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({double pipe}x{double pipe}) ≤ V (t, x) ≤ b({double pipe}x{double pipe}) for some a, b ∈ K such that V(t, x) ≤ -c({double pipe}x{double pipe}) for some c ∈ K. In this paper, we prove that if V (t, x) is bounded on some set [tk - T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that V(t, x) ≤ -c({double pipe}x{double pipe}) can be weakened and replaced by that, for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1-10] © 2011 Springer Science+Business Media, Inc.