Differential Games (Differential Games)

Author: 天疆说

This article is compiled from 张乘铭 (2021) "航天器追逃博弈制导策略研究"

Definition

Differential games is a branch of operations research and control theory that studies confrontation or cooperation among multiple decision-makers (players) in continuous-time dynamic systems. When the objectives of opposing parties are completely opposite (one party's gain is the other party's loss), it is called a zero-sum differential game. Differential game theory provides a rigorous mathematical modeling and solution framework for spacecraft pursuit-evasion, missile interception, and UAV air combat problems.

Historical Development

• 1965: American RAND Corporation mathematician Isaacs published a monograph, establishing the mathematical theoretical foundation of differential games
• 1971: Friedman used discrete game approximation sequences to define differential games, proving saddle point existence conditions
• 1970s-1980s: Differential games were widely applied in military fields (air combat, missile interception)
• Recent years: Combined with fuzzy control, reinforcement learning, and deep neural networks to solve high-dimensional complex game problems

Core Elements

Basic Classification

Differential games can be classified according to different dimensions:

Key Concepts

• Saddle-point strategy: In zero-sum differential games, neither party can unilaterally improve their strategy at the saddle point
• Payoff function: A performance metric measuring game outcomes; in zero-sum games, one party's loss equals the other's gain
• Co-state variables: Auxiliary variables accompanying state variable changes, used to derive necessary conditions for optimality
• Two-point boundary value problem: The mathematical problem form to which pursuit-evasion games are transformed, representing the core difficulty in solving saddle-point strategies

Relationship with Optimal Control

Differential games are closely related to optimal control. When there is only one decision-maker in a game, differential games degenerate into optimal control problems. The Maximum Principle of optimal control can be viewed as a special case of differential games. In pursuit-evasion problems, saddle-point guidance laws derived from differential game theory are formally consistent with guidance laws designed based on optimal control theory.

Application Domains

• Spacecraft pursuit-evasion games: Design of adversarial maneuvering strategies in orbital rendezvous and proximity operations
• Missile interception: Guidance law design for air-to-air and air-to-ground missiles
• UAV swarm coordination: Game decision-making for multi-agent systems
• Deep space exploration: Mission planning for spacecraft approaching asteroids and comets

References

• Isaacs R. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Optimization and Control[M]. John Wiley & Sons, 1965.
• Friedman A. Differential Games[M]. American Mathematical Society, 1971.
• 张乘铭. 航天器追逃博弈制导策略研究[D]. 国防科技大学, 2021.