Abstract
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.
Highlights
We essentially improve the theory of secondorder sufficient optimality conditions for state-constrained optimal control problems of elliptic and parabolic type
We are able to complete the theory of second-order sufficient conditions for this class of problems, if the dimension of the spatial domain is sufficiently small
We study the following optimal control problem:
Summary
We essentially improve the theory of secondorder sufficient optimality conditions for state-constrained optimal control problems of elliptic and parabolic type. This theorem shows why we assume n ≤ 3: To prove Theorem 4.1 on second-order sufficient conditions, we need the operator G to be differentiable from L2(Ω) to C(Ω ) This result holds true only for n ≤ 3. Maurer and Zowe [21] used first-order sufficient conditions to consider strict positivity of Lagrange multipliers Inspired by their approach, in [12] an application to state-constrained elliptic boundary control was suggested. We recall that the convexity of L with respect to u was necessary to prove the existence of an optimal control Under this strict convexity assumption, the sufficient second-order optimality conditions are reduced to (4.5).
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