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A Differential Game with Exit Costs

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Abstract

We study a differential game where two players separately control their own dynamics, pay a running cost, and moreover pay an exit cost (quitting the game) when they leave a fixed domain. In particular, each player has its own domain and the exit cost consists of three different exit costs, depending whether either the first player only leaves its domain, or the second player only leaves its domain, or they both simultaneously leave their own domain. We prove that, under suitable hypotheses, the lower and upper values are continuous and are, respectively, the unique viscosity solution of a suitable Dirichlet problem for a Hamilton–Jacobi–Isaacs equation. The continuity of the values relies on the existence of suitable non-anticipating strategies respecting the domain constraint. This problem is also treated in this work.

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Acknowledgements

This work was partially supported by the Italian INDAM-GNAMPA project 2017.

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Authors and Affiliations

  1. Department of Mathematics, University of Trento, Trento, Italy

    Fabio Bagagiolo

  2. Department of Management, Ca’ Foscari University of Venice, Venice, Italy

    Rosario Maggistro

  3. Department of Mathematical Sciences, Politecnico di Torino, Turin, Italy

    Marta Zoppello

Authors
  1. Fabio Bagagiolo
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  2. Rosario Maggistro
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  3. Marta Zoppello
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Correspondence to Fabio Bagagiolo.

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Bagagiolo, F., Maggistro, R. & Zoppello, M. A Differential Game with Exit Costs. Dyn Games Appl 10, 297–327 (2020). https://doi.org/10.1007/s13235-019-00334-y

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