时滞系统中若干非Lyapunov方法和非传统Lyapunov方法研究

时滞存在于各类实际系统中。时滞系统具有非常复杂的动力学性质，是控制理论研究的重要对象之一，有大量问题亟待解决。本项目主要研究时滞系统的动力学性质，着眼于探索非Lyapunov方法和非传统Lyapunov方法，提出若干新思路和新方法，包括以下四个方面的内容。1 采用包络法分析受扰时滞非线性系统和时滞切换非线性系统的鲁棒稳定性，建立时滞非线性级联系统稳定的充要条件。2 揭示对偶时滞系统间稳定性的等价关系。3 揭示时滞一般齐次系统和时滞非线性系统渐近稳定和指数稳定间的等价关系和等价条件。4 采用高次齐次Lyapunov泛函方法建立时滞系统的稳定性判据。本项目力图从新的角度展开研究，为时滞系统的分析与综合提供新的工具和途径，为研究时滞系统提供思想和方法上的启迪。本项目联系该领域的研究前沿，从事具有挑战性的工作，具有重要的理论意义。由于时滞系统有着深刻的实际背景，本项目的结果具有广阔的应用前景。

Delays extensively exist in various practical systems. Delayed systems exhibit complicated dyanmics and are one of the most important fields in the control community. It is well-know that there remain many unsolved problems in this aera. This proposal mainly addresses dynamical properties of dynamic systems with delays, tries to propose some new ideas and methods, and focuses the following four aspects: 1. By means of covering method, we propose several robust stability conditions for delayed nonlinear systems with disturbances and delayed switched nonlinear systems with disturbances, and provide necessary and sufficient asymptotic and exponential stability criteria for cascade nonlinear systems with time-varying delays. 2. Stability equivalence between a delayed linear system and its dual system is revealed. 3. The equivalent relationships between two properties (asymptotic stability and exponential stability) of a generalized homogeneous system with delays or of nonlinear system with delays are established, and the condition guaranteeing such equivalences is also explored. 4. By using higher order homogeneous Lyapunov functions and functionals, some stability criteria are established for systems with uncertainties and systems with delays, focusing on the continuous-time systems. The project attempts to investigate the dynamic properties of delayed systems from some new points of view, exploring some novel ways or tools and helping improve the ideas for further investigation. According to the state of the art in the area of study, this project is challenging and of theoretically significant. Due to the profound practical background of systems with delays, the project will have broad applications in the future.