On the Existence of Optimal Solutions for Time-Optimal Semilinear Parabolic Boundary Control Problems

Abstract

A class of time-optimal semilinear parabolic control problems with state constraints and a generalized terminal condition is considered. For the derivation of solvability results, which is the main aim of the paper, we present two methods. The first method works with the complete continuity of the state mapping, whereas in the second one theorems about the separation of convex sets and measurable selection are applied for overcoming the complete continuity of the state mapping. The connection with a family of associated fixed-time problems is very helpful in both cases. This connection between time-optimal control problems and related problems with fixed time are also interesting in their own right, especially they can be used to get optimality conditions for the original time-optimal control problem.