A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations

In this paper, we study optimal control problems of semilinear elliptic and parabolic equations. A tracking cost functional, quadratic in the control and state variables, is considered. No control constraints are imposed. We prove that the corresponding state equations are well posed for controls in L2. However, it is well known that in the L2 framework the mappings involved in the control problem are not Frechet differentiable in general, which makes any analysis of the optimality conditions challenging. Nevertheless, we prove that every L2 optimal control belongs to L∞, and consequently standard optimality conditions are available.