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http://dx.doi.org/10.4134/JKMS.2012.49.1.099

JACOBI DISCRETE APPROXIMATION FOR SOLVING OPTIMAL CONTROL PROBLEMS  

El-Kady, Mamdouh (Department of Mathematics Faculty of Science Helwan University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.1, 2012 , pp. 99-112 More about this Journal
Abstract
This paper attempts to present a numerical method for solving optimal control problems. The method is based upon constructing the n-th degree Jacobi polynomials to approximate the control vector and use differentiation matrix to approximate derivative term in the state system. The system dynamics are then converted into system of algebraic equations and hence the optimal control problem is reduced to constrained optimization problem. Numerical examples illustrate the robustness, accuracy and efficiency of the proposed method.
Keywords
Jacobi polynomials; differentiation and integration matrices; optimal control problem;
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1 D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977.
2 IPOPT open source NLP solver. https://projects.coin-or.org/Ipopt.
3 H. Jadu, Spectral method for constrained linear-quadratic optimal control, Math. Comput. Simulation 58 (2002), no. 2, 159-169.   DOI   ScienceOn
4 W. Kang and N. Bedrossian, Pseudospectral optimal control theory makes debut fiight, saves nasa 1m in under three hours, SIAM News 40 (2007).
5 W. Kang, Q. Gong, I. M. Ross, and F. Fahroo, On the Convergence of Nonlinear Optimal Control Using Pseudospectral Methods for Feedback Linearizable Systems, Internat. J. Robust Nonlinear Control 17 (2007), no. 14, 1251-1277.   DOI   ScienceOn
6 H. T. Rathod, B. Venkatesudu, K. V. Nagaraja, and Md. S. Islam, Gauss Legendre-Gauss Jacobi quadrature rules over a tetrahedral region, Appl. Math. Comput. 190 (2007), no. 1, 186-194.   DOI   ScienceOn
7 I. M. Ross and F. Fahroo, Pseudospectral knotting methods for solving nonsmooth optimal control problems, Journal of Guidance Control and Dynamics 27 (2004), 397-405.   DOI   ScienceOn
8 J. Shen and L.Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys. 5 (2009), no. 2-4, 195-241.
9 G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, 1985.
10 M. Urabe, Numerical solution of multi-point boundary value problems in Chebyshev series. Theory of the method, Numer. Math. 9 (1967), 341-366.   DOI   ScienceOn
11 J. Vlassenbroeck, A Chebyshev polynomial method for optimal control with state constraints, Automatica J. IFAC 24 (1988), no. 4, 499-506.   DOI   ScienceOn
12 W. W. Bell, Special function for scientists and engineers, D. Van Nostrand Co., Ltd., London-Princeton, N.J.-Toronto, Ont., 1968.
13 R. Bhattacharya, A MATLAB Toolbox for Optimal Trajectory Generation, 2006.
14 A. H. Bhrawy and S. I. El-Soubhy, Jacobi spectral Galerkin method for the integrated forms of second-order differential equations, Applied Mathematics and Computation 217 (2010), 2684-2697.   DOI   ScienceOn
15 E. H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A 37 (2004), no. 3, 657-675.   DOI   ScienceOn
16 E. H. Doha and H. M. Ahmed, Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of quantum classical orthogonal polynomials, Journal of Advanced Research - Cairo Univ. 1, 193-207 (2010).
17 E. H. Doha and A. H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math. 58 (2008), no. 8, 1224-1244.   DOI   ScienceOn
18 E. H. Doha and A. M. Waleed, Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. Sci. Comput. 24 (2003), no. 2, 548-571.
19 T. M. El-Gindy and M. S. Salim, Penalty function with partial quadratic interpolation technique in the constrained optimization problems, J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), no. 1, 85-90.