Uniform asymptotic stability implies exponential stability for nonautonomous half-linear differential systems

摘要

The present paper is considered a two-dimensional half-linear differential system:x′=a11(t)x+a12(t)ϕp∗(y)$x' = a_{11}(t)x+a_{12}(t)phi_{p^{*}}(y)$,y′=a21(t)ϕp(x)+a22(t)y$y' = a_{21}(t)phi_{p}(x)+a_{22}(t)y$, where all time-varying coefficients are continuous; p andp∗$p^{*}$are positive numbers satisfying1/p+1/p∗=1$1/p + 1/p^{*} = 1$; andϕq(z)=|z|q−2z$phi_{q}(z) = |z|^{q-2}z$forq=p$q = p$orq=p∗$q = p^{*}$. In the special case, the half-linear system becomes the linear systemx′=A(t)x$mathbf{x}' = A(t)mathbf {x}$whereA(t)$A(t)$is a2×2$2 imes2$continuous matrix and x is a two-dimensional vector. It is well known that the zero solution of the linear system is uniformly asymptotically stable if and only if it is exponentially stable. However, in general, uniform asymptotic stability is not equivalent to exponential stability in the case of nonlinear systems. The aim of this paper is to clarify that uniform asymptotic stability is equivalent to exponential stability for the half-linear differential system. Moreover, it is also clarified that exponential stability, global uniform asymptotic stability, and global exponential stability are equivalent for the half-linear differential system. Finally, the converse theorems on exponential stability which guarantee the existence of a strict Lyapunov function are presented.