随机时滞微分方程解的矩稳定性和有界性

项目名称： 随机时滞微分方程解的矩稳定性和有界性

项目编号： No.11426080

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 王珍

作者单位： 合肥工业大学

项目金额： 3万元

中文摘要： 本项目将利用Laplace变换研究线性随机时滞微分方程解的矩稳定性和有界性。对于具有离散时滞的线性随机微分方程和具有分布时滞的线性随机微分方程，Lei等和Wang等已经分别建立了其解的二阶矩对应的特征方程，并利用该特征方程给出了二阶矩有界和无界的充分条件，但是他们都没有详细分析特征方程根的分布情况，给出的二阶矩有界和无界的充分条件都不方便应用。为此，本项目将对具有离散时滞和分布时滞的这两类线性随机时滞微分方程的特殊情形，深入细致地分析其对应的特征方程，研究特征方程根的分布情况，建立便于应用的二阶矩有界和无界的充分条件。作为理论结果的应用，本项目还将研究随机时滞logistic模型和具有随机扰动的白细胞增殖的外围控制模型的矩稳定性和有界性。

中文关键词： 时滞微分方程；年龄结构模型；半线性随机发展方程；随机稳定性；Lyapunov 泛函

英文摘要： This project will study the stability and boundedness of moments of the solutions to linear stochastic delay differential equations through techniques of the Laplace transform. For the linear stochastic differential equation with discrete delay and the linear stochastic differential equation with distributed delay, although Lei et al and Wang et al established the characteristic equations for the second moments of their solutions and gave the sufficient conditions for boundedness and unboundedness of the moments, respectively, they did not analysize the distribution of the roots of the characteristic equations and the sufficient conditions were not applicable easily. Thus, in this project, for the special cases of two kinds of equations: the linear stochastic differential equations with discrete delay and with distributed delay, we will analysize the roots of the corresponding characteristic equations detailedly and will establish the sufficient conditions for the moments to be bounded and unbounded for application. As the application of theoretic results, we will investigate the moment stability and boundedness for the stochastic logistic model and the peripheral control of white blood cell production with stochastic perturbation.

英文关键词： Delay differential equation；age-structured model；semilinear stochastic evolution equation；stochastic stability；Lyapunov functional