A weaker criterion of asymptotic stability

作者：Gao Bo; Zhang Weinian^*

来源：Computers & Mathematics with Applications, 2008, 56(7): 1693-1698.

DOI：10.1016/j.camwa.2008.04.013

• asymptotic stability
• Liapunov's direct method
• uniform continuity
• negative semi-definite
• boundedness

The classical criterion of asymptotic stability for differential equations requires the existence of a Liapunov function V with negative definite dV/dt. Successive efforts have been made to weaken the negative definiteness of dV/dt to semi-negative definiteness. Recently, it was given an interesting result that, under the boundedness of d(m+1)/dt(m+1), the negative definiteness can be weakened to that dV/dt <= 0 together with that -(vertical bar dV/dt vertical bar + vertical bar d(2)V/dt(2)vertical bar + ...+ vertical bar d(m)V/dt(m)vertical bar + vertical bar d(m+P)V/dt (m+P)vertical bar) is negative definite. Unfortunately, its basic lemma is proved to be false by a counter example and cannot support this interesting result. In this paper we re-establish the weak criterion for asymptotic stability with less requirements.

• 出版日期2008-10
• 单位四川大学