Operator-Theoretic Methods for Differential Games

TL;DR

This paper introduces two Koopman Operator-based methods for solving zero-sum differential games, providing global feedback and trajectory-specific control solutions, demonstrated on a turret defense game.

Contribution

It presents novel KO-based approaches for differential games, including a global feedback method and a data-driven control policy method, with practical demonstration and analysis.

Findings

Methods replicate analytical solutions accurately.

Global feedback approach offers continuous-time solutions.

Data-driven approach computes open-loop controls effectively.

Abstract

Differential game theory offers an approach for modeling interactions between two or more agents that occur in continuous time. The goal of each agent is to optimize its objective cost functional. In this paper, we present two different methods, based on the Koopman Operator (KO), to solve a zero-sum differential game. The first approach uses the resolvent of the KO to calculate a continuous-time global feedback solution over the entire domain. The second approach uses a discrete-time, data-driven KO representation with control to calculate open-loop control policies one trajectory at a time. We demonstrate these methods on a turret defense game from the literature, and we find that the methods' solutions replicate the behavior of the analytical solution provided in the literature.. Following that demonstration, we highlight the relative advantages and disadvantages of each method and discuss potential future work for this line of research.