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

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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.

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Acknowledgements

The authors gratefully acknowledge support for this research from the U.S. Office of Naval Research under Grants N00014-15-1-2048 and N00014-19-1-2543, and from the U.S. National Science Foundation under Grants CBET-1404767, DMS-1522629, DMS-1819002, CMMI-1563225, and CMMI-2031213.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 32611, USA

    Joseph D. Eide & Anil V. Rao

  2. Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA

    William W. Hager

  3. Society for Industrial and Applied Mathematics, Philadelphia, USA

    William W. Hager

  4. AIAA, Reston, USA

    Anil V. Rao

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  1. Joseph D. Eide
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Eide, J.D., Hager, W.W. & Rao, A.V. Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions. J Optim Theory Appl 191, 600–633 (2021). https://doi.org/10.1007/s10957-021-01810-5

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