稳定是系统能正常运转的前提。许多实际问题难以采用连续或离散系统进行数学建模，亟需更广泛的理论分析框架，时标系统的引入为此类问题的解决提供了可能。时标系统不仅包含连续、离散系统，也涵盖混杂系统、Delta算子系统等其他复杂情形。分析时标时滞系统稳定性常用技术均依赖严格Lyapunov函数或泛函，而在某些情形下，此方法并不有效，因而较为保守，有待提出新型稳定性判据。本项目拟引入非严格Lyapunov泛函方法，提出低保守性稳定性定理。首先，针对时标非线性时滞系统，分别考虑时滞有界与无界情形，构建依赖于非严格Lyapunov泛函的稳定性定理。其次，综合考虑子系统稳定与不稳定情形，运用非严格Lyapunov泛函方法和驻留时间技术，研究时标切换时滞系统的稳定性。最后，针对时标脉冲时滞系统，根据脉冲为固定时刻和随机两种情形，分析其对系统稳定性的影响。本项目将极大程度上推动时标时滞系统的稳定性研究进展。

Stability is the basic premise for the good-operating control systems. When handling some practical engineering questions, the mathematical models based on continuous or discrete systems are no longer effective. Thus, a more general theoritical framework is needed to address such questions. Time-scale systems or systems on time scales make it possible. Time-scale systems include not only continuous and discrete systems, but also hybrid systems and Delta-operator systems, etc. The usual approaches to stability analysis of time-delay systems on time scales always depend on the strict Lyapunov function or functional. However, in some cases, these methods are not so effective due to their conservatism. Therefore, some new stability criteria should be proposed. This project plans to introduce the non-strict Lyapunov function or functional technique, and propose some less conservative stability theorems. Firstly, when investigating nonlinear time-delay systems on time scales, bounded delays and unbounded delays are considered, respectively. Then, some stability theorems, which are based on the non-strict Lyapunov function or functional, are constructed. Secondly, the non-strict Lyapunov functional approach and the dwell-time technique are applied to stability analysis of switched time-delay systems on time scales with modes stable or unstable. Finally, stability problems of time-delay systems with fixed-time and stochastic impulses on time scales are considered, respectively. For the most part, this project will promote the research advance of stability of time-delay systems on time scales.