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1. Adaptive Mesh Refinement and Error Estimation Method for Optimal Control Using Direct Collocation

   Haman, George V_III; Rao, Anil V (February 2025, ASME Journal of Dynamic Systems, Measurement, and Control)

   An adaptive mesh refinement method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not been met, the mesh is refined by either increasing the degree of the approximating polynomial in a mesh interval or dividing a mesh interval into subintervals. In regions of the solution where the desired accuracy tolerance has been met, the mesh size may be reduced by either merging adjacent mesh intervals or decreasing the degree of the approximating polynomial in a mesh interval. Coupled with the mesh refinement method described in this paper is a newly developed relative error estimate that is based on the differences between solutions obtained from the collocation method and those obtained by solving initial-value and terminal-value problems in each mesh interval using an interpolated control obtained from the collocation method. Because the error estimate is based on explicit simulation, the solution obtained via collocation is in close agreement with the solution obtained via explicit simulation using the control on the final mesh, which ensures that the control is an accurate approximation of the true optimal control. The method is demonstrated on three examples from the open literature, and the results obtained show an improvement in final mesh size when compared against previously developed mesh refinement methods. 

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2. AdaptiveMesh RefinementMethod for Optimal Control Using Nonsmoothness Detection andMesh Size Reduction

   https://doi.org/DOI: 10.1016/j.franklin.2015.05.028  

   Liu, F. (January 2015, Journal of the Franklin Institute)

   An adaptive mesh refinement method for solving optimal control problems is developed. The method employs orthogonal collocation at Legendre–Gauss–Radau points, and adjusts both the mesh size and the degree of the approximating polynomials in the refinement process. A previously derived convergence rate is used to guide the refinement process. The method brackets discontinuities and improves solution accuracy by checking for large increases in higher-order derivatives of the state. In regions between discontinuities, where the solution is smooth, the error in the approximation is reduced by increasing the degree of the approximating polynomial. On mesh intervals where the error tolerance has been met, mesh density may be reduced either by merging adjacent mesh intervals or lowering the degree of the approximating polynomial. Finally, the method is demonstrated on two examples from the open literature and its performance is compared against a previously developed adaptive method. 

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3. CGPOPS: A C++ Software for Solving Multiple-Phase Optimal Control Problems Using Adaptive Gaussian Quadrature Collocation and Sparse Nonlinear Programming

   https://doi.org/10.1145/3390463  

   Agamawi, Yunus M.; Rao, Anil V. (September 2020, ACM Transactions on Mathematical Software)

   null (Ed.)

   A general-purpose C++ software program called CGPOPS is described for solving multiple-phase optimal control problems using adaptive direct orthogonal collocation methods. The software employs a Legendre-Gauss-Radau direct orthogonal collocation method to transcribe the continuous optimal control problem into a large sparse nonlinear programming problem (NLP). A class of hp mesh refinement methods are implemented that determine the number of mesh intervals and the degree of the approximating polynomial within each mesh interval to achieve a specified accuracy tolerance. The software is interfaced with the open source Newton NLP solver IPOPT. All derivatives required by the NLP solver are computed via central finite differencing, bicomplex-step derivative approximations, hyper-dual derivative approximations, or automatic differentiation. The key components of the software are described in detail, and the utility of the software is demonstrated on five optimal control problems of varying complexity. The software described in this article provides researchers a transitional platform to solve a wide variety of complex constrained optimal control problems. 

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4. Mesh refinement method for solving optimal control problems with nonsmooth solutions using jump function approximations

   https://doi.org/10.1002/oca.2719  

   Miller, Alexander T.; Hager, William W.; Rao, Anil V. (March 2021, Optimal Control Applications and Methods)

   Abstract A mesh refinement method is described for solving optimal control problems using Legendre‐Gauss‐Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targetedh‐refinement approach whereby the discontinuity locations are bracketed with mesh points. The remaining smooth portions of the mesh are refined using previously developed techniques. The method is demonstrated on two examples, and results indicate that the method solves optimal control problems with discontinuous control solutions using fewer mesh refinement iterations and less computation time when compared with previously developed methods. 

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

   https://doi.org/10.1007/s10589-019-00100-1  

   Hager, William W.; Hou, Hongyan; Mohapatra, Subhashree; Rao, Anil V.; Wang, Xiang-Sheng (May 2019, Computational Optimization and Applications)

   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. 

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