Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state

In this paper we are concerned with optimal control problems governed by an elliptic semilinear equation, the control being distributed in . The existence of constraints on the control as well as pointwise constraints on the gradient of the state is assumed. A convenient choice of the control space permits us to derive the optimality conditions and study the adjoint state equation, which has derivatives of measures as data. In order to carry out this study, we prove a trace theorem and state Green's formula by using the transposition method.This research was partially supported by Dirección General de Investigation Cientifica y Técnica (Madrid).