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Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. (English) Zbl 1161.49019

Summary: Second-order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In contrast to former publications on this subject, the cone of critical directions is the smallest possible in the sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one, respectively.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
90C48 Programming in abstract spaces
35B37 PDE in connection with control problems (MSC2000)
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