Some contributions to degenerate state-dependent sweeping processes

Abstract

This dissertation aims to study differential inclusions related to normal cones driven by nonregular moving sets (subsmooth and positively α-far) on Hilbert spaces. Specially, we are interested in a particular variant of the classical sweeping process known as the dege- nerate state-dependent sweeping process, which corresponds to the case where a linear/non- linear operator is incorporated into the classical sweeping process. This differential inclusion has been motivated by quasi-variational inequalities that arise, for instance, in problems of quasistatic evolution with friction, the evolution of sandpiles, and micromechanical damage models for iron materials, among others. It is worth to mention that degenerate sweeping process has been studied only in the framework of regular (convex/prox-regular) sets that vary in a Lipschitz or absolutely continuous way with respect to the Hausdorff distance. We introduce the fundamental definitions for the study of the degenerate state-dependent sweeping processes in Chapter 1. We focus on the Clarke normal cone, Clarke subdifferential, proximal subdifferential, truncated Hausdorff distance, and ρ-uniformly prox-regular sets together with some of the most useful properties to show the existence and uniqueness of solutions to the variants of the classic sweeping process. In Chapter 2, we focus on the study of the degenerate state-dependent sweeping processes problem. For this purpose, using the Moreau-Yosida regularization technique, we guarantee the existence of solutions under the assumption that moving sets varies of a Lipschitz way with respect to truncated Hausdorff distance. All the results from this section can be found in [56]. In Chapter 3, we obtain results on the existence of solutions of the so called perturbed degenerate state-dependent sweeping processes, which is based on [57]. Using an appropriate existence result for differential inclusions with single-valued perturbation, and together to a suitable adaptation of the Moreau-Yosida regularization technique, we show the existence of solutions under the assumption that moving sets varies of a Lipschitz way with respect to truncated Hausdorff distance. Consequently, the existence of solutions for the integrally perturbed degenerate state-dependent sweeping processes can be obtained. Our results from Chapter 3, can be applied to study of the so-called online mirror descent method, which is related to the k-server problem. This work ends with some conclusions and open questions.