Lyapunov stability, semistability, and asymptotic stability of matrix second-order systems

This paper provides a self-contained, unified and extended treatment of the stability of matrix second-order systems. The results we obtain encompass numerous results from prior literature in addition\´to several new results. Specifically, in addition to obtaining necessary and sufficient conditions for Lyapunov and asymptotic stability, we consider the case of semistability. Semistability is of particular interest in the analysis of vibrating systems in that it represents the case of "damped rigid body modes", that is, systems that eventually come to rest, although not necessarily at a specified equilibrium point. This paper presents the first treatment of semistability for matrix second-order systems.