Abstract
The present paper is considered a two-dimensional half-linear differential system: $x' = a_{11}(t)x+a_{12}(t)\phi_{p^{*}}(y)$ , $y' = a_{21}(t)\phi_{p}(x)+a_{22}(t)y$ , where all time-varying coefficients are continuous; p and $p^{*}$ are positive numbers satisfying $1/p + 1/p^{*} = 1$ ; and $\phi_{q}(z) = |z|^{q-2}z$ for $q = p$ or $q = p^{*}$ . In the special case, the half-linear system becomes the linear system $\mathbf{x}' = A(t)\mathbf {x}$ where $A(t)$ is a $2 \times2$ 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.
Highlights
1 Introduction We consider a system of differential equations of the form x = a (t)x + a (t)φp∗ (y), ( . )
The following result is a converse theorem on exponential stability, which guarantees the existence of a Lyapunov function estimated by quadratic form x
We present the converse theorems for half-linear system ( . ), for comparison with Theorems B, C, and E
Summary
The following result is a converse theorem on (global) exponential stability, which guarantees the existence of a Lyapunov function estimated by quadratic form x (see [ , , , ]). ) is globally exponentially stable, there exist three positive constants β (α), β (α), β and a Lyapunov function V (t, x) defined on I × Sα which satisfies the following conditions:. ) is globally exponentially stable, there exists a Lyapunov function V (t, x) defined on I × Sα which satisfies the following conditions:. ) is uniformly asymptotically stable, there exist a λ > and a β > such that t ∈ I and (ξ , η) ∈ R imply x(t; t , ξ , η), φp∗ y(t; t , ξ , η) p ≤ βe–λ(t–t ) ξ , φp∗ (η) p for all t ≥ t.
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