时间周期线性扰动系统零解的稳定性分析

Stability analysis of the zero solution for time-periodic linear system with small perturbations

研究周期线性扰动系统的零解稳定性。通过对周期线性系统性质的研究,可使用寻找非奇异可微周期矩阵的方法,将对周期线性系统性质的研究转化为对常系数线性系统性质的研究,并将常系数线性扰动系统的零解稳定性结论推广至周期线性扰动系统。在一定条件下,当周期线性扰动系统的扰动项是关于||x||的高阶无穷小时,周期线性扰动系统与未扰动系统的零解具有相同的稳定性。最后,给出了文中所得结果的一个应用举例。

In this paper,we studied the stability of the zero solution for time-periodic linear perturbed systems. After investigating the properties of periodic linear systems,we applied the method of finding nonsingular and differentiable periodic matrices into the study of the properties of linear systems with constant coefficients. Then we extended the zero solution's stability of the constant coefficient linear system with small perturbation to the periodic linear system with small perturbations. Under the condition that the perturbation term of a periodic linear disturbance system is of a higher order than ||x||, the periodic linear disturbance system has the same stability as the zero solution of the disturbed system. Finally, the result was illustrated.