A stability condition for a time-varying system represented by a couple of a second- and a first-order differential equations

In this paper, we analyze stability of a time-varying system represented by a second-order vector differential equation based on the characteristics of its coefficient matrices. The objective system has a singular coefficient matrix M(t), so the system is composed by a couple of a second- and a first-order vector differential equations. New sufficient conditions for asymptotic stability of the equilibrium points are derived. We also discuss the relations between our result and that for time-invariant system, and that for non-singular coefficient matrix M(t).