混杂动力系统的回复性及相关问题研究

项目名称： 混杂动力系统的回复性及相关问题研究

项目编号： No.11471227

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 李洪旭

作者单位： 四川大学

项目金额： 70万元

中文摘要： 混杂动力系统的回复性及其相关问题是现代数学中出现的一个崭新的研究课题，其结果可以广泛应用于自动控制论、电子学、计算机科学、生物医学和种群生态学等众多应用科学。本项目将结合拓扑动力系统理论、微分方程的技巧和泛函分析的方法，对此问题进行系统深入的研究，主要涵盖以下内容：以具逐段常量微分方程为基本模型的混杂动力系统的一般回复性、概周期性和概自守性，特别是高阶混杂动力系统的概周期性和概自守性，包括连续和脉冲混杂动力系统的连续和逐段连续的概周期性及其推广回复性态，涉及相关解的存在性、唯一性、解集的结构、稳定性和渐近性等问题，并将上述研究结果应用于种群生态学和生物医学等相关学科领域。其中许多问题的研究目前还处于初始阶段，有的甚至处于空白状态，亟待全面深入展开。本项目力争获得一系列有重要意义的研究结果，建立系统深入的理论体系，使混杂动力系统的回复性及相关问题在理论和应用研究中获得实质性的进展。

中文关键词： 混杂动力系统；回复性；概周期；脉冲；稳定性

英文摘要： The topic of recurrence and relative problems for the hybrid dynamical systems is a new area in mathematics, and can be extensively applied in lots of other applied science such as control theory, electronics, computer science, biomedicine, population ecology, etc. The main purpose of this project is to study intensively in this area by combining the theory of toplogical dynamics, the methods of differential equations and functional analysis. The contents include the general recurrence, almost periodicity, almost automorphy and other recurrent properties of hybrid dynamical systems, especially higher order hybrid dynamical systems, whose mathematical models are differential equations with picewise constant arguments. We will consider the continuous and piecewise almost periodicity as well as the generalized properties for the continuous and impulsive hybrid dynamical systems. Our research involves the existence, uniqueness, the structure of the set of the solutions, asymptotic behaviors and stability of the solutions for the hybrid dynamical systems. Moreover, we will also apply the theoretical results of our studies in the applied science such as population ecology and biomedicine. Some of the problems considered in this project are quite new and seldom considered in the literature. We will try our best to made significant progress in the studies of both theory and applications for the recurrence and relative problems of hybrid dynamical systems.

英文关键词： hybrid dynamical systems;recurrence;almost periodicity;impulsive;stablity