A new class of evolution multivalued quasi-variational inequalities I: existence and nonsmooth optimal control

TL;DR

This paper introduces a new class of evolution multivalued quasi-variational inequalities, establishing existence results and analyzing a related nonsmooth optimal control problem with applications in distributed-boundary control and parameter identification.

Contribution

It develops a novel framework for existence and compactness of solutions and studies a nonlinear nonsmooth optimal control problem governed by these inequalities.

Findings

Proved existence and compactness of solutions under mild assumptions.

Established sufficient conditions for solvability of the optimal control problem.

Provided a model for distributed-boundary control and parameter identification.

Abstract

In this paper, we consider a new kind of evolution multivalued quasi-variational inequalities with feedback effect and a nonlinear bifunction which contain several (evolution) quasi-variational/hemivariational inequalities as special cases. The main contribution of this paper is twofold. The first goal is to establish a novel framework for proving the existence of solutions and the compactness of solution set to the evolution multivalued quasi-variational inequalities, under quite mild assumptions. Whereas, the second contribution is to introduce and study a nonlinear and nonsmooth optimal control problem governed by an evolution multivalued quasi-variational inequality, and then to obtain the sufficient conditions for guaranteeing the solvability of the nonlinear and nonsmooth optimal control problem under consideration. Such nonlinear and nonsmooth optimal control problem could as a useful model to explore the simultaneous distributed-boundary optimal control problems driven by evolution multivalued quasi-variational inequalities, and optimal parameters identification for evolution multivalued quasi-variational inequalities.