subset of convex polynomial Lyapunov functions whose convexity is certified through an explicit algebraic identity. One can search over sos-convex Lyapunov functions of a given degree by solving a single semidefinite program whose size is polynomial in the description size of the input dynamical system. The methodology can directly handle the linear switched system in (\[eq:switched.linear.system\]) or its nonlinear counterpart in (\[eq:switched.nonlinear.system\])-(\[eq:ftilda=conv\]), if the maps $f_1,\ldots,f_m$ are *polynomial functions*.[^3]

We will review some results from the thesis of the first author which show that for certain dimensions and degrees, the set of convex and sos-convex Lyapunov functions coincide. In fact, in relatively low dimensions and degrees, it is quite challenging to find convex polynomials that are not sos-convex [@AAA_PP_not_sos_convex_journal]. This is evidence for the strength of this semidefinite relaxation and is encouraging from an application viewpoint. Nevertheless, since sos-convex polynomials are in general a strict subset of the convex ones, a more refined (and perhaps more computationally relevant) converse Lyapunov question for switched linear systems is to see whether their stability guarantees existence of an sos-convex Lyapunov function. This question is also addressed in this paper.

We shall remark that there are other classes of convex Lyapunov functions whose construction 