Spectral and Pseudospectral Optimal Control Over Arbitrary Grids

Spectral and Pseudospectral Optimal Control Over Arbitrary Grids

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Article ID: iaor20162411
Volume: 169
Issue: 3
Start Page Number: 759
End Page Number: 783
Publication Date: Jun 2016
Journal: Journal of Optimization Theory and Applications
Authors: , ,
Keywords: optimization
Abstract:

In advancing our prior work on a unified theory for pseudospectral (PS) optimal control, we present the mathematical foundations for spectral collocation over arbitrary grids. The computational framework is not based on any particular choice of quadrature nodes associated with orthogonal polynomials. Because our framework applies to non‐Gaussian grids, a number of hidden properties are uncovered. A key result of this paper is the discovery of the dual connections between PS and Galerkin approximations. Inspired by Polak’s pioneering work on consistent approximation theory, we analyze the dual consistency of PS discretization. This analysis reveals the hidden relationship between Galerkin and pseudospectral optimal control methods while uncovering some finer points on covector mapping theorems. The new theory is used to demonstrate via a numerical example that a PS method can be surprisingly robust to grid selection. For example, even when 60 % of the grid points are chosen to be uniform–the worst possible selection from a pseudospectral perspective–a PS method can still produce satisfactory result. Consequently, it may be possible to choose non‐Gaussian grid points to support different resolutions over the same grid.

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