Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function Vu N. Phat, + Phan T. Nam. * + Institute of Mathematics 18 Hoang Quoc Viet Road, Hanoi, Vietnam * Department of Mathematics Quinhon University, Quinhon City, Vietnam In this paper, the problem of stability and stabilization for a class of uncertain linear time-varying systems is considered. The system matrix belongs to a polytope and the time-varying parameter as well as its time derivative are bounded. Based on a time-varying version of Lyapunov stability theorem, new sufficient conditions for the exponential stability and stabilization are given. Using parameter dependent Lyapunov function, the conditions are formulated in terms of two linear matrix inequalities without introducing extra useless decision variables and hence are simply verified. The results are illustrated by numerical examples. Key Words: Exponential stabilization, polytopic systems, parameter-dependent Lyapunov functional, time-varying Lyapunov equations. 1 Introduction Lyapunov stability of linear time-varying (LTV) systems and applications to control theory have re- ceived considerable attention for the past decades, see; e.g. de La Sen (2002), Haurani et al. (2004), Kharitonov(2004, 2005), Kolmanovskii et al. (2003), Phat et al. (2003, 2006), Piri et al. (2006), Xie et al. (1995). As a special case of LTV systems, linear parameter dependent (LPD) systems are now established as one of important representations of uncertain LTV systems. When the system matrices of uncertain systems are formulated by a polytope of matrices, a stability problem of the LPD systems naturally arises and the use of Lyapunov functions is certainly the main tool for solving this problem. The simplest approach consists in looking for a common fixed Lyapunov function that proves robust stability of linear time-invariant (LTI) systems. In order to provide less conservative results, parameter-dependent Lyapunov functions have been recently employed and several techniques involving parameter-dependent Lyapunov functions have been proposed for the robust stability and stabilizability, see Montagner et al. (2004), Spark (1997). Recently, the stability problem has been developed to time-varying polytopic sys- tems by Colaneri et al. (2005), Karimi et al. (2005), where the conditions were derived in terms of a set of linear matrix inequalities, being thus solvable by numerical methods available in Boyd et al. (1994). In this paper, starting from the work done in Colaneri et al. (2005) and Ramos et al. (2002), we consider exponential stability and stabilization problem for uncertain linear time-varying polytopic systems. The system matrix belongs to a polytope and the time-varying parameter as well as its time derivative are bounded. In contrast to the approach of Colaneri et al. (2005) and Tanaka et al. (2003), by proving a time-varying version of the Lyapunov stability theorem, we establish new conditions for the exponential stability. Our main contribution consists of sufficient conditions for both the exponential stability and stabilizations. The conditions are formulated in terms of two linear matrix inequalities 1