Optimal control of an ill-posed elliptic semilinear equation with an exponentialnon linearity

Abstract

We study here an optimal control problem for a semilinear elliptic equation with an exponential nonlinearity, such that we cannot expect to have a solution of the state equation for any given control. We then have to speak of pairs (control, state). After having defined a suitable functional class in which we look for solutions, we prove existence of an optimal pair for a large class of cost functions using a non standard compactness argument. Then, we derive a first order optimality system assuming the optimal pair is slightly more regular. © 1998 Société de Mathématiques Appliquées et Industrielles.