the restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is a weak Pareto optimal solution of the finite-horizon problem (F M T k ).

Proof. We will prove assertion (i) first.

(De) case: We will prove for each case of (P M j e ) where j ∈ {1, 2, 3}. -For (PM 1 e ): We proceed by contradiction. Assume that ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is not Pareto optimal for (F M T k ). Then there exists ((x 0 , . . . , x T +1 ), (u 0 , . . . , u T )) which is admissible for (F M T k ) such that

J T ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) > J T ((x t ) 0≤t≤T +1 , (û t ) 0≤t≤T )
This inequality means that for all j ∈ {1, . . . , �}, J T j ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) ≥ J T j ((x t ) 0≤t≤T +1 , (û t ) 0≤t≤T ) and for some k ∈ {1, . . . , �},

J T k ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) > J T k ((x t ) 0≤t≤T , (û t ) 0≤t≤T -1
). When t > T + 1, we set x t := xt and when t ≥ T + 1, we set u t := ût . From the admissibility and this setting, it is obvious that x ∈ � t∈N X t and u ∈ � t∈N U t . Also, x t+1 = xt+1 = f t (x t , ût ) = f t (x t , u t ) when t ≥ T + 1. It implies that (x, u) belongs to the admissible set of (P M 1 e ) i.e. (x, u) ∈ Adm e . Now, we have for all j ∈ {1, . . . , �},

+∞ � t=T +1 φ j t (x t , u t ) = +∞ � t=T +1 φ j t (x t , ût ) < +∞ then +∞ � t=0
φ j t (x t , u t ) < +∞ or J j (x, u) < +∞. And so, (x, u) ∈ Dom e (J j ) for all j ∈ {1, . . . , �} which implies that (x, u)

∈ � � � j=1 Dom e J j �
= DOM e (J).

Reduction to Finite Horizon

97

Now, for all j ∈ {1, . . . , �}, we have

J j (x, u) = +∞ � t=0 φ j t (x t , u t ) = T � t=0 φ j t (x t , u t ) + +∞ � t=T +1 φ j t (x t , u t ) = J T j ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) + +∞ � t=T +1 φ j t (x t , ût ) ≥ J T j ((x t ) 0≤t≤T +1 , (û t ) 0≤t≤T ) + +∞ � t=T +1 φ j t (x t , ût ) = T � t=0 φ j t (x t , u t ) + +∞ � t=T +1 φ j t (x t , ût )=J j (x, û).
Besides, for some k ∈ {1, . . . , �}, we have

J k (x, u) = +∞ � t=0 φ k t (x t , u t ) = T � t=0 φ k t (x t , u t ) + +∞ � t=T +1 φ k t (x t , u t ) = J T k ((x t ) 0≤t≤T +1 , (u t ) 0≤t≤T ) + +∞ � t=T +1 φ k t (x t , ût ) > J T k ((x t ) 0≤t≤T +1 , (û t ) 0≤t≤T ) + +∞ � t=T +1 φ k t (x t , ût ) = T � t=0 φ k t (x t , ût ) + +∞ � t=T +1 φ k t (x t , ût )=J k (x, û).
And so, J(x, u) > J(x, û). This is contradiction since (x, û) is the Pareto optimal solution for Problem (P M 1 e ). Hence, ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) must be Pareto optimal for (F M T k ). -For (PM 3 e ): Let (x, û) be a solution of (P M 3 e ). Assume that ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is not Pareto optimal for (F M T k ). By realizing analogous proceedings like in the previous case, we can build a process (x, u) ∈ Adm e such that for all j ∈ {1, . . . , �},

T � t=0 φ j t (x t , u t ) ≥ T � t=0 φ j t (x t , ût ) and for some k ∈ {1, . . . , �}, T � t=0 φ k t (x t , u t ) > T � t=0 φ k t (x t , ût ). Then we have when h ≥ T + 1, lim inf h→+∞ ( h � t=0 φ j t (x t , u t ) - h � t=0 φ j t (x t , ût )) = T � t=0 φ j t (x t , u t ) - T � t=0 φ j t (x t , ût ) ≥ 0, for all j ∈ {1, . . . , �} and lim inf h→+∞ ( h � t=0 φ k t (x t , u t ) - h � t=0 φ k t (x t , ût )) = T � t=0 φ k t (x t , u t ) - T � t=0 φ k t (x t , ût ) > 0,
for some k ∈ {1, . . . , �} which is a contradiction since (x, û) is a solution of (P M 3 e ). -For (PM 2 e ): Let (x, û) be a solution of (P M 2 e ). It is clear that (x, û) is also a solution of (P M 3 e ) and hence, its restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is a Pareto optimal solution for (F M T k ). (Di) case: the proof is completely similar. Assertion (i) is proven. The proof of assertion (ii) is analogous to the one for assertion (i).

Theorem 5.7. Let x be a solution of (M m ). We assume that the following assumptions are fulfilled.

(i) For all i ∈ {1, . . . , �}, ϑ i is Fréchet differentiable at x. (ii) For all i ∈ {1, . . . , p}, g i is Fréchet differentiable at x when g i (x) = 0.

(iii) For all i ∈ {1, . . . , p}, g i is Gâteaux differentiable at x and lower semicontinuous at

x when g i (x) > 0.

(iv) For all i ∈ {1, . . . , q}, h i is continuous on a neighborhood of x and Fréchet differentiable at x.

Then there exist θ 1 , . . . , θ � , λ 1 , . . . , λ p ∈ R + and µ 1 , . . . , µ q ∈ R such that the following conditions are satisfied.

(a) (θ 1 , . . . , θ � , λ 1 , . . . , λ p , µ 1 , . . . , µ q ) � = (0, . . . , 0).

(b) For all i ∈ {1, . . . , p}, λ i g i (x) = 0. (c) � � i=1 θ i D G ϑ i (x) + � p i=1 λ i D G g i (x) + � q i=1 µ i D G h i (x) = 0.
Moreover, under the additional assumption (v) Dh 1 (x), . . . , Dh q (x) are linearly independent, we can take

(d) (θ 1 , . . . , θ � , λ 1 , . . . , λ p ) � = (0, . . . , 0).
Proof. Let x be a solution of (M m ). Using Lemma 5.5, x is also an optimal solution of Problem (5.2) for any given k ∈ {1, . . . , �}. Let k = 1 then x solves following problem.

Maximize ϑ 1 (x) when x ∈ Ω when ∀i ∈ {2, . . . , �}, ϑ i (x) ≥ ϑ i (x) when ∀i ∈ {1, . . . , p}, g i (x) ≥ 0, and when ∀i ∈ {1, . . . , q}, h i (x) = 0.

             (5.5)
We set β i (x) := ϑ i (x)-ϑ i (x) for all i ∈ {2, . . . , �}. Then the above-mentioned problem is rewritten as follows.

Maximize ϑ 1 (x) when x ∈ Ω when ∀i ∈ {2, . . . , �}, β i (x) ≥ 0 when ∀i ∈ {1, . . . , p}, g i (x) ≥ 0, and when ∀i ∈ {1, . . . , q}, h i (x) = 0.

             (5.6)
which has the same form of Problem (M). Under the assumptions of this theorem, we will verify that all the assumptions of Theorem 4.5 are fulfilled.

(i) Under assumption (i), ϑ 1 is Fréchet differentiable at x.

(ii) Under assumption (i) and (ii), for all i ∈ {2, . . . , �}, β i is Fréchet differentiable at x when β i (x) = 0 and for all i ∈ {1, . . . , p}, g i is Fréchet differentiable at x when g i (x) = 0.

(iii) Under assumption (iii), for all i ∈ {1, . . . , p}, g i is Gâteaux differentiable at x and lower semicontinuous at x when g i (x) > 0. (Here, notice that for i ∈ {2, . . . , �} we do not care about the Gâteaux differentiability and lower semicontinuity of β i at x because β i (x) = ϑ i (x)ϑ i (x) = 0).

New Pontryagin Principles for Multiobjective Optimal Control

Problems in Finite-Horizon Setting 101

(iv) Under assumption (iv), for all i ∈ {1, . . . , q}, h i is continuous on a neighborhood of x and Fréchet differentiable at x.

Then we can apply Theorem of New Multiplier Rule in previous chapter (Theorem 4.5) to obtain the existence of θ 1 , . . . , θ � , λ 1 , . . . , λ p ∈ R + and µ 1 , . . . , µ q ∈ R such that the following conditions are satisfied.

(a) (θ 1 , . . . , θ � , λ 1 , . . . , λ p , µ 1 , . . . , µ q ) � = (0, . . . , 0). (b) For all i ∈ {1, . . . , p}, λ i g i (x) = 0.

For all i ∈ {2, . . . , �}, θ i β i (x) = 0 (this is obvious since β i (x) = 0).

(c)

� � i=1 θ i D G ϑ i (x) + � p i=1 λ i D G g i (x) + � q i=1 µ i D G h i (x) = 0.
Then conclusions (a), (b) and (c) of this theorem hold. Now, under the additional assumption (v) which assumes that Dh 1 (x), . . . , Dh q (x) are linearly independent, we can choose (θ 1 , . . . , θ � , λ 1 , . . . , λ p ) � = (0, . . . , 0). This is easily verified using results (a), (c) and by contradiction and we obtain conclusion (d). The proof is complete.

New Pontryagin Principles for Multiobjective Optimal

Control Problems in Finite-Horizon Setting

In this section, we present Pontryagin principles in both weak and strong forms for multiobjective optimal control problems in a finite horizon framework, namely problems (F M T k ) when k ∈ {e, i}. They differ from the existing results since they use lighter smoothness assumptions.

Weak Pontryagin Principles

To obtain weak Pontryagin principles for multiobjective optimal control problems in finite-horizon framework (F M T k ) when k ∈ {e, i}, we will rely on New multiplier Rules established in Section 5.3. To do this, we will translate these problems into static optimization problems and we will apply New Multiplier Rules. Note that, in these problems, x 0 and x T +1 are fixed and so they are not unknown variables. Assume that for all t ∈ N, X t is open and the sets of admissible controls U t , with t ∈ N are defined by equalities and inequalities as follows

U t (x t ) = � m i � i=1 {u ∈ R d : g i t (x t , u) ≥ 0 � ∩ � me � k=1 {u ∈ R d : h k t (x t , u) = 0 � (5.7)
and we also assume that for all t ∈ N, U t (x t ) � = 0.

For Problem (F M T i ), we have the following theorem:

Theorem 5.8. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a Pareto optimal solution of Problem (F M T i ) when η and xT +1 are fixed vectors in R n . We assume that for all t ∈ {0, . . . , T }, X t is open and U t (x t ) is defined by (5.7). We also assume that the following conditions are fulfilled.

(i) For all j ∈ {1, . . . , �}, for all t ∈ {0, . . . , T }, φ j t is Fréchet differentiable at (x t , ût ). (ii) For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, g i t is Fréchet differentiable at (x t , ût ) when g i t (x t , ût ) = 0. For all j ∈ {1, . . . , n}, for all t ∈ {0, . . . , T }, f j t is Fréchet differentiable at (x t , ût ) when f j t (x t , ût ) = xj t+1 .

(iii) For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, g i t is Gâteaux differentiable at (x t , ût ) and lower semicontinuous at (x t , ût ) when g i t (x t , ût ) > 0. For all j ∈ {1, . . . , n}, for all t ∈ {0, . . . , T }, f j t is Gâteaux differentiable and and lower semicontinuous at (x t , ût ) when f j t (x t , ût ) > xj t+1 . (iv) For all k ∈ {1, . . . , m e }, for all t ∈ {0, . . . , T }, h k t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

Then there exist θ T 1 , . . . , θ T � ∈ R + , λ T i,t ∈ R + , µ T k,t ∈ R and p T t+1 ∈ R n * + where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i } and k ∈ {1, . . . , m e } such that the following conditions are satisfied.

(a) The multipliers are not simultaneously equal to zero.

(b) For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, λ T i,t g i t (x t , ût ) = 0. For all t ∈ {0, . . . , T },

� p T t+1 , f t (x t , ût ) -xt+1 � = 0. (c) ∀t ∈ {1, . . . , T }, p T t = � � j=1 � θ T j .D 1 φ j t (x t , ût ) � + p T t+1 • D G,1 f t (x t , ût ) + m i � i=1 λ T i,t D G,1 g i t (x t , ût ) + me � k=1 µ T k,t D 1 h k t (x t , ût ). (d) ∀t ∈ {0, . . . , T }, � � j=1 � θ T j .D 2 φ j t (x t , ût ) � + p T t+1 • D G,2 f t (x t , ût ) + m i � i=1 λ T i,t D G,2 g i t (x t , ût ) + me � k=1 µ T k,t D 2 h k t (x t , ût ) = 0.
Proof. We arbitrarily fix T ∈ N * . Let (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a solution of Problem (F M T i ) where η and xT +1 be given.

Step 1: We rewrite Problem (F M T i ) under model of (M m ). Let Ω = T � t=1 X t × � R d � T +1
, then Ω is open. Let z = (x 1 , . . . , x T , u 0 , . . . , u T ) ∈ Ω. We set φ(z) = (φ 1 (z), . . . , φ � (z)) where for all j ∈ {1, . . . , �}, φ j (z) := φ j 0 (η, u 0 ) + � T t=1 φ j t (x t , u t ); and for all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, gi t (z) := g i t (x t , u t ); and for all k ∈ {1, . . . , m e }, for all t ∈ {0, . . . , T }, hk t (z) := h k t (x t , u t ); and for all j ∈ {1, . . . , n}, ψ j 0 (z) =:= f j 0 (η, u 0 )x j 1 ; and for all j ∈ {1, . . . , n}, for all t ∈ {1, . . . , T -1}, ψ j t (z) := f j t (x t , u t )x j t+1 ; and, finally, for all j ∈ {1, . . . , n}, ψ j T (z) := f j T (x T , u T ) -xj T +1 . So we have translated Problem (F M T i ) into this form Maximize φ(z) = φ(x 1 , . . . , x T , u 0 , . . . , u T ); when z ∈ Ω, ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, gi t (z) ≥ 0, ∀k ∈ {1, . . . , m e }, ∀t ∈ {0, . . . , T }, hk t (z) = 0, ∀j ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ j t (z) ≥ 0,

             (F M T )
which has the same form of Problem (M m ).

Step 2: We will prove that with this problem, all conditions of Theorem 5.7 are fulfilled. It is obvious that ẑ = (x 1 , . . . , xT , û0 , . . . , ûT ) is a Pareto optimal solution of the above-mentioned problem. Under (i), (ii), (iii) and (iv) we obtain the following statements: For all j ∈ {1, . . . , �} function φ j is Fréchet differentiable at ẑ as a sum of T Fréchet differentiable functions; For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, functions gi t is Fréchet differentiable at ẑ when gi t (ẑ) = 0; For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, functions gi t is Gâteaux differentiable at ẑ and lower semicontinuous at ẑ when gi t (ẑ) > 0; For all k ∈ {1, . . . , m e }, for all t ∈ {0, . . . , T }, hi t is continuous on a neighborhood of ẑ and Fréchet differentiable at ẑ; For all j ∈ {1, . . . , n}, for all 5.5. New Pontryagin Principles for Multiobjective Optimal Control Problems in Finite-Horizon Setting 103 t ∈ {0, . . . , T }, ψ j t is Fréchet differentiable at ẑ when ψ j t (ẑ) = 0; For all j ∈ {1, . . . , n}, for all t ∈ {0, . . . , T }, functions ψ j t is Gâteaux differentiable at ẑ and lower semicontinuous at ẑ when ψ j t (ẑ) > 0.

Step 3: Application of Theorem 5.7. Now for Problem (F M i 1 ), all the conditions of Theorem 5.7 are satisfied, by applying it, we obtain the multiplier θ T = (θ T 1 , . . . , θ T � ) ∈ R � + for the criterion function φ(z), multipliers λ T i,t ∈ R + for inequality constraints gi t (z) ≥ 0, multipliers µ T k,t ∈ R for equality constraints hi t (z) = 0 and multipliers p T t+1,j ∈ R + for the equality constraints ψ j t (z) ≥ 0 where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i }, j ∈ {1, . . . , n} and k ∈ {1, . . . , m e } such that all the conclusions of Theorem 5.7 hold. From conclusion (a) of Theorem 5.

L(z, θ T 1 , . . . , θ T � , λ T 1,0 , . . . , λ T m i ,T , µ T 1,0 , . . . , µ T me,T , p T 1,1 , . . . , p T T +1,n ) = � � j=1 θ T j .φ j (z) + T � t=0 m i � i=1 λ T i,t gi t (z) + T � t=0 me � k=1 µ T k,t hi t (z) + T � t=0 n � j=1 p T t+1,j ψ j t (z) = � � j=1 � θ T j (φ j 0 (η, u 0 ) + � T t=1 φ j t (x t , u t )) � + T � t=0 m i � i=1 λ T i,t g i t (x t , u t ) + T � t=0 me � k=1 µ T k,t h k t (x t , u t ) + T � t=0 � p T t+1 , f t (x t , u t ) -x t+1 � .
As preliminary calculations, since all the functions in L are Gâteaux differentiable, then for all t ∈ {1, . . . , T }, the partial Gâteaux differential of the generalized Lagrangian with respect to x t is

∂L ∂xt (z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = D G,xt L(z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = � � j=1 � θ T j .D G,1 φ j t (x t , u t ) � + m i � i=1 λ T i,t D G,1 g i t (x t , u t ) + me � k=1 µ T k,t D G,1 h k t (x t , u t ) +p T t+1 • D G,1 f t (x t , u t ) -p T t ,
and for all t ∈ {0, . . . , T } the partial Gâteaux differential of the generalized Lagrangian with respect to u t is

∂L ∂ut (z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = D G,ut L(z, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = � � j=1 � θ T j .D G,2 φ j t (x t , u t ) � + p T t+1 • D G,2 f t (x t , u t ) + m i � i=1 λ T i,t D G,2 g i t (x t , u t ) + me � k=1 µ T k,t D G,2 h k t (x t , u t ).
From the conclusion (c) of Theorem 5.

� j=1 � θ T j .D G,1 φ j t (x t , ût ) � + p T t+1 • D G,1t f t (x t , ût ) + m i � i=1 λ T i,t D G,1 g i t (x t , ût ) + me � k=1 µ T k,t D G,1 h k t (x t , ût ), ∀t ∈ {0, . . . , T -1}, � � j=1 � θ T j .D G,2 φ t (x t , ût ) � + p T t+1 • D G,2 f t (x t , ût ), + m i � i=1 λ T i,t D G,2 g i t (x t , ût ) + me � k=1 µ T k,t D G,2 h k t (x t , ût ) = 0.                   
(5.8)

We have yet seen that conclusions (a) and (b) are satisfied. Conclusions (c) and (d) are obtained after identifying Gâteaux differential with Fréchet differential of Fréchet differentiable functions in (5.8).

In the special case where for every t ∈ N, U t is an arbitrary subset of R d and u t belongs to the interior of U t , Problem (F M T ) now contains only inequality constraints as follows

    
Maximize φ(z); when z ∈ Ω; ∀j ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ j t (z) ≥ 0. This problem has the form of Problem (I m ). Then we obtain the following corollary after applying Theorem 5.6.

Corollary 5.9. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a Pareto optimal solution of Problem (F M T i ) when η and xT +1 are fixed vectors in R n . We assume that for all t ∈ {0, . . . , T }, X t are open and ût ∈ int(U t ). We also assume that the following assumptions are fulfilled (i) For all j ∈ {1, . . . , �}, for all t ∈ {0, . . . , T }, φ j t is Gâteaux differentiable at (x t , ût ). For all t ∈ {0, . . . , T }, f t is Gâteaux differentiable at (x t , ût ).

(ii) For all j ∈ {1, . . . , n}, for all t ∈ {0, . . . , T }, f j t is lower semicontinuous at (x t , ût ) when f j t (x t , ût ) > xj t+1 . Then there exist θ T 1 , . . . , θ T � ∈ R + and p T t+1 ∈ R n * + where t ∈ {0, . . . , T } such that the following conditions are satisfied (a) θ T 1 , . . . , θ T � , p T t+1 , t ∈ {0, . . . , T } are not all zeros. (b) For all t ∈ {0, . . . , T },

� p T t+1 , f t (x t , ût ) -xt+1 � = 0. (c) ∀t ∈ {1, . . . , T }, p T t = � � j=1 � θ T j .D G,1 φ j t (x t , ût ) � + p T t+1 • D G,1 f t (x t , ût ). (d) ∀t ∈ {0, . . . , T }, 0 = � � j=1 � θ T j .D G,2 φ j t (x t , ût ) � + p T t+1 • D G,2 f t (x t , ût ).
For Problem (F M T e ): the way we treat it is almost similar to what we did with Problem (F M T i ) case. The difference is that the greater or equal sign in ψ j t (x 1 , . . . , x T , u 0 , . . . , u T ) ≥ 0 will be replaced by the equal sign since now the phase constraint has the form of (De). Apply Theorem 5.7 for this problem with a notice that now the inequality constraints group includes only functions gi t and the equality constraints group includes functions hk t and ψ j t , we obtain the weak Pontryagin principle for finite horizon problem with (De) as follows.

Theorem 5.10. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a Pareto optimal solution of Problem (F M T e ) when η and xT +1 are fixed vectors in R n . We assume that for all t ∈ {0, . . . , T }, X t is open and U t (x t ) is defined by (5.7). We also assume that the following conditions are fulfilled. (i) For all j ∈ {1, . . . , �}, for all t ∈ {0, . . . , T }, φ j t is Fréchet differentiable at (x t , ût ). (ii) For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, g i t is Fréchet differentiable at (x t , ût ) when g i t (x t , ût ) = 0. (iii) For all i ∈ {1, . . . , m i }, for all t ∈ {0, . . . , T }, g i t is Gâteaux differentiable at (x t , ût ) and lower semicontinuous at ût when g i t (x t , ût ) > 0. (iv) For all k ∈ {1, . . . , m e }, for all t ∈ {0, . . . , T }, h k t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

For all t ∈ {0, . . . , T }, f t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

Then there exist θ T 1 , . . . ,

θ T � ∈ R + , λ T i,t ∈ R + , µ T k,t ∈ R and p T t+1 ∈ R n *
where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i } and k ∈ {1, . . . , m e } such that conclusions (a), (c), (d) of Theorem 5.8 are satisfied (in which, D G,α f t (x t , ût ) is replaced by D α f t (x t , ût ) where α ∈ {1, 2}) together with the following one:

(b) ∀i ∈ {1, . . . , m i }, ∀t ∈ {0, . . . , T }, λ T i,t g i t (x t , ût ) = 0.
In the special case where for every t ∈ N, U t is an arbitrary subset of R d and u t belongs to the interior of U t , Problem (F M T ) is reduced to the following simpler form of Problem (M m ):

     Maximize φ(z);
when z ∈ Ω; ∀j ∈ {1, . . . , n}, ∀t ∈ {0, . . . , T }, ψ j t (z) = 0. Then applying Theorem 5.7, we obtain the following simpler statement Corollary 5.11. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a Pareto optimal solution of Problem (F M T e ) when η and xT +1 are fixed vectors in R n . We assume that for all t ∈ {0, . . . , T }, X t are open and ût ∈ int(U t ). We also assume that the following conditions are fulfilled:

(i) For all j ∈ {1, . . . , �}, for all t ∈ {0, . . . , T }, φ j t is Fréchet differentiable at (x t , ût ). (ii) For all t ∈ {0, . . . , T }, f t are continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

Then there exist θ T 1 , . . . , θ T � ∈ R + and p T t+1 ∈ R n * where t ∈ {0, . . . , T } such that the conclusions (a), (c), (d) of Corollary 5.9 are satisfied in which D G,α f t (x t , ût ) is replaced by D α f t (x t , ût ) where α ∈ {1, 2}. Remark 5.12. Notice that in the absence of constraints, one can weaken the assumptions on objective functions φ j t and on f t as can be seen in the above-mentioned corollaries.

Remark 5.13. When all the sets of state and control variables are convex and when all the functions are convex (or concave), Gâteaux differentiability and Fréchet differentiability are identified.

Remark 5.14. One can obtain analogous results with the assumption of weak Pareto optimality by applying other multiplier rules (for instance, multiplier rule of Novo and Jimenez (see Theorem 3.10 in [START_REF] Novo | Lagrange Multipliers in Multiobjective Optimization under Mixed Assumptions of Fréchet and Directional Differentiability[END_REF])). However, the required assumptions for the multiobjective optimal control problems will become very difficult to write due to the complication of qualification constraints needed in such rules to obtain a Fritz John like multiplier rule when the functions are not Fréchet differentiable.

Strong Pontryagin Principles

In this section, firstly, we recall a theorem which provides a necessary condition of optimality in a strong form for multiobjective static optimization problems. Let X, Y, Z, W be Banach spaces. Let Y and Z be ordered by cones K and M , respectively. Let U be a set equipped with the trivial topology (containing only U and ∅), so that X ×U is a topological space. Let mappings J : X × U → Y, F : X × U → Z, H : X × U → W be given. Consider the following problem of vector optimization:

Maximize J(x, u) F (x, u) ≥ 0, H(x, u) = 0, x ∈ X, u ∈ U.         
(5.9)

Let us define the following generalized Lagrangian of Problem (5.9):

L(x, u, λ, µ, υ) := �λ, J(x, u)� + �µ, F (x, u)� + �υ, H(x, u)� .

We introduce the multiplier rule for this problem which is proven by Khanh and Nuong in [START_REF] Khanh | On Necessary Optimality Conditions in Vector Optimization Problems[END_REF] and is described in the following theorem Theorem 5.15. Assume that Problem (5.9) satisfies the following conditions:

(i) intK � = 0 and intM � = 0.

(ii) For each u ∈ U , H(., u) is continuously differentiable at x.

(iii) J(., û) and F (., û) have directional derivatives at x which are concave.

(iv) For each u ∈ U, J(., u) and F (., u) are continuous at x in any direction h, in the sense that J(x + λh, u) and F (x + λh, u) tend to J(x, u) and F (x, u), respectively, as λ → 0 + .

(v) For each x in a neighborhood V of x, the following convexity condition is satisfied: if

u 1 ∈ U, u 2 ∈ U, 0 ≤ α ≤ 1, one can find u ∈ U such that J(x, u) ≥ αJ(x, u 1 ) + (1 -α)J(x, u 2 ), F (x, u) ≥ αF (x, u 1 ) + (1 -α)F (x, u 2 ), H(x, u) = αH(x, u 1 ) + (1 -α)H(x, u 2 ). (vi) codimD 1 H(x, û) is finite.
Then, if (x, û) is a weak Pareto optimal solution of Problem (5.9), there exist λ ∈ K * , µ ∈ M * , υ ∈ W * , not all zero, such that

(a) � λ, � D 1 J(x, û; h) � + � µ, � D 1 F (x, û; h) � + �υ, D 1 H(x, û).h� ≥ 0, for all h ∈ X, i.e. 0 ∈ ∂ 1 L(x, û, λ, µ, υ).
(b) L(x, û, λ, µ, υ) = max u∈U L(x, u, λ, µ, υ).

(c) �µ, F (x, û)� = 0.

We will use this theorem to establish a strong Pontryagin principles for finite-horizon multiobjective optimal control problems (F M T k ). To obtain strong Pontryagin principles for multiobjective optimal control problems in finite horizon framework (F M T k ) when k ∈ {e, i}, we will rely on the abovementioned multiplier rule for multiobjective static optimization problems. To do this, 5.5. New Pontryagin Principles for Multiobjective Optimal Control Problems in Finite-Horizon Setting 107 we will translate the multiobjective optimal control problems into multiobjective static optimization problems as we did for the weak principles. In this section we will consider Problems (F M T k ) when k ∈ {e, i} in the case where there exist inequality constraints on the optimal solution. Assume that for each t ∈ N, the sets of controls are defined by inequalities as follows

W t (x t ) = � 1≤k≤m {u ∈ R d : g k t (x t , u) ≥ 0} (5.10)
where g k t : R n × R d → R. Let T ∈ N * be a fixed number. Assume that (x 0 , . . . xT +1 , û0 , . . . ûT ) is a weak Pareto optimal solution of Problem (F M T k ) when k ∈ {e, i}. In this problem, x 0 and x T +1 are given so we can rewrite (F M T k ) as follows (5.11) where:

Maximize J T (x T , u T ) F T (x T , u T ) ≥ 0 when k = i, (or F T (x T , u T ) = 0 when k = e) G T (x T , u T ) ≥ 0, x T ∈ X(T ), u T ∈ U (T ) = (R d ) T +1 ,             
-X(T ) = X 1 × • • • × X T , x T = (x 1 , . . . , x T ), u T = (u 0 , . . . , u T ); -J T (x T , u T ) = (J T 1 (x T , u T ), . . . , J T � (x T , u T ))
where for all j ∈ {1, . . . , �}, J T j (x T , u T ) = φ j 0 (η, u 0 ) + � T t=1 φ j t (x t , u t ); -F T (x T , u T ) = (F T 0 (x T , u T ), . . . , F T T (x T , u T )) where

F T 0 (x T , u T ) = f 0 (η, u 0 ) -x 1 , for all t ∈ {1, . . . , T -1}, F T t (x T , u T ) = f t (x t , u t ) -x t+1 , F T T (x T , u T ) = f T (x T , u T ) -xT +1 ; -G T (x T , u T ) = (G T 0 (x T , u T ), . . . , G T T (x T , u T ))
where for all t ∈ {1, . . . , T }, G T t (x T , u T ) = (g 1 t (x t , u t ), . . . , g m t (x t , u t )) and G T 0 (x T , u T ) = (g 1 0 (η, u 0 ), . . . , g m 0 (η, u 0 )). We can see that X(T ) is an open subset of (R n ) T , U (T ) is the whole space (R d ) T +1 . The vector function J T : X(T ) × (R d ) T +1 → R � is ordered by the cone R � + ; the mapping

F T : X(T ) × (R d ) T +1 → (R n ) T +1 is ordered by the cone (R n + ) T +1 ; and the mapping G T : X(T ) × (R d ) T +1 → (R m ) T +1 is ordered by the cone (R m + ) T +1 . Obviously, intR � + � = ∅, int(R n + ) T +1 � = ∅ and int(R m + ) T +1 � = ∅.
The generalized Lagrangian of this problem has this form

L(x T , u T , θ T , p T , µ T ) := � θ T , J T (x T , u T ) � + � p T , F T (x T , u T ) � + � µ T , G T (x T , u T ) � , where θ T = (θ T 1 , . . . , θ T � ) ∈ R � , p T = (p T 1 , . . . , p T T +1 ) ∈ (R n * ) T +1 and µ T = (µ T 0 , . . . , µ T T ) ∈ (R m * ) T +1
. Now, all the elements depend on T . The augmented Hamiltonian is now defined for all t ∈ {0, . . . , T } as follows

H T t (x t , u t , θ T , p T t+1 , µ T t ) = � � j=1 θ T j φ j t (x t , u t ) + � p T t+1 , f t (x t , u t ) � + � m k=1 µ k,T t .g k t (x t , u t ),
where µ 1,T t , . . . , µ m,T t are the coordinates of vector µ T t in R m * .

(Di) case

We present a strong Pontryagin principle for the multiobjective optimal control problems with (Di) in finite horizon as follows.

Theorem 5.16. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a weak Pareto optimal solution of Problem (F M T i ) where η and xT +1 are fixed vectors in R n . Assume that for all t ∈ {0, . . . , T }, the control sets are defined as in (5.10) and that the following conditions are fulfilled (i) For all t ∈ {1, . . . , T }, for all j ∈ {1, . . . , �} and for all k ∈ {1, . . . , m}, φ j t (., ût ), f t (., ût ) and g k t (., ût ) have directional derivatives at xt which are concave. (ii) For all t ∈ {1, . . . , T }, for all j ∈ {1, . . . , �}, for all k ∈ {1, . . . , m}, for all u ∈ W t (x t ), φ j t (., u) , f t (., u) and g k t (., u) are continuous at xt in any direction h.

(iii) For all t ∈ {0, . . . , T }, for each x t in a neighborhood V t of xt such that x 0 = η, the following convexity condition is satisfied:

if u � ∈ W t (x t ), u �� ∈ W t (x t ), 0 ≤ α ≤ 1, one can find u ∈ W t (x t ) such that ∀j ∈ {1, . . . , �}, φ j t (x t , u) ≥ αφ j t (x t , u � ) + (1 -α)φ j t (x t , u �� ), f t (x t , u) ≥ αf t (x t , u � ) + (1 -α)f t (x t , u �� ), ∀k ∈ {1, . . . , m}, g k t (x t , u) ≥ αg k t (x t , u � ) + (1 -α)g k t (x t , u �� ).
Then there exist θ T = (θ T 1 , . . . , θ T � ) ∈ R � + , p T = (p T 1 , . . . , p T T +1 ) ∈ (R n * + ) T +1 and µ T = (µ T 0 , . . . , µ T T ) ∈ (R m * + ) T +1 not all zero which satisfy the following conclusions:

(a) For all t ∈ {0, . . . , T }, for all k ∈ {1, . . . , m},

� p T t+1 , f t (x t , ût ) -xt+1 � = 0 and µ k t g k t (x t , ût ) = 0. (b) For all t ∈ {1, . . . , T }, p T t ∈ ∂ 1 H T t (θ T , xt , ût , p T t+1 , µ T t ). (c) For all t ∈ {0, . . . , T }, H T t (x t , ût , θ T , p T t+1 , µ T t ) = max u∈R d H T t (x t , u, θ T , p T t+1 , µ T t )
. Firstly, we introduce the following lemmas: Lemma 5.17. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a weak Pareto optimal solution of Problem (F M T k ) where η and xT +1 are fixed vectors in R n . Assume that condition (i) in Theorem 5.16 is fulfilled. Then, for all h T ∈ (R n ) T , J T (., ûT ), F T (., ûT ) and G T (., ûT ) have directional derivatives at xT in the direction h T , which are concave.

Proof. We will prove the existence of � D 1 J T (x T , ûT ; h T ), i.e. the existence of the following limit

lim λ→0 + J T (x T + λh T , ûT ) -J T (x T , ûT ) λ ,
where h T = (h 1 , . . . , h T ) is an arbitrary vector from (R n ) T . Since J T is a vector mapping, it is equivalent to prove the existence of each element of the above-mentioned limit. For each j ∈ {1, . . . , �}, we consider the following relation:

J T j (x T + λh T , ûT ) -J T j (x T , ûT ) λ = φ j 0 (η, û0 ) + � T t=1 φ j t (x t + λh t , ût ) -(φ j 0 (η, û0 ) + � T t=1 φ j t (x t , ût )) λ = � T t=1 φ j t (x t + λh t , ût ) -φ j t (x t , ût ) λ .
Using the hypothesis of this lemma, it is obvious that And so, for all j ∈ {1, . . . , �}, J T j (., ûT ) has directional derivative at xT and it is concave since it is a sum of T concave mappings. Thus, � D 1 J T (x T , ûT ; h T ) exists for all h T ∈ (R n ) T and it is concave. Now, we do the similar process for F T (., ûT ) = (F T 0 (., ûT ), . . . , F T T (., ûT )), i.e. we will prove the existence of � D 1 F T t (x T , ûT ; h T ) where h T is an arbitrary vector in (R n ) T and where t ∈ {0, . . . , T }. When t = 0, consider the following relation:

lim λ→0 + � � T t=1 φ j t (xt+λht,ût)-φ j t (xt,ût) λ � = � T t=1 � lim λ→0 + φ j t (xt+λht,ût)-φ j t (xt,ût) λ � = � T t=1 � D 1 φ j t (x t , ût ; h t ).
F T 0 (x T + λh T , ûT ) -F T 0 (x T , ûT ) λ = f 0 (η, û0 ) -(x 1 + λh 1 ) -(f 0 (η, û0 ) -x1 ) λ = -λh 1 λ = -h 1 .
Since the result does not depend on λ, we obtain lim

λ→0 + F T 0 (x T + λh T , ûT ) -F T 0 (x T , ûT ) λ = -h 1
Therefore, F T 0 (., ûT ) has directional derivatives at xT in the direction h T and obviously, it is linear and thus, concave. Now for t ∈ {1, . . . , T -1}, we have

F T t (x T + λh T , ûT ) -F T t (x T , ûT ) λ = f t (x t + λh t , ût ) -(x t+1 + λh t+1 ) -(f t (x t , ût ) -xt+1 ) λ = -h t+1 + f t (x t + λh t , ût ) -f t (x t , ût ) λ .
Using hypothesis of the lemma, we know that lim

λ→0 + f t (x t + λh t , ût ) -f t (x t , ût ) λ = � D 1 f t (x t , ût ; h t ).
Therefore, we can assert that lim

λ→0 + F T t (x T + λh T , ûT ) -F T t (x T , ûT ) λ = -h t+1 + � D 1 f t (x t , ût ; h t ).
Then for t ∈ {1, . . . , T -1}, F T t (., ûT ) has directional derivatives at xT in the direction h T and it is concave since it is a sum of a linear mapping and a concave mapping. Finally, when t = T, we have

F T T (x T + λh T , ûT ) -F T T (x T , ûT ) λ = f T (x T + λh T , ûT ) -xT +1 -(f T (x T , ûT ) -xT +1 ) λ = f T (x T + λh T , ûT ) -f T (x T , ûT ) λ . Since lim λ→0 + f T (x T + λh T , ûT ) -f T (x T , ûT ) λ = � D 1 f T (x T , ûT ; h T ),
we can assert that F T t (., ûT ) has directional derivatives at xT in the direction h T and it is concave since � D 1 f T (x T , ûT ; h T ) is concave. And so, all the element mappings of F T (., ûT ) have directional derivatives at xT in the direction h T which are concave. Therefore, F T (., ûT ) has directional derivative at xT in the direction h T and it is concave. Using the result on directional derivatives of J T (., ûT ), F T (., ûT ) and G T (., ûT ) at xT which are proven in Lemma 5.17, we can rewrite (a) as follows

∀h T ∈ (R n ) T , � � j=1 θ j � T t=1 � D 1 φ j t (x t , ût ; h t ) - � p T 1 , h 1 � + � T -1 t=1 � p T t+1 , � D 1 f t (x t , ût ; h t ) -h t+1 � + � p T +1 , � D 1 f T (x T , ûT ; h T ) � + T � t=1 m � k=1 µ k,T t . � D 1 g k t (
x t , ût ; h t ) ≥ 0 Now, we consecutively choose h T = (0, . . . , h t , . . . , 0), in which all the elements are zero except for the t-th position to obtain the following:

-When t = 1, h T = (h 1 , 0, . . . , 0),

� � j=1 θ T j � D 1 φ j 1 (x 1 , û1 ; h 1 ) - � p T 1 , h 1 � + � p T 2 , � D 1 f 1 (x 1 , û1 ; h 1 ) � + m � k=1 µ k 1 . � D 1 g k 1 (x 1 , û1 ; h 1 ) ≥ 0. -For t ∈ {2, . . . , T -1}, h T = (0, ., 0, h t , 0, . . . , 0), i.e. h t � = 0 at the t-th position, � � j=1 θ T j � D 1 φ j t (x t , ût ; h t ) + � p T t+1 , � D 1 f t (x t , ût ; h t ) � - � p T t , h t � + m � k=1 µ k t . � D 1 g k t (x t , ût ; h t ) ≥ 0.
-When t = T , h T = (0, . . . , 0, h T ),

� � j=1 θ T j � D 1 φ j T (x T , ûT ; h T ) - � p T T , h T � + � p T T +1 , � D 1 f T (x T , ûT ; h T ) � + m � k=1 µ k T . � D 1 g k T (x T , ûT ; h T ) ≥ 0.
And so, for all t ∈ {1, . . . , T } and for all h t ∈ R n , the following condition holds:

� � j=1 θ T j � D 1 φ j t (x t , ût ; h t ) + � p T t+1 , � D 1 f t (x t , ût ; h t ) � + m � k=1 µ k t . � D 1 g k t (x t , ût ; h t ) ≥ � p T t , h t � ⇐⇒ � D 1 H T t (θ T , xt , ût , p T t+1 , µ T t ; h t ) ≥ � p T t , h t � , i.e. p T t ∈ ∂ 1 H T t (θ T , xt , ût , p T t+1 , µ T t
) and (b) holds. Now we expand the formula of L(x T , u T , θ T , p T , µ T ) as follows:

L(x T , u T , θ T , p T ) = � θ T , J T (x T , u T ) � + � p T , F T (x T , u T ) � + � µ T , G T (x T , u T ) � = � � j=1 θ T j � φ j 0 (η, u 0 ) + � T t=1 φ j t (x t , u t ) � + � p T 1 , f 0 (η, u 0 ) -x1 � + + � T t=1 � p T t+1 , f t (x t , u t ) -xt+1 � + � T t=0 m � k=1 µ k,T t .g k t (x t , u t ) = � T t=0 � H T t (x t , u t , θ T , p T t+1 , µ T t ) - � p T t+1 , xt+1 �� . From (5.13), we have T � t=0 H T t (x t , ût , θ T , p T t+1 µ T t ) = max u T ∈U (T ) � T � t=0 H T t (x t , u t , θ T , p T t+1 µ T t ) � .
For this equality, for each t ∈ {0, . . . , T }, we consecutively choose u T = (û 0 , . . . , ût-1 , u t , ût+1 , . . . , ûT ), i.e. only at t-th position, u t � = ût , and we obtain

H T t (x t , ût , θ T , p T t+1 , µ T t ) = max ut∈R d H T t (x t , u t , θ T , p T t+1 , µ T t
). So we have proven (c). Finally, from (5.14), we imply that Chapter 5. Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems having non-zero limit. We will use some particular assumptions like an invertibility assumption or a positivity assumption used by Blot for single-objective optimal control to overcome this difficulty. The control sets are considered in some specific cases: the ones with both inequality and equality constraints, the special case when control sets are independent with state variables and the case of interior optimal controls.

Control sets with both inequality and equality constraints

Consider Problem (P M j k ) where k ∈ {i, e} and where the control sets are defined as in (5.7), i.e. the control sets are defined by inequalities and equalities and for each t ∈ N, U t depends on x t . We will establish weak Pontryagin principles for such problems. Notice that in the next theorems and corollaries, we will use the notations span for linear span and conv for convex hull of a set of vectors, which were introduced in previous chapter.

Theorem 5.22. Let (x, û) be a Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 i ), (P M 3 i )) where the control sets are defined in (5.7). For all t ∈ N, we assume that the following conditions are fulfilled:

(i) For all j ∈ {1, . . . , �}, φ j t is Fréchet differentiable at (x t , ût ). (ii) For all α ∈ {1, . . . , n}, f α t is Fréchet differentiable at (x t , ût ) when f α t (x t , ût ) = xα t+1 . (iii) For all α ∈ {1, . . . , n}, f α t is lower semicontinuous and Gâteaux differentiable at (x t , ût ) when f α t (x t , ût ) > xα t+1 . (iv) For all k ∈ {1, . . . , m i }, g k t is Fréchet differentiable at (x t , ût ) when g k t (x t , ût ) = 0. (v)
For all k ∈ {1, . . . , m i }, g k t is lower semicontinuous and Gâteaux differentiable at (x t , ût ) when g k t (x t , ût ) > 0. (vi) For all i ∈ {1, . . . , m e }, h i t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

We set D G,α ĝk t := D G,α g k t (x t , ût ), D G,α ft := D G,α f t (x t , ût ), D α ĥi t := D α h i t (x t , ût )
for each α ∈ {1, 2}. Assume that the following conditions are fulfilled for all t ∈ N * :

(vii) D G,1 ft is invertible. (viii) Let v k t := D G,1 ĝk t • D G,1 f -1 t • D G,2 ft -D G,2 ĝk t for all k ∈ {1, . . . , m i } and w i t := D 1 ĥk t • D G,1 f -1 t • D G,2 ft -D 2 ĥk t for all i ∈ {1, . . . , m e }. Then span{w i t : i ∈ {1, . . . , m i }} ∩ conv{v k t : k ∈ I s t } = ∅, where I s t := {k ∈ {1, . . . , m i } : v k t = 0}. (ix) The family (w i t ) 1≤i≤me defined in (viii) is linearly independent. Then there exist θ 1 , . . . ., θ � ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ 1,t ) t∈N ∈ R N ,. . . ., (µ me,t ) t∈N ∈ R N , (λ 1,t ) t∈N ∈ R N ,. . . ., (λ m i ,t ) t∈N ∈ R N
which satisfy the following conditions.

(a) (θ 1 , . . . , θ � , p 1 , λ 1 0 , . . . λ m i 0 , µ 1 0 , . . . , µ me 0 ) � = (0, . . . , 0). (b) θ j ≥ 0 for all j ∈ {1, . . . , �}, p t ≥ 0 for all t ∈ N * , λ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }.

(c) For all t ∈ N, for all α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût )x α t+1 ) = 0, and for all k ∈ {1, . . . , m i }, λ k,t • g k t (x t , ût ) = 0. Proof. Let (x, û) be a Pareto optimal solution of Problem (P M j i ) when j ∈ {1, 2, 3}. Using Theorem 5.3, the restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is a Pareto optimal solution of the finite-horizon problem (F M T i ). Our assumptions (i-vi) imply that the assumptions of Theorem 5.8 in Section 5.4 are fulfilled and so we know that, for all T ∈ N * , there exist θ

(d) For all t ∈ N * , p t = p t+1 •D G,1 ft + � � j=1 θ j D 1 φj t + m i � k=1 λ k,t D G,1 ĝk t + me � i=1 µ i,t D 1 ĥi t ,
T 1 , . . . , θ T � ∈ R, λ T i,t ∈ R, µ T k,t ∈
p T t+1 , f t (x t , ût ) -xt+1 � = 0, λ T i,t g k t (x t , ût ) = 0. (5.18) ∀t ∈ {1, . . . , T }, p T t = � � j=1 θ T j .D 1 φj t + p T t+1 • D G,1 ft + m i � k=1 λ T k,t D G,1 ĝk t + me � i=1 µ T i,t D 1 ĥi t . (5.19) ∀t ∈ {0, . . . , T }, 0 = � � j=1 θ T j .D 2 φj t + p T t+1 • D G,2 ft + m i � k=1 λ T k,t D G,2 ĝk t + me � i=1 µ T i,t D 2 ĥi t . (5.20)
We will prove that (θ T 1 , . . . , θ T � , p T 1 , λ 1,T 0 , . . . λ m i ,T 0 , µ 1,T 0 , . . . , µ me,T 0 ) � = (0, . . . , 0) by contradiction. Notice that here, for the multipliers associated with the constraints, only the 0-indexed ones appear. Assume that (θ T 1 , . . . , θ T � , p T 1 , λ 1,T 0 , . . . λ m i ,T 0 , µ 1,T 0 , . . . , µ me,T 0 ) = (0, . . . , 0). When t = 1, using assumption (vii), we can formulate (5.19) as follows

-p T 2 = � -p T 1 + � � j=1 θ T j .D 1 φj 1 + � m i k=1 λ T k,1 D G,1 ĝk 1 + � me i=1 µ T i,1 D 1 ĥi 1 � • D G,1 f1 -1 = � � m i k=1 λ T k,1 D G,1 ĝk 1 + � me i=1 µ T i,1 D 1 ĥi 1 � • D G,1 f1 -1 ,   
(5.21) and using (5.20), we have

-p T 2 • D G,2 f1 = � � j=1 θ T j .D 2 φj 1 + � m i k=1 λ T k,1 D G,2 ĝk 1 + me � i=1 µ T i,1 D 2 ĥi 1 = � m i k=1 λ T k,1 D G,2 ĝk 1 + me � i=1 µ T i,1 D 2 ĥi 1 .            (5.22)
Using (5.21) and (5.22), we have

� � m i k=1 λ T k,1 D G,1 ĝk 1 + � me i=1 µ T i,1 D 1 ĥi 1 � • D G,1 f1 -1 • D G,2 f1 - � m i k=1 λ T k,1 D G,2 ĝk 1 - me � i=1 µ T i,1 D 2 ĥi 1 =
0, which can be rewritten as follows

� m i k=1 λ T k,1 � D G,1 ĝk 1 • D G,1 f1 -1 • D G,2 f1 -D G,2 ĝk 1 � + � me i=1 µ T i,1 � D 1 ĥi 1 • D G,1 f1 -1 • D G,2 f1 -D 2 ĥi 1 � = 0    ⇐⇒ m i � k=1 λ T k,1 v k 1 + me � i=1 µ T i,1 w i 1 = 0. (
(ix) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible. Then there exist θ 1 , . . . , θ � ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ 1,t ) t∈N ∈ R N ,. . . ., (µ me,t ) t∈N ∈ R N , (λ 1,t ) t∈N ∈ R N ,. . . ., (λ m i ,t ) t∈N ∈ R N
which satisfy the following conditions.

(a) (θ 1 , . . . , θ � , p 1 ) � = (0, . . . , 0, 0).

(b) θ j ≥ 0 for all j ∈ {1, . . . , �},, p t ≥ 0 for all t ∈ N * , λ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }.

(c) For all t ∈ N, for all α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût )x α t+1 ) = 0, and for all

k ∈ {1, . . . , m i }, λ k,t • g k t (û t ) = 0. (d) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + � � j=1 θ j D 1 φ j t (x t , ût ). (e) For all t ∈ N, p t+1 • D G,2 f t (x t , ût ) + � � j=1 θ j D 2 φ j t (x t , ût ) + m i � k=1 λ k,t D G g k t (û t ) + me � i=1 µ i,t Dh i (û t ) = 0.
Corollary 5.25. In the setting of Corollary 5.24, assume that conditions (i-viii) of Corollary 5.24 and the following one are fulfilled (ix') For all t ∈ N * , for all α, β ∈ {1, . . . , n},

∂f α t (xt,ût) ∂x β
≥ 0 and for all α ∈ {1, . . . , n},

∂f α t (xt,ût) ∂x α > 0.
Then all the conclusions of Corollary 5.24 hold.

We will prove Corollaries 5.24 and 5.25 simultaneously as follows.

Proof. Proceeding as in the proof of Theorem 4.25, with a notice that now there are � multipliers for multiobjective criterion instead of one multiplier in that theorem, we obtain the proof for these corollaries. Now, we state a weak maximum principle for multiobjective optimal control problems with (De) and with the control sets defined by both inequality and equality constraints.

Corollary 5.26. Let (x, û) be a Pareto optimal solution of Problem (P M 1 e ) (respectively, solution of (P M 2 e ), (P M 3 e )) where the sets U t are defined in (5.26). We assume that the following assumptions are fulfilled for all t ∈ N.

(i) For all j ∈ {1, . . . , �}, φ j t is Fréchet differentiable at (x t , ût ). (ii) f t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

(iii) For all k ∈ {1, . . . , m i }, g k t is Fréchet differentiable at ût when g k t (û t ) = 0. (iv) For all k ∈ {1, . . . , m i }, g k t is lower semicontinuous and Gâteaux differentiable at ût when g k t (û t ) > 0. Then there exist θ 1 , . . . , θ � ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ t,1 ) t∈N ∈ R N ,. . . ., (µ t,me ) t∈N ∈ R N , (λ t,1 ) t∈N ∈ R N ,. . . ., (λ t,m i ) t∈N ∈ R N which satisfy conclusions (a,d,e) of Theorem 5.24 and the following conditions (b) θ j ≥ 0 for all j ∈ {1, . . . , �}, λ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }.

(c) For all t ∈ N, for all k ∈ {1, . . . , m i }, λ k,t • g k t (û t ) = 0. The difference is the replacement of inequality constraints by equality constraints in the problem issued from the reduction to finite horizon, the consequence of this difference is the lost of the sign of the adjoint variable p t . Remark 5.27. In Theorems 5.22, if control sets are independent of state variables then its assumptions (viii,ix) are reduced to assumptions (vii,viii) of Corollary 5.24.

Finally, we consider the case when the control sets are described for each t ∈ N as follows

U t = � 1≤k≤m i {u ∈ R d : g k t (u) ≥ 0}) (5.27)
Then, for multiobjective optimal control problems with (Di), we have the following corollaries in which, the assumptions are only related to Gâteaux differentiability and lower semicontinuity.

Corollary 5.28. Let (x, û) be a Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 i ), (P M 3 i )) where the sets U t are defined by (5.27). We assume that the following assumptions are fulfilled (i) For all t ∈ N, for all j ∈ {1, . . . , �}, φ j t and f t are Gâteaux differentiable at (x t , ût ). (ii) For all t ∈ N, for all k ∈ {1, . . . , m i }, g k t is Gâteaux differentiable at ût . (iii) For all t ∈ N, for all α ∈ {1, . . . , n}, f α t is lower semicontinuous at (x t , ût ) when and(λ m t ) t∈N ∈ R N which satisfy the following conditions. (a) (θ 1 , . . . , θ � , p 1 ) � = (0, 0).

f α t (x t , ût ) > xα t+1 . (iv) For all t ∈ N, for all k ∈ {1, . . . , m i }, g k t is lower semicontinuous at ût when g k t (û t ) > 0. (v) For all t ∈ N, 0 / ∈ conv{D G g k t (û t ) : k ∈ I s t } where I s t := {k ∈ {1, . . . , m i } : g k t (û t ) = 0}. (vi) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible. Then, under (i-vi) there exist θ 1 , . . . , θ � ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (λ 1 t ) t∈N ∈ R N ,. . . .,
(b) θ j ≥ 0 for all j ∈ {1, . . . , �} , p t ≥ 0 for all t ∈ N * , and λ k t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }.

(c) For all t ∈ N, for all α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût ) -xα t+1 ) = 0, and for all k ∈ {1, . . . , m i }, Then, all the conclusions of Corollary 5.28 hold.

λ k t • g k t (û t ) = 0. (d) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + � � j=1 θ j .D G,
The proof for Corollary 5.28 can be realized as in Theorem 5.22. However, in this case, by using an existing result in Chapter 4, we can prove Corollaries 5.28 and 5.29 as follows Proof. Proceeding as in the proof of Theorem 4.21, with the same notice as in previous corollaries' proofs, we obtain the proof for Corollaries 5.28 and 5.29.

The case of interior optimal controls

In this subsection, we consider the case where for every t ∈ N, U t is an arbitrary subset of R d and u t belongs to the interior of U t and where the system is governed by the difference inequation (Di). For such problems, we can lighten the assumption of smoothness and continuity of the objective functions φ j t and functions f t . This will be shown in the following corollaries: Corollary 5.30. Let (x, û) be a Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 i ), (P M 3 i )). We also assume that the following assumptions are fulfilled: (i) For all t ∈ N, ût ∈ intU t .

(ii) For all j ∈ {1, . . . , �}, for all t ∈ N, φ j t and f t are Gâteaux differentiable at (x t , ût ). (iii) For all t ∈ N, for all α ∈ {1, . . . , n}, f α t is lower semicontinuous at (x t , ût ) when

f α t (x t , ût ) > xα t+1 . (iv) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible.
Then there exist θ 1 , . . . , θ � ∈ R + and p t+1 ∈ R n * + where t ∈ N which satisfy the following conclusions:

(a) (θ 1 , . . . , θ � , p 1 ) � = (0, . . . , 0).

(b) For all t ∈ N, �p t+1 , f t (x t , ût ) -xt+1 � = 0. (c) For all t ∈ N * , p t = � � j=1 � θ j .D G,1 φ j t (x t , ût ) � + p t+1 • D G,1 f t (x t , ût ). (d) For all t ∈ N, � � j=1 (θ j .D G,2 φ t (x t , ût )) + p t+1 • D G,2 f t (x t , ût ) = 0.
Proof. Although this is a corollary of Theorem 5.22, however, in this case we can realize a much simpler proof. By proceeding as in the proof of Theorem 4.14, with a notice that now there are � multipliers for multiobjective criterion instead of one multiplier in that theorem, we obtain the proof for this corollary.

Corollary 5.31. Let (x, û) be a Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 i ), (P M 3 i )). We assume that the assumptions (i,ii,iii) of Corollary 5.30 are fulfilled. Moreover, we assume that the following assumption is fulfilled (iv') For all t ∈ N * , for all α, β ∈ {1, . . . , n},

∂f α t (x t , ût ) ∂x β t ≥ 0 and ∂f α t (x t , ût ) ∂x α t > 0.
Then the conclusions of Corollary 5.30 hold.

New Pontryagin Principles for Multiobjective Optimal Control

Problems in Infinite-Horizon Setting 121

Proof. The proof is obtained by proceeding as in the proof of Theorem 4.16, with the same notice as in previous corollary's proof.

Remark 5.32. All the results in Section 5.1 can be considered as generalizations of Theorem 3.1, Theorem 3.2 and Theorem 3.3 in [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF] if we replace weak Pareto optimality in those theorems by Pareto optimality. Generalization here can be understood under two meanings: using lighter smoothness assumptions and considering multiobjective optimal control problems in the presence of constraints.

Strong Pontryagin Principles

In this section, we will establish strong Pontryagin principles for the considered problems with infinite-horizon settings. To do that, we will use the existing results in finite horizon case, which are presented in Section 4, and add some particular assumptions beside the invertibility assumption to assure that all the multipliers are not simultaneously equal to zero. We recall the definition of control sets:

W t (x t ) = � 1≤k≤m {u ∈ R d : g k t (x t , u) ≥ 0} (5.28) (Di)

with inequality constraints

Firstly, we state the theorem for multiobjective optimal control problems with (Di) and with control sets defined by inequality constraints.

Theorem 5.33. Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 � i ), (P M 3 � i )) where the control sets are defined as in (5.28). We assume that the following conditions are fulfilled (i) For all t ∈ N * , for all j ∈ {1, . . . , �} and for all k ∈ {1, . . . , m}, φ j t (., ût ) and g k t (., ût ) have directional derivatives at xt which are concave.

(ii) For all t ∈ N * , for all j ∈ {1, . . . , �}, for all k ∈ {1, . . . , m}, for all u ∈ W t (x t ), φ j t (., u) , f t (., u) and g k t (., u) are continuous at xt in any direction h. (iii) For all t ∈ N, for each x t in a neighborhood V t of xt such that x 0 = η, the following convexity condition is satisfied:

if u � ∈ W t (x t ), u �� ∈ W t (x t ), 0 ≤ α ≤ 1, one can find u ∈ W t (x t ) such that ∀j ∈ {1, . . . , �}, φ j t (x t , u) ≥ αφ j t (x t , u � ) + (1 -α)φ j t (x t , u �� ), f t (x t , u) ≥ αf t (x t , u � ) + (1 -α)f t (x t , u �� ), ∀k ∈ {1, . . . , m}, g k t (x t , u) ≥ αg k t (x t , u � ) + (1 -α)g k t (x t , u �� ).
(iv) For all t ∈ N * , f t (., ût ) is Gâteaux differentiable at xt and D G,1 f t (x t , ût ) is invertible.

(v) For all t ∈ N, there exists ǔt ∈ W t (x t ) such that f t (x t , ǔt ) = f t (x t , ût ) and g k t (x t , ǔt ) > 0 for all k ∈ {1, . . . , m}. Then there exist θ = (θ 1 , . . . , θ � ) ∈ R � + , p t+1 ∈ R n * + and µ t ∈ R m * + where t ∈ N which satisfy the following conclusions:

(a) (θ 1 , . . . , θ � , p 1 ) � = (0, . . . , 0).

(b) For all t ∈ N, �p t+1 , f t (x t , ût ) -xt+1 � = 0 and �µ t , g t (x t , ût )� = 0.

(c) For all t ∈ N * , p t ∈ ∂ 1 H t (x t , ût , θ, p t+1 , µ). (d) For all t ∈ N, H t (x t , ût , θ, p t+1 , µ) = max u∈R d H t (x t , u, θ, p t+1 , µ). 1 , . . . , θ T � , p T 1 ) � � � = � � j=1 � � �θ T j � � � + � � �p T 1 � � � = 1 (5.30)
Since the values of the sequence (θ T 1 , . . . , θ T � , p T 1 ) T ∈N * belong to the unit sphere of R � × R n * which is compact, using the Bolzano-Weierstrass theorem we can say that there exist an increasing function r

1 : N * → N * and (θ 1 , . . . , θ � , p 1 ) ∈ R � × R n * such that � � j=1 |θ j | + �p 1 � = 1, lim T →+∞ θ r 1 (T ) j = θ j for all j ∈ {1, . . . , �} and lim T →+∞ p r 1 (T ) 1 = p 1 .
Since for all t ∈ {1, . . . , T }, for all j ∈ {1, . . . , �} and for all k ∈ {1, . . . , m}, ϕ j,T t ∈ ∂ 1 φ j t (x t , ût ) and ψ k,T t ∈ ∂ 1 g k t (x t , ût ), then they are compact. Using Theorem A.1 in the appendix A of [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF], there exist an increasing function r 2 : N * → N * , mappings

ϕ j t ∈ ∂ 1 φ j t (x t , ût ) and ψ k t ∈ ∂ 1 g k t (x t , ût ) such that lim T →+∞ ϕ j,r 2 (T ) t
= ϕ j t for all j ∈ {1, . . . , �} and lim T →+∞ ψ k,r 2 (T ) t = ψ t for all k ∈ {1, . . . , m}. We set r = r 2 • r 1 . We have Now we consider the multiobjective optimal control problems with (De) and with control sets defined by inequality constraints. For these problems, we have the following strong Pontryagin principles which provide a necessary condition of optimality in strong form.

Theorem 5.39. Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 e ) (respectively, solution of (P M 2 � e ), (P M 3 � e )) when the control sets are defined as in (5.28). We assume that conditions (i,ii,v) in Theorem 5.33 hold and the following conditions are fulfilled (iii) For all t ∈ N, for each x t in a neighborhood V t of xt such that x 0 = η, the following convexity condition is satisfied: if u � ∈ W t (x t ), u �� ∈ W t (x t ), 0 ≤ α ≤ 1, one can find u ∈ W t (x t ) such that ∀j ∈ {1, . . . , �}, φ j t (x t , u) ≥ αφ j t (x t , u � ) + (1α)φ j t (x t , u �� ), f t (x t , u) = αf t (x t , u � ) + (1α)f t (x t , u �� ), ∀k ∈ {1, . . . , m}, g k t (x t , u) ≥ αg k t (x t , u � ) + (1α)g k t (x t , u �� ).

(iv) For all t ∈ N * , for all u t ∈ W t (x t ), f t (., u t ) is continuously differentiable at xt and moreover, D 1 f t (x t , ût ) is invertible.

Then there exist θ = (θ 1 , . . . , θ � ) ∈ R � + and p t+1 ∈ R n * , t ∈ N which satisfy conclusions (a,c,d) of Theorem 5.33 and the following one (b) For all t ∈ N, for all k ∈ {1, . . . , m}, µ k t .g k t (x t , ût ) = 0.

The proof of this theorem is similar to the proof of Theorem 5.33. The difference is the lost of sign of p t+1 since now the dynamical system is defined by difference equations.

In the special case when the sets of controls are defined by inequalities but only depend on control variable, we can state a corollary as follows.

Corollary 5.40. Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 e ) (respectively, solution of (P M 2 � e ), (P M 3 � e )) when the control sets are defined as in Corollary 5.37. We assume that condition (ii) in Theorem 5.33 and conditions (iii,iv) in Theorem 5.39 are fulfilled. Moreover, assume that the additional assumptions (v',vi) in Corollary 5.37 are fulfilled. Then there exist θ = (θ 1 , . . . , θ � ) ∈ R � + , p t+1 ∈ R n * and µ t ∈ R m * + where t ∈ N such that all the conclusions of Theorem 5.39 hold.

The proof for this theorem is similar to that of Corollary 5.37.

Remark 5.41. Theorem 5.39 and Corollary 5.40 can be considered as generalizations of Theorem 4.1 in [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF] since now there exist the inequality constraints in the control sets. Besides, in these results, we use weaker assumptions compared to the condition of partially continuous differentiability of Theorem 4.1 in [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF].

Remark 5.42. All the results for multiobjective problems in this chapter become Pontryagin principles for single-objective optimal control problems in Chapter 4 when � = 1. Therefore, we also obtain strong Pontryagin principles and more general weak Pontryagin principles for single-objective optimal control problems in finite or infinite horizon.

Transversality Condition as a Necessary Condition of Optimality

Using Corollary 5.30 and Corollary 5.31, we analyse the transversality condition in the form lim t→+∞ p t = 0 in two cases: with invertibility condition and with positivity condition. In each case, we will add more assumptions in order to obtain the transversaltity condition.

The transversality condition for the problem with invertibility assumption From Corollary 5.30, we know that there exist θ 1 , . From this, we can assert that, for all t ∈ N * ,

�p t+1 � ≤ �p t � � � �D G,1 f t (x t , ût ) -1 � � � + � � j=1 θ j . � � �D G,1 φ j t (x t , ût ) � � � � � �D G,1 f t (x t , ût ) -1 � � � . (5.33)
We set for all t ∈ N, A t := D G,1 f t (x t , ût ), A -1 t := (D G,1 f t (x t , ût )) -1 and b j t := D G,1 φ j t (x t , ût ). Then, we have for all t ∈ N, A t , b j t are bounded linear operators. Since A -1 t exists for all t ∈ N * (from the invertibility assumption), we can assert that, for all t ∈ N, A -1 t is also a bounded linear operator (because A t .A -1 t = E n , the unit matrix in R n×n ). We will find an expression that describes p t+1 by p 1 and θ 1 , . . . , θ � (notice that (θ 1 , . . . , θ � , p 1 ) � = (0, . . . , 0, 0)).

∀t ∈ N * , �p t+1 � ≤ �p t � � � �A -1 t � � � + � � j=1 θ j . � � �b j t � � � � � �A -1 t � � �
(5.34) While t > 1 we have:

�p t+1 � ≤   �p t-1 � � � �A -1 t-1 � � � + � � j=1 θ j . � � �b j t-1 � � � � � �A -1 t-1 � � �   � � �A -1 t � � � + � � j=1 θ j . � � �b j t � � � � � �A -1 t � � � ⇔ �p t+1 � ≤ �p t-1 � � � �A -1 t-1 � � � � � �A -1 t � � � + � � j=1 θ j �� � �b j t-1 � � � � � �A -1 t-1 � � � � � �A -1 t � � � + � � �b j t � � � � � �A -1 t � � � � .
By induction we obtain That means lim t→+∞ α t = 0. Thus, (α t ) t∈N is bounded. For β t , we have

�p t+1 � ≤ �p 1 � � � �A -1 1 � � � . . .
β t = t � s=1 ( � � �b j s � � � t � k=s � � �A -1 s � � �) ≤ t � s=1 γ s M t-s+1 ≤ � � �γ � � � t � s=1 M t-s+1 = M t+1 � � �γ � � � t � s=1 M -s = � � �γ � � � M 1 -M t 1 -M ≤ � � �γ � � � M 1 -M < +∞.
Then, (β t ) t∈N is also bounded. From this, we can assert that lim sup t→+∞ �p t � = 0 since M < 1. Thus, lim t→+∞ �p t � = 0 i.e. lim t→+∞ p t = 0. And so, we have proven the following corollary Then, all the conclusions of Corollary 5.30 hold and moreover, lim t→+∞ p t = 0. Now we consider the special case when (γ t ) t∈N = γ ∈ � 1 (N, R), i.e.

� +∞ t=0 |γ t | < +∞. This case is usual in Macroeconomic Theory. For example, φ j t (x, u) = β t ψ j t (x, u) for all t ∈ N and for all j ∈ {1, . . . , �} where β ∈ (0, 1) and ψ j t is a mapping satisfying sup t∈N � � �D G,1 ψ j t (x t , ût ) � � � := K < +∞ for all j ∈ {1, . . . , �}. Here we can define γ t := β t K for all t ∈ N and then,

� +∞ t=0 |γ t | = K � +∞ t=0 β t = K 1-β t .
Another example is φ j t (x t , u t ) = ae -ct ψ j t (x t , u t ) for all t ∈ N and for all j ∈ {1, . . . , �} where a, c > 0 and ψ j t is a same mapping in the previous example. Here we can define γ t := acKe -ct for all t ∈ N and then, Then, all the conclusions of Corollary 5.30 hold and moreover, lim t→+∞ p t = 0.

Proof. From Corollary 5.30, we know that there exist θ 1 , . . . , θ � ∈ R + and p t+1 ∈ R n * + where t ∈ N which satisfy conclusions (a-d) of this corollary. Using conclusion (c) of Corollary 5.30, we have: From the last inequality, we obtain (p t ) t∈N * ∈ � 1 (N * , R n * ) which assures that lim t→+∞ p t = 0.

∀t ∈ N * , p t = � � j=1 � θ j .D G,

The transversality condition for the problem with positivity assumption

From the proof of Corollary 5.31, the following condition holds for all t ∈ N * and for all T ≥ t ) lim sup t→+∞ �p t � ≤ 0. Since δ > 1 then (1 -1 δ ) > 0 and from the last inequality, we can assert that lim sup t→+∞ �p t � = 0 since M < 1. Thus, lim t→+∞ �p t � = 0 i.e. lim t→+∞ p t = 0. We have proven the following corollary: Then, all the conclusions of Corollary 5.31 hold and moreover, lim t→+∞ p t = 0.

� � �p T t+1 � � � ≤ 1 ρ t   � � �p T t � � � + � � j=1 � � �D G,

Sufficient Condition for Multiobjective Optimal Control Problem

In this section we establish results of sufficient condition of optimality which uses the adjoint equation and the maximum principles. All the results will use the concavity assumption on the Hamiltonian. Some results need its concavity with respect to both state variable and control variable. Others need its concavity with respect to only state variable.

Firstly, we introduce the scalarization technique, namely the weighting method. For k ∈ {e, i}, consider the following single-objective control problems

(Q 1 k ) Maximize � � j=1
θ j J j (x, u) when (x, u) ∈ Dom k (J).

(Q θ j (φ j t (x t , u t )φ j,t (x t , ût )) > 0.

We have the following lemma where k ∈ {e, i} and j ∈ {1, 3}. The following lemmas are taken from [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF].

Lemma 5.47. Assume that (x, û) is a solution of Problem (Q 1 k ). The two following assertions hold:

-If θ j > 0 for all j ∈ {1, . . . , �} then (x, û) is a Pareto optimal solution of Problem (P M 1 k ). -If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ = (θ 1 , . . . , θ � ) � = 0 then (x, û) is a weak Pareto optimal solution of Problem (P M 1 k ).

Proof. If θ j > 0 for all j ∈ {1, . . . , �} and (x, û) is not a Pareto optimal solution of Problem (P M 1 k ); thus there exists a process (x, u) ∈ Dom k (J) such that for all j ∈ {1, . . . , �}, J j (x, u) ≥ J j (x, û) and for some i ∈ {1, . . . , �}, J i (x, u) > J i (x, û). And so 5.7. Sufficient Condition for Multiobjective Optimal Control Problem131 � � j=1 θ j J j (x, u) > � � j=1 θ j J j (x, û) which is a contradiction since (x, û) is the solution of (Q 1 k ). Hence, (x, û) is a Pareto optimal solution of Problem (P M 1 k ). Now if θ j ≥ 0 for all j ∈ {1, . . . , �} and θ � = 0. Assume that (x, û) is not a weak Pareto optimal solution of Problem (P M 1 k ). Thus there exists a process (x, u) ∈ Dom k (J) such that for all j ∈ {1, . . . , �}, J j (x, u) > J j (x, û). And so, since θ � = 0, � � j=1 θ j J j (x, u) > � � j=1 θ j J j (x, û) which is a contradiction since (x, û) is the solution of (Q 1 k ). Hence, (x, û) is a weak Pareto optimal solution of Problem (P M 1 k ).

Lemma 5.48. Assume that (x, û) is a solution of Problem (Q 3 k ). The two following assertions hold.

-If θ j > 0 for all j ∈ {1, . . . , �} then (x, û) is a solution of Problem (P M 3 k ). -If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ � = 0 then (x, û) is a solution of Problem (P M 3 � k ).

Proof. If θ j > 0 for all j ∈ {1, . . . , �} and (x, û) is not a solution of Problem (P M 3 k ); thus there exists a process (x, u) ∈ Adm k such that for all j ∈ {1, . . which contradicts the assumption that (x, û) is a solution of Problem (Q 3 k ). Hence, (x, û) is a solution of Problem (P M 3 � k ). Now we provide a theorem which uses the concavity assumption to establish sufficient condition for multiobjective problems.

Theorem 5.49. Let (x, û) ∈ Dom e (J). Assume that there exist θ = (θ 1 , . . . , θ � ) ∈ (R � ), p = (p t ) t∈N * ∈ (R n * ) N * , not all zero such that the following conditions hold.

(i) For all t ∈ N, X t × U t is convex. The end of this proof goes like that of the previous theorem. (θ j φ j t (x t , u t )θ j φ j t (x t , ût )) ≤ 0.

And so, (x, û) is a solution of

Sufficient Condition for Multiobjective Optimal Control Problem135

Theorem 5.53. Let (x, û) ∈ Dom i (J). Assume that there exist θ = (θ 1 , . . . , θ � ) ∈ R � , p = (p t ) t∈N * ∈ (R n * ) N * , not all zero such that the conditions (i-vi) of Theorem 5.49 and moreover, the following conditions hold.

(ix) For all t ∈ N * , p t ≥ 0. (x) For all t ∈ N, �p t+1 , f t (x t , ût ) -xt+1 � = 0.

Then

-If θ j > 0 for all j ∈ {1, . . . , �} then (x, û) is a Pareto optimal solution of Problem (P M 1 i ).

-If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ = (θ 1 , . . . , θ � ) � = 0 then (x, û) is a weak Pareto optimal solution of Problem (P M 1 i ). Proof. Let (x, û), (x, u) ∈ Dom i (J), i.e. xt , x t ∈ X t , ût , u t ∈ U t , xt+1 ≤ f t (x t , ût ), x t+1 ≤ f t (x t , u t ) for all t ∈ N, x0 = x 0 = η and J j (x, û) = Notice that �p t+1 , f t (x t , ût ) -xt+1 � = 0 from (x). Moreover, since f t (x t , u t )x t+1 ≥ 0, from (ix) we obtain �p t+1 , f t (x t , u t )x t+1 � ≥ 0, or equivalently, �p t+1 , f t (x t , u t ) ≥ �p t+1 , x t+1 �. Hence, we have Now by a similar interpretation as in Theorem 5.49 and using Lemma 5.47, we obtain the announced conclusion.

Theorem 5.54. Let (x, û) ∈ Dom i (J). Assume that there exist θ = (θ 1 , . . . , θ � ) ∈ (R � ), p = (p t ) t∈N * ∈ (R n * ) N * , not all zero and such that all the hypotheses of Theorem 5.51 together with hypotheses (ix) and (x). Then the conclusion of Theorem 5.53 hold.

Proof. The proof is similar to that of Theorem 5.51.

Corollary 5.55. If (x, û) satisfies all hypotheses of Theorem 5.53 or Theorem 5.54 except for hypothesis (iii), which is replaced by (vii) or (viii), then -If θ j > 0 for all j ∈ {1, . . . , �} then (x, û) is a solution of Problem (P M 3 i ).

-If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ = (θ 1 , . . . , θ � ) � = 0 then (x, û) is a solution of Problem (P M 3 � i ). Proof. The proof is similar to the proof of Corollary 5.52.

Remark 5.56. In case of bounded sequences, i.e. x ∈ � ∞ (N, R n ) and u ∈ � ∞ (N, R d ), it is clear that (x t -xt ) t∈N is a bounded sequence. Moreover, under the setting and hypotheses showed in [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF], [START_REF] Blot | Infinite Horizon Discrete Time Control Problems for Bounded Processes[END_REF], (p t ) t∈N ∈ � 1 (N, R n * ) and then the condition lim t→+∞ p t = 0 automatically holds.

Notations

∀ universal quantifier, "for every" ∃ existential quantifier, "there exists" ⇒ sign of implication, "... implies ..." ⇔ sign of equivalence := by definition, is equal to x ∈ X the element x belongs to the set X x / ∈ X the element x does not belong to the set X |X| the cardinality of set X when it is a finite set, i.e. the number its elements

Notation 1 . 1 .

 11 (Basic sets and spaces) -N is the set of all nonnegative integers. N * = N\{0} = {1, 2, . . .}.

(

  N1) �λx� = |λ| �x� for all λ ∈ R, x ∈ E; (N2) �x� = 0 if and only if x = 0; (N3) �x + y� ≤ �x� + �y� . Chapter 1. Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces

  lim n→+∞ x n = x, or equivalently: lim n→+∞ �x n -x� = 0. Notation 1.4. (norm and norm dual on R k ) -When k ∈ N * , on R k we consider the norm �v� := max �� � �v j � � � : j ∈ {1, . . . , k} � , where v = (v 1 , . . . , v k ) ∈ R k . For simplicity, from now on we denote the norm on R k by |•| . -The canonical norm on the dual R k * is |p| * := sup{|�p, v�| : v ∈ R k , |v| ≤ 1} where �p, v� := p(v) is the duality bracket. Note that when R k is endowed with the norm |•|, |p| * = � k j=1 |p j | , where p j := �p, e j � .

Definition 1 . 5 .

 15 (Basic sequence spaces) -Space � p (N, E) : For all p ∈ [1, +∞), � p (N, E) := {x ∈ E N : � +∞ t=0 |x t | p < +∞}. Endowed with the norm �x� p := � � +∞ t=0 |x t | p � 1/p , it is a Banach space. -Space � ∞ (N, E) : � ∞ (N, E) := {x ∈ E N :sup t∈N |x t | < +∞}. Endowed with the norm �x� ∞ := sup t∈N |x t | , it is Banach space. -Space c(N, E) : c(N, E) := {x ∈ E N : lim t→+∞ x t exists in E}. Endowed with the norm �x� ∞ , it is a Banach subspace of � ∞ (N, E). -Space c 0 (N, E) : c 0 (N, E) := {x ∈ E N : lim t→+∞ x t = 0}. Endowed with the norm �x� ∞ , it is a Banach subspace of � ∞ (N, E).

Chapter 1 .

 1 Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces Definition 1.7. (real convex function)

Theorem 1 . 8 .Example 1 . 9 .

 1819 Let f be a real function which is differentiable on the open interval (a, b).Then f is convex on (a, b) if and only if its derivative f � is increasing on (a, b). After the previous theorem, the exponential function e x is convex since (e x ) � = e x > 0 for all x ∈ R. From the definition of convex function we have the following inequality e B+α(A-B) ≤ e B + α(e Ae B ) for all A, B ∈ R and for all α ∈ [0, 1].

3

 3 Duality of Sequence Spaces Definition 1.15. (Dual space of a sequence space)

Chapter 1 .

 1 Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces (i) The subdifferential ∂F (x) is a closed convex set (possibly empty).

  [a, b] = {x : x = a + t(ab), 0 ≤ t ≤ 1}, and the open line segment (a, b) is defined as folows: (a, b) = {x : x = a + t(ab), 0 < t < 1}. Theorem 1.36. (Mean Value Theorem) Let X and Y be normed linear spaces, and let an open set U ⊂ X contains a closed line segment [a, b].

1. 2 . 5

 25 Differentiation in a Product Space. Partial Derivatives. Theorem on The Total Differential.

Chapter 2 .

 2 Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle

. 2 )

 2 Since (x, û) is a solution of the initial problem, û is a solution of the following problem� Maximize B(u) when u ∈ N û ∩ U Chapter 2.Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle

Chapter 2 .

 2 Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle

Chapter 2 .

 2 Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle

(β. 4 )

 4 For all B bounded nonempty in U, lim x→0 (sup u∈B �D 1 ζ(x, u)�) = 0. (β.5) D 1 ζ transforms the nonempty bounded subsets of R n × U into bounded subsets of R m . (β.6) For all u ∈ U, ζ(0, u) = 0. Remark 3.1. Under conditions (β.3), (β.5) and (β.6), condition (β.2) holds.

3. 3 .

 3 Some Useful Properties of Nemytskii Operators 55

Chapter 3 .

 3 Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality �D

Lemma 3 . 5 .

 35 Under the conditions (C1), (C2) and (C4), for all u

3. 5 . 59 Lemma 3 . 6 .

 55936 Strong Pontryagin Principle for Problem (Ps) Under the conditions (C1-C6), for all u

  û), and consequently, contains a neighborhood of the origin of c 0 (N, R n ) × R n . Hence, assumption (d) of Lemma 3.4 is fulfilled.

)

  For all B bounded, nonempty set in U we have lim z→0 (sup u∈B �D 1 �(z, u)�) = 0.

  D6) For all B bounded, nonempty set in U we have lim y→y∞ (sup u∈B �D 1 g(y, u)�) = 0.

Chapter 3 .

 3 Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality

Chapter 3 .

 3 Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality Theorem 3.14. Let U be a nonempty convex subset of R d , β ∈ (0, 1), η, y ∞ ∈ R n , and two mappings ψ : R n × U → R and g : R n

Chapter 4 .

 4 Lightenings of Assumptions for Pontryagin Principles in Finite Horizon and Discrete Time 4.3 The New Multiplier Rule 4.3.1 Recall of the Multiplier Rule of Halkin

Theorem 4 . 6 .

 46 Let m, n ∈ N * , ϕ 1 , . . . , ϕ m ∈ R n * , and a ∈ R n * . The two following assertions are equivalent.

  m}, and we obtain A e � = ∅ =⇒ ((a), (b), (c) hold). (4.6) Then with (4.3) and (4.6) the conclusions (a), (b), (c) are proven.

Chapter 4 .

 4 Lightenings of Assumptions for Pontryagin Principles in Finite Horizon and Discrete Time ∂L ∂xt

Lemma

. 26 ) 4 .

 264 Chapter Lightenings of Assumptions for Pontryagin Principles inFinite Horizon and Discrete Time

4. 5 .

 5 Weak Pontryagin Principles for Infinite-Horizon Problems with Constrained Controls 91

5. 5 .

 5 New Pontryagin Principles for Multiobjective Optimal Control Problems in Finite-Horizon Setting 105

5. 5 .

 5 New Pontryagin Principles for Multiobjective Optimal Control Problems in Finite-Horizon Setting 109

Chapter 5 .

 5 Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems � p T , F T (x T , ûT ) � = 0 and � µ T , G T (x T , ûT )

5. 6 .

 6 New Pontryagin Principles for Multiobjective Optimal Control Problems in Infinite-Horizon Setting 125 (De) with inequality constraints

5 . 6 .j=1 θ j + �p 1 ���

 561 New Pontryagin Principles for Multiobjective Optimal Control Problems in Infinite-Horizon Setting 127 ⇔ �p t+1 � ≤ �p 1 � = 1, then ∀t ∈ N * , �p t+1 � ≤ = M < 1 and for all t ∈ N, for all j ∈ {1, . . . , �}, ≤ γ t where (γ t ) t∈N = γ ∈ c 0 (N, R). From this, we have 0 ≤ lim t→+∞ α t = lim t→+∞ M t = 0 (since 0 ≤ M < 1).

Corollary 5 . 43 .

 543 If (x, û) satisfies all hypotheses of Corollary 5.30 and the following condition(vi) sup t∈N * � � �A -1 t � � � = M < 1 and � � �D G,1 φ j t (x t , ût ) � � � ≤ |γ t |for all j ∈ {1, . . . , �}, for all t ∈ N where (γ t ) t∈N = γ ∈ c 0 (N, R).

Chapter 5 .

 5 Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems

�

  +∞ t=0 |γ t | = acK � +∞ t=0 e -ct = acK1-e -c . For this case, we will provide another condition which assures the validation of transversality condition. We have the following corollary Corollary 5.44. If (x, û) satisfies all hypotheses of Corollary 5.30 together with the following condition(vii) sup t∈N * �D G,1 f t (x t , ût )� = M < 1 and � � �D G,1 φ j t (x t , ût ) � � � ≤ |γ t |for all j ∈ {1, . . . , �}, for all t ∈ N where (γ t ) t∈N = γ ∈ � 1 (N, R).

  t ) t∈N is bounded. For β t , we have

Corollary 5 . 45 .

 545 If (x, û) satisfies all hypotheses of Corollary 5.31 and the following condition(viii) δ := inf t∈N * ρ t > 1 and � � �D G,1 φ j t (x t , ût ) � �� ≤ γ t for all j ∈ {1, . . . , �}, for all t ∈ N where (γ t ) t∈N = γ ∈ c 0 (N, R).

Lemma 5 . 46 . 2 k� 3 k�. 2 k�.

 546232 If (x, û) is a solution of Problem� We prove this lemma by contradiction. Suppose that (x, û) is a solution of Problem � Q but not a solution of Problem � Q Then there exists (x, u) ∈ Adm k θ j (φ j t (x t , u t )φ j,t (x t , ût )) > 0. This inequality implies that limsup θ j (φ j t (x t , u t )φ j t (x t , ût )) > 0 and this contradicts to the fact that (x, û) is a solution of Problem � Q The lemma is proven. Now we introduce the lemmas that show the relationship between the solutions of the single-objective weighted problems �

Corollary 5 . 52 .

 552 If (x, û) satisfies all hypotheses of Theorem 5.49 or Theorem 5.51 except for hypothesis (iii), which is replaced by (vii) limsupt→+∞ �p t , x t -xt � = 0 or (viii) liminf h→+∞ �p t , x t -xt � = 0, then -If θ j > 0 for all j ∈ {1, . . . , �} then (x, û) is a solution of Problem (P M 3 e ).-If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ � = 0 then (x, û) is a solution of Problem (P M 3 � e ). Proof. It is analogous to that of Theorem 5.49 or Theorem 5.51 until we obtain limsup φ j t (x t , ût )θ j φ j t (x t , u t ))

� Q 3 e� 3 e�

 33 then using Lemma 5.48, we obtain the announced conclusion.Now under (viii), by analogous realization, we obtain (x, û) is a solution of � also. Then using Lemma 5.48 again to obtain the conclusion of this corollary.

  ∅ empty set A ∪ B the union of the sets A and B A ∩ B the intersection of the sets A and B A \ B the difference of the sets A and B, i.e., the set of elements that belong to the set A but do not belong to the set B A ⊂ B the set A is contained in the set B A × B the Cartesian product of the sets A and B {x : P (x)} the set of those elements x that possess the property P (.) {x 1 , ..., x n , ...} the set which consists of elements x 1 , ..., x n , ... F : X → Y the mapping F of the set X into the set Y ; the function F with domain X whose values belong to the set Y x � → F (x) the mapping (function) F assigns the element F (x) to an element x; the notation of the mapping (function) F in the case when it is desirable to indicate the notation of its argument F (.) the notation which stresses that F is a mapping (function) F (A) the image of the set A under the mapping F imF = {y : y = F (x), x ∈ X} the image of the mapping F : X → Y F -1 (A) the inverse image of the set A under the mapping F F |A the restriction of the mapping F to the set A F • G the composition of the mappings G and F : (F • G)(x) = F (G(x)) R the set of all real numbers N the set of all nonnegative integers; N * = N\{0} inf A (sup A) the infimum (supremum) of the numbers which belong to the set A ∈ R R n the arithmetical n-dimensional space endowed with the standard Euclidean structure R n + = {x = (x 1 , ..., x n ) ∈ R n : x i ≥ 0} the nonnegative orthant of R n e 1 , ..., e n the vectors of the standard orthonormal basis in R n ; e 1 = (1, 0, ..., 0), ..., e n = (0, ..., 0, 1) [x, y] = {z : z = αx + (1α)y, 0 ≤ α ≤ 1} the line segment joining the points x and y R n * the arithmetical n-dimensional space conjugate to R n px = �p, x� = � n i=1 p i x i for all p ∈ R n * and for all x ∈ R n �x� the norm of element x in normed space

  Chapter 1. Preliminary on Sequence Spaces and Differential Calculus in Normed Spaces -(D1) For each y ∈ � 1 (N, E * ), we consider series � t∈N �y t , x t � where x = (x t ) t∈N ∈ c 0 (N, E) arbitrary. We have | � t∈N �y t , x t �| ≤

	� t∈N |�y t , x t �| = lim

  1.2. Recall of Differential Calculus in Normed Linear SpacesProof. Let P be a Banach space and Q is a closed subspace of P . Consider an arbitrary Cauchy sequence (x n ) n∈N in Q then (x n ) n is a Cauchy sequence in P as well since Q ⊂ P . Now P is Banach, hence, (x n ) n is convergent and lim n→+∞ x n = x ∈ P . Since Q is closed then if lim n→+∞ x n exists, it must belongs to Q. Hence, x ∈ Q. And so, every Cauchy sequence in Q is convergent within Q. It means that Q is a Banach space.

Corollary 1.24. c(N, E) and c 0 (N, E) are Banach spaces.

  [START_REF] Giannessi | Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, chapter On the Theory of Vector Optimization and Variational Inequalities[END_REF]. If F is Fréchet differentiable at the point x then F is continuous at x.

	1.2. Recall of Differential Calculus in Normed Linear Spaces	29
	Now with strictly differentiable mapping, we have a similar but stronger result:
	Theorem 1.31. If F is strictly differentiable at the point x then F is continuous on a
	neighborhood of x.

  X, sup u∈B �ζ(x, u)� < +∞) . Therefore, for all x ∈ X, the partial mapping u → ζ(x, u) transforms the bounded subsets of U into bounded subsets of Y. 2. The reasoning is similar to 1. 3. Since B = {u} is a bounded set for all u ∈ U, we obtain �ζ(0, u)� = lim x→0 �ζ(x, u)� = lim x→0 (sup u∈{u} �ζ(x, u)�) = 0. Hence, for all u ∈ U, ζ(0, u) = 0. In the setting of Theorem 2.2, if in addition, we assume that dim V < +∞ and U is closed then if ζ(x, .) ∈ C 0 (U, Y ), ζ (x, .) transforms the bounded subsets of U into bounded subsets of Y . For all (u t ) t ∈ � ∞ (N, U ), For all (x t ) t ∈ c 0 (N, X), lim t→+∞ ζ(x t , u t ) = 0. (iii) For all B bounded, nonempty in U , for all (x t ) t ∈ c 0 (N, X),

	Chapter 2. Infinite-Horizon Optimal Control Problem in Presence of
	Asymptotical Constraint and a Weak Pontryagin Principle
	2. The assertion (i) of Lemma 2.1 is equivalent to
					�	�
	∀R ∈ (0, +∞), lim x→0	u∈U ∩B R sup	�ζ(x, u)�	= 0.
	3. It is also noticed that assertion (i) of Lemma 2.1 and the continuity of [x � → ζ(x, u)]
	for all u ∈ U imply ζ(0, u) = 0 for all u ∈ U.
	Proof. We will prove these assertions one after another.
	1. Let ζ is a function that satisfies the assumption of Theorem 2.2. It is obvious that
	B R := {v ∈ V : �v� ≤ R} is bounded for all R ∈ (0, +∞). Then B := U ∩ B R is a bounded subset of U and hence, for all x ∈ X, ζ(x, .) transforms B into bounded
	subset of Y which means that for all x ∈ X, sup u∈U ∩B R �ζ(x, u)� < +∞. On the contrary, let B be a bounded subset of U then there exists R ∈ (0, +∞) such
	that B ⊂ (B R ∩ U ) . Hence, from	�	∀x ∈ X, sup u∈U ∩B R �ζ(x, u)� < +∞	�	we obtain
	(∀x ∈ Remark 2.5. Proof. Let B be a bounded subset of U . Since U is closed, B, the closure of B, is also a
	subset of U and since B is bounded, B is compact. Now if ζ(x, .) ∈ C 0 (U, Y ), then ζ(x, .)
	is continuous on B. Using Theorem 4.14 on page 89 of [53], we obtain {ζ(x, u), u ∈ B} is
	also compact in Y . Therefore, its subset {ζ(x, u), u ∈ B} is bounded.
	Remark 2.6. Let ζ ∈ C 0 (X × V, Y ) where X, V and Y are finite-dimensional Banach
	spaces and U be a nonempty closed subset of V . Then the following statement are
	equivalent:
	(i) For all B bounded, nonempty in U , lim x→0 (sup u∈B �ζ(x, u)�) = 0.
	(ii) lim t→+∞	(sup u∈B	�ζ(x t , u)�) = 0.
						�ζ(x, u)� < +∞.
					u∈U ∩B R

  Weak Pontryagin Principle for Main ProblemSo far in Section 1 we have introduced the main problem as follows:

	2.7. Weak Pontryagin Principle for Main Problem	51
	2.7

Remark 2.

[START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]

. In Theorem 2.22, (AE1) means Adjoint Equation, (WM1) means weak Maximum Principle. Since p = (p t+1 ) t∈N ∈ � 1 (N * , R n * ), note that the transversality condition at infinity for Problem (P s ), lim t→+∞ p t = 0, is satisfied.

  2. Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Weak Pontryagin Principle -Study the conclusions We can easily rewrite conclusions (i) and (ii) for Problem (P m ) as follows

  Chapter 3. Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality We consider the following conditions:

.2)

Note that the condition (WM1) of Theorem 2.22 in Chapter 2 is equivalent to the following condition ∀u ∈ U, ∀t ∈ N, �D 2 H t (x t , ût , p t+1 ), u -ût � ≤ 0.

In this section, we will replace (WM1) by the strengthened condition H t (x t , ût , p t+1 ) = max u∈U H t (x t , u, p t+1 ) for all t ∈ N. Note that (WM1) can be considered as a first-order necessary condition of the optimality of H t (x t , ., p t+1 ) at ût on U .

  3. Strong Pontryagin Principle for Infinite-Horizon Optimal Control Problem in Presence of Asymptotical Constraint and a Sufficient Condition of Optimality where T

  Then using Theorem of New Multiplier Rule for Problem (R T e ) as we did before the statement of this proposition, we easily obtain the multipliersλ T o , λ T i,t ∈ R + , µ T k,t ∈R and p T t+1 ∈ R n * where t ∈ {0, . . . , T }, i ∈ {1, . . . , m i } and k ∈ {1, . . . , m e } that satisfy results (a) and (b). Results (c) and (d) are obtained after identifying Gâteaux differential with Fréchet differential of Fréchet differentiable functions in (4.16).

  Finite Horizon and Discrete TimeUsing assumption (iv) we can formulate (4.20) as follows.∀t ∈ {1, . . . , T }, p T t+1

	R × (R n * + ) T +1 which satisfies the following conditions.	T 1 , . . . , p T T +1 ) ∈
	(λ T o , p T 1 , . . . , p T T +1 ) � = (0, 0, . . . , 0).	(4.17)
	λ T

o ≥ 0 and ∀t ∈ {1, . . . , T + 1}, p T t ≥ 0. (4.18) ∀t ∈ {0, . . . , T }, � p T t+1 , f t (x t , ût ) -xt+1 � = 0. (4.19) ∀t ∈ {1, . . . , T }, p T t = λ T o .D 1 φ t (x t , ût ) + p T t+1 • D 1 f t (x t , ût ). (4.20) ∀t ∈ {0, . . . , T }, 0 = λ T o .D 2 φ t (x t , ût ) + p T t+1 • D 2 f t (x t , ût ). (4.21)

  4.17. Under assumption (v) of Theorem 4.16, setting ρ t := min For all y∈ R n + , D 1 f t (x t , ût ).y ≥ ρ t .y. (ii) For all π ∈ R n * + , π • D 1 f t (x t , ût ) ≥ ρ t .π.Proof. (i) Let y ∈ R n + arbitrary. For all α ∈ {1, . . . , n} we have That means D 1 f t (x t , ût ).y ≥ ρ t .y.(ii) Let π ∈ R n * + . Using (i) we know for all y ∈ R n + , D 1 f t (x t , ût ).y ≥ ρ t .y. Then we have

	4.4. Weak Pontryagin Principles for Infinite-Horizon Problems with
	Interior Optimal Controls				85
	(i) n � β=1	∂f α t (x t , ût ) ∂x β t	.y β ≥	∂f α t (x t , ût ) t ∂x α	.y α ≥ ρ t .y α .
	a real positive number. Then the following assertions hold:	1≤α≤n	∂f α t (x t , ût ) ∂x α t	-

  {1, . . . , T + 1}, p t ≥ 0, and ∀t ∈ {0, . . . , T }, ∀k ∈ {1, . . . , m}, λ k,t t ≥ 0. {0, . . . , T }, ∀k ∈ {1, . . . , m}, λ k,T

			satisfies the following
	assertions.	(λ T 0 , p T 1 , . . . , p T T +1 , λ 1,T 1 , . . . , λ T,T m ) � = 0.	(4.28)
		λ T 0 ≥ 0, ∀t ∈

� (4.29) ∀t ∈ {0, . . . , T }, ∀α ∈ {1, . . . , n}, p α t+1 • (f α t (x t , ût ) -xα t+1 ) = 0, and ∀t ∈ t • g k t (û t ) = 0. � (4.30) ∀t ∈ {1, . . . , T },

  [START_REF] Blot | Infinite-Horizon Discrete-Time Pontryagin Principles via Results of Michel[END_REF], we know that the multipliers are not simultaneously equal to {1, . . . , m i }, ∀t ∈ {0, . . . , T }, λ T

	zero. We set p T t+1 :=	n � j=1	p T t+1,j e * j ∈ R n * . From conclusion (b) of Theorem 5.7, we
	gi t (ẑ) = 0 i.e. ∀i ∈ {1, . . . , m i }, ∀t ∈ t (x t , ût ) = 0; and ∀t ∈ {0, . . . , T }, ∀j ∈ {1, . . . , n}, p T have: ∀i ∈ i,t {0, . . . , T }, λ T i,t g i t+1,j ψ j t (ẑ) =
	p T t+1,j (f j t (x t , ût ) -xj t+1 ) = 0. i.e. ∀t ∈ {0, . . . , T }, the generalized Lagrangian of this problem is	�	p T t+1 , f t (x t , ût ) -xt+1	�	= 0. Then

  [START_REF] Blot | Infinite-Horizon Discrete-Time Pontryagin Principles via Results of Michel[END_REF], the partial Gâteaux differential of L with respect to z vanishes at ẑ. That means

	⇐⇒
	∀t ∈ {1, . . . , T }, p T t =	�
	�	∀t ∈ {1, . . . , T }, D G,xt L(ẑ, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = 0, ∀t ∈ {0, . . . , T }, D G,ut L(ẑ, λ T o , p T 1,1 , . . . , p T 1,n , . . . , p T T +1,1 , . . . , p T T +1,n ) = 0.

  R and p T t+1 ∈ R n * where t ∈ {0, . . . , T }, k ∈ {1, . . . , m i } and i ∈ {1, . . . , m e } which satisfy the following conditions.

	All the multipliers are not simultaneously equal to zero.	(5.16)
	∀j ∈ {1, . . . , �}, θ T

j ≥ 0, ∀t ∈ {0, . . . , T }, ∀i ∈ {1, . . . , m i }, p T t+1 ≥ 0 and λ T i,t ≥ 0. (5.17)

∀t ∈ {0, . . . , T }, ∀k ∈ {1, . . . , m i }, �

  iii) For all α ∈ {1, . . . , n}, f α t is lower semicontinuous and Gâteaux differentiable at(x t , ût ) when f α t (x t , ût ) > xα t+1 . (iv) For all k ∈ {1, . . . , m i }, g k t is Fréchet differentiable at ût when g k t (û t ) = 0. (v) For all k ∈ {1, . . . , m i },g k t is lower semicontinuous and Gâteaux differentiable at ût when g k t (û t ) > 0. (vi) For all i ∈ {1, . . . , m e }, h i t is continuous on a neighborhood of ût and Fréchet differentiable at ût . (vii) span{Dh i t (û t ) : i ∈ {1, . . . , m e }} ∩ conv{D G g k t (û t ) : k ∈ I s t } = ∅, where I s t := {k ∈ {1, . . . , m i } : g k t (û t ) = 0}. (viii) Dh 1 t (û t ),. . . ., Dh me (û t ) are linearly independent.

  For all i ∈ {1, . . . , m e }, h i t is continuous on a neighborhood of ût and Fréchet differentiable at ût . (vi) span{Dh i t (û t ) : i ∈ {1, . . . , m e }} ∩ conv{D G g k t (û t ) : k ∈ I s t } = ∅, where I s t := {k ∈ {1, . . . , m i } : g k t (û t ) = 0}. (vii) Dh 1 t (û t ),. . . ., Dh me t (û t ) are linearly independent. (viii) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible.

	5.6. New Pontryagin Principles for Multiobjective Optimal Control
	Problems in Infinite-Horizon Setting	119
	(v)

  For all t ∈ N, p t+1 • D G,2 f t (x t , ût ) + Corollary 5.29. In the setting of Corollary 5.28, assume that assumptions (i-v) of Corollary 5.28 and the following assumption are fulfilled (vi') For all t ∈ N * , for all α, β ∈ {1, . . . , n},

	Chapter 5. Pontryagin Principles for Infinite-Horizon Discrete-Time
			Multiobjective Optimal Control Problems
	(e) � � j=1 θ j .D G,2 φ j t (x t , ût ) +	� m i k=1 λ k t D G g k t (û) = 0.
	∂f α t (xt,ût) ∂x α	> 0.	∂f α t (xt,ût) ∂x β	≥ 0 and for all α ∈ {1, . . . , n},

1 φ j t (x t , ût ).

  . . , θ � ∈ R + and p t+1 ∈ R n * + where t ∈ N which satisfy conclusions (a-d) of this corollary. Using conclusion (c) of Corollary 5.30, we have:∀t ∈ N * , p t = t+1 • D G,1 f t (x t , ût ).⇒ ∀t ∈ N * , p t+1 = (p t -

	� �	�	θ j .D G,1 φ j t (x t , ût )
	j=1

� + p � � j=1 θ j D G,1 φ j t (x t , ût )) • D G,1 f t (x t , ût ) -1 .

  Therefore, (p t ) t∈N * is bounded i.e. lim sup t→+∞ �p t � < +∞. Take lim sup both side of (5.34) with a notice that lim sup t→+∞ �p t � = lim sup t→∞ �p t+1 � and for all j ∈ {1, . . . , �}, lim sup t→+∞ ≤ lim sup t→+∞ |γ t | = 0 to obtain lim sup t→+∞ �p t � ≤ lim sup t→+∞ �p t � =⇒ lim sup t→+∞ �p t � ≤ M lim sup t→+∞ �p t � =⇒ (1 -M ) lim sup t→+∞ �p t � ≤ 0.

	� � �b	j t	� � � � � �A -1 t	� � �

  1 φ j t (x t , ût ) + �p t+1 � . �D G,1 f t (x t , ût )� =⇒ �p t � ≤ |γ t | � � j=1 θ j + M �p t+1 � =⇒ �p t � ≤ |γ t | + M �p t+1 �

			� � j=1	�	θ j .	� � �D G,1 φ j t (x t , ût ) � � � �
	since	� � j=1 θ j ≤ 1. Thus,
		=⇒ =⇒	� +∞ t=1 �p t � ≤	� +∞ t=1 |γ t | + M (1 -M ) � +∞ t=1 �p t � ≤ � +∞ t=1 �p t+1 � ≤ � +∞ t=1 �p t � ≤ � +∞ � +∞ t=1 |γ t | + M t=1 |γ t | � +∞ t=1 |γt| 1-M < +∞.	� +∞ t=1 �p t �

� + p t+1 • D G,1 f t (x t , ût ).

Then, we can assert that for all t ∈ N * , �p t � ≤

  1 φ j t (x t , ût ) t > 1 and for all t ∈ N, for all j ∈ {1, . . . , �}, ≤ γ t where (γ t ) t∈N = γ ∈ c 0 (N, R).

	5.6. New Pontryagin Principles for Multiobjective Optimal Control
	Problems in Infinite-Horizon Setting	129
	By induction, we have
	�p t+1 � ≤	1 s=1 ρ t � t	�p 1 � +	  � t s=1	 	1 k=s ρ t � t	j=1 � �	 � � s (x t , ût ) �D G,1 φ j � � � 	 
			≤	1 s=1 ρ s � t	+	 	� t s=1	 	1 k=s ρ k � t	j=1 � �	  � � s (x t , ût ) �D G,1 φ j � � �  
			= α t + β t ,
	where α t := 1 t �	ρs	> 0 and β t :=	t � s=1	1 � t	ρ k	� � j=1	� � D G,1 φ j s (x t , ût ) � � ≥ 0.
	s=1 Now we make an assumption that δ := inf k=s t∈N *
	� � �D G,1 φ j t (x t , ût ) � � � =	� � �b	j t	� � � From here,
	we have
		0 ≤ lim
										
										�
										� �  ,
	where ρ t = min
										
										� � �  .	(5.35)

1≤α≤n ∂f α t (x t , ût ) ∂x α t > 0.

In this inequality, we let T → +∞ and obtain

�p t+1 � ≤ 1 ρ t   �p t � + � � j=1 � � �D G,1 φ j t (x t , ût ) ρ t→+∞ α t = lim

  Then, (β t ) t∈N is also bounded. Thus, (p t ) t∈N * is bounded i.e. lim sup t→+∞ �p t � < +∞. Take lim sup both side of (5.35) with a notice that lim sup t→+∞ �p t � = lim sup t→∞ �p t+1 � and for all j ∈ {1, . . . , �}, lim sup t→+∞ ≤ lim sup t→+∞ γ t = 0 to obtain lim sup t→+∞ �p t � ≤ lim sup t→+∞1 ρt �p t � =⇒ lim sup t→+∞ �p t � ≤ 1 δ lim sup t→+∞ �p t � =⇒ (1 -1 δ

	� � �b	j t	� � �

  2 k ) Find (x, û) ∈ Adm k such that, there does not exist a process (x, u) ∈ Adm k , satisfying (x t , u t )φ j,t (x t , ût )) > 0.(Q 3 k ) Find (x, û) ∈ Adm k such that, there does not exist a process (x, u) ∈ Adm k satisfying

	limsup h→+∞ t liminf h � t=0 � � j=1 θ j (φ j h � � � h→+∞ t=0 j=1

  . , �}, which contradicts the assumption that (x, û) is a solution of Problem (Q 3 k ). Hence, (x, û) is a solution of Problem (P M 3 k ). If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ � = 0. Assume that (x, û) is not a solution of Problem (P M 3 � k ); thus there exists a process (x, u) ∈ Adm k such that for all j ∈ {1, . . . , �}, liminf (x t , u t )φ j t (x t , ût )) ≥

	liminf h→+∞ t (x So we have ( h � t=0 φ j
	liminf h→+∞	h � t=0	� � j=1 t (x h→+∞ θ j (φ j ( � h t=0 φ j t (x t , u t ) -	� h t=0 φ j t (x t , ût )) > 0 . So we have
	liminf h→+∞	h � t=0	� � j=1	θ j (φ j t � � j=0	θ j liminf h→+∞	t=0 h � (	φ j t (x t , u t ) -	t=0 h �	φ j t (x

t , u t ) -h � t=0 φ j t (x t , ût )) ≥ 0

and for some i ∈ {1, . . . , �},

liminf h→+∞ ( h � t=0 φ i,t (x t , u t ) -h � t=0 φ i,t (x t , ût )) > 0. t , u t )φ j t (x t , ût )) ≥ � � j=0 θ j liminf h→+∞ ( h � t=0 φ j t (x t , u t ) -h � t=0 φ j t (x t , ût )) > 0, t , ût )) > 0,

  Chapter 5. Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems by the definition of H * t and noticing that H t (θ, xt , ût , p t+1 ) = H * t (θ, xt , p t+1 ), then we have-H t (θ, x t , u t , p t+1 ) ≥ -H t (θ, x t , ût , p t+1 ) = -H * t (θ, x t , p t+1 ). Take T ∈ N * . From (5.40), one has * t (θ, xt , p t+1 ) -H * t (θ, x t , p t+1 ) -�p t+1 , xt+1x t+1 �) .The right hand side of this inequation can be written asRHS = H * 0 (x 0 , p 1 ) -H * 0 (x 0 , p 1 ) + * t (θ, xt , p t+1 ) -H * t (θ, x t , p t+1 ) -�p t , xtx t �) -�p T +1 , xT +1x T +1 � = * t (θ, xt , p t+1 ) -H * t (θ, x t , p t+1 ) -�D 1 H t (θ, xt , ût , p t+1 ), xtx t �) -�p T +1 , xT +1x T +1 �.Now using the assumptions on H * t and Lemma 5.50, for all t ∈ {1, . . . , T }, one can obtain H * t (θ, xt , p t+1 ) -H * t (θ, x t , p t+1 ) -�D 1 H t (θ, xt , ût , p t+1 ), xtx t � ≥ 0. Thus, we have

			T � t=0	� � j=1	θ j .	�	φ j t (x t , ût ) -φ j t (x t , u t )
	T	�
	�	�	θ j .
	t=0	j=1

� ≥ T � t=0 (H T � t=1 (H T � t=1 (H � φ j t (x t , ût )φ j t (x t , u t ) � ≥ �p T +1 , xT +1x T +1 �.

�

  +∞ t=0 φ j t (x t , ût ), J j (x, u) = � +∞ t=0 φ j t (x t , u t ) converge in R for all j ∈ {1, . . . , �}. By doing like in Theorem 5.49, we obtain (5.36), (5.37) and (5.38) hold. For all t ∈ N, we have (θ, xt , ût , p t+1 ) -�p t+1 , f t (x t , ût )�-H t (θ, x t , u t , p t+1 ) + �p t+1 , f t (x t , u t ) = H t (θ, xt , ût , p t+1 ) -H t (θ, x t , u t , p t+1 ) -�p t+1 , f t (x t , ût ) -xt+1 � -�p t+1 , xt+1 � +�p t+1 , f t (x t , u t )�.

� � j=1 θ j . � φ j t (x t , ût )φ j t (x t , u t ) � = H t

is bounded for all t ∈ N. Using (vii), (viii) and Lemma 4. [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF] we can assert that there exist an increasing function r 1 : N * → N * , µ t,j ∈ R (for all t ∈ N and for all j ∈ {1, . . . , m e }), λ t,k ∈ R (for all t ∈ N and for all k ∈ {1, . . . , m i }) such that lim Theorem 4.26. Let (x, û) be a solution of (P j e ) where j ∈ {1, 2, 3} and where the sets U t are defined in (4.27). We assume that the following assumptions are fulfilled for all t ∈ N.

(i) φ t is Fréchet differentiable at (x t , ût ).

(ii) f t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

(iii) For all k ∈ {1, . . . , m i }, g k t is Fréchet differentiable at ût when g k t (û t ) = 0. (iv) For all k ∈ {1, . . . , m i }, g k t is lower semicontinuous and Gâteaux differentiable at ût when g k t (û t ) > 0. (v) For all j ∈ {1, . . . , m e }, e j t is continuous on a neighborhood of ût and Fréchet differentiable at ût .

(vi) span{De j t (û t ) : j ∈ {1, . . . , m e }} ∩ conv{D G g k t (û t ) : k ∈ I s t } = ∅, where I s t := {k ∈ {1, . . . , m i } : g k t (û t ) = 0}. (vii) De 1 t (û t ), . . . , De me t (û t ) are linearly independent. (viii) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible.

Then under (i-viii) there exist λ 0 ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ t,1 ) t∈N ∈ R N , . . . , (µ t,me ) t∈N ∈ R N , (λ t,1 ) t∈N ∈ R N , . . . , (λ t,m i ) t∈N ∈ R N which satisfy the following conditions.

(NN) (λ 0 , p 1 ) � = (0, 0).

(Si) λ 0 ≥ 0, λ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }.

(S�) For all t ∈ N, for all k ∈ {1, . . . , m i }, λ k,t • g k t (û t ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D 1 φ t (x t , ût ).

(WM) For all t ∈ N,

The proof of this theorem is similar to the one of Theorem 4.25. The difference is the replacement of inequality constraints by equality constraints in the problem issued from the reduction to finite horizon. The consequence of this difference is the lost of the sign of the adjoint variable p t .

Chapter 5

Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems

Introduction

Multiobjective optimal control is an important branch of Optimal Control Theory. Multiobjective optimal control problems naturally arise, for example, in economics ( [START_REF] Dockner | Differential Games in Economics and Management Science[END_REF][START_REF] Dockner | Strategic Growth[END_REF][START_REF] Sorger | A Dynamic Common Property Resource Problem with Amenity Value and Extraction Costs[END_REF] and references therein), in aerospace, mechanical and chemical engineering ( [START_REF] Bhaskar | Multiobjective Optimization of an Industrial Wiped Film Pet Reactor[END_REF][START_REF] Bhaskar | Applications of Multiobjective Optimization in Chemical Engineering[END_REF] and references therein), in multiobjective control design ( [START_REF] Vroemen | Multiobjective Control: An Overview[END_REF] and references therein), in environmental studies ( [START_REF] Dockner | International Pollution Control: Cooperative versus Noncooperative Strategies[END_REF] and references therein)... Multiobjective optimal control was first studied by Zadeh [START_REF] Zadeh | Optimality and Non-scalar-valued Performance Criteria[END_REF]. Some works followed like Salukvadze [START_REF] Salukvadze | On the Existence of Solutions in Problems of Optimization under Vector-valued Criteria[END_REF], Yu and Leitmann [START_REF] Yu | Nondominated Decision and Cone Convexity in Dynamic Multicriteria Decision Problems[END_REF], Toivonen [START_REF] Toivonen | A Multiobjective Linear Quadratic Gaussian Control Problem[END_REF], Ishizuka and Shimizu [START_REF] Ishizuka | Necessary and Sufficient Conditions for the Efficient Solutions of Nondifferentiable Multiobjective Problems[END_REF], Khanh and Nuong [START_REF] Khanh | On Necessary Optimality Conditions in Vector Optimization Problems[END_REF], Yang and Teo [START_REF] Yang | Necessary Optimality Conditions for Bicriterion Discrete Optimal Control Problems[END_REF], Giannessi et al. [START_REF] Giannessi | Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, chapter On the Theory of Vector Optimization and Variational Inequalities[END_REF] and references therein who developed necessary and sufficient conditions as well as various methods for multiobjective optimal control. Multiple linear quadratic control problems can be found in Li [START_REF] Li | On General Multiple Linear-quadratic Control Problems[END_REF], Liao and Li [START_REF] Liao | Successive Method for General Multiple Linear-quadratic Control Problem in Discrete Time[END_REF], Liu [START_REF] Liu | Characterization of Nondominated Controls in Terms of Solutions of Weighting Problems[END_REF].

The first works on infinite-horizon single-objective optimal control problems are due to Pontryagin and his school [START_REF] Pontryagin | Méthodes Mathématiques des Processus Optimaux[END_REF] and Halkin [START_REF] Halkin | Necessary Conditions for Optimal Control Problems with Infinite Horizons[END_REF]. Other works followed as Carlson et al. [START_REF] Carlson | Infinite Horizon Optimal Control: Deterministic and Stochastic Systems[END_REF], Zaslavski [START_REF] Zaslavski | Turnpike Properties in the Calculus of Variations and Optimal Control[END_REF][START_REF] Zaslavski | Turnpike Phenomenon and Infinite Horizon Optimal Control[END_REF][START_REF] Zaslavski | Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems[END_REF], Blot and Chebbi [START_REF] Blot | Discrete Time Pontryagin Principle in Infinite Horizon[END_REF], Blot and Hayek [START_REF] Blot | Infinite-Horizon Pontryagin Principles with Constraints[END_REF][START_REF] Blot | Infinite Horizon Discrete Time Control Problems for Bounded Processes[END_REF][START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF], Blot [START_REF] Blot | Infinite-horizon Pontryagin Principles without Invertibility[END_REF][START_REF] Blot | A Pontryagin Principle for Infinite-horizon Problems under Constraints[END_REF], Blot et al. [START_REF] Blot | Pontryagin Principles for Bounded Discrete-time Processes[END_REF], Blot and Ngo [START_REF] Blot | Lightenings of Assumptions for Pontryagin Principles in Infinite Horizon and Discrete Time[END_REF][START_REF] Blot | Pontryagin Principles in Infinite Horizon in the Presence of Asymptotical Constraints[END_REF].

Infinite-horizon multiobjective optimal control problems in the continuous-time framework can be found in Bellaassali and Jourani [START_REF] Bellaassali | Necessary Optimality Conditions in Multiobjective Dynamic optimization[END_REF], in Zhu [START_REF] Zhu | Hamiltonian Necessary Conditions for a Multiobjective Optimal Control Problem with Endpoint Constraints[END_REF] and in Reddy and Engwerda [START_REF] Reddy | Necessary and Sufficient Conditions for Pareto Optimality in Infinite Horizon Cooperative Differential Games[END_REF] and references therein.

Infinite-horizon multiobjective optimal control problems in the discrete-time framework can be found in Hayek [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF][START_REF] Hayek | A Generalization of Mixed Problems with an Application to Multiobjective Optimal Control[END_REF] and in Blot and Hayek [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF] and in references therein.

In this chapter, necessary conditions of Pareto optimaltity under the form of Pontryagin principles for finite horizon and infinite-horizon multiobjective optimal control problems in discrete-time framework are studied. The aim of this chapter is to establish weak and strong maximum principles of Pontryagin for problems in the presence of constraints and under assumptions which are weaker than the usual ones. In this way, this chapter generalizes existing results for single-objective optimal control problems and for multiobjective optimal control problems with or without constraints. The general method we follow is to reduce the infinite-horizon problems into finite-horizon problems and Chapter 5. Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems

New Multiplier Rules for Multiobjective Problem

Let Ω be a nonempty open subset of R n , let φ : Ω → R � , ϑ : Ω → R � be mappings, let f i : Ω → R (when i ∈ {1, . . . , m}) be functions, let g i : Ω → R (when i ∈ {0, . . . , p}) and h i : Ω → R (when i ∈ {1, . . . , q}) be functions. With these elements, we consider the two following problems:

when x ∈ Ω and when ∀i ∈ {1, . . . , m}, f i (x) ≥ 0,

when x ∈ Ω when ∀i ∈ {1, . . . , p}, g i (x) ≥ 0 and when ∀i ∈ {1, . . . , q}, h i (x) = 0.

Notice that Problem (I m ) is a special case of Problem (M m ) when the equality constraints are omitted. The Pareto optimal solutions the above-mentioned problems are understood under the same meaning like those in the previous sections.

Before stating New Multiplier Rules for those multiobjective static optimization problems, we introduce the following lemmas: Lemma 5.4. If x is a Pareto optimal solution of Problem (I m ) then it is also a solution of the following problem Maximize φ k (x) when x ∈ Ω when ∀i ∈ {1, . . . , �}, i � = k, φ i (x) ≥ φ i (x) and when ∀i ∈ {1, . . . , m}, f i (x) ≥ 0,

for any given k ∈ {1, . . . , �}.

Proof. Let x be a Pareto optimal solution of Problem (I m ) and let k ∈ {1, . . . , �} be given. We will prove that x is also a solution of Problem (5.1) by contradiction.

If x is not a solution of Problem (5.1) then there exists x which is admissible for Problem (5.1) and φ k (x) > φ k (x). From the admissibility of x we know that ∀i ∈ {1, . . . , m}, f i (x) ≥ 0. Hence, x is admissible for Problem (I m ). Besides, for all i ∈ {1, . . . , �}, i � = k, we have φ i (x) ≥ φ i (x) (from the admissibility of x) and φ k (x) > φ k (x). Therefore, we can conclude that x is admissible for Problem (I m ) and φ(x) > φ(x) which is a contradiction. And so, x is an optimal solution of (5.1) for any given k ∈ {1, . . . , �}.

By a similar argument, we obtain the following lemma. Lemma 5.5. If x is a Pareto optimal solution of Problem (M m ) then it is also a solution of the following problem Maximize ϑ k (x) when x ∈ Ω when ∀i ∈ {1, . . . , �}, i � = k, ϑ i (x) ≥ ϑ i (x) when ∀i ∈ {1, . . . , p}, g i (x) ≥ 0, and when ∀i ∈ {1, . . . , q}, h i (x) = 0.

for any given k ∈ {1, . . . , �}.

New Multiplier Rules for Multiobjective Problem

99

Now we introduce the New Multiplier Rules based on the ones for single-objective static optimization problems of Blot that were introduced in previous chapter.

Theorem 5.6. Let x be a Pareto optimal solution of (I m ). We assume that the following assumptions are fulfilled.

(i) For all i ∈ {1, . . . , �}, φ i is Gâteaux differentiable at x. For all i ∈ {1, . . . , m}, f i is Gâteaux differentiable at x.

(ii) For all i ∈ {1, . . . , m}, f i is lower semicontinuous at x when f i (x) > 0.

Then there exist θ 1 , . . . , θ � , λ 1 , . . . , λ m ∈ R + such that the following conditions hold.

(a) (θ 1 , . . . , θ � , λ 1 , . . . , λ m ) � = (0, . . . , 0).

Proof. Let x be a Pareto optimal solution of (I m ). Using Lemma 5.4, x is also an optimal solution of Problem (5.1) for any given k ∈ {1, . . . , �}. Let k = 1 then x solves following problem.

Maximize φ 1 (x) when x ∈ Ω when ∀i ∈ {2, . . . , �}, φ i (x) ≥ φ i (x) and when ∀i ∈ {1, . . . , m}, f i (x) ≥ 0.

We set α i (x) := φ i (x)φ i (x) for all i ∈ {2, . . . , �}. Then the above-mentioned problem is rewritten as follows.

Maximize φ 1 (x) when x ∈ Ω when ∀i ∈ {2, . . . , �}, α i (x) ≥ 0 and when ∀i ∈ {1, . . . , m}, f i (x) ≥ 0

which has the same form of Problem (I) in Chapter 4. The assumptions of this theorem can be rewritten as follows.

For all i ∈ {2, . . . , �}, α i is Gâteaux differentiable at x.

For all i ∈ {1, . . . , m}, f i is Gâteaux differentiable at x.

(ii) For all i ∈ {1, . . . , m}, f i is lower semicontinuous at x when f i (x) > 0.

Now, all the assumptions of Theorem 4.4 are fulfilled (Here, notice that for i ∈ {2, . . . , �} we do not care about the lower semicontinuity of α i at x because α i (x) = φ i (x)φ i (x) = 0). Then we can apply Theorem 4.4 to obtain the multipliers θ 1 ∈ R + for the objective function φ 1 (x) and θ 2 , . . . , θ � , λ 1 , . . . , λ m ∈ R + for the inequality constraints which satisfy the following results.

(a) (θ 1 , . . . , θ � , λ 1 , . . . , λ m ) � = (0, . . . , 0).

For all i ∈ {2, . . . , �}, θ i α i (x) = 0 (this is obvious since α i (x) = 0).

(c)

Hence, we have obtained the desired results. Finally, we prove the existence of � D 1 G T (x T , ûT ; h T ), where h T = (h 1 , . . . , h T ) is an arbitrary vector from (R n ) T . Since G T (., u T ) is a vector mapping which contains T elements, we will prove the existence of � D 1 G T t (x T , ûT ; h T ) where t ∈ {0, . . . , T }. Now, each element G T t is a m-dimensional vector mappings and each of its element g k t , where k ∈ {1, . . . , m}, has directional derivative and it is concave. Therefore, � D 1 G T t (x T , ûT ; h T ) exists for all t ∈ {0, . . . , T } and it is concave. Thus, � D 1 G T (x T , ûT ; h T ) exists for all h T ∈ (R n ) T and it is concave. The lemma is proven. Lemma 5.18. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a weak Pareto optimal solution of Problem (F M T k ) where η and xT +1 are fixed vectors in R n . Assume that condition (ii) in Theorem 5.16 is fulfilled.

Then, for each u T ∈ U (T ), J T (., u T ), F (., u T ) and G(., u T ) are continuous at xT in any direction h.

We know that J T (., u T ), F (., u T ) and G(., u T ) are vector mappings. Their elements are elementary expressions that relate to φ j t (., u) , f t (., u) and g k t (., u) where t ∈ {0, . . . , T }, j ∈ {1, . . . , �} and k ∈ {1, . . . , m} which were defined before in Problem (5.11). Using the hypothesis of this lemma the following facts: (1) The composition of continuous mappings are continuous; (2) Linear mappings are continuous; (3) Bilinear mappings are continuous and ( 4) Classical functions of one variable are continuous, we can assert that J T (., u T ), F T (., u T ) and G T (., u T ) are continuous at xT in any direction h.

T ) ∈ U (T ) and 0 ≤ α ≤ 1. Let x T be in a neighborhood V T of xT . Then using (C3), for all t ∈ {0, . . . , T }, one can find u t ∈ U t such that the following convexity condition is satisfied.

We set u T := (u 0 , . . . , u T ) ∈ U (T ). The first inequality implies that for all j ∈ {1, . . . , �},

New Pontryagin Principles for Multiobjective Optimal Control

Problems in Finite-Horizon Setting 111

When k = i, the second inequality implies

and when t ∈ {1, . . . , T -1},

and,

When k = e by doing similarly, we obtain

Finally, the third inequality implies that for all t ∈ {1, . . . , T },

This inequality holds for each t ∈ {1, . . . , T }, thus,

The lemma is proven. Now we move to the proof of Theorem 5.16.

Proof. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a weak Pareto optimal solution of Problem (F M T i ), then (x T , ûT ) = (x 1 , . . . , xT , û0 , . . . , ûT ) is a weak Pareto optimal solution of Problem (5.11) with k = i. Now, since conditions (i-iii) are fulfilled, after Lemma 5.17, Lemma 5.18 and Lemma 5.19, all the conditions of Theorem 5.15 are satisfied. Then, there exist

T +1 not all zero, such that the conclusions of Theorem 5.15 hold and thus, we obtain the following statements

x t , ût ) = 0 for all t ∈ {0, . . . , T } and for all k ∈ {1, . . . , m} which gives (a). Thus, the theorem is proven.

(De) case

Now we state a strong Pontryagin principle for the multiobjective optimal control problems with (De) in finite horizon as follows that for all t ∈ {0, . . . , T }, the control sets are defined as in (5.10). We also assume that conditions (i,ii) in Theorem 5.16 hold for φ j t and g k t and moreover, assume that the following conditions holds: (iii') For all t ∈ {0, . . . , T }, for each x t in a neighborhood V t of xt such that x 0 = η, the following convexity condition is satisfied:

Then for all T ∈ N * , there exist

, not all zero which satisfy the conclusions (b,c) in Theorem 5.16 and the following one (a') For all t ∈ {0, . . . , T }, for all k ∈ {1, . . . , m}, µ k t .g k t (x t , ût ) = 0. Before proving this theorem, we introduce the following lemma.

Lemma 5.21. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a weak Pareto optimal solution of Problem (F M T e ) where η and xT +1 are fixed vectors in R n . Assume that condition (iv) in Theorem 5.20 is fulfilled. Then, for each u T ∈ U (T ), F T (., u T ) is continuously differentiable at xT . Proof. Let T ∈ N * and u T ∈ U (T ). We know that F T (., u T ) = (F T 0 (., u T ), ..., , F T T (., u T )). And so, to prove that F T (., u T ) is of class C 1 at xT , it is equivalent to prove that for all t = 0, ..., T , F T t (., u T ) is of class C 1 at xT . Following the definition of F T (., u T ) in Problem (5.11), we have that for all t = 1, ..., T -1,

]. Thus, for all t = 1, ..., T -1, F T t (., u T ) is continuously differentiable at xT since it is a composition of C 1 mappings under hypothesis (iv) of Theorem 4.9. A similar argument is used for

]. Thus, F T (., u T ) is continuously differentiable at xT . Now we prove the theorem.

Proof. Let T ∈ N * be given and (x 0 , . . . , xT +1 , û0 , . . . , ûT ) be a weak Pareto optimal solution of Problem (F M T e ), then (x T , ûT ) = (x 1 , . . . , xT , û0 , . . . , ûT ) is a weak Pareto optimal solution of Problem (5.11) with k = e. Now, under the assumptions of this theorem, all the conditions of Theorem 5.15 are fulfilled. Therefore, we can apply it to obtain

T +1 not all zero, such that conclusions (1,2) in the proof of Theorem 5.16 and the following one are fulfilled

(5.15)

The rest of the proof goes like in proof of Theorem 5.16 with only a small change that now, �

New Pontryagin Principles for Multiobjective Optimal

Control Problems in Infinite-Horizon Setting

Weak Pontryagin Principles

In this section, we give weak Pontryagin principles for the considered problems with an infinite-horizon setting. The difficulty is in the extraction of subsequences of multipliers Using assumption (viii) and (ix), after Lemma 4.23, we obtain that λ T k,1 = 0 for all k ∈ I s t , and consequently we have λ T k,1 = 0 for all k ∈ {1, . . . , m i }. Then, using assumption (ix), we obtain µ T i,1 = 0 for all i ∈ {1, . . . , m e } and thus, using (5.21) again, we have p T 2 = 0. Repeat this procedure for each t ∈ {2, . . . , T }, we obtain p T t+1 = 0, λ T k,t = 0 and µ T i,t = 0 for all t ∈ {0 . . . , T }, k ∈ {1, . . . , m i } and i ∈ {1, . . . , m e }. Thus, all the multipliers are zero which is a contradiction. And so, we have proven that (θ T 1 , . . . , θ T � , p T 1 , λ 1,T 0 , . . . λ m i ,T 0 , µ 1,T 0 , . . . , µ me,T 0 ) � = (0, . . . , 0). Since the set of the lists of multipliers of Problem is a cone, we can normalize the multipliers by setting

(5.23)

Since the values of the sequence (θ T 1 , . . . , θ T � , p T 1 , λ T 1,0 , . . . λ T m i ,0 , µ T 1,0 , . . . , µ T me,0 ) T ∈N * belong to the unit sphere of R � × R n * which is compact, using the Bolzano-Weierstrass theorem we can say that there exist an increasing function ϕ :

Now we will prove that for all t ∈ N * , the sequences T → p T t+1 , T → λ T k,t and T → µ T i,t are bounded for all k ∈ {1, . . . , m i } and for all i ∈ {1, . . . , m e }. After (5.23), it is clear that the sequences (θ T j ) T , (p T 1 ) T , (λ T k,0 ) T and (µ T i,0 ) T are bounded for all j ∈ {1, . . . , �}, k ∈ {1, . . . , m i } and i ∈ {1, . . . , m e }. When t = 1, using (5.21) and (5.22), we have

Since all the elements in the first line of the last equation are bounded, we can assert that

is also bounded. Using Lemma 4.24, we obtain (λ T k,1 ) T is bounded in R + for all k ∈ {1, . . . , m i } and (µ T i,1 ) T is bounded in R for all i ∈ {1, . . . , m e }. Using (5.20) with t = 1, we obtain that (p T 2 ) T is bounded in R n * . Repeat this procedure, for each t = 2, 3, . . . , by induction, we obtain that for all t ∈ N * , (λ T k,t ) T ≥t is bounded in R + for all k ∈ {1, . . . , m i }, (µ T i,t ) T ≥t is bounded in R for all i ∈ {1, . . . , m e } and (p T t+1 ) T ≥t+1 is bounded in R n * . Then using diagonal process of Cantor which is formulated in [START_REF] Blot | Infinite-Horizon Optimal Control in the Discrete-Time FrameWork[END_REF] (Theorem A.1, p.94), we know that there exist an increasing function δ :

= µ i,t for all i ∈ {1, . . . , m e }. And so we have built (θ 1 , . . . , θ � ) ∈ R � , sequences (p t ) t∈N * ∈ (R n * ) N * , (λ k,t ) t∈N ∈ R N for all k ∈ {1, . . . , m i } and (µ i,t ) t∈N ∈ R N for all i ∈ {1, . . . , m e } such that when T → +∞, from (5.23), (5.17), (5.18), (5.19) and (5.20) we obtain conclusions (a), (b), (c), (d) and (e), respectively (notice that for all the multipliers in these relations, we replace their upper index T by ϕ • δ(T )).

New Pontryagin Principles for Multiobjective Optimal Control Problems in Infinite-Horizon Setting 117

By a similar realization, we can propose a weak Pontryagin principle for Problem (P M j e ) when j ∈ {1, 2, 3}. The differences in assumptions and conclusions compared to the previous theorem are showed in the following theorem.

Theorem 5.23. Let (x, û) be a Pareto optimal solution of Problem (P M 1 e ) (respectively, solution of (P M 2 e ), (P M 3 e )) where the control sets are defined in (5.7). We assume that assumptions (i,iv-ix) of Theorem 5.22 are satisfied for all t ∈ N together with the following one.

(ii') f t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ).

Then there exist θ 1 , . . . , θ � ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ t,1 ) t∈N ∈ R N ,. . . ., (µ t,me ) t∈N ∈ R N , (λ t,1 ) t∈N ∈ R N ,. . . ., (λ t,m i ) t∈N ∈ R N which satisfy conclusions (a,d,e) of Theorem 5.22 and the following conditions (b) θ j ≥ 0 for all j ∈ {1, . . . , �}, λ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, . . . , m i }.

(c) For all t ∈ N, for all k ∈ {1, . . . , m i }, λ k,t • g k t (x t , ût ) = 0.

Proof. Let (x, û) be a Pareto optimal solution of Problem (P M j e ) when j ∈ {1, 2, 3}. Using Theorem 5.3, the restriction ((x 0 , . . . , xT +1 ), (û 0 , . . . , ûT )) is a Pareto optimal solution of Problem (F M T e ). Our assumptions imply that the assumptions of Theorem 5.10 in Section 4 are fulfilled and so we know that, for all T ∈ N * , there exist θ

. . , m i } and i ∈ {1, . . . , m e } which satisfy conditions (5.16,5.19,5.20) and the following ones:

∀t ∈ {0, . . . , T }, ∀k ∈ {1, . . . , m i }, λ T i,t g k t (x t , ût ) = 0.

(5.25)

The rest of this proof goes completely like that of the previous one in which relations (5.17) and (5.18) are replaced by relations (5.24) and (5.25), respectively.

Control sets that are independent of state variables

In this subsection, we will provide weak Pontryagin principles for the problems in which the control sets are independent of the state variable. Although these results are the corollaries of the theorems in Section 5.6.1, we give here simpler proofs for some of them thanks to the similar results for single-objective optimal control problems in infinite horizon setting in [START_REF] Blot | Lightenings of Assumptions for Pontryagin Principles in Infinite Horizon and Discrete Time[END_REF].

Firstly, we consider the case when both inequality and equality constraints appear in the problem as follows.

Then we have the following corollaries for multiobjective optimal control problems with (Di).

Corollary 5.24. Let (x, û) be a Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 i ), (P M 3 i )) where the sets U t are defined in (5.26). For all t ∈ N, we assume that the following assumptions are fulfilled (i) For all j ∈ {1, . . . , �}, φ j t is Fréchet differentiable at (x t , ût ). (ii) For all α ∈ {1, . . . , n}, f α t is Fréchet differentiable at (x t , ût ) when

Chapter 5. Pontryagin Principles for Infinite-Horizon Discrete-Time Multiobjective Optimal Control Problems Proof. Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 � i ), (P M 3 � i )). Using Theorem 5.3, for all T ∈ N * , we know that (x 0 , . . . , xT +1 , û0 , . . . , ûT +1 ) is a weak Pareto optimal solution of Problem (F M T i ). Under conditions (i-iv), after Theorem 5.16, for each T ∈ N * , there exist

, not all zero which satisfy the conclusions (a-c) of Theorem 5.16. From (b) of Theorem 5.16, using the assumption (iv), we know that for all t ∈ {1, . . . , T }, for all j ∈ {1, . . . , �} and for all k ∈ {1, . . . , m}, there exist ϕ j,T t ∈ ∂ 1 φ j t (x t , ût ) and

(5.29)

Assume that (θ T 1 , . . . , θ T � , p T 1 ) = (0, . . . , 0, 0) then from conclusion (c) of Theorem 5.16, for all t ∈ {0, . . . , T }, we have:

for all u ∈ U t . Now for each t ∈ {0, . . . , T }, take u = ǔt satisfying (v) then we have

Using (a) of Theorem 5.16, we can assert that for all t ∈ {0, . . . , T }, 0 ≥ � m k=1 µ k,T t g k t (x t , ǔt ) but from (ii), since g k t (x t , ǔt ) > 0, we must have µ k,T t = 0 for all t ∈ {0, . . . , T } and for all k ∈ {1, . . . , m}. Now apply those into the backward recursive equation (5.29), we have p T t+1 = p T t • D G,1 f t (x t , ût ) -1 for all t ∈ {1, . . . , T } with the initial condition p T 1 = 0. Therefore, it is obvious that p T t = 0 for all t ∈ {1, . . . , T + 1} and thus all the multipliers are zero which is a contradiction. Hence, (θ T 1 , . . . , θ T � , p T 1 ) � = (0, . . . , 0, 0). Since the set of the lists of multipliers is a cone, we can normalize the multipliers by setting

Then, for all t ∈ N * , we have proven that there exist ϕ j t ∈ ∂ 1 φ j t (x t , ût ) for all j ∈ {1, . . . , �} and ψ k t ∈ ∂ 1 g k t (x t , ût ) for all k ∈ {1, . . . , m} such that

+ since they are the limits of θ T , p T t and µ T t , respectively, when T → ∞. From conclusion (a) of Theorem 5.16 we obtain (b). From conclusion (c) of Theorem 5.16 we obtain (d).

Remark 5.34. Condition (v) is taken from Theorem 4.1 in [START_REF] Hayek | A Generalization of Mixed Problems with an Application to Multiobjective Optimal Control[END_REF] and it is useful to prove that all the multipliers are not simultaneously equal to zero.

In the previous result, we use the additional condition (iv) on f t in order to obtain the backward recursive equation which expresses p t+1 through p t . In fact, we can lighten the assumption on f t from Gâteaux differentiable down to directional differentiable. However, by doing that, it will be necessary to have the invertibility of all the mappings belonging to the partial subdifferential of f t with respect to x t . The next theorem will clarify this point.

Theorem 5.35. Under the settings of Theorem 5.33, we assume that conditions (ii,iii) and (v) of Theorem 5.33 hold together with the following assumption (i') For all t ∈ N * , for all j ∈ {1, . . . , �} and for all k ∈ {1, . . . , m}, φ j t (., ût ), g k t (., ût ) and f t (., ût ) have directional derivatives at xt which are concave.

Then there exist θ = (θ 1 , . . . , θ � ) ∈ R � + , p t+1 ∈ R n * + and µ t ∈ R m * + where t ∈ N such that all the conclusions of Theorem 5.33 hold.

Proof. Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 � i ), (P M 3 � i )). Using an analogous argument like in previous theorem, for each T ∈ N * , there exist

, not all zero which satisfy the conclusions (a-c) of Theorem 5. [START_REF] Blot | Pontryagin Principles for Bounded Discrete-time Processes[END_REF].

From conclusion (b) of Theorem 5.16 we know that for all t ∈ {1, . . . , T }, for all j ∈ {1, . . . , �} and for all k ∈ {1, . . . , m}, there exist ϕ j,T t

. By doing an similar procedure like the previous proof, we obtain (θ T 1 , . . . , θ T � , p T 1 ) � = (0, . . . , 0, 0) and we can normalize it as in (5.30). Now using the same argument like before, there exist an increasing function r :

The rest of the proof goes like that of the previous one.

Remark 5.36. Notice that when n = 1, i.e. the space of state variables is identified with R, then condition (iv') in the previous theorem can be rewritten as follows: For all t ∈ N * , 0 / ∈ ∂ 1 f t (x t , ût ). This implies for all t ∈ N * , xt is not an extremum of f t (., ût ).

In the special case when the sets of controls are defined by inequalities but only depend on control variables:

, we can take advantage of special assumption of Corollary 5.30 to state a strong principle for multiobjective optimal control problems with (Di) as in the following corollary.

Corollary 5.37. Let (x, û) be a weak Pareto optimal solution of Problem (P M 1 i ) (respectively, solution of (P M 2 � i ), (P M 3 � i )) when the control sets are defined as above. We assume that conditions (ii,iii) of Theorem 5.33 are satisfied. Moreover, assume that the following additional assumptions are fulfilled (i") For all t ∈ N * , for all j ∈ {1, . . . , �}, φ j t has directional derivative at (x t , ût ) and it is concave.

(iv") For all t ∈ N * , f t is Gâteaux differentiable at (x t , ût ) and D G,1 f t (x t , ût ) is invertible. (v') For all t ∈ N, for all k ∈ {1, . . . , m}, g k t is Gâteaux differentiable at ût and moreover, 0 / ∈ conv{D G g k t (û t ) : k ∈ I s t } where I s t := {k ∈ {1, . . . , m} : g k t (û t ) = 0}. Then there exist θ = (θ 1 , . . . , θ � ) ∈ R � + , p t+1 ∈ R n * + and µ t ∈ R m * + , where t ∈ N, such that all the conclusions of Theorem 5.33 hold.

Proof. The proof of this corollary is almost the same like that of Theorem 5.33, except for the step of proving (θ T 1 , . . . , θ T � , p T 1 ) � = (0, . . . , 0, 0). By contradiction, assume that (θ T 1 , . . . , θ T � , p T 1 ) = (0, . . . , 0, 0) then from conclusion (b) of Theorem 5.16, using assumption (iv"), we know that for all t ∈ {1, . . . , T }, for all j ∈ {1, . . . , �}, there exists ϕ j,T t ∈ ∂ 1 φ j t (x t , ût ) such that

(5.31)

Then, since (θ T 1 , . . . , θ T � , p T 1 ) = (0, . . . , 0, 0), p T t = 0 for all t ∈ {1, . . . , T + 1}. From conclusion (c) of Theorem 5.16, we have 0 ∈ ∂ 2 H t (x t , ût , θ, p t+1 , µ t ) which means that for all t ∈ {0, . . . , T }, for all j ∈ {1, . . . , �}, there exists α j,T t

(5.32)

From this equation we obtain

By the same argument as in the proof of Corollary 5.28, we have µ k,T t = 0 for all t ∈ {0, . . . , T }, for all k ∈ {1, . . . , m}. Thus, all the multipliers are zero which is a contradiction. Then (θ T 1 , . . . , θ T � , p T 1 ) � = (0, . . . , 0, 0). Remark 5.38. Theorem 5.33, Theorem 5.35 and Corollary 5.37 can be considered as generalizations of Theorem 4.3 in [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF] since now there exist the inequality constraints in the control sets. Besides, in these results, we use weaker assumptions compared to the condition of partial continuous differentiability in Theorem 4.3 in [START_REF] Hayek | Infinite Horizon Multiobjective Optimal Control Problems in the Discrete Time[END_REF]. (ii) ∀j = 1, . . . , �, ∀t ∈ N, (x, u) � → φ j t (x, u) and (x, u) � → f t (x, u) are Fréchet differentiable at (x t , ût ).

Then -If θ j > 0 for all j ∈ {1, . . . , �} then (x, û) is a Pareto optimal solution of Problem (P M 1 e ). -If θ j ≥ 0 for all j ∈ {1, . . . , �} and θ = (θ 1 , . . . , θ � ) � = 0 then (x, û) is a weak Pareto optimal solution of Problem (P M 1 e ). Proof. Let (x, û), (x, u) ∈ Dom e (J), i.e. xt ,

Then, using (5.36) and (5.38) we obtain

Therefore, using (5.36), we obtain, for all T ∈ N * ,

Using (iii), we have lim T →+∞ (-�p T +1 , xT +1x T +1 �) = 0, and then, from (5.39), doing

. Finally, using Lemma 5.47 obtain the conclusion of this theorem.

One can weaken the hypothesis of concavity of H t with respect to x t and u t and replace it by the concavity of H * t with respect to x t as the following theorem shows. Let U t be compact for all t ∈ N and let H * t (θ, x t , p t+1 ) = max ut∈Ut H t (θ, x t , u t , p t+1 ). The maximum is atteined since U t is compact. The following lemma is useful before stating the sufficient condition with the concavity of H * t .

Lemma 5.50. Let A be a convex subset of R n and γ a real concave function defined on A. Let (z) be an interior point of A. Let φ be a real function defined on a ball B(ẑ, δ) such that φ is differentiable at ẑ, φ(ẑ) = γ(ẑ) and φ(z) ≤ γ(z) for all z ∈ B(ẑ, δ).

Then for all z ∈ B(ẑ, δ), γ(ẑ)γ(z) ≥ Dφ(ẑ)(ẑz).

The proof of this lemma can be found in Chapter 5, §23 of [START_REF] Rockafellar | Convex Analysis[END_REF].

Theorem 5.51. Let (x, û) ∈ Dom e (J). Assume that there exist θ = (θ 1 , . . . , θ � ) ∈ (R � ), p = (p t ) t∈N * ∈ (R n * ) N * , not all zero such that the conditions (iii) and (iv) of Theorem 5.49 hold together with the following hypotheses (i') ∀t ∈ N, X t is convex and U t is compact.

(ii') ∀j = 1, . . . , �, ∀t ∈ N, ∀u ∈ U t , x � → φ j t (x, u) and x � → f t (x, u) are Fréchet differentiable at xt . (v') ∀t ∈ N, H t (θ, xt , ût , p t+1 ) = max ut∈Ut H t (θ, x t , u t , p t+1 ).

(vi') The function H * t is concave with respect to x t on X t for all t ∈ N. Then the conclusion of Theorem 5.49 holds.

Proof. Let (x, û), (x, u) ∈ Dom e (J), i.e. xt , x t ∈ X t , ût , u t ∈ U t , xt+1 = f t (x t , ût ), x t+1 = f t (x t , u t ) for all t ∈ N, x0 = x 0 = η and J j (x, û) = (5.40)

Notations

x the sequence (x t ) t∈N � p (N, U ), p ≥ 1 space of all sequences x from U N satisfying � +∞ t=0 |x t | p < +∞ � ∞ (N, U ) space of all sequences x from U N satisfying � +∞ t=0 |x t | < +∞ c(N, U ) space of all sequences x from U N which converge in U c 0 (N, U ) space of all sequences x from U N which converge to 0 c 00 (N, U ) space of all sequences x from U N satisfying x t = 0 except for finitely many indexes t A the closure of the set A intA the interior of the set A spanA the linear span of the vectors of set A convA the convex hull of the set A X * the conjugate space of X x * an element of the conjugate X * �x * , x� the value of the linear functional x * ∈ X * on the element x ∈ X dimL the dimension of the space L L(X, Y ) the space of continuous linear mappings of the space X into the space Y . When X = R n , Y = R m the mappings belonging to the space L(R n , R m ) can be identified with their matrices relative to the standard bases in R n and R m . KerΛ = {x : Λx = 0} the kernel of the linear operator Λ ImΛ = {y : y = Λx} the image of the linear operator Λ -→ DF (x; h) the directional derivative of the function F at the point x in the direction of the vector h D G F (x) the Gâteaux differential of the mapping F at the point x DF (x) the Fréchet differential of the mapping F at the point x ∂F (x) the subdifferential of the function F at the point x � D 1 F (x 1 , x 2 ; h) ( � D 2 F (x 1 , x 2 ; h)) the partial directional derivative of F w.r.t x 1 (x 2 ) at (x 1 , x 2 ) in the direction of h D G,1 F (x 1 , x 2 ) (D G,2 F (x 1 , x 2 )) the partial Gâteaux differential of F w.r.t x 1 (x 2 ) at (x 1 , x 2 ) D 1 F (x 1 , x 2 ) (D 2 F (x 1 , x 2 )) the partial Fréchet differential of F w.r.t x 1 (x 2 ) at (x 1 , x 2 ) ∂ 1 F (x 1 , x 2 ) (∂ 2 F (x 1 , x 2 )) the partial subdifferential of F w.r.t x 1 (x 2 ) at (x 1 , x 2 )