Multi-time scale zero-sum differential games with perfect state measurements
The paper is devoted to three multi-time scale zero-sum differential games. Its open-loop dynamics are described by a singularly perturbed differential equation which depends on two small positive parameters \(\varepsilon_1\), \(\varepsilon_2\). Given a quadratic cost functional, to be minimized the Player 1 and maximized by Player 2, for each fixed \(\varepsilon_1\), \(\varepsilon_2\) the well-known approach based on a generalized Riccati equation can be used. However, its realization for small values of the above parameters is prone to numerical errors and inaccuracies. The notion of ``approximate saddle- point solution'' is used in the paper to refer to strategies that do not depend on \(\varepsilon_1\) and \(\varepsilon_2\), while providing cost levels within \(O(\varepsilon_1)\) of the full-order game. Necessary and sufficient conditions for the existence of such ``approximate'' solutions are obtained. It is shown that the original game can be decomposed into three subgames -- slow, fast and fastest -- the composite solution of which make up the approximate saddle-point solution of the original game.
multi-time scale zero-sum differential games
scientific article; zbMATH DE number 729845