B空间上中立型随机泛函微分方程的动力学研究

Neutral stochastic functional differential equations have been applied widely in many fields such as chemistry, ecology and control. It is a prerequisite to study dynamic behaviors for applications. It is crucial to choose an appropriate phase space for dynamic analysis of neutral stochastic functional differential equations with infinte delays. In most existing literature, the BC space has been used extensively. In this project, the more general phase space B will be introduced to investigate dynamic behaviors of such equations. Based on the properties of the B space and the theory of semigroups of operators, we will investigate the existence, uniqueness and some relevant problems for such equations by constructiing a new successive approximation of stochastic process squence and a new contracting mapping. By using stochastic version of Razumikhin technique with neutral term in phase space B, several criteria for stability will be established for neutral stochastic functional differential equations with infinite delays by combining the upper bound of diffusion operator of energy function which is not only in a more general nonlinear form but also sign-indefinite. The properties of limit set and existence of attractor of such equations will be studied by establishing a generalized stochastic version of the LaSalle theorem in B space with multiple energy functions and decay functions. In addition, several sufficient conditions about boundedness and attracting for solutions will be established. The obtained achievements in this project can enrich and develop the dynamic theory of stochastic functional differential equation, and provide the theoretical basis for practical applications of neutral stochastic functional differential equations.

中立型随机泛函微分方程已广泛应用于化学、生态学和控制科学等诸多领域，其动力学分析是应用的前提条件。对于无限时滞中立型随机泛函微分方程，选择合适的相空间对其动力学研究起着至关重要的作用。已有的工作大多数都选择BC空间。本项目将在更一般的抽象B空间上，研究其动力学行为。我们将基于B空间的性质和算子半群理论，构造新的连续近似随机过程序列和压缩映射，来研究方程解的存在唯一性及相关问题；利用B空间上的中立型随机Razumikhin技术，结合能量函数扩散算子的非线性不定号上界估计，建立解过程的一系列稳定性判据；通过引入衰减函数和多个能量函数，建立B空间上的广义随机LaSalle型定理，结合轨道预紧性和奇异系统比较原理，研究随机极限集的性质和随机吸引子的存在性，建立解的有界性和吸引性充分条件。本项目的研究成果将丰富和完善随机泛函微分方程的动力学研究内容，为中立型随机泛函微分方程的广泛应用提供理论基础。

中立型随机泛函微分方程已广泛应用于化学、生态学和控制科学等诸多领域，其动力学分析是应用的前提条件。本项目针对无限时滞中立型随机泛函微分方程，选择B空间作为初始函数的相空间。在较弱的系数条件下，利用B空间上的压缩映射原理、截断技巧以及能量函数方法，建立了B空间上中立型随机泛函微分方程局部解和全局解的存在性、唯一性条件。利用李雅普诺夫函数方法和L算子的非线性不定号上界估计技巧，建立了一系列解的稳定性和有界性判据。通过引入增广系统和有界扰动估计，结合随机Lasalle定理和随机小增益方法，获得了随机系统不变集和吸引集的存在性判据，并估计了解在全局吸引集中的最终边界。利用所得到的随机系统动力学分析理论成果，结合现代控制方法和技巧，我们进一步研究了一系列人工神经网络模型、复杂网络化系统以及生物动力系统的动力学行为和相关控制问题，得到了若干有意义的新结果。本项目所取得的研究成果，进一步丰富和发展了中立型随机泛函微分方程的理论研究，有利于形成统一的随机泛函微分方程解的定性理论研究框架，为人工神经网络、复杂网络以及生物动力系统的广泛应用提供了重要的理论支撑。本项目执行期间发表SCI论文16篇，EI会议论文2篇，出版学术专著1部。

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