Numerical analysis of a family of optimal distributed control problems governed by an elliptic variational inequality

TL;DR

This paper conducts a detailed numerical analysis of a family of optimal control problems governed by elliptic variational inequalities, focusing on finite element discretization and parameter limits.

Contribution

It introduces a comprehensive analysis of the finite element method for these control problems, including limit behaviors as parameters tend to infinity and convergence results.

Findings

Finite element approximation converges as mesh size h approaches zero.

Optimal controls and states converge as the parameter α tends to infinity.

Double convergence occurs when both h approaches zero and α tends to infinity.

Abstract

The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $\alpha >0$) is obtained through the finite element method when its parameter $h\rightarrow 0$. We also obtain the limit of the discrete optimal control and the associated state system solutions when $\alpha\rightarrow \infty$ (for each $h>0$) and a commutative diagram for two continuous and two discrete optimal control and its associated state system solutions is obtained when $h\rightarrow 0$ and $\alpha\rightarrow \infty$. Moreover, the double convergence is also obtained when $(h, \alpha)\rightarrow(0, \infty)$.