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State Constrained Two Player Differential Games with Decoupled Dynamics

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  • Piernicola Bettiol

    (CNRS, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique)

  • Jérémy Rouot

    (CNRS, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique)

Abstract

We consider a two player zero-sum differential game with state constraints, in which the dynamics is decoupled: each player has to stay in a closed (nonempty) set. We prove that, under suitable assumptions, the lower and the upper values are locally Lipschitz continuous and we establish that they are solutions, in the viscosity sense, of the Hamilton–Jacobi–Isaacs equation, which involves an appropriate Hamiltonian, called inner Hamiltonian. We finally provide a comparison theorem. It follows that the differential game under consideration admits a value (which coincides with the lower and the upper values). A key step in our analysis is a new nonanticipative Filippov-type theorem, which is valid for general closed sets.

Suggested Citation

  • Piernicola Bettiol & Jérémy Rouot, 2025. "State Constrained Two Player Differential Games with Decoupled Dynamics," Dynamic Games and Applications, Springer, vol. 15(2), pages 610-636, May.
    • Handle: RePEc:spr:dyngam:v:15:y:2025:i:2:d:10.1007_s13235-024-00589-0
    DOI: 10.1007/s13235-024-00589-0
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    References listed on IDEAS

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    1. Piernicola Bettiol & Pierre Cardaliaguet & Marc Quincampoix, 2006. "Zero-sum state constrained differential games: existence of value for Bolza problem," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(4), pages 495-527, November.
    2. Fabio Bagagiolo & Rosario Maggistro & Marta Zoppello, 2020. "A Differential Game with Exit Costs," Dynamic Games and Applications, Springer, vol. 10(2), pages 297-327, June.
    3. M. Falcone, 2006. "Numerical Methods For Differential Games Based On Partial Differential Equations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 231-272.
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