Asymptotic stability and instability criteria for nonautonomous systems.

The classical criterion of asymptotic stability of the zero solution of equations x′ = f( t, x) is that there exists a function V ( t, x), a(∥ x∥) ≤ V ( t, x) ≤ b(∥ x∥) for some a, b ∈ K such that $$ \dot{V} $$( t, x) ≤ −c(∥ x∥) for some c ∈ K. In this paper, we prove that if $$ \mathop {V}\limits^{(m + {1})} $$( t, x) is bounded on some set [ tk − T, tk + T] × BH( tk → + ∞ as k → ∞), then the condition that $$ \dot{V} $$( t, x) ≤ −c(∥ x∥) can be weakened and replaced by that $$ \dot{V} $$( t, x) ≤ 0 and − (− $$ \dot{V} $$( tk, x)| + − $$ \ddot{V} $$( tk, x)| + ⋯ + − $$ \mathop {V}\limits^{(m)} $$( tk, x)|) ≤ −c′(∥ x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [-] [ABSTRACT FROM AUTHOR]