Multi-time scale zero-sum differential games with perfect state measurements

We obtain necessary and sufficient conditions for the existence of "approximate" saddle-point solutions in linear-quadratic zero-sum differential games when the state dynamics are defined on multiple (three) time scales. These different time scales are characterized in terms of two small positive parameters ∈₁, and ∈₂, and the terminology "approximate saddle-point solution" is used to refer to saddle-point policies that do not depend on ∈₁ and ∈₂, while providing cost levels within O(∈₁) of the full-order game. It is shown in the paper that, under perfect state measurements, the original game can be decomposed into three subgames-slow, fast and fastest, the composite saddle-point solution of which make up the approximate saddle-point solution of the original game. Specifically, for the minimizing player, it is necessary to use a composite policy that uses the solutions of all three subgames, whereas for the maximizing player, it is sufficient to use a slow policy. In the finite-horizon case this slow policy could be a feedback policy, whereas in the infinite-horizon case it has to be chosen as an open-loop policy that is generated from the solution and dynamics of the slow subgame. These results have direct applications in the H^∞-optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.