Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints

Eduardo Casas

doi:10.1051/cocv:2002049

Abstract

The goal of this paper is to derive some error estimates for the numerical discretization of some optimal control problems governed by semilinear elliptic equations with bound constraints on the control and a finitely number of equality and inequality state constraints. We prove some error estimates for the optimal controls in the L∞ norm and we also obtain error estimates for the Lagrange multipliers associated to the state constraints as well as for the optimal states and optimal adjoint states.

Highlights

In this paper we study an optimal control problem governed by a semilinear elliptic equation, the control being distributed in Ω
Bound constraints on the control and finitely many equality and inequality state constraints are included in the formulation of the problem
The aim is to consider the numerical approximation of this problem by using finite element methods

Summary

Introduction

In this paper we study an optimal control problem governed by a semilinear elliptic equation, the control being distributed in Ω. There are no many papers devoted to the study of error estimates for the numerical discretization of control problems governed by partial differential equations. A significant change when studying control problems with a nonlinear equation or a non quadratic functional is the necessity of using the sufficient second order optimality conditions to derive these error estimates. Arada et al [1] followed this procedure to get the error estimates for the same problem studied in this paper except by the fact that there were no state constraints. They derived the same L∞ error estimates than we obtain here.

The control problem

First and second order optimality conditions

Findings

Convergence results