几类随机微分方程数值方法的稳定性分析

This project is concerned with the stability of numerical methods for several classes of stochastic ordinary differential equations and stochastic delay differential equations. For multi-dimensional stochastic ordinary differential equations, we will try to investigate the mean square stability and almost sure stability of stochastic linear multi-step methods and high order stochastic Runge-Kutta methods when they are applied to some special linear test equations and more general stochastic ordinary differential systems. For nonlinear stochastic ordinary differential equations, we will try to seek some more weak stability conditions for the true solution. And based on these, we will investigate the nonlinear stability of the numerical methods. In addition, we also concern with the delay dependent stability of the true solution and numerical solution for stochastic delay differential equations. For the linear scalar stochastic delay differential equations, we try to look for the sufficient and necessary conditions of the delay dependent stability and analyze the delay dependent stability of numerical methods by boundary locus method. For nonlinear case, we will also seek some more weak conditions to obtain the delay dependent stability and get the corresponding results of the numerical delay dependent stability. This project will develop some new research ideas and analysis techniques. All of the results will enhance and develop the stability theory of the true solution and the numerical solution of stochastic differential equations.

本项目旨在研究几类随机常微分方程以及随机延迟微分方程数值方法的稳定性。对于多维随机常微分方程，我们尝试研究随机线性多步方法以及高阶随机Runge-Kutta方法应用到一类特殊的线性测试方程以及更为一般的随机常微系统时的均方稳定性以及几乎必然稳定性。对于非线性随机常微分方程，我们尝试寻找真解更弱的稳定性条件，并在此基础上研究数值方法的非线性稳定性。除此之外，我们还关注随机延迟微分方程真解和数值解的延迟依赖稳定性。针对线性标量随机延迟微分方程，我们尝试给出其真解延迟依赖的充分必要条件，并利用边界轨迹法研究数值方法的延迟依赖稳定性。对于非线性随机延迟微分方程，我们将寻求比现有条件更弱的延迟依赖稳定性条件，并得到相应的数值延迟依赖稳定性结果。本项目将发展新的研究思路和分析手段，所获成果将进一步丰富和发展随机微分方程真解与数值解稳定性理论。

本项目重点研究几类随机微分方程解析解和数值解均方稳定性。对于随机Volterra积分微分方程，给出了其解析解均方指数稳定的充分条件，并研究了随机Theta方法应用到该类型方程上的均方收敛性和稳定性。对于带有多个变延迟线性中立型随机延迟系统，通过构造合适的Lyapunov–Krasovski泛函，利用线性矩阵不等式的方法得到了系统解析解均方指数稳定的充分条件。除此之外，我们还增加了部分研究内容，研究了Fisher方程反馈控制问题，针对其平衡解局部指数稳定化在L^2 (0,1)以及H^1 (0,1)空间中设计一个显式有限维反馈控制，该反馈控制由线性化方程不稳定特征值所对应的特征函数表示。我们基本完成研究计划，发表期刊论文3篇。所获结果在自动控制和计算生物等领域也具有广泛应用前景。。