Contrôlabilité approchée pour certaines équations intégrodifférentielles non locales dans les espaces de Banach

Control theory is an interdisciplinary field that addresses the behavior of dynamical systems with the primary goal of managing their output. A specialized subset of this is mathematical control theory, which focuses on utilizing mathematical methods to analyze system behavior and design controllers. This involves applying differential equations, linear algebra, optimization, and various mathematical tools to comprehend, model, and regulate system behavior. These systems have extensive applications across robotics, automation, aerospace, electrical engineering, mechanical systems, robotics, biological and social systems, among others. Described by complex models such as partial differential equations, functional differential equations, and other infinite-dimensional models, these systems pose intricate challenges, rendering the analysis of their behavior a pivotal and intricate area of research. In recent years, the application of control theory to analyze and regulate the behavior of these systems has attracted significant attention. This thesis aims to investigate the approximate controllability of certain infinite-dimensional dynamical systems described by integrodifferential equations. The thesis is structured into three chapters, each addressing the problem of achieving approximate controllability in integrodifferential evolution equations equipped with nonlocal conditions. The first chapter introduces fundamental tools critical to establishing our main findings, including the theory of resolvent operators, multi-valued maps, duality mapping, mathematical control theory, and other essential concepts. Chapter 2 specifically focuses on the approximate controllability of semilinear integrodifferential evolution equations with nonlocal conditions of the form w(0)=w0+g(w). Here, assuming the linear part is precisely null and approximately controllable, we employ resolvent operator theory to present our main results. Chapter 3 centers on investigating the existence of mild solutions and the approximate controllability of integrodifferential evolution systems with multi-valued nonlocal conditions (w(0) belongs w0+g(w)). By using resolvent operator theory, we establish sufficient conditions for both existence and controllability. Introducing a general Kalman controllability criterion, we examine approximate controllability in linear cases and subsequently demonstrate it in nonlinear cases. Throughout these chapters, we provide illustrative examples to support our main findings.