World Applied Sciences Journal 5 (4): 517-521, 2008 ISSN 1818-4952 © IDOSI Publications, 2008 Corresponding Author: M. Mehrabinezhad, Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran 517 A Discretisation Method for Solving Time Optimal Control Problems M.H. Farahi, A.V. Kamyad, M. Mehrabinezhad and M.R. Zarrabi Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran Abstract: Classical methods are mostly deficient for solving nonlinear time optimal control problems. In this paper an approach to solve this kind of problems is considered. This method was first presented by Badakhshan et al . in [1] and in this paper, this method is expanded to solve time optimal control problems. In this approach the time optimal control problem is changed to a problem in calculus of variations and then is solved by a discretisation method. Key words: Nonlinear programming • optimal control • time optimal control • discretisation INTRODUCTION Optimal control problems are widely used in industry and the goal is to control a dynamical system from a given initial point, to a given target, which one may try to minimize energy, time or any indicated costs. In time optimal control problems, a dynamical system is going to be controlled in a minimum part of time. A neural network approach for controlling such systems is proposed in [2] and also time-optimal control of disturbance-rejection tracking systems is considered in [3]. The Bang-Bang principle of time optimal controls for the heat equation is presented in [4] and a discussion on time optimal control of integrator switched systems is considered in [5]. This paper deals with this class of problems and a discrete method is proposed to solve such problems. Definition 1: A classical control problem has the following general form, a b x(t) g(x(t),u(t),t) s.t. x(a) x, x(b) x = = = (1) where g is a nonlinear continuous functional such that [ ] n g:A U a,b R, × × → t (a,b) R , n x(t) A R , m u(t) U R. Sets A and U are given compact sets, 1 n x(t) (x(t), ,x (t)), = K is the continuous state function and 1 m u(t) (u(t), ,u (t)), = K is the control function, which is assumed to be a measurable function on [a,b]. Also x a (initial point) and x b (end point), are given [6, 7]. Definition 2: A classical optimal control problem has the following general form b 0 a n m a b Min J(x(t),u(t),t) f (x(t),u(t),t)dt s.t. x(t) g(x(t),u(t),t) x(t) A R, x(a) x, u(t) U R , x(b) x, t (a,b) R, = = = = ∫ (2) where [ ] 0 f :A U a,b R × × → is a continuous function and all the other functions and variables are as the same as defined previously in Definition 1. PRELIMINARIES Consider again the nonlinear system (1). One define an error function as follow p E(x(t),x(t),u(t),t) x (t) g(x(t),u(t),t) , = - where the norm function || || p is defined as: 1 n p p i p i1 f f , p 1 = = ≥ ∑ By the use of the above error function and considering the control problem (1), one may write the following optimization problem ( b a m n a n b Min J(x(t),x(t),u(t),t) E(x(t),x(t),u(t),t )dt s.t. x(a) x , u(t) U R , x(t) A R, x(b) x, x(t) B R, t a,b R. = = = ∫ (3) Lemma 1: If E(x(t),x(t),u(t),t) be a continuous function on B× A× U× [a,b], then