by an *sos-convex Lyapunov function* is a polynomial function which is sos-convex and satisfies all other required Lyapunov inequalities with sos certificates.[^5] When the Lyapunov function can be taken to be homogeneous—as is the case when the dynamics are homogeneous [@HomogHomog]—then the following lemma establishes that the convexity requirement of the polynomial automatically meets its nonnegativity requirement.

Recall that a *homogeneous polynomial* (or a *form*) is a polynomial whose monomials all have the same degree.

Convex forms are nonnegative and sos-convex forms are sos.

See [@Helton_Nie_SDP_repres_2 Lemma 8] or [@AAA_PP_table_sos-convexity Lemma 3.2].

For stability analysis of the switched linear system in (\[eq:switched.linear.system\]), the requirements of a (common) sos-convex Lyapunov function $V$ are therefore the following: $$\label{eq:sos-convex.Lyap.requirements}
\begin{array}{ll}
V(x) & \mbox{sos-convex}\\
V(x)-V(A_ix) & \mbox{sos for}\ i=1,\ldots,m.
\end{array}$$

Given a set of matrices $\{A_1,\ldots,A_m\}$ with rational entries, the search for the coefficients of a fixed-degree polynomial $V$ satisfying the above conditions amounts to solving an SDP whose size is polynomial in the bit size of the matrices. If this SDP is (strictly) feasible, the switched system in (\[eq:switched.linear.system\]) is stable under arbitrary switching. We remark that the same implication is true if the sos-convexity requirement of $V$ is replaced with the requirement 