OPTIMAL CONTROL OF SOME CLASSES OF SWEEPING PROCESSES AND APPLICATIONS

Abstract

This thesis investigates the optimal control of dynamical systems governed by sweeping processes, with a particular focus on both implicit and explicit formulations involving convex and polyhedral geometries. These systems, characterized by highly nonsmooth differential inclusions with state-dependent constraints, arise naturally in models of elasto-plasticity, hysteresis, and contact mechanics. The primary challenge in controlling such systems arises from the discontinuous nature of the normal cone operator, which obstructs the direct application of classical optimal control tools.

We begin by establishing a Pontryagin-type maximum principle for a class of optimal control problems governed by implicit sweeping processes with general endpoint constraints. The sweeping set is assumed to be polyhedral and control-dependent, and the dynamics include state and control variables in both the inclusion and perturbation terms. The analysis relies on precise coderivative estimates for the metric projection mapping onto polyhedral convex sets. Applications include variational inequalities and projected dynamical systems, such as generalized Lotka–Volterra models.

To support this framework, we conduct a detailed study of the coderivative geometry of projection operators. We present new estimates for the Dini and Mordukhovich coderivatives of the metric projection mapping, focusing on two important classes of convex sets: those with strictly Hadamard differentiable boundaries and polyhedral sets. These results are instrumental in handling the nonsmooth structure of sweeping processes and form a critical component in deriving necessary optimality conditions.

We further propose a novel approximation scheme for solving optimal control problems governed by explicit sweeping processes, which are inherently more challenging due to their nonsmoothness. By introducing a family of implicit sweeping dynamics, we regularize the system in a way that enables the application of classical maximum principle techniques. We establish convergence results and validate the method through an application to the optimal control of a single degree-of-freedom elasto-plastic oscillator, a benchmark model in structural mechanics and seismic engineering.

Overall, the dissertation provides a unified theoretical and methodological framework for the optimal control of sweeping processes, contributing new insights to nonsmooth analysis, variational inequalities, and control theory.