On a class of linear constant systems which have a common quadratic Lyapunov function

The common Lyapunov function problem to be studied here, is to find a set of systems which have a common quadratic Lyapunov function guaranteeing asymptotic stability of its all member systems. This problem comes from diverse fields of system stability analysis. Several sets of linear systems which have a common quadratic Lyapunov function are known. But these results are still restrictive. In this paper, we provide one more new class of such systems. We first show a necessary and sufficient condition for a set of systems to have a common quadratic Lyapunov function. By using this condition and a property of M-matrices, we obtain a class of systems having a common quadratic Lyapunov function. A numerical example is presented to show that the obtained class is in fact a new one