This paper studies the stabilization and safety problems of nonlinear time-delay systems. Following both Razumikhin and Krasovskii approaches, we propose novel control Lyapunov functions/functionals for the stabilization problem and novel control barrier functions/functionals for the safety problem. The proposed control Lyapunov and barrier functions/functionals extend the existing ones from the delay-free case to the time-delay case, and allow for designing the stabilizing and safety controllers in closed-form. Since analytical solutions to time-delay optimal control problems are hard to be achieved, a sliding mode control based approach is developed to merge the proposed control Lyapunov and barrier functions/functionals. Based on the sliding surface functional, a feedback control law is established to investigate the stabilization and safety objectives simultaneously. In particular, the properties of the sliding surface functional are analyzed, and further how to construct the sliding surface functional is discussed. Finally, the proposed approaches are illustrated via two numerical examples from the connected cruise control problem of automotive systems and the synchronization problem of multi-agent systems.

本文研究非线性时滞系统的稳定性和安全性问题。遵循 Razumikhin 和 Krasovskii方法，我们提出了用于稳定问题的新型控制 Lyapunov 函数/泛函和用于安全问题的新型控制屏障函数/泛函。所提出的控制 Lyapunov 和障碍函数/功能将现有的从无延迟情况扩展到时间延迟情况，并允许以封闭形式设计稳定和安全控制器。由于时滞最优控制问题难以得到解析解，因此采用滑模控制开发了基于方法来合并提出的控制 Lyapunov 和障碍函数/泛函。基于滑动面泛函，建立反馈控制律，同时研究稳定和安全目标。特别是分析了滑动面泛函的性质，并进一步讨论了如何构造滑动面泛函。最后，通过汽车系统的连接巡航控制问题和多智能体系统的同步问题的两个数值示例来说明所提出的方法。