tailieunhanh - Báo cáo hóa học: "LYAPUNOV FUNCTIONS FOR LINEAR NONAUTONOMOUS DYNAMICAL EQUATIONS ON TIME SCALES"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: LYAPUNOV FUNCTIONS FOR LINEAR NONAUTONOMOUS DYNAMICAL EQUATIONS ON TIME SCALES | LYAPUNOV FUNCTIONS FOR LINEAR NONAUTONOMOUS DYNAMICAL EQUATIONS ON TIME SCALES PETER E. KLOEDEN AND ALEXANDRA ZMORZYNSKA Received 25 January 2006 Revised 23 March 2006 Accepted 13 April 2006 The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess. Copyright 2006 P. E. Kloeden and A. Zmorzynska. 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. 1. Introduction Lyapunov functions are a very useful tool for investigating the behaviour of dynamical equations. They have been used now for over a century for differential equations of many types 15 as well as difference equations 1 . They were first used in the context of time scales in 12 . See 16 for a more recent and an extensive investigation of Lyapunov functions on time scales. Much of the literature on Lyapunov functions especially applications oriented deals with sufficient conditions assuming that a Lyapunov function is known. An important theoretical issue with practical implications is whether or not a Lyapunov function characterizing a particular dynamical property actually exists such results are known as necessary conditions. In this paper we establish the existence of a Lyapunov function characterising the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamic equation x A t x on a time scale T with a bounded graininess where the matrix-valued mapping t A t is right dense continuous rd-continuous on T that is A e Tjrd T Rnxn . Such linear dynamical equations and their inhomogeneous variants play an important role in investigations of the dynamical behaviour both in themselves and also as Hindawi Publishing .