ACTA MATHEMATICA VIETNAMICA Volume 25, Number 2, 2000, pp. 231–249 231 LYAPUNOV STABILITY OF NONLINEAR TIME-VARYING DIFFERENTIAL EQUATIONS TRAN TIN KIET AND VU NGOC PHAT Abstract. The paper studies asymptotic stability of nolinear time- varying differential equations by Lyapunov’s direct method. Sufficient conditions for asymptotic stability are given in terms of nondifferentiable Lyapunov-like functions. An application to stabilizability of a class of nonlinear control systems with feedback controls is also given. 1. Introduction Consider a nonlinear time-varying differential equation of the form (1) ˙ x(t)= f (t, x(t)), t ≥ 0 It is well known that there are two major approaches to the Lyapunov stability analysis of system (1): the first linearization method and the second direct method. Stability of system (1) can be investigated via the first linearization method, but in general and the most powerful technique is the second direct method. For this method one usually assumes the existence of, so called Lyapunov function, a positive definite function with negative derivative along the trajectory of the system. In the last decade the Lyapunov direct second method has been a fruitful technique in sta- bility analysis of nonlinear differential equations and has gained increasing significance in the development of qualitative theory of dynamical systems [5, 6, 9, 18]. There are a number of books and papers available expouding the extensions and generalizations of Lyapunov-like functions, see, e.g., [2, 7, 8, 14, 16, 19] and references therein. It is recognized that the Lyapunov- like functions serve as a main tool to reduce a given complicated system into a relatively simpler system and provide useful applications to control systems [1, 3, 10, 11, 15, 17]. Received May 10, 1999; in revised form March 20, 2000. 1991 Mathematics Subject Classification. 34K20, 34D10; 49M7, 93C10 Key words and phrases. Nonlinear system, time-varying system, asymptotic stability, stabilizability, feedback control, nondifferentiable function, Lyapunov function.