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Numerical Solutions to the Bellman Equation of Optimal Control

Author

Listed:
  • Cesar O. Aguilar

    (California State University)

  • Arthur J. Krener

    (Naval Postgraduate School)

Abstract

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.

Suggested Citation

  • Cesar O. Aguilar & Arthur J. Krener, 2014. "Numerical Solutions to the Bellman Equation of Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 527-552, February.
    • Handle: RePEc:spr:joptap:v:160:y:2014:i:2:d:10.1007_s10957-013-0403-8
    DOI: 10.1007/s10957-013-0403-8
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    References listed on IDEAS

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    1. S. C. Beeler & H. T. Tran & H. T. Banks, 2000. "Feedback Control Methodologies for Nonlinear Systems," Journal of Optimization Theory and Applications, Springer, vol. 107(1), pages 1-33, October.
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    Cited by:

    1. Askhat Diveev & Elizaveta Shmalko, 2022. "Machine Learning Feedback Control Approach Based on Symbolic Regression for Robotic Systems," Mathematics, MDPI, vol. 10(21), pages 1-32, November.

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