A Differential Game with Exit Costs

  • Fabio BagagioloEmail author
  • Rosario Maggistro
  • Marta Zoppello


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.


Differential games Exit costs Dirichlet problems for Hamilton–Jacobi–Isaacs equations Viscosity solutions Uniqueness Non-anticipating strategies 

Mathematics Subject Classification

49N70 49L25 



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


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Fabio Bagagiolo
    • 1
    Email author
  • Rosario Maggistro
    • 2
  • Marta Zoppello
    • 3
  1. 1.Department of MathematicsUniversity of TrentoTrentoItaly
  2. 2.Department of ManagementCa’ Foscari University of VeniceVeniceItaly
  3. 3.Department of Mathematical SciencesPolitecnico di TorinoTurinItaly

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