On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form

In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices A₁ and 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.