Copositive Lyapunov functions for switched systems over cones

We answer two open questions on copositive Lyapunov functions which were recently posed by M.K. Çamlıbel and J.M. Schumacher in the book Unsolved Problems in Mathematical Systems and Control Theory, edited by V.D. Blondel and A. Megretski [M.K. Çamlibel, J.M. Schumacher, Copositive Lyapunov functions, in: V.D. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press, 2004, pp. 189–193. Available online at http://press.princeton.edu/math/blondel/]. These questions are: what are necessary and sufficient conditions for the existence of a Lyapunov function for a linear system which is defined over a cone? How can this be extended to switched linear systems where the system matrix varies over time? sent conditions answering these questions. Our conditions amount to checking feasibility or infeasibility of a system of linear inequalities.