Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper 'Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints'. It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone-Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem