Abstract
In this paper, optimal control problems for semilinear parabolic equationswith distributed and boundary controls are considered. Pointwise constraints onthe control and on the state are given. Main emphasis is laid on the discussionof second order sufficient optimality conditions. Sufficiency for local optimality isverified under different assumptions imposed on the dimension of the domain andon the smoothness of the given data.
Highlights
This paper is a further contribution to the theory of second order su cient optimality conditions for optimal control problems governed by nonlinear partial di erential equations
We consider the control of semilinear parabolic equations with pointwise constraints on the control and the state
This fact is crucial in the analysis of second order su cient optimality conditions
Summary
This paper is a further contribution to the theory of second order su cient optimality conditions for optimal control problems governed by nonlinear partial di erential equations. Casas, Troltzsch and Unger 6] have discussed second order su cient conditions for the boundary control of semilinear elliptic equations with pointwise state-constraints. It is convenient, to formulate this class of constraints in spaces of continuous functions, the associated Lagrange multipliers are Borel measures. (v; u)) 0 for all (v; u) 2 Vad Uad: nuity4.inRYe2guVlariUtyorfetshuelqtusafdorrattihcefosrtmatLe00a(yn;dv;tuh;ep;ad)jo(yi1n;tve1;quu1a);t(ioy2n;sv.2;Tuh2e)]cpolnatyisa crucial role in the second order analysis (see section 6) This continuity property depends on regularity results for the adjoint state p and for solutions of the linearized state equation with source terms belonging to V and U. The proof is similar to the one of Theorem 4.1
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