Application Of Pseudo-spectral Method For The Optimal Control

In this paper, we consider the direct method of the optimal control problem.The methods mainly are the pseudo-spectral method based on the orthogonalpolynomials, which is different from the traditional difference method based on thelocal properties. The pseudo-spectral method is based on the global properties of thefunction, using a series of orthogonal polynomials to approximate the objectfunction and structuring the formula of derivation. Thus, we could transform thecontinuous system into a discrete system, which is called the nonlinear problem,and then the optimal control problem can be solved by the nonlinear programmingtheory. The key of the method is how to select the configure points and how toconstruct differential equations.This article is mainly composed of the following there parts:In Chapter1, we mainly introduce the background, practical application anddevelopment prospects of the optimal control problem.In Chapter2, we introduce some basic knowledge, including the basic conceptsof optimal control problem, the Bolza problem of continuous-time and thefirst-order necessary optimality conditions, the basic idea of the numericalalgorithms; Then we introduce the nature of the orthogonal polynomials and twobasic pseudo-spectral methods. Finally, consider for the less-smooth case, wepresent a new direct method based on the spline interpolation and analyze theresults.In Chapter3, based on the knowledge, we analyze and explore the directmethod deeply. First, we analyze the principle of the Gauss pseudo-spectral methodand its co-state variables theorem and give a method improving the Gauss pseudo-spectral method in computational efficiency. The improved Gauss pseudo-spectralmethod not only reduces the amount of computation, but also avoides thepolynomial shock caused by polynomial interpolation function.Finally, thenumerical results proved that the improved new method is indeed reducedcomputation.