A Class of Continuous-Time Linear Systems Having a Common Lyapunov Function

Abstract

The common Lyapunov function problem arises in association with stability analysis of diverse fields of systems. The problem is numerically solvable. However, it is not an easy task to fully characterize such a class of systems which have a common quadratic Lyapunov function. It is thus far known that a set of linear stable systems has a common quadratic Lyapunov function if their system matrices are in a commuting family. But this condition is considerably restrictive. Our objective here is to find another class of systems which has a common quadratic Lyapunov function. It turns out that if all systems have stable system matrices which have an upper (lower) triangular structure, then they have a common quadratic Lyapunov function. Note that such system matrices are not necessarily in a commuting family. The obtained results would give an insight into further exploration of the common Lyapunov function problem. Numerical examples are worked out for illustration.