Mason et al. further implies existence of a Lyapunov function that can be found with sum of squares techniques. In [@Pablo_Jadbabaie_JSR_journal], Parrilo and Jadbabaie prove that stable switched linear systems in discrete time always admit a (not necessarily convex) polynomial Lyapunov function which can be found with sum of squares techniques. Blanchini and Franco show in [@blanchini_no_convex_Lyap] that in contrast to the case of uncontrolled switching (our setting), controlled linear switched systems, both in discrete and continuous time, can be stabilized by means of a suitable switching strategy without necessarily admitting a convex Lyapunov function.

In [@Chesi_Hung_journal], [@Chesi_Hung_conf], Chesi and Hung motivate several interesting applications of working with convex Lyapunov functions or Lyapunov functions with convex sublevel sets. These include establishing more regular behavior of the trajectories, ease of optimization over sublevel sets of the Lyapunov function, stability of recurrent neural networks, etc. The authors in fact propose sum of squares based conditions for imposing convexity of polynomials. However, it is shown in [@AAA_PP_CDC10_algeb_convex Sect. 4] that these conditions lead to semidefinite programs of larger size than those of sos-convexity, while at the same time being at least as conservative. Moreover, the works in [@Chesi_Hung_journal], [@Chesi_Hung_conf] do not offer an 