%0 Journal Article
%T Direct Method for Resolution of Optimal Control Problem with Free Initial Condition
%A Louadj Kahina
%A Aidene Mohamed
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/173634
%X 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¨C4]. 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¨C13] 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, we use a finite set of switching points of a control [11, 14¨C16]. We solve the same problem in the article [17], we transform a problem initial to a problem of linear programming by carrying changes of variables in three procedures: change of control, change of support, and final procedure, in our paper, a solution of this problem, we discretize a problem initial to find an optimal support by using change of control and change of support, and we present the final procedure which uses this solution as an initial approximation for solving problem in the class of piecewise continuous function. We explain below that the realizations of the adaptive method [18] described in the paper possess the following advantages.(1)Size of the support (the main tool of the method), which mainly influences the complexity of an iteration of the method, does not depend on all general
%U http://www.hindawi.com/journals/ijde/2012/173634/