Distributed optimal control problems for a class of elliptic hemivariational inequalities with a parameter and its asymptotic behavior

TL;DR

This paper investigates optimal control problems for elliptic hemivariational inequalities with a parameter, focusing on existence and asymptotic behavior of controls and states in a heat conduction context.

Contribution

It establishes existence of optimal controls and analyzes their asymptotic behavior as the parameter approaches infinity.

Findings

Existence of optimal controls proven.

Asymptotic behavior of controls and states characterized.

Results applicable to heat conduction problems with complex boundary conditions.

Abstract

In this paper, we study optimal control problems on the internal energy for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system has been originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the optimal controls and we show an asymptotic result for the optimal controls and the system states, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.