Spectral and Pseudospectral Optimal Control Over Arbitrary Grids

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• 作者：Qi Gong ; Isaac Michael Ross ; Fariba Fahroo
• 关键词：Optimal control ; Pseudospectral ; Computation ; 49J15
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：169
• 期：3
• 页码：759-783
• 全文大小：1,596 KB
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• 作者单位：Qi Gong (1)
  Isaac Michael Ross (2)
  Fariba Fahroo (3)
  
  1. Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA, 95064, USA
  2. Department of Mechanical and Aerospace Engineering, Naval Postgraduate School, Monterey, CA, 93943, USA
  3. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943, USA
  
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878
• 卷排序：169

文摘

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. Keywords Optimal control Pseudospectral Computation