| Issue |
ESAIM: COCV
Volume 30, 2024
|
|
|---|---|---|
| Article Number | 91 | |
| Number of page(s) | 37 | |
| DOI | https://doi.org/10.1051/cocv/2024081 | |
| Published online | 10 December 2024 | |
Viscosity solutions of centralized control problems in measure spaces
1
INSA Rouen Normandie, Normandie Univ, LMI UR 3226, 76000 Rouen, France
2
Université de Rennes, INSA Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France
* Corresponding author: averil.aussedat@insa-rouen.fr
Received:
4
March
2024
Accepted:
7
November
2024
This work focuses on a control problem in the Wasserstein space of probability measures over ℝd. Our aim is to link this control problem to a suitable Hamilton–Jacobi–Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland’s principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations.
Mathematics Subject Classification: 35F21 / 35R06 / 49Lxx
Key words: Hamilton–Jacobi / Wasserstein / viscosity solutions
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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