A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems

We present a conjecture concerning necessary and sufficient conditions for the existence of a common quadratic Lyapunov function (CQLF) for a switched linear system obtained by switching between two positive linear time-invariant (LTI) systems. We conjecture that these conditions are also necessary and sufficient for the exponential stability of such switched linear systems; namely, the existence of a CQLF is a non-conservative stability condition in this case. A number of new results supporting this conjecture are described.