A Differential Game with Exit Costs

Fabio Bagagiolo (), Rosario Maggistro () and Marta Zoppello ()
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Fabio Bagagiolo: University of Trento
Rosario Maggistro: Ca’ Foscari University of Venice
Marta Zoppello: Politecnico di Torino

Dynamic Games and Applications, 2020, vol. 10, issue 2, No 1, 297-327

Abstract: 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.

Keywords: Differential games; Exit costs; Dirichlet problems for Hamilton–Jacobi–Isaacs equations; Viscosity solutions; Uniqueness; Non-anticipating strategies; 49N70; 49L25 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13235-019-00334-y

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