具有加性时变时滞和马尔科夫跳跃的中立型动力系统的稳定与控制

This project is based on the basic knowledge of the stability theory of functional differential equation and the nonlinear control theory, combining with practicalbackground of additive time-varying delays and Markovian jump, using optimization theory and methods as well as relevant analyzing skills, we can reach the goal of improving the conservatism of the existing results as well as complete the researching contents of neutral dynamical system by doing some study about the problem of stability analysis and control design for the neutral dynamical systems with additive time-varying delays and Markovian jump in a systematic way.. The major studying contents of the project including the followings. Getting the Jensens’ inequalities based on the theory of optimization, combining the property of the computation for definite integral and the inequality technique; Improving the delay-dependent stability conditions for retard systems with additive time-varying delays, neural networks with additive time-varying delays, neural networks with neutral type delay and additive time-varying delays; Discussing the delay-dependent stability conditions for neutral delay systems with additive time-varying delays, neutral delay systems with additive time-varying delays and Markovian jump, neural networks with additive time-varying delays and Markovian jump; Studying the problem of the stability and control design for neutral complex networks with both Markovian jump and additive time-varying delays.. After the study of this project, we are expecting to promote the theoretical system of neutral dynamical systems, as well as to enrich practical field of neutral dynamical systems and to enhance the analysis and design for hybrid dynamical systems.

本项目以泛函微分方程稳定性理论和非线性控制理论为基础，结合加性时变时滞和马尔科夫跳跃现象广泛存在的客观事实，拟用最优化理论方法及相关分析技巧，系统开展对具有加性时变时滞和马尔科夫跳跃的中立型动力系统稳定性与控制设计问题的研究，达到改进现有研究结果的保守性和完善中立型动力系统的研究内容的目的。.项目主要研究内容包括：利用最优化理论，结合定积分计算性质及不等式处理技巧，研究得到新詹森类不等式；改进具有加性时变时滞的滞后型时滞系统、神经网络及具有加性时变时滞的中立型神经网络的时滞依赖稳定性条件；研究具有加性时变时滞的中立型时滞系统、马尔科夫跳跃中立型时滞系统、马尔科夫跳跃中立型神经网络的稳定性问题；研究具有加性时变时滞和马尔科夫跳跃的中立型复杂网络的稳定性和控制器设计问题。.通过项目的研究，有望能够完善中立型动力系统的理论体系，拓展中立型动力系统的应用领域，推动混杂动力学系统的分析与设计研究。

本项目以泛函微分方程稳定性理论和非线性控制理论为基础，结合加性时变时滞和马尔科夫跳跃现象，运用最优化理论方法及相关分析技巧，系统地开展了对具有加性时变时滞和马尔科夫跳跃的中立型动力系统的稳定性与控制器设计问题的研究，得到了一系列新的研究成果，其中的许多结果降低了现有相关条件的保守性。. 项目主要研究内容包括：基于最优化理论，构造正交函数或非正交函数，适当引用自由矩阵，利用不等式处理技巧，根据研究系统的本质属性，研究得到了一类保守性较低的新积分不等式，降低了现有对应不等式的保守性；为了线性化处理非线性问题，基于凸优化思想，研究得到了凸组合优化不等式。改进了具有加性时变时滞的滞后型时滞系统、时滞神经网络、具有加性时变时滞的中立型神经网络等系统的时滞依赖稳定性条件；研究了具有加性时变时滞的中立型时滞系统、马尔科夫跳跃中立型时滞系统、马尔科夫跳跃中立型神经网络等系统的稳定性问题；研究了具有加性时变时滞和不确定半马尔科夫跳跃的中立型神经网络的随机稳定性问题；研究了具有加性时变时滞和半马尔科夫跳跃的不确定中立型神经网络的有限时间有界问题。研究了具有加性时变时滞和马尔科夫跳跃的中立型神经网络的稳定性和控制器设计问题。研究了具有半马尔科夫跳跃和加性时变时滞的中立型神经网络的事件驱动同步控制问题。研究了传输率不确定情形下的稳定性、控制器设计等相关准则。 . 本项目的研究，进一步完善了中立型动力系统的理论体系，拓展了中立型动力系统的研究领域，推动了随机动力系统的分析与设计研究。