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

William W. Hager (), Hongyan Hou (), Subhashree Mohapatra (), Anil V. Rao () and Xiang-Sheng Wang ()
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William W. Hager: University of Florida
Hongyan Hou: Minnesota State University Moorhead
Subhashree Mohapatra: University of Florida
Anil V. Rao: University of Florida
Xiang-Sheng Wang: University of Louisiana at Lafayette

Computational Optimization and Applications, 2019, vol. 74, issue 1, No 11, 275-314

Abstract: 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.

Keywords: hp Collocation; Radau collocation; Convergence rate; Optimal control; Orthogonal collocation; 49M25; 49M37; 65K05; 90C30 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10589-019-00100-1

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