Abstract

We present methods for proving new sufficient stability and asymptotic stability tests for the zero solution of a time-varying differential equation using nonmonotone sign-indefinite Lyapunov functions. As an example of application of our statements, we establish new stability tests for a gradient system in which the equilibrium is not isolated and the function defining the right-hand side does not have a minimum at the equilibrium point. A stability test for a Hamiltonian system is indicated for the case of a nonstrict minimum of potential energy at the equilibrium point.

中文翻译：

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具有非单调Lyapunov函数的微分方程的解稳定性条件

摘要

我们提出了使用非单调符号不定Lyapunov函数证明时变微分方程零解的新的足够的稳定性和渐近稳定性测试的方法。作为应用我们的陈述的一个例子，我们为梯度系统建立了新的稳定性测试，在该系统中，平衡不是孤立的，并且定义右侧的函数在平衡点处没有最小值。针对平衡点处势能的非严格最小值的情况，指示了哈密顿系统的稳定性测试。