Abstract
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.
Highlights
Lyapunov functions are a very useful tool for investigating the behaviour of dynamical equations. They have been used 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]
An important theoretical issue with practical implications is whether or not a Lyapunov function characterizing a particular dynamical property exists—such results are known as necessary conditions
Background concepts and results on time scales are taken from Bohner and Peterson [4] and, for brevity, will not be stated explicitly here in general
Summary
Lyapunov functions are a very useful tool for investigating the behaviour of dynamical equations They have been used for over a century for differential equations of many types [15] as well as difference equations [1]. 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 ∈ Ꮿrd(T, Rn×n). Background concepts and results on time scales are taken from Bohner and Peterson [4] (see [2, 5, 8, 9]) and, for brevity, will not be stated explicitly here in general
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