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Approximate Solutions to the Time-Invariant Hamilton–Jacobi–Bellman Equation

Author

Listed:
  • R. W. Beard

    (Brigham Young University)

  • G. N. Saridis

    (Rensselaer Polytechnic Institute)

  • J. T. Wen

    (Rensselaer Polytechnic Institute)

Abstract

In this paper, we develop a new method to approximate the solution to the Hamilton–Jacobi–Bellman (HJB) equation which arises in optimal control when the plant is modeled by nonlinear dynamics. The approximation is comprised of two steps. First, successive approximation is used to reduce the HJB equation to a sequence of linear partial differential equations. These equations are then approximated via the Galerkin spectral method. The resulting algorithm has several important advantages over previously reported methods. Namely, the resulting control is in feedback form and its associated region of attraction is well defined. In addition, all computations are performed off-line and the control can be made arbitrarily close to optimal. Accordingly, this paper presents a new tool for designing nonlinear control systems that adhere to a prescribed integral performance criterion.

Suggested Citation

  • R. W. Beard & G. N. Saridis & J. T. Wen, 1998. "Approximate Solutions to the Time-Invariant Hamilton–Jacobi–Bellman Equation," Journal of Optimization Theory and Applications, Springer, vol. 96(3), pages 589-626, March.
    • Handle: RePEc:spr:joptap:v:96:y:1998:i:3:d:10.1023_a:1022664528457
    DOI: 10.1023/A:1022664528457
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    Cited by:

    1. Angelo Alessandri & Patrizia Bagnerini & Roberto Cianci & Mauro Gaggero, 2019. "Optimal Propagating Fronts Using Hamilton-Jacobi Equations," Mathematics, MDPI, vol. 7(11), pages 1-10, November.
    2. Goodarzi, Milad & Meinerding, Christoph, 2023. "Asset allocation with recursive parameter updating and macroeconomic regime identifiers," Discussion Papers 06/2023, Deutsche Bundesbank.
    3. 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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