A Relaxation Approach to Discretization of Boundary Optimal Control Problems of Semilinear Parabolic Equations

Abstract

We consider an optimal boundary control problem described by a semilinear parabolic\r partial differential equation, with control and state constraints. Since this problem may have no\r classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized\r by using a finite element method in space and a partially implicit scheme in time, while the\r controls are approximated by piecewise constant relaxed controls. We first state the necessary\r conditions for optimality for the continuous problem and the discrete relaxed problem. Next,\r under appropriate assumptions, we prove that accumulation points of sequences of optimal (resp.\r admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal)\r for the continuous relaxed problem.