非线性时滞随机混杂系统的稳定性与镇定问题研究

This project is concerned with the stability and stabilization of nonlinear stochastic hybrid systems with time-delays, including the pantograph ones. Three system models are involved. The first one is the hybrid stochastic system with asynchronous Markov switch, the second one is the stochastic complex networks with time-varying delays and heterogeneous impulse effect, the third one is the neutral stochastic systems driven by Lévy noise with pantograph delay. The project is associated with a series of complex system models and complex factors, for example, the asynchronous switch, the heterogeneous impulses, the neutral operator and the unbounded delay, which lead to a series of difficulties for analysis. To this end, we firstly investigate the essential attribute of the Markov chain for the first objective to overcome the difficulty brought about by the asynchrony and to avoid the transition estimates among the parameters for stability criteria with less conservativeness. For the second objective, a new method, namely the auxiliary monotone function method, is proposed to deal with the time-delays to avoid the requirement for the differentiability of time-varying delays applied in the literature. For the third objective, an estimate method with maximum value is proposed to deal with the difficulty associated to the neutral operator and achieve the polynomial stability under the same criteria for the systems with bounded delay. The investigations are based on the insight into the objectives, the problems and related phenomena and benefit by the proposed methods. It is expected that the project provides some theoretical reference and basis for the analysis and design of control systems with randomness, hysteresis, nonlinearity and hybridity by the methods proposed and the advanced results established with the project.

本项申请研究具有时滞（包括比例滞后）的非线性随机混杂系统的稳定性与镇定问题，涉及三类模型，异步Markov切换的随机混杂系统、具有时变时滞和非均匀脉冲影响的随机复杂网络及由Lévy噪声驱动的中立型随机比例时滞系统，模型复杂，需要处理的因素多，包括异步切换、非均匀脉冲、中立算子、无界滞后等，难点较多，为此，对第一类问题先研究Markov切换信号的本质特性，以克服异步带来的困难，同时避免参数过度估计进而减少稳定性判据的保守性，对第二类问题提出处理时变滞后新方法：辅助单调函数法，以避免对时变滞后可微性要求，对第三类问题提出最大值估计法以解决中立型算子带来的困难，在与有界时滞微分系统相同稳定性判据下得到多项式稳定性。基于对研究对象、问题和现象的洞察，希望项目所提炼的方法具有参考价值，得到的结果具有先进性，为具有随机性、滞后性、非线性及混杂等复杂因素的控制系统的分析与设计提供理论参考和依据。