Stability, Bifurcation, and Continuation Theory for Perturbed Sweeping Processes

Abstract

This dissertation is devoted to the development of a qualitative theory of perturbed sweeping processes, which are a combination of differential equations and a moving constraint. The differential equations involved are always assumed Lipschitz continuous. As for the moving constraint, several different situations are addressed: Lipschitz continuous in time, BV-continuous in time, state-dependent, state-independent, with convex interior, with prox-regular interior, bounded in time, periodic in time, almost periodic in time. We prove the existence of local and global solutions as well as boundedness, periodicity, almost periodicity, and asymptotic stability of solutions. Furthermore, we establish results on the occurrence of periodic solutions from a switched boundary equilibrium and on bifurcation of cycles from a regular boundary equilibrium. Concrete examples illustrate the main results of the dissertation.