大系统指数稳定性的某些结果

SOME RESULTS ABOUT EXPONENTIAL STABILITY OF LARGE SCALE DYNAMIC SYSTEMS

中文摘要

      Lyapunov直接方法的关键在于寻求一个满足一定条件的Lyapunov函数（简称为V函数）。而大系统指数稳定性对V函数的要求往往比较苛刻。专著[1]中要求V函数‖x‖2同量级（即存在K1，K2>0，使得满足K1‖x‖2《V（t，x）《K2‖x‖2）；专著[2]中要求V函数与某个K类函数同量级，后者的条件虽然比前者的弱一些，但并未给出论证。本文通过放宽对V函数的要求而给出了几个较一般的结果，使得[1]中的一些主要定理相应地成为它们的简单推论。

英文摘要

      The key to the Lyapunov`s direct method is how to find a Lyapunov function or V function, which must satisfy certain conditions. It is difficult to find such a V function for determining the exponential stability of large scale systems, for the strict condition must be satisfied by V function. It is required that V function should be the same order of magitude as ‖x‖2 in [1] (i.e , there exist two positive numbers K1 and K2 so that K1‖x‖2《V（t，x）《K2‖x‖2）. While in [2] it is required that V function should be the same order of magnitude as the function belonged to the Class K. Although the condition of the latter is not so strict as that of the former, its proofs have not been given. This paper gives some general results by deducing the condition for V function. As a result, some main theorems in [1] have become their corollaries.