SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 223-241, January 2021.
This paper presents criteria for the convergence of trajectories of time-varying nonlinear systems in terms of Lyapunov densities. The results are provided without assuming local stability and forward completeness of trajectories. As well as general time-varying nonlinear systems, periodically time-varying systems are also considered in this paper, where a weaker criterion is proposed for periodically time-varying systems. Also the existence of Lyapunov densities is proved for general and periodic time-varying nonlinear systems under the asymptotic stability of the equilibrium.

中文翻译：

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具有相反结果的时变和周期时变非线性系统的Lyapunov密度准则

SIAM控制与优化杂志，第59卷，第1期，第223-241页，2021年1月。
本文根据Lyapunov密度提出了时变非线性系统轨迹收敛的标准。在不假设局部稳定性和轨迹正向完整性的情况下提供了结果。除了一般的时变非线性系统外，本文还考虑了周期性时变系统，其中针对周期性时变系统提出了一个较弱的判据。同时证明了一般和周期时变非线性系统在平衡点的渐近稳定性下的Lyapunov密度的存在性。