Xing Lu

Keywords: Exact boundary controllability, exact boundary synchronization, exact boundary synchro-

the life of young people in France. Christophe Kugler is an 'ambitious' prof who continuously encourages me to challenge new things. And Kamil, he promised to join me in China.

I am immensely

Tatsien Li and Prof. Bopeng Rao. They have been supportive since I first started my study and helped me a lot in the process of writing this thesis. I am deeply grateful to Tatsien for his continuous instruction and encouragement. He is an excellent and respectful mathematician who keeps reminding me to do research with patience and imagination. We have spent a lot of time in discussing my work. He can always give me helpful advises in how to improve the results, also in how to write a paper and make a presentation. He witnesses my progress not only in scientific research, but also in becoming a better person.

I also owe my deepest gratitude to Bopeng, he made me feel at home when I was far away from home alone in France. He kindly managed accommodation for me, picked me up at the trains station, even helped me deal with complex administrative procedures when I first arrived in Strasbourg. I met Bopeng in the third year of my study in Fudan, since then we have discussed frequently. He is very patient and knowledgeable.

From our discussion I have learned and understood more about control theory and synchronization. Under his direction, we obtained very good results in exact boundary synchronization for a kind of coupled system with coupled Robin boundary controls.

Then, my sincere thanks goes to the joint Ph.D. program supported by University of Strasbourg and Fudan University, which gives me this opportunity to study in France and experience a different culture.

My special thanks are owe to IRMA for having accepted me, providing me with not only good working condition, but also chances to discuss with talented researchers. In addition, I also thank the group of PDE which is a supportive and instructive work group.

Then, I would like to show my deep appreciation to professors on the jury, Jean-Michel Coron, Olive Glass, Vilmos Komornik, Jean-Pierre Puel, Xu Zhang, for their support and their helpful advises. Thank Vilmos for being a good professor and friend, although with high level of academic attainment, he is very amiable.

I also would like to express my gratitude to my colleagues and friends who share my precious time in Strasbourg. Frederic Valet is a truly and industrious friend, we have talked about plenty of interesting subjects. Amandine Pierrot is a talkative girl who speaks extremely fast, which apparently inherited from her father. She is a good dancer and archer. Rafael Porto is one of the best friends, we have so many terrific stories together. I spent Christmas, New Year's Eve and many other 'festivals' with him and his group of Brazilians. Louis Dufour is an interesting French friend who invited me to eat dumplings at his home. Dans le chapitre 3, nous considérons la contrôlabilité exacte et la synchronisation exacte (par groupes) pour le système couplé d'équations des ondes avec des contrôles frontières couplés de Robin. A cause du manque de régularité de la solution, nous rencontrons beaucoup plus de difficultés que dans les cas de Dirichlet ou de Neumann. Afin de surmonter ces difficultés, on transforme le problème de Robin en un problème de Neumann. Puis, grâce aux résultats de régularité optimale de Lasiecka-Triggiani sur le problème de Neumann, on obtient un résultat sur la trace de la solution faible du problème de Robin. Ceci nous a permis d'établir la contrôlabilité exacte du système s'il y a suffisamment de contrôles, et, par la méthode de la perturbation compacte, la non-contrôlabilité frontière exacte du système quand il n'y a pas assez de contrôle frontière. En se basant sur ces résultats, on étudie la synchronisation frontière exacte (par groupes) du même système, la détermination de l'état de synchronisation par groupes, ainsi que la nécessité des conditions de compatibilité des matrices de couplage. Pour le système hyperbolique étudié dans cette thèse, l'étude du problème de contrôle remonte au travail de David L. Russell dans les années 1960. Il a synthétisé les résultats sur la contrôlabilité et la stabilité pour l'équation aux dérivées partielles linéaire dans [START_REF] Russell | Controllability and Stabilization theory for linear partial differetial equations: Recent progress and open questions[END_REF]. Il a également mentionné beaucoup de problèmes ouverts et intéressants. Dans les années 1980, J.-L. Lions a proposé la Hilbert Uniqueness Method (HUM).

Mots

Cette méthode fournit un cadre pour étudier la contrôlabilité d'équation hyperbolique linéaire, en particulier l'équation des ondes linéaire ( [START_REF] Lions | Perturbations et Stabilisation de Systmes Distribués[END_REF], [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]). Cette méthode permet d'établir la contrôlabilité du système grâce à une inégalité d'observabilité du système adjoint. En combinant la HUM et le théorème du point fixe de Schauder, on peut aussi étudier l'equation des ondes semi-linéaire ([45]- [START_REF] Zuazua | Exact controllability for semilinear wave equations in one space dimension[END_REF])

u tt -∆u = f (u). (0.1.2)
Pour le système hyperbolique quasi-linéaire, actuellement il y a peu de résultats sur la contrôlabilité (Cirinà [2]- [START_REF] Cirinà | Nonlinear hyperbolic problems with solutions on preassigned sets[END_REF]). Depuis 2002, Tatsien Li et Bopeng Rao ont proposé une méthode de construction directe avec une structure modulaire ([21]- [START_REF] Li | Exact boundary controllability for quasi-linear hyperbolic systems[END_REF]). Ils ont établi une théorie complète sur la contrôlabilité frontière exacte pour le système hyperbolique quasi-linéaire de premier ordre

∂u ∂t + A(u) ∂u ∂x = F (u) (0.1.3)
avec des conditions frontières non-linéaires basé sur la théorie de la C 1 solution semi-globale ( [START_REF] Li | Controllability and Observability for Quasilinear Hyperbolic Systems[END_REF] et [START_REF] Li | Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF]).

Pour un système hyperbolique, sous la condition des multiplicateurs du domaine et avec suffisamment Ce modèle décrit les interactions dépendantes sinusoïdalement des différences de phase entre N oscillateurs couplés:

dθ i dt = ω i + j K ij sin(θ j -θ i ), i = 1, • • • , N,
où θ i est la phase du i-ème oscillateur, ω i est la fréquence propres du i-ème oscillateur, et K ij indique la force de couplage. Des restrictions adéquates sur ces paramètres nous permettent d'utiliser ce modèle pour discuter divers phénomènes. On a une compréhension plus approfondie de la synchronisation à partir de ce modèle:

1. Ce n'est que lorsque le couplage K ij est assez fort et que toutes les fréquences propres ω i sont presque identiques, qu'une solution de synchronisation est possible. Sinon, lorsque le couplage est faible et que les fréquences propres sont clairement différentes, la synchronisation ne peut pas être réalisée.

2. Inspirée par la sociologie, la synchronisation complète ne peut pas être réalisée immédiatement, l'état de synchronisation se développe au fil du temps, par exemple, certaines parties des variables d'état peuvent être synchronisées avant la synchronisation complète entre toutes les variables d'état.

3. La structure topologique du système détermine les différents modes de développement de la synchroni-sation. Au contraire, les différents modes de la synchronisation reflètent la structure topologique du système.

Il est alors facile de voir l'application de la théorie des graphes.

Synchronisation pour un système couplé d'équations aux dérivées partielles

Comme de plus en plus de résultats sur la synchronisation sont acquis pour les systèmes gouvernés par des équations différentielles ordinaires, Tatsien Li et Bopeng Rao ont pour la première fois étudié la synchronisation de systèmes décrits par des équations aux dérivées partielles en 2012-2013. Prenant un système couplé d'équations des ondes avec des contrôles frontières de Dirichlet comme exemple, ils ont proposé le concept de synchronisation frontière exacte dans le cadre de solutions faibles, à savoir, par des contrôles frontières, le système peut réaliser la synchronisation dans un temps fini ( [START_REF] Li | Local exact boundary controllability for a class of quasilinear hyperbolic systems[END_REF], [START_REF] Li | Synchronisation exacte d'un système couplé d'équations des ondes par des contrôles frontières de Dirichlet[END_REF], [START_REF] Li | Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls[END_REF], [START_REF] Li | On the exactly synchronizable state to a coupled system of wave equations[END_REF], [START_REF] Li | Exact boundary synchronization for a coupled system of 1-D wave equations[END_REF], [START_REF] Lu | Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its applications[END_REF]).

∂ ν U = DH sur (0, +∞) × Γ 1 , (0.1.6) ∂ ν U + BU = DH sur (0, +∞) × Γ 1 , (0.1.7)
et la condition initiale

t = 0 : U = U 0 , U = U 1 dans Ω, (0.1.8) où Ω ⊂ R n est un ouvert borné avec la frontière régulière Γ = Γ 1 ∪ Γ 0 (Γ 1 ∩ Γ 0 = ∅), ∆ = ∂ 2 ∂x 2 1 + • • • + ∂ 2 ∂x 2 n (0.1.9)
est le Laplacien n-dimensionnel, ∂ ν désigne la dérivée normale à l'extérieur sur la frontière, la matrice de couplage A = (a ij ) est une matrice d'ordre N avec les éléments constants, la N × M (M ≤ N ) matrice D de contrôle est de rang plein avec les éléments constants, U = (u (1) ,

• • • , u (N ) ) T et H = (h (1) , • • • , h (M ) ) T sont la variable d'état et le contrôle frontière respectivement.
On donne la définition de la contrôlabilité frontière exacte et celle de la synchronisation frontière exacte pour le système ci-dessus avec des contrôles frontières de Neumann. La définition est similaire pour le système avec des contrôles frontières couplés de Robin.

On définit les espaces

H 0 = L 2 (Ω), H 1 = H 1 Γ0 (Ω), L = L 2 loc (0, +∞; L 2 (Γ 1 )), (0.1.10)
où H 1 Γ0 (Ω) est le sous-espace de H 1 (Ω) composé des fonctions de H 1 (Ω) avec la trace nulle sur Γ 0 .

Definition 0.1. S'il existe un T > 0, tel que pour toutes les données initiales Si la solution U = (u (1) ,

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N , on peut trouver une fonction H ∈ L M à support compact dans [0, T ], telle que le problème (1.4.1)-(1.4.4) et (1.4.6) admet une solution U = U (t, x) ∈ C 0 loc ([0, +∞); H 1-s ) N ∩ C 1 loc ([0, +∞); H -s ) N unique, où s > 1 2 , qui satisfait la condition finale t ≥ T U (t, x) ≡ 0, x ∈ Ω, ( 0
• • • , u (N ) ) T satisfait t ≥ T : u (1) (t, x) ≡ • • • ≡ u (N ) (t, x) def.
= u(t, x), x ∈ Ω, (0.1.12) alors le système (1.4.1)-(1.4.4) possède la synchronisation frontière exacte au moment T > 0, où u = u(t, x) inconnu a priori, est l'état de synchronisation.

Posant 0 = m 0 < m 1 < m 2 < • • • < m p = N , si la solution U = U (t, x) satisfait t ≥ T u (k) ≡ u (l) def. = u s , m s-1 + 1 ≤ k, l ≤ m s , 1 ≤ s ≤ p, x ∈ Ω, ( 0
T > 0, où (u 1 , • • • , u p ) T ,
C 1 =         1 -1 1 -1 . . . 1 -1         (N -1)×N . (0.1.14)
Supposons que la matrice A de couplage satisfait la C 1 -condition de compatibilité:

AKer(C 1 ) ⊆ Ker(C 1 ), (0.1.15)
ce qui équivaut au fait qu'il existe une unique matrice A d'ordre (N -1), telle que

C 1 A = AC 1 . (0.1.16)
Posant W = C 1 U , on obtient un système réduit:

         W -∆W + AW = 0 dans (0, +∞) × Ω, W = 0 sur (0, +∞) × Γ 0 , ∂ ν W = DH sur (0, +∞) × Γ 1 (0.1.17)
avec la condition initiale

t = 0 : W = W 0 , W = W 1 dans Ω, (0.1.18) où D = C 1 D, W 0 = C 1 U 0 , W 1 = C 1 U 1 .
(U 0 , U 1 ) ∈ (H 1 × H 0 ) N , on peut trouver un contrôle frontière H ∈ L M à support compact dans [0, T ], et H L M ≤ c (U 0 , U 1 ) (H1×H0) N , ( 0
t ≥ T : (u, u )(t) -(ψ, ψ )(t) H2-s×H1-s ≤ c C 1 (U 0 , U 1 ) (H1×H0) N -1 , (0.1.22)
où ψ est la solution du problème

                   ψ -∆ψ + aψ = 0 dans (0, +∞) × Ω, ψ = 0 sur (0, +∞) × Γ 0 , ∂ ν ψ = 0 sur (0, +∞) × Γ 1 , t = 0 : ψ = (E, U 0 ), ψ = (E, U 1 ) dans Ω, (0.1.23) et s > 1 2 .
Il est clair que la solution du problème, qui détermine l'état de synchronisation, possède une plus forte régularité, malgré que la solution du problème initial soit de régularité faible.

De plus, on peut aussi obtenir des résultats similaires pour la synchronisation frontière exacte par groupes de la même façon. A ce moment, la (N -p) × N matrice de synchronisation par p-groupes correspondante est:

C p =         S 1 S 2 . . . S p         , (0.1.24)
où S s est la (m s -m s-1 -1) × (m s -m s-1 ) matrice de rang plein:

S s =         1 -1 1 -1 . . . 1 -1         , 1 ≤ s ≤ p. ( 0
(U, U ) ∈ C 0 ([0, T ]; (H α (Ω) × H α-1 (Ω)) N ) (0.1.27) et U | Γ1 ∈ (H 2α-1 (Σ 1 )) N , (0.1.28) où Σ 1 = (0, T ) × Γ 1 . D'ailleurs, l'application (U 0 , U 1 , H) → (U, U )
est continue pour les topologies correspondantes.

De même, nous avons la contrôlabilité frontière exacte pour le système couplé avec des contrôles frontières couplés de Robin par suffisamment de contrôles frontières.

Théorème 0.4. On suppose que M = N . Alors, il existe un T > 0, tel que pour toutes les données initiales

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N ,
H (L 2 (0,T ;L 2 (Γ1))) N ≤ c (U 0 , U 1 ) (H1×H0) N . ( 0
H (L 2 (0,T,L 2 (Γ1))) N -1 ≤ c C 1 (U 0 , U 1 ) (H1×H0) N -1 . ( 0
(u, u )(T ) -(φ, φ )(T ) H α+1 (Ω)×H α (Ω) ≤ c C 1 (U 0 , U 1 ) (H1×H0) N -1 , (0.1.34)
où φ est la solution du problème:

               φ -∆φ + λφ = 0 dans (0, +∞) × Ω, φ = 0 sur (0, +∞) × Γ 0 , ∂ ν φ + µφ = 0 sur (0, +∞) × Γ 0 , t = 0 : φ = (U 0 , E), φ = (U 1 , E) dans Ω, (0.1.35)
où λ et µ sont définis par Ae = λe, Be = µe.

En projetant le système sur un sous-espace, on peut obtenir un système réduit, qui détermine l'état de synchronisation. De cette façon, on peut simplifier largement la détermination de l'état de synchronisation.

Dans la thèse, on a beaucoup développé la synchronisation frontière exacte par groupes pour le système couplé avec des contrôles frontières couplés de Robin. Cette partie est plus difficile à cause du manque de régularité de la solution faible. La nécessité des conditions de compatibilité des matrices de couplage a été établie seulement dans des domaines spécifiques, par exemple, parallélépipèdes. Il reste encore beaucoup de problèmes intéressants et prometteurs à considérer.

Chapter 1

Introduction

From controllability to synchronization

Consider a finite dimensional dynamical system

ẋ(t) = f (x(t), h(t)), (1.1.1)
where x(t) is the state variable of the system at the time t, and h(t) denotes the control function. The corresponding control problem is to choose a proper control function h such that the dynamic behavior of the system satisfies a given requirement under the action of control h. Controllability is an important property of a system, and plays a vital role in many mathematical and practical problems, such as the stability of feed back systems, optimal control problems and so on.

For the distributed parameter system described by partial differential equations, the control problem is also an important research subject. For general hyperbolic systems, the exact controllability is defined as follows: for any given initial data ϕ and final data ψ, there exists a T > 0 and a suitable control (boundary control or internal control), such that the system can drive any given initial data ϕ at the time t = t 0 to any given final data ψ at the time t = t 0 + T by means of the control. When the control is applied only on the boundary, we have the exact boundary controllability. When the control acts on a part of the domain, we have the exact internal controllability. In general, there is also the exact controllability, where the control acts on both the boundary and an internal part of the domain. Besides, for non-linear systems, such as quasi-linear hyperbolic systems, in the sense of classical solutions, if the exact controllability can be achieved only for sufficiently small initial data and final data, we have the local exact controllability, otherwise, the global exact controllability. In this thesis we mainly consider the exact boundary controllability.

For the hyperbolic system studied in this thesis, the research on its control problem traces back to the work of David L. Russell in the 1960s, who systematically summarized the results on controllability and stability for linear partial differential equations in [START_REF] Russell | Controllability and Stabilization theory for linear partial differetial equations: Recent progress and open questions[END_REF], and mentioned many problems that are worth for further study. After that, in the 1980s, J.-L. Lions proposed the pioneering Hilbert Uniqueness Method (HUM), which provides a general framework on the study of controllability for linear hyperbolic equations, and in particular for the linear wave equations (see [START_REF] Lions | Perturbations et Stabilisation de Systmes Distribués[END_REF], [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]). For linear hyperbolic systems, controllability and observability are dual respects of the same problem, and the HUM obtains the controllability by proving the corresponding observability inequality. By combining HUM and Schauder fixed-point theorem, some results on the exact boundary controllability for semi-linear wave equations

u tt -∆u = f (u) (1.1.2)
have also been obtained ( [START_REF] Zuazua | Exact controllability for the semilinear wave equation[END_REF]- [START_REF] Zuazua | Exact controllability for semilinear wave equations in one space dimension[END_REF]).

Introduction

For quasi-linear hyperbolic systems, there were only few work on the study of controllability (Cirinà [2]- [START_REF] Cirinà | Nonlinear hyperbolic problems with solutions on preassigned sets[END_REF]). Since 2002, Tatsien Li and Bopeng Rao proposed a direct constructive method with modular structure (see [START_REF] Li | Local exact boundary controllability for a class of quasilinear hyperbolic systems[END_REF]- [START_REF] Li | Exact boundary controllability for quasi-linear hyperbolic systems[END_REF]), and established a complete theory on the exact boundary controllability for the general first order quasi-linear hyperbolic system

∂u ∂t + A(u) ∂u ∂x = F (u) (1.1.3)
with general non-linear boundary conditions based on the semi-global C 1 solution theory (see [START_REF] Li | Controllability and Observability for Quasilinear Hyperbolic Systems[END_REF] and [START_REF] Li | Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF]).

For a coupled hyperbolic system, under the usual multiplier geometric condition on the domain, the exact boundary controllability can be always realized by adequate boundary controls in a suitable large control time. However, when any one of the conditions mentioned above fails, we can not get the exact boundary controllability in general. For example, in practical applications, some boundary conditions have clear and definite physical meanings, on which boundary controls can not be acted. This means that there will be a lack of controls on the boundary, so that the system can not obtain the exact boundary controllability. In this case, we need to consider whether the system possesses some kind of boundary controllability in a weaker sense or not. The answer is positive, and according to different requirements, the corresponding problems and research methods will be different. In this direction, Tatsien Li and Bopeng Rao made a pioneering contribution in the following two aspects. On one hand, by fewer boundary controls, they proved that system may achieve the exact boundary synchronization (see [START_REF] Li | Synchronisation exacte d'un système couplé d'équations des ondes par des contrôles frontières de Dirichlet[END_REF]- [START_REF] Li | Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls[END_REF]), in which all the components of the state variable reach a same state of synchronization, but the state of synchronization is a priori unknown. On the other hand, they defined the approximate boundary null controllability (see [START_REF] Li | Asymptotic controllability for linear hyperbolic systems[END_REF]) for the hyperbolic system: under suitable boundary controls, there exists a sequence of boundary control functions, such that the corresponding sequence of solutions approaches to zero as n → +∞, but the sequence of boundary control functions does not necessarily converge. Similarly, the approximate boundary synchronization was also taken into consideration for the system. Their research led to a combination of synchronization and controllability, then the study of synchronization became a part of the control theory, and at the same time the research of synchronization for systems governed by partial differential equations rose and developed.

Synchronization phenomena and their study

Synchronization is a widespread natural phenomenon. It was first observed by Huygens in 1665 ( [START_REF] Huygens | Oeuvres Complètes[END_REF]), who found that two pendulum clocks hanging on the wall synchronized in phase. Then, scientists gradually started the observation and research on these interesting phenomena, including the synchronization of organ pipes, firefly synchronous flashing in tropical forests, crickets chirping synchronously, synchronization in the neural system, and synchronization of heart peacemaker cells and so on ( [START_REF] Strogatz | SYNC: The Emerging Science of Spontaneous Order[END_REF]).

Introduction

In the 1950s, N. Wiener first began a theoretical research on synchronization from a mathematical point of view ( [START_REF] Wiener | Cybernetics, or control and communication in the animal and the machine[END_REF]). Biological models were then abstracted into a model of oscillators by A.T. Winfree in 1967.

This model considers only the phase rather than the amplitude of the oscillators, and this milestone work largely promoted the development of research on synchronization in mathematics. In 1990, L. M.Pecora and T.L. Carroll from the US found that synchronization also occurs in chaotic systems. However, in nature the coupling form between components is not always all-to-all, that is to say, not every two elements have an impact on each other. For this reason, mathematicians tried to use the graph theory to solve related problems, and in particular, the random graph was introduced into the study of synchronization. D. J. Watts and S. Strogatz proposed the small-world network in 1998, R. Albert and A.L. Barabasi put forward the scale-free network in 1999, which indicates an increasing significance of synchronization in complex networks.

Kuramoto model proposed by Kuramoto in 2003 is the most successful one to depict synchronization, since a large number of phenomena can be essentially described by this model. This model deals with the interactions that depend sinusoidally on the phase difference between N coupled oscillators:

dθ i dt = ω i + j K ij sin(θ j -θ i ), i = 1, • • • , N,
where θ i is the phase of the i-th oscillator, ω i is the intrinsic natural frequency of the i-th oscillator, and K ij denotes the coupling strength. Proper restrictions on these parameters allow us to use this model to discuss diverse phenomena. A deeper understanding on synchronization is gained from this model:

1. Only when the coupling coefficients K ij are sufficiently strong and all the natural frequencies ω i are almost uniform, a fully synchronized solution is possible. Otherwise, when the coupling is weak and the natural frequencies are obviously different, synchronization can not be realized.

2. Inspired by sociology, the full synchronization can not be achieved immediately, the state of synchronization develops as time goes on, for example, some parts of the state variables of the system may be synchronized before the full synchronization between all the state variables.

3. The topologic structure of the system determines different developing modes of synchronization.

Conversely, different developing modes of synchronization reflects the topologic structure of the system. It is then easy to see the application of graph theory.

4. Furthermore, in the case of variable coefficients:

ω i = ω i (t), K ij = K ij (t)
, the system is an adaptive network, the structure of which develops with the time and forms a feed-back loop, so that it is possible for the system to continuously observe and adjust its structure to obtain synchronization or avoid it.

On the other hand, not only the developing procedure is strongly related to the network structure, but also the topologic structure of the network determines to a large degree the stability of the state of synchronization, which is a parallel research subject corresponding to synchronization. In practical problems, and especially in chaotic systems, solutions are very sensitive to the initial data, which prompts us to raise the question: after adding perturbation at the initial time, does the extra signal disappear with time or not?

In 1983, H. Fujisaka and T. Yamada proposed an extended Lyapunov matrix method to study the synchronization stability for chaotic systems. They assume that the system starts from a state of synchronization at the initial time, and the states of synchronization form an invariant set. They added perturbation to the system, decomposed the solution into a synchronized part and a perturbed part:

θ i (t) = s(t)(synchronized part) + δθ i (t)(perturbed part), i = 1, • • • , N,

Synchronization for a coupled system of partial differential equations

As more and more results on the research of synchronization are acquired for systems governed by ordinary differential equations, Tatsien Li and Bopeng Rao for the first time studied the synchronization for systems described by partial differential equations in 2012-2013. Taking a coupled system of wave equations with Dirichlet boundary controls as an example, they proposed the concept of exact boundary synchronization in the framework of weak solutions, namely, by boundary controls, the system can realize the synchronization in a finite time, and switching off the controls after, the state of synchronization remains. After that, they and their collaborators successively got quite a lot of results ( [START_REF] Li | Local exact boundary controllability for a class of quasilinear hyperbolic systems[END_REF], [START_REF] Li | Synchronisation exacte d'un système couplé d'équations des ondes par des contrôles frontières de Dirichlet[END_REF], [START_REF] Li | Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls[END_REF], [START_REF] Li | On the exactly synchronizable state to a coupled system of wave equations[END_REF], [START_REF] Li | Exact boundary synchronization for a coupled system of 1-D wave equations[END_REF], [START_REF] Lu | Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its applications[END_REF]). In 2014, Tatsien

Li, Bopeng Rao and Long Hu discussed the exact boundary synchronization for a coupled system of wave equations with various boundary controls (Dirichlet type, Neumann type, coupled Robin type and coupled dissipative type) in one-space-dimensional case in the framework of classical solutions, and deeply studied the synchronization by groups ( [START_REF] Li | A note on the exact synchronization by groups for a coupled system of wave equations[END_REF], [START_REF] Li | Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls[END_REF]) and the determination of the state of synchronization ( [START_REF] Hu | Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type[END_REF], [START_REF] Li | Exact boundary synchronization for a coupled system of 1-D wave equations[END_REF]).

Besides, Tatsien Li, Bopeng Rao and Yimin Wei expanded the definition of synchronization in a work of 2014, put forward the concept of generalize synchronization.

Furthermore, when the added assumptions are further weakened, namely, when the usual multiplier geometric condition on the domain fails, and (or) there is a further lack of boundary control functions, we should consider the synchronization in a much weaker sense, that is to say, we can find a sequence of solutions, the differences between components of the state variable tend to zero, which is called to be the approximate boundary synchronization, see the work of Tatsien Li and Bopeng Rao in 2014 ( [START_REF] Li | Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF], [START_REF] Li | Criteria of Kalman's type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF]), where they also mentioned Kalman's criterion as a necessary condition for the approximate boundary synchronization.

Based on the research of boundary synchronization for the system with Dirichlet boundary controls, it is a necessary and challenging stage to further study the synchronization for system with other boundary controls. Different types of boundary controls correspond to different physical models, and give different systems in essence. We should figure out whether the change of the property of solution resulting from the change of boundary conditions will impact the synchronization of the system, and whether can similar results be obtained to that with Dirichlet boundary controls? These will be the key ingredients of the study in this thesis.

Main results

In this thesis, we consider the exact boundary controllability and the exact boundary synchronization for the following coupled system of wave equations with different types of boundary controls:

U -∆U + AU = 0 in (0, +∞) × Ω (1.4.1)
with the homogeneous Dirichlet boundary condition on Γ 0 :

U = 0 on (0, +∞) × Γ 0 , (1.4.2)
the boundary condition of Dirichlet type, Neumann type and coupled Robin type, respectively, on Γ 1 :

U = DH on (0, +∞) × Γ 1 , (1.4.3)
∂ ν U = DH on (0, +∞) × Γ 1 , (1.4.4) ∂ ν U + BU = DH on (0, +∞) × Γ 1 , (1.4.5)
and the corresponding initial condition

t = 0 : U = U 0 , U = U 1 in Ω, (1.4.6)
where Ω ⊂ R n is a bounded domain with smooth boundary

Γ = Γ 1 ∪ Γ 0 (Γ 1 ∩ Γ 0 = ∅), ∆ = ∂ 2 ∂x 2 1 + • • • + ∂ 2 ∂x 2 n (1.4.7)
is the n-dimensional Laplace operator, ∂ ν denotes the outward normal derivative on the boundary, the coupling matrix A = (a ij ) is a matrix of order N with constant elements, the boundary control matrix D is a full column-rank matrix of order N × M (M ≤ N ) with constant elements, U = (u (1) ,

• • • , u (N ) ) T and H = (h (1) , • • • , h (M )
) T are the state variable and the boundary control function, respectively.

As an example, we give the exact boundary controllability and the exact boundary synchronization for the above system with Neumann boundary controls. Definitions can be given in a similar way for the system with coupled Robin boundary controls.

Denote

H 0 = L 2 (Ω), H 1 = H 1 Γ0 (Ω), L = L 2 loc (0, +∞; L 2 (Γ 1 )), (1.4.8)
where H 1 Γ0 (Ω) is the subspace of H 1 (Ω) composed of all the functions with the null trace on Γ 0 .

Definition 1.1. If there exists a T > 0, such that for any given initial data If the solution U = (u (1) ,

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N ,
• • • , u (N ) ) T satisfies t ≥ T : u (1) (t, x) ≡ • • • ≡ u (N ) (t, x) def.
= u(t, x), x ∈ Ω, (1.4.10)

then system (1.4.1)-(1.4.2) and (1.4.4) is exactly boundary synchronizable at the time T > 0, where u = u(t, x), being a priori unknown, is called to be the state of synchronization. for all the initial data in L 2 (0, L) × H -1 (0, L) by boundary control functions in L 2 (0, T ), however, under Nuemann boundary controls, the observability inequality for the corresponding adjoint problem to this coupled system is valid only in a weaker norm, therefore, the controllable space of the system is required to have a higher regularity. Moreover, the controllable space can not be precisely described as in the case with Dirichlet boundary controls.

Let 0 = m 0 < m 1 < m 2 < • • • < m p = N . If the solution U = U (t, x) satisfies t ≥ T u (k) ≡ u (l) def. = u s , m s-1 + 1 ≤ k, l ≤ m s , 1 ≤ s ≤ p, x ∈ Ω, ( 1
In order to get the exact boundary synchronization for the original system, we define the matrix of synchronization as

C 1 =         1 -1 1 -1 . . . 1 -1         (N -1)×N
.

(1.4.12)

Assume that the coupling matrix A satisfies the following C 1 -compatibility condition:

AKer(C 1 ) ⊆ Ker(C 1 ), (1.4.13)
which is equivalent to the fact that there exists a unique (N -1) matrix A, such that

C 1 A = AC 1 . (1.4.14)
Let W = C 1 U . The original system of U can be reduced to a self-closed system of W :

         W -∆W + AW = 0 in (0, +∞) × Ω, W = 0 on (0, +∞) × Γ 0 , ∂ ν W = DH on (0, +∞) × Γ 1 (1.4.15)
Introduction with the corresponding initial condition

t = 0 : W = W 0 , W = W 1 in Ω, (1.4.16)
where

D = C 1 D, W 0 = C 1 U 0 , W 1 = C 1 U 1 .
C p =         S 1 S 2 . . . S p         , (1.4.17)
where S s is the following (m s -m s-1 -1) × (m s -m s-1 ) full row-rank matrix:

S s =         1 -1 1 -1 . . . 1 -1         , 1 ≤ s ≤ p. (1.4.18)
Correspondingly, we denote W = C p U .

We can prove that the C 1 -compatibility condition (1.4.13) is sufficient to gurantee the exact boundary synchronization. The necessity of the C 1 -compatibility condition is another important subject in the research of synchronization.

Theorem 1.1. Assume that Ω satisfies the usual multiplier geometric condition. Without loss of generality, we assume that there exists an x 0 ∈ R n , such that for m = x -x 0 , we have

(m, ν) ≤ 0, ∀x ∈ Γ 0 , (m, ν) > 0, ∀x ∈ Γ 1 , (1.4.19)
where (•, •) denotes the inner product in R n . Assume furthermore that M = N -1. Under the C 1 -compatibility condition (1.4.13), if the boundary control matrix D satisfies rank(C 1 D) = N -1, then there exists a constant T > 0 so large that for any given initial data (U 0 , U 1 ) ∈ (H 1 × H 0 ) N , we can find a boundary control function

H ∈ L M with compact support in [0, T ],
satisfying

H L M ≤ c (U 0 , U 1 ) (H1×H0) N , (1.4.20)
where c is a positive constant, such that the mixed initial-boundary value problem (1.4.1)-(1.4.2), (1.4.4) and

(1.4.6) admits a unique weak solution

U = U (t, x) ∈ C 0 loc ([0, +∞); H 1-s ) N ∩ C 1 loc ([0, +∞); H -s ) N on t ≥ 0, where s > 1 2
, which is exactly boundary synchronizable at the time T . On the contrary, as In particular, as M < N -1, no matter how large T > 0 is taken, the system is not exactly boundary synchronizable.

M = N -1, if system (1.
Similar results hold for the synchronization by groups, and in this situation, at least (N -p) boundary controls are obligatory for the synchronization by p-groups.

AKer(C p ) ⊆ Ker(C p ), (1.4.21)
where C p is defined by (1.4.17

a ij = α rs , m r-1 + 1 ≤ i ≤ m r , 1 ≤ r, s ≤ p, (1.4.22)
which indicates that the coupling matrix satisfies the row-sum condition by blocks.

Moreover, the determination of the state of synchronization is discussed with details for the exact boundary synchronization and the exact boundary synchronization by 2 and 3-groups, respectively. Generally speaking, the state of synchronization by p-groups depends on both the initial data (U 0 , U 1 ) and the applied boundary control function H. But when the coupling matrix A possesses certain properties, the state of synchronization by p-groups is independent of the applied boundary control function and determined entirely by the solution to a coupled system of wave equations with homogeneous boundary conditions. For 1 ≤ s ≤ p, let e s be vectors defined by where ψ = (ψ 1 , • • • , ψ p ) T is the solution to the following problem with homogeneous boundary conditions:

(e s ) j =      1, m s-1 + 1 ≤ j ≤ m s , 0, others.
                     ψ r -∆ψ r + p s=1 α rs ψ s = 0 in (0, +∞) × Ω, ψ r = 0 on (0, +∞) × Γ 0 , ∂ ν ψ r = 0 on (0, +∞) × Γ 1 , t = 0 : ψ r = (E r , U 0 ), ψ r = (E r , U 1 ) in Ω, (1.4.25)
where α rs (1 ≤ r, s ≤ p) are given by (1.4.22).

Even if the coupling matrix A does not satisfy the above assumption, the solution to problem (1.4.25) can be still used to estimate the state of synchronization by p-groups.

Theorem 1.4. Under the C p -compatibility condition (1.4.21), assume that

{E 1 , E 2 , • • • , E p } is bi-orthonormal to {e 1 , • • • , e p }.
Then there exist a boundary control matrix D and a positive constant c independent of the initial data, such that the state of synchronization by p-groups u = (u 1 , • • • , u p ) T satisfies the following estimate:

t ≥ T : (u, u )(t) -(ψ, ψ )(t) (H2-s×H1-s) p ≤ c C(U 0 , U 1 ) (H1×H0) N -p , (1.4.26
)

where ψ = (ψ 1 , • • • , ψ p )
is the solution to problem (1.4.25), and s > 1 2 .

From above, it is clear that although the solution to the problem with Neumann boundary controls possesses a weaker regularity, the solution to the mixed problem which determines the state of synchronization by p-groups possesses a higher regularity than the original problem itself, thus the regularity of the state of synchronization by p-groups is relatively improved, which makes it possible to approach the state of synchronization by p-groups by a solution to a more regular problem. present situation, we have not only a coupling matrix A in the coupled system of wave equations, but also a coupling matrix B on the coupled Robin boundary conditions. The interaction between these two coupling matrices makes the problem more complicated ( [START_REF] Li | Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls[END_REF]).

Here and hereafter, according to different situations, we define α, β, respectively, as

     α = 3/5 -, β = 3/5
, Ω is a general bounded smooth domain,

α = β = 3/4 -, Ω is a parallelepiped, (1.4.27)
where > 0 is an arbitrarily given small constant.

Theorem 1.5. For any given H ∈ (L 2 (0, T ; L 2 (Γ 1 ))) M and any given

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N ,
(U, U ) ∈ C 0 ([0, T ]; (H α (Ω) × H α-1 (Ω)) N ) (1.4.28)
and

U | Γ1 ∈ (H 2α-1 (Σ 1 )) N , (1.4.29)
where Σ 1 = (0, T ) × Γ 1 , α is defined by (1.4.27). Furthermore, the mapping

(U 0 , U 1 , H) → (U, U )
is continuous respect to the corresponding topologies.

Similarly to proving the exact boundary controllability for the corresponding system with Neumann boundary controls, we have the exact boundary controllability for the coupled system with coupled Robin boundary controls with enough boundary controls.

Theorem 1.6. Assume that M = N . Then there exists a T > 0, such that for any given initial data

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N ,
H (L 2 (0,T ;L 2 (Γ1))) N ≤ c (U 0 , U 1 ) (H1×H0) N , (1.4.30)
where c > 0 is a positive constant.

To obtain the non exact boundary controllability for the system lacking boundary controls, we apply the method of compact perturbation. Nevertheless, only in the case that the domain Ω is a parallelepiped, the trace U | Γ1 can almost have an optimal regularity (H

data (U 0 , U 1 ) ∈ (H 1 × H 0 ) N .
Based on this, by the method of compact perturbation, we get We also study the determination of the state of synchronization (see §3.7). Similarly, under certain algebraic conditions satisfied by the coupling matrices A and B, the state of synchronization is independent of boundary control function which realizes the synchronization. In general, the state of synchronization depends not only on the initial data, but also on the applied boundary control function, in this case, an estimate on the state of synchronization can be still given.

H (L 2 (0,T,L 2 (Γ1))) N -1 ≤ c C 1 (U 0 , U 1 ) (H1×H0) N -1 . ( 1
(u, u )(T ) -(φ, φ )(T ) H α+1 (Ω)×H α (Ω) ≤ c C 1 (U 0 , U 1 ) (H1×H0) N -1 , (1.4.35)
where φ is the solution to the following mixed initial-boundary value problem:

               φ -∆φ + λφ = 0 in (0, +∞) × Ω, φ = 0 on (0, +∞) × Γ 0 , ∂ ν φ + µφ = 0 on (0, +∞) × Γ 0 , t = 0 : φ = (U 0 , E), φ = (U 1 , E) in Ω, (1.4.36)
where λ and µ are defined by Ae = λe, Be = µe.

Generally speaking, in order to determine the state of synchronization, denote V as the minimum invariant subspace of both A and B, containing e. Correspondingly, let W be an invariant subspace of both A T and B T , being bi-orthonormal to V . We can get a subsystem by projecting the system into the subspace W , so that the state of synchronization can be accurately determined (see Theorem 3.13).

Therefore, when the problem possesses a large scale, we can determine the state of synchronization by a problem with smaller dimension, so that we can reduce the computational complexity to a large degree.

Nevertheless, the dimension of the small-scale problem is determined by the property of the coupling matrices

A and B, namely, the dimension of their common invariant subspace.

For the determination of the state of synchronization by groups, we have similar results (see §3.10).

Exact boundary synchronization for a coupled system of wave equations with Neumann boundary controls

Exact synchronization with Neumann boundary controls

Introduction

In this paper, we consider the following coupled system of wave equations with Neumann boundary controls:

           U -∆U + AU = 0 in (0, +∞) × Ω, U = 0 on (0, +∞) × Γ 0 , ∂ ν U = DH on (0, +∞) × Γ 1 (2.1.1)
and the corresponding initial data

t = 0 : U = U 0 , U = U 1 , (2.1.2)
where Ω ⊂ R n is a bounded domain with smooth boundary Γ = Γ 1 ∪ Γ 0 such that Γ 1 ∩ Γ 0 = ∅ and mes(Γ 0 ) > 0, ∂ ν denotes the outward normal derivative on the boundary, the coupling matrix A = (a ij ) is of order N , the boundary control matrix D is a full column-rank matrix of order N × M (M ≤ N ), both

A and D have real constant elements, U = (u (1) , • • • , u (N ) ) T and H = (h (1) , • • • , h (M ) ) T denote the state variables and the boundary controls, respectively.

Denote

H 0 = L 2 (Ω), H 1 = H 1 Γ0 (Ω), L = L 2 (0, +∞; L 2 (Γ 1 )), (2.1.3)
where H 1 Γ0 (Ω) is the subspace of H 1 (Ω), composed of all the functions with the null trace on Γ 0 , and T > 0 is a given constant.

We assume that Ω satisfies the usual multiplier geometric condition ( [START_REF] Lions | Perturbations et Stabilisation de Systmes Distribués[END_REF]). Without loss of generality, we assume that there exists an x 0 ∈ R n , such that setting m = x -x 0 , we have

(m, ν) ≤ 0, ∀x ∈ Γ 0 ; (m, ν) > 0, ∀x ∈ Γ 1 , (2.1.4)
where (•, •) denotes the inner product in R n .

Define the linear unbounded operator -∆ in H 0 by

D(-∆) = {Φ ∈ H 2 (Ω) : Φ| Γ0 = 0, ∂ ν Φ| Γ1 = 0}.
Clearly, -∆ is a positively definite self-adjoint operator with a compact resolvent. Then, for any given s ∈ R, we can define the operator (-∆) s 2 with the domain

H s = D((-∆) s 2 ),
which, endowed with the norm Φ s = (-∆) s 2 Φ L 2 (Ω) constitutes a Hilbert space, and its dual space is

H s = H -s .
In particular, we have (see [START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF])

H 1 = D( √ -∆) = {Φ ∈ H 1 (Ω) : Φ| Γ0 = 0}.
Lemma 2.1 (See [START_REF] Li | Exact boundary controllability for a coupled system of wave equations with Neumann controls[END_REF]). For any given initial data (U 0 , U 1 ) ∈ (H 1-s × H -s ) N with s >

H 1-s )) N ∩ (C 1 loc ([0, +∞); H -s )) N with s > 1/2, satisfying t ≥ T : U = U ≡ 0. (2.1.5)
Moreover, we have the continuous dependence:

H L M ≤ C (U 0 , U 1 ) (H1×H0) N , (2.1.6)
where C is a positive constant.

For the exact null controllability and the non-exact null controllability of system (2.1.1), the following results have been proved in [START_REF] Li | Exact boundary controllability for a coupled system of wave equations with Neumann controls[END_REF].

Lemma 2.2. When M = N , there exists a constant T > 0, such that system (2.1.1) is exactly null controllable at the time T for any given initial data

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N .
Remark 2.1. In fact, the controllable space is not a usual function space. However, it contains the subspace (H 1 × H 0 ) N . The choice of (H 1 × H 0 ) N as the controllable space is convenient for the application.

However, if there is a lack of boundary controls, we have Lemma 2.3. When M < N , no matter how large T > 0 is, system (2.1.1) is not exactly null controllable at the time T for any given initial data

(U 0 , U 1 ) ∈ (H 1 × H 0 ) N .
Therefore, it is necessary to discuss whether system (2.1.1) is controllable in some weaker senses when there is a lack of boundary controls, namely, when M < N . Although the results are similar to those for the coupled system of wave equations with Dirichlet boundary controls, since the solution to a coupled system of wave equations with Neumann boundary condition has a relatively weaker regularity, in order to realize the desired result, we need stronger function spaces, and the corresponding adjoint problem is also different.

satisfies t ≥ T : u (1) (t, x) ≡ • • • ≡ u (N ) (t, x) def. = u(t, x), (2.2.1)
where, u = u(t, x), being unknown a priori, is called the corresponding state of exact synchronization.

The above definition requires that system (2.1.1) maintains the state of synchronization even after canceling the boundary control since the time T .

Theorem 2.1. Assume that M < N . If system (2.1.1) is exactly synchronizable in the space (H 1 × H 0 ) N , then the coupling matrix A = (a ij ) should satisfy the following condition of compatibility (the sum of elements in every row is equal to each other):

N j=1 a ij def. = a (i = 1, • • • , N ), (2.2.2)
where a is a constant independent of i = 1, • • • , N .

Proof By Lemma 2.3, since M < N , system (2.1.1) is not exactly null controllable, then there exists an initial data (U 0 , U 1 ) ∈ (H 1 × H 0 ) N , such that for any given boundary control H, the corresponding state of synchronization u(t, x) ≡ 0. Then, noting (2.2.1), the solution to problem (2.1.1) corresponding to this initial data satisfies

u -∆u + N j=1 a ij u = 0, i = 1, • • • , N in D ((T, +∞) × Ω). (2.2.3) Then, for i, k = 1, • • • , N , we have N j=1 a kj - N j=1 a ij u = 0 in D ((T, +∞) × Ω), (2.2.4) therefore N j=1 a kj = N j=1 a ij , i, k = 1, • • • , N, (2.2.5)
which is just the required condition of compatibility (2.2.2).

Now, let

C 1 =         1 -1 1 -1 . . . . . . 1 -1         (N -1)×N (2.2.6)
be the corresponding matrix of synchronization. C 1 is a full row-rank matrix, and Ker(C 1 ) = Span{e 1 },

where e 1 = (1, 1, • • • , 1) T .
Clearly, the synchronization (2.2.1) can be equivalently written as

t ≥ T : C 1 U (t, x) ≡ 0 in Ω. (2.2.7)
By Lemma 4.7 in Appendix, we have Lemma 2.5. The following properties are equivalent:

(1) The condition of compatibility (2.2.2) holds;

(2) e 1 = (1, 1, • • • , 1) T is a right eigenvector of A, corresponding to the eigenvalue a given by (2.2.2);

(3) Ker(C 1 ) is an one-dimensional invariant subspace of A:

AKer(C 1 ) ⊆ Ker(C 1 ); (2.2.

8)

(4) There exists a unique matrix A 1 of order (N -1), such that

C 1 A = A 1 C 1 .
(2.2.9) is invertible, namely, rank(C 1 D) = N -1, then there exists a constant T > 0 so large that system (2.1.1) is exactly synchronizable at the time T in the space (H 1 × H 0 ) N , moreover, we have the continuous dependence:

A 1 = (ā ij ) is called the reduced matrix of A by C 1 , where āij = N p=j+1 (a i+1,p -a ip ) = j p=1 (a ip -a i+1,p ), i, j = 1, • • • , N -1. ( 2
H L N -1 ≤ C C 1 (U 0 , U 1 ) (H1×H0) N -1 , (2.2.11)
where C is a positive constant.

On the other hand, when rank(C 1 D) < N -1 (especially, when M < N -1), no matter how large T > 0 is, system (2.1.1) is not exactly synchronizable at the time T .

Proof Under the condition of compatibility (2.2.2), let

W = C 1 U, W 0 = C 1 U 0 , W 1 = C 1 U 1 .
Noting (2.2.9), it is easy to see that the original mixed problem (2.1.1)-(2.1.2) for U can be reduced to the following self-closed mixed problem for W :

               W -∆W + A 1 W = 0 in (0, +∞) × Ω, W = 0 on (0, +∞) × Γ 0 , ∂ ν W = DH on (0, +∞) × Γ 1 , t = 0 : W = W 0 , W = W 1 in Ω, (2.2.12)
where On the other hand, when rank(C 1 D) < N -1, by Lemma 2.3, the reduced system (2.2.12) is not exactly null controllable, then system (2.1.1) is not exactly synchronizable.

D = C 1 D. Noting that C 1 is a surjection from (H 1 × H 0 ) N onto (H 1 × H 0 ) N -1 ,

Exact boundary synchronization by p-groups

When there is a further lack of boundary controls, we consider the exact boundary synchronization by p-groups ( p ≥ 1; when p = 1, it becomes the exact boundary synchronization). This indicates that the components of U are divided into p groups:

(u (1) , • • • , u (m1) ), (u (m1+1) , • • • , u (m2) ), • • • , (u (mp-1+1) , • • • , u (mp) ), (2.3.1)
where

0 = m 0 < m 1 < m 2 < • • • < m p = N,
(k) ≡ u (l) def. = u s , m s-1 + 1 ≤ k, l ≤ m s , 1 ≤ s ≤ p, (2.3.2)
where, (u 1 , • • • , u p ) T , being unknown a prior, is called the corresponding state of synchronization by p-groups.

Let S s be a (m s -m s-1 -1) × (m s -m s-1 ) full row-rank matrix:

S s =         1 -1 1 -1 . . . 1 -1         , 1 ≤ s ≤ p, (2.3.3)
and let C p be the following (N -p) × N matrix of synchronization by p-groups:

C p =         S 1 S 2 . . . S p         . (2.3.4)
Obviously, we have

Ker(C p ) = Span{e 1 , • • • , e p }, (2.3.5)
where for 1 ≤ s ≤ p,

(e s ) j =      1, m s-1 + 1 ≤ j ≤ m s , 0, otherwise.
( Setting W = Cr U in (2.1.1), we get the following reduced problem:

               W -∆W + ÃW = 0 in (0, +∞) × Ω, W = 0 on (0, +∞) × Γ 0 , ∂ ν W = DH on (0, +∞) × Γ 1 , t = 0 : W = Cr U 0 , W = Cr U 1 in Ω, (2.3.11)
where D = Cr D. Moreover, by (2.3.9) we have

t ≥ T : W ≡ 0. (2.3.12)
Noting that Cr is a (N -p + r) × N full row-rank matrix, the linear mapping

(U 0 , U 1 ) → ( Cr U 0 , Cr U 1 ) (2.3.13) is a surjection from (H 1 × H 0 ) N onto (H 1 × H 0 ) N -p+r ,
( Cr D) = N -p + r, then M = rank(D) ≥ rank( Cr D) = N -p + r ≥ N -p. (2.3.14)
In particular, when M = N -p, we have r = 0, namely, the condition of compatibility ( 2

a ij = α rs , m r-1 + 1 ≤ i ≤ m r , 1 ≤ r, s ≤ p. ( 2
H L N -p ≤ C C p (U 0 , U 1 ) (H1×H0) N -p , (2.3.17)
where C is a positive constant.

On the other hand, when rank(C p D) < N -p (especially, when M < N -p), no matter how large T > 0 is, system (2.1.1) is not exactly synchronizable by p-groups at the time T .

Proof Assume that the coupling matrix A = (a ij ) satisfies the condition of compatibility (2.3.8). By Lemma 4.7 in Appendix, there exists a unique matrix A p of order (N -p), such that

C p A = A p C p . (2.3.18) Setting W = C p U, D = C p D.
We can similarly get the following reduced system for W :

               W -∆W + A p W = 0 in (0, +∞) × Ω, W = 0 on (0, +∞) × Γ 0 , ∂ ν W = DH on (0, +∞) × Γ 1 , t = 0 : W = C p U 0 , W = C p U 1 in Ω, (2.3.19)
where W is a vector valued function of (N -p) components. By the assumption that rank(D) = rank(C p D) = N -p and Lemma 2.2, system (2.2.12) is exactly null controllable. Also, by (2.1.6) in the Definition 2.1, we get the continuous dependence of (2.3.17). Then the original system (2.1.1) for U is exactly synchronizable by p-groups.

On the other hand, when rank(C p D) < N -p, by Lemma 2.3, the reduced system (2.2.12) is not exactly null controllable, then system (2.1.1) is not exactly synchronizable by p-groups.

Determination of the state of synchronization by p-groups

Now, we are going to discuss the determination of the state of synchronization of system (2.1.1). Generally speaking, the state of synchronization should depend on the initial data (U 0 , U 1 ) and the applied boundary control H. However, when the coupling matrix A possesses some good properties, the state of synchronization is independent of the applied boundary control, and can be determined entirely by the solution to a system of wave equations with homogeneous boundary condition.

First, by Lemma 2.1 and noting that the space (

H 1 × H 0 ) N given in Definition 2.3 is included in (H 1-s × H -s ) N (s > 1 2
), differently from the case of Dirichlet boundary controls, the attainable set of states of exact boundary synchronization by p-groups for the system with Neumann boundary controls is not the whole space (H 1-s × H -s ) p . Besides, as in the case of Dirichlet boundary controls ( [START_REF] Li | On the exactly synchronizable state to a coupled system of wave equations[END_REF]), the choice of boundary controls is not unique. We have the following Theorem 2.5. Let H denote the set of all the boundary controls H which can realize the exact boundary synchronization by p-groups at the time T for system (2.1.1). If the condition of compatibility (2.3.8) holds, then for > 0 small enough, the value of H ∈ H on (0, ) × Γ 1 can be arbitrarily chosen.

Proof First of all, there exists a T 0 > 0 independent of the initial data, such that, when T > T 0 , the reduced problem (2.2.12) is exactly null controllable at the time T . According to the proof of Theorem 2.2, the exact synchronization of system (2.1.1) is equivalent to the exact null controllability of the reduced system (2.2.12). Therefore, taking an > 0 so small that T -> T 0 , system (2.1.1) is still exactly synchronizable at the time T -.

Assuming firstly that (U 0 , U 1 ) ∈ (C ∞ 0 (Ω) × C ∞ 0 (Ω)) N , and choosing arbitrarily

H ∈ (C ∞ 0 ([0, ] × Γ 1 )) N -p ,
we solve the forward problem (2.1.1) on [0, ] with H = H , and get the solution ( U , U )

∈ C 0 ([0, ]; (H 1 × H 0 ) N ). Taking ( U ( , •), U ( , •)) ∈ (H 1 × H 0 )
N as initial data, by Theorem 2.2, for system (2.1.1), there exists a boundary control

H ∈ L 2 ( , T ; L 2 (Γ 1 )) N -p
such that the corresponding solution U satisfies exactly the initial condition

t = : U = U ( , x), U = U ( , x)
and realizes the synchronization at the time t = T . Let

H =      H t ∈ (0, ), H t ∈ ( , T ), U =      U t ∈ (0, ), U t ∈ ( , T ).
It can be verified that U is the solution to the mixed problem (2.1.1) with boundary control H, and it is exactly synchronizable at the time T . By this way, we get an infinity of boundary controls H, the values of which on (0, ) × Γ 1 can be taken arbitrarily. Finally, by the denseness of C ∞ 0 (Ω) in H 1 and H 0 , we can get the desired result.

In fact, the state of synchronization is closely related to the properties of the coupling matrix A. When

A T possesses an invariant subspace Span{E 1 , E 2 , • • • , E p } such that (E i , e j ) = δ ij , 1 ≤ i, j ≤ p, Span{E 1 , E 2 , • • • , E p } and Ker(C p ) = Span{e 1 , • • • , e p } are called to be bi-orthonormal. Let D N -p = {D ∈ M N ×(N -p) (R) : rank(D) = rank(C p D) = N -p}.
By [START_REF] Li | Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls[END_REF], D ∈ D N -p if and only if it can be expressed by

D = (C T p + (e 1 , • • • , e p )D 0 ) D, (2.4.1)
where D 0 is a p × (N -p) matrix, and D is a reversible matrix of order (N -p). where ψ = (ψ 1 , • • • , ψ p ) T is the solution to the following problem with homogeneous boundary condition:

                     ψ r -∆ψ r + p s=1 α rs ψ s = 0 in (0, +∞) × Ω, ψ r = 0 on (0, +∞) × Γ 0 , ∂ ν ψ r = 0 on (0, +∞) × Γ 1 , t = 0 : ψ r = (E r , U 0 ), ψ r = (E r , U 1 ) in Ω, (2.4.3)
where α rs (1 ≤ r, s ≤ p) are given by (2.3.16). On the other hand, since Span{E 1 , E 2 , • • • , E p } is an invariant subspace of A T , we may denote

Proof Noting that Span{E 1 , E 2 , • • • , E p } is bi-orthonormal to Ker(C p ) = Span{e 1 , • • • , e p },

and taking

D 0 = -E T C T p , E = (E 1 , E 2 , • • • , E p ) (2.
A T E r = p s=1 β sr E s , 1 ≤ r ≤ p,
where β sr are some constants. By

(A T E r , e s ) = (E r , Ae s ), 1 ≤ r, s ≤ p,
and noticing (3.9.10), we have

( p t=1 β tr E t , e s ) = (E r , p t=1
α ts e t ), 1 ≤ r, s ≤ p.

Then by bi-orthonormality, we get

β sr = α rs ,
namely,

A T E r = p s=1 α rs E s , 1 ≤ r ≤ p. ( 2

.4.6)

Let ψ r = (E r , U ), taking the inner product with E r on both sides of (2.1.1), we get (2.4.3). Finally, for the state of synchronization by p-groups, by (2.3.7) we have

t ≥ T : ψ r (t) = (E r , U ) = p s=1 (E r , e s )u s = u r , 1 ≤ r ≤ p. (2.4.7)
When the assumptions in Theorem 2.6 fail, we can use the solution of (2.4.3) to give an estimate on the state of synchronization by p-groups.

Theorem 2.7. Under the condition of compatibility (2.3.8), assume that

{E 1 , E 2 , • • • , E p } is bi-orthonormal to {e 1 , • • • , e p }.
Then there exist a boundary control matrix D ∈ D N -p and a constant c independent of the initial data, such that the state of synchronization by p-groups u = (u 1 , • • • , u p ) satisfies the following estimate:

t ≥ T : (u, u )(t) -(ψ, ψ )(t) (H2-s×H1-s) p ≤ c C p (U 0 , U 1 ) (H1×H0) N -p , ( 2
φ r -∆φ r + (E r , AU ) = 0. Since (E r , AU ) = (A T E r , U ) = ( p s=1 α rs E s + A T E r - p s=1 α rs E s , U ) = p s=1 α rs (E s , U ) + (A T E r - p s=1 α rs E s , U ) = p s=1 α rs φ s + (A T E r - p s=1 α rs E s , U ), (2.4.9)
and for any given k ∈ {1, • • • , p}, we have

(A T E r - p s=1 α rs E s , e k ) = (E r , Ae k ) - p s=1 α rs (E s , e k ) = (E r , p s=1 α sk e s ) -α rk = p s=1 α sk (E r , e s ) -α rk = α rk -α rk = 0, (2.4.10)
we get

A T E r - p s=1 α rs E s ∈ {Ker(e 1 , • • • , e p )} ⊥ = Im(C T p ).
Therefore, there exists an vector R r ∈ R N -p such that

A T E r - p s=1 α rs E s = C T p R r . (2.4.11)
Thus, for r = 1, • • • , p, we have

                     φ r -∆φ r + p s=1 α rs φ s = (R r , C p U ) in (0, +∞) × Ω, φ r = 0 on (0, +∞) × Γ 0 , ∂ ν φ r = 0 on (0, +∞) × Γ 1 , t = 0 : φ r = (E r , U 0 ), φ r = (E r , U 1 ) in Ω,
(2.4.12)

where α rs (1 ≤ r, s ≤ p) are defined by (2.3.16), and

U = U (t, x) ∈ C(0, T ; (H 1-s ) N ) ∩ C 1 (0, T ; (H -s ) N ) is
the solution to the mixed initial-boundary value problem (2.1.1)-(2.1.2). Moreover, we have

t ≥ T : φ r (t) = (E r , U ) = p s=1 (E r , e s )u s = u r , r = 1, • • • , p. (2.4.13)
Noting that (2.4.3) and (2.4.12) possess the same initial data and boundary condition, by the well-posedness for a system of wave equations with Neumann boundary condition, we have (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]) that, when t ≥ 0,

(ψ, ψ )(t) -(φ, φ )(t) 2 (H2-s×H1-s) p ≤ c T 0 C p U
T 0 C p U 2 (H1-s) N -p ds ≤ c( C p (U 0 , U 1 ) 2 (H1×H0) N -p + DH L N -p ). (2.4.15)
Moreover, by (2.3.17) we have

DH L N -p ≤ c C p (U 0 , U 1 ) 2 (H1×H0) N -p . (2.4.16)
Substituting it into (2.4.15), we have In order to exactly express the state of synchronization by p-groups, we can extend the subspace

T 0 C p U 2 (H1-s) N -p ds ≤ c C p (U 0 , U 1 ) 2 (H1×H0) N -p , ( 2
(ψ, ψ )(t) -(φ, φ )(t) 2 (H2-s×H1-s) p ≤ c C p (U 0 , U 1 ) 2 (H1×H0) N -p . ( 2
Span{e 1 , • • • , e p } to an invariant subspace Span{e 1 , • • • , e p , • • • , e q } of A, such that A T possesses an in- variant subspace Span{E 1 , • • • , E p , • • • , E q }, which is bi-orthonormal to Span{e 1 , • • • , e p , • • • , e q }. Let P = q s=1 e s ⊗ E s , (2.4.19)
in which the tensor product is defined by

(e ⊗ E)U = (E, U )e = E T U e, ∀ U ∈ R N .
P can be represented by a matrix of order N . It is easy to see that

Im(P ) = Span{e 1 , e 2 , • • • , e q }, Ker(P ) = Span{E 1 , E 2 , • • • , E q } ⊥ (2.4.20)
and

P A = AP. (2.4.21)
Let U = U (t, x) be the solution to the mixed initial-boundary value problem (2.1.1)-(2.1.2). We define its synchronizable part U s and controllable part U c , respectively, as follows:

U s := P U, U c := (I -P )U. (2.4.22)
If system (2.1.1) is exactly synchronizable by p-groups, then

t ≥ T : U ∈ Span{e 1 , • • • , e p } ⊆ Span{e 1 , • • • , e p , • • • , e q } = Im(P ), (2.4.23)
then we have

t ≥ T : U s = P U = U, U c = (I -P )U = 0.
Noting (2.4.21), multiplying P and (I -P ) from the left on both sides of (2.1.1) respectively, we see that the synchronizable part U s of U satisfies the following system:

                   U s -∆U s + AU s = 0 in (0, +∞) × Ω, U s = 0 on (0, +∞) × Γ 0 , ∂ ν U s = P DH on (0, +∞) × Γ 1 , t = 0 : U s = P U 0 , U s = P U 1 in Ω, (2.4.24)
while, the controllable part U c of U satisfies the following system:

                   U c -∆U c + AU c = 0 in (0, +∞) × Ω, U c = 0 on (0, +∞) × Γ 0 , ∂ ν U c = (I -P )DH on (0, +∞) × Γ 1 , t = 0 : U c = (I -P )U 0 , U c = (I -P )U 1 in Ω.
(

In fact, under the boundary control H, U c with the initial data ((I -P )U 0 , (I -P )U 1 ) ∈ Ker(P ) × Ker(P ) is exactly null controllable, while, U s with the initial data (P U 0 , P U 1 ) ∈ Im(P ) × Im(P ) is exactly synchronizable.

Theorem 2.8. Let P be defined by (2.4.19). If system (2.1.1) is exactly synchronizable by p-groups, and the synchronizable part U s is independent of the applied boundary control H, then we have p = q and P D = 0.

(2.4.26)

In particular, if P U 0 = P U 1 = 0, then, for such initial data (U 0 , U 1 ), system (2.1.1) is exactly null controllable.

Proof By Theorem 2.5, the value of H on (0, ) × Γ 1 can be arbitrarily taken. If the synchronizable part U s is independent of the applied boundary control H, then we have

P D = 0, hence Im(D) ⊆ Ker(P ).
Noting (2.4.20), we have dim Ker(P ) = N -q, dim Im(D) = N -p, then p = q.

State of exact boundary synchronization

By Lemma 2.5, e = (1, 1, • • • , 1)
T is a right eigenvector of A, corresponding to the eigenvalue a, defined by (

. Let e 1 , e 2 , • • • , e r and E 1 , E 2 , • • • , E r with r ≥ 1 be the Jordan chains of A and A T , respectively, corresponding to the eigenvalue a, and

Span{e 1 , e 2 , • • • , e r } is bi-orthonormal to Span{E 1 , E 2 , • • • , E r }.
Thus we have

                   Ae l = ae l + e l+1 , 1 ≤ l ≤ r, A T E k = aE k + E k-1 , 1 ≤ k ≤ r, (E k , e l ) = δ kl , 1 ≤ k, l ≤ r, e r = (1, 1 • • • , 1)
T , e r+1 = 0, E 0 = 0.

(2.5.1)

Let U = U (t, x) be the solution to the mixed initial-boundary value problem (2.1.1)-(2.1.2). If system (2.1.1) is exactly synchronizable, then

t ≥ T : U = ue r , (2.5.2)
where u = u(t, x) is the corresponding state of synchronization. The synchronizable part and the controllable part are, respectively,

t ≥ T : U s = ue r , U c = 0.
If the synchronizable part is independent of the applied boundary control H, by Theorem 2.8, we have r = 1, then A possesses a left eigenvector E such that (E, e) = 1.

Generally speaking, when r ≥ 1, setting

ψ k = (E k , U ), 1 ≤ k ≤ r, noting (2.4
U s = r k=1 (E k , U )e k = r k=1 ψ k e k .
Thus, (ψ 1 , • • • , ψ r ) are the coordinates of U s under the basis (e 1 , e 2 , • • • , e r ).

Taking the inner product with E k on both sides of (2.4.24), we get the following Theorem 2.9. Let e 1 , e 2 , • • • , e r and E 1 , E 2 , • • • , E r be the Jordan chains of A and A T , respectively, corresponding to the eigenvalue a, in which e r = (1,

• • • , 1) T . Then the synchronizable part U s = (ψ 1 , • • • , ψ r ) is
determined by the following system (1 ≤ k ≤ r):

                   ψ k -∆ψ k + aψ k + ψ k-1 = 0 in (0, +∞) × Ω, ψ k = 0 on (0, +∞) × Γ 0 , ∂ ν ψ k = h k on (0, +∞) × Γ 1 , t = 0 : ψ k = (E k , U 0 ), ψ k = (E k , U 1 ) in Ω, (2.5.3)
where

ψ 0 = 0, h k = E T k DH. (2.5.4)
Noting (2.5.2), we have

t ≥ T : ψ k = (E k , U ) = (E k , ũe r ) = ũδ kr , 1 ≤ k ≤ r.
Thus, the state of synchronization ũ is determined by

t ≥ T : u = u(t, x) = ψ r (t, x).
However, in order to get the state of synchronization ũ, we must solve the whole coupled problem (2.5.3)-(2.5.4).

State of exact boundary synchronization by 2-groups

In this section, we will discuss the case p = 2 for the state of exact boundary synchronization by p-groups of system (2.1.1).

Assume that, when t ≥ T , we have

u (1) (t, x) ≡ • • • ≡ u (m) (t, x) def. = u 1 (t, x), (2.6.1)
u (m+1) (t, x) ≡ • • • ≡ u (N ) (t, x) def.
= u 2 (t, x).

(2.6.2)

Let C 2 be the matrix of synchronization by 2-groups, defined by (2.3.4). Obviously,

Ker(C 2 ) = Span{ẽ 1 , ẽ2 }, (2.6.3)
where

ẽ1 = (1, • • • , 1 m , 0, • • • , 0 N -m ) T , ẽ2 = (0, • • • , 0 m , 1, • • • , 1 N -m ) T ,
and the state of synchronization (2.6.1)-(2.6.2) means that t ≥ T : U = ũ1 ẽ1 + ũ2 ẽ2 .

(2.6.4)

Assume that the subspace Span{ẽ 1 , ẽ2 } contains two right eigenvectors e r and f s of A, corresponding to eigenvalues λ and µ, respectively. Let e 1 , e 2 , • • • , e r and f 1 , f 2 , • • • , f s be the Jordan chains corresponding to these two right eigenvectors, respectively:

     Ae i = λe i + e i+1 , 1 ≤ i ≤ r, e r+1 = 0, Af j = µf j + f j+1 , 1 ≤ j ≤ s, f s+1 = 0.
(2.6.5)

Accordingly, let ξ 1 , ξ 2 , • • • , ξ r and η 1 , η 2 , • • • , η s be the Jordan chains to the corresponding left eigenvectors, respectively:

     A T ξ i = λξ i + ξ i-1 , 1 ≤ i ≤ r, ξ 0 = 0, A T η j = µη j + η j-1 , 1 ≤ j ≤ s, η 0 = 0, (2.6.6)
such that

(e i , ξ l ) = δ il (i, l = 1, • • • r), (f j , η m ) = δ jm (j, m = 1, • • • s) (2.6.7)
and

(e i , η j ) = (f j , ξ i ) = 0 (i = 1, • • • r; j = 1, • • • s). (2.6.8)
Let

φ i = (U, ξ i ), ψ j = (U, η j ).
Taking the inner product with ξ i and η j on both sides of (2.1.1)-(2.1.2), respectively, we get

                   φ i -∆φ i + λφ i + φ i-1 = 0 in (0, +∞) × Ω, φ i = 0 on (0, +∞) × Γ 0 , ∂ ν φ i = ξ T i DH on (0, +∞) × Γ 1 , t = 0 : φ i = (ξ i , U 0 ), φ i = (ξ i , U 1 )
in Ω (2.6.9)

and

                   ψ j -∆ψ j + µψ j + ψ j-1 = 0 in (0, +∞) × Ω, ψ j = 0 on (0, +∞) × Γ 0 , ∂ ν ψ j = η T j DH on (0, +∞) × Γ 1 , t = 0 : ψ j = (η j , U 0 ), ψ j = (η j , U 1 ) in Ω, (2.6.10)
where i = 1, • • • , r; j = 1, • • • , s, and
φ 0 = ψ 0 = 0. (2.6.11)
Once we get the solutions φ 1 • • • , φ r and ψ 1 , • • • , ψ s , the corresponding state of synchronization by 2groups can be determined. Since the sum of the numbers of right eigenvectors is bigger than or equal to 2, we discuss the following two cases, respectively.

(1) When r ≥ 1 and s ≥ 1, we have

     ẽ1 = α 1 e r + α 2 f s , ẽ2 = β 1 e r + β 2 f s .
(2.6.12) By (2.6.4), when t ≥ T , we have

U = ( ũ1 α 1 + ũ2 β 1 )e r + ( ũ1 α 2 + ũ2 β 2 )f s . (2.6.13)
Noting (2.6.7)-(2.6.8), we have

t ≥ T :      φ r = ũ1 α 1 + ũ2 β 1 , ψ s = ũ1 α 2 + ũ2 β 2 .
(2.6.14)

Since ẽ1 , ẽ2 are linearly independent, by (2.6.12) and noticing (2.6.7)-(2.6.8), we have det

  α 1 α 2 β 1 β 2   = 0. (2.6.15)
Then, we can get the state ( ũ1 , ũ2 ) T of synchronization by 2-groups by solving the linear system (2.6.14).

When r = 1 and s = 1, there exists a boundary control matrix D ∈ D N -2 , such that ξ T 1 D = η T 1 D = 0, hence the state ( ũ1 , ũ2 ) T of synchronization by 2-groups is independent of the applied boundary controls.

(2) When r ≥ 2 and s = 0 (or r = 0 and s ≥ 2), we have

     ẽ1 = α 1 e r + α 2 e r-1 , ẽ2 = β 1 e r + β 2 e r-1 .
(2.6.16) Similarly to (2.6.13), when t ≥ T , we have

U = ( ũ1 α 1 + ũ2 β 1 )e r + ( ũ1 α 2 + ũ2 β 2 )e r-1 .
(2.6.17)

Similarly, noting (2.6.7)-(2.6.8), (2.6.15) holds, thus, the state ( ũ1 , ũ2 ) T of synchronization by 2-groups can be determined by solving the following linear system:

     φ r = ũ1 α 1 + ũ2 β 1 , φ r-1 = ũ1 α 2 + ũ2 β 2 .
(2.6.18)

If r = 2, then Ker(C 2 ) = Span{e 1 , e 2 }, and {ξ 1 , ξ 2 } is an invariant subspace of A T , which is biorthonormal to Ker(C 2 ). By Theorem 2.6, similarly to (2.4.4), there exists a boundary control matrix

D ∈ D N -2 , such that ξ T 1 D = ξ T 2 D = 0,
then the state ( ũ1 , ũ2 ) T of synchronization by 2-groups is independent of the applied boundary controls.

State of exact boundary synchronization by 3-groups

Finally, we discuss the case p = 3 for the state of exact boundary synchronization by p-groups of system (2.1.1).

Assume that, when t ≥ T , we have

u (1) (t, x) ≡ • • • ≡ u (m1) (t, x) def. = u 1 (t, x), (2.7.1)
u (m1+1) (t, x) ≡ • • • ≡ u (m2) (t, x) def. = u 2 (t, x), (2.7.2)
u (m2+1) (t, x) ≡ • • • ≡ u (N ) (t, x) def. = u 3 (t, x). (2.7.3)
Let C 3 be the matrix of synchronization by 3-groups, defined by (2.3.4). Obviously,

Ker(C 3 ) = Span{ẽ 1 , ẽ2 , ẽ3 }, (2.7.4)
where

ẽ1 = (1, • • • , 1 m1 , 0, • • • , 0 m2-m1 , 0, • • • , 0 N -m2 ) T , ẽ2 = (0, • • • , 0 m1 , 1, • • • , 1 m2-m1 , 0, • • • , 0 N -m2 ) T , ẽ3 = (0, • • • , 0 m1 , 0, • • • , 0 m2-m1 , 1, • • • , 1 N -m2
) T , and the state of synchronization (2.7.1)-(2.7.3) means that t ≥ T : U = ũ1 ẽ1 + ũ2 ẽ2 + ũ3 ẽ3 .

(2.7.5)

Assume that there exist three right eigenvectors e r , f s and g t of A in the invariant subspace Span{ẽ 1 , ẽ2 , ẽ3 } of C 3 , corresponding to eigenvalues λ, µ and ν, respectively. Let

e 1 , e 2 , • • • , e r ; f 1 , f 2 , • • • , f s and g 1 , g 2 , • • • , g t
be the Jordan chains corresponding to these three right eigenvectors, respectively:

           Ae i = λe i + e i+1 , 1 ≤ i ≤ r, e r+1 = 0, Af j = µf j + f j+1 , 1 ≤ j ≤ s, f s+1 = 0, Ag k = νg k + g k+1 , 1 ≤ k ≤ t, g t+1 = 0.
(2.7.6)

Correspondingly, let ξ 1 , ξ 2 , • • • , ξ r ; η 1 , η 2 , • • • , η s and ζ 1 , ζ 2 , • • • , ζ t be
the Jordan chains to the corresponding left eigenvectors, respectively:

           A T ξ i = λξ i + ξ i-1 , 1 ≤ i ≤ r, ξ 0 = 0, A T η j = µη j + η j-1 , 1 ≤ j ≤ s, η 0 = 0, A T ζ k = νζ k + ζ k-1 , 1 ≤ k ≤ t, ζ 0 = 0, (2.7.7)
such that

(e i , ξ l ) = δ il , (f j , η m ) = δ jm , (g k , ζ n ) = δ kn (i, l = 1, • • • r; j, m = 1, • • • s; k, n = 1, • • • t) (2.7.8)
and

(e i , η j ) = (e i , ζ k ) = (f j , ξ i ) = (f j , ζ k ) = (g k , ξ i ) = (g k , η j ) = 0 (i = 1, • • • r; j = 1, • • • s; k = 1, • • • t). (2.7.9)
Taking the inner product with ξ i , η j ζ k on both sides of (2.1.1)-(2.1.2), respectively, and denoting

φ i = (U, ξ i ), ψ j = (U, η j ), θ k = (U, ζ k ), we get                    φ i -∆φ i + λφ i + φ i-1 = 0 in (0, +∞) × Ω, φ i = 0 on (0, +∞) × Γ 0 , ∂ ν φ i = ξ T i DH on (0, +∞) × Γ 1 , t = 0 : φ i = (ξ i , U 0 ), φ i = (ξ i , U 1 ) in Ω, (2.7.10)
                   ψ j -∆ψ j + µψ j + ψ j-1 = 0 in (0, +∞) × Ω, ψ j = 0 on (0, +∞) × Γ 0 , ∂ ν ψ j = η T j DH on (0, +∞) × Γ 1 , t = 0 : ψ j = (η j , U 0 ), ψ j = (η j , U 1 ) in Ω (2.7.11) and                    θ k -∆θ k + νθ k + θ k-1 = 0 in (0, +∞) × Ω, θ k = 0 on (0, +∞) × Γ 0 , ∂ ν θ k = ζ T k DH on (0, +∞) × Γ 1 , t = 0 : θ k = (ζ k , U 0 ), θ k = (ζ k , U 1 ) in Ω,
(2.7.12)

where i = 1, • • • , r; j = 1, • • • , s; k = 1, • • • , t, and
φ 0 = ψ 0 = θ 0 = 0. (2.7.13)
Once we get the solutions

φ 1 • • • , φ r ; ψ 1 , • • • , ψ s and ϕ 1 , • • • , ϕ t ,
the state of synchronization by 3-groups can be determined. Since the sum of the numbers of right eigenvectors is bigger than or equal to 3, we discuss the following three cases, respectively.

(1) None of r, s, t is equal to 0 ( r ≥ 1, s ≥ 1 and t ≥ 1). We have

           ẽ1 = α 1 e r + α 2 f s + α 3 g t , ẽ2 = β 1 e r + β 2 f s + β 3 g t , ẽ3 = γ 1 e r + γ 2 f s + γ 3 g t .
(2.7.14) By (2.7.5), when t ≥ T , we have

U = ( ũ1 α 1 + ũ2 β 1 + ũ3 γ 1 )e r + ( ũ1 α 2 + ũ2 β 2 + ũ3 γ 2 )f s + ( ũ1 α 3 + ũ2 β 3 + ũ3 γ 3 )g t . ( 2
           φ r = ũ1 α 1 + ũ2 β 1 + ũ3 γ 1 , ψ s = ũ1 α 2 + ũ2 β 2 + ũ3 γ 2 , θ t = ũ1 α 3 + ũ2 β 3 + ũ3 γ 3 .
(

     α 1 α 2 α 3 β 1 β 2 β 3 γ 1 γ 2 γ 3      = 0.
(2.7.17)

Then, the state ( ũ1 , ũ2 , ũ3 ) T of synchronization by 3-groups can be determined by solving the linear system (2.7.16).

When r = 1, s = 1 and t = 1, there exists a boundary control matrix D ∈ D N -3 , such that ξ T 1 D = η T 1 D = ζ T 1 D = 0, then the state ( ũ1 , ũ2 , ũ3 ) T of synchronization by 3-groups is independent of the applied boundary controls.

(2) One of r, s, t is equal to 0. Without loss of generality, we may assume that r ≥ 2, t ≥ 1 and s = 0 and let

           ẽ1 = α 1 e r + α 2 e r-1 + α 3 g t , ẽ2 = β 1 e r + β 2 e r-1 + β 3 g t , ẽ3 = γ 1 e r + γ 2 e r-1 + γ 3 g t .
(2.7.18) Similarly, when t ≥ T , we have

U = ( ũ1 α 1 + ũ2 β 1 + ũ3 γ 1 )e r + ( ũ1 α 2 + ũ2 β 2 + ũ3 γ 2 )e r-1 + ( ũ1 α 3 + ũ2 β 3 + ũ3 γ 3 )g t .
(2.7.19) Noting (2.7.8)-(2.7.9), (2.7.17) holds. Thus, the state ( ũ1 , ũ2 , ũ3 ) T of synchronization by 3-groups can be determined by solving the following linear system:

           φ r = ũ1 α 1 + ũ2 β 1 + ũ3 γ 1 , φ r-1 = ũ1 α 2 + ũ2 β 2 + ũ3 γ 2 , θ t = ũ1 α 3 + ũ2 β 3 + ũ3 γ 3 .
(2.7.20)

If r = 2 and t = 1, there exists a boundary control matrix D ∈ D N -3 , such that ξ T 1 D = ξ T 2 D = ζ T 1 D = 0, then the state ( ũ1 , ũ2 , ũ3 ) T of synchronization by 3-groups is independent of the applied boundary controls.

(3) Two of r, s, t are equal to 0. Without loss of generality, we may assume that r ≥ 3, s = t = 0. Then we have:

           ẽ1 = α 1 e r + α 2 e r-1 + α 3 e r-2 , ẽ2 = β 1 e r + β 2 e r-1 + β 3 e r-2 , ẽ3 = γ 1 e r + γ 2 e r-1 + γ 3 e r-2 .
(2.7.21) Similarly, when t ≥ T , we have U = ( ũ1 α 1 + ũ2 β 1 + ũ3 γ 1 )e r + ( ũ1 α 2 + ũ2 β 2 + ũ3 γ 2 )e r-1 + ( ũ1 α 3 + ũ2 β 3 + ũ3 γ 3 )e r-2 .

(2.7.22) Noting (2.7.8)-(2.7.9), (2.7.17) holds. Thus, the state ( ũ1 , ũ2 , ũ3 ) T of synchronization by 3-groups can be determined by solving the following linear system:

           φ r = ũ1 α 1 + ũ2 β 1 + ũ3 γ 1 , φ r-1 = ũ1 α 2 + ũ2 β 2 + ũ3 γ 2 , φ r-2 = ũ1 α 3 + ũ2 β 3 + ũ3 γ 3 .
(2.7.23)

If r = 3, then Ker(C 3 ) = Span{ẽ 1 , ẽ2 , ẽ3 }, and {ξ 1 , ξ 2 , ξ 3 } is an invariant subspace of A T , which is bi-orthonormal to Ker(C 3 ). By Theorem 2.6, similarly to (2.4.4), there exists a boundary control matrix

D ∈ D N -3 , such that ξ T 1 D = ξ T 2 D = ξ T 3 D = 0
, then the state ( ũ1 , ũ2 , ũ3 ) T of synchronization by 3-groups is independent of the applied boundary controls.

The state of synchronization by p-groups can be discussed in a similar way. Lemma 3.1. Assume that M = N . Then there exists a T > 0, for any given initial data (U 0 , U 1 ) ∈ (H 1 ) N × (H 0 ) N , where

H 0 = L 2 (Ω), H 1 = H 1 Γ0 (Ω), (3.1.3)
in which H 1 Γ0 (Ω) is the subspace of H 1 (Ω), composed of all the functions with the null trace on Γ 0 , there exists a boundary control function H ∈ L 2 loc (0, +∞; L 2 (Γ 1 )) N with compact support in [0, T ], such that system (3.1.2) is exactly boundary null controllable at the time T , namely, the corresponding solution U = U (t, x) satisfies t ≥ T : U (t, x) ≡ 0, x ∈ Ω.

(3.1.4) Remark 3.1. By the method given in [START_REF] Li | Exact boundary controllability for a coupled system of wave equations with Neumann controls[END_REF], boundary control function H can be chosen to continuously depend on the initial data:

H (L 2 (0,T,L 2 (Γ1))) N ≤ c (U 0 , U 1 ) (H1) N ×(H0) N , (3.1.5)
here and hereafter, c is a positive constant independent of the initial data.

On the other hand, when there is a lack of boundary controls, we have Lemma 3.2. When M < N , no matter how large T > 0 is, system (3.1.2) is not exactly boundary controllable at the time T for any given initial data (U 0 , U 1 ) ∈ (H 1 ) N × (H 0 ) N .

With more complicated boundary conditions, the study of synchronization will be more difficult. In this paper, we consider the following coupled system of wave equations with coupled Robin boundary controls:

           U -∆U + AU = 0 in (0, +∞) × Ω, U = 0 on (0, +∞) × Γ 0 , ∂ ν U + BU = DH on (0, +∞) × Γ 1 (3.1.6)
and corresponding initial condition

t = 0 : U = U 0 , U = U 1 in Ω, (3.1.7)
where B = (b ij ) is the boundary coupling matrix of order N with constant elements.

Regularity of solutions with Neumann boundary conditions

As a problem with Neumann boundary conditions, a problem with Robin boundary conditions no longer enjoys the hidden regularity as in the case with Dirichlet boundary conditions. In particular, in higher dimensional space, the solution to the problem with Robin boundary condition is not smooth enough for the proof of the non-exact boundary controllability of the system, which makes a huge trouble.

Consider the following second order hyperbolic problem on a bounded domain Ω ⊂ R n (n ≥ 2) with boundary Γ:

           y tt + A(x, ∂)y = f in (0, T ) × Ω = Q, ∂y ∂ν A | Σ = g on (0, T ) × Γ = Σ, t = 0 : y = y 0 , y t = y 1 in Ω, (3.2.1)
where

A(x, ∂) = - n i,j=1 a ij (x) ∂ 2 ∂x i ∂x j + n j=1 b j (x) ∂ ∂x j + c 0 (x), (3.2.2)
in which a ij (x) with a ij (x) = a ji (x), b j (x) and c 0 (x) are smooth real coefficients, and the principal part of A(x, ∂) is supposed to be uniformly strong elliptic in Ω:

n i,j=1 a ij (x)η i η j ≥ c n j=1 η 2 j , ∀ x ∈ Ω, ∀ η = (η 1 , • • • , η n ) ∈ R n , (3.2.3)
where, c > 0 is a positive constant, ∂y ∂ν A is the outward normal derivative of A: In [START_REF] Lasiecka | Regularity theory of hyperbolic equaitons with non-homogeneous Neumann boundary conditions. II. General boundary data[END_REF], Lasiecka and Triggiani got the optimal regularity of the solution to problem (3.2.1) by means of the theory of cosine operator. In particular, more regularity results can be obtained when the domain is a parallelepiped. For conciseness and clarity, we list only those results needed in this paper.

∂y ∂ν A = N i=1 N j=1 a ij (x) ∂y ∂x i ν j , (3.2
Let > 0 be an arbitrarily small number. Here and hereafter, we always assume that α, β are given respectively as follows: Theorem 3.1. Assume that B is similar to a real symmetric matrix. Then for any given (Φ 0 , Φ 1 ) ∈ (H 1 ) N × (H 0 ) N , the adjoint problem

           α = 3/
                   Φ -∆Φ + A T Φ = 0 in (0, +∞) × Ω, Φ = 0 on (0, +∞) × Γ 0 , ∂ ν Φ + B T Φ = 0 on (0, +∞) × Γ 1 , t = 0 : Φ = Φ 0 , Φ = Φ 1 in Ω (3.3.6)
admits a unique weak solution

(Φ, Φ ) ∈ C 0 loc ([0, +∞); (H 1 ) N × (H 0 ) N ) (3.3.7)
in the sense of C 0 semigroup, where H 1 and H 0 are defined by (3.1.3).

Proof. Since B is similar to a real symmetric matrix, there exists an invertible matrix P such that B = P -1 BP is symmetric. Let Φ = P T Φ and  = P -1 AP . Problem (3.3.6) can be transformed to and

                   Φ -∆ Φ + ÂT Φ = 0 in (0, +∞) × Ω, Φ = 0 on (0, +∞) × Γ 0 , ∂ ν Φ + B Φ = 0 on (0, +∞) × Γ 1 , t = 0 : Φ = P T Φ 0 = Φ0 , Φ = P T Φ 1 = Φ1 in Ω, (3.3.8
F ( Φ) = A T Φ -λ(N + λ|m| 2 ) Φ -2λm ⊗ ∇ Φ, (3.3.10)
where ⊗ denotes the tensor product. A direct calculation leads to

                   Φ -∆ Φ + F ( Φ) = 0 in (0, +∞) × Ω, Φ = 0 on (0, +∞) × Γ 0 , ∂ ν Φ + B Φ = 0 on (0, +∞) × Γ 1 , t = 0 : Φ = 1 h Φ0 , Φ = 1 h Φ1 in Ω. (3.3.11)
By the multiplier geometric condition (3.1.1), for any given θ > 0, there exists a positive constant λ > 0 so large that

( Bξ, ξ) ≥ θ|ξ| 2 , ∀x ∈ Γ 1 , ∀ξ ∈ R n , (3.3.12)
then B is a positive definite symmetric matrix.

Let

X = (H 1 ) N × (H 0 ) N . (3.3.13)
For any given (Φ i , Ψ i ) T ∈ X(i = 1, 2), we define the inner product in X as and

  Φ 1 Ψ 1   ,   Φ 2 Ψ 2   X = Ω ∇Φ 1 • ∇Φ 2 dx + Γ1 ( BΦ 1 , Φ 2 )d Γ + Ω Ψ 1 Ψ 2 dx. ( 3
D(A) = (H 2 (Ω) ∩ H 1 ) N × (H 1 ) N ⊆ X. (3.3.16)
Obviously, D(A) is dense in X. We now prove that A is the infinitesimal generator of a C 0 contractive semigroup on X.

Firstly, we prove that A is dissipative. In fact, it is easy to see from (3.3.14) that

A   Φ Ψ   ,   Φ Ψ   X =   Ψ ∆Φ   ,   Φ Ψ   X = Ω ∇Φ • ∇Ψdx + Γ1 ( BΨ, Φ)d Γ + Ω ∆Φ • Ψdx = Ω ∇Φ • ∇Ψdx + Γ1 ( BΨ, Φ)d Γ + Γ1 ∂ ν Φ • Ψd Γ - Ω ∇Φ • ∇Ψdx = Γ1 (Ψ, BΦ)d Γ - Γ1 ( BΦ, Ψ)d Γ = 0, ∀(Φ, Ψ) ∈ D(A). (3.3.17)
Then, we prove that 0 ∈ ρ(A). For any given (Φ i , Ψ i ) ∈ X (i = 1, 2), we define the following bilinear functional:

a(   Φ 1 Ψ 1   ,   Φ 2 Ψ 2   ) = A   Φ 1 Ψ 1   ,   Φ 2 Ψ 2   def = Ω Ψ 1 • Ψ 2 dx + Γ1 ( BΦ 1 , Φ 2 )d Γ + Ω ∇Φ 1 • ∇Φ 2 dx. (3.3.18)
By Lemma 3.7, (3.3.1) holds. Obviously, for any given (Φ, Ψ) ∈ X, we have

a(   Φ Ψ   ,   Φ Ψ   ) = Ω |∇Φ| 2 dx + Γ1 ( BΦ, Φ)d Γ + Ω |Ψ| 2 dx = (Φ, Ψ) 2 X . (3.3.19)
Hence, it follows from Lemma 3.6 that for any given F ∈ X , there exists a unique (Φ 1 , Ψ 1 ) ∈ X, such that a(

  Φ 1 Ψ 1   ,   Φ 2 Ψ 2   ) = A   Φ 1 Ψ 1   ,   Φ 2 Ψ 2   = F,   Φ 2 Ψ 2   , ∀   Φ 2 Ψ 2   ∈ X (3.3.20)
and

(Φ 1 , Ψ 1 ) X = A -1 F X ≤ 1 δ F X , (3.3.21)
where X is the dual space of X. It shows that A -1 is a bounded linear operator from X to X, namely 0 ∈ ρ(A). Thus, by Lemma 3.8, A is the infinitesimal generator of a C 0 contractive semigroup on X.

Finally, problem (3.3.11) can be written as Noting that F ( Φ) is a bounded linear mapping in X, B is a bounded linear operator in X, by Lemma 3.9, A + B is the infinitesimal generator of a C 0 semigroup on X. The proof is complete.

  Φ Φ   = A   Φ Φ   + B   Φ Φ   , (3.3
Remark 3.3. From now on, in order to guarantee the well-posedness of the problem with coupled Robin boundary condition, we always assume that B is similar to a real symmetric matrix.

U, U ) ∈ C 0 ([0, T ]; (H 0 ) N × (H 1 ) N ), (
such that for any given (Φ 0 , Φ 1 ) ∈ (H 1 ) N × (H 0 ) N and t (0 ≤ t ≤ T ), we have the following equality:

(U (t), -U (t)), (Φ(t), Φ (t)) (H 1 ) N ×(H0) N ;(H1) N ×(H0) N = (U 1 , -U 0 ), (Φ 0 , Φ 1 ) (H 1 ) N ×(H0) N ;(H1) N ×(H0) N + t 0 Γ1 (DH(τ ), Φ(τ ))dxdt, ( 3
(U 0 , U 1 , H) → (U, U )
is continuous with respect to the corresponding topology.

Proof. Taking the inner product with Φ on both sides of (3.1.6) and integrating by parts, we have

Ω (U (t), Φ(t))dx - Ω (U (t), Φ (t))dx = Ω (U 1 , Φ 0 )dx - Ω (U 0 , Φ 1 )dx + t 0 Γ1
(DH(τ ), Φ(τ ))dxdt, (3.3.27) which can be written as

(U (t), -U (t)), (Φ(t), Φ (t)) (H 1 ) N ×(H0) N ;(H1) N ×(H0) N = (U 1 , -U 0 ), (Φ 0 , Φ 1 ) (H 1 ) N ×(H0) N ;(H1) N ×(H0) N + t 0 Γ1 (DH(τ ), Φ(τ ))dxdt. (3.3.28)
Define a linear form as follows:

L t (Φ 0 , Φ 1 ) = (U 1 , -U 0 ), (Φ 0 , Φ 1 ) (H 1 ) N ×(H0) N ;(H1) N ×(H0) N + t 0 Γ1 (DH(τ ), Φ(τ ))dxdt. ( 3
(t)) ∈ (H 1 ) N × (H 0 ) N , such that L t • S -1 t (Φ(t), Φ (t)) = (U (t), -U (t)), (Φ(t), Φ (t)) (H 1 ) N ×(H0) N ;(H1) N ×(H0) N . (3.3.30) By L t • S -1 t (Φ(t), Φ (t)) = L t (Φ 0 , Φ 1 ), (3.3.31)
for any given (Φ 0 , Φ 1 ) ∈ (H 1 ) N × (H 0 ) N , (3.3.25) holds, and we have

(U (t), -U (t)) (H 1 ) N ×(H0) N = L t • S -1 t ≤ c( (U 1 , U 0 ) (H 1 ) N ×(H0) N + H (L 2 (0,T ;L 2 (Γ1))) M ), ∀ t ∈ [0, T ], (3.3.32)
where L t • S -1 t is the operator norm of L t • S -1 t . At last, by the classical dense approximation method, we obtain the regularity desired by (3.3.26). Problem (3.1.6) can be rewritten to the following problem with Neumann boundary conditions:

               ψ -∆ψ + 2λ∇h • ∇ψ + λ(∆h -λ|∇h| 2 )ψ = -e λh (e, AU ) in (0, T ) × Ω, ψ = 0 on (0, T ) × Γ 0 , ∂ ν ψ = e λh (e, DH) on (0, T ) × Γ 1 , t = 0 : ψ = 0, ψ = 0 in Ω. ( 3
               ψ -∆ψ + 2λ∇h • ∇ψ + λ(∆h -λ|∇h| 2 )ψ = -e λh (e, AU ) in (0, T ) × Ω, ψ = 0 on (0, T ) × Γ 0 , ∂ ν ψ = 0 on (0, T ) × Γ 1 , t = 0 : φ = 0, φ = 0 in Ω (3.4.8) satisfies (ψ, ψ ) ∈ C 0 ([0, T ]; H 1 (Ω) × L 2 (Ω)) ⊂ C 0 ([0, T ]; H α (Ω) × H α-1 (Ω)). (3.4.9)
Next, we consider the following problem with inhomogeneous Neumann boundary conditions but without internal force terms:

               ψ -∆ψ + 2λ∇h • ∇ψ + λ(∆h -λ|∇h| 2 )ψ = 0 in (0, T ) × Ω, ψ = 0 on (0, T ) × Γ 0 , ∂ ν ψ = e λh (e, DH) on (0, T ) × Γ 1 , t = 0 : φ = 0, φ = 0 in Ω. ( 3
ψ, ψ ) ∈ C 0 ([0, T ]; H α (Ω) × H α-1 (Ω)) (3.4.11)
and Remark 3.4. By Lemma 3.4, it is easy to check that the solution U to the following problem with any given internal force term F ∈ L 2 (0, T ; L 2 (Ω)):

ψ| Γ1 ∈ H 2α-1 (Σ 1 ) = H 2α-1 (0, T ; L 2 (Γ 1 )) ∩ L 2 (0, T ; H 2α-1 (Γ 1 )). ( 3
                   U -∆U + AU = F in (0, +∞) × Ω, U = 0 on (0, +∞) × Γ 0 , ∂ ν U + BU = DH on (0, +∞) × Γ 1 , t = 0 : U = U 0 , U = U 1 in Ω ( 3

Exact boundary controllability and non-exact boundary controllability

In this section, we study the exact boundary controllability and non-exact boundary controllability for the coupled system (3.1.6) of wave equations with coupled Robin boundary controls. We will prove that when M = N , namely, when D is invertible, system (3.1.6) is exactly boundary controllable for any given initial data (U 0 , U 1 ) ∈ (H 1 ) N × (H 0 ) N . However, when M < N and Ω is a parallelepiped, system (3.1.6) is not exactly boundary controllable in (H 1 ) N × (H 0 ) N .

Theorem 3.4. Assume that M = N . Then there exists a T > 0, such that for any given initial data (U 0 , U 1 ) ∈ (H 1 ) N × (H 0 ) N , there exists a boundary control function H ∈ (L 2 loc (0, +∞; L 2 (Γ 1 ))) N with compact support in [0, T ], such that system (3.1.6) is exactly boundary controllable at the time T , and the control function continuously depends on the initial data:

H (L 2 (0,T ;L 2 (Γ1))) N ≤ c (U 0 , U 1 ) (H1) N ×(H0) N , (3.5.1)
where, c > 0 is a positive constant.

Proof. By Lemma 3.1 and Remark 3.1, for any given initial data (U 0 , U 1 ) ∈ (H

H (L 2 (0,T ;L 2 (Γ1))) N ≤ c (U 0 , U 1 ) (H1) N ×(H0) N . ( 3
∂ ν U = DH (3.5.3)
We then prove that L and R are compact mappings, which contradicts Theorem 3.5. In fact, by Theorem

3.4, θ → H is a continuous mapping from L 2 (Ω) to (L 2 (0, T ; L 2 (Γ 1 ))) M . By Theorem 3.3, (θ, H) → U is a continuous mapping from L 2 (Ω) × (L 2 (0, T ; L 2 (Γ 1 ))) M to (C 0 (0, T ; H α (Ω)) ∩ C 1 (0, T ; H α-1 (Ω))) N .
Besides, by Lions's compact embedding theorem ( [START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF]), the mapping from {ψ ∈ (L 2 (0, T ;

H α (Ω))) N , ∂ t ψ ∈ (L 2 (0, T ; H α-1 (Ω))) N } to (L 2 (0, T ; L 2 (Ω))) N is compact, hence L is a compact mapping from L 2 (Ω) to L 2 (0, T ; L 2 (Ω)). By (3.4.2), H → U | Σ1 is a continuous mapping from (L 2 (0, T ; L 2 (Γ 1 ))) M to (H 2α-1 (Σ 1 )) N , then, R : θ → -(e, BU )| Σ1 is a continuous mapping from L 2 (Ω) to H 2α-1 (Σ 1 ). When Ω is a parallelepiped, α = 3/4 -, then 2α -1 > 1 -α, and H 2α-1 (Σ 1 ) → H 1-α (Σ 1 ) is a compact embedding, therefore, R is a compact mapping from L 2 (Ω) to H 1-α (Σ 1

Exact boundary synchronization

Based on the results on the exact boundary controllability and the non-exact boundary controllability, we continue to study the exact boundary synchronization for system (3.1.6) under coupled Robin boundary controls.

Let

C 1 =         1 -1 1 -1 . . . . . . 1 -1         (N -1)×N (3.6.1)
be the corresponding full row-rank matrix of synchronization. We have

Ker(C 1 ) = Span{e}, (3.6.2)
where e = (1, 1, • • • , 1) T . By the definition of synchronization (see [START_REF] Li | Exact boundary synchronization for a coupled system of wave equations with Neumann boundary controls[END_REF] and [START_REF] Li | Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls[END_REF]), the system is exactly boundary synchronizable at the time T > 0, which means that for any given initial data (U 0 , U 1 ) ∈ (H 1 ) N × (H 0 ) N , there exists a boundary control function H ∈ (L 2 loc (0, +∞; L 2 (Γ 1 ))) N with compact support in [0, T ], such that the corresponding solution U = U (t, x) to the mixed problem (3.1.6)-(3.1.7) satisfies t ≥ T : C 1 U ≡ 0 in Ω.

(3.6.3)

Let S s be a (m s -m s-1 -1) × (m s -m s-1 ) full row-rank matrix:

S s =         1 -1 1 -1 . . . . . . 1 -1         , 1 ≤ s ≤ p, (3.8.3)
and let C p be the following (N -p) × N matrix of synchronization by p-groups:

C p =         S 1 S 2 . . . S p         . (3.8.4)
Evidently, we have

Ker(C p ) = Span{e 1 , • • • , e p }, (3.8.5)
where for 1 ≤ s ≤ p, Assume that both A and B satisfy the following C p -conditions of compatibility:

(e s ) j =      1, m s-1 + 1 ≤ j ≤ m s , 0
AKer(C p ) ⊆ Ker(C p ), BKer(C p ) ⊆ Ker(C p ). ( 3
C p A = A p C p , C p B = B p C p . (3.8.16) Let W = C p U, D p = C p D. (3.8.17) We have                W -∆W + A p W = 0 in (0, +∞) × Ω, W = 0 on (0, +∞) × Γ 0 , ∂ ν W + B p W = D p H on (0, +∞) × Γ 1 , t = 0 : W = C p U 0 , W = C p U 1 in Ω. ( 3

C p -conditions of compatibility

In this section, we discuss the necessity of the C p -conditions of compatibility. This is a problem closely related to the number of boundary control functions. We first study the C p -condition of compatibility for the coupling matrix A. In particular, when α rs = 0, 1 ≤ r, s ≤ p, (3.9.12)

we say that A satisfies the zero-sum condition by blocks. In this case, we have Ae s = 0, 1 ≤ s ≤ p. By Holmgren's uniqueness theorem ( [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]), u ≡ 0 on the whole domain Ω, then system (3.1.6) is exactly boundary null controllable. However, rank(D) = N -2, and Ω is a parallelepiped, it contradicts Theorem 3.6.

We now prove that it is also impossible to have rank(C 2 Be 1 , C 2 Be 2 ) = 1. Otherwise, since where m = x -x 0 .

Denote the energy function of system (3.11.1) as Integrating by parts the second term on the right-hand side of (3.11.9), and using Rellich's identity (3.11.5) to the third term on the right-hand side of (3.11.9), we get Dans cette thèse, nous étudions la synchronisation, qui est un phénomène bien répandu dans la nature. Elle a été observé pour la première fois par Huygens en 1665. En se basant sur les résultats de la contrôlabilité frontière exacte, pour un système couplé d'équations des ondes avec des contrôles frontières de Neumann, nous considérons la synchronisation frontière exacte (par groupes), ainsi que la détermination de l'état de synchronisation. Ensuite, nous considérons la contrôlabilité exacte et la synchronisation exacte (par groupes) pour le système couplé avec des contrôles frontières couplés de Robin. A cause du manque de régularité de la solution, nous rencontrons beaucoup plus de difficultés. Afin de surmonter ces difficultés, on obtient un résultat sur la trace de la solution faible du problème de Robin grâce aux résultats de régularité optimale de Lasiecka-Triggiani sur le problème de Neumann. Ceci nous a permis d'établir la contrôlabilité exacte, et, par la méthode de la perturbation compacte, la non-contrôlabilité exacte du système. De plus, nous allons étudier la détermination de l'état de synchronisation, ainsi que la nécessité des conditions de compatibilité des matrices de couplage.

E(t

2 0. 1 . 1 2 0. 1 . 2 3 0. 1 . 3 6 0. 1 . 4 6 0. 1 . 5 8 0. 1 . 6 1 15 1. 1

 2112123136146158161151 Thanks to Paul Espinasse, I gradually joined a group of dancers. It was with these people that I experienced 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . La contrôlabilité et la synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . La synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synchronisation pour un système couplé d'équations aux dérivées partielles . . . . . . Résultats principaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . La synchronisation frontière exacte pour le système couplé d'équations des ondes avec des contrôles frontières de Neumann . . . . . . . . . . . . . . . . . . . . . . . . . . . . La contrôlabilité frontière exacte et la synchronisation frontière exacte pour un système couplé d'équations des ondes avec des contrôles frontières couplés de Robin . . . . . . Introduction From controllability to synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Synchronization phenomena and their study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Synchronization for a coupled system of partial differential equations . . . . . . . . . . . . . . 1.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Exact boundary synchronization for a coupled system of wave equations with Neumann boundary controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 4 4 Appendix 4 . 1

 4441 Regularity of solutions to the mixed initial-boundary value problem with coupled Robin boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Exact boundary controllability and non-exact boundary controllability . . . . . . . . . . . . 3.6 Exact boundary synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Determination of the state of synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Exact boundary synchronization by groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 C p -conditions of compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Determination of the state of synchronization by groups . . . . . . . . . . . . . . . . . . . . . 3.11 Continuous uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some preliminary algebraic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography viii nization by groups, coupled system of wave equations, Neumann boundary control, coupled Robin boundary control. 2000 MR Subject Classification: 93B05, 93B07, 93C20 Chinese Library Classification: O177 x La contrôlabilité frontière exacte et la synchronisation frontière exacte pour un système couplé d'équations des ondes avec des contrôles frontières de Neumann et des contrôles frontières couplés de Robin RESUME Pour un système couplé d'equations des ondes, la contrôlabilité frontière exacte et la synchronisation frontière exacte peuvent être réalisées par des contrôles frontières de Dirichlet, de Neumann ou de Robin, etc. A ce jour, il existe beaucoup de résultats pour le système couplé d'équations des ondes avec des contrôles frontières de Dirichlet. Sur la base de ces résultats, dans cette thèse, nous considérons la synchronisation frontière exacte pour le système couplé d'équations des ondes avec des contrôles frontières de Neumann et des contrôles frontières couplés de Robin. Cette thèse est organisée comme suit: Dans le premier chapitre, nous donnerons une courte introduction sur la synchronisation et l'état de l'art en recherche. Elle comprend la définition du sujet, la situation actuelle de la recherche, et la relation entre la synchronisation et la contrôlabilité. Dans le deuxième chapitre, en se basant sur les résultats de la contrôlabilité frontière exacte, pour un système couplé d'équations des ondes avec des contrôles frontières de Neumann, nous considérons la synchronisation frontière exacte et la synchronisation frontière exacte par groupes. De plus, nous allons détailler la détermination de l'état de synchronisation exacte et synchronisation exacte par 2 et 3 groupes respectivement.

Figure 1 :

 1 Figure 1: Manuscrit de Huygens, expérience des pendules.

0. 1 . 4

 14 Résultats principauxDans cette thèse, on considère la contrôlabilité frontière exacte et la synchronisation frontière exacte pour le système couplé d'équations des ondes U -∆U + AU = 0 dans (0, +∞) × Ω (0.1.4) avec la condition aux bords de Dirichlet sur Γ 0 : U = 0 sur (0, +∞) × Γ 0 , (0.1.5) la condition aux bords de Neumann ou de Robin couplé sur Γ 1 :

.1. 11 )

 11 alors le système (1.4.1)-(1.4.4) possède la contrôlabilité frontière exacte au moment T > 0.

.1. 13 )

 13 alors le système (1.4.1)-(1.4.4) possède la synchronisation frontière exacte par p-groupes au moment

  then studied the differential equation satisfied by the perturbed part of the solution, and put forward necessary conditions of stability.In 1998, L.M. Pecora and T.L.Carroll further developed the Lyapunov matrix method. They constructed a Master Stability Function to discuss the relationship between the topologic structure and stability, in which not only the stable case, but also the unstable case were taken into consideration. This method is called the Master Stability Function Method. However, this method can only solves the problem of linear and local stability. In 2007, D. A. Wiley, S. H. Strogatz and M. Girvan suggested a new method, known as the Basin Stability Method, to study the non-linear global stability of the system. The Basin Stability Method can obtain the attraction domain of attractors by the numerical simulation method. With this method, we can not only calculate the probability of the solution returning back to a state of synchronization after a stochastic disturbance, but also obtain the dependance of the state of synchronization to the topologic structure.

  4.1)-(1.4.2) and (1.4.4) is exactly boundary synchronizable, then the coupling matrix A should satisfy the C 1 -compatibility condition (1.4.13).

Theorem 1 . 2 .

 12 Assume that system (1.4.1)-(1.4.2) and (1.4.4) is exactly boundary synchronizable by pgroups, then M ≥ N -p. In particular, as M = N -p, the coupling matrix A = (a ij ) satisfies the following C p -compatibility condition:

  ). Under the C p -compatibility condition (1.4.21), if rank(C p D) = N -p, then system (1.4.1)-(1.4.2) and (1.4.4) is exactly boundary synchronizable by p-groups under the action of boundary control function H ∈ L N -p . The C p -compatibility condition (1.4.21) is equivalent to the fact that there exists constants α rs (1 ≤ r, s ≤ p) such that ms j=ms-1+1

( 1 . 4 . 23 ) 1 . 3 .

 142313 Theorem Under the C p -compatibility condition(1.4.21), assume that A T possesses an invariant sub-space Span{E 1 , E 2 , • • • , E p } which is bi-orthonormal to Ker(C p ) = Span{e 1 , • • • , e p }.Then there exists a boundary control matrix D, such that the state of synchronization by p-groups u = (u 1 , • • • , u p ) T is independent of the applied boundary controls, and can be determined as follows: t ≥ T : u = ψ, (1.4.24)

Introduction 1 . 4 . 2

 142 Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controlsIn Chapter 3, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave equations with coupled Robin boundary controls. Due to difficulties from the lack of regularity of the solution, we have to confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In particular, in higher-space-dimensional case, the regularity of the solution in the general case is not enough to be used to prove the non exact boundary controllability for the system lacking boundary controls. In order to overcome this difficulty, we transform the mixed initialboundary problem with coupled Robin boundary conditions into a mixed problem with Neumann boundary conditions, then by the results on the regularity of the solution to the mixed problem with Neumann boundary conditions(Lasiecka and Triggiani), as well as the method of compact perturbation, we can discuss the regularity of the solution to the mixed problem with coupled Robin boundary conditions. Moreover, in the

  the weak solution U to the problem (1.4.1)-(1.4.2) and (1.4.5)-(1.4.6) satisfies

  we can find a boundary function H ∈ (L 2 loc (0, +∞; L 2 (Γ 1 ))) N with compact support on [0, T ], such that system (1.4.1)-(1.4.2) and (1.4.5) is exactly boundary controllable at the time T , and the control function H continuously depends on the initial data:

1 2 (

 2 Σ 1 )) N , and the proof can be accomplished with such a regularity. In general, under fewer boundary controls, the non exact boundary controllability for the coupled system with coupled Robin boundary controls is still an open problem ([20]). Theorem 1.7. Assume that rank(D) = M < N and Ω ⊂ R n is a parallelepiped. Then no matter how large T > 0 is, system (1.4.1)-(1.4.2) and (1.4.5) is not exactly boundary null controllable for any given initial

Theorem 1 . 8 .

 18 Assume that Ω ⊂ R n is a parallelepiped. If the coupled system of wave equations with coupled Robin boundary controls (1.4.1)-(1.4.2) and (1.4.5) is exactly boundary synchronizable, then rank(C 1 D) = N -1. (1.4.31) Under certain conditions, the coupling matrices A and B satisfy the corresponding C 1 -compatibility conditions similar to (1.4.13). In fact, we have Theorem 1.9. Let Ω ⊂ R n be a parallelepiped. Assume that rank(D) = N -1. If the coupled system (1.4.1)-(1.4.2) and (1.4.5) is exactly boundary synchronizable, then the following C 1 -compatibility conditions hold: AKer(C 1 ) ⊆ Ker(C 1 ), BKer(C 1 ) ⊆ Ker(C 1 ). (1.4.32) Similarly to the case with Neumann boundary controls, the exact boundary synchronization (by groups) for the coupled system with coupled Robin boundary controls can be realized from the exact boundary null controllability of the corresponding reduced system. Theorem 1.10. Assume that Ω satisfies the usual multiplier geometric condition. Assume furthermore that both A and B satisfy the C 1 -compatibility conditions (1.4.32). Then we can find a boundary control matrix D satisfying rank(D) = rank(C 1 D) = N -1, (1.4.33) such that system (1.4.1)-(1.4.2) and (1.4.5) is exactly boundary synchronizable, and the applied boundary control function H continuously depends on the initial data:

.4. 34 )

 34 For the exact boundary synchronization by groups, we can get similar results. At least (N -p) boundary controls are obligatory for the synchronization by p-groups (see §3.8). Here, the coupling matrix A in the coupled system of wave equations satisfies the C p -compatibility condition(1.4.21). However, compared with the internal coupling matrix A, to study the necessity of the C p -compatibility condition of the coupling matrix B on the boundary is more complicated. This concerns the regularity of the solution to the problem with coupled Robin boundary conditions (see §3.9). We have obtained the C p -compatibility condition of B only under certain restricted conditions (see Theorem 3.17 and Theorem 3.18).

Theorem 1 . 11 . 1 . 1 . 12 .

 1111112 Assume that both A and B satisfy the C 1 -compatibility condition (1.4.32). Assume furthermore that A T and B T possess a common eigenvector E ∈ R N with (E, e) = 1, where e = (1, • • • , 1) T . Then we can find a boundary control matrix D, rank(D) = N -1, such that system (1.4.1)-(1.4.2) and (1.4.5) is exactly boundary synchronizable, and the state of synchronization is independent of the applied boundary control function. On the contrary, assume that under the C 1 -compatibility condition (1.4.32), system (1.4.1)-(1.4.2) and(1.4.5) is exactly boundary synchronizable. Assume furthermore that there exists a non-trivial vector E ∈ R N , such that the project φ = (E, U ) is independent of the applied boundary control function H in (0, T ) × Ω, then E is a common eigenvector of A T and B T , E ∈ Ker(D T ), and, without loss of generality, we may assume that (E, e) = Theorem Assume that both A and B satisfy the C 1 -compatibility condition(1.4.32). Let E ∈ R N be a eigenvector of B T , satisfying (E, e) = 1. Then there exists a boundary control matrix D such that system (1.4.1)-(1.4.2) and (1.4.5) is exactly boundary synchronizable, and the state of synchronization u satisfies the following estimate:

Theorem 2 . 6 .

 26 Under the condition of compatibility (2.3.8), assume that A T possesses an invariant subspace Span{E 1 , E 2 , • • • , E p } which is bi-orthonormal to Ker(C p ) = Span{e 1 , • • • , e p }. Then there exists a boundary control matrix D ∈ D N -p , such that the state of synchronization by p-groups u = (u 1 , • • • , u p ) T is independent of the applied boundary controls, and can be determined as follows: t ≥ T : u = ψ, (2.4.2)

  4.4) in (2.4.1), we get a boundary control matrix D ∈ D N -p , such that E r ∈ Ker(D T ), 1 ≤ r ≤ p.(2.4.5)

. 4 )

 4 ν = (ν 1 , • • • , ν n ) T being the unit normal vector.Define the operator A byA = A(x, ∂), D(A) = y ∈ H 2 (Ω) : ∂y ∂ν A | Γ = 0 . (3.2.5)

5 -

 5 , β = 3/5, for a general smooth bounded domain Ω, and A(x, ∂)is defined by (3.2.2), α = β = 3/4 -, for a parallelepiped Ω, A(x, ∂) = -∆.

  ) therefore, it is only necessary to prove that problem (3.3.8) is well-posed. Let x 0 ∈ R n and λ be a constant large enough. Define m(x) = x -x 0 , h(x) = e λ|m(x)| 2 /2 , Φ = h Φ. Let B = B + λ(m, ν)I (3.3.9)

Definition 3 . 1 .

 31 U is a weak solution to the mixed problem (3.1.6)-(3.1.7), if

.4. 12 )

 12 Since this regularity result holds for all the eigenvectors of B T , and all the eigenvectors of B T constitute a set of basis in R N , we get the desired (3.4.1) and (3.4.2).

.4. 13 )

 13 satisfies (3.4.1)-(3.4.2), too.

.5. 2 )

 2 Noting that M = N , D is invertible, then the boundary condition in(3.1.2)

Theorem 3 . 14 . 8 ) 5 . 3 . 15 .

 31485315 Let Ω ⊂ R n be a parallelepiped. Assume that system (3.1.6) is exactly synchronizable by p-groups. Then we have rank(C p D) = N -p.(3.8.In particular, we have rank(D) ≥ N -p. (3.8.9) Proof. If Ker(D T ) ∩ Im(C T p ) = {0}, by Lemma 4.10 in Appendix, we have rank(C p D) = rank(D T C T p ) = rank(C T p ) = N -p. (3.8.10) Next, we prove that it is impossible to have Ker(D T ) ∩ Im(C T p ) = {0}. Otherwise there exists a a vector E = 0, such that D T C T p E = 0. (3.8.11) Let w = (E, C p U ), Lθ = -(E, C p AU ), Rθ = -(E, C p BU ). (3.8.12) We get again problem (3.5.6) for w. Besides, the exact boundary synchronization by p-groups for system (3.1.6) indicates that the final condition (3.5.7) holds. Similarly to the proof of Theorem 3.7, we get a contradiction to Theorem 3.Theorem Let C p be the (N -p) × N matrix of synchronization by p-groups defined by (3.8.3)-(3.8.4).

.8. 13 )

 13 Then there exists a boundary control matrix D satisfyingrank(D) = rank(C p D) = N -p,(3.8.14)such that system (3.1.6) is exactly boundary synchronizable by p-groups, and the corresponding boundary control function H possesses the following continuous dependence:H (L 2 (0,T,L 2 (Γ1))) N -p ≤ c C p (U 0 , U 1 ) (H1) N -p ×(H0) N -p . (3.8.15) Proof. Since both A and B satisfy the C p -conditions of compatibility (3.8.13), by Lemma 4.7 in Appendix, there exist matrices A p and B p of order (N -p), such that

Let D be defined

  by Ker(D T ) = Span{e 1 , • • • , e p } = Ker(C p ). (3.8.19)We have rank(D) = N -P , andKer(C p ) ∩ Im(D) = Ker(C p ) ∩ {Ker(C p )} ⊥ = {0}. (3.8.20) By Lemma 4.10 in Appendix, we get rank(C p D) = rank(D) = N -p, thus D p is an invertible matrix of order (N -p). By Theorem 3.4, the reduced system (3.8.18) is exactly boundary controllable, then system (3.1.6) is exactly boundary synchronizable by p-groups. By (3.5.1), we get (3.8.15).

Theorem 3 . 16 .Cp 5 . 3 . 10 .

 3165310 Let Ω ⊂ R n be a parallelepiped. Assume that M = rank(D) = N -p. If system (3.1.6) is exactly boundary synchronizable by p-groups, then the coupling matrix A = (a ij ) satisfies the following C p condition of compatibility: AKer(C p ) ⊆ Ker(C p ). (3.9.1) Proof. It suffices to prove that C p Ae s = 0, 1 ≤ s ≤ p. By (3.8.7), taking the inner product with C p on both sides of (3.1.6), we get t ≥ T : p s=1 Ae s u s = 0 in Ω. (3.9.2) If there exists an s such that C p Ae s = 0, then there exist constant coefficients α s (1 ≤ s ≤ p) such that p s=1 α s u s = 0 in Ω, (3.9.3) in which not all the α s (1 ≤ s ≤ p) are equal to zero. Let r , e s ) = e r 2 δ rs , we have t ≥ T : c p+1 U = 8.19) and (3.9.4), it is easy to see that c T p+1 ∈ Im(C T p ), then, rank( C p-1 ) = N -p + 1. By rank(D) = N -p, we have Ker(D T ) ∩ Im( C T p-1 ) = {0}, then there exists a vector E = 0, such thatD T C T p-1 E = 0. (3.9.7) Let w = (E, C p-1 U ), Lθ = -(E, C p-1 AU ), Rθ = -(E, C p-1 BU ). (3.9.8) We get again problem (3.5.6) for w. Noting (3.8.6) and (3.9.5), we have t = T : w(T ) = (E, C p-1 U ) = (E, have w (T ) = 0, then (3.5.7) holds. Noting that Ω ⊂ R n is a parallelepiped, similarly to the proof of Theorem 3.7, we can get a conclusion that contradicts Theorem 3.Remark The C p -condition of compatibility (3.9.1) is equivalent to the fact that there exist constants α rs (1 ≤ r, s ≤ p) such that Ae s = p r=1 α rs e r , 1 ≤ s ≤ p, (3.9.10) or, A satisfies the following row-sum condition by blocks: ms j=ms-1+1 a ij = α rs , m r-1 + 1 ≤ i ≤ m r , 1 ≤ r, s ≤ p. (3.9.11) When p = 1, this condition of compatibility becomes (4.1.19) below.

2

 2 (3.9.13) On the other hand, taking the inner product with C 2 on both sides of the boundary condition of(3.1.6) on Γ 1 , we gett ≥ T : C Be 1 u 1 + C 2 Be 2 u 2 ≡ 0 on Γ 1 . (3.9.36) If rank(C 2 Be 1 , C 2 Be 2 ) = 0, then C 2 Be 1 = C 2 Be 2 = 0, namely, B satisfies the C 2 -condition of compatibility.If rank(C 2 Be 1 , C 2 Be 2 ) = 2, then we have u ≡ 0 on Γ 1 . By the boundary condition on Γ 1 in (3.9.31), we get∂ ν u ≡ 0 on Γ 1 .

C 2 1 2 2 ΩΩ |∇u| 2 dx + 2 Γ

 2122 Be 1 = 0 or C 2 Be 2 = 0, it follows from (3.9.36) that there exists a non-zero vector D 2 ∈ R 2 , such that t ≥ T : u. By Theorem 3.22 and Remark 3.12, the following Hautus's criterion rank( Λ -µI 2 , D) = 2, ∀µ ∈ R (3.9.39) guarantees the uniqueness of the trivial solution to system (3.9.35) in the infinite time interval [T, ∞) under observation (3.9.38). Then by the non-exact boundary null controllability of system (3.1.6), condition (3.9.39) does not hold. Thus, there exists a vector E = 0 in R 2 , such that ΛT E = ΛE = µE, DT E = 0. (3.9.40) Noting (3.9.38) and the second formula of (3.9.40), we have E and w| Γ1 ∈ Ker( D). Since Dim Ker( D) = 1, there exists a constant α such that we have w = αE on Γ 1 . Therefore, noting the first formula of (3.9.40), we have Λw = ΛαE = µαE = µw on Γ 1 . (3.9.41)Proof. First we recall Green's formulaΩ ∆uvdx = -Ω ∇u • ∇vdx + Γ ∂u ∂ν vd Γ , ∀ u ∈ H 2 (Ω), v ∈ H 1 (Ω) (3.11.4) and Rellich's identity ([9]) ∆u(m • ∇u)dx = (n -2) ∂u ∂ν (m•∇u)d Γ -Γ (m, ν)|∇u| 2 d Γ , ∀u ∈ H 2 (Ω), (3.11.5)

8 ) 2 Ω

 82 Taking the inner product with 2m • ∇u on both sides of the first equation of (3.11.1) and integrating by parts, we get0 = [u t (m • ∇u)]| T t=0 dx -T 0 Ω m • ∇|u t | 2 dxdt -2 T 0 Ω∆u(m • ∇u)dxdt.(3.11.9)

0 = 2 Ω(|u t | 2 +(|u| 2 +

 222 [u t m • ∇u]| T t=0 dx -ν)|u t | 2 d Γ dt + n ν)|u t | 2 d Γ dt +2 T 0 Γ ∂u ∂ν (m • ∇u)d Γ dt -T 0 Γ (m, ν)|∇u| 2 d Γ dt. (3.11.11) Noting the multiplier geometric condition (3.1.1), there exist R and δ > 0, such that(m, ν) ≥ δ > 0, m ∞ = R < +∞, then on Γ 1 we have ν)|∇u| 2 = (m, ν)| ∂u ∂ν | 2 . (3.11.14)Thus, noting the multiplier geometric condition (3.1.1), we have ν)|u t | 2 d Γ dt + 4R ν)|u t | 2 d Γ dt + 4R 2 a 2 δ T 0 Γ1 |u| 2 d Γ dt. (3.11.16)Taking the inner product with u on both sides of the first equation of (3.11.1) and integrating by parts, |∇u| 2 )dxdt≤ ( 4R 2 a 2 δ -a(n -1)) T 0 Γ1 |u| 2 d Γ dt + T 0 Γ1 (m, ν)|u t | 2 d Γ dt + cE(0). (3.11.19) ν)|v t | 2 d Γ dt + cE(0). |v| 2 + |u t | 2 + |v t | 2 )d Γ dt + cE(0). (3.11.21) Taking the inner product with v on both sides of the first equation of (3.11.1) and integrating by parts, the inner product with u on both sides of the second equation of (3.11.1) and integrating by partsv -v t u]| T t=0 dx + (a -b) observation (3.11.2), we have v = -α β u on Γ 1 , then it comes from (3.11.24) that T 0 Γ1 |u| 2 d Γ dt ≤ cE(0).

  a = b, it follows from (3.11.45) that λ a m = λ b n , ∀m, n ∈ Z. (3.11.55) Besides, by the monotonicity of the function λ → tan λ + λ a , we have λ a m = λ a n , λ b m = λ b n , ∀m, n ∈ Z. (3.11.56) On the other hand, we have λ a n -λ b n = a -b nπ + O(1) n 2 . (3.11.57)

INSTITUT

  .u-strasbg.fr irma@math.unistra.fr IRMA 2018/002 https ://hal.archives-ouvertes.fr/tel-01816413 ISSN 0755-3390 Ins titut de R echerche Mathématique Avancée

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Exact boundary synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Exact boundary synchronization by p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Determination of the state of synchronization by p-groups . . . . . . . . . . . . . . . . . . . . 2.5 State of exact boundary synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 State of exact boundary synchronization by 2-groups . . . . . . . . . . . . . . . . . . . . . . 2.7 State of exact boundary synchronization by 3-groups . . . . . . . . . . . . . . . . . . . . . . 3 Exact controllability and exact synchronization with coupled Robin boundary controls 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.2 Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls . . . . . . . . . . . . . . vii 2.1 3.2 Regularity of solutions with Neumann boundary conditions . . . . . . . . . . . . . . . . . . . 3.3 Existence and uniqueness of solutions to the mixed initial-boundary value problem with coupled Robin boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  -clés: Contrôlabilité frontière exacte, synchronisation frontière exacte, synchronisation frontière exacte par groupes, système couplé d'équations des ondes, contrôle frontière de Neumann, contrôle frontière couplé de Robin. est la variable d'état, et h désigne la fonction de contrôle. Le problème de contrôle consiste à trouver une fonction de contrôle h convenable telle que le comportement dynamique du système satisfait une condition finale sous l'action du contrôle h. La contrôlabilité est une propriété importante, elle joue un rôle crucial dans beaucoup de problèmes mathématiques.

	0.1 Introduction
	2000 MR Subject Classification: 93B05, 93B07, 93C20 0.1.1 La contrôlabilité et la synchronisation
	Chinese Library Classification: O177
	Résumé en français On considère un système dynamique d'une équation différentielle ordinaire:
	ẋ(t) = f (x(t), h(t)),	(0.1.1)
	où x
	xii

Pour les systèmes gouvernés par les équations aux dérivées partielles, le problème de contrôle est un sujet très important. Pour un système hyperbolique, la contrôlabilité exacte signifie que pour toutes les données initiales et finales, il existe un T > 0 et un contrôle convenable sur l'intervalle [0, T ] (contrôle frontière ou contrôle interne), tels que le système satisfait exactement la condition initiale et la condition finale. Quand le contrôle est appliqué sur la frontière, on a la contrôlabilité frontière exacte. Quand le contrôle est appliqué sur le domaine, on a la contrôlabilité exacte interne. Pour le système hyperbolique quasi-linéaire, la contrôlabilité exacte ne peut être réalisée que pour les données initiales et données finales assez petites, on a la contrôlabilité exacte locale, sinon la contrôlabilité exacte globale. Dans cette thèse nous considérons principalement la contrôlabilité frontière exacte.

  de contrôles frontières, la contrôlabilité frontière exacte peut être réalisée si le temps est suffisamment grand. Toutefois, quand l'une des conditions mentionnées ci-dessus n'est pas satisfaite, nous ne pouvons pas obtenir la contrôlabilité frontière exacte en général. Par exemple, dans des applications spécifiques, certaines conditions frontières ont des significations physiques clairement définies, sur lesquelles les contrôles frontières ne peuvent pas être effectuées. Cela signifie qu'il y aura un manque de contrôles frontières, de

sorte que le système n'a pas de contrôlabilité frontière exacte. Dans ce cas, nous devons examiner si le système possède une sorte de contrôlabilité frontière dans un sens plus faible. Tatsien Li et Bopeng Rao ont apporté une contribution pionnière dans les deux aspects suivants. D'une part, par moins de contrôles frontières, ils ont prouvé que le système peut atteindre la synchronisation frontière exacte ([24]-

[START_REF] Li | Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls[END_REF]

), dans lequel toutes les composantes de la variable d'état atteignent un même état de synchronisation, mais l'état de synchronisation est a priori inconnu. D'autre part, ils ont défini la contrôlabilité frontière approchée (

[START_REF] Li | Asymptotic controllability for linear hyperbolic systems[END_REF]

) pour le système hyperbolique: il existe une suite de contrôles frontières, telle que la suite des solutions correspondantes s'approche de zéro quand n → +∞. Mais la suite des contrôles frontières ne converge pas nécessairement. De même, la synchronisation frontière approchée a également été prise en compte. La recherche a conduit à une combinaison de synchronisation et de contrôlabilité, puis l'étude de la synchronisation a développé la théorie du contrôle pour les systèmes gouvernés par des équations aux dérivées partielles.

0.1.2 La synchronisation

La synchronisation est un phénomène bien répandu dans la nature. Elle a été observé pour la première fois par Huygens en 1665 ([7]), qui a trouvé que deux pendules sur le mur se synchronisent en phase. Ensuite, les scientifiques ont progressivement commencé l'observation et l'étude sur ces phénomènes intéressants, en particulier, la synchronisation de tuyaux d'orgue, le clignotement synchrone des lucioles dans les forêts tropicales, les cris synchrones des grenouilles après une pluie d'été, la synchronisation dans le système neuronal, et la synchronisation des pacemakers cardiaques, etc ([43]). Dans les années 1950, N. Wiener a commencé une recherche théorique sur la synchronisation d'un point de vue mathématique ([44]). Les modèles biologiques ont été proposés par A.T. Winfree en 1967. Ce modèle ne considère que la phase plutôt que l'amplitude des oscillateurs, et ce travail a largement favorisé le

  Après cela, eux et leurs collaborateurs, ils ont successivement obtenu beaucoup de résultats. En 2014, Tatsien Li, Bopeng Rao et Long Hu ont étudié la synchronisation frontière exacte pour un système couplé d'équations des ondes avec

divers contrôles frontières (type Dirichlet, Neumann, Robin couplé et dissipatif couplé) en cas unidimensionnel dans le cadre de solutions classiques, et profondément étudié la synchronisation par groupes (

[START_REF] Li | A note on the exact synchronization by groups for a coupled system of wave equations[END_REF]

,

[START_REF] Li | Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls[END_REF]

) et la détermination de l'état de synchronisation (

[START_REF] Hu | Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type[END_REF]

,

[START_REF] Li | Exact boundary synchronization for a coupled system of 1-D wave equations[END_REF]

). En outre, Tatsien Li, Bopeng Rao et Yimin Wei ont étendu la définition de la synchronisation dans un travail en 2014, dans lequel ils ont proposé le concept de synchronisation généralisée.

De plus, lorsque la condition des multiplicateurs échoue, et (ou) qu'il y a un manque de contrôles frontières, nous devrions considérer la synchronisation dans un sens plus faible, c'est-à-dire, lorsque les différences entre les composantes de la variable d'état tendent vers zéro, ce qui est appelé la synchronisation frontière approchée. Dans le travail de Tatsien Li et Bopeng Rao en 2014 (

[START_REF] Li | Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF]

,

[START_REF] Li | Criteria of Kalman's type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF]

), ils ont également mentionné le critère de Kalman comme une condition nécessaire pour la synchronisation frontière approchée.

En se basant sur la recherche de la synchronisation frontière pour le système avec des contrôles frontières de Dirichlet, c'est une étape nécessaire et difficile pour étudier plus loin la synchronisation pour le système avec d'autres contrôles frontires. Différents types de contrôles frontières correspondent à différents modèles physiques et donnent des systèmes différents en substance. Nous devrions étudier si le changement de la solution résultant du changement des conditions frontières aura un impact sur la synchronisation, et si des résultats similaires peuvent être obtenus comme ceux avec des contrôles frontières de Dirichlet. Ce sera l'étude principal dans cette thèse.

  inconnu a priori, est l'état de synchronisation par p-groupes.

	Il existe déjà des résultats complets sur la contrôlabilité frontière exacte et la synchronisation frontière
	exacte pour le système couplé d'équations des ondes avec des contrôles frontières de Dirichlet. Sur cette
	base, nous étudions la contrôlabilité frontière exacte et la synchronisation frontière exacte pour le système
	couplé d'équations des ondes avec d'autres types de contrôles frontières.
	0.1.5 La synchronisation frontière exacte pour le système couplé d'équations
	des ondes avec des contrôles frontières de Neumann
	Le deuxième chapitre concerne la synchronisation frontière exacte et la synchronisation frontière exacte par
	groupes pour un système couplé d'équations des ondes avec des contrôles frontières de Neumann. Dans
	le cas avec des contrôles frontières de Dirichlet, nous savons que le système est exactement contrôlable
	et exactement synchronisable pour toutes les données initiales dans L 2 (0, L) × H -1 (0, L) par un contrôle
	frontière dans L 2 (0, T ), cependant, avec des contrôles frontières de Neumann, l'inégalité d'observabilité pour
	le problème adjoint correspondant n'est valide que sous une norme plus faible, donc il faut une plus forte
	régularité pour l'espace contrôlable.
	De plus, l'espace contrôlable ne peut pas être exactement décrit comme celui sous les contrôles frontières
	de Dirichlet.
	Afin d'obtenir la synchronisation frontière exacte pour le système initial, nous définissons la matrice de
	synchronisation suivante:

  On transforme la synchronisation frontière exacte du système initial en la contrôlabilité frontière exacte du système réduit en W . On essaie de trouver un contrôle frontière adéquat pour obtenir la contrôlabilité frontière exacte du système réduit, et ce contrôle est exactement celui qui réalise la synchronisation frontière exacte du système initial. De cette manière, on peut étudier la synchronisation par la contrôlabilité. Par ailleurs, avec assez de contrôles frontières, la contrôlabilité frontière exacte du système réduit est réalisable, de même que la synchronisation frontière exacte du système initial.

	On prouvera que la C 1 -condition de compatibilité (1.4.13) est suffisante pour garantir la synchronisation
	frontière exacte. La nécessité de la C 1 -condition de compatibilité est un autre sujet important dans la
	recherche de la synchronisation.
	Théorème 0.1. Supposons qu'on a la condition des multiplicateurs. Soit x 0 ∈ R n , tel que en posant
	m = x -x 0 , on a
	(m, ν) ≤ 0, ∀x ∈ Γ 0 ,	(m, ν) > 0, ∀x ∈ Γ 1 ,	(0.1.19)
	où (•, •) désigne le produit scalaire dans R n . Sous la C 1 -condition de compatibilité (1.4.13), si la matrice D
	du contrôle satisfait rang(C 1 D) = N -1, alors il existe un T > 0, tel que pour toutes les données initiales

  Ensuite, on étudie la détermination de l'état de synchronisation. Si la matrice A possède certaines propriétés, l'état de synchronisation est indépendant du contrôle frontière. En général, l'état de synchronisation dépend de la donnée initiale (U 0 , U 1 ) et aussi du contrôle H appliqué.

	Théorème 0.2. Sous la C 1 -condition de compatibilité (1.4.13), il existe une constante a indépendante de
	i = 1, • • • , N , telle que
	N
	a ij	def. = a (i = 1, • • • , N ).	(0.1.21)
	j=1
	Soit e = (1, • • • , 1) T le vecteur propre de A correspondant à la valeur caractéristique a. On suppose que E
	est orthogonal à e. Alors, il existe une matrice D et une constante c > 0 indépendante de la donnée initiale,
	telles que l'état de synchronisation u satisfait l'estimation suivante:
			.1.20)
	où c est une constante positive, telle que le problème (1.4.1)-(1.4.4) et (1.4.6) soit exactement synchronisable
	au moment T > 0.
	Réciproquement, si M = N -1, si le système (1.4.1)-(1.4.4) est exactement synchronisable, alors la
	matrice A satisfait la C 1 -condition de compatibilité (1.4.13).
	En revanche, si M < N -1, le système n'est pas exactement synchronisable.

  .1.25) Posant W = C p U , la synchronisation frontière exacte du système initial est transformée en la contrôlabilité frontière exacte du système réduit en W . Le reste est similaire, cependant, il faut au moins (N -p) contrôles frontières pour réaliser la synchronisation frontière exacte par p-groupes.

	0.1.6 La contrôlabilité frontière exacte et la synchronisation frontière exacte
		pour un système couplé d'équations des ondes avec des contrôles fron-
		tières couplés de Robin
	Au chapitre 3, nous considérons la contrôlabilité frontière exacte et la synchronisation frontière exacte (par
	groupes) pour un système couplé d'équations des ondes avec des contrôles frontières couplés de Robin. Le
	problème est plus difficile à cause du manque de régularité de la solution faible. On transforme le problème
	aux conditions frontières de Robin en un problème aux conditions frontières de Neumann. Puis, on utilise
	les résultats de la régularité optimale du problème de Neumann de Lasiecka-Triggiani et la méthode de la
	perturbation compacte pour établir la contrôlabilité exacte et la non-contrôlabilité exacte selon le nombre
	de contrôles. D'ailleurs, dans cette situation, nous avons non seulement une matrice de couplage A dans le
	système, mais aussi une matrice de couplage B sur les conditions frontières couplées de Robin. L'interaction
	entre ces deux matrices complique le problème ([20]).
	Ici et après, selon les cas différents, nous définissons α, β, respectivement, comme suit:
	
	 	α = 3/5 -, β = 3/5,	si Ω est un ouvert borné général avec la frontière régulière,	(0.1.26)
	 	α = β = 3/4 -,	si Ω est un parallélépipède,
	où > 0 est un réel arbitrairement petit.
	Théorème 0.3. Pour tout H ∈ (L 2 (0, T ; L 2 (Γ 1 ))) M et tous (U 0 , U 1 ) ∈ (H 1 × H 0 ) N , la solution faible U du
	problème (1.4.1)-(1.4.2)et (1.4.5)-(1.4.6) satisfait

  on peut trouver un contrôle frontière H ∈ (L 2 loc (0, +∞; L 2 (Γ 1 ))) N à support compact dans [0, T ], tel que le système (1.4.1)-(1.4.2) et (1.4.5) est exactement contrôlable au moment T . Le contrôle H dépend continûment de la donnée initiale:

  .1.33) Ensuite, on considère la détermination de l'état de synchronisation. Sous certaines conditions algébriques des matrices A et B, l'état de synchronisation est indépendant du contrôle appliqué. En général, l'état de synchronisation dépend non seulement de la donnée initiale (U 0 , U 1 ), mais aussi du contrôle H appliqué. On suppose que A et B satisfont les C 1 -conditions de compatibilité(1.4.32). Soit E ∈ R N un vecteur propre du B T , tel que (E, e) = 1. Alors il existe une matrice D du contrôle, telle que le système

	Théorème 0.9. On suppose que A et B satisfont les C 1 -conditions de compatibilités (1.4.32). On suppose
	que A T et B T possèdent un vecteur propre commun E ∈ R N avec (E, e) = 1, où e = (1, • • • , 1) T . Alors on
	peut trouver une matrice D du contrôle, rang(D) = N -1, telle que le système (1.4.1)-(1.4.2) et (1.4.5) est
	exactement synchronisable, et l'état de synchronisation est indépendant du contrôle H appliqué.
	Réciproquement, on suppose que sous les C 1 -conditions de compatibilité (1.4.32), le système (1.4.1)-
	(1.4.2) et (1.4.5) est exactement synchronisable. On suppose qu'il existe un vecteur non-trivial E ∈ R N , tel
	que la projection φ = (E, U ) est indépendante du contrôle, alors E est un vecteur propre commun de A T et
	B T et E ∈ Ker(D T ), tel que (E, e) = 1.
	Théorème 0.10. (1.4.1)-(1.4.2) et (1.4.5) est exactement synchronisable, et que l'état de synchronisation satisfait

  There are already complete results on the exact boundary controllability and the exact boundary synchronization for the coupled system of wave equations with Dirichlet boundary controls. Based on this, we study the exact boundary controllability and the exact boundary synchronization for the coupled system of wave equations with other types of boundary controls.Chapter 2 concerns the exact boundary synchronization and the exact boundary synchronization by groups for a coupled system of wave equations with Nuemann boundary controls. In the case with Dirichlet boundary controls, we know that the system is exactly boundary controllable and exactly boundary synchronizable

	1.4.1 Exact boundary synchronization for a coupled system of wave equations
	with Neumann boundary controls

.4.11) then system(1.4.1)-(1.4.2) and (1.4.4) is exactly boundary synchronizable by p groups at the time T > 0, and (u 1 , • • • , u p ) T , being a priori unknown, is called to be the state of synchronization by pgroups.

  Thus, the exact boundary synchronization for the original system (1.4.1)-(1.4.2) and (1.4.4) of U is equivalent to the exact boundary null controllability of the reduced system (1.4.15) of W . Hence, if we can find a proper boundary control function such that the reduced system is exact boundary null controllable, we can correspondingly find the boundary control function that realizes the exact boundary synchronization for the original system. It then turns out that it is possible to study the synchronization problem by means of studying the controllability. We know that the reduced system is exactly boundary controllable with enough boundary controls, hence, the original system should be exactly boundary synchronizable.

The exact boundary synchronization by groups can be discussed in a similar method. The corresponding (N -p) × N matrix of synchronization by p-groups is given by

  1 2 , and any given boundary function H ∈ L M , the mixed initial-boundary value problem (2.1.1)-(2.1.2) admits a unique weak solution U ∈ (C loc ([0, +∞); H 1-s )) N ∩ (C 1 loc ([0, +∞); H -s )) N with continuous dependance. Definition 2.1. System (2.1.1) is exactly null controllable at the time T > 0 in the space (H 1 × H 0 ) N , if for any given initial data (U 0 , U 1 ) ∈ (H 1 × H 0 ) N , there exists a boundary control H ∈ L M with compact support

	in [0, T ], such that the corresponding mixed initial-boundary value problem (2.1.1)-(2.1.2) admits a unique
	weak solution U ∈ (C loc ([0, +∞);

  System (2.1.1) is exactly synchronizable at the time T > 0 in the space (H 1 × H 0 ) N , if for any given initial data (U 0 , U 1 ) ∈ (H 1 × H 0 ) N , there exists a boundary control H ∈ L M with compact support

	Then we have either
	AKer(C) ⊆ Ker(C)	(2.1.8)
	or there exists a full row-rank (N -p + 1) × N matrix Ĉ such that
	t ≥ T :	ĈU = 0 in Ω.	(2.1.9)
	2.2 Exact boundary synchronization
	Definition 2.2.
	Since Lemma 2.1 in [29] is independent of the type of boundary conditions, we still have
	Lemma 2.4. Assume that U is the solution to the mixed problem (2.1.1)-(2.1.2). Let C be a full row-rank
	(N -p) × N matrix (where p ≥ 1) such that
	t ≥ T :	CU = 0 in Ω.	(2.1.7)

in [0, T ], such that the weak solution U = U (t, x) to the mixed initial-boundary value problem (2.1.1)-(2.1.2)

  and each group is required to possess the exact boundary synchronization, respectively, and, in the meantime, every group is independent of each other.

Definition 2.3. System (2.1.1) is exactly synchronizable by p-groups at the time T > 0 in the space (H 1 × H 0 ) N , if for any given initial data (U 0 , U 1 ) ∈ (H 1 × H 0 ) N , there exists a boundary control H ∈ L M with compact support in [0, T ], such that the weak solution U = U (t, x) to the mixed initial-boundary value problem (2.1.1)-(2.1.2) satisfies t ≥ T : u

  Proof By (2.3.7), we have C p U = 0 in Ω when t ≥ T . If AKer(C p ) ⊆ Ker(C p ), by Lemma 2.4, we can construct a full row-rank (N -p + 1) × N matrix C1 such that C1 U = 0 in Ω when t ≥ T . If AKer( C1 ) ⊆ Ker( C1 ), still by Lemma 2.4, we can construct another full row-rank (N -p + 2) × N matrix C2 such that C2 U = 0 in Ω when t ≥ T , • • • . This procedure should stop at the r th step, where 0 ≤ r ≤ p.

	Thus, we get a full row-rank (N -p + r) × N matrix Cr such that
	t ≥ T :	Cr U = 0	in Ω		(2.3.9)
	and
	AKer( Cr ) ⊆ Ker( Cr ).		(2.3.10)
					2.3.6)
	Thus, (2.3.2) can be equivalently written as
			p
			u s e s	in Ω.	(2.3.7)
			s=1
	Theorem 2.3. Assume that system (2.1.1) is exactly synchronizable by p-groups. Then we necessarily have

t ≥ T : C p U ≡ 0 or U = M ≥ N -p. Especially, when M = N -p, the coupling matrix A = (a ij ) should satisfy the following condition of compatibility: AKer(C p ) ⊆ Ker(C p ). (2.3.8) By Lemma 4.7 in Appendix, there exists a unique matrix à of order (N -p + r), such that Cr A = à Cr .

  then, (2.3.11) is exactly null controllable at the time T in the space (H 1 × H 0 ) N -p+r . By Lemma 2.2 and Lemma 2.3, it is necessary that rank

  Remark 2.2. The condition of compatibility (2.3.8) is equivalent to the fact that there exist some constants α rs (1 ≤ r, s ≤ p) such that

	p
	Ae s =	α rs e r , 1 ≤ s ≤ p,	(2.3.15)
	r=1
	or, noting (2.3.6), A satisfies the following row-sum condition by blocks:
	ms
	j=ms-1+1

.3.8) holds.

  .3.16) Especially, this condition of compatibility becomes (2.2.2) when p = 1. Theorem 2.4. Let C p be the (N -p) × N matrix of synchronization by p-groups defined by (2.3.3)-(2.3.4). Under the condition of compatibility (2.3.8), assume that the N × (N -p) boundary control matrix D has full column-rank and satisfies rank(C

p D) = N -p. Then system (2.1.1) is exactly synchronizable by p-groups by means of boundary control H ∈ L N -p , moreover, we have the continuous dependence:

  Proof Since {E 1 , E 2 , • • • , E p } is bi-orthonormal to {e 1 , • • • , e p },similarly to (2.4.4), there exists a boundary control matrix D ∈ D N -p , such that (2.4.5) holds. Let φ r = (E r , U ). Taking the inner product with E r on both sides of (2.1.1)-(2.1.2), we get

.4.8) where ψ = (ψ 1 , • • • , ψ p ) is the solution to problem (2.4.3), and s > 1 2 .

  .[START_REF] Li | Approximate boundary null controllability and approximate boundary synchronization for a coupled system of wave equations with Neumann boundary controls[END_REF]) and (2.4.22), we have

  Theorem 3.2. For any given H ∈ (L 2 (0, T ; L 2 (Γ 1 ))) M and (U 0 , U 1 ) ∈ (H 0 ) N × (H 1 ) N , problem (3.1.6)-(3.1.7) admits a unique weak solution(U, U ) ∈ C 0 ([0, T ]; (H 0 ) N × (H 1 )

	N ),	(3.3.26)
	moreover, the mapping

.3.25)

in which Φ(t) is the solution to the adjoint problem (3.3.6).

  1 ) N × (H 0 ) N , there exists a boundary control function H ∈ (L 2 loc (0, +∞; L 2 (Γ 1 ))) N with compact support in [0, T ], such that problem (3.1.2) with Neumann boundary controls is exactly boundary controllable at the time T , and the control function H continuously depends on the initial data:

  ). The proof is complete.Remark 3.6. We obtain the non-exact boundary controllability for system (3.1.6) with coupled Robin boundary controls on a parallelepiped Ω in the case of lack of boundary controls. The main idea is to use the compact perturbation theory which has a higher requirement on the regularity of the solution to the problem with coupled Robin boundary condition. How to generalize this result to the general domain is still an open problem.

  Thus, the exact boundary snchronization by p-groups for system (3.1.6) is equivalent to the exact boundary controllability for the reduced system (3.8.18), and the boundary control H, which realizes the exact boundary controllability for the reduced system (3.8.18), must be the boundary control which realizes the exact boundary synchronization for system(3.1.6).

.8.18)

Noting that C p is a surjection from R N to R N -p , any given initial data (U 0 , U 1 ) ∈ (H 1 ) N × (H 0 ) N corresponds to a unique initial data (C p U 0 , C p U 1 ) for the reduced system

(3.8.18)

.

  ) = 1 2 Ω (|u t | 2 + |∇u| 2 + |v t | 2 + |∇v| 2 )dx +

	a 2 Γ1	|u| 2 d Γ +	b 2 Γ1	|v| 2 d Γ .	(3.11.6)
	By system (3.11.1) and integrating by parts, we have
	E (t) =

Ω (u t u tt + ∇u • ∇u t + v t v tt + ∇v • ∇v t )dx + a Γ1 uu t d Γ + b Γ1 vv t d Γ = Ω (u t ∆u + ∇u • ∇u t + v t ∆v + ∇v • ∇v t )dx + a Γ1 uu t d Γ + b Γ1 vv t d Γ = 0,

(3.11.

  |v| 2 d Γ dt ≤ cE(0).(3.11.26) Noting that the equations in (3.11.1) do not change after having taken derivative with respect to t, we obtain the following estimates of u t , v t on Γ 1 :+ |∇u t | 2 + |∆v| 2 + |∇v t | 2 )dx + a 2 Γ1 |u t | 2 d Γ + b 2 Γ1 |v t | 2 d Γ . (3.11.29) By e 2iλ = cos 2λ + i sin 2λ = cos 2 λ -sin 2 λ + 2i sin λ cos λ, (3.11.46) and noting (3.11.45), we have cos 2 λ -sin 2 λ = 1 -tan 2 λ 1 + tan 2 λ = a 2 -λ 2 a 2 + λ 2 ,

	T T 0 0 sin λ cos λ = e 2iλ = a 2 -λ 2 1 + tan 2 λ Γ1 |u t | 2 d Γ dt ≤ c Ê(0) Γ1 |v t | 2 d Γ dt ≤ c Ê(0), tan λ = -aλ a 2 + λ 2 , a 2 + λ 2 -2aλi a 2 + λ 2 = -λ + ai λ -ai . The asymptotic expansion of e 2iλ at λ = ∞ gives and where Ê(t) = 1 2 Ω then e 2iλ = -1 -2ai λ + O(1) λ 2 . Taking the logarithm on both sides, we get 2iλ = ln(-1 -2ai λ + O(1) λ 2 ) + 2nπi = iπ + ln 1 + 2ai λ + O(1) λ 2 + 2nπi = iπ + 2ai λ + O(1) λ 2 + 2nπi, then λ = λ a n = (n + 1 2 )π + 2a λ + O(1) λ 2 . Noting λ a n ∼ nπ, we get λ a n = (n + 1 2 )π + a nπ + O(1) n 2 . (|∆u| 2 (3.11.47) (3.11.27) (3.11.28) (3.11.48) (3.11.49) (3.11.50) (3.11.51) (3.11.52) (3.11.53) Similarly, by the second formula of (3.11.44), we have

D'ailleurs, en cas de coefficients variables: ω i = ω i (t), K ij = K ij (t), le système est un réseau adaptatif, la structure se développe au fil du temps et forme une boucle de rétroaction, de sorte que le système peut observer et ajuster continuellement sa structure pour obtenir une synchronisation ou l'éviter. D'un autre côté, non seulement la procédure de développement est fortement liée à la structure du réseau, mais la structure topologique du réseau détermine aussi la stabilité de l'état de synchronisation dans une large mesure.Dans les problèmes pratiques, en particulier dans les systèmes chaotiques, les solutions sont sensibles aux données initiales, ce qui nous amène à poser la question suivante: après l'ajout de la perturbation au moment initial, le signal supplémentaire disparaît-il avec le temps? En 1983, H. Fujisaka et T. Yamada ont proposé une méthode de matrice Lyapunov étendue pour étudier la stabilité de synchronisation pour les systèmes chaotiques. Ils supposent que le système part d'un état de synchronisation au moment initial, et que les états de synchronisation forment un ensemble invariant. Ils ont ajouté une perturbation au système, décomposé la solution en une partie synchronisée et une partie perturbée: θ i (t) = s(t)(partie synchronisée) + δθ i (t)(partie perturbée), i = 1, • • • , N, puis étudié ensuite l'équation différentielle de la partie perturbée, et proposé les conditions nécessaires pour sa stabilité. En 1998, L.M. Pecora et T.L.Carroll ont développé la méthode de matrice Lyapunov. Ils ont construit une fonction de stabilité maîtresse pour discuter la relation entre la structure topologique et la stabilité, non seulement le cas stable, mais aussi le cas instable ont été pris en considération. Cette méthode s'appelle la méthode de fonction de stabilité maîtresse. Cependant, cette méthode ne peut que traiter la stabilité linéaire et locale. En 2007, D. A. Wiley, S. H. Strogatz et M. Girvan ont proposé une nouvelle méthode, connue sous le nom de méthode de stabilité du bassin, pour étudier la stabilité globale non-linéaire. La méthode de stabilité du bassin peut obtenir le domaine d'attraction des attracteurs par une méthode de simulation numérique. Avec cette méthode, nous pouvons non seulement calculer la probabilité que la solution retourne à un état de synchronisation après une perturbation stochastique, mais aussi obtenir la dépendance de l'état de synchronisation à la structure topologique.

ACKNOWLEDGEMENTS

First and foremost, I would like to express my deepest and sincere gratitude to my two advisors, Prof.

Chapter 3

Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls Exact controllability and exact synchronization with coupled Robin boundary controls

Introduction

Synchronization is a widespread natural phenomenon. It was first observed by Huygens in 1665 ( [START_REF] Huygens | Oeuvres Complètes[END_REF]). The theoretical research on synchronization from mathematical point of view dates back to N. Wiener in 1950s ( [START_REF] Wiener | Cybernetics, or control and communication in the animal and the machine[END_REF]). Since 2012, Li and Rao started the research on synchronization for coupled systems governed by PDEs, meanwhile, synchronization can be realized in a finite time by means of proper boundary controls.

Consequently, the study of synchronization becomes a part of research in control theory. Precisely speaking, Li and Rao considered the exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in any given space dimensions in the framework of weak solutions ( [START_REF] Li | Synchronisation exacte d'un système couplé d'équations des ondes par des contrôles frontières de Dirichlet[END_REF], [START_REF] Li | Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls[END_REF], [START_REF] Li | On the exactly synchronizable state to a coupled system of wave equations[END_REF]) and in one-space-dimensional case in the framework of classical solutions ( [START_REF] Hu | Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type[END_REF], [START_REF] Li | Exact boundary synchronization for a coupled system of 1-D wave equations[END_REF], [START_REF] Lu | Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its applications[END_REF]). Corresponding results have been expanded to the exact boundary synchronization by p(≥ 1) groups ( [START_REF] Li | A note on the exact synchronization by groups for a coupled system of wave equations[END_REF], [START_REF] Li | Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls[END_REF]). Moreover, Li and Rao proposed the concept of approximate boundary null controllability and approximate boundary synchronization in [START_REF] Li | Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF] and [START_REF] Li | Criteria of Kalman's type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF] and further studied them.

Let Ω ⊂ R n be a bounded domain with sufficiently smooth boundary Γ = Γ 1 ∪ Γ 0 (Γ 1 ∩ Γ 0 = ∅), ∂ ν denotes the outward normal derivative on the boundary, the coupling matrix A = (a ij ) is of order N , the boundary control matrix D is a full column-rank matrix of order N × M (M ≤ N ), both A and D having real constant elements, U = (u (1) , • • • , u (N ) ) T and H = (h (1) , • • • , h (M ) ) T denote the state variables and the boundary controls, respectively. The discussion on the control problem will become more flexible because of the introduction of the boundary control matrix D.

In this paper, we always assume that Ω satisfies the usual multiplier geometric condition ( [START_REF] Lions | Perturbations et Stabilisation de Systmes Distribués[END_REF]). Without loss of generality, we assume that there exists an x 0 ∈ R n , such that by setting m = x -x 0 , we have (m, ν) ≤ 0, ∀x ∈ Γ 0 ;

(m, ν) > 0, ∀x ∈ Γ 1 ,

where (•, •) denotes the inner product in R n .

Inspired by the synchronization of the system with Dirichlet boundary controls, Li, Lu and Rao studied the following coupled system of wave equations with Neumann boundary controls:

and the corresponding results on the exact boundary synchronization and the approximate boundary synchronization have been obtained ( [START_REF] Li | Exact boundary synchronization for a coupled system of wave equations with Neumann boundary controls[END_REF], [START_REF] Li | Approximate boundary null controllability and approximate boundary synchronization for a coupled system of wave equations with Neumann boundary controls[END_REF], [START_REF] Li | Exact boundary controllability for a coupled system of wave equations with Neumann controls[END_REF]). In particular, we have and

where Σ = (0, T ) × Γ, (D(A γ )) (γ > 0) is the dual space of D(A γ ) with respect to L 2 (Ω). (1) If (y 0 , y 1 ) ∈ H 1 (Ω) × L 2 (Ω), then problem (3.2.1) admits a unique solution y such that (y, y ) ∈ C 0 ([0, T ]; H 1 (Ω) × L 2 (Ω)) (3.2.12) and y| Γ ∈ H β (Σ).

(3.2.13)

(2) If (y 0 , y 1 ) ∈ L 2 (Ω) × (H 1 (Ω)) , where (H 1 (Ω)) is the dual space of H 1 (Ω) with respect to L 2 (Ω), then problem (3.2.1) admits a unique solution y such that

and y| Γ ∈ H α-1 (Σ).

(3.2.15)

Remark 3.2. In the results mentioned above, the mappings of solutions are continuous with respect to the corresponding topologies.

Existence and uniqueness of solutions to the mixed initialboundary value problem with coupled Robin boundary conditions

Lemma 3.6 (Lax-Milgram Theorem [START_REF] Lax | Functional Analysis[END_REF]). Assume that a(u, v) is a bounded coercive bilinear functional on the Hilbert space X, namely, there exists an M > 0, such that

and there exists a δ > 0, such that a(u, u) ≥ δ u 2 X , ∀u ∈ X.

(3.3.2)

Then for any given f ∈ X , there exists a unique u ∈ X, satisfying

where X is the dual space of X, and < •, • > is the dual product of X and X .

Lemma 3.7 ( [START_REF] Liu | Semigroups associated with dissipative systems[END_REF]). Assume that Ω is a bounded domain in R n with C 1 boundary Γ. Then, for any given function u ∈ H 1 (Ω), we have the following interpolation inequality:

where c is a positive constant independent of u.

Lemma 3.8 ([39]

). Let X be a Hilbert space and let A be an unbounded linear operator, the domain D(A) of which is dense in X. If A is dissipative and 0 ∈ ρ(A), ρ(A) being the resolvent of A, then A is the infinitesimal generator of a C 0 contractive semigroup on X.

Lemma 3.9 ( [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]). Let X be a Hilbert space and let A be the infinitesimal generator of a C 0 semigroup S(t) on X, satisfying S(t) ≤ M e ωt . If B is a bounded linear operator on X, then A + B is the infinitesimal generator of a C 0 semigroup T (t) on X, satisfying T (t) ≤ M e (ω+M B )t , ∀ t ≥ 0.

and

where Σ 1 = (0, T ) × Γ 1 , α is defined by (3.2.6). Moreover, the mapping

is continuous with respect to the corresponding topology.

Proof. Noting (3.2.12) and (3.2.13) in Lemme 3.5, we need only to consider the case when

Assume that Ω is sufficiently smooth, for example, with C we have can be rewritten as

By the regularity result in Theorem 3.3 (in which we take B=0), the trace U | Γ1 ∈ (H 2α-1 (Σ 1 )) N , where α is defined by (3.2.6), and we have 3, for some special domains, the solution may possess better regularity. In particular, when Ω is a parallelepiped, the optimal regularity of trace

This benefits a lot in the proof of the non-exact boundary controllability for the system with fewer boundary controls.

When Ω is a parallelepiped with Γ 0 = {∅}, the boundary Γ of Ω is piecewise smooth. By Fourier analysis, the solution to the dual problem of problem (3.1.2) is still sufficiently smooth. Moreover, Lemma 3.1 and Lemma 3.2 still hold, provided that the initial space (H 1 ) N × (H 0 ) N satisfies the additional condition:

Theorem 3.5. Let L be a compact linear mapping from L 2 (Ω) to L 2 (0, T ; L 2 (Ω)), and let R be a compact

, where α is defined by (3.2.6). Then we can not find a T > 0, such that for any given θ ∈ L 2 (Ω), the solution to the following mixed problem:

satisfies the final condition

Proof. Let φ be the solution to the following problem:

and

(3.5.10) By (3.5.7), taking the inner product with φ on both sides of (3.5.8) and integrating by parts, we get

RθφdΓ.

(3.5.11) Noting (3.5.9)-(3.5.10), we then have

which contradicts the compactness of L and R. The proof is complete.

Theorem 3.6. Assume that rank(D) = M < N , and Ω ⊂ R n is a parallelepiped with Γ 0 = {∅}. Then, no matter how large T > 0 is, system (3.1.6) is not exactly boundary null controllable for any given initial data

Proof. Since M < N , there exists an e ∈ R N , such that D T e = 0. Take the special initial data t = 0 : U = 0, U = eθ (3.5.13)

for system (3.1.6). If the system is exactly boundary controllable for any given θ ∈ L 2 (Ω), then there exists a boundary control function H ∈ (L 2 loc (0, +∞, L 2 (Γ 1 ))) M with compact support in [0, T ], such that the corresponding solution satisfies Otherwise, if Ker(D T ) ∩ Im(C T 1 ) = {0}, then there exists a vector E = 0, such that where u is the corresponding state of synchronization.

Taking the inner product with C 1 on both sides of (3.1.6), we get t ≥ T :

Proof. Since both A and B satisfy the C 1 -conditions of compatibility (3.6.8), by Lemma 4.7 in Appendix, there exist (N -1) matrices A 1 and B 1 such that

(3.6.17)

) to the reduced system (3.6.17). Then, it follows that the exact boundary synchronization for system (3.1.6) is equivalent to the exact boundary controllability for the reduced system (3.6.17), and then the boundary control H, which realizes the exact boundary controllability for the reduced system (3.6.17), is just the boundary control which realizes the exact boundary synchronization for system (3.1.6).

By Lemma 4.9 in Appendix, when B is similar to a symmetric matrix, its reduced matrix B 1 is also similar to a symmetric matrix, which guarantees the well-posedness of the reduced system (3.6.17).

Defining the boundary control matrix D as follows:

is an invertible matrix of order (N -1), by Theorem 3.4, the reduced system (3.6.17) is exactly boundary controllable, hence system (3.1.6) is exactly boundary synchronizable. Moreover, by (3.5.1), we get (3.6.14).

Determination of the state of synchronization

When there is a lack of boundary controls, although under certain conditions, the system can realize exact boundary synchronization by fewer boundary controls, the state of synchronization is a priori unknown, which depends not only on the given initial data, but also on the applied boundary controls. In this section, we will discuss the determination and estimate of the state of synchronization.

Theorem 3.10. Assume that both A and B satisfy the C 1 -conditions of compatibility (3.6.8), and A T and Then, by Theorem 3.9, system (3.1.6) is exactly boundary synchronizable.

Next, we prove that the state of synchronization is independent of the boundary control function which realizes the synchronization. Noting that E is a common eigenvector of A T and B T , and B is similar to a real symmetric matrix, there exist λ ∈ C and µ ∈ R, such that

Obviously, the solution φ to this problem is independent of the boundary control H.

On the other hand, noting

, where α is defined by (3.2.6). Let Û be the Gâteaux derivative of U in the direction of Ĥ:

By linearity, Û satisfies the same problem as U :

We first prove that E ∈ Im(C T 1 ). Otherwise, there exists a vector

Noting that C 1 Û is the solution to the reduced system (3.6.17) with the zero initial data, by the exact boundary synchronization of system (3.1.6), the reduced system (3.6.17) has the exact boundary controllability, then, the value of C 1 Û at the time T can be arbitrarily chosen, as a result, from (3.7.10) we get R = 0, which contradicts the fact that E = 0, hence E ∈ Im(C T 1 ). Then, noting (3.6.2), we can choose E such that (E, e) = 1.

Thus, {E, C T 1 } consists of a basis in R N , hence there exist a λ ∈ C and a vector Q ∈ R N -1 , such that

By (3.7.10), taking the inner product with E on both sides of (3.7.8), and noting (3.7.9), we get

Similarly, by the exact boundary controllability for the reduced system (3.6.17), we can get Q = 0, then, it follows from (3.7.11) that

On the other hand, by (

We claim D T E = 0, namely, E ∈ Ker(D T ). Otherwise, taking Ĥ = D T Ev, we get

Thus, we have

Similarly, since {E, C T 1 } consists of a basis in R N , there exist a µ ∈ R and a vector P ∈ R N -1 , such that

Substituting it into (3.7.16) and noting (3.7.9), we have

then by the exact boundary controllability for the reduced system (3.6.17), we get P = 0, hence we have

which indicates that E is a common eigenvector of A T and B T . The proof is complete.

Theorem 3.12. Assume that both A and B satisfy the C 1 -conditions of compatibility (3.6.8). Let E ∈ R N be an eigenvector of B T , satisfying (E, e) = 1. Then there exists a boundary control matrix D such that system (3.1.6) is exactly boundary synchronizable, and the state of synchronization u satisfies the following estimate:

where φ is the solution to the following problem: Thus, by Theorem 3.9, system (3.1.6) is exactly boundary synchronizable.

Taking the inner product with E on both sides of problem (3.1.6)-(3.1.7) and denoting ψ = (E, U ), we Therefore, there exists a vector R ∈ R N -1 , such that

(3.7.29) By Remark 3.4, we get

) .

(3.7.30)

Noting then W = C 1 U , we use Theorem 3.3 for the reduced problem (3.6.17) to get

then, by (3.5.1) we have

Thus, we get

On the other hand, we have In the general situation, in order to efficiently determine the state of synchronization, combining Lemma such that

β jk E j .

Let

We have

in Ω, Proof. Noting (E 1 , e 1 ) = 1 and t ≥ T :

the state of synchronization u is determined by

Exact boundary synchronization by groups

When there is a further lack of boundary controls, similarly to the case with Dirichlet or Neumann boundary controls, we consider the exact boundary synchronization by p-groups for system (3.1.6) ( p ≥ 1; the special case p = 1 means nothing but the exact boundary synchronization ). This indicates that the components of U are divided into p groups:

where

and each group is required to possess the exact boundary synchronization, respectively, and the requirement of synchronization for every group is independent of each other:

where the corresponding state of synchronization by p-groups

Comparing with the internal coupling matrix A, the study on the necessity of the C p -condition of compatibility for the boundary coupling matrix B is more complicated. It concerns the regularity of solution to the problem with coupled Robin boundary conditions.

Let

be a set of classical orthogonal basis in R N , and let

In what follows, we discuss the necessity of the C p -condition of compatibility for the boundary coupling matrix B under the assumption that

Theorem 3.17. Let Ω ⊂ R n be a parallelepiped. Assume that M = rank(D) = N -p and

(3.9.16)

If system (3.1.6) is exactly boundary synchronizable by p-groups, then the boundary coupling matrix B satisfies the following C p -condition of compatibility:

(3.9.17)

Proof. Noting (3.8.7), as t ≥ T we have

(3.9.18) Noticing (3.9.16) and the fact that subspaces

(3.9.19)

Taking the inner product with C p on both sides of the boundary condition in (3.9.19), and noting (3.8.5), we get

(3.9.20)

We claim that C p Be k = 0 (k = 1, • • • , p), which just mean that B satisfies the corresponding C p -condition of compatibility. Otherwise, we may assume that there exists a k (1 ≤ k ≤ p), such that

then, by the boundary condition in system (3.9.19), we get ), then it is easy to show that rank( C p-1 ) = N -p + 1. Noting rank(Ker(D T )) = p, we have Ker(D T ) ∩ Im( C T p-1 ) = {0}, then there exists a vector E = 0, such that

We get again problem (3.5.6) for w, and (3.5.7) holds. Therefore, noting that Ω is a parallelepiped, similarly to the proof of Theorem 3.7, we get a contradiction to Theorem 3.5. Proof. First, since B is similar to a real symmetric matrix, there exists an invertible real matrix P and a real symmetric matrix B, such that B = P -1 BP .

By the exact boundary synchronization by 2-groups, we have t ≥ T : U = e 1 u 1 + e 2 u 2 in Ω.

(3.9.28) Noting (3.9.13), as t ≥ T we have

Ae i u i = 0, then it is easy to see that t ≥ T :

(3.9.29) Let u = (u 1 , u 2 ) T , êi = P T P e i (i = 1, 2). (3.9.30)

Taking the inner product on both sides of (3.9.29) with êj , we get t ≥ T :

where L = (e i , êj ) and Λ = (Be i , êj ) are 2 × 2 matrices.

We can prove that L is a positive definite symmetric matrix and Λ is a symmetric matrix. In fact, since (e i , êj ) = (e i , P T P e j ) = (P e i , P e j ), (3.9.32)

it is easy to see that L is symmetric. Besides, for any given non-zero vector X = (x, y) T ∈ R 2 , we have

thus L is positive definite. Besides, since (Be i , êj ) = (Be i , P T P e j ) = (P Be i , P e j ) = ( BP e i , P e j ), (3.9.34)

Λ is symmetric.

Taking the inner product on both sides of (3.9.31) with L -1 2 and denoting w = L 1 2 u, we get t ≥ T :

where Λ = L -1 2 ΛL -1 2 is a symmetric matrix.

Thus (3.9.35) can be rewritten as

(3.9.42)

Let z = DT w. Noting (3.9.38), we have

(3.9.43)

Then, by Holmgren's uniqueness theorem, we have

. Define the following row vector

(3.9.45)

Noting (e 1 , e 2 ) = 0 and (3.9.44), we have

Let

(3.9.47)

, it is easy to see that rank( C 1 ) = N -1, and Ker(D T ) ∩ Im( C T 1 ) = {0}, thus, there exists a vector Ẽ = 0, such that

(3.9.49)

We get again problem (3.5.6) for w, satisfying (3.5.7). Noting that Ω is a parallelepiped, similarly to the proof of Theorem 3.7, we get a contradiction to Theorem 3.5.

Determination of the state of synchronization by groups

When the coupling matrices A and B satisfy certain algebraic conditions, the state of synchronization by groups will be independent of applied boundary control functions. In the general situation, the state of synchronization by groups depends not only on the initial data, but also on the applied boundary control functions. In this section, we first discuss the determination of the state of synchronization by p-groups in the former situation, then we present the estimate on the state of synchronization by p-groups in the latter situation.

Since V is a common invariant subspace of A T and B T , there exist constants α ij , β ij such that

β ij E j .

(3.10.5)

in Ω.

(3.10.7)

On the other hand, we have t ≥ T :

are independent of applied boundary control function H in (0, T ) × Ω, which realizes the exact boundary synchronization by p-groups, then V is a common invariant subspace of A T and B T , and biorthogonal to Ker(C p ).

Proof. Similarly to the proof of Theorem 3.11, assuming that (U 0 , U 1 ) = (0, 0), by Theorem 3.3, we have that

, where α is defined by (3.2.6).

Let Û be the Gâteaux derivative of U in the direction of Ĥ, defined by (3.7.7). Û satisfies a similar system to that of U :

(3.10.10) Since the projection functions

Otherwise, there exist an i and a vector

(3.10.12)

Since C p Û is the solution to the reduced problem (3.8.18), noting the equivalence between the exact boundary synchronization by p-groups for the original system and the exact boundary controllability for the reduced system, by the exact boundary synchronization by p-groups for system (3.1.6), we know that the reduced system (3.8.18) is exactly boundary controllable, then the value of C p Û at the time T can be chosen arbitrarily, thus we get

It contracidts E i = 0, then, we have

, then, by Lemma 4.2 and Lemma 4.3 in Appendix, we get that V is biorthogonal to Ker(C p ), and (V, C T p ) consists of a set of basis in R N . Hence there exist constant coefficients α ij (i, j = 1, • • • , p) and vectors

Noting (3.10.12) and taking the inner product with E i on both sides of (3.10.10), we get

Similarly, by the exact boundary controllability for the reduced system (3.8.18), we get

thus we have

which means that V is an invariant subspace of A T .

On the other hand, noting (3.10.12) and taking the inner product with E i on both sides of the boundary condition on Γ 1 in (3.10.10), we get

By Theorem 3.3, for i = 1, • • • , p we have

We claim that

Thus, by (3.10.15) we have

Similarly, there exist constants

Similarly, by the exact boundary controllability for the reduced system (3.8.18), we get

then we have

which indicates that V is also an invariant subspace of B T . The proof is complete.

When A and B do not satisfy all the conditions mentioned in Theorem 3.19, the state of synchronization by p-groups depends on applied boundary control functions. We have the following

where α is defined by (3.2.6), while φ = (φ 1 , • • • , φ p ) is the solution to the following problem (1

in which

by

we have (see Lemma 4.11 in Appendix) Denoting

By the first formula of (3.10.23), noting the assumption that V is biorthogonal to Ker(C p ), without loss of generality, we may assume that

then for any given k ∈ {1, • • • , p}, we get

hence

Taking the inner product on both sides of (3.1.6) with E i , and noting (3.10.25), we have

in Ω, (3.10.33) Similarly to the proof of Theorem 3.12, we get

Continuous uniqueness theorem

In the proof or Theorem 3.18, we claim that under Hautus's criterion (3.9.39), the observation (3.9.38) on the infinite time interval [T, +∞) guarantees a unique trivial solution to problem (3.9.35). In this section, we further discuss this continuous uniqueness problem.

Noting that Λ in problem (3.9.35) is a symmetric matrix, without loss of generality, we consider the following coupled system of wave equations with Robin boundary conditions: In the one-space-dimensional case, the result mentioned above can be further improved. In fact, under where αβ = 0, then we have u(t, x) = v(t, x) ≡ 0 in (0, +∞) × (0, 1). (3.11.37) We first recall a generalized Ingham's Inequality ( [START_REF] Li | Criteria of Kalman's type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls[END_REF]). Let Z denote the set of all the integers, and let

n } 1≤l≤m,n∈Z be a strictly increasing sequence of real numbers: then, when T > 2πD + , the sequence {e iβ (l) n t } 1≤l≤m;n∈Z is ω-linearly independent in L 2 (0, T ), where D + is the upper density of the sequence {β Exact controllability and exact synchronization with coupled Robin boundary controls Without loss of generality, assuming a > b > 0, we can arrange {λ a n } ∪ {λ b n } into a monotonic increasing sequence

and, by a = b, there exist positive constants c, γ > 0, such that

.11.58) Thus, we get the corresponding eigenvectors

and {E a n , E b n } n∈Z consists a set of Hilbert basis in (H 1 (Ω)) N × (L 2 (Ω)) N (see [START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators[END_REF]), then any given solution to system (3.11.35) can be represented by

Especially, W and V ⊥ (or W ⊥ and V ) have the same dimension.

Lemma 4.2 ([32]

). Let V and W be two subspaces of R N . Then

where δ ij is the Kronecker symbol.

Lemma 4.3 ([32]

). Assume that V and W are non-trivial subspaces of R N , then V and W are biorthogonal if and only if

and Lemma 4.6. There exists a unique minimum subspace V 0 containing e, such that V 0 possesses a complement W ⊥ , and V 0 and W ⊥ are invariant for A and B.

Proof. Let V be the set of all the subspaces V containing e, and V possesses a complement W ⊥ such that V and W ⊥ are invariant for A and B.

Assuming that V 1 , V 2 ∈ V, by Lemma 4.5 in Appendix, there exist two subspaces W 1 , W 2 biorthogonal to V 1 and V 2 , respectively, and W 1 , W 2 are invariant for A T and B T . By (

Since W 1 , W 2 are biorthogonal to V 1 and V 2 , respectively, by Lemma 4.3 in Appendix we have

Similarly, we have possesses a complement (W 1 ∩W 2 ) ⊥ , which is invariant for A and B, and V 1 ∩V 2 contains e, then V 1 ∩V 2 ∈ V.

At last, define V 0 by

Obviously, V 0 is unique, namely, it is the desired unique minimum subspace. In particular, if