Optimality conditions for state-constrained PDE control problems with time-dependent controls

The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are R m -vector- valued functions of time, while they are vectors of R m in elliptic problems. Under natural assumptions, first- and second-order suffi- cient optimality conditions are derived. The main result is the ex- tension of second-order sufficient conditions to semilinearparabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semi-infinite optimiza- tion. Here, different examples are discussed that exhibit different forms of active sets and where second-order sufficient conditions are satisfied at the optimal solution.