Necessary and sufficient conditions for common quadratic Lyapunov functions for a pair of stable LTI systems whose system matrices are in companion form

In this paper the problem of determining necessary and sufficient conditions for the Lyapunov function for a pair of stable linear time-invariant systems whose system matrices, A₁, A₂, are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product A₁A₂ does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.