常微分方程部分变元稳定性理论中的矩阵问题

将Lyapunov第二方法用于研究常系数线性系统零解的部分变元渐近稳定性,可引出相关的一些特殊矩阵问题,包括部分稳定矩阵、部分位正定矩阵、矩阵方程的可解性等.给出了部分稳定矩阵的几种判据与部分位正定矩阵的标准型,研究了矩阵方程可解性、唯一性的几种条件,提出一些可供进一步研究的问题.

Matrix problem in partial stability theory of ordinary differential equation

Lyapunov second method is applied to analysis of partial stability of equilibrium of linear ordinary di?erential equation with constant coe?cients, the problems on partial stable matrix, partial positive de?nite matrix and Lyapunov matrix equation are studied. Criterions of partial stable matrix and normal form of partial positive de?nete matrix are given, some conditions of solvability and uniqueness of Lyapunov matrix equation are discussed. Furthermore, some prob- lems for further study are presented.