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Strict Sense Minimizers Which Are Not Extended Minimizers and Abnormality
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: Received: 22 February 2024 / Approved: 22 February 2024 / Online: 23 February 2024 (08:33:55 CET)
A peer-reviewed article of this Preprint also exists.
Fusco, G.; Motta, M. Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers. Mathematics 2024, 12, 943. Fusco, G.; Motta, M. Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers. Mathematics 2024, 12, 943.
Abstract
We consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control, used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. It is therefore relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in non-smooth analysis, we investigate the relation between the occurrence of an infimum gap and abnormality of necessary conditions. For a notion of local minimizer based on control distance and an extension including the impulsive one, we prove that (i) a local extended minimizer which is not a local minimizer of the original problem and (ii) a local strict-sense minimizer which is not a local minimizer of the extended problem, both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now it had only been shown that a strict-sense minimizer which is not an extended minimizer is abnormal for an `averaged version’ of the maximum principle.
Keywords
Optimal control problems; Maximum principle; State constraints; Gap phenomena; Impulsive optimal control
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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