Convergence Results for Optimal Control Problems Governed by Elliptic Quasivariational Inequalities

TL;DR

This paper establishes convergence results for optimal control problems governed by elliptic quasivariational inequalities, demonstrating how solutions to perturbed problems approach solutions of the original, with applications to elastic contact and heat transfer models.

Contribution

It introduces a new framework for analyzing the convergence of solutions in optimal control problems involving elliptic quasivariational inequalities, including sufficient conditions and practical applications.

Findings

Proved convergence of solutions under perturbations.

Established existence and uniqueness for specific boundary value problems.

Applied theoretical results to elastic contact and heat transfer models.

Abstract

We consider an optimal control problem $\cQ$ governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see \cite{SS}, for instance. We associate to $\cQ$ a new optimal control problem $\wQ$, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem $\wQ$ to a solution of Problem $\cQ$. The proofs are based on convergence results for elliptic quasivariational inequalities, obtained by using arguments of compactness, lower semicontinuity, monotonicity, penalty and various estimates. Finally, we illustrate the use of the abstract convergence results in the study of optimal control associated with two boundary value problems. The first one describes the equilibrium of an elastic body in frictional contact with an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated to a version of Coulomb's law of dry friction. The second one describes a stationary heat transfer problem with unilateral constraints. For the two problems we prove existence, uniqueness and convergence results together with the corresponding physical interpretation.