A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations
Eduardo Casas, Daniel Wachsmuth · SIAM Journal on Control and Optimization · 2023
Abstract. 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 [Formula: see text]. However, it is well known that in the [Formula: see text] 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 [Formula: see text] optimal control belongs to [Formula: see text], and consequently standard optimality conditions are available.