a convex polynomial Lyapunov functions is always guaranteed, we prove that in worst case, the degree of such a Lyapunov function can be arbitrarily higher than that of a non-convex polynomial Lyapunov function (Theorem \[thm:degree.higher\]).

In Section \[section:nonlinear\], we study nonlinear switched systems. We show that stability of these systems cannot be inferred from the existence of a common Lyapunov function for the corner systems (Example \[ex:nonconvex.fails\]). However, we prove that this conclusion can be made if the common Lyapunov function is convex (Proposition \[prop:convex.lyap\]). We also give a lemma that shows that the radial unboundedness requirement of a Lyapunov function is implied by its convexity (Lemma \[lem:convex.coercive\]). We then provide an algorithm based on semidefinite programming that under mild conditions finds a full-dimensional inner approximation to the region of attraction of a locally stable equilibrium point of a polynomial switched system (Theorem \[thm:nl.beta.guarantee\]). This algorithm is based on a search for an sos-convex polynomial whose sublevel set is proven to be in the region of attraction via a sum of squares certificate coming from Stengle’s Positivstellensatz. Some examples are provided in Section \[subsec:examples\].

Finally, in Section \[sec:future\], we briefly describe some future directions and extensions of our framework to 