Numerical analysis of distributed optimal control problems governed by elliptic variational inequalities

TL;DR

This paper performs a numerical analysis of an optimal control problem governed by elliptic variational inequalities, demonstrating convergence of finite element solutions to the continuous problem.

Contribution

It introduces a finite element discretization approach for the problem and proves convergence of discrete solutions to the continuous optimal control and state system.

Findings

Existence of discrete optimal control and state system for each mesh size h.

Convergence of discrete solutions to continuous solutions as h approaches zero.

Validation of finite element method for solving elliptic variational inequality-based control problems.

Abstract

A continuous optimal control problem governed by an elliptic variational inequality was considered in Boukrouche-Tarzia, Comput. Optim. Appl., 53 (2012), 375-392 where the control variable is the internal energy $g$. It was proved the existence and uniqueness of the optimal control and its associated state system. The objective of this work is to make the numerical analysis of the above optimal control problem, through the finite element method with Lagrange's triangles of type 1. We discretize the elliptic variational inequality which define the state system and the corresponding cost functional, and we prove that there exists a discrete optimal control and its associated discrete state system for each positive $h$ (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter $h$ goes to zero.