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Convergence rate for a Radau hp collocation method applied to constrained optimal control

Author

Listed:
  • William W. Hager

    (University of Florida)

  • Hongyan Hou

    (Minnesota State University Moorhead)

  • Subhashree Mohapatra

    (University of Florida
    SRM Institute of Science and Technology)

  • Anil V. Rao

    (University of Florida)

  • Xiang-Sheng Wang

    (University of Louisiana at Lafayette)

Abstract

For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.

Suggested Citation

  • William W. Hager & Hongyan Hou & Subhashree Mohapatra & Anil V. Rao & Xiang-Sheng Wang, 2019. "Convergence rate for a Radau hp collocation method applied to constrained optimal control," Computational Optimization and Applications, Springer, vol. 74(1), pages 275-314, September.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:1:d:10.1007_s10589-019-00100-1
    DOI: 10.1007/s10589-019-00100-1
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    References listed on IDEAS

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    1. Wanchun Chen & Wenhao Du & William W. Hager & Liang Yang, 2019. "Bounds for integration matrices that arise in Gauss and Radau collocation," Computational Optimization and Applications, Springer, vol. 74(1), pages 259-273, September.
    2. William W. Hager & Hongyan Hou & Anil V. Rao, 2016. "Convergence Rate for a Gauss Collocation Method Applied to Unconstrained Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 801-824, June.
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    Cited by:

    1. Wanchun Chen & Wenhao Du & William W. Hager & Liang Yang, 2019. "Bounds for integration matrices that arise in Gauss and Radau collocation," Computational Optimization and Applications, Springer, vol. 74(1), pages 259-273, September.
    2. Elisha R. Pager & Anil V. Rao, 2022. "Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation," Computational Optimization and Applications, Springer, vol. 81(3), pages 857-887, April.
    3. Joseph D. Eide & William W. Hager & Anil V. Rao, 2021. "Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions," Journal of Optimization Theory and Applications, Springer, vol. 191(2), pages 600-633, December.
    4. Zhang, Zemian & Chen, Xuesong, 2021. "A conjugate gradient method for distributed optimal control problems with nonhomogeneous Helmholtz equation," Applied Mathematics and Computation, Elsevier, vol. 402(C).

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    1. Joseph D. Eide & William W. Hager & Anil V. Rao, 2021. "Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions," Journal of Optimization Theory and Applications, Springer, vol. 191(2), pages 600-633, December.
    2. Elisha R. Pager & Anil V. Rao, 2022. "Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation," Computational Optimization and Applications, Springer, vol. 81(3), pages 857-887, April.
    3. Wanchun Chen & Wenhao Du & William W. Hager & Liang Yang, 2019. "Bounds for integration matrices that arise in Gauss and Radau collocation," Computational Optimization and Applications, Springer, vol. 74(1), pages 259-273, September.

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