A Priori Error Analysis for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

TL;DR

This paper develops nearly optimal a priori error estimates and second order optimality conditions for an elliptic variational inequality-based control problem, enabling precise numerical solutions despite restrictive conditions.

Contribution

It introduces a quadratic growth condition framework and constructs a one-dimensional locally optimal solution for numerical validation.

Findings

Derived nearly optimal a priori error estimates

Established second order sufficient optimality conditions

Constructed an exact solution for numerical experiments

Abstract

We consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. Based on a quadratic growth condition we derive nearly optimal a priori error estimates. Moreover, we establish second order sufficient optimality conditions that ensure a quadratic growth condition. These conditions are rather restrictive, but allow us to construct a one-dimensional locally optimal solution with reduced regularity, which serves as an exact solution for numerical experiments.