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Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential for compactly supported probability measures, and investigate the intrinsic linearised Cauchy problems associated to non-local continuity equations. In particular, we show that when the velocity perturbations belong to the tangent cone to the convexification of the set of admissible velocities, the solutions of these linearised problems are tangent to the solution set of the corresponding continuity inclusion. We then make use of these novel concepts to provide a synthetic and geometric proof of the celebrated Pontryagin Maximum Principle for an optimal control problem with inequality final-point constraints. In addition, we propose sufficient conditions ensuring the normality of the maximum principle. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-021-09772-w
Publisher site
See Article on Publisher Site

Abstract

In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential for compactly supported probability measures, and investigate the intrinsic linearised Cauchy problems associated to non-local continuity equations. In particular, we show that when the velocity perturbations belong to the tangent cone to the convexification of the set of admissible velocities, the solutions of these linearised problems are tangent to the solution set of the corresponding continuity inclusion. We then make use of these novel concepts to provide a synthetic and geometric proof of the celebrated Pontryagin Maximum Principle for an optimal control problem with inequality final-point constraints. In addition, we propose sufficient conditions ensuring the normality of the maximum principle.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: May 19, 2021

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