Bounds for integration matrices that arise in Gauss and Radau collocation

Chen, Wanchun; Du, Wenhao; Hager, William W.; Yang, Liang

doi:10.1007/s10589-019-00099-5

Cited by 14 publications

(10 citation statements)

References 12 publications

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“…Convergence of the p method was then achieved by increasing the degree of the polynomial approximation. For problems whose solutions are smooth and well behaved, a Gaussian quadrature orthogonal collocation method converges at an exponential rate [17,[36][37][38][39]. The most well-developed p Gaussian quadrature methods are those that employ either Legendre-Gauss (LG) points [8,54], Legendre-Gauss-Radau (LGR) points [24,25,45], or Legendre-Gauss-Lobatto (LGL) points [19].…”

Section: Introductionmentioning

confidence: 99%

Cgpops

Agamawi

Rao

2020

ACM Trans. Math. Softw.

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A general-purpose C++ software program called CGP OP S 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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Section: Introductionmentioning

confidence: 99%

Cgpops

Agamawi

Rao

2020

ACM Trans. Math. Softw.

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“…In a p method, the number of intervals is fixed, and convergence is achieved by increasing the degree of the approximation in each interval. To achieve maximum effectiveness, p methods have been developed using orthogonal collocation at Gaussian quadrature points [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. For problems whose solutions are smooth and well-behaved, a Gaussian quadrature orthogonal collocation method converges at an exponential rate [20,21,22,23,24].…”

mentioning

confidence: 99%

“…To achieve maximum effectiveness, p methods have been developed using orthogonal collocation at Gaussian quadrature points [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. For problems whose solutions are smooth and well-behaved, a Gaussian quadrature orthogonal collocation method converges at an exponential rate [20,21,22,23,24]. Gauss quadrature collocation methods use either Legendre-Gauss (LG) points [8,12,13,20,21,22,23], Legendre-Gauss-Radau (LGR) points [11,12,13,14,20,23,24], or Legendre-Gauss-Lobatto (LGL) points [7].…”

mentioning

confidence: 99%

“…For problems whose solutions are smooth and well-behaved, a Gaussian quadrature orthogonal collocation method converges at an exponential rate [20,21,22,23,24]. Gauss quadrature collocation methods use either Legendre-Gauss (LG) points [8,12,13,20,21,22,23], Legendre-Gauss-Radau (LGR) points [11,12,13,14,20,23,24], or Legendre-Gauss-Lobatto (LGL) points [7].…”

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confidence: 99%

“…In this paper a new direct collocation mesh refinement method is developed for solving optimal control problems whose solutions have a bang-bang structure. While in principle the method of this paper can be used with any collocation method, in this paper the Legendre-Gauss-Radau collocation method [11,12,13,14,20,23,24] is employed because it produces accurate state, control, and costate approximations [11,12,13,14,24]. Moreover, the approach of this paper is designed to be used in conjunction with a previously developed hp mesh refinement method such as those described in Refs.…”

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confidence: 99%

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Mesh Refinement Method for Solving Bang-Bang Optimal Control Problems Using Direct Collocation

Agamawi

Hager

Rao

2020

AIAA Scitech 2020 Forum

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A mesh refinement method is developed for solving bang-bang optimal control problems using direct collocation. The method starts by finding a solution on a coarse mesh. Using this initial solution, the method then determines automatically if the Hamiltonian is linear with respect to the control, and, if so, estimates the locations of the discontinuities in the control. The switch times are estimated by determining the roots of the switching functions, where the switching functions are determined using 1 arXiv:1905.11895v3 [math.OC]

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

et al. 2019

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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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Bounds for integration matrices that arise in Gauss and Radau collocation

Cited by 14 publications

References 12 publications

Cgpops

Cgpops

Mesh Refinement Method for Solving Bang-Bang Optimal Control Problems Using Direct Collocation

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

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