DI-型模糊微分方程若干定解问题的研究

Integrated with the functional space theory, the differential inclusion theory, set-valued analysis, Sobolev space theory and the ordinary differential equation qualitative theory, this project utilizes these methods to choose an appropriate functional space and its weak topology, then the fuzzy differential equations based on differential inclusions are developed. That's the related theory and method of DI-type fuzzy differential equation. The main research contents are as follows. (1) Establish the criterion for the global solution's stability, uniform stability and asymptotic stability to the initial value problem of DI-type fuzzy differential equations; (2) Establish the criterion for the existence of the solution and the big solution to the two-point boundary value problem of DI-type fuzzy differential equations, and to obtain the inclusion relation between the solution and the big solution; (3) Establish the criterion for the existence and the structure stability of the solution to the periodic problem, the times-periodic problem and the anti-periodic problem of DI-type first order nonlinear and semi-linear fuzzy differential equations. The research of this project not only establishes the fairly complete qualitative theory for DI-type fuzzy differential equations, but also explores some new attempts in the fields of water quality model of water supply network, fuzzy local control and fuzzy optimization.

本项目综合应用泛函空间理论、微分包含理论、集值分析、Sobolev空间理论、常微分方程定性理论等手段，通过选择适当的泛函空间及其弱拓扑，发展基于微分包含意义的模糊微分方程，即（DI）型（也称DI-型）模糊微分方程的相关理论和方法。主要研究内容包括：(1)建立（DI）型模糊微分方程初值问题全局解的稳定性、一致稳定性及渐近稳定性判据；(2) 建立（DI）型模糊微分方程两点边值问题解、大解的存在性判据及解与大解的包含关系；(3) 建立（DI）型一阶非线性和半线性模糊微分方程周期问题、倍周期问题和反周期问题解的存在性和结构稳定性判据。本项目的研究不仅建立（DI）型模糊微分方程较完整的定性理论，而且为（DI）型模糊微分方程在给水管网水质模型、模糊局部控制、模糊优化等方面的应用提供新的尝试。

众所周知，在汽车发动机变速箱等切片系统、模糊局部控制系统等研究中需要处理基于微分包含意义的模糊微分方程即（DI）型模糊微分方程问题。本项目综合应用泛函空间理论、微分包含理论、集值分析、Sobolev空间理论、常微分方程定性理论等手段，通过选择适当的泛函空间及其弱拓扑，发展（DI）型模糊微分方程的相关理论和方法。主要研究内容包括：(1)建立（DI）型模糊微分方程初值问题全局解的稳定性、一致稳定性及渐近稳定性判据；(2)建立（DI）型模糊微分方程两点边值问题解、大解的存在性判据及解与大解的包含关系；(3)建立（DI）型一阶非线性和半线性模糊微分方程周期问题、倍周期问题和反周期问题解的存在性和结构稳定性判据。本项目组获得了如下重要结果：(A)对（DI）型通用振荡模糊微分方程两点边值问题的三种类型的阻尼振荡，得到了解、大解的存在性判据及解与大解的包含关系；(B)对（DI）型一阶非线性和半线性模糊微分方程周期问题，证明了周期解的存在唯一性，并建立了周期解的结构稳定性判据；(C)对（DI）型无阻尼模糊微分方程两点边值问题，证明了解和大解的存在唯一性，并建立了解和大解的结构稳定性判据。