Hindawi Publishing Corporation International Journal of Differential Equations Volume 2012, Article ID 173634, 18 pages doi:10.1155/2012/173634 Research Article Direct Method for Resolution of Optimal Control Problem with Free Initial Condition Louadj Kahina 1 and Aidene Mohamed 2 1 Department of Mathematics, Faculty of Sciences, University Mouloud Mammeri of Tizi-Ouzou, Tizi-Ouzou, Algeria 2 Laboratoire de Conception et Conduite de Syst` emes de Production (L2CSP), UMMTO Tizi-Ouzou, Algeria Correspondence should be addressed to Louadj Kahina, louadj kahina@yahoo.fr Received 22 September 2011; Accepted 3 November 2011 Academic Editor: Sabri Arik Copyright q 2012 L. Kahina and A. Mohamed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The theory of control analyzes the proprieties of commanded systems. Problems of optimal control OC have been intensively investigated in the world literature for over forty years. During this period, series of fundamental results have been obtained, among which should be noted the maximum principle Pontryagin et al., 1962 and dynamic programming Bellman, 1963. For many of the problems of the optimal control theory OCT, adequate solutions are found Bryson and Yu-chi, 1969, Lee and Markus, 1967, Gabasov and Kirillova, 1977, 1978, 1980. Results of the theory were taken up in various fields of science, engineering, and economics. The present paper aims at extending the constructive methods of Balashevich et al., 2000 that were developed for the problems of optimal control with the bounded initial state is not fixed are considered. 1. Introduction Problems of optimal control OC have been intensively investigated in the world literature for over forty years. During this period, a series of fundamental results have been obtained, whose majority is based on the maximum principle 1 and dynamic programming 2– 4. Currently there exist two types of methods of resolution: direct methods and indirect methods. The indirect methods are based on the maximum principle 1 and the methods of the shooting 5. The direct methods are based on the discretization of the initial problem, but here we obtain an approximate solution. The aim of this paper is to apply an adaptive method of linear programming 6–13 for an optimal control problem with a free initial condition. Here we use a final procedure based on a resolution of linear system with the Newton method to obtain a optimal solution. Here,