State-Constrained Relaxed Problems for Semilinear Elliptic Equations

In this paper we study optimal control problems governed by semilinear elliptic equations in the presence of pointwise state constraints. Since no convexity condition is assumed on data of the problem, we define a relaxed control problem, prove the existence of relaxed solutions, and give some relaxation results. By adapting the penalty method of Berkovitz, we prove a Pontryagin’s minimum principle for relaxed solutions in nonqualified form and in qualified form under a stability condition.