Existence, Approximation, and Suboptimality Conditions for Minimax Control of Heat Transfer Systems with State Constraints

In this paper, we consider a minimax control problem for heat transfer systems with uncertain perturbations and pointwise state constraints. We study properties of mild solutions to such parabolic systems with Dirichlet boundary controls and prove an existence theorem for optimal solutions to the minimax problem. We develop penalized procedures to approximate state constraints and then establish convergence results for these approximations. Using a variational analysis, we obtain necessary optimality conditions for approximating solutions which ensure suboptimality conditions in the original minimax problem with state constraints.