Qualitative sign stability of linear time invariant descriptor systems

This article discusses assessing the instability of a continuous linear homogeneous timeinvariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed