Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints

Eduardo Casas,Mariano Mateos

doi:10.1137/s0363012900382011

Abstract

This paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints. Some recent results on this topic for optimal control problems based upon results for abstract optimization problems are compared with some new results using methods adapted to the control problems. Meanwhile, the Lagrangian formulation is followed to provide the optimality conditions in the first case; the Lagrangian and Hamiltonian functions are used in the second statement. Finally, we prove the equivalence of both formulations.

Highlights

The first goal of this paper is to provide some new second order optimality conditions for control problems of semilinear elliptic partial differential equations with finitely many state constraints
While there exists a very extensive literature about first order optimality conditions for control problems of partial differential equations, only a few papers are devoted to second order conditions
Such sufficient optimality conditions are useful for carrying out the numerical analysis of a control problem, for obtaining error estimates in the numerical discretization, and for analyzing the sequential quadratic programming algorithms applied to control problems

Summary

Introduction

The first goal of this paper is to provide some new second order optimality conditions for control problems of semilinear elliptic partial differential equations with finitely many state constraints. These conditions involve the Lagrangian and the Hamiltonian functions. We have to study the second order necessary conditions and compare them with the sufficient ones in order to check if there is a reasonable gap between them. This was studied by Casas and Troltzsch [7], [8] and Casas, Mateos, and Fernandez [5] for some control problems. The gap between the established necessary and the sufficient conditions was very small

EDUARDO CASAS AND MARIANO MATEOS

Then given

Let us now define the function

Then there exist ε and α such that

Let us denote yk