Minimax boundary control problem for parabolic systems with state constraints

We study a minimax Dirichlet boundary control problem with pointwise state constraints for a class of parabolic systems under unknown distributed perturbations. Our approach to optimality conditions is based on splitting the original minimax problem into two interrelated optimal control problems for distributed and boundary controllers with moving state constraints. Then we approximate the state-constrained linear systems by families of nonlinear systems with no state constraints by using effective penalization procedures. Based on the properties of mild solutions, we prove the variational convergence of approximations and obtain necessary optimality conditions for approximating solutions which ensure suboptimality conditions for the original minimax problems.