is amenable to convex optimization. The main examples include polytopic Lyapunov functions, and piecewise quadratic Lyapunov functions that are a geometric combination of several quadratics [@gwz05; @protasov2; @jungersprotasov09; @protasov-jungers-blondel09; @BlNes05; @dual_LMI_diff_inclusions; @LeeD06]. These Lyapunov functions are mostly studied for the case of linear switched systems, where they are known to be necessary and sufficient for stability. The extension of their applicability to polynomial switched systems should be possible via the sum of squares relaxation. Our focus in this paper however is solely on studying the power of sos-convex polynomial Lyapunov functions. Only in our last section, do we briefly comment on extensions to piecewise sos-convex Lyapunov functions.

Related work
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The literature on stability of switched systems is too extensive for us to review. We simply refer the interested reader to [@Shorten05stabilitycriteria; @gst-book; @jungers_lncis] and the references therein. Closer to the specific focus of this paper is the work of Mason et al. [@mason-boscain-chitour06], where the authors prove existence of polynomial Lyapunov functions for switched linear systems in *continuous time*. Our proof of the analogous statement in discrete time closely follows theirs. In [@ahmadi2017sum], Ahmadi and Parrilo show that in the continuous time case, existence of the Lyapunov function of 