Asymptotic Solution of a Cheap Control Game with Slow and Fast State Variables

TL;DR

This paper develops a novel asymptotic analysis method for a zero-sum linear-quadratic differential game with slow and fast state variables, where control costs are asymmetrically scaled.

Contribution

It introduces a new approach to asymptotic analysis of matrix Riccati equations in complex differential games with multiple time scales.

Findings

Derived an asymptotic solution for the differential game.

Presented an illustrative example demonstrating the approach.

Abstract

A finite-horizon zero-sum linear-quadratic differential game is considered. Its features are: (i) the control cost of the minimizing player in the game's cost functional is much smaller than the control cost of the maximizing player and the state cost; (ii) the cost of the fast state variable in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. These features require developing a significantly novel approach to asymptotic analysis of the matrix Riccati differential equation associated with the considered game. Using this analysis, an asymptotic solution of the game is derived. An illustrative example is presented.