Limit Hamilton-Jacobi–Isaacs Equations for Singularly Perturbed Zero-Sum Dynamic (Dicrete Time) Games

In this paper we study a singularly perturbed zero-sum dynamic game with full information. We introduce the upper (lower) value function of the dynamic game, in which the minimizer (maximizer) can be guaranteed if at the beginning of each interval his move (the choice of decision) precedes the move of the maximizer (minimizer). We show that when the singular perturbations parameter tends to zero, the upper (lower) value function of the dynamic game has a limit which coincides with a viscosity solution of a Hamilton–Jacobi–Isaacs-type equation. Two examples are given to demonstrate the potential of the proposed technique.