Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An... Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control. Applied Mathematics and Optimization Springer Journals

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

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Springer US
Copyright © 2014 by Springer Science+Business Media New York
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
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