非线性与时滞随机系统的稳定性分析及镇定方法研究

There always exist random factors in the real systems and their external environments, which influence the dynamical behavior of the systems. In fact, sometimes, random models can accurately reflect the dynamical characteristics of natural, social and engineering systems. At the time being, the control theory for stochastic systems with various complicated factors such as nonlinearities, time-delays, varying coefficients, Markov jumps, impulses, distributed parameters and fuzziness are the current research hotspots. By this project, the stability and stabilization will be investigated for the nonlinear or delayed stochastic systems. Novel methods and approaches for the stability analysis, stabilization and stabilization by noise will be developed in order that less conservativeness and more features on randomness are implied by the stability criteria obtained. The methods for stability analysis by moment equations for nonlinear stochastic systems and those for stability analysis by Lyapunov functions combined with the system equations for the delayed stochastic systems will be mainly explored, and the Lyapunov stability theorem with more features on randomness, new type Razumikhin theorem and the operator approach for the stability theorem of delayed stochastic systems will be also established. We will bring some new visions and theoretical methods for the classic stability analysis and stabilization problems of nonlinear or delayed stochastic systems, which will further improve and develop the theory for stochastic systems and provide theoretical basis and consultation for the engineering and social practices.

在任何实际系统及其外部环境中都存在着随机因素，影响系统的动态。实际上，随机模型有时更能准确反映自然与社会工程系统动态特性。含有非线性、时滞、变系数、Markov跳变、脉冲、分布参数、奇异性、模糊性等复杂因素的随机系统的控制理论是当前的研究热点。本项目以非线性、时滞随机系统为研究对象，探讨系统的稳定性与镇定问题。以体现随机系统特色、减小稳定性判据的保守性为追求目标，在非线性与时滞随机系统稳定性分析方法、镇定、噪声镇定等方面探索新的方法与途径。主要探索非线性随机系统稳定性的矩方程法、时滞随机系统稳定性分析的Lyapunov函数法+系统方程法；建立具有随机系统特色的Lyapunov稳定性定理、新型Razumikhin定理、时滞随机系统的算子型稳定性定理等，为非线性与时滞随机系统稳定性分析、镇定控制这一经典问题带来一些新的视野和理论方法，进一步完善和发展随机系统理论，为工程和社会实践提供理论依据。

通过本项目，我们研究了非线性和滞后随机系统的稳定性、镇定与控制。主要进展包括：.1. 围绕“时滞”主题，我们获得了对滞后随机系统稳定性理论成立但对非滞后系统不成立的独特结果；提出了一类构造滞后随机系统Lyapunov函数方法的方法，建立了相应的稳定性判据，并进行了应用研究；通过比较原理，建立了滞后随机系统的新型Razumikhin定理；建立系统时变时滞微分不等式；.2. 围绕“非线性”主题，我们去掉线性增长，仅在局部Lipschitz条件下，建立了随机系统的稳定性定理，其结果几乎允许系统模型具有任意的时变性和任意的非线性动力学特征，因此，研究结果具有一般性；.3. 围绕“时变性”主题，我们取得了对时变系统成立但对非时变系统不成立的结果，取得了对具有时变时滞的（随机）系统成立但对相应的常时滞系统不成立的结果；.4. 围绕“随机性”主题，我们建立了随机系统矩稳定性与几乎必然稳定性之间的关系，分析了两类稳定性对应的系统机理；.5. 围绕“随机系统稳定性”主题，我们建立了算子型滞后随机系统的稳定性定理，并研究了滞后随机系统的控制问题；.6. 围绕“随机系统的镇定与控制”主题，我们在极弱假设下、对一般模型，建立了随机系统噪声镇定的结果，提出了设计方法；提出了随机系统的分时反馈镇定控制技术。.围绕上述内容，在IEEE Transactions on Automatic Control、Automatica、SIAM Journal on Control and Optimization、International Journal of Systems Science等刊物发表系列论文，组织了四届TCCT Workshop on Stochastic Systems and Control等系列学术会议，围绕随机系统控制主题，为学术刊物组织了两期特辑与专刊，培养了十名博士研究生。.结合该课题的研究，我们也研究了非线性和滞后随机系统的基本理论、数值计算。在上述研究的基础上，我们拓展了新的研究课题：随机系统的采样控制、复杂网络控制等。