中立型随机微分方程中若干问题的研究
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摘要
本篇博士论文由四部分组成，主要讨论了具无限时滞中立型随机泛函微分方程解的存在惟一性，L~p(p≥2)指数估计和局部解的存在惟一性；考虑了具变时滞中立型随机神经网络的二阶矩指数稳定性和渐近稳定性和研究了具变时滞中立型随机偏微分方程Mild解的存在惟一陛，p(p≥2)阶矩指数稳定性和p阶矩渐近稳定性．
第一章简单地介绍问题产生的背景知识，发展概况，本文的主要工作以及预备知识．
第二章在有界连续函数空间中，研究了具无限时滞中立型随机泛函微分方程解的存在惟一性，L~p指数估计和局部解的存在惟一性．首先，在一致Lipschitz条件，线性增长条件和压缩性条件下，直接得到了具无限时滞中立型随机泛函微分方程解的存在惟一性，并给出了解的矩估计，近似解与精确解之间的误差估计；将一致Lipschitz条件替换为局部Lipschitz条件，也得到了具无限时滞中立型随机泛函微分方程解的存在惟—性，同时，也给出了在整个区间[0，+∞)上具无限时滞中立型随机泛函微分方程解的存在惟一性定理；其次，也讨论了具无限时滞中立型随机泛函微分方程解的L~p指数估计；最后，在局部Lipschitz条件和压缩性条件下，建立了具无限时滞中立型随机泛函微分方程局部解的存在惟一性定理。
第三章首先，利用线性矩阵不等式(LMI)方法和半鞅收敛定理，考虑了一类具变时滞中立型随机神经网络的二阶矩指数稳定性和几乎必然指数稳定性，所得到的条件保守性要好；其次，采用不动点方法，给出了具变时滞中立型随机神经网络的二阶矩渐近稳定性的充要条件．
第四章首先，在非Lipschitz条件下，研究了具变时滞中立型随机偏微分方程Mild解的存在惟一性；其次，通过建立一引理，给出了变时滞中立型随机偏微分方程Mild解的p(p≥2)阶矩指数稳定性和几乎必然指数稳定性的充分条件；特别地，当中立项消失时，也建立一引理，所得到的Mild解的p阶矩指数稳定性和几乎必然指数稳定性的充分条件比已有文献的条件要弱，因而，我们改进了原有的结果；最后，也通过建立一引理，给出了具变时滞中立型随机偏微分方程Mild解的p阶矩渐近稳定性的充分条件；当方程为与其对应的具变时滞随机偏微分方程时，也建立一引理，也能得到的Mild解的p阶矩渐近稳定性的充分条件，这二者结果都是新的。
This Ph. D. thesis is composed of four Chapters. The existence and uniqueness, L~p (p≥2)-exponential estimate and the local existence and uniqueness theorem for the solution of neutral stochastic functional differential equations with infinite delay are mainly discussed; The exponential stability and asymptotic stability in mean square for neutral stochastic neural networks with time-varying delays are investigated, respectively; And the existence and uniqueness, exponential stability and asymptotic stability in p (p≥2)-moment for mild solution of neutral stochastic partial differential equations with delays are also studied.
In Chapter 1 The history background and developments about problems above, some main works and some preliminary knowledges in this context are briefly introduced.
In Chapter 2 The existence and uniqueness, L~p (p≥2)-exponential estimate and the local existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay are mainly studied under the space of bounded and continuous functions. Firstly, the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay under the uniformly Lipschitz condition, linear grown condition and contractive condition can be directly derived; And the moment estimate of the solution and the estimate for error between the approximate solution and the accurate solution can be both given; If the uniformly Lipschitz condition is replaced by the local Lipschitz condition, the existence and uniqueness theorem can be gained; Meanwhile, the existence and uniqueness of the global solution in the interval [0, +∞) can also be obtained; Secondly, L~p-exponential estimate of the solution for neutral stochastic functional differential equations with infinite delay can be studied; At length, the theorem of the local solution about neutral stochastic functional differential equations with infinite delay only under the local Lipschitz condition and the contractive condition can be established.
In Chapter 3 Firstly, the exponential stability in mean square and almost sure exponential stability for neutral stochstic neural networks with time-varying delay can be discussed by using the linear matrix inequality (LMI) and the semimartingale convergence theorem; And our sufficient conditions are less conserative; Secondly, by utilizing the fixed point theorem, some sufficient and necessary conditions ensuring the asymp- totic stability in mean square for neutral stochastic neural networks with time-varying delays can be given.
In Chapter 4 In first quarter, the existence and uniqueness of mild solution for stochastic partial differential equations with delays under the non-Lipschitz condition are mainly discussed; In second quarter, by establishing a Lemma, some sufficient conditions ensuring p (p≥2)-moment exponential stability and almost surely exponential stability for mild solution to neutral stochastic partial differential equations with delays can be given; Specially, when the neutral item is removed, some sufficient conditions about p-moment exponential stability and almost sure exponential stability for mild solution of stochastic partial differential equations with delays are weaker than some existing literatures by establishing a Lemma, too; Thus, we improve some results; In third quarter, similarly, by constructing a Lemma, some sufficient conditions about asymptotic stability in p-moment of mild solution for neutral stochastic partial differential equations with delays can be given; If the neutral item is dropped out, some sufficient conditions about asymptotic stability in p-moment of mild solution for stochastic partial differential equations with delays can be derived; These two results are new.
第一章简单地介绍问题产生的背景知识，发展概况，本文的主要工作以及预备知识．
第二章在有界连续函数空间中，研究了具无限时滞中立型随机泛函微分方程解的存在惟一性，L~p指数估计和局部解的存在惟一性．首先，在一致Lipschitz条件，线性增长条件和压缩性条件下，直接得到了具无限时滞中立型随机泛函微分方程解的存在惟一性，并给出了解的矩估计，近似解与精确解之间的误差估计；将一致Lipschitz条件替换为局部Lipschitz条件，也得到了具无限时滞中立型随机泛函微分方程解的存在惟—性，同时，也给出了在整个区间[0，+∞)上具无限时滞中立型随机泛函微分方程解的存在惟一性定理；其次，也讨论了具无限时滞中立型随机泛函微分方程解的L~p指数估计；最后，在局部Lipschitz条件和压缩性条件下，建立了具无限时滞中立型随机泛函微分方程局部解的存在惟一性定理。
第三章首先，利用线性矩阵不等式(LMI)方法和半鞅收敛定理，考虑了一类具变时滞中立型随机神经网络的二阶矩指数稳定性和几乎必然指数稳定性，所得到的条件保守性要好；其次，采用不动点方法，给出了具变时滞中立型随机神经网络的二阶矩渐近稳定性的充要条件．
第四章首先，在非Lipschitz条件下，研究了具变时滞中立型随机偏微分方程Mild解的存在惟一性；其次，通过建立一引理，给出了变时滞中立型随机偏微分方程Mild解的p(p≥2)阶矩指数稳定性和几乎必然指数稳定性的充分条件；特别地，当中立项消失时，也建立一引理，所得到的Mild解的p阶矩指数稳定性和几乎必然指数稳定性的充分条件比已有文献的条件要弱，因而，我们改进了原有的结果；最后，也通过建立一引理，给出了具变时滞中立型随机偏微分方程Mild解的p阶矩渐近稳定性的充分条件；当方程为与其对应的具变时滞随机偏微分方程时，也建立一引理，也能得到的Mild解的p阶矩渐近稳定性的充分条件，这二者结果都是新的。
This Ph. D. thesis is composed of four Chapters. The existence and uniqueness, L~p (p≥2)-exponential estimate and the local existence and uniqueness theorem for the solution of neutral stochastic functional differential equations with infinite delay are mainly discussed; The exponential stability and asymptotic stability in mean square for neutral stochastic neural networks with time-varying delays are investigated, respectively; And the existence and uniqueness, exponential stability and asymptotic stability in p (p≥2)-moment for mild solution of neutral stochastic partial differential equations with delays are also studied.
In Chapter 1 The history background and developments about problems above, some main works and some preliminary knowledges in this context are briefly introduced.
In Chapter 2 The existence and uniqueness, L~p (p≥2)-exponential estimate and the local existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay are mainly studied under the space of bounded and continuous functions. Firstly, the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay under the uniformly Lipschitz condition, linear grown condition and contractive condition can be directly derived; And the moment estimate of the solution and the estimate for error between the approximate solution and the accurate solution can be both given; If the uniformly Lipschitz condition is replaced by the local Lipschitz condition, the existence and uniqueness theorem can be gained; Meanwhile, the existence and uniqueness of the global solution in the interval [0, +∞) can also be obtained; Secondly, L~p-exponential estimate of the solution for neutral stochastic functional differential equations with infinite delay can be studied; At length, the theorem of the local solution about neutral stochastic functional differential equations with infinite delay only under the local Lipschitz condition and the contractive condition can be established.
In Chapter 3 Firstly, the exponential stability in mean square and almost sure exponential stability for neutral stochstic neural networks with time-varying delay can be discussed by using the linear matrix inequality (LMI) and the semimartingale convergence theorem; And our sufficient conditions are less conserative; Secondly, by utilizing the fixed point theorem, some sufficient and necessary conditions ensuring the asymp- totic stability in mean square for neutral stochastic neural networks with time-varying delays can be given.
In Chapter 4 In first quarter, the existence and uniqueness of mild solution for stochastic partial differential equations with delays under the non-Lipschitz condition are mainly discussed; In second quarter, by establishing a Lemma, some sufficient conditions ensuring p (p≥2)-moment exponential stability and almost surely exponential stability for mild solution to neutral stochastic partial differential equations with delays can be given; Specially, when the neutral item is removed, some sufficient conditions about p-moment exponential stability and almost sure exponential stability for mild solution of stochastic partial differential equations with delays are weaker than some existing literatures by establishing a Lemma, too; Thus, we improve some results; In third quarter, similarly, by constructing a Lemma, some sufficient conditions about asymptotic stability in p-moment of mild solution for neutral stochastic partial differential equations with delays can be given; If the neutral item is dropped out, some sufficient conditions about asymptotic stability in p-moment of mild solution for stochastic partial differential equations with delays can be derived; These two results are new.
引文
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