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    Area of Science:

    • Robotics
    • Game Theory
    • Control Theory

    Background:

    • Reach-avoid games are crucial for autonomous systems.
    • Analyzing games with differing player speeds and capture radii presents challenges.
    • Existing numerical methods have limitations in computational complexity and real-time application.

    Purpose of the Study:

    • To develop geometric methods for analyzing reach-avoid games with heterogeneous speeds.
    • To construct optimal analytical strategies and defensive barriers.
    • To enable precise, real-time calculations for nonpoint capture scenarios.

    Main Methods:

    • Geometric analysis of state space division into defensive and offensive advantage regions.
    • Development of optimal strategies based on payoff functions and player positions.
    • Construction of barriers and calculation of unique optimal target points.

    Main Results:

    • Demonstration of optimal analytical strategies for various speed ratios.
    • Identification of a unique optimal target point within the offensive advantage region.
    • Validation of constructed barriers' effectiveness in multiplayer simulations.

    Conclusions:

    • The proposed geometric method provides precise and real-time solutions for reach-avoid games.
    • This approach overcomes limitations of traditional numerical methods.
    • The findings are applicable to complex, multiplayer defensive scenarios.