Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality

Mariela Olguín,Domingo A Tarzia

doi:10.1155/2015/407930

Abstract

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energyg. The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positiveh(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 parameterhgoes to zero.

Highlights

IntroductionIn [2], the following continuous distributed optimal control problem associated with (S) or the elliptic variational inequality (3) was considered as follows
We consider a bounded domain Ω ⊂ Rn whose regular boundary ∂Ω = Γ1 ∪ Γ2 consists of the union of two disjoint portions Γ1 and Γ2 with meas(Γ1) > 0
We discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive h

Summary

Introduction

In [2], the following continuous distributed optimal control problem associated with (S) or the elliptic variational inequality (3) was considered as follows. With M > 0, a given constant, and ug is the corresponding solution of the elliptic variational inequality (3) associated with the control g. The objective of this work is to make the numerical analysis of the optimal control problem (P) which is governed by the elliptic variational inequality (3) by proving the convergence of a discrete solution to the continuous optimal control problems. Let g ∈ H, b > 0, and q ∈ Q; there exists unique solution of the problem (Sh) given by the elliptic variational inequality (10). If ug and uhg are the solutions of the elliptic variational inequalities (3) and (10), respectively, for the control g ∈ H, uhg → ug in V strong when h → 0+.

Discretization of the Optimal Control Problem

Conclusions