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Numerical Methods For Differential Games Based On Partial Differential Equations

Author

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  • M. FALCONE

    (Dipartimento di Matematica, Università di Rome, "La Sapienza", P. Aldo Moro 2, 00185 Rome, Italy)

Abstract

In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls and optimal trajectories. The approximation schemes also have an interesting control interpretation since the time-discrete scheme stems from a dynamic programming principle for the associated discrete time dynamical system. The general framework for convergence results to the value function is the theory of viscosity solutions. Numerical experiments are presented solving some classical pursuit-evasion games.

Suggested Citation

  • M. Falcone, 2006. "Numerical Methods For Differential Games Based On Partial Differential Equations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 231-272.
    • Handle: RePEc:wsi:igtrxx:v:08:y:2006:i:02:n:s0219198906000886
    DOI: 10.1142/S0219198906000886
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    Citations

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    Cited by:

    1. Adriano Festa & Richard B. Vinter, 2016. "Decomposition of Differential Games with Multiple Targets," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 848-875, June.
    2. Alexander Moll & Meir Pachter & Eloy Garcia & David Casbeer & Dejan Milutinović, 2020. "Robust Policies for a Multiple-Pursuer Single-Evader Differential Game," Dynamic Games and Applications, Springer, vol. 10(1), pages 202-221, March.
    3. Herbert Dawid & Maurizio Falcone, 2017. "Preface: DGAA Special Issue on Numerical Methods for Dynamic Games," Dynamic Games and Applications, Springer, vol. 7(4), pages 531-534, December.
    4. E. Cristiani & P. Martinon, 2010. "Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 321-346, August.
    5. Elisabetta Carlini & Adriano Festa & Francisco J. Silva & Marie-Therese Wolfram, 2017. "A Semi-Lagrangian Scheme for a Modified Version of the Hughes’ Model for Pedestrian Flow," Dynamic Games and Applications, Springer, vol. 7(4), pages 683-705, December.

    More about this item

    Keywords

    Pursuit-evasion games; numerical methods; dynamic programming; Isaacs equation; Primary 65M12; Secondary 49N25; Secondary 49L20;
    All these keywords.

    JEL classification:

    • B4 - Schools of Economic Thought and Methodology - - Economic Methodology
    • C0 - Mathematical and Quantitative Methods - - General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics

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