不连续泛函微分方程的Lyapunov稳定性研究

Due to delays and discontinuous factors, discontinuous functional differential equations are numerously present in dynamic modeling in many fields such as control engineering, neural networks and economics. The research object of this project is Lyapunov stability of discontinuous functional differential equations, and the contents include two parts as follows: (1) Base on the theory of set-valued map and differential inclusions, establish properties such as exsistence for solutions of discontinuous functional differential equations and give a solution to the basic theory for the solutions; (2) By theory of non-smooth analysis and convex analysis, give a generalized chain rule for non-differentiable Lyapunov functions, solve the problem of derivative of non-differentiable Lyapunov funcitons along the solutions of discontinuous systems. Then by means of the generalized chain rule, give the results of stability、asymototical stability and convergence in finite time for isotated equilibrium. Through the studies in this project, basic theory for solutions and stability theory are gradually established and strong theory foundatiion is provided for application research for discontinuous functional differential equations.

由于时滞和不连续因素的影响，不连续泛函微分方程大量出现在控制工程、神经网络和经济学等领域的动力学建模中。本项目以不连续泛函微分方程的Lyapunov稳定性理论为研究对象，研究内容包括以下两个方面：（1）基于集值映射和微分包含理论，建立不连续泛函微分方程解的存在性等性质，解决解的基本理论问题；（2）利用非光滑分析和凸分析理论，建立非可微Lyapunov泛函的广义链式法则，解决非可微Lyapunov泛函沿不连续系统解的求导问题。在此基础上，建立不连续泛函微分方程孤立平衡点的稳定、渐近稳定以及有限时间收敛性等结果。通过本项目的研究，逐步建立不连续泛函微分方程解的基本理论和稳定性理论，为不连续泛函微分方程的应用研究提供坚实的理论基础。

本项目以不连续泛函微分方程为研究对象，在不连续泛函微分方程解的基本理论方面，基于微分包含理论给出了不连续泛函微分方程解的局部存在性结果，并结合生物学背景给出了一类时滞泛函微分方程多个周期解的存在性结果。在不连续泛函微分方程稳定性方面，利用非光滑分析和凸分析理论，本项目在去掉Lyapunov函数的光滑性条件下，得到了非时滞不连续微分方程奇异平衡点集的稳定和渐近稳定性结果，并解决了Gouzé等在2006年提出的一个重要猜想。而且，在去掉Lyapunov泛函的可微性和无穷小上界的基础上，建立了时滞不连续泛函微分方程平衡解的稳定性和渐近稳定性的一个重要结果，该结果是连续泛函微分方程相应结果的推广，具有重要的应用价值。