Bang-bang principle for state-constrained parabolic systems with dirichlet boundary controls

In this paper we study a class of state-constrained boundary control problems for linear parabolic equations. Principal complications come from discontinuous and bounded Dirichlet boundary controllers which are motivated by applications to some robust control problems under uncertain perturbations. We develop an effective procedure to approximate state constraints and prove desirable convergences of such approximations. Using variational analysis, we establish necessary optimality conditions for approximation problems and then, passing to the limit under a proper constraint qualification, we prove necessary optimality conditions for the original state-constrained problem in the bang-bang principle form