Lamsade Lamsade

Stefano Moretti
email: stefano.moretti@dauphine.fr

An application of the inverse power index problem to the design of the PSL electoral college

Keywords: apportionment, coalitional games, Banzhaf index, inverse power index problem

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Paris Sciences & Lettres (PSL) is a federal university that brings together 25 education and research institutions in Paris (to be hereafter denominated the PSL Institution Members or, simply, the Institutions ). Founded in 2010, the organisation of PSL is based on two main bodies 2 : the PSL Foundation for Scientic Cooper- ation (Fondation de Coopération scientique ), which is mainly responsible for the management of key actions of the PSL project (e.g., the recruitment of chairs of excellence, the development of strategic international partnerships, the implementation of innovative programs in research and training, etc.) and the Community of Universities and Institutions (Communauté d'Universités et Etablissements, also denominated ComUE ) which is responsible of the decisions concerning the training and graduation policy of PSL, and of other decisions over the common actions related with the educational and research community (e.g., the joint coordination of research policies and international projects for knowledge dissemination, the activation of digital actions, the implementation of joint strategies concerning students' life, etc.).

The ComUE is governed by an Administration Board (AB) (Conseil d'Administration ) of 30 members, assisted by an Academic Senate (AS) (Conseil Académique ) of 120 members with a consultative role. The members of the AB and the AS of the ComUE are representatives of the dierent Institutions but, for statutory reasons, only 16 Institutions (see Table 1) out of the 25 of PSL participate to the electoral process for their designation. Notice that the PSL Foundation and the ComUE itself are both represented as independent establishment within the AB and the AS of the ComUE.

The members of the AB and the AS are indirectly elected, among the candidates of the dierent Institutions, by a college of great electors designated by the Institutions according to their own statutes (usually, via general elections within their respective Institutions). Moreover, the members of the AB and the AS must be appointed in the respect of their professional categories (e.g., teachers, researchers, administrative and technical sta, etc.) according to the proportion specied in the Internal Regulations [START_REF] Psl | Internal Regulations of the ComUE (Reglement Interieur de la ComUE )[END_REF], addressing the general recommendations of the relevant national legislation. The same internal rules also specify that the great electors must have dierent amounts of say (weights) and should appoint candidates members for the AB and the AS using a simple majority mechanism.

The ComUE is also provided with a Steering Committee (SC) (Comité des Membres ) formed by the PSL institutions heads, the president and the vice-president of PSL, and the deans of the main departments. The objective of the SC is to ensure the proper functioning of PSL and to address the implementation of the guidelines provided by the Internal Regulations [START_REF] Psl | Internal Regulations of the ComUE (Reglement Interieur de la ComUE )[END_REF].

In this paper, we shortly introduce an approach based on coalitional games and on power indices to establish a fair distribution of the weights among the great electors within the PSL electoral college for the appointment of the AB and the AS members. In the game theoretic literature, several power indices (e.g. the well known Banzhaf index [START_REF] Banzhaf | Weighted voting doesn't work: A mathematical analysis[END_REF] and the Shapley-Shubik [START_REF] Shapley | A method for evaluating the distribution of power in a committee system[END_REF] index), have been proposed to evaluate the (a priori ) distribution of power on committee decisions (see for instance [START_REF] Felsenthal | The Measurement of Voting Power[END_REF] or [START_REF] Laruelle | Voting and collective decision-making: bargaining and power[END_REF] for a complete introduction to the literature). For instance, the Banzhaf index, that will play a central role in this paper, measures the ability of a voter to cast the decisive vote in winning coalitions when all coalitions have the same probability to form (see Section 2 for a more detailed discussion). Applications of power indices to existing electoral systems include the analysis of the United Nations Security Council [START_REF] Shapley | A method for evaluating the distribution of power in a committee system[END_REF][START_REF] Shapley | Simple games: an outline of the descriptive theory[END_REF], the International Monetary Fund [START_REF] Leech | Voting power in the governance of the International Monetary Fund[END_REF], the European Union Council of Ministers [START_REF] Algaba | The distribution of power in the European Constitution[END_REF], the Electoral College of the United States [START_REF] Owen | Evaluation of a presidential election game[END_REF] and many others.

Another very interesting and related issue is the inverse power index problem : if a vector of desired individual powers is given (for instance, as the outcome of a negotiation process), can we determine a voting method where a certain power index yields a good approximation of the desired vector? For instance, assuming that the Banzhaf index is used to evaluate the power of n voters, the inverse power index problem can be formulated as follows: given a vector P = (p 1 , . . . , p n ) of n real numbers and an appropriate metric to evaluate the distance between real-valued vectors, nd a voting system with n voters such that the distance (or error ) between the Banzhaf index computed on such a voting system and the vector P is smaller than a predened value (see [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Alon | The inverse Banzhaf problem[END_REF] for further details, and [START_REF] Kurz | On the inverse power index problem[END_REF] for a recent review of the existing methods to solve this problem).

According to the recommendation of the SC [START_REF]Reporting of the PSL Steering Committee meeting held on 21[END_REF], an ideal voting system should take into account the following three criteria: 1) all the Institutions should participate with a non-negligible power to the process of taking decisions in the AB and the AS of the ComUE ; 2) larger Institutions (in terms of number of sta employed) should play a more relevant role; 3) Institutions with a major academic oer should be fairly represented. Because of an important disproportion of students over the dierent Institutions (many Institutions have no student at all), the SC of the Co-mUE considers this last principle less relevant than the rst two, in order not to destabilise the global economy of the decision-making process within PSL.

Consequently, a triple majority system (with thresholds over the total number of Institutions, sta and students) has been adopted as an ideal voting system (see Section 3 for more details on the three respective quotas). Our goal, is then to solve an inverse Banzaf index problem [START_REF] Alon | The inverse Banzhaf problem[END_REF] with the objective to compute the weights of the great electors of PSL such that the Banzhaf index of the electoral college (based on a simple majority mechanism) is as close as possible to the Banzhaf index of the ideal triple majority system recommended by the SC [START_REF]Reporting of the PSL Steering Committee meeting held on 21[END_REF].

We start with some preliminary notations and denitions in the next section. Section 3 is devoted to the discussion of the method used to solve the inverse Banzhaf index problem in our setting. Section 4 concludes.

Preliminaries and notations

A coalitional game, also known as game in characteristic function form or Transferable Utility (TU) game, is a pair (N, v), where N denotes a nite set of players and v is the characteristic function, assigning to each S ⊆ N , a real number v(S) ∈ R, with v(∅) = 0 by convention. If the set N of players is xed, we identify a coalitional game (N, v) with the corresponding characteristic function v. A group of players S ⊆ N is called a coalition and v(S) is called the worth of this coalition.

A coalitional game (N, v) such that v(S) ∈ {0, 1} (i.e., the worth of every coalition is either 0 or 1) for each S ∈ 2 N \ {∅} and v(N ) = 1 is said a simple game.

The standard interpretation for these games is to consider coalitions as winning (v(S) = 1) or losing (v(S) = 0).

A particular class of simple games is the one of weighted majority games, where the players in N are associated to a vector of n = |N | weights (w 1 , . . . , w n ) and a majority quota q is given. A weighted majority game (N, v w ) on the weight w and the quota q is such that for each S ∈ 2 N \ {∅}: This paper mainly deals with a very famous index for coalitional games: the Banzhaf index [START_REF] Banzhaf | Weighted voting doesn't work: A mathematical analysis[END_REF]. The Banzhaf index of a game v is denoted by B(v) and can be obtained directly from the following relation:

v w (S) =    1 if i∈S w i > q, 0 otherwise.
B i (v) = S∈2 N \{i} 1 2 n-1 m i (S), (2)
where m i (S) = v(S ∪ {i}) -v(S) is the marginal contribution of i entering in S, for each i ∈ N and S ∈ 2 N \{i} . So, B i (v) is the expected marginal contribution of player i ∈ N in game v over all possible coalitions (not containing i) in which i can enter, and assuming that all coalitions not containing i have the same probability to form (i.e., probability equal to 1 2 n-1 ). The Banzhaf index [4] of coalitional games arising from voting situations (e.g., the simple majority games) has been widely used in the literature as a power index : the vector of values assigned by the Banzhaf index to the players of a (monotonic) simple game is often interpreted as the probability to cast a decisive vote or, more in general, as the ability of players to inuence a voting mechanism.

A simple game (N, v) is said monotonic if v(S) = 1 implies that v(T ) = 1 for each S, T ∈ 2 N with S ⊆ T . For instance, every weighted majority game is a monotonic simple game. For a monotonic simple game (N, v) and a player i ∈ N , a coalition S ∈ 2 N \ {∅} with i ∈ S and such that v(S) = 1 and v(S \ {i}) = 0 is said a swing for i and we denote by s i (v) the number of swings for player i in game v. It is easy to check that the normalized Banzhaf index on v can be dened for monotonic simple games as follows:

β i (v) = B i (v) i∈N B i (v) = s i (v) i∈N s i (v) , (3)
for each i ∈ N .

Example 1. Consider a parliament with three political parties 1, 2 and 3 with weights (i.e., seats) 47, 33 and 20, respectively, and suppose that to pass any motion, a qualied majority of two-third of the seats is needed. This situation can be described as a simple majority game (N, v w ) with N = {1, 2, 3}, w = (47, 33, 20) and q = 200 3 .

As a consequence, v w ({1, 2, 3}) = v w ({1, 2}) = v w ({1, 3}) = 1 ({1, 2, 3},{1, 2} and {1, 3} are winning coalitions) and v w ({1, 2, 3}) = 0 for all the remaining coalitions.

Notice that the number of swings for player 1 is s 1 (v w ) = 3 ({1, 2, 3},{1, 2} and {1, 3} are all swing for player 1), whereas for 2 and 3 we have

s 2 (v w ) = s 3 (v w ) = 1
({1, 2} is the only swing for 2 and {1, 3} is the only swing for 3). So the normalized

Banzhaf index is β(v w ) = ( 3 5 , 1 5 , 1 5
). The reader can immediately verify that the weight w i does not represent at all the eective power of players i ∈ N .

The proposed apportionment

In this section, we introduce the inverse Banzhaf index problem used for the apportionment of the weights of the great electors within the electoral college for the appointment of the AB members (the analysis for the AS is very similar and therefore omitted from these notes). To be more specic, we consider the inverse Banzhaf index problem as introduced in the next denition.

Denition 1 (Inverse Banzhaf index problem). Let N = {1, . . . , n} be a nite set of players, let λ ∈ [0, 1] be a predened quota (expressed as a fraction of the total weight) and let ∈ [0, 1] be the maximum tolerated error.

Given another vector P = (p 1 , ..., p n ) ∈ [0, 1] N of n real numbers on the interval [0, 1] and such that i∈N p i = 1 (i.e., P is normalized), nd a vector of non-negative integer numbers w = (w 1 , . . . , w n ) ∈ Z N with w i ≥ 0 for each i ∈ N and such that

i∈N |β i (v w ) -p i | < , (4)
where v w is the weighted majority game on N dened according to relation [START_REF] Algaba | The distribution of power in the European Constitution[END_REF] with weights w and a quota q = λ i∈N w i , and |β i (v w ) -p i | is the absolute value of the dierence between the normalized Banzhaf index β i (v w ) of player i ∈ N and in game v w and the ideal value p i .

Notice that in the above denition, it is asked to nd a vector of integer weights and the quota is provided a priori (for dierent formulations of the inverse Banzhaf index problem see [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Alon | The inverse Banzhaf problem[END_REF]).

According to the aforementioned ideal triple-majority rule recommended by the SC [START_REF]Reporting of the PSL Steering Committee meeting held on 21[END_REF] A simple game (N, v t ), with N = {1, . . . , 16} as the set of players representing the 16 Institutions Members of PSL, was dened according to the above triple majority mechanism. Precisely, let e i and s i , be, respectively, the number of employees and of students of each Institution i ∈ N , and take a coalition of Institutions S ⊆ N , S = {∅}, then we have that:

v t (S) =    1 if |S| > 8
and i∈S e i > 2 3 ρ and i∈S s i > 1 4 σ, 0 otherwise, [START_REF] Bilbao | Nice rules and voting power in the 27-European Union[END_REF] where ρ = i∈N e i is the total size of personnel aliated to the 16 Institutions of the ComUE and σ = i∈N s i is the total number of enrolled students. The normalized Banzhaf index β(v t ) of game v t has been computed according to relation [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF] and by means of the computer program introduced in [START_REF] Leech | Computer Algorithms for Voting Power Analysis[END_REF] and the Mathematica [START_REF]Mathematica, Version 10[END_REF] functions introduced in [START_REF] Bilbao | Nice rules and voting power in the 27-European Union[END_REF] (see the Appendix for the Mathematica instructions).

The vector β(v t ) is shown in the rst column of Table 3.

A weighted majority game (N, v w ) is also dened according to relation [START_REF] Algaba | The distribution of power in the European Constitution[END_REF], where the weights are computed solving the inverse Banzhaf index problem dened in Denition 1 with P = β(v t ) and λ = 1 2 (i.e., according to the guidelines of the Internal Regulations [START_REF] Psl | Internal Regulations of the ComUE (Reglement Interieur de la ComUE )[END_REF] we focus on a simple-majority systems with quota q = i∈N w i 2

). The total error introduced by the approximation of the Banzhaf index computed on the ideal game v t , is dened as follows:

δ(β(v w ), β(v t )) = i∈N |β i (v w ) -β i (v t )|,
which is bounded by the the vector in the problem of Denition 1 (for our purposes, we dene equal to 5%).

In order to nd a solution of the previously introduced instance of the inverse Banzhaf index problem (specically, the one introduced in Denition 1 with P := β(v t ), λ = 1 2 and = 0.05), we apply a trial-and-error procedure. At the initial step of the procedure, we dene an initial vector w = (w 1 , . . . , w n ) of n integer weights ordered according to β i (v t ), i.e. such that w i ≥ w j :⇔ β i (v t ) ≥ β j (v t ), for each i, j ∈ N . Using this vector of initial integer weights w, we compute the associated weighted majority game v w and the corresponding Banzhaf index β i (v w ) together with the initial total error δ(β(v w ), β(v t )). If this initial total error δ(β(v w ), β(v t )) is smaller than , then the procedure stops and the vector w is a solution of the inverse Banzhaf problem; otherwise, the vector of integer weights is adjusted as follows: each w i such that player i has the largest absolute dierence

|β i (v w ) -β i (v t )| is decreased by 1, if β i (v w ) -β i (v t
) > 0, and increased by 1, otherwise. A new weighted majority game v w associated to the adjusted weight vector is computed, together with the corresponding Banzhaf index β i (v w ) and the new total error δ(β(v w ), β(v t )). Again, if the new total error δ(β(v w ), β(v t )) is smaller than , then the procedure stops and the new vector w is a solution of the inverse Banzhaf problem; otherwise, a new adjustment of the weights occurs, and the process described above is repeated till nding a vector of integer such that the corresponding total error δ(β(v w ), β(v t )) ≤ .

Of course, using this iterative procedure, in general we cannot guarantee to obtain a solution of the inverse Banzhaf index problem as specied in Denition 1 (see [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF] for a more detailed discussion of the problem). In our case, however, the procedure yielded the vector of integer weights shown in the last column of Table 3 together with the Banzhaf value β i (v w ) of the corresponding weighted majority game (see also Figure 3 for a comparison between β i (v w ) and β i (v t ) of PSL members ordered according to their Banzhaf index in the AB). = 0.05). The vector of weights is shown in the last column (the quota is then xed at q = 77 2 ).

been introduced in [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF]. Using the implementation of Algorithm 1 in [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF] to solve our instance of the inverse Banzhaf index problem, we obtain the vector of weights shown in Table 10 in the Appendix (see the Appendix also for the Mathematica code adapted from [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF] to our problem). The vector of weights found with this algorithm is very close to the one we obtain by our procedure, and the total error with respect to the Banzhaf index of the ideal game is the same (δ(β(v w ), β(v t )) = 0.049).

Concluding remarks

In this report we shortly introduce an application of the inverse Banzhaf index problem to the apportionment of the weights of great electors in the electoral college for the appointment of the AB and AS members of the ComUE PSL [START_REF] Psl | Internal Regulations of the ComUE (Reglement Interieur de la ComUE )[END_REF]. Game theoretic power indices [START_REF] Felsenthal | The Measurement of Voting Power[END_REF][START_REF] Laruelle | Voting and collective decision-making: bargaining and power[END_REF] are often used in literature to describe the a priori power of players in existing voting systems [START_REF] Shapley | A method for evaluating the distribution of power in a committee system[END_REF][START_REF] Shapley | Simple games: an outline of the descriptive theory[END_REF][START_REF] Algaba | The distribution of power in the European Constitution[END_REF][START_REF] Owen | Evaluation of a presidential election game[END_REF]. The objective of this report is to show that the inverse power index problem [START_REF] Kurz | On the inverse power index problem[END_REF] can also provide the basis to prescribe a good electoral system. We are aware that many aspects of our analysis can be improved and embedded in a more general framework. For instance, the choice of the appropriate power index to evaluate the inuence of the players in the ideal game is a delicate issue that deserves further discussion. Another interesting issue that should be further investigated is related to the algorithmic aspects of the inverse power index problem (for more sophisticated algorithms see, for instance, [START_REF] Leech | Power indices as an aid to institutional design: the generalised apportionment problem[END_REF][START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Alon | The inverse Banzhaf problem[END_REF]). With respect to the inverse Banzhaf index problem, we would like to conclude with some open questions raised by [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF] that to the best of our knowledge are still without a denitive answer:

[3]: ...The questions we are interested in exploring are: is there a way of directly transforming a multiple majority weighted game into a weighted voting game with the same voting powers. Is there any loss of information in the transformation? Is it possible to identify and remove redundant weighted games from the multiple majority weighted voting game ?

In our opinion, the method we propose for the design of the electoral college of the ComUE should be seen as an attempt to provide an objective basis for the negotiation of the weights among the PSL Institutions. This is at least, we believe, how it has been perceived by the members of the SC of the ComUE [START_REF]Reporting of the PSL Steering Committee meeting held on 21[END_REF], where our proposition has been debated and nally approved under few modications during the PSL-SC meeting held in Paris on April 21st, 2015. institutions16PSL = {"ENS", "Université Paris-Dauphine", "Institut Curie", "Mines ParisTech", "CNSMDP", "Observatoire de Paris", "ESPCI", "ENSBA", "ENSAD", "Chimie ParisTech", "La Fémis", "CNRS", "ComUE PSL", "CNSAD", "INRIA", "FCS PSL"}; stapermil = {223, TableHeadings -> {institutions16PSL, {"Sta", "Students", "Membership", "Swing", "Banzhaf Normalized"}}] Table 4: Computation of the Banzhaf index β(v t ) of the ideal game v t (code adapted from [START_REF] Bilbao | Nice rules and voting power in the 27-European Union[END_REF]). TableHeadings -> {institutions16PSL, {"Weight", "Swing", "Banzhaf Normalized" }}]

Table 5: Computation of the Banzhaf index β(v w ) of the ideal game v w (code adapted from [START_REF] Bilbao | Nice rules and voting power in the 27-European Union[END_REF]).

0.0125

Table 8: The output of the instructions in Table 5.

Table 9: The output of the instructions in Table 6. Table 10: The weight of vector obtained using Algorithm 1 in [START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF] and using the code in Table 6.

Institution

10 Fondation

 10 des beaux-arts ENSBA 9 École supérieure de physique et de chimie industrielles de la ville de Paris ESPCI

Figure 1 :

 1 Figure 1: Comparison of the normalized Banzhaf index of the ideal triple-majority game v t (left column) and the Banzhaf index of the weighted majority game v w (right column) for each Institution of the ComUE.

weight16s = 15 ,

 15 [START_REF] Shapley | Simple games: an outline of the descriptive theory[END_REF][START_REF] Kurz | On the inverse power index problem[END_REF][START_REF] Kurz | On the inverse power index problem[END_REF][START_REF] Felsenthal | The Measurement of Voting Power[END_REF][START_REF] Felsenthal | The Measurement of Voting Power[END_REF][START_REF] Banzhaf | Weighted voting doesn't work: A mathematical analysis[END_REF][START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Aziz | Ecient algorithm for designing weighted voting games[END_REF][START_REF] Alon | The inverse Banzhaf problem[END_REF], 1, 1, 1, 1 swingssimple = banzhafPower[weight16s, 39] ban3aPSL17simple = SetPrecision[swingssimple/Plus @@ swingssimple, 3] TableForm[Transpose[{weight16s, swingssimple, ban3aPSL17simple}],

Table 1 :

 1 The 16 Institutions taking part to the electoral college for the appointment of the AB and AS members.

Table 2 :

 2 a subset of the 16 Institutions of the ComUE shown in Table1is a winning coalition if it satises the following three criteria: i) it is formed by a simple majority (> 50%) of the Institutions of the ComUE, 66%) of the total number of employees working in the Institutions of the ComUE ; iii) it represents at least one-fourth (> 25%) of the entire population of students enrolled in the Institutions of the ComUE ;The data concerning the number of employees and students of each Institution of the ComUE are shown in Table2. Relevant number of employees (e i ) and students (s i ) of each Institution i of the ComUE in the electoral college for the appointment of the AB members.

	all in favour;
	ii) it represents a qualied majority (>

Table 3 :

 3 A more sophisticated algorithm to solve the inverse Banzhaf index problem has Institutionβ(v t ) β(v w ) |β(v w ) -β(v t )| Weight Normalized Banzhaf indexof the ideal triple-majority game v t and of the weighted majority game v w . The absolute value of the dierences between the two Banzhaf indices |β(v w ) -β(v t )| is shown in the third column (notice that the total error is less than the maximum tolerated one

	Chimie ParisTech	0.039	0.037	0.002	3
	CNRS	0.023	0.024	0.001	2
	CNSAD	0.016	0.013	0.003	1
	CNSMDP	0.073	0.076	0.003	6
	ComUE PSL	0.018	0.013	0.005	1
	ENS	0.213	0.208	0.005	15
	ENSAD	0.041	0.037	0.004	3
	ENSBA	0.042	0.037	0.005	3
	ESPCI	0.049	0.050	0.001	4
	FCS PSL	0.011	0.013	0.002	1
	INRIA	0.013	0.013	0	1
	Institut Curie	0.088	0.090	0.002	7
	La Fémis	0.039	0.037	0.002	3
	Mines ParisTech	0.079	0.090	0.011	7
	Observatoire de Paris	0.071	0.076	0.005	6
	Université Paris-Dauphine	0.186	0.186	0	14
	Total	1	1	0.049	77

Table 6 :

 6 Weights computation using the algorithm introduced in [3] (code adapted

	from [3]

I am very grateful to Jean-Claude Petit for his valuable comments about the PSL University's organization, functions and duties and his useful suggestions to improve this paper.

For a more detailed description of the governance bodies of PSL, see the PSL ocial website: https://www.univ-psl.fr/fr


Loïc Bourdin
email: loic.bourdin@unilim.fr.

Note on Pontryagin maximum principle with running state constraints and smooth dynamics -Proof based on the Ekeland variational principle

In this note our aim is to give a proof of the Pontryagin maximum principle for a general optimal control problem with running state constraints and smooth dynamics. Our proof is based on the classical Ekeland variational principle.

The main result (and its proof) of this note are not new and are already well-known in the literature.

Main result

We first introduce some notations available throughout the paper. Let T > 0 be fixed. For every n ∈ N * and every 1 ≤ r ≤ +∞, we denote by • BF n := BF([0, T ], R n ) the classical space of bounded functions endowed with the classical uniform norm • ∞ ;

• C n := C([0, T ], R n ) the classical space of continuous functions endowed with • ∞ ;

• AC n := AC([0, T ], R n ) the classical space of absolutely continuous functions;

• BV n := BV([0, T ], R n ) the classical space of functions with bounded variations endowed with • BV n (see Appendix C for some recalls);

• L r n := L r ([0, T ], R n ) the classical Lebesgue space of r-integrable functions endowed with its usual norm • L r n . In the whole paper, when no confusion is possible, we remove the subscript n and we just denote by BF, C, AC, BV or L r . We denote by BF + 1 := BF([0, T ], R + ) and C + 1 := C([0, T ], R + ) where R + = [0, +∞).

Then, η ∈ BV n is said to be normalized if η(0) = 0 and η is left-continuous on (0, T ). The subspace of normalized functions with bounded variations will be denoted by NBV n .

Finally, the classical Lebesgue measure on [0, T ] will be denoted by λ.

A state constrained optimal control problem

Let m, n and j ∈ N * be fixed. In this paper we consider the optimal control problem (OCP) given by minimize Ψ(q(T )), subject to q ∈ AC n , u ∈ L ∞ m , q(t) = f (q(t), u(t), t), a.e. t ∈ [0, T ], q(0) = q 0 , u(t) ∈ Ω, a.e. t ∈ [0, T ], G i (q(t), t) ≤ 0, ∀t ∈ [0, T ], ∀i = 1, . . . , j, (OCP) where Ψ : R n → R is of class C 1 , where f : R n × R m × [0, T ] → R n is continuous and of class C 1 in its two first variables, where G = (G i ) i=1,...,j : R n × [0, T ] → R j is continuous and of class C 1 in its first variable, and where q 0 ∈ R n is fixed and Ω ⊂ R m is a nonempty closed subset.

Since Ψ, f and G are all regular, Problem (OCP) is said to be an optimal control problem with smooth dynamics. The last constraint corresponds to running state constraints.

We now introduce the Hamiltonian H : R n × R m × R n × [0, T ] → R associated to Problem (OCP) defined by H(q, u, p, t) := p, f (q, u, t) R n ×R n .

Pontryagin maximum principle

Our main result in this note is given by the following theorem.

Theorem 1 (Pontryagin maximum principle). Let (q * , u * ) ∈ AC n × L ∞ m be an optimal solution of Problem (OCP). There exists a nontrivial couple (ψ, η) where ψ ≥ 0 and η = (η i ) i=1,...,j ∈ NBV j such that u * (t) ∈ arg min v∈Ω H(q * (t), v, p(t), t)

for a.e. t ∈ [0, T ], where p ∈ BV n is the unique global solution of the backward linear Cauchy-Stieltjes problem given by -dp = ∂ 1 H(q * , u * , p, •) dt + j i=1 ∂ 1 G i (q * , •) dη i , on [0, T ], p(T ) = ψ∇Ψ(q * (T )).

In addition, it holds that η i is monotically increasing on [0, T ] and T 0 G i (q * (τ ), τ ) dη i (τ ) = 0, for every i = 1, . . . , n.

We refer to Appendix C for some recalls about functions of bounded variations and to Appendix D for details on linear Cauchy-Stieltjes problems.

Proof

This section is entirely devoted to the proof of Theorem 1. This proof is based on the classical Ekeland variational principle and is inspired from several references like [START_REF] Bonnans | Optimal control of state constrained integral equations[END_REF][START_REF] Li | Optimal control theory for infinite dimensional systems[END_REF].

Preliminaries

Let u ∈ L ∞ . In this preliminary section we focus on the forward (nonlinear) Cauchy problem (CP u ) given by q(t) = f (q(t), u(t), t), a.e. t ∈ [0, T ], q(0) = q 0 . (CP u )

A couple (q, I) is said to be a (local) solution of (CP u ) if 1. I ⊂ [0, T ] is an interval with nonempty interior such that min I = 0;

2. q : I → R n is absolutely continuous on I and q satisfies q(t) = f (q(t), u(t), t), a.e. t ∈ I, q(0) = q 0 , or equivalently, q : I → R n is continuous on I and q satisfies q(t) = q 0 + t 0 f (q(τ ), u(τ ), τ ) dτ, for every t ∈ I.

The couple (q, I) is said to be a global solution of (CP u ) if

I = [0, T ].
Let (q, I) and (q ′ , I ′ ) be two local solutions of (CP u ). We say that (q, I) is an extension of (q ′ , I ′ ) if I ′ ⊂ I and q(t) = q ′ (t) for every t ∈ I ′ . We say that (q, I) is a maximal solution of (CP u ) if it extends all other local solutions of (CP u ).

Some Cauchy-Lipschitz and continuous dependence results

Recall the two following classical Cauchy-Lipschitz (or Picard-Lindelöf) results.

Lemma 1. For every u ∈ L ∞ , there exists a unique maximal solution of (CP u ).

In the sequel we denote by (q(•, u), I(u)) the maximal solution of (CP u ) associated to u ∈ L ∞ .

Lemma 2. Let u ∈ L ∞ . If (q(•, u), I(u)) is not global (that is, T / ∈ I(u)) then I(u) is not closed and q(•, u) is unbounded on I(u).
In the sequel we denote by U ⊂ L ∞ the set of controls u ∈ L ∞ such that T ∈ I(u). A control u ∈ U is usually said to be admissible. For every u ∈ U and every R > u L ∞ , we introduce

K u,R := {(x, v, t) ∈ R n × R m × [0, T ] | x -q(t, u) R n ≤ 1 and v R m ≤ R}. From continuity of q(•, u) on [0, T ], K u,R is a compact subset of R n ×R m ×[0, T ]. As a consequence, f , ∂ 1 f and ∂ 2 f are bounded on K u,R by some L u,R ≥ 0 and it holds that f (x 2 , v 2 , t) -f (x 1 , v 1 , t) R n ≤ L u,R ( x 2 -x 1 R n + v 2 -v 1 R m ), (1)
for all (x 1 , v 1 , t), (x 2 , v 2 , t) ∈ K u,R . Proposition 1. Let u ∈ U . For every R > u L ∞ , there exists ν u,R > 0 such that E u,R := B L ∞ (0, R) ∩ B L 1 (u, ν u,R ) is contained in U . Moreover, for every u ′ ∈ E u,R , it holds that (q(τ, u ′ ), u ′ (τ ), τ ) ∈ K u,R for a.e. τ ∈ [0, T ]. Proof. Let R > u L ∞ and let ν u,R > 0 be such that ν u,R L u,R e T Lu,R < 1. Let u ′ ∈ E u,R .
Our aim is to prove that T ∈ I(u ′ ). By contradiction, let us assume that the set

A := {t ∈ I(u ′ ) | q(t, u ′ ) -q(t, u) R n > 1}
is not empty and let t 0 := inf A. From continuity, it holds that q(t 0 , u ′ )q(t 0 , u) R n ≥ 1.

Moreover, one has t 0 > 0 since q(0, u ′ ) = q(0, u) = q 0 . Hence, q(τ, u ′ )q(τ, u) R n ≤ 1 for every τ ∈ [0, t 0 ). Therefore (q(τ, u ′ ), u ′ (τ ), τ ) and (q(τ, u), u(τ ), τ ) belong to K u,R for a.e. τ ∈ [0, t 0 ). Since one has

q(t, u ′ ) -q(t, u) = t 0 f (q(τ, u ′ ), u ′ (τ ), τ ) -f (q(τ, u), u(τ ), τ ) dτ,
for every t ∈ I(u ′ ), it follows from (1) that

q(t, u ′ ) -q(t, u) R n ≤ L u,R t 0 u ′ (τ ) -u(τ ) R m dτ + L u,R t 0 q(τ, u ′ ) -q(τ, u) R n dτ,
for every t ∈ [0, t 0 ], which implies from the classical Gronwall lemma that

q(t, u ′ ) -q(t, u) R n ≤ L u,R e T Lu,R u ′ -u L 1 ≤ ν R L u,R e T Lu,R < 1,
for every t ∈ [0, t 0 ]. This raises a contradiction at t = t 0 . Therefore A is empty. We conclude that q(•, u ′ ) is bounded on I(u ′ ), then T ∈ I(u ′ ). Moreover, since A is empty, we also conclude that q(t, u ′ )q(t, u) R n ≤ 1 for every t ∈ [0, T ], and thus (q(τ,

u ′ ), u ′ (τ ), τ ) ∈ K u,R for a.e. τ ∈ [0, T ].
We conclude this section with the following continuous dependence result.

Proposition 2. Let u ∈ U and R > u L ∞ . The mapping F u,R : (E u,R , • L 1 ) -→ (C n , • ∞ ) u ′ -→ q(•, u ′ ) is C u,R -Lipschitz continuous for some C u,R ≥ 0.
Proof. Let u ′ and u ′′ be two elements of E u,R ⊂ U . We know that (q(τ, u ′′ ), u ′′ (τ ), τ ) and (q(τ, u ′ ), u ′ (τ ), τ ) are elements of K u,R for a.e. τ ∈ [0, T ]. Following the same arguments as in the previous proof, it follows that

q(t, u ′′ ) -q(t, u ′ ) R n ≤ L u,R e T Lu,R u ′′ -u ′ L 1 , for every t ∈ [0, T ]. The lemma follows with C u,R := L u,R e T Lu,R ≥ 0.

Implicit spike variations and a differentiable dependence result

Before introducing the concept of implicit spike variations, we first need to recall the following lemma (see [START_REF] Li | Optimal control theory for infinite dimensional systems[END_REF]Paragraph 3.2 p.143]). The proof is recalled in Appendix A.

Lemma 3. Let h ∈ L 1 n .
Then, for all ρ ∈ (0, 1), there exists a measurable set

Q ρ ⊂ [0, T ] such that λ(Q ρ ) = ρT and sup t∈[0,T ] t 0 1 - 1 ρ 1 Qρ (s) h(s) ds R n ≤ ρ.
Note that Q ρ depends on h.

Let u ∈ U and u ′ ∈ L ∞ . For every ρ ∈ [0, 1), we introduce the so-called implicit spike variation u(•, ρ) of u associated to u ′ as

u(τ, ρ) := u ′ (τ ) if τ ∈ Q ρ , u(τ ) if τ / ∈ Q ρ , for a.e. τ ∈ [0, T ], where Q ρ is defined in Lemma 3 associated to h u,u ′ ∈ L ∞ n ⊂ L 1 n defined by h u,u ′ (τ ) := f (q(τ, u), u ′ (τ ), τ ) -f (q(τ, u), u(τ ), τ ), for a.e. τ ∈ [0, T ]. 1
Finally, we introduce the so-called variation vector w(•, u, u ′ ) associated to (u, u ′ ) as the unique maximal solution, which is moreover global (see Appendix D.2), of the forward linear Cauchy problem given by ẇ(t) = ∂ 1 f (q(t, u), u(t), t) × w(t) + h u,u ′ (t), a.e. t ∈ [0, T ],

w(0) = 0.
Let us prove the following differentiability dependence result.

Proposition 3. The mapping F u,u ′ defined by

F u,u ′ (ρ) := q(•, u(•, ρ)) ∈ C n ,
for sufficiently small ρ ≥ 0, is Fréchet-differentiable at ρ = 0, with DF u,u ′ (0) = w(•, u, u ′ ). ). As a consequence, for sufficiently small ρ ≥ 0, u(•, ρ) ∈ E u,R ⊂ U and then F u,u ′ (ρ) is well-defined. Moreover, it follows from Proposition 2 that q(•, u(•, ρ))q(•, u) ∞ ≤ 2RT C u,R ρ, and consequently q(•, u(•, ρ)) uniformly converges on [0, T ] to q(•, u).

Proof. Let R := max( u L ∞ + 1, u ′ L ∞ ). Since λ(Q ρ ) = ρT (see Lemma 3), it holds that u(•, ρ) -u L 1 ≤ 2RT ρ for every ρ ∈ [0, 1
Let us assume by contradiction that F u,u ′ is not Fréchet-differentiable at ρ = 0 with DF u,u ′ (0) = w(•, u, u ′ ). Then, there exists ε > 0 et (ρ k ) k a positive sequence such that (ρ k ) k tends to zero and such that

F u,u ′ (ρ k ) -F u,u ′ (0) ρ k -w(•, u, u ′ ) ∞ ≥ ε
for all k ∈ N. In this proof, for the ease of notations, we denote by w := w(•, u, u ′ ), q := q(•, u) and by q k := q(•, u(•, ρ k )), u k := u(•, ρ k ) for every k ∈ N. Since the sequence (u k ) k converges to u in L 1 , we deduce from the (partial) converse of the classical Lebesgue dominated convergence theorem that there exists a subsequence (that we do not relabel) such that (u k ) k tends to u a.e. on [0, T ].

For every k ∈ N and every t ∈ [0, T ], we define z k (t) := q k (t)-q(t) ρ k w(t). From our assumption, it holds that z k ∞ ≥ ε for all k ∈ N. On the other hand, we have

z k (t) = t 0 f (q k (τ ), u k (τ ), τ ) -f (q(τ ), u(τ ), τ ) ρ k -∂ 1 f (q(τ ), u(τ ), τ ) × w(τ ) -h u,u ′ (τ ) dτ, that is, z k (t) = t 0 f (q k (τ ), u k (τ ), τ ) -f (q(τ ), u k (τ ), τ ) ρ k -∂ 1 f (q(τ ), u(τ ), τ ) × w(τ ) + f (q(τ ), u k (τ ), τ ) -f (q(τ ), u(τ ), τ ) ρ k -h u,u ′ (τ ) dτ,
for every t ∈ [0, T ]. From the classical Taylor formula with integral rest, we obtain that

z k (t) = t 0 ∂ 1 f (q(τ ), u(τ ), τ ) × z k (τ ) dτ + t 0 1 ρ k 1 Qρ k (τ ) -1 h u,u ′ (τ ) dτ + t 0 1 0 ∂ 1 f (q(τ ) + θ(q k (τ ) -q(τ )), u k (τ ), τ ) dθ -∂ 1 f (q(τ ), u(τ ), τ ) q k (τ ) -q(τ ) ρ k dτ,
for every t ∈ [0, T ]. Hence, from Lemma 3, we get that

z k (t) R n ≤ ρ k + 2RT C u,R κ k + L u,R t 0 z k (τ ) R n dτ,
for every t ∈ [0, T ], where

κ k := T 0 1 0 ∂ 1 f (q(τ ) + θ(q k (τ ) -q(τ )), u k (τ ), τ ) -∂ 1 f (q(τ ), u(τ ), τ ) R n ×R n dθdτ.
From the continuity and the boundedness of ∂ 1 f on K u,R , since (u k ) k tends to u a.e. on [0, T ] and from the classical Lebesgue dominated convergence theorem, one can easily prove that (κ k ) k tends to zero. Finally, from the classical Gronwall lemma, we obtain that

z k (t) R n ≤ (ρ k + 2RT C u,R κ k )e T Lu,R for every t ∈ [0, T ].
This raises a contradiction with the inequality z k ∞ ≥ ε for all k ∈ N. The proof is complete.

Application of the Ekeland variational principle

Let us introduce g = (g i ) i=1,...,j : C n → C j the application defined by g(q) := G(q, •) for every q ∈ C n , and let S be the nonempty closed convex cone of C j defined by S := C([0, T ], (R -) j ). Thus the running state constraints in Problem (OCP) can equivalently be replaced by

g(q) ∈ S.
Note that g is of class C 1 with Dg(q)(w) = ∂ 1 G(q, •) × w for every q, w ∈ C n , and that S has a nonempty interior.

Since (C j , • ∞ ) is a separable Banach space, we endow C j with an equivalent norm • Cj such that the associated dual norm • C * j is strictly convex (see Proposition 4 in Appendix B.1). Then, we denote by d S the 1-Lipschitz continuous distance function to S defined by d S (q) := inf z∈S qz Cj for every q ∈ C j . Since the dual norm • C * j is strictly convex, we know that d S is strictly Hadamard-differentiable on C j \S with Dd S (q) C * j = 1 for every q ∈ C j \S (see Proposition 5 in Appendix B.2). As a consequence, d 2 S is also strictly Hadamard-differentiable on C j \S with Dd 2 S (q) = 2d S (q)Dd S (q) for every q ∈ C j \S. We also recall that d 2 S is Fréchet-differentiable on S with Dd 2 S (q) = 0 for every q ∈ S (see Remark 6 in Appendix B.2).

In the whole section, let q(•, u * ) ∈ AC and u * ∈ L ∞ be an optimal solution of Problem (OCP). Let (R ℓ ) ℓ be a positive sequence such that R ℓ > u * L ∞ for every ℓ ∈ N and such that lim ℓ R ℓ = +∞. Let (ε k ) k be a positive sequence such that lim k ε k = 0. For every ℓ, k ∈ N, we consider the penalized functional given by

J ℓ k : E Ω u * ,R ℓ -→ R + * u -→ Ψ(q(T, u)) -Ψ(q(T, u * )) + ε k + 2 + d 2 S g q(•, u) , where E Ω u * ,R ℓ := {u ∈ E u * ,R ℓ | u(τ ) ∈ Ω for a.e. τ ∈ [0, T ]}. Note that J ℓ k is a positive functional because of the optimality of u * . We endow E Ω u * ,R ℓ with the classical norm • L 1 . Since Ω is a nonempty closed subset of R m , it follows from the (partial) con- verse of the classical Lebesgue dominated convergence theorem that (E Ω u * ,R ℓ , • L 1 ) is a nonempty closed subset of (L 1 , • L 1 ) and consequently (E Ω u * ,R ℓ , • L 1
) is a complete metric space. Moreover, from the continuities of Ψ, F u * ,R ℓ (see Proposition 2), d S and of g, one can easily see that

J ℓ k is continuous on (E Ω u * ,R ℓ , • L 1 ).
Moreover it holds that J ℓ k (u * ) = ε k . As a consequence, from the classical Ekeland variational principle, we conclude that for every ℓ, k ∈ N, there exists

u ℓ k ∈ E Ω u * ,R ℓ such that u ℓ k -u * L 1 ≤ √ ε k and - √ ε k u -u ℓ k L 1 ≤ J ℓ k (u) -J ℓ k (u ℓ k ), (2)
for all u ∈ E Ω u * ,R ℓ . In particular, for a fixed ℓ ∈ N, note that the sequence (u ℓ k ) k converges to u * in L 1 and consequently, the sequence (q(•,

u ℓ k )) k uniformly converges on [0, T ] to q(•, u * ) (see Proposition 2).
For every ℓ, k ∈ N, we introduce

ψ ℓ k := 1 J ℓ k (u ℓ k ) Ψ(q(T, u ℓ k )) -Ψ(q(T, u * )) + ε k + ≥ 0,
and

ϕ ℓ k :=        1 J ℓ k (u ℓ k ) d S g(q(•, u ℓ k )) Dd S g q(•, u ℓ k ) ∈ C * j if g(q(•, u ℓ k )) / ∈ S, 0 ∈ C * j if g(q(•, u ℓ k )) ∈ S. In particular it holds that |ψ ℓ k | 2 + ϕ ℓ k 2 C * j = 1 for every ℓ, k ∈ N.
In the sequel our aim is to derive some important inequalities from Inequality (2) using implicit spike variations on u ℓ k .

Remark 1. In this remark (and in Remarks 2 and 3), our aim is to provide two crucial inequalities satisfied by ϕ ℓ k . In the case g(q(•, u ℓ k )) / ∈ S, recall that Dd S (g(q(•, u ℓ k ))) belongs to the subdifferential of d S at the point g(q(•, u ℓ k )). As a consequence, in both cases g(q(•, u ℓ k )) / ∈ S and g(q(•,

u ℓ k )) ∈ S, it holds that ϕ ℓ k , z -g(q(•, u ℓ k )) C * j ×Cj ≤ 0, (3)
for every z ∈ S. Since S has a nonempty interior, there exists ξ ∈ S and δ > 0 such that ξ + δz ∈ S for every z ∈ B (Cj , • C j ) (0, 1). As a consequence, we obtain that

δ ϕ ℓ k , z C * j ×Cj ≤ ϕ ℓ k , g(q(•, u ℓ k )) -ξ C * j ×Cj , for every z ∈ B (Cj , • C j ) (0, 1). We deduce that δ ϕ ℓ k C * j = δ 1 -|ψ ℓ k | 2 ≤ ϕ ℓ k , g(q(•, u ℓ k )) -ξ C * j ×Cj . (4)

First inequality depending on ℓ fixed

In this section, we fix ℓ ∈ N. Recall that the sequence (u ℓ k ) k converges to u * in L 1 . Using compactness arguments, we infer the existence of a subsequence of (ε k ) k (that we do not relabel) 2 such that (u ℓ k ) k converges to u * a.e. on [0, T ], (ψ ℓ k ) k converges to some ψ ℓ ≥ 0 and (ϕ ℓ k ) k weakly* converges to some ϕ ℓ ∈ C * j . In particular, it holds that

|ψ ℓ | 2 + ϕ ℓ 2 C * j ≤ 1.
In the whole section, for the ease of notations, we denote by

q ℓ k := q(•, u ℓ k ) for every k ∈ N. Let u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω ∩ B R m (0, R ℓ ) for a.e. τ ∈ [0, T ].
For every ρ ∈ [0, 1), we consider the implicit spike variation

u ℓ k (τ, ρ) := u ′ (τ ) if τ ∈ Q ρ , u ℓ k (τ ) if τ / ∈ Q ρ , for a.e. τ ∈ [0, T ], where Q ρ is defined in Lemma 3 associated to h u ℓ k ,u ′ ∈ L ∞ n ⊂ L 1 n defined by h u ℓ k ,u ′ (τ ) := f (q ℓ k (τ ), u ′ (τ ), τ ) -f (q ℓ k (τ ), u ℓ k (τ ), τ ), for a.e. τ ∈ [0, T ]. 3 First of all, note that u ℓ k (•, ρ) -u * L 1 ≤ u ℓ k (•, ρ) -u ℓ k L 1 + u ℓ k -u * L 1 ≤ 2R ℓ T ρ + √ ε k < ν u * ,
R ℓ for sufficiently small ρ and sufficiently large k and then

u ℓ k (•, ρ) ∈ E Ω u * ,R ℓ .
For such a sufficiently small ρ and sufficiently large k, we apply Inequality (2) with u = u ℓ k (•, ρ) and we obtain that

-2R ℓ T √ ε k ≤ J ℓ k (u ℓ k (•, ρ)) -J ℓ k (u ℓ k ) ρ = J ℓ k (u ℓ k (•, ρ)) 2 -J ℓ k (u ℓ k ) 2 ρ × 1 J ℓ k (u ℓ k (•, ρ)) + J ℓ k (u ℓ k )
.

From the continuity of J ℓ k , we get that lim ρ→0

J ℓ k (u ℓ k (•, ρ)) + J ℓ k (u ℓ k ) = 2J ℓ k (u ℓ k ). From the differen- tiabilities of the application x → (x + ) 2 for x ∈ R, of Ψ, of g, of d 2 S and of F u ℓ k ,u ′ (see Proposition 3), we obtain that lim ρ→0 J ℓ k (u ℓ k (•, ρ)) 2 -J ℓ k (u ℓ k ) 2 ρ = 2 Ψ(q ℓ k (T )) -Ψ(q(T, u * )) + ε k + ∇Ψ(q ℓ k (T )), w(T, u ℓ k , u ′ ) R n ×R n + 2d S (g(q ℓ k ))Dd S (g(q ℓ k )), Dg(q ℓ k )(w(•, u ℓ k , u ′ )) C * j ×Cj
, with the convention that the second term is zero if g(q ℓ k ) ∈ S. Finally, we have obtained that

-2R ℓ T √ ε k ≤ ψ ℓ k ∇Ψ(q ℓ k (T )), w(T, u ℓ k , u ′ ) R n ×R n + ϕ ℓ k , Dg(q ℓ k )(w(•, u ℓ k , u ′ )) C * j ×Cj . (5)
To conclude this section, we need the following result. 4Lemma 4. The sequence (w(•, u ℓ k , u ′ )) k uniformly converges on [0, T ] to w(•, u * , u ′ ). Proof. In this proof, for the ease of notations, we denote by q * := q(•, u * ), w := w(•, u * , u ′ ) and by Proposition 1) and (q ℓ k (τ ), u ′ (τ ), τ ) ∈ K u * ,R ℓ for a.e. τ ∈ [0, T ] and recall that f and ∂ 1 f are bounded on K u * ,R ℓ by L u * ,R ℓ ≥ 0. Recall also that (u ℓ k ) k tends to u * a.e. on [0, T ]. Finally, using similar arguments than in the proof of Proposition 3 and the classical Gronwall lemma, one can easily conclude the proof.

w k := w(•, u ℓ k , u ′ ) for all k ∈ N. It holds that w k (t) -w(t) = t 0 ∂ 1 f (q ℓ k (τ ), u ℓ k (τ ), τ ) × w k (τ ) + h u ℓ k ,u ′ (τ ) -∂ 1 f (q * (τ ), u * (τ ), τ ) × w(τ ) -h u * ,u ′ (τ ) dτ, that is, w k (t) -w(t) = t 0 ∂ 1 f (q ℓ k (τ ), u ℓ k (τ ), τ ) × (w k (τ ) -w(τ )) dτ + t 0 h u ℓ k ,u ′ (τ ) -h u * ,u ′ (τ ) dτ + t 0 ∂ 1 f (q ℓ k (τ ), u ℓ k (τ ), τ ) -∂ 1 f (q * (τ ), u * (τ ), τ ) × w(τ ) dτ, for every t ∈ [0, T ]. Recall that u ℓ k ∈ E u * ,R ℓ and u ′ ∈ B L ∞ (0, R ℓ ), then (q ℓ k (τ ), u ℓ k (τ ), τ ) ∈ K u * ,R ℓ (see
Using the above lemma and the C 1 -regularity of Ψ and g, by letting k tend to +∞ in Inequality [START_REF] Burk | A garden of integrals[END_REF], we obtain that

0 ≤ ψ ℓ ∇Ψ(q(T, u * )), w(T, u * , u ′ ) R n ×R n + ϕ ℓ , Dg(q(•, u * ))(w(•, u * , u ′ )) C * j ×Cj . ( 6
)
Remark 2. Letting k tend to +∞ in Remark 1, one can easily obtain the two following crucial inequalities:

ϕ ℓ , z -g(q(•, u * )) C * j ×Cj ≤ 0, (7)
for every z ∈ S, and

δ 1 -|ψ ℓ | 2 ≤ ϕ ℓ , g(q(•, u * )) -ξ C * j ×Cj . (8)

Second inequality independent of ℓ

In the previous section, we have obtained Inequality (6) that is valid for a fixed ℓ ∈ N and for every u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω ∩ B R m (0, R ℓ ). Our aim in this section is to remove the dependence in R ℓ (in order to cover the case where Ω is unbounded).

Since |ψ ℓ | 2 + ϕ ℓ 2 C * j ≤ 1 for every ℓ ∈ N and from compactness arguments, we infer the existence of a subsequence of (R ℓ ) ℓ (that we do not relabel) such that (ψ ℓ ) ℓ converges to some ψ ≥ 0 and (ϕ ℓ ) ℓ weakly* converges to some ϕ ∈ C * j .

Let u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω for a.e. τ ∈ [0, T ]. Let ℓ ∈ N be sufficiently large in order to have

R ℓ > u ′ L ∞ . From Inequality (6), it holds that 0 ≤ ψ ℓ ∇Ψ(q(T, u * )), w(T, u * , u ′ ) R n ×R n + ϕ ℓ , Dg(q(•, u * ))(w(•, u * , u ′ )) C * j ×Cj
.

Letting ℓ tend to +∞, we prove that

0 ≤ ψ ∇Ψ(q(T, u * )), w(T, u * , u ′ ) R n ×R n + ϕ, Dg(q(•, u * ))(w(•, u * , u ′ )) C * j ×Cj , (9)
for every u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω for a.e. τ ∈ [0, T ].

Remark 3. Letting ℓ tend to +∞ in Remark 2, one can easily obtain the two following crucial inequalities:

ϕ, z -g(q(•, u * )) C * j ×Cj ≤ 0, (10)
for every z ∈ S, and

δ 1 -|ψ| 2 ≤ ϕ, g(q(•, u * )) -ξ C * j ×Cj . (11)
Inequality [START_REF] Li | Optimal control theory for infinite dimensional systems[END_REF] proves that the couple (ψ, ϕ) is not trivial.

2.3 Introduction of the adjoint vector p

Introduction of η

Let us denote by ϕ = (ϕ i ) i=1,...,j where ϕ i ∈ C * 1 and let us apply the classical Riesz theorem (see Proposition 6 in Appendix C.1). For every i = 1, . . . , j, there exists a unique η i ∈ NBV 1 such that

ϕ i , z C * 1 ×C1 = T 0 z(τ ) dη i (τ ),
for every z ∈ C 1 . Recall that ϕ i = 0 if and only if η i = 0. As a consequence, from Remark 3, the couple (ψ, η) is not trivial, where η := (η i ) i=1,...,j ∈ NBV j .

Taking z = g 1 (q(•, u * )), . . . , g i-1 (q(•, u * )), 0, g i+1 (q(•, u * )), . . . , g j (q(•, u * )) ∈ S and

z = g 1 (q(•, u * )), . . . , g i-1 (q(•, u * )), 2g i (q(•, u * )), g i+1 (q(•, u * )), . . . , g j (q(•, u * )) ∈ S
in Inequality [START_REF] Limaye | Functional analysis[END_REF], we obtain that ϕ i , g i (q(•, u * ))

C * 1 ×C1 = 0, that is, T 0 G i (q(τ, u * ), τ ) dη i (τ ) = 0,
for every i = 1, . . . , n.

Moreover, it follows that ϕ i , z C * 1 ×C1 ≥ 0 for every z ∈ C + 1 . From the classical Riesz theorem (see Proposition 6 in Appendix C.1), we deduce that η i is monotically increasing on [0, T ] for every i = 1, . . . , j.

Definition of p

Using notations introduced in Appendix C.2, one has

ϕ, z C * j ×Cj = T 0 z(τ ), dη(τ )
for every z ∈ C j . From Inequality (9) and since Dg(q)(w) = ∂ 1 G(q, •) × w for every q, w ∈ C n , we have proved that

0 ≤ ψ ∇Ψ(q(T, u * )), w(T, u * , u ′ ) R n ×R n + T 0 ∂ 1 G(q(τ, u * ), τ ) × w(τ, u * , u ′ ), dη(τ ) , (12)
for every u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω for a.e. τ ∈ [0, T ].

Let Z(•, •) be the state-transition matrix associated to 12), using first the Fubini-type formula (18) and then Equality (17), one can obtain that

∂ 1 f (q(•, u * ), u * , •) ∈ L ∞ ([0, T ], R n,n ) (
0 ≤ T 0 h u * ,u ′ (s), ψZ(T, s) ⊤ × ∇Ψ(q(T, u * )) + T s Z(τ, s) ⊤ × ∂ 1 G(q(τ, u * ), τ ) ⊤ × dη(τ ) R n ×R n ds, (13)
for every u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω for a.e. τ ∈ [0, T ].

Let p ∈ BV n be the unique global solution of the backward linear Cauchy-Stieltjes problem given by

-dp = ∂ 1 f (q(•, u * ), u * , •) ⊤ × p dt + j i=1 ∂ 1 G i (q(•, u * ), •) dη i , on [0, T ], p(T ) = ψ∇Ψ(q(T, u * )).
We refer to Proposition 9 in Appendix D.3 for the existence and uniqueness of p. Note that p is independent of u ′ . From the Duhamel-type formula (see Proposition 10 in Appendix D.3), it holds that

p(s) = ψZ(T, s) ⊤ × ∇Ψ(q(T, u * )) + T s Z(τ, s) ⊤ × ∂ 1 G(q(τ, u * ), τ ) ⊤ × dη(τ ) ∈ R n ,
for every s ∈ [0, T ]. It follows from the above expression of p and from Inequality (13) that T 0 h u * ,u ′ (s), p(s) R n ×R n ds ≥ 0, that is, T 0 H(q(s, u * ), u ′ (s), p(s), s) -H(q(s, u * ), u * (s), p(s), s) ds ≥ 0, [START_REF] Sierpinski | Sur les fonctions d'ensemble additives et continues[END_REF] for every u ′ ∈ L ∞ such that u ′ (τ ) ∈ Ω for a.e. τ ∈ [0, T ].

End of the proof

Let v ∈ Ω be fixed. Let t ∈ [0, T ) be a continuity point of p ∈ BV n and be a Lebesgue point of the application s → H(q(s, u * ), u * (s), p(s), s) which belongs to L ∞ 1 . Let α ∈ (0, Tt) and let us consider

u ′ (τ ) := v if τ ∈ [t, t + α), u * (τ ) if τ / ∈ [t, t + α),
for a.e. τ ∈ [0, T ]. From Inequality [START_REF] Sierpinski | Sur les fonctions d'ensemble additives et continues[END_REF], it holds that t+α t H(q(s, u * ), v, p(s), s) -H(q(s, u * ), u * (s), p(s), s) ds ≥ 0.

Dividing by α > 0 and letting α → 0 + , we obtain that H(q(t, u * ), v, p(t), t) -H(q(t, u * ), u * (t), p(t), t) ≥ 0.

Since the last inequality is true for every v ∈ Ω and for a.e.

A Proof of Lemma 3

Recall that the classical Lesbesgue measure λ is a nonatomic measure (see, e.g., [8, Remark 1.161 p.111]). As a consequence, from the classical Sierpinski (or Lyapunov) theorem (see [START_REF] Sierpinski | Sur les fonctions d'ensemble additives et continues[END_REF] or [9, p.37]), for all measurable set R ⊂ [0, T ], there exists a measurable set R ρ ⊂ R such that λ(R ρ ) = ρλ(R) for all ρ ∈ (0, 1).

The whole section is dedicated to the proof of Lemma 3. Let ρ ∈ (0, 1).

Lemma 5. Let b : [0, T ] 2 → R n be defined by b(t, s) := h(s)1 [0,t] (s). Then, b ∈ C([0, T ], L 1 n ).
Proof. Let t ∈ [0, T ] and let (t n ) ⊂ [0, T ] be a decreasing sequence such that t n → t. Then, it holds that

b(t n , •) -b(t, •) L 1 = T 0 |b(t n , s) -b(t, s)| ds = t 0 |b(t n , s) -b(t, s)| ds + tn t |b(t n , s) -b(t, s)| ds = tn t |h(s)| ds → 0.
Similarly, we prove that b(t n , •)b(t, •) L 1 → 0 for any increasing sequence (t n ) ⊂ [0, T ] such that t n → t. The proof is complete.

Since [0, T ] is compact, there exists δ > 0 such that b(t, •)b( t, •) L 1 ≤ ρ 2 2(ρ+1) for all t, t ∈ [0, T ] satisfying |t -t| < δ. In the sequel, we fix 0 = t 0 < t 1 < . . . < t N = T such that |t r+1t r | < δ for all r = 0, . . . , N -1 and we define

B(•) := b(t 0 , •), b(t 1 , •), . . . , b(t N , •) ∈ L 1 ([0, T ], (R n ) N +1 ). Lemma 6. There exists a measurable set Q ρ ⊂ [0, T ] such that λ(Q ρ ) = ρT and T 0 1 - 1 ρ 1 Qρ (s) B(s) ds (R n ) N +1 ≤ ρ 2 . Proof. Since B ∈ L 1 ([0, T ], (R n ) N +1 ), there exists a simple function J : [0, T ] → (R n ) N +1 such that T 0 B(s) -J(s) (R n ) N +1 ds ≤ ρ 2 2(ρ+1) . Let us denote by J := K i=1 a i 1 R i , where a i ∈ (R n ) N +1 and R i ⊂ [0, T ] are measurable sets such that K i=1 R i = [0, T ]. Since λ is nonatomic, there exist R i ρ ⊂ R i such that λ(R i ρ ) = ρλ(R i ) for all i = 1, . . . , K. Let us define Q ρ := K i=1 R i ρ ⊂ [0, T ].
Note that λ(Q ρ ) = ρT . Moreover, it holds that

T 0 1 - 1 ρ 1 Qρ (s) B(s) ds (R n ) N +1 ≤ T 0 1 - 1 ρ 1 Qρ (s) J(s) ds (R n ) N +1 + T 0 1 - 1 ρ 1 Qρ (s) (B(s) -J(s)) ds (R n ) N +1
.

The second integral in the left term can be easily bounded by ρ 2 and the first one is equal to

K i=1 a i T 0 1 R i (s) - 1 ρ 1 Qρ∩R i (s) ds (R n ) N +1 = K i=1 a i λ(R i ) - 1 ρ λ(R i ρ ) (R n ) N +1 = 0.
The proof is complete.

Let us now conclude the proof of Lemma (see remark after Lemma 5). It holds that

t 0 1 - 1 ρ 1 Qρ (s) h(s) ds R n = T 0 1 - 1 ρ 1 Qρ (s) b(t, s) ds R n ≤ T 0 1 - 1 ρ 1 Qρ (s) (b(t, s) -b(t r , s)) ds R n + T 0 1 - 1 ρ 1 Qρ (s) b(t r , s) ds R n .
The first term can be bounded by (1

+ 1 ρ ) b(t, •) -b(t r , •) L 1 ≤ ρ
2 and the second one can be bounded by Lemma 6). Finally, we have proved that

T 0 1 -1 ρ 1 Qρ (s) B(s)ds (R n ) N +1 ≤ ρ 2 (see
t 0 1 - 1 ρ 1 Qρ (s) h(s) ds R n ≤ ρ.
The proof of Lemma 3 is complete.

B Some recalls about Banach spaces geometry

B.1 Renorming a separable Banach space

Let (X, • ) be a normed linear space. The dual space of (X, • ) is X * := L((X, • ), R) and X * can be endowed with the dual norm of • defined by

• * : X * -→ R + f -→ sup x∈X x ≤1 | f, x X * ×X |.
In this case, we denote (X * , • * ) = dual((X, • )). Recall that (X * , • * ) is a Banach space, even if (X, • ) is not.

Lemma 7. Let (X, • ) be a Banach space5 and (X * , • * ) = dual((X, • )). Let N * be a norm on X * equivalent to • * . Then, the following properties are equivalent:

1. N * is weak* lower semicontinuous on (X * , • * );

2. There exists N a norm on X equivalent to • such that (X * , N * ) = dual((X, N )).

Remark 4. This lemma is given in [START_REF] Penot | Calculus without derivatives[END_REF]Lemma 3.94

* (f ) ≤ lim inf N * (f n ).
1 ⇒ 2. We know that there exist 0

< µ 1 ≤ µ 2 such that µ 1 f * ≤ N * (f ) ≤ µ 2 f * for every f ∈ X * .
Our proof is based on two steps.

First step -Definition of N and equivalence to • . We define

N : X -→ R + x -→ sup f ∈X * N * (f )≤1 | f, x X * ×X |.
First of all, let us note that N is well-

defined since | f, x X * ×X | ≤ f * x ≤ 1 µ1 N * (f ) x ≤ 1 µ1
x for every x ∈ X and every f ∈ X * such that N * (f ) ≤ 1. In particular it holds that N (x) ≤ 1 µ1 x for every x ∈ X. Let us prove that N is a norm on X. Clearly we have N (λx) = |λ|N (x) and N (x + y) ≤ N (x) + N (y) for every x, y ∈ X and every λ ∈ R. Moreover, it holds that N (0) = 0. Now let us consider x = 0. From the classical Hahn-Banach theorem (see [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Corollary 1.3 p.3]) applied to (X, • ) (with dual (X * , • * )), there exists f ∈ X * such that f, x X * ×X = x 2 and f * = x . Let g :=

f µ2 x ∈ X * . It holds that N * (g) = N * (f ) µ2 x ≤ µ2 f * µ2 x = 1 and | g, x X * ×X | = | f,x X * ×X | µ2 x = x µ2 . Thus N (x) ≥ x µ2 > 0.
As a consequence, we have proved that N (x) = 0 if and only if x = 0. We conclude that N is a norm on X. Moreover, we have also proved that 1 µ2 x ≤ N (x) ≤ 1 µ1 x for every x ∈ X (the case x = 0 is obvious). As a consequence, N is equivalent to • .

Second step -(X * , N * ) = dual((X, N )). Since N is equivalent to • , L((X, N ), R) = L((X, •
), R) = X * , i.e. X * is the dual space of (X, N ). Let us introduce Ñ the dual norm of N on X * given by Ñ :

X * -→ R + f -→ sup x∈X N (x)≤1 | f, x X * ×X |.
In particular, we have (X * , Ñ ) = dual((X, N )). Our aim is to prove that

N * = Ñ . Firstly, let f ∈ X * such that f = 0. Let x ∈ X such that N (x) ≤ 1. From the definition of N , since N * ( f N * (f ) ) = 1, we have | f N * (f ) , x X * ×X | ≤ N (x) ≤ 1. Thus, | f, x X * ×X | ≤ N * (f ) for every x ∈ X such that N (x) ≤ 1.
As a consequence, from the definition of Ñ , we obtain that Ñ (f ) ≤ N * (f ) for every f ∈ X * (the case f = 0 is obvious). Secondly, let us assume by contradiction that there exists f 0 ∈ X * such that Ñ (f 0 ) < N * (f 0 ). Thus f 0 = 0 and we can define g := f0 Ñ (f0) ∈ X * satisfying Ñ (g) = 1 < N * (g). Hence g / ∈ B (X * ,N * ) (0, 1). From Hypothesis 1, one can easily see that B (X * ,N * ) (0, 1) is a nonempty weakly* closed convex of (X * , • * ). Then, recall that the dual of (X * , • * ) = dual((X, • )) endowed with the classical weak* topology is the set ( •, x X * ×X ) x∈X (see [4, Proposition 3.14 p.64]). Finally, from the classical Hahn-Banach theorem 6(see [4, Theorem 1.7 p.7]), there exists x ∈ X, α ∈ R and ε > 0 such that f, x X * ×X ≤ αε < α + ε ≤ g, x X * ×X for every f ∈ B (X * ,N * ) (0, 1). As a consequence, from the definition of N , we obtain that N (x) < g, x X * ×X ≤ Ñ (g)N (x) and finally Ñ (g) > 1. This is a contradiction.

Proposition 4. Let (X, • ) be a separable Banach space and (X * , • * ) = dual((X, • )). There exists a norm N on X equivalent to • such that:

1. N * is equivalent on X * to • * ; 2. N * is strictly convex on X * ;
where (X * , N * ) = dual((X, N )).

Remark 5. This proposition is given in [11, Theorem 2.18 p.42].

Proof. Our proof is based on four steps, including the application of Lemma 7.

First step (definition of N * and equivalence to • * ). Let (e k ) k∈N * ⊂ X be a sequence dense in B (X, • ) (0, 1). Then, let us consider the linear operator given by

A : X * -→ ℓ 2 (N * , R) f -→ 1 2 k/2 f, e k X * ×X k∈N * . Let us note that A(f ) 2 ℓ 2 = k∈N * 1 2 k | f, e k X * ×X | 2 ≤ f 2 * ,
for every f ∈ X * . Now we define

N * : X * -→ R + f -→ f * + A(f ) ℓ 2 .
Clearly N * is a norm on X * that is equivalent to

• * since f * ≤ N * (f ) ≤ 2 f * for every f ∈ X * .
Second step (strict convexity of N * ). Let f , g ∈ X * and let λ ∈ (0, 1) such that N * (λf +(1-λ)g) = λN * (f ) + (1λ)N * (g). Our aim is to prove that f = µg for some µ ≥ 0. Since • * and A(•) ℓ 2 are convex, we get that A(λf

+ (1 -λ)g) ℓ 2 = λ A(f ) ℓ 2 + (1 -λ) A(g) ℓ 2 .
Computing the square of the previous equality and using the linearity of A and the bilinearity of

•, • ℓ 2 , one can easily obtain that A(f ), A(g) ℓ 2 = A(f ) ℓ 2 A(g) ℓ 2 .
From the classical Cauchy-Schwarz inequality in ℓ 2 (N * , R), there exists µ ≥ 0 such that A(f ) = µA(g), that is, A(fµg) = 0 and fµg, e k X * ×X = 0 for every k ∈ N * . From density and homogeneity, we easily obtain that fµg, x X * ×X = 0 for every x ∈ X, i.e. fµg = 0.

Third step (N * is weak* lower semicontinuous on (X * , • * )). Let (f n ) ⊂ X * and f ∈ X * such that (f n ) weak* converges to f in (X * , • * ) = dual((X, • )), i.e. f n , x X * ×X tends to f, x X * ×X for every x ∈ X (see [4, Proposition 3.13 p.63]). Let us prove that N * (f ) ≤ lim inf N * (f n ).
First of all, recall that ( f n * ) is bounded by some M ≥ 0 and that f * ≤ lim inf f n * (see [4, Proposition 3.13 p.63]). Thus, lim inf

N * (f n ) ≥ lim inf f n * + lim inf A(f n ) ℓ 2 ≥ f * + lim inf A(f n ) ℓ 2 . Let us recall that A(f n ) 2 ℓ 2 = k∈N * 1 2 k f n , e k 2 X * ×X . Since ( 1 2 k f n , e k 2 X * ×X ) converge to 1 2 k f, e k 2 X * ×X and since | 1 2 k f n , e k 2 X * ×X | ≤ 1 2 k f n 2 * e k 2 ≤ M 2 2 k ∈ ℓ 1 (N * , R) for every n ∈ N, we obtain from the classical Lebesgue dominated convergence theorem that lim A(f n ) ℓ 2 = A(f ) ℓ 2 . Finally, we have proved that lim inf N * (f n ) ≥ f * + A(f ) ℓ 2 = N * (f ).
Fourth step (conclusion). We conclude the proof by applying Lemma 7.

B.2 The distance function

For this section, we essentially refer to [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I[END_REF][START_REF] Penot | Calculus without derivatives[END_REF].

In the whole section, (X, • ) denotes a normed linear space and (X * , • * ) := dual((X, • )). In this section, we denote by •, • := •, • X * ×X . Lemma 8. Let S ⊂ X be a convex subset such that Int(S) = ∅. Then, Adh(Int(S)) = Adh(S).

Proof. Let us prove that Adh(S) ⊂ Adh(Int(S)). Let x ∈ Adh(S) and let y ∈ Int(S) such that y = x. There exists ε > 0 such that y + B X (0, ε) ⊂ S. Let us prove that the line segment (x, y) ⊂ Int(S). Let w := tx+ (1t)y for some t ∈ (0, 1). Since x ∈ Adh(S), there exists x ′ ∈ S such that x-x ′ < ε 1-t t . Let us denote by z := xx ′ . Finally, we have w = tx ′ + (1t)y + tz = tx ′ + (1t)(y + ξ) where ξ := t 1-t z ∈ B X (0, ε). As a consequence, since x ′ ∈ S, since y + B X (0, ε) ⊂ S and since S is convex, w ∈ tx ′ + (1t)(y + B X (0, ε)) ⊂ S, that is, w belongs to an open subset included in S, i.e. w ∈ Int(S). Finally, we have proved that (x, y) ⊂ Int(S) and thus x ∈ Adh(Int(S)).

Let ϕ : X → R be a convex function and x ∈ X. Recall that the subdifferential of ϕ at x is defined by ∂ϕ(x) := {f ∈ X * | f, yx ≤ ϕ(y)ϕ(x), ∀y ∈ X}.

Note that ∂ϕ(x) is a weakly* closed convex subset of X * .

Lemma 9. If ϕ : X → R is continuous, then ∂ϕ(x) = ∅ for every x ∈ X.

Proof. Recall that the epigraph of ϕ is defined by

Epi ϕ := {(y, λ) ∈ X × R | ϕ(y) ≤ λ} ⊂ X × R.
Since ϕ is convex and continuous on X, Epi ϕ is clearly a nonempty closed convex subset of X × R. Let x ∈ X. It is clear that (x, ϕ(x)) / ∈ Int(Epi ϕ ) (since (x, ϕ(x)ε) / ∈ Epi ϕ for all ε > 0) and that Int(Epi ϕ ) is a nonempty7 open convex 8 subset of X × R. From the classical Hahn-Banach theorem (see [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Lemma 1.3 p.6]), there exists (f, c) ∈ (X × R) * = X * × R such that (f, c), (x, ϕ(x)) < (f, c), (w, r) , i.e. f, wx + c(rϕ(x)) > 0 for all (w, r) ∈ Int(Epi ϕ ). From the continuity of ϕ at x, it follows that (x, ϕ(x)+1) ∈ Int(Epi ϕ ). Thus, taking (w, r) = (x, ϕ(x)+1), we obtain that c > 0. From Lemma 8, we also obtain that f, wx + c(rϕ(x)) ≥ 0 for all (w, r) ∈ Epi ϕ . Finally, taking r = ϕ(w), we conclude that -1 c f, wx ≤ ϕ(w)ϕ(x) for all w ∈ X, i.e. -1 c f ∈ ∂ϕ(x). The proof is complete. Lemma 10. If ϕ : X → R is L-Lipschitz continuous on X for some L ≥ 0, then ∂ϕ(x) = ∅ and ∂ϕ(x) ⊂ B X * (0, L) for every x ∈ X.

Proof. Let x ∈ X and f ∈ ∂ϕ(x) = ∅ (see Lemma 9). One can easily obtain that

| f, z | = f, ±z = f, ±z + x -x ≤ ϕ(±z + x) -ϕ(x) ≤ |ϕ(±z + x) -ϕ(x)| ≤ L ± z + x -x = L z for every z ∈ X. Thus f * ≤ L. Lemma 11. If ϕ : X → R is Gâteaux-differentiable at some x ∈ X, then ∂ϕ(x) = {Dϕ(x)}.
Proof. Firstly, let us prove that Dϕ(x) ∈ ∂ϕ(x). From the convexity of ϕ, it holds that ϕ(x + λ(yx)) ≤ ϕ(x) + λ(ϕ(y)ϕ(x)), i.e. ϕ(x+λ(y-x))-ϕ(x) λ ≤ ϕ(y)ϕ(x) for any y ∈ X and any λ ∈ (0, 1). Passing to the limit λ → 0 + , we obtain that Dϕ(x), yx ≤ ϕ(y)ϕ(x) for every y ∈ X. Thus Dϕ(x) ∈ ∂ϕ(x). Secondly, let us prove that ∂ϕ(x) ⊂ {Dϕ(x)}. Let f ∈ ∂ϕ(x). It holds that f, λ(yx) ≤ ϕ(x + λ(yx))ϕ(x), i.e. f, yx ≤ ϕ(x+λ(y-x))-ϕ(x) λ for every y ∈ X and λ ∈ (0, 1). Passing to the limit λ → 0 + , we get that f, yx ≤ Dϕ(x), yx for every y ∈ X. From linearity, we conclude that f, y = Dϕ(x), y for all y ∈ X, i.e. f = Dϕ(x).

Lemma 12. If ϕ : X → R is Lipschitz continuous around x ∈ X and if ∂ϕ(x) is reduced to a singleton {f }, then ϕ is strictly Hadamard-differentiable at x with Dϕ(x) = f . Proof. See [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I[END_REF]Theorem 3.54 p.313].

Lemma 13. Let S ⊂ X be a nonempty subset and let d S : X → R be the distance function to S. The following properties hold:

1. d S is 1-Lipschitz continuous; 2. If S is convex, d S is convex;
3. If S is closed and convex, ∂d S (x) = ∅ and ∂d S (x) ⊂ Sph X * (0, 1) for every x ∈ X \ S.

Proof. 1. Let x, y ∈ S. Let (y n ) ⊂ S be a sequence such that yy n → d S (y). Then it holds that d S (x) ≤ xy n ≤ xy + yy n → xy + d S (y). In a very similar way we obtain that d S (y) ≤ xy + d S (x). We conclude that |d S (y) Lemma 10). From Lemma 10, we already know that f * ≤ 1. Since S is closed, note that d S (x) > 0. There exists (x n ) ⊂ S such that x-x n → d S (x). Moreover, it holds that f,

-d S (x)| ≤ y -x . 2. Let x, y ∈ S and λ ∈ [0, 1]. Let (x n ), (y n ) ⊂ S such that x-x n → d S (x) and y -y n → d S (y). Note that (1 -λ)x n + λy n ∈ S since S is convex. Then, d S ((1 -λ)x + λy) ≤ [(1 -λ)x + λy] - [(1 -λ)x n + λy n ] ≤ (1 -λ) x -x n + λ y -y n → (1 -λ)d S (x) + λd S (y). Thus, d S is convex. 3. Let x ∈ X \ S and f ∈ ∂d S (x) = ∅ (see
x n -x ≤ d S (x n ) -d S (x) = -d S (x). Thus, d S (x) ≤ f, x -x n ≤ f * x -x n → f * d S (x)
. Thus, f * ≥ 1. We conclude that ∂d S (x) ⊂ Sph X * (0, 1). Remark 6. Since d S is 1-Lipschitz continuous on X, one can easily prove that d 2 S is Fréchetdifferentiable on S, with Dd 2 S (x) = 0 for every x ∈ S.

Proposition 5. Let us assume that • * is strictly convex on X * . Let S ⊂ X be a nonempty closed and convex subset. Then, d S is strictly Hadamard-differentiable on X \ S with Dd S (x) * = 1 for every x ∈ X \ S.

Proof. Let x ∈ X \ S. From Lemma 13, it follows that ∂d S (x) = ∅ and ∂d S (x) ⊂ Sph X * (0, 1). Since ∂d S (x) = ∅ is a convex subset included in Sph X * (0, 1) and since • * is strictly convex, it follows that ∂d S (x) is necessarily reduced to a singleton {f } satisfying f * = 1. We conclude from Lemma 12.

C Functions of bounded variations and Stieltjes integrals

For this section, we essentially refer to [START_REF] Bachman | Functional analysis[END_REF][START_REF] Burk | A garden of integrals[END_REF][START_REF] Conway | A course in abstract analysis[END_REF][START_REF] Faraut | [END_REF][START_REF] Limaye | Functional analysis[END_REF][START_REF] Wheeden | Measure and integral: an introduction to real analysis[END_REF]. and ε > 0 such that ν(t -) ≤ ν(t)ε. Finally, from [START_REF] Wheeden | Measure and integral: an introduction to real analysis[END_REF], we obtain that

η t - 2 k = η t - 1 k - 1 k ≤ η t - 1 k - = ν t - 1 k ≤ ν(t -) ≤ ν(t) -ε = η(t -) -ε,
for every k ∈ N * . This raises a contradiction when k tends to +∞. Hence ν ∈ NBV 1 . Now let us prove that

T 0 z(τ ) dη(τ ) = T 0 z(τ ) dν(τ )
for every z ∈ C 1 . Since η and ν can be different only at discontinuity points of η (which are at most countable), we consider a sequence of partitions ((t ℓ k ) k ) ℓ of [0, T ] such that the length of the partitions tends to zero when ℓ tends to +∞ and such that no point t ℓ k is a discontinuity point of η. As a consequence, it holds that

T 0 z(τ ) dη(τ ) = lim ℓ→∞ k z(t k )(η(t k+1 ) -η(t k )) = lim ℓ→∞ k z(t k )(ν(t k+1 ) -ν(t k )) = T 0 z(τ ) dν(τ )
for every z ∈ C 1 .

In Appendix D.3, we will need the two following results.

Lemma 14. If h ∈ AC 1 , then h ∈ BV 1 .
Proof. One can easily get that Proof. One can easily get that

k |h(t k+1 ) -h(t k )| = k t k+1 t k ḣ(τ ) dτ ≤ T 0 | ḣ(τ )| dτ = ḣ L 1 , for every partition (t k ) k of [0, T ].
k |h(t k+1 ) -h(t k )| = k t k+1 t k z(τ ) dη(τ ) ≤ k t k+1 t k |z(τ )| dη(τ ) ≤ T 0 |z(τ )| dη(τ ), for every partition (t k ) k of [0, T ].
We conclude this section with some recalls about the Lebesgue-Stieltjes integral. Let η ∈ NBV 1 be monotically increasing. for every z ∈ L ∞ ([0, T ] 2 , R) such that z is continuous in its first variable.

Remark 7. Note that the Stieltjes integrals in (16) are both well-defined in terms of Riemann-Stieltjes integration 9 and the classical integrals in (16) have to be understood in the Lebesgue sense. Actually, one can easily see that the double integrals in (16) both exist.

C.2 Notations and Fubini-type formulas

For any η = (η i ) i=1,...,j ∈ NBV j such that η i is monotically increasing and for any z = (z i ) i=1,...,j ∈ C j , we denote by

where Φ ∈ L ∞ ([0, T ] 2 , R j ) and A ∈ L ∞ ([0, T ] 2 , R r,j ) are continuous in their first variable.

D State-transition matrices and linear Cauchy-Stieltjes problems

In the whole section A ∈ L ∞ ([0, T ], R n,n ).

9 Indeed, one can easily prove from the classical Lebesgue dominated convergence theorem that the function τ → τ 0 z(τ, s) ds is continuous on [0, T ].

D.1 Recalls on state-transition matrices

For every s ∈ [0, T ], the backward/forward linear Cauchy problem (BFCP A,s ) given by Ż(t) = A(t) × Z(t), a.e. t ∈ [0, T ],

Z(s) = Id n , (BFCP A,s ) admits a unique maximal solution that is moreover global. 10 We denote this solution by Z(

D

  State-transition matrices and linear Cauchy-Stieltjes problems 22 D.1 Recalls on state-transition matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 D.2 Recalls on linear Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 D.3 Linear Cauchy-Stieltjes problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Lemma 15 .

 15 Let η ∈ NBV 1 be monotically increasing and z ∈ C 1 . The function h : [0, T ] → R defined by h(t) := t 0 z(τ ) dη(τ ), is of bounded variations, i.e. h ∈ BV 1 .

  Then η induces a finite nonnegative measure on the Borel set of [0, T ] denoted by dη. This measure is constructed from the equalities dη([a, b)) = η(b)η(a) for every 0 ≤ a ≤ b ≤ T and extended from the classical Carathéodory extension theorem. For every z ∈ C 1 , the Riemann-Stieltjes integral of z with respect to η and the Lebesgue-Stieltjes integral of z with respect to dη (that corresponds to the classical Lebesgue integral of z with respect to the measure dη) coincide. We refer to[7, p.83] or[15, p.288] for more details. Finally, the following Fubini-type formula holds:

0 Aa 0 A

 00 τ ) dη i (τ ) ∈ R.Let r ∈ N * . We denote byT (τ ) × dη(τ ) := ki (τ ) dη i (τ ) k=1,...,r ∈ R r , and T 0 z(τ ), A(τ ) × dη(τ ) := T 0 A(τ ) ⊤ × z(τ ), dη(τ ) ∈ R, for every continuous matrices A(•) = (a ki (•)) ki : [0, T ] → R r,j and every z ∈ C r . Moreover, one can easily prove that if z ∈ R r (i.e. z ∈ C r constant) then T 0 z, A(τ ) × dη(τ ) = z, T (τ ) × dη(τ ) R r ×R r .(17)Finally, one can prove from Equality (16) that the following Fubini-type formulas both hold: , s) × dη(τ ) ds,

2 . 8 .

 28 ) × Z(ξ, τ ) × A(τ ) dξ dτ, for every (t, s) ∈ [0, T ] 2 . Using the classical Fubini formula (and inversing the roles of τ and ξ), we obtain thatt s Z(t, τ ) × A(τ ) dτ = t s A(τ ) Id n + τ s Z(τ, ξ) × A(ξ) dξ dτ.Finally, adding Id n in the above equality, we obtain that T (•, •) satisfiesT (t, s) = Id n + t s A(τ ) × T (τ, s) dτ, for every (t, s) ∈ [0, T ] 2 .From uniqueness of the global solution of (BFCP A,s ), we obtain thatT (t, s) = Z(t, s). To conclude, from the definition of Z(•, •), it is clear that Z(•, •) is (absolutely) continuous inits first variable. Using the second equality and the classical Lebesgue dominated convergence theorem, one can easily prove that Z(•, •) is continuous on [0, T ] Remark From Lemma 16, note that Z(t, •) is the unique global solution of the backward/forward linear Cauchy problem given by Ż(s) = -Z(s) × A(s), a.e. s ∈ [0, T ], Z(t) = Id n , for every t ∈ [0, T ].

  see Appendix D.1). From the classical Duhamel formula (see Proposition 8 in Appendix D.2), it holds that w(t, u * , u ′ ) =

t 0 Z(t, s) × h u * ,u ′ (s) ds for every t ∈ [0, T ]. Replacing w(•, u * , u ′ ) in (

  t ∈ [0, T ], we obtain the maximization condition u

* (t) ∈ arg min v∈Ω H(q(t, u * ), v, p(t), t) for a.e. t ∈ [0, T ].

  3. Let t ∈ [0, T ]. There exists r ∈ {0, . . . , N -1} such that t ∈ [t r , t r+1 ]. In particular, it holds that |tt r | < δ and thus b(t, •)b(t r , •) L 1 ≤ ρ 2

	2(ρ+1)

  p.251].Proof. 2 ⇒ 1. Since (X, • ) is a Banach space and since N is equivalent to • , note that (X, N ) is a Banach space. Let us prove that N * is weak* lower semicontinuous on (X * , • * ). Let (f n ) ⊂ X * and f ∈ X * . From [4, Proposition 3.13 p.63], (f n ) weak* converges to f in (X * , • * ) = dual((X, • )) if and only if f n , x X * ×X tends to f, x X * ×X for every x ∈ X if and only if (f n ) weak* converges to f in (X * , N * ) = dual((X, N )). Since (X, N ) is a Banach space, we obtain from [4, Proposition 3.13 p.63] that (N * (f n )) is bounded and N

  •, s) : [0, T ] → R n,n . The matrix function Z(•, •) is the so-called state-transition matrix associated to A. ) × A(τ ) dτ,for every (t, s) ∈ [0, T ] 2 . In particular, Z(•, •) : [0, T ] 2 → R n,n is continuous.Proof. The first equality is obvious since it corresponds to the definition of a global solution of (BFCP A,s ). From this equality and from the classical Gronwall lemma, one can easily prove that Z(•, •) is bounded on [0, T ] 2 . For every (t, s) ∈ [0, T ] 2 , we introduce

	Lemma 16. The following equalities both hold
		t
	Z(t, s) = Id n +	s	A(τ ) × Z(τ, s) dτ,
		t
	= Id n +		Z(t, τ
		s

T (t, s) := Id n + t s Z(t, τ ) × A(τ ) dτ, that is well-defined since Z(•, •) is bounded on [0, T ] 2 . Our aim is to prove that Z(t, s) = T (t, s).

From the first equality, it holds that

For ρ = 0, we fix Qρ = ∅.

The subsequence of (ε k ) k is not relabel. However, it is worth to note that the extracted subsequence depends on ℓ fixed.

For ρ = 0, we fix Qρ = ∅.

This result requires to fix ℓ ∈ N. Indeed, one needs a bound on u ℓ k L ∞ in order to conclude from the classical Lebesgue dominated convergence theorem.

The Banach assumption is necessary in order to apply [4, Proposition 3.13 p.63] deriving from the classical Banach-Steinhaus theorem.

The classical Hahn-Banach theorem can be applied here since (X * , • * ) = dual((X, • )) endowed with the classical weak* topology is a topological vector space that is locally convex.

Int(Epi ϕ ) is nonempty since, from Lemma

[START_REF] Fonseca | Modern methods in the calculus of variations: L p spaces[END_REF], Adh(Int(Epi ϕ )) = Adh(Epi ϕ ) = Epi ϕ is nonempty.[START_REF] Fonseca | Modern methods in the calculus of variations: L p spaces[END_REF] One can easily prove that the interior of a convex set is a convex set.

This results follows from the classical linear version of the Cauchy-Lipschitz (or Picard-Lindelöf) theorem.

C.1 Some recalls

Recall that a function η : [0, T ] → R is said to be of bounded variations if V(η) := sup

where the supremum is taken over all partitions (t k ) k of [0, T ]. In that case, we denote by η ∈ BV 1 .

Recall that η : [0, T ] → R belongs to BV 1 if and only if η is equal to the difference of two monotically increasing functions. In particular, if η ∈ BV 1 , then η admits a left-limit denoted by η(t -) (resp. a right-limit denoted by η(t + )) at every point t ∈ (0, T ] (resp. t ∈ [0, T )) and the set of discontinuity points of η is at most countable.

Recall that the application • BV1 given by

defines a norm on BV 1 . Moreover, recall that (BV 1 ,

Recall that if z ∈ C 1 and η ∈ BV 1 , then the classical Riemann-Stieltjes integral defined by

exists. In the above equality, the limit means that the length of the partition (t k ) k tends to zero.

Let us recall the following classical Riesz theorem.

Proposition 6 (Riesz theorem). Let ϕ ∈ C * 1 . There exists a unique η ∈ NBV 1 such that

, then η is monotically increasing. We refer to [10, p.245] for a complete proof of Proposition 6. In this section we will only detail the proof of the following weaker result, which is sufficient for the completeness of this note.

There exists η ∈ NBV 1 such that η is monotically increasing on [0, T ] and

for every z ∈ C 1 . Moreover, from construction of η, ϕ = 0 if and only if η = 0.

Proof. If ϕ = 0, it is sufficient to consider η = 0 ∈ NBV 1 that is monotically increasing. Now let us consider that ϕ = 0. In the sequel we simply denote by 1

From the classical Hahn-Banach theorem (see [4, Corollary 1.2 p.3]), there exists a linear continuous application φ : BF 1 → R such that φ extends ϕ to BF 1 and φ

Now we introduce η(t) := φ, 1 (0,t] for every t ∈ [0, T ]. In particular η(0) = 0. Moreover, since

Then, we introduce

As a consequence it holds that

In particular, it holds that η(T

To conclude the proof, let us prove that η can be chosen left-continuous on (0, T ). First of all, since η(0) = 0 and η is monotically increasing on [0, T ], η(t -) exists for all 0 < t ≤ T and it holds that

for every 0 ≤ s < t ≤ T . Let us define

In particular ν = 0. Using [START_REF] Wheeden | Measure and integral: an introduction to real analysis[END_REF], one can easily prove that ν is monotically increasing on [0, T ]. Let us assume by contradiction that ν is not left-continuous on (0, T ), that is, there exists t ∈ (0, T )

D.2 Recalls on linear Cauchy problems

Let B ∈ L ∞ ([0, T ], R n ) and q 0 , p T ∈ R n . From the classical linear version of the Cauchy-Lipschitz (or Picard-Lindelöf) theorem, the forward linear Cauchy problem (FCP A,B ) given by q(t) = A(t) × q(t) + B(t), a.e. t ∈ [0, T ], q(0) = q 0 , (FCP A,B ) admits a unique maximal solution that is moreover global. Similarly, the backward linear Cauchy problem (BCP A,B ) given by

also admits a unique maximal solution that is moreover global.

Proposition 8 (Duhamel formulas). The global solutions of (FCP A,B ) and (BCP A,B ) are given by

for every t ∈ [0, T ], where Z(•, •) is the state-transition matrix associated to A.

Proof. Let q : [0, T ] → R n be defined by

for every t ∈ [0, T ]. Replacing the value of Z(•, •) by the first equality given in Lemma 16 and using the classical Fubini formula, one can easily prove that

for every t ∈ [0, T ]. As a consequence, q is the unique global solution of (FCP A,B ). Similarly, let p : [0, T ] → R n be defined by

Replacing the value of Z(•, •) by the second equality given in Lemma 16 and using the classical Fubini formula, one can easily prove that

for every t ∈ [0, T ]. As a consequence, p is the unique global solution of (BCP A,B ).

D.3 Linear Cauchy-Stieltjes problems

Let q 0 , p T ∈ R n . Let B := (B i ) i=1,...,j and η := (η i ) i=1,...,j where B i ∈ C n and η i ∈ NBV 1 is monotically increasing for every i = 1, . . . , j. We say that q is a global solution of the forward linear Cauchy-Stieltjes problem (FCSP A,B ) given by dq = A × q dt + j i=1 B i dη i , on [0, T ], q(0) = q 0 , (FCSP A,B ) if q ∈ BF n and q satisfies

for every t ∈ [0, T ]. In such a case, it follows from Lemmas 14 and 15 that q ∈ BV n .

Similarly, we say that p is a global solution of the backward linear Cauchy-Stieltjes problem (BCSP A,B ) given by

for every t ∈ [0, T ]. In such a case, it follows from Lemmas 14 and 15 that p ∈ BV n . Proof. In this proof, we only treat Problem (FCSP A,B ). Let us consider the functional given by

Note that G is well-defined from Lemmas 14 and 15. Our aim is to prove that G admits a unique fixed point. To do so, we will prove that G admits a contractive iterate. One can easily prove by induction on k ∈ N * that

for every q 1 , q 2 ∈ BF n , every t ∈ [0, T ] and every k ∈ N * . As a consequence, it holds that

for every q 1 , q 2 ∈ BF n and every k ∈ N * . Taking k ∈ N * sufficiently large to get ( A L ∞ T ) k k!

< 1, we obtain that G k is a contractive iterate of G. Since (BF n , • ∞ ) is a Banach space, we conclude that G admits a unique fixed point.

Proposition 10 (Duhamel-type formulas). The global solutions of (FCSP A,B ) and (BCSP A,B ) are given by q(t) = Z(t, 0) × q 0 + j i=1 t 0 Z(t, s) × B i (s) dη i (s), and p(t) = Z(T, t) ⊤ × p T + j i=1 T t Z(τ, t) ⊤ × B i (τ ) dη i (τ ), for every t ∈ [0, T ], where Z(•, •) is the state-transition matrix associated to A.

Proof. From the Fubini-type formulas provided in Appendix C for Stieltjes integrals, the proof is exactly the same than in Proposition 8.


Mohamad Koteich

Gilles Duc

Abdelmalek Maloum

Guillaume Sandou

A unified model for low-cost high-performance AC drives: the Equivalent Flux concept

This paper presents a unified modeling approach of alternating current (AC) machines for low-cost high-performance drives. The Equivalent Flux concept is introduced. Using this concept, all AC machines can be seen as a non-salient synchronous machine with modified (equivalent) rotor flux. Therefore, complex salient-rotor machines models are simplified, and unified shaft-sensorless AC drives can be sought. For this purpose, a unified observer-based structure for rotor-flux position and speed estimation is proposed. The equivalent flux concept generalizes the existing concepts, such as the extended back-electromotive force, the fictitious flux and the active flux.

I. INTRODUCTION

Alternating current (AC) drives control has been an active research topic, since the invention of AC machine more than a century ago [START_REF] Tesla | A new system of alternate current motors and transformers[END_REF]. High-performance control of AC drives can be achieved using vector-control techniques, which requires an accurate knowledge of the rotor flux position and/or speed. One recent challenge in this field is the so-called sensorless (or self-sensing) control that consists of replacing the shaft sensors by estimators (observers) using the available information on the machine currents and voltages [START_REF] Holtz | Sensorless vector control of induction motors at very low speed using a nonlinear inverter model and parameter identification[END_REF].

Sensorless control techniques have witnessed several achievements in the last two decades [START_REF] Schauder | Adaptive speed identification for vector control of induction motors without rotational transducers[END_REF]- [START_REF] Holtz | Sensorless control of induction machines -with or without signal injection?[END_REF]. Yet, some challenging topics still need further development in the AC machine theory [START_REF] Boldea | Active flux concept for motionsensorless unified AC drives[END_REF]- [START_REF] Koteich | Discussion on "AC drive observability analysis[END_REF]. The history of AC machines tells us that the most successful achievements rely on finding an adequate machine model. This is the case of the well-known two-circuit model developed in the first half of the XXth century [START_REF] Park | Two-reaction theory of synchronous machines generalized method of analysis-part i[END_REF], [START_REF] Stanley | An analysis of the induction machine[END_REF]. This model has been the driving force behind the innovation in AC machine drives.

The two-circuit model of AC machines in the rotating reference frame (called the Park's model) is very well-suited for control purposes, since it transforms the AC machine into an equivalent DC machine model, where the flux and torque can be controlled independently. Nevertheless, this model requires an accurate knowledge of the rotor (or the rotor flux) position, which makes it not well-suited for position estimation purpose.

A. Towards a unified model for sensorless AC drives

There have been several attempts to unify the models of AC machines (or some of them) for sensorless control purposes. To the best knowledge of the authors, the first attempt is reported by Chen et al. [START_REF] Chen | Sensorless control of interior permanent magnet synchronous motor by estimation of an extended electromotive force[END_REF], where the concept of the extended back-electromotive force (EMF) is introduced. This concept transforms the salient interior (I) permanent magnet (PM) synchronous machine (SM) model into a non-salient PMSM model, with a stator inductance L d and a modified (extended) back-EMF (the equations are detailed in the appendix). This concept has been used and proved to be useful in observerbased sensorless IPMSM drives [START_REF] Morimoto | Sensorless control strategy for salient-pole pmsm based on extended emf in rotating reference frame[END_REF]- [START_REF] Akrad | Design of a fault-tolerant controller based on observers for a pmsm drive[END_REF].

Another attempt to unify the models of PMSMs was reported in [START_REF] Koonlaboon | Sensorless control of interior permanent-magnet synchronous motors based on a fictitious permanent-magnet flux model[END_REF], where the fictitious flux concept has been introduced. Using this concept, the salient PMSM model can be written as an SPMSM model, with a stator inductance L q and a modified (fictitious) rotor flux. This new concept seems to be more physically insightful as it is, by definition, the torque-producing flux, which multiplies the stator quadrature current, i sq , in the torque equation in the rotating reference frame.

The fictitious flux concept was generalized by Boldea et al. [START_REF] Boldea | Active flux concept for motionsensorless unified AC drives[END_REF] by introducing the active-flux concept, which is the torqueproducing flux in all AC machines, including the wound-rotor synchronous machine (WRSM), the synchronous reluctance machines (SyRM) and the induction machine (IM). Nevertheless, the active flux definition for the IM is based on the steady-state equations of the machine.

B. Paper objectives

This paper proposes a unified model of all AC machines. The hybrid-excited synchronous machines (HESM), whose rotor contains both DC-excited electromagnets and permanent magnets, is considered to be the general synchronous machines. Furthermore, the IMs are studied without restriction to the steady-state operation. The generalized flux that provides a unified modeling of AC machines will be called Equivalent Flux. It is believed that this terminology is more accurate that the aforementioned terms [START_REF] Boldea | Active flux concept for motionsensorless unified AC drives[END_REF], [START_REF] Chen | Sensorless control of interior permanent magnet synchronous motor by estimation of an extended electromotive force[END_REF], [START_REF] Koonlaboon | Sensorless control of interior permanent-magnet synchronous motors based on a fictitious permanent-magnet flux model[END_REF], since the unified model describes a machine similar to the non-salient PMSM with a flux equivalent to the permanent-magnets rotor flux.

In the next section, the equations of the SPMSM are presented as a source of inspiration. The Equivalent Flux concept is introduced in Section III, and the demonstration is made for both SMs and IMs. Section IV describes a unified state-observer structure for sensorless control of AC drives, based in the Equivalent Flux concept.

C. Notations

Throughout this paper, the complex space vector notation is used to represent the rotating electromagnetic quantities. Let x be a stator quantity in the rotor reference frame, it can be expressed as

x r s = x s s e -jθ or equivalently

x sd + jx sq = (x sα + jx sβ )(cos θ -j sin θ)
where subscripts s and r stand for stator and rotor quantities respectively, whereas superscripts s and r tell whether the quantity is expressed in the stator (αβ) or the rotor (dq) reference frame. θ is the angle between the stator and the rotor reference frames. The fluxes, currents, voltages and back EMFs are denoted by ψ, i, v and e, respectively. The inductance and the resistance of a given circuit are denoted by L and R respectively, and M denotes the mutual inductance between two circuits.

II. THE SPMSM: A SOURCE OF INSPIRATION

In an SPMSM, the stator and rotor flux equations are given by ( 1) and ( 2), respectively, in the stationary reference frame.

ψ s s = L s i s s + ψ s r (1)
ψ s r = ψ r e jθ (2)
The voltage model equation is

v s s = R s i s s + dψ s s dt (3)
Using the equations ( 1)-( 3), the rotor flux can be estimated, from the knowledge of the stator currents and voltages, using the following equation:

ψ s r = -L s i s s + (v s s -R s i s s ) dt (4)
Therefore, the rotor position and speed estimates are given by

θ = ∠ ψ s r = arctan ψ rβ ψ rα (5)
ω = d θ dt = ψ rα dψ rβ dt -ψ rβ dψ rα dt ψ 2 rα + ψ 2 rβ ( 6
)
In addition, the electromagnetic torque estimate is given by

T m = p ψ s r × i s s = pψ r i sβ cos θ -i sα sin θ (7)
The transformation of the SPMSM model into the rotor (dq) reference frame, gives the following flux equations

ψ r s = L s i r s + ψ r r (8) ψ r r = ψ r (9)
and the voltage model

v r s = R s i r s + dψ r s dt + jωψ r s ( 10
)
The torque equation in the dq reference frame is

T m = p ψ r r × i r s = pψ r i sq (11)
The steady-state vector diagram of the SPMSM is shown in Fig. 1. Based on the SPMSM model, various flux observers have been proposed in the literature [START_REF] Matsui | Sensorless PM brushless DC motor drives[END_REF]- [START_REF] Ortega | Estimation of rotor position and speed of permanent magnet synchronous motors with guaranteed stability[END_REF]. The simplicity of these equations, and the intuitive estimation approach, motivate us to seek a similar model formulation for all AC machine. In this section, the unified AC machine model is formulated using the equivalent flux concept. In the sequel, it will be demonstrated that all AC machines models can be written similarly to the non-salient PMSM model, with an equivalent rotor flux ψ eq and an equivalent stator inductance L eq , as illustrated in Fig. 2. Thus, the flux equation, in the stator coordinates, can be written as

d q i sd i sq i s ψ r L s i s ψ s jωψ s R s i s v s
ψ s s = L eq i s s + ψ s eq ( 12
)
ψ s eq = ψ eq e jθ [START_REF] Morimoto | Sensorless control strategy for salient-pole pmsm based on extended emf in rotating reference frame[END_REF] with L eq = L q for synchronous machines and L eq = σL s for inductions machines, and the equivalent flux ψ eq is expressed for each machine as

HESM : ψ eq = L ∆ i sd + M f i f + ψ r WRSM : ψ eq = L ∆ i sd + M f i f IPMSM : ψ eq = L ∆ i sd + ψ r SyRM : ψ eq = L ∆ i sd SPMSM : ψ eq = ψ r IM : ψ eq = k r ψ rd (14)
with: = R s i r s + L eq di r s dt + dψ eq dt + jω (L eq i r s + ψ eq )( 16)

L ∆ = L d -L q ; σ = 1 -k r k s ; k r = M L r ; k s = M L s L s L s d q θ ψ r α β (
The steady state vector diagram of the equivalent AC machine, shown on Fig. 3, is similar to the vector diagram of the SPMSM (Fig. 1).

d q i sd i sq i s ψ eq L eq i s ψ s jωψ s R s i s v s

Fig. 3: Vector diagram of the equivalent AC machine

The electromagnetic torque equation of the equivalent machine can be formulated similarly to that of the SPMSM using the equivalent flux:

T m = p ψ s eq × i s s = pψ eq (i sβ cos θ -i sα sin θ) (17)
= p ψ r eq × i r s = pψ eq i sq

Remark: The equivalent flux is aligned to the rotor direct (d-)axis in synchronous machines, and to the rotor flux vector in induction machines, that is

ψ r s = L eq i r s + ψ eq (19)
Furthermore, the equivalent flux is the torque-producing flux, since it multiplies the current i sq in the torque equation in the synchronous reference frame. Hence, the Equivalent flux amplitude and orientation are extremely important in both field-oriented control and direct-torque-and-flux control strategies.

The following paragraphs demonstrate the equivalent flux formulation for SMs and IMs.

A. Synchronous machines

The equations are developed for the salient-type HESM, they can be easily extended to the other SMs . The stator flux equation in the stator reference frame can be written as [START_REF] Krause | Analysis of electric machinery and drive systems[END_REF]:

ψ s s = L 0 i s s + L 2 i s * s e j2θ + (M f i f + ψ r )e jθ (20)
Adding and subtracting L 2 i s s , we get:

ψ s s = (L 0 -L 2 )i s s + L 2 i s s + i s * s e j2θ + (M f i f + ψ r )e jθ
1 For WRSM ψr = 0, for PMSM i f = 0, for SyRM both ψr and i f are null. For non-salient SMs L 2 = 0.

Knowing that L q = L 0 -L 2 , the above equation becomes ψ s s = L q i s s + L 2 i s s e -jθ + i s s e -jθ * e jθ + (M f i f + ψ r )e jθ Then ψ s s = L q i s s + (2L 2 Re (i r s ) + M f i f + ψ r ) e jθ Knowing that L ∆ = 2L 2 , the stator flux equation finally becomes:

ψ s s = L q i s s + (L ∆ i sd + M f i f + ψ r ) e jθ = L q i s
s + ψ eq e jθ where ψ eq is the equivalent flux defined in [START_REF] Sayeef | Rotor position and speed estimation of a variable structure direct-torque-controlled ipm synchronous motor drive at very low speeds including standstill[END_REF].

Remark: In steady-state operation, the time derivative of the equivalent flux is nothing but the excitation voltage [START_REF] Krause | Analysis of electric machinery and drive systems[END_REF], a known concept in the analysis of synchronous machines in steady-state operation.

B. Induction machines

The equations of the stator and rotor fluxes of the IM in the stationary reference frame are:

ψ s s = L s i s s + M i s r (21) ψ s r = M i s s + L r i s r ( 22
)
By replacing the following rotor current equation (from ( 22))

i s r = 1 L r ψ s r -M i s s ( 23
)
in the stator flux equation ( 21), the equivalent flux formulation prevails:

ψ s s = σL s i s s + k r ψ s r = L eq i s s + ψ s eq ( 24
)
The stator flux equation in the dq reference frame is then:

ψ r s = L σ i r s + k r ψ r r = L σ i r s + k r ψ rd (25)
Remark: In steady-state operating condition, the rotor flux is constant ( ψrd ). Therefore:

i rd = - 1 R r dψ rd dt = 0 (26)
which gives

k r ψ rd = k r (M i sd + L r i rd ) = k r M i sd (27)
In this case, the definition of the equivalent flux matches the active flux concept [START_REF] Boldea | Active flux concept for motionsensorless unified AC drives[END_REF]:

ψ eq = (L s -L σ ) i sd (28)
It can be noticed that in steady state, the IM behaves as a SyRM with L d = L s and L q = σL s .

IV. UNIFIED OBSERVER/ESTIMATOR FOR SENSORLESS

CONTROL

High performance torque-controlled AC drives require the knowledge of the machine torque and the rotor (or the rotor flux) angular position and speed. In this section, it is shown that these required variables can be estimated using a unified observer-based estimation algorithm (Fig. 4), thanks to the equivalent flux concept.

z -1 R s L eq v cor p ψ eq α k i sβ k -ψ eq β k i sα k ψ eq α k-1 ψ eq β k -ψ eq β k-1 ψ eq α k T e (ψ eq α k ) 2 +T e (ψ eq

A. Open-loop estimation

The equivalent flux can be estimated based on the following model (Fig. 5): Consequently, the angular position and speed of the estimated equivalent flux can be calculated using the following equations:

ψ s eq = -L eq i s s + (v s s -R s i s s ) dt (29)
θ = ∠ ψ s eq = arctan ψ eq β ψ eq α (30) ω = d θ dt = ψ eq α d ψ eq β dt -ψ eq β d ψ eq α dt ( ψ eq α ) 2 + ( ψ eq β ) 2 (31)
The electromagnetic torque can be evaluated using the following equation:

T m = p ψ s eq × i s = p ψ eq α i sβ -ψ eq β i sα (32)
For real-time implementation, the following discrete-time equations can be used (Fig. 4):

θ k = arctan ψ eq β k ψ eq α k (33)
ω k = 1 T e ψ eq α k-1 ψ eq β k -ψ eq β k-1 ψ eq α k ( ψ eq α k ) 2 + ( ψ eq β k ) 2 (34)
T m k = p ψ eq α k i sβ k -ψ eq β k i sα k (35)

B. Equivalent flux observer

The stator flux estimation using the voltage-model pureintegration is prone to drift problems due to noise, measurement errors, stator resistance uncertainty or unknown initial conditions [START_REF] Koteich | Flux estimation algorithms for electric drives: a comparative study[END_REF]. Thus, a correction term, v cor , is added to the voltage model, similarly to the correction term in a traditional state-observer structure, to get the following equivalent flux observer:

ψ s eq = -L eq i s s + (v s s -R s i s s + v cor ) dt (36)
The tuning of the correction term v cor is still an open topic. Various correction terms have been reported in the literature. They can be classified into two main categories: current-based correction and flux-based correction (Fig. 6). In a currentbased correction structure, the measured stator currents are compared with the currents estimated using the estimated stator flux. On the other hand, in a flux-based correction observer, the estimated stator flux based on the current model is compared with the flux estimated using the voltage model. For instance, the observers in [START_REF] Koteich | A real-time observer for UAV's brushless motors[END_REF], [START_REF] Foo | Sensorless sliding-mode mtpa control of an ipm synchronous motor drive using a sliding-mode observer and hf signal injection[END_REF] use current-based correction, and the observers in [START_REF] Lascu | A modified direct torque control for induction motor sensorless drive[END_REF]- [START_REF] Boldea | Active flux DTFC-SVM sensorless control of ipmsm[END_REF] use flux-based correction for different AC machines. The angular speed estimation using numerical derivation of the estimated position is prone to noises. Therefore, a phasedlocked loop (PLL) technique can be used to estimate the equivalent flux angular speed and position [START_REF] Boldea | Active flux concept for motionsensorless unified AC drives[END_REF].

V. CONCLUDING REMARKS

A unified model of AC machines was proposed in this paper by introducing the equivalent flux concept. This torqueproducing flux concept emulates, and generalizes, the rotor flux in a non-salient permanent-magnet synchronous machine. It is particularly useful for low-cost (shaft-sensorless) highperformance (vector and field oriented) control of AC machines. Based on this concept, a unified observer-based estimation structure was presented. The only required parameters for the rotor position and speed estimation are the stator resistance R s and the equivalent stator inductance L eq . At the end of this paper, two topics are to be addressed in order to design a high-performance unified observer for AC drives, 1) the The SPMSM model can be written in the stationary reference frame as follows:

v sα v sβ = R s i sα i sβ + L s d dt i sα i sβ + e sα e sβ (37)
with

e sα e sβ = ωψ r C (θ) = ωψ r -sin(θ) cos θ (38)
The Laplace transformation of this model gives (Fig. 7.a):

v(s) = R s i(s) A + sL s i(s) B + e(s) C (39)
The rotor position and speed can be estimated using the following equations:

θ = arctan - e sα e sβ (40)
ω = 1 ψ r e 2 sα + e 2 sβ ( 41
)
This simple scheme inspires the authors of [START_REF] Chen | Sensorless control of interior permanent magnet synchronous motor by estimation of an extended electromotive force[END_REF] to elaborate a similar model for the IPMSM by introducing the extended back-EMF concept. The equations of the IPMSM in the stator reference frame are given by: v(s) = R s i(s) The dq model of the voltage equation is:

v sd = R s i sd + L d di sd dt -ωL q i sq (43)
v sq = R s i sq + L q di sq dt + ω (L d i sd + ψ r ) (44)
By adding and subtracting the following term

ωL q i sd -ωL q i sd + L d di sq dt -L d di sq dt = 0 (45)
to the v sq equation we get

v sd v sq = R s -ωL q ωL q R s i sd i sq + L d d dt i sd i sq + 0 (L d -L q ) ωi sd - disq dt + ωψ r (46)
The above equation describes a salient PMSM, with a stator inductance L d , and with the following extended back-EMF:

E e sdq = 0 E e = 0 (L d -L q ) ωi sd - disq dt + ωψ r (47)
The equations (46) gives the following αβ model (Fig.

Fig. 1 :

 1 Fig. 1: Vector diagram of the SPMSM

Fig. 2 :

 2 Fig. 2: Symbolic representation

Fig. 4 :

 4 Fig. 4: Equivalent flux observer for sensorless control

Fig. 5 :

 5 Fig. 5: Equivalent flux estimator

Fig. 6 :

 6 Fig. 6: Observer structure

  7.b):v(s) = R s i(s) A + (sL d I 2 -ω(L d -L q )J 2 ) i(s) X + e e (s) Y (48)where e e (s) is the Laplace transform of E e C (θ).

Fig. 7 :v 1 L ∆ i sd + ψ r e e sα 2 + e e sβ 2 (

 7122 Fig. 7: Vector diagrams of the PMSMs

Renault Group, Technocentre, 78288 Guyancourt, France

L2S -CentraleSupélec -CNRS -Paris-Saclay University, 91192 Gif-sur-Yvette, France. E-mail: mohamad.koteich@renault.com