Lévy过程驱动随机系统的稳定、控制及应用研究

Stochastic systems are general in the actual projects, and they have attracted many scholars' attention owing to their complicated evolution modes and rich dynamics behaviors as well as closely related to the practical problems of production and life. As a consequence, stochastic systems have been an important research topic in the world. Stability problem is the primary problem in the design of modern control systems, and all the control systems must be stable before they have normal operation. Different from the previous literature stability research in which the system has a simple and continuous sample path, this project will further study the numerical solution、stability and control problem of stochastic system driven by Lévy processes, which is a typo of discontinuous stochastic systems. We try to establish some new models of stochastic systems driven by Lévy processes according to actual problems and study the numerical solution of the system、stable principles、stable methods and stability control strategies. We aim to reveal that some factors such as Markov switching、 delays and Lévy noises have some effects on and how to affect the stability of the system, and obtain stability control methods、 sufficient conditions of the realization of the stable or stability control, and a number of verifiable stability criteria. Also, we study the convergence and stability of the system's numerical solution, and its specific application to the finance、 economics and complex network, which promotes the application and development of some theories such as the information science、 computer science、 finance、 complex network and other related subject areas.

随机系统普遍存在于实际工程,由于其复杂的演化方式,丰富的动力学内涵,以及与生产、生活中的实际问题紧密联系，吸引着众多学者的广泛关注，是国际上研究的重要课题。稳定性问题是现代控制系统设计中的首要问题，一切控制系统能够正常运行的必要前提是稳定。不同于以往文献研究简单连续随机系统的稳定性，本项目将深入研究Lévy过程驱动随机系统这类不连续系统的数值解、稳定与控制问题。拟结合实际问题建立Lévy过程驱动随机系统新模型，并研究这些系统的数值解、稳定原理、稳定方法及稳定控制策略，以揭示Markov切换、时滞和Lévy噪声等因素对系统稳定的具体影响及作用规律, 获得其稳定控制方法及实现稳定或稳定控制的充分性条件，建立若干可验证的稳定准则,并探讨系统数值解的收敛性和稳定性，同时探索其在金融、经济和复杂网络中的具体应用，为促进信息科学、计算机科学、金融学、复杂网络等相关学科领域理论的应用与发展。

四年来,我们对Lévy过程驱动随机系统的稳定与控制这个国际重要课题进行了系统、深入的研究,并取得了如下成果：（1）Lévy过程驱动复杂随机系统的稳定性、 Levy过程和马氏过程共同驱动随机泛函微分方程的著名拉斯米斯定理、具有泊松跳随机偏微分方程的p阶矩稳定性等重要问题，揭示了 Markov 切换、时滞和 Lévy 噪声等因素对系统稳定的具体影响及作用规律, 获得了其稳定控制方法及实现稳定或稳定控制的充分性条件，建立了若干可验证的稳定准则。（2）随机非线性系统解的存在唯一性、渐近稳定、均方指数稳定、有限时稳定、风险灵敏与连续可微控制器的设计。（3）随机时滞 Hopfield神经网络的数值解稳定性、中立型随机时滞微分系统中两类θ数值方法的均方指数稳定性。（4）随机复杂网络的稳定性，提出了比较原理、输入状态稳定、时滞分割、leakage 时滞、系数含有数学期望等新的理论、概念和方法，并考虑了金融学中的最优控制和期权定价问题。成果发展了许多新的理论和方法,并解决了相关开问题。在Automatica、IEEE TAC、IEEE TNNLS、Syst. Control Lett.等国际权威刊物发表论文 83篇，其中 SCI 占 74 篇，ESI高被引论文7篇，远远超出了原计划 15 篇 SCI 的预期目标。特别，论文得到大量广泛引用与好评,朱全新2014-2016连续三年获得爱思维尔中国高被引者、江苏省高等学校自然科学奖一等奖、德国洪堡基金高访学者等多项科研成果奖励。