Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints

Dpt. Matemática Aplicada y Ciencias de la Computación, E.T.S.I.I y T., Universidad de Cantabria, Av. Los Castros s/n 39005 Santander, Spain; eduardo.casas@unican.es

Received: 3 May 2006
Revised: 20 November 2006

Abstract

The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal control is Lipschitz in the whole domain. Necessary and sufficient second order conditions are proved with a minimal gap between them.