Differential games of survival
Differential games of survival are studied using the definitions of strategy and payoff introduced recently by the author [SIAM J. Control Optimization 23, 172-196 (1985; Zbl 0572.90108)]. The Isaacs condition is assumed to hold at all points and a capturability condition is assumed to hold at boundary points of the terminal set. The dynamics and the integrand of the payoff are assumed to be Lipschitz continuous in the state, while the terminal payoff function is assumed to be Lipschitz in time and state. Under these hypotheses, the value exists, is continuous on the closure of the set \({\mathcal R}\) of initial positions (t,x) and is Lipschitz continuous on compact subsets of \({\mathcal R}\). The upper and lower directional derivates of the value satisfy a condition which can be viewed as a generalization of the Isaacs equation. These relations imply that the value is a viscosity solution of the Isaacs equation.
scientific article; zbMATH DE number 4045530