一类非线性微分方程的指数稳定性

A proof of the exponential stability in the nonlinear differential equation′s null solution

在非线性微分方程x′=A(t)x+f(t,x)中,假定对所有的t∈R+,A(t)的特征值的实部都不大于某个负常数α,那么在某些给定条件下,利用指数型二分法等相关理论,可以证明这样的微分方程的零解是指数型渐近稳定的,且推广了Coppel的相关结论。

Suppose that every eigenvalue of A(t) has real part≤α<0 for all t∈R+ in the nonlinear differential equation x′=A(t)x+f(t,x),we can prove that the null solution of this differential equation is exponential stable under given conditions using the exponential dichotomy theory.The result extends relevant conclusions of the Coppel.