Stability results for constrained dynamical systems

Differential-Algebraic Equations (DAE) provide an appropriate framework to model and analyse dynamic systems with constraints. This framework facilitates modelling of the system behaviour through natural physical variables of the system, while preserving the topological constraints of the system. The main purpose of this dissertation is to investigate stability properties of two important classes of DAEs. We consider some special cases of Linear Time Invariant (LTI) DAEs with control inputs and outputs, and also a special class of Linear switched DAEs. In the first part of the thesis, we consider LTI systems, where we focus on two properties: passivity and a generalization of passivity and small gain theorems called mixed property. These properties play an important role in the control design of large-scale interconnected systems. An important bottleneck for a design based on the aforementioned properties is their verification. Hence we intend to develop easily verifiable conditions to check passivity and mixedness of Single Input Single Output (SISO) and Multiple Input Multiple Output (MIMO) DAEs. For linear switched DAEs, we focus on the Lyapunov stability and this problem forms the basis for the second part of the thesis. In this part, we try to find conditions under which there exists a common Lyapunov function for all modes of the switched system, thus guaranteeing exponential stability of the switched system. These results are primarily developed for continuous-time systems. However, simulation and control design of a dynamic system requires a discrete-time representation of the system that we are interested in. Thus, it is critical to establish whether discrete-time systems, inherit fundamental properties of the continuous-time systems from which they are derived. Hence, the third part of our thesis is dedicated to the problems of preserving passivity, mixedness and Lyapunov stability under discretization. In this part, we examine several existing discretization methods and find conditions under which they preserve the stability properties discussed in the thesis

Stability results for constrained dynamical systems

Differential-Algebraic Equations (DAE) provide an appropriate framework to model and analyse dynamic systems with constraints. This framework facilitates modelling of the system behaviour through natural physical variables of the system, while preserving the topological constraints of the system. The main purpose of this dissertation is to investigate stability properties of two important classes of DAEs. We consider some special cases of Linear Time Invariant (LTI) DAEs with control inputs and outputs, and also a special class of Linear switched DAEs. In the first part of the thesis, we consider LTI systems, where we focus on two properties: passivity and a generalization of passivity and small gain theorems called mixed property. These properties play an important role in the control design of large-scale interconnected systems. An important bottleneck for a design based on the aforementioned properties is their verification. Hence we intend to develop easily verifiable conditions to check passivity and mixedness of Single Input Single Output (SISO) and Multiple Input Multiple Output (MIMO) DAEs. For linear switched DAEs, we focus on the Lyapunov stability and this problem forms the basis for the second part of the thesis. In this part, we try to find conditions under which there exists a common Lyapunov function for all modes of the switched system, thus guaranteeing exponential stability of the switched system. These results are primarily developed for continuous-time systems. However, simulation and control design of a dynamic system requires a discrete-time representation of the system that we are interested in. Thus, it is critical to establish whether discrete-time systems, inherit fundamental properties of the continuous-time systems from which they are derived. Hence, the third part of our thesis is dedicated to the problems of preserving passivity, mixedness and Lyapunov stability under discretization. In this part, we examine several existing discretization methods and find conditions under which they preserve the stability properties discussed in the thesis

Analyse des équations de transport avec vitesses non-locales et applications à la commande des foules

The study of the collective behavior of a large number of interacting agents, often called a “crowd”, has drawn a growing attention from several scientific communities. This research topic has an impact on civil engineering (problems of emergency egress and road traffic), robotics (coordination of robots and networked control), computer science and sociology (social networks), and biology (groups, herds and flocks).Some major difficulties are deeply related to the use of such models. First of all, from the theoretical viewpoint, standard analytical tools are not very useful in this context, since the presence of a large number of agents corresponds to a state space of big dimension. Moreover, for human and animal crowds, the dynamics of each agent is not clearly identified, since it is highly sensitive to internal and external perturbations (such as stress, panic and presence of obstacles).For these reason, stimulated by the numerous theoretical and applied challenges, a large number of researcher currently works on crowd models.The first question to address, in this context, is the choice of a mathematical framework for the description of the dynamics of agents. The crowd can be described with three frameworks: microscopic, macroscopic or multi-scale. In the microscopic approach, the crowd is represented by the position of each agent, and its dynamics is a system of ordinary differential equations of very big dimension. In the macroscopic approach, the crowd is given by the density of agents, and its dynamics is a partial differential equation (PDE in the following), often of transport type. In the multi-scale approach, also called “granular”, the population is composed both of a microscopic part of “remarkable” agents (such as leaders), and of a macroscopic part for the rest of the crowd. In this third framework, in which I develop most of my researches, measures are the main tool.In Section 1, I present my results of analysis in this context. I focus on a particular class of PDEs for dynamics of measures. This class of transport equations with non-local velocities is at the core of models for crowds. Indeed, each agent in a crowd interacts with its neighbors, generating a dynamics depending not only on its position (local term), but also on positions of others (non-local term). In Section 1.1, I describe a rigorous and mathematically rich framework in which transport equations with non-local velocities enjoy good properties: existence and uniqueness of solutions, continuous dependance, etc. I then study some numerical schemes for such equations, described in Section 1.2, for which I also prove convergence. After this, I define in Section 1.3 a generalization of the Wasserstein distance to mass-varying measures. For this distance, I prove interesting properties related to the transport PDE with source, and in particular a generalization of the Benamou-Brenier formula. I finally present in Section 1.4 some specific results for the transport equation in a non-smooth setting, useful for models of road traffic.Beside the problem of analyzing the collective behavior of crowds, it is interesting to understand what behavior changes can be induced on them by an external agent – e.g. a policy maker or some leaders.The research mainstream for such problems has been focused on creating efficient facilities or rules, with a static point of view. Recently, such setting has been challenged by a more dynamic and time-dependent point of view. This implies a change of approach: we pass from a static optimization problem to a control problem, depending on time and on configurations. For example, if we consider emergency egress problems, with a statical point of view the infrastructure is designed with a given configuration and it cannot be modified. With the new, dynamical point of view, one can introduce light signals or portable devices to drive the crowd to the best direction.This innovative approach leads to the problem of control of crowds: one wants to understand what changes of behavior can be induced to the crowd by some leaders or by an external policy maker. Such problems have already been studied in control theory for coordination of agents of very different natures, such as flying drones formations, routing in telecommunication networks, smart grids and power network control.In this general framework, I worked on the control of the transport equation with non-local velocities. My results are presented in Section 2. I first discuss in Section 2.1 a theoretical problem for crowd control. When one passes from a microscopic to a macroscopic model, the indistinguishability of agents is a necessary requirement: this is in sharp contrast with the fact that controls are applied to specific agents. For this reason, I present two specific control problems. In Section 2.2, I describe the control the kinetic Cucker-Smale problem to a flocking configuration. This is one of the few known results of control of the transport equation with non-local velocity, and the only one with localized control. Section 2.3 is focused on an optimal control problem for a system coupling a controlled dynamical system for leaders and a transport PDE for the rest of the crowd. The main result is a generalization of the Pontryagin Maximum Principle to this measure setting, in which the Hamiltonian equation is written in terms of a Wasserstein gradient.This thesis also contains some other research results in control and system theory, as well as analysis of PDEs. They are shortly described in Section 3. These results are quite independent with respect to topics in previous sections. Nevertheless, research tools used in the three sections are common, being based on geometric control. Moreover, I focus on PDE models, and in particular on limits permitting to pass from a finite-dimensional system to the associated PDE. See some examples in Sections 1.1, 2, 3.1 and 3.2.Finally, a Curriculum Vitae is presented in Section 4.L’étude du comportement collectif d’un grand nombre d’agents en interaction, souvent dénommé “foule”, a suscité un grand intérêt de la part des communautés scientifiques. Ce sujet de recherche touche aussi bien le Génie civil (évacuation de bâtiments et problèmes du trafic routier), la Robotique (coordination de robots volants), l’Informatique et la Sociologie (réseaux sociaux), que la Biologie (groupes, troupeaux et vols d’oiseaux).Des difficultés majeures sont intimement liées à l’utilisation de ces modèles. En effet, d’un point de vue théorique, la présence d’un grand nombre d’agents rend les outils classiques de l’analyse mathématique peu utiles, car l’espace d’état est de très grande dimension. De plus, pour les foules humaines ou d’animaux, la dynamique de chaque agent n’est pas clairement identifiée, car elle est très sensible aux facteurs intérieurs et extérieurs (comme le stress, la panique, la présence d’obstacles). C’est pour cela que, stimulés par les multiples défis théoriques et applicatifs, de nombreux chercheurs travaillent sur les modèles de foules.La première question qu’il est nécessaire de se poser, en ce contexte, concerne le choix d’un cadre mathématique pour la description de la dynamique des agents. La population peut se décrire de trois façons : microscopique, macroscopique ou multiéchelle. Dans l’approche microscopique, la foule est représentée par la position de chaque agent, et sa dynamique est un système d’équations aux dérivées ordinaires de dimension très grande. Dans l’approche macroscopique, la foule est donnée par la densité d’agents, et sa dynamique est une équation aux dérivées partielles (EDP dans la suite), souvent de type transport. Dans l’approche “multi-échelle”, dite aussi “granulaire”, la population se compose tout autant d’une partie microscopique d’agents “significatifs” (tels que les leaders) que d’une partie macroscopique pour le reste de la foule. Dans cette troisième approche, dans laquelle je développe la plupart de mes recherches, les mesures sont l’outil principal utilisé.La Section 1 présente mes résultats d’analyse dans ce contexte. Je m’intéresse à une classe particulière d’EDP pour la dynamique des mesures. Cette classe d’équations de transport avec vitesses non-locales est au coeur des modèles pour les foules. En effet, chaque agent dans une foule est en interaction avec ses voisins, engendrant ainsi une dynamique qui ne dépend pas seulement de sa position (terme locale), mais aussi des positions des autres (terme non-local).En Section 1.1, j’expose d’abord un cadre assez rigoureux et riche dans lequel les équations de transport avec vitesses non-locales ont de bonnes propriétés : existence et unicité de la solution, dépendance continue, etc. Puis, j’étudie certains schémas numériques pour ces équations, décrits en Section 1.2, et je démontre leur convergence. Je définis ensuite en Section 1.3 une généralisation de la distance de Wasserstein aux mesures de masse variable. Pour cette distance, je prouve des propriétés intéressantes en lien avec l’EDP de transport avec source, et notamment une généralisation de la formule de Benamou-Brenier. Enfin, je présente en Section 1.4 certains résultats spécifiques pour l’équation de transport dans un cadre non-lisse, qui est utilisée dans des modèles de trafic routier.Au-delà de la description et de l’analyse du comportement collectif, il est intéressant de se demander quels changements un agent extérieur – un gouvernement régulateur ou des leaders, par exemple – peut induire sur une foule. La plupart des recherches dans ce domaine ont été consacrées à la création de structures ou de règles efficientes, avec un point de vue statique. Aujourd’hui cependant, ce point de vue est remis en question par une vision dynamique et variable dans le temps. On assiste à un changement de paradigme : d’un problème d’optimisation statique on passe à un problème de commande, dépendant du temps et des configurations. Si l’on considère les problèmes d’évacuation, par exemple, avec un point de vue statique, l’infrastructure est conçue dans une configuration donnée et elle ne peut subir aucune modification. Avec le nouveau paradigme dynamique, quand une sortie de secours est congestionnée, on peut introduire des signaux lumineux ou des dispositifs portables pour envoyer la foule vers la direction la plus convenable.Cette approche introduit le problème de la commande des foules : on souhaite comprendre quels changements de comportement peuvent être produits sur la foule par des leaders ou par un régulateur extérieur. Ces problèmes ont déjà été étudiés en automatique pour la coordination d’agents dans des situations très variées, comme les formations de drones en vol, le routing dans les réseaux de télécommunication, les problèmes d’énergie avec les “smart grids”.Dans ce cadre très général, j’ai travaillé à la commande de l’équation de transport avec vitesse non-locale. La Section 2 en présente les résultats. Je discute d’abord en Section 2.1 d’un problème conceptuel pour la commande des foules. Pour passer d’un modèle microscopique à un modèle macroscopique, l’indistinguabilité des agents est nécessaire : cette propriété s’oppose au fait que les commandes agissent normalement sur des agents précis. Pour cette raison, je présente des problèmes de commande dans deux cas particuliers. En Section 2.2, je montre la commande du modèle de champ moyen de Cucker et Smale vers une configuration d’alignement. C’est l’un des rares résultats en littérature de commande de l’équation de transport avec vitesse non-locale, et le seul avec commande localisée. En Section 2.3, j’étudie un problème de commande optimale où la dynamique est donnée par le couplage d’un système contrôlé pour des leaders avec une EDP de transport pour le reste de la foule. Le résultat principal est la généralisation du Principe de Maximum de Pontryaguine à ce problème de mesures, dans lequel l’équation de Hamilton est écrite comme un gradient de Wasserstein.Ce mémoire contient aussi plusieurs autres résultats de recherche dans le domaine de l’automatique, de la commande et de l’analyse des EDP. La Section 3 les décrit plus brièvement. Ces résultats sont assez indépendants par rapport aux sujets présentés dans les sections précédentes. Il est à noter, cependant, que les instruments de recherche utilisés dans les trois sections relèvent tous de la commande géométrique. Je focalise également mon attention sur les modèles d’EDP, et en particulier sur les méthodes de limite permettant de passer d’un système en dimension finie à une EDP associée. Voire des exemples en Sections 1.1, 2, 3.1 et 3.2.Enfin, un CV détaillé est présenté en Section 4

New approach to the stability and control of reaction networks

A new system-theoretic approach for studying the stability and control of chemical reaction networks (CRNs) is proposed, and analyzed. This has direct application to biological applications where biochemical networks suffer from high uncertainty in the kinetic parameters and exact structure of the rate functions. The proposed approach tackles this issue by presenting "structural" results, i.e. results that extract important qualitative information from the structure alone regardless of the specific form of the kinetics which can be arbitrary monotone kinetics, including Mass-Action. The proposed method is based on introducing a class of Lyapunov functions that we call Piecewise Linear in Rates (PWLR) Lyapunov functions. Several algorithms are proposed for the construction of these functions. Subject to mild technical conditions, the existence of these functions can be used to ensure powerful dynamical and algebraic conditions such as Lyapunov stability, asymptotic stability, global asymptotic stability, persistence, uniqueness of equilibria and exponential contraction. This shows that this class of networks is well-behaved and excludes complicated behaviour such as multi-stability, limit cycles and chaos. The class of PWLR functions is then shown to be a subset of larger class of Robust Lyapunov functions (RLFs), which can be interpreted by shifting the analysis to reaction coordinates. In the new coordinates, the problem transforms into finding a common Lyapunov function for a linear parameter varying system. Consequently, dual forms of the PWLR Lyapunov functions are presented, and the interpretation in terms of the variational dynamics and contraction analysis are given. An other class of Piecewise Quadratic in Rates Lyapunov function is also introduced. Relationship with consensus dynamics are also pointed out. Control laws for the stabilization of the proposed class of networks are provided, and the concept of control Lyapunov function is briefly discussed. Finally, the proposed framework is shown to be widely applicable to biochemical networks.Open Acces