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Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

Author

Listed:
  • Joseph D. Eide

    (University of Florida)

  • William W. Hager

    (University of Florida
    Society for Industrial and Applied Mathematics)

  • Anil V. Rao

    (University of Florida
    AIAA)

Abstract

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:191:y:2021:i:2:d:10.1007_s10957-021-01810-5
    DOI: 10.1007/s10957-021-01810-5
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    References listed on IDEAS

    as
    1. 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.
    2. William W. Hager & Hongyan Hou & Subhashree Mohapatra & Anil V. Rao & Xiang-Sheng Wang, 2019. "Correction to: Convergence rate for a Radau hp collocation method applied to constrained optimal control," Computational Optimization and Applications, Springer, vol. 74(1), pages 315-316, September.
    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.
    4. 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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