Abstract
We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using the formalism of subdifferential calculus in Wasserstein spaces. We show that the geometric approach based on needle variations and on the evolution of the covector (here replaced by the evolution of a mesure on the dual space) can be translated into this formalism.
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Notes
We omit the dependence on p for clarity and conciseness.
Namely, \(\nu _T^*(t)\) is defined as the \(\mu ^*(T)\)-almost uniquely determined measure which has \(\mu ^*(T)\) as its first marginal and which disintegration is given by \(\{ \sigma ^*_x(t)\}_x\) (see Definition 4).
Recall that \(\bar{\varvec{\gamma }}^{\circ }_{\varphi }(\cdot )\) is a continuous map by hypothesis (B).
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Acknowledgements
This work has been carried out in the framework of Archimde Labex (ANR-11-LABX-0033) and of the A*MIDEX project (ANR- 11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR).
The authors also thank the reviewer for the several useful comments that he/she provided.
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Communicated by L. Ambrosio.
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