摘要
在文[1]中Lehnigk证明了对于实方阵A存在严格半正定阵C使系统x=Ax之零解渐近稳定的充分必要条件是Lyapunov矩阵方程A'B+BA=-C有正定矩阵解B.但由于C的构造复杂且对于给定的A只能得到一个C,这个结果事实上是难于应用的. 本文得到:若A没有这样的特征向量,其第l_1,…,l_m位分量为零,对于任意第l_1,…,1_m位的部分正定阵C,系统x=Ax之零解渐近稳定的充要条件是上述矩阵方程有正定矩阵解B.由于不须求解矩阵A的特征问题,且部分正定阵C的类型广泛,易于构造,从而圆满地解决了这一问题。
In the paper[1], Lehnigk discussed the problem on the existence of a quadratic form as a Lyapunovfunction for a Linear system with constant coefficients such that the total derivative is strictly negative semi-definite and not identically equal to zero for every non-trivial solution of the given system. Butthc results which was proved by Lehnigk is difficult in application. This paper proves that there exists the aforesaid quadratic form as aLyapunov function such that its total derivative equal to some given partial negative definite quadratic form respectively. The partial positive definite matrix has many category, which is easy to construct and there is not need of solving the eigenvalue problem of coefficient matrix of the system.
出处
《工程数学学报》
CSCD
1990年第3期39-44,共6页
Chinese Journal of Engineering Mathematics
