A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations
Eduardo Casas, Daniel Wachsmuth
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 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.
Topics & Concepts
MathematicsDifferentiable functionOptimal controlState (computer science)Applied mathematicsPartial differential equationElliptic partial differential equationQuadratic equationControl (management)Mathematical analysisMathematical optimizationComputer scienceAlgorithmArtificial intelligenceGeometryAdvanced Mathematical Modeling in EngineeringStability and Controllability of Differential EquationsAdvanced Numerical Methods in Computational Mathematics