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Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

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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.

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Acknowledgements

This material is based upon work supported by the Air Force Office of Scientific Research under Award Number FA9550-18-1-0254.

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  1. CNRS, IMJ-PRG, UMR 7586, Sorbonne Université, 4 place Jussieu, 75252, Paris, France

    Benoît Bonnet & Hélène Frankowska

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  1. Benoît Bonnet
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  2. Hélène Frankowska
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Correspondence to Benoît Bonnet.

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Bonnet, B., Frankowska, H. Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces. Appl Math Optim 84 (Suppl 2), 1281–1330 (2021). https://doi.org/10.1007/s00245-021-09772-w

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