On the Optimal Control of Variational-Hemivariational Inequalities

TL;DR

This paper extends previous work on elliptic variational-hemivariational inequalities by establishing more general continuous dependence results and proving the existence of optimal control pairs, with applications to elastic contact problems.

Contribution

It provides a more general continuous dependence result and proves the existence of optimal pairs for a broad class of control problems involving variational-hemivariational inequalities.

Findings

Proved continuous dependence of solutions on parameters.

Established existence of optimal control pairs.

Applied results to elastic contact equilibrium problem.

Abstract

The present paper represents a continuation of our previous one. There, a continuous dependence result for the solution of an elliptic variational-hemivariational inequality was obtained and then used to prove the existence of optimal pairs for two associated optimal control problems. In the current paper we complete this study with more general results. Indeed, we prove the continuous dependence of the solution with respect to a parameter which appears in all the data of the problem, including the set of constraints, the nonlinear operator and the two functionals which govern the variational-hemivariational inequality. This allows us to consider a general associated optimal control problem for which we prove the existence of optimal pairs, together with a new convergence result. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to elliptic variational-hemivariational inequalities. To provide an example, we illustrate our results in the study of an inequality which describes the equilibrium of an elastic body in frictional contact with a foundation made of a rigid body covered by a layer of soft material.