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.

1 Introduction

The problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function (CQLF) for a set of stable linear time-invariant (LTI) systems $$\Sigma_{A_{i}}:\dot{x}(t)=A_{i}x(t), A_{i}\in {\BBR}^{n\times n}, 1\leq i\leq k$$ plays an important role in the study of switched linear systems of the form: $$\dot{x}(t)=A(t)x(t), A(t)\in\{A_{1}, \ldots, A_{k}\}. \eqno{\hbox{(1)}}$$