Decomposition of Differential Games

TL;DR

This paper introduces a decomposition method for zero-sum differential games with union-of-targets, simplifying their solution by analyzing smaller, more manageable subgames and combining their results.

Contribution

It presents a novel decomposition technique for differential games with union targets, providing criteria for validity and illustrating applications to pursuit/evasion scenarios.

Findings

Decomposition simplifies solving complex differential games.

The value of the original game is the lower envelope of subgame values.

Application examples demonstrate effectiveness in pursuit/evasion games.

Abstract

This paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets; the decomposition consists of replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games, where the decomposition arises from considering the interaction of individual pursuer/evader pairs.