<p>Cyber-physical systems (CPS) are a class of complicated systems integrating cyber components with physical components. Although such a cyber-physical interaction improves the system performance and intelligence, it increases the system complexity and makes the system vulnerable to various types of faults, failures, and cyber-attacks. To assure the security and improve the resiliency of CPS, it is found that the hybrid system model can be a powerful tool in the domain of fault detection and isolation, cyber-attack diagnosis and containment, as well as resilient control and reconfiguration. Several problems are concerned in this dissertation. For situational awareness, \textit{mode discernibility}, which stands for whether the discrete state of a hybrid system can be correctly identified, is characterized and discussed with potential applications to monitoring system design. For CPS vulnerability analysis, the problem of stealthy attack design for systems with switching structures is investigated, which is motivated by the recent literature. To further understand and remedy for the vulnerabilities, the detectability and identifiability for severe cyber-attacks are defined and characterized, which are followed by the discussions on the methodologies for cyber-attack detection and identification. Last but not least, based on the understanding of identifiability, a framework of resilient control design is proposed to mitigate the impact of cyber-attacks, which can be generalized in future to account for additional design criteria.</p>

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Cette thèse est divisée en trois parties. Dans la première partie, nous commençons par décrire des résultats très connus en théorie du contrôle géométrique tels que le théorème de Chow-Rashevsky, la condition de rang de Kalman, l'application Entrée-Sortie et le test linéaire. De plus, nous définissons et nous étudions brièvement la contrôlabilité locale au voisinage d'un contrôle de référence au premier et au second ordre. Dans la deuxième partie, nous donnons une preuve élémentaire du lemme de Franks linéaire pour les flots géodésiques qui utilise des techniques basiques de théorie du contrôle géométrique. Dans la dernière partie, étant donnée une variété Riemanienne compacte, nous prouvons un lemme de Franks uniforme au second ordre pour les flots géodésiques et on applique le résultat à la théorie de la persistance. Dans cette partie, nous introduisons avec plus de détails les notions de contrôlabilité locale au premier et au second ordre. En effet, nous donnons un résultat de contrôlabilité au second ordre dont la preuve est longue et technique. / This thesis is devided into three parts. In the first part we begin by describing some well known results in geometric control theory such as the Chow Rashevsky Theorem, the Kalman rank condition, the End-Point Mapping and the linear test. Moreover, we define and study briefly local controllability around a reference control at first and second order. In the second part we provide an elementary proof of the Franks lemma for geodesic flows using basic tools of geometric control theory. In the last part, given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory. In this part we introduce with more details notions of local controllability at first and second order. In fact, we provide a second order controllability result whose proof is long and technical.

Dans cette thèse nous considérons l’équation des ondes amorties vectorielle sur une variété riemannienne compacte, lisse et sans bord. L’amortisseur est ici une fonction lisse allant de la variété dans l’espace des matrices hermitiennes de taille n. Les solutions de cette équation sont donc à valeurs vectorielles. Nous commençons dans un premier temps par calculer le meilleur taux de décroissance exponentiel de l’énergie en fonction du terme d’amortissement. Ceci nous permet d’obtenir une condition nécessaire et suffisante la stabilisation forte de l’équation des ondes amorties vectorielle. Nous mettons aussi en évidence l’apparition d’un phénomène de sur-amortissement haute fréquence qui n’existait pas dans le cas scalaire. Dans un second temps nous nous intéressons à la répartition asymptotique des fréquences propres de l’équation des ondes amorties vectorielle. Nous démontrons que, à un sous ensemble de densité nulle près, l’ensemble des fréquences propres est contenu dans une bande parallèle à l’axe imaginaire. La largeur de cette bande est déterminée par les exposants de Lyapunov d’un système dynamique défini à partir du coefficient d’amortissement. / In this thesis we are considering the vectorial damped wave equation on a compact and smooth Riemannian manifold without boundary. The damping term is a smooth function from the manifold to the space of Hermitian matrices of size n. The solutions of this équation are thus vectorial. We start by computing the best exponential energy decay rate of the solutions in terms of the damping term. This allows us to deduce a sufficient and necessary condition for strong stabilization of the vectorial damped wave equation. We also show the appearance of a new phenomenon of high-frequency overdamping that did not exists in the scalar case. In the second half of the thesis we look at the asymptotic distribution of eigenfrequencies of the vectorial damped wave equation. Were show that, up to a null density subset, all the eigenfrequencies are in a strip parallel to the imaginary axis. The width of this strip is determined by the Lyapunov exponents of a dynamical system defined from the damping term.

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático. / The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático. / The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.

Dans cette thèse nous étudions la dynamique engendrée par une famille de flots Hamiltoniens. Un tel système dynamique à plusieurs générateurs est aussi appelé ‘polysystème’. Motivés par des questions liées au phénomène de la diffusion d’Arnold, notre objectif est de construire des trajectoires du polysystème qui relient deux régions lointaines de l’espace des phases. La thèse est divisée en trois parties.Dans la première partie, nous considérons le polysystème engendré par les flots discrétisés d’une famille d’Hamiltoniens Tonelli. En utilisant une approche variationnelle issue de la théorie KAM faible, nous donnons des conditions suffisantes pour l’existence des trajectoires souhaitées.Dans la deuxième partie, nous traitons le cas d’un polysystème engendré par un couple de flots Hamiltoniens à temps continu, dont l’étude rentre dans le cadre de la théorie géométrique du contrôle. Dans ce contexte, nous montrons dans certains cas la transitivité d’un polysystème générique, à l’aide du théorème de transversalité de Thom.La dernière partie de la thèse est dédiée à obtenir une nouvelle version du théorème de transversalité de Thom s’exprimant en termes d’ensembles rectifiables de codimension positive. Dans cette partie il n’est pas question de polysystèmes, ni d’Hamiltoniens. Néanmoins, les résultats obtenus ici sont utilisés dans la deuxième partie de la thèse / In this thesis we study the dynamics generated by a family of Hamiltonian flows. Such a dynamical system with several generators is also called ‘polysystem’.Motivated by some questions related to the phenomenon of Arnold diffusion, our aim is to construct trajectories of the polysystem which connect two far-apart regions of the phase space.The thesis is divided into three parts.In the first part, we consider the polysystem generated by the time-onemaps of a family of Tonelli Hamiltonians. By using a variational approach falling within the framework of weak KAM theory, we give sufficient conditions for the existence of the desired trajectories.In the second part, we address the case of a polysystem generated by twocontinuous-time Hamiltonian flows. This problem fits into the framework of geometriccontrol theory. In this context, we show in some cases the transitivity of a generic polysystem, by means of Thom’s transversality theorem.The third and last part of the thesis is devoted to the proof of a newversion of Thom’s transversality theorem, formulated in terms of rectifiable sets of positive codimension. Neither polysystems nor Hamiltonians are explicitly involved in this part. However, the results obtained here are used in the second part of the thesis.

A presente tese aborda dois problemas distintos e independentes: triangularização de sistemas não-lineares com duas entradas e controle de sistemas sem arrasto que evoluem no grupo especial unitário SU(n). Em relação ao primeiro, estabeleceu-se, através da generalização de resultados bem conhecidos, condições geométricas para que um sistema com duas entradas seja descrito por uma forma triangular específica após uma mudança de coordenadas e uma realimentação de estado estática regular. Para o segundo problema, desenvolveu-se uma estratégia de controle que força o estado do sistema a rastrear assintoticamente uma trajetória de referência periódica que passa por um estado objetivo arbitrário. O método de controle proposto utiliza os resultados de convergência de tipo- Lyapunov que foram estabelecidos pela presente pesquisa e que tiveram como inspiração uma versão periódica do princípio da invariância de LaSalle. Apresentou-se, ainda, os resultados de simulação obtidos com a aplicação da técnica de controle desenvolvida a um sistema quântico consistindo de duas partículas de spin-1/2, com o objetivo de gerar a porta lógica quântica C-NOT. / This thesis treats two distinct and independent problems: triangularization of nonlinear systems with two inputs and control of driftless systems which evolve on the special unitary group SU(n). Concerning the first, one has established, by means of the generalization of well-known results, geometric conditions for a system with two inputs to be described by a specific triangular form after a change of coordinates and a regular static state feedback. For the second problem, one has developed a control strategy that forces the state of the system to track in an asymptotic manner a periodic reference trajectory which passes by an arbitrary goal state. The proposed control method uses Lyapunovlike convergence results that were established in this research and which were inspired in a periodic version of LaSalles invariance principle. Furthermore, one has shown the simulation results obtained from the application of the developed control technique to a quantum system consisting of two spin-1/2 particles, with the aim of generating the C-NOT quantum logic gate.

A presente tese aborda dois problemas distintos e independentes: triangularização de sistemas não-lineares com duas entradas e controle de sistemas sem arrasto que evoluem no grupo especial unitário SU(n). Em relação ao primeiro, estabeleceu-se, através da generalização de resultados bem conhecidos, condições geométricas para que um sistema com duas entradas seja descrito por uma forma triangular específica após uma mudança de coordenadas e uma realimentação de estado estática regular. Para o segundo problema, desenvolveu-se uma estratégia de controle que força o estado do sistema a rastrear assintoticamente uma trajetória de referência periódica que passa por um estado objetivo arbitrário. O método de controle proposto utiliza os resultados de convergência de tipo- Lyapunov que foram estabelecidos pela presente pesquisa e que tiveram como inspiração uma versão periódica do princípio da invariância de LaSalle. Apresentou-se, ainda, os resultados de simulação obtidos com a aplicação da técnica de controle desenvolvida a um sistema quântico consistindo de duas partículas de spin-1/2, com o objetivo de gerar a porta lógica quântica C-NOT. / This thesis treats two distinct and independent problems: triangularization of nonlinear systems with two inputs and control of driftless systems which evolve on the special unitary group SU(n). Concerning the first, one has established, by means of the generalization of well-known results, geometric conditions for a system with two inputs to be described by a specific triangular form after a change of coordinates and a regular static state feedback. For the second problem, one has developed a control strategy that forces the state of the system to track in an asymptotic manner a periodic reference trajectory which passes by an arbitrary goal state. The proposed control method uses Lyapunovlike convergence results that were established in this research and which were inspired in a periodic version of LaSalles invariance principle. Furthermore, one has shown the simulation results obtained from the application of the developed control technique to a quantum system consisting of two spin-1/2 particles, with the aim of generating the C-NOT quantum logic gate.