Journal of Optimization Theory and Applications

, Volume 130, Issue 2, pp 209–229 | Cite as

Representation Formulas for Solutions of the HJI Equations with Discontinuous Coefficients and Existence of Value in Differential Games

  • M. Garavello
  • P. Soravia


In this paper, we study the Hamilton-Jacobi-Isaacs equation of zerosum differential games with discontinuous running cost. For such class of equations, the uniqueness of the solutions is not guaranteed in general. We prove principles of optimality for viscosity solutions where one of the players can play either causal strategies or only a subset of continuous strategies. This allows us to obtain nonstandard representation formulas for the minimal and maximal viscosity solutions and prove that a weak form of the existence of value is always satisfied. We state also an explicit uniqueness result for the HJI equations for piecewise continuous coefficients, in which case the usual statement on the existence of value holds.


Differential games viscosity solutions existence of value discontinuous Lagrangians 


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© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • M. Garavello
    • 1
  • P. Soravia
    • 2
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPadovaItaly

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