Báo Cáo Báo cáo hóa học EXPONENTIAL STABILITY OF DYNAMIC EQUATIONS ON TIME SCALES ALLAN C PETERSON AND YOUSS

EXPONENTIAL STABILITY OF DYNAMIC EQUATIONS ON TIME SCALES ALLAN C. PETERSON AND YOUSSEF N. RAFFOUL Received 6 July 2004 and in revised form 16 December 2004

We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given.

1. Introduction

This paper considers the exponential stability of the zero solution of the first-order vector dynamic equation x∆ = f (t,x), t ≥ 0. (1.1) Throughout the paper, we let x(t,t0 ,x0 ) denote a solution of the initial value problem (IVP) (1.1), x t0 = x0 , t0 ≥ 0, x0 ∈ R. (1.2) (For the existence, uniqueness, and extendability of solutions of IVPs for (1.1)-(1.2), see [2, Chapter 8].) Also we assume that f : [0, ∞) × Rn → Rn is a continuous function and t is from a so-called “time scale” T (which is a nonempty closed subset of R).