Spectral and Pseudospectral Optimal Control Over Arbitrary Grids

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.