Representation Formulas for Solutions of the HJI Equations with Discontinuous Coefficients and Existence of Value in Differential Games
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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.
KeywordsDifferential games viscosity solutions existence of value discontinuous Lagrangians
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- 8.GRUNE, L., Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1783, 2002.Google Scholar
- 14.SORAVIA, P., Degenerate Eikonal Equations with Discontinuous Refraction Index,ESAIM Control, Optimization, and Calculus of Variations, Vol. 12, pp. 216–230, 2006.Google Scholar
- 19.SUBBOTIN, A. I., Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective, Systems and Control: Foundations and Applications, Birkhäuser, Basel, Switzerland, 1995.Google Scholar
- 20.KRASSOWSKI, N., and SUBBOTIN, A. I., Jeux Différentiels, Mir, Moskow, Russia, 1977.Google Scholar
- 24.SORAVIA, P., Uniqueness Results for Viscosity Solutions of Fully Nonlinear Degenerate Elliptic Equations with Discontinuous Coefficients, Communications in Pure and Applied Analysis, Vol. 5, pp. 213–240, 2006.Google Scholar