Dominated splitting and quantization of hybrid systems : towards efficient control of cyber-physical systems and networked control systems

The goal of systems and control theory is to study natural phenomena (e.g., biological processes), technological devices (e.g., robots) or combinations of both (e.g., medical devices like pacemakers, etc.), and to design strategies to control them so that they behave in some intended way. For that, we rely on mathematical models describing the evolution of these phenomena/devices (called systems) and their reaction to external inputs. The challenge with modern systems is that these systems are becoming immensely complex. We think for instance to "cyber-physical systems", which result from the interaction of physical components and computerized components (e.g., self-driving cars where the dynamics of the car is governed by embedded or decentralized micro-controllers controlling it). These systems have become pervasive in our technological world, but they have also many non-standard characteristics (e.g., hybrid behavior, networked components), which preclude the use of classical control techniques and thus call for the development of new mathematical and algorithmic tools for their analysis and control. In this thesis, we study these two fundamental and challenging aspects of modern control systems (hybrid behavior and networked systems). For that, we leverage several tools from classical control theory and generalize them to switched or hybrid systems. In particular, we focus on the property that the dynamics of these systems can often be divided into several components, which grow at different speeds. This property, called "dominance", allows to study the convergence properties of these systems to low-dimensional attractors (with application for instance in population dynamics where the stability of the population composition amounts to the convergence of the system to a subspace of dimension one); or to study the stability of complex attractors for these systems (with application for instance in physics to study chaotic behaviors of electrical or meteorological systems). It is also highly relevant in networked control because dynamics that grow at different speeds generally do not require the same information flow (to be sent across the network) to be controlled satisfactorily. We provide both theoretical and algorithmic frameworks for the study of these questions for switched and networked systems, and we demonstrate their applicability on various numerical examples and concrete modern control problems.