Polytope Lyapunov functions for stable and for stabilizable LSS

We consider switched linear systems of odes, $\dot{x}(t) = A\left( u(t) \right) x(t)$ where $A(u(t)) \in \cA$, a compact set of matrices. In this paper we propose a new method for the approximation of the upper Lyapunov exponent and lower Lyapunov exponent of the LSS when the matrices in $\cA$ are Metzler matrices (or the generalization of them for arbitrary cone), arising in many interesting applications (see e.g. \cite{FR}). % The method is based on the iterative construction of invariant positive polytopes for a sequence of discretized systems obtained by forcing the switching instants to be multiple of $\Delta^{(k)} t$ where $\Delta^{(k)} t \to 0$ as $k \to \infty$. These polytopes are then used to generate a monotone piecewise-linear joint Lyapunov function on the positive orthant, which gives tight upper and lower bounds for the Lyapunov exponents. As a byproduct we % are able to detect whether the considered system is stabilizable or uniformly stable. The efficiency of this approach is demonstrated in numerical examples, including some of relatively large dimensions.