J. Korean Math. Soc. 49 (2012), No. 1, pp. 99–112 http://dx.doi.org/10.4134/JKMS.2012.49.1.099 JACOBI DISCRETE APPROXIMATION FOR SOLVING OPTIMAL CONTROL PROBLEMS Mamdouh El-Kady Abstract. This paper attempts to present a numerical method for solv- ing 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 equa- tions and hence the optimal control problem is reduced to constrained optimization problem. Numerical examples illustrate the robustness, ac- curacy and efficiency of the proposed method. 1. Introduction Bellman’s dynamic programming and Pontryagin’s maximum principle pres- ent the most known methods for solving optimal control problems. The com- putational approaches are considered a very important part of the solution of those problems [19]. In recent years, considerable attention has been given to the use of spectral methods for solving optimal control problems [2, 12, 14, 15]. Part of the difficulty of optimal control is that the first order conditions yield differential equation, which we have to solve to obtain a closed form solution [13]. Hence, to solve optimal control problems we have to study the numerical solutions of differential equations. The same previous attention, the spectral methods play an important role for solving differential equations [9, 17, 16]. The Jacobi polynomials P (α,β) n (x) play important roles in approximation theory and its applications, see [3, 4]. In his recent work [7, 5, 6] Doha develops a class of spectral-Galerkin methods for the direct solution of higher order differential equations. One of particular interest here is the Jacobi formula, based on finite Jacobi expansion in terms of power of x. In this paper, a numerical solution for solving optimal control problem is presented. Spectral method is a family of techniques for solving optimal control problems in which the summation in the numerical derivative is accelerated to Received August 26, 2010. 2010 Mathematics Subject Classification. Primary 65N35, 65L05, 65J10, 49J15. Key words and phrases. Jacobi polynomials, differentiation and integration matrices, op- timal control problem. c ⃝2012 The Korean Mathematical Society 99