Abstract
We study Dirichlet boundary control problem for semilinear parabolic equations, with pointwise state constraints. By penalizing the Dirichlet boundary condition, we define a family of control problems with Robin boundary conditions. We study the asymptotic behavior of solutions of these penalized problems, when the penalty parameter tends to infinity. We also prove that optimality conditions for the Dirichlet boundary control problem can be obtained by passage to the limit in the optimality conditions of the penalized problem.