Contrôle optimal et ses applications

Abstract

Optimal control is a fundamental area of applied mathematics, situated at the in- tersection of dynamical systems theory, analysis, and optimization. Its objective is to determine a control that allows a system to evolve from an initial state to a desired state while minimizing a cost functional representing, depending on the context, energy expenditure, trajectory error, transfer time, or economic cost. In this thesis, we first present the mathematical foundations necessary for the study of optimal control, including dynamical systems, existence and uniqueness results for solutions, and some notions of convexity. We then formulate the general problem of optimal control by introducing state variables, admissible controls, constraints, and the cost functional. Particular attention is paid to the main theoretical methods in the field, especially Pontryagin’s maximum principle, dynamic programming, the Hamilton- Jacobi-Bellman equation, and linear quadratic control based on the Riccati equation. Finally, a numerical section using MATLAB is devoted to illustrating the theore- tical results on linear and non-linear systems. The simulations obtained demonstrate the effectiveness of the methods studied for stabilizing dynamic systems and improving their performance while limiting the control effort. This work thus highlights the relevance of optimal control both theoretically and in its numerous applications in automation, engineering, economics, and energy systems.