On the choice of the function spaces for some state-constrained control problems

We study a quadratic parabolic control problem with pointwise final state constraints. As the set of admissible states has an empty interior, the existence of Lagrange multipliers cannot be proved directly. We obtain, however some optimality conditions by expressing the fact that among a space of regular perturbations of the optimal control, the null perturbation is optimal. We show that the qualification hypothesis can be effectively checked in some examples and that the information given by the optimality conditions is useful because it allows to get some regularity results for the optimal control.