Viscosity solutions of the Isaacs equation οn an attainable set
We apply a modification of the viscosity solution concept introduced in  to the Isaacs equation defined on the set attainable from a given set of initial conditions. We extend the notion of a lower strategy introduced by us in  to a more general setting to prove that the lower and upper values of a differential game are subsolutions (resp. supersolutions) in our sense to the upper (resp. lower) Isaacs equation of the differential game. Our basic restriction is that the variable duration time of the game is bounded above by a certain number T>0. In order to obtain our results, we prove the Bellman optimality principle of dynamic programming for differential games.
-  E. Barron, L. Evans and R. Jensen, Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls, J. Differential Equations 53 (1984), 213-233.
-  M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42.
-  R. Elliott and N. Kalton, Cauchy problems for certain Isaacs-Bellman equations and games of survival, ibid. 198 (1974), 45-72.
-  L. Evans and H. Ishii, Differential games and nonlinear first order PDE on bounded domains, Manuscripta Math. 49 (1984), 109-139.
-  A. Friedman, Differential Games, Wiley, New York, 1971.
-  W. Fleming, The Cauchy problem for degenerate parabolic equations, J. Math. Mech. 13 (1964), 987-1008.
-  H. Ishii, Remarks on existence of viscosity solutions of Hamilton-Jacobi equations, Bull. Fac. Sci. Engrg. Chuo Univ. 26 (1983), 5-24.
-  H. Ishii, J.-L. Menaldi and L. Zaremba, Viscosity solutions of the Bellman equation on an attainable set, Problems Control Inform. Theory 20 (1991), 317-328.
-  N. Krasovskiĭ and A. Subbotin, Positional Differential Games, Nauka, Moscow, 1974 (in Russian).
-  N. Krasovskiĭ and A. Subbotin, An alternative for the game problem of convergence, J. Appl. Math. Mech. 34 (1971), 948-965.
-  P. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, 1982.
-  O. Oleĭnik and S. Kruzhkov, Quasi-linear second order parabolic equations with several independent variables, Uspekhi Mat. Nauk 16 (5) (1961), 115-155 (in Russian).
-  P. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations 56 (1985), 345-390.
-  A. Subbotin, A generalization of the fundamental equation of the theory of differential games, Dokl. Akad. Nauk SSSR 254 (1980), 293-297 (in Russian).
-  A. Subbotin, Existence and uniqueness results for Hamilton-Jacobi equation, Nonlinear Anal., to appear.
-  A. Subbotin and A. Taras'ev, Stability properties of the value function of a differential game and viscosity solutions of Hamilton-Jacobi equations, Problems Control Inform. Theory 15 (1986), 451-463.
-  L. S. Zaremba, Optimality principles of dynamic programming in differential games, J. Math. Anal. Appl. 138 (1989), 43-51.