A differential game with graph constrained dynamic switching strategies

A two-person zero-sum differential game of finite horizon with graph-constrained control strategies is addressed where both players are constrained control switches selected from a finite control graph which may depend on the state of the system. The control graph is a directed graph where the vertices define pairs of control values for both players, and the edges define the allowable control switches. A positive switching cost is associated with each edge, being the payoff function defined by the sum of the switching costs associated with each player. Issues concerning both the existence of value of the game and optimality conditions are addressed. in terms of value functions. Optimal strategies are derived in terms of a value functions which solve a coupled system of quasi-variational inequalities.