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1 2008

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

Los Reyes J. C. de, Merino P., Rehberg F., Troltzsch F.

Control and Cybernetics

2008

Vol. 37, no 1

5-38

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 Rm -vector-valued functions of time, while they are vectors of Rm in elliptic problems. Under natural assumptions, first- and second-order sufficient optimality conditions are derived. The main result is the extension of second-order sufficient conditions to semilinear parabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semi-infinite optimization. Here, different examples are discussed that exhibit different forms of active sets and where second-order sufficient conditions are satisfied at the optimal solution.