In this paper, we establish a new evolution property of Lyapunov functions that satisfy the Razumikhin Stability Theorem conditions. The derived property is shown to be a valid alternative criterion to prove the asymptotic stability of a delay system without calculating the derivative of the Lyapunov function, and it is especially useful in solving stabilization problems for time delay systems when the delay is unknown. As an example application of this derived property, we investigate the stabilization of continuous-time linear systems with an unknown time-varying delay, by proposing a switched low gain feedback control technique that utilizes the derived property. Existing literature on stabilization of continuous time delay systems all require some knowledge, such as an upper bound, of the delay. In this work, by proposing a switched low gain feedback control law utilizing the derived property, we are able to achieve asymptotic stabilization of the system without any knowledge of the delay. Furthermore, the proposed switched low gain feedback law also significantly improves the closed-loop system performance in terms of overshoot and convergence speed, in comparison with the traditional fixed gain low gain feedback design. Simulation study verifies the theoretical results.

在本文中，我们建立了满足Razumikhin稳定性定理条件的Lyapunov函数的新演化性质。证明了导出的性质是证明延迟系统的渐近稳定性而无需计算Lyapunov函数导数的有效替代准则，当延迟未知时，它对于解决时滞系统的稳定问题特别有用。作为此派生特性的示例应用，我们通过提出一种利用派生特性的开关式低增益反馈控制技术，研究具有未知时变延迟的连续时间线性系统的稳定性。现有的有关连续时间延迟系统稳定的文献都需要一些知识，例如延迟的上限。在这项工作中 通过提出利用导出属性的切换低增益反馈控制律，我们能够在不了解任何延迟的情况下实现系统的渐近稳定。此外，与传统的固定增益低增益反馈设计相比，拟议的开关低增益反馈定律在过冲和收敛速度方面也显着提高了闭环系统性能。仿真研究验证了理论结果。