bounded.

Existence of sos-convex polynomial Lyapunov functions {#subsec:sos-convex}
-----------------------------------------------------

We now strengthen the converse result of the previous subsection by showing that asymptotically stable switched linear systems admit an *sos-convex* Lyapunov function. This in particular implies that such a Lyapunov function can be found with semidefinite programming.

Recall that a homogeneous polynomial $h$ is said to be *positive definite (pd)* if $h(x)>0$ for all $x\neq 0$.

\[thm:existence-sos-convex-poly\] If the switched linear system (\[eq:switched.linear.system\]) is asymptotically stable under arbitrary switching, then there exists a homogeneous polynomial $q$ that satisfies the sum of squares constraints $$\nonumber
\begin{array}{ll}
q(x) & \mbox{sos-convex},\\
q(x)-q(A_jx) & \mbox{sos for}\ j=1,\ldots,m.
\end{array}$$ Moreover, this polynomial $q$ is positive definite and is such that the $m$ polynomials $q(x)-q(A_jx)$ are also positive definite.

Our proof will make crucial use of the following Positivstellensatz due to Scheiderer.

\[thm:claus\] Given any two positive definite homogeneous polynomials $h$ and $g$, there exists a positive integer $N$ such that $hg^N$ is sos.

(of Theorem \[thm:existence-sos-convex-poly\]). We have already shown in the proof of Theorem \[thm:existence-convex-poly\] that under our assumptions, there exist vectors $a_1,\ldots,a_N\in\mathbb{R}^n$ and a positive integer $d$ such that the convex form $p(x)=\sum_{i=1}^N (a_i^Tx)^{2d}$ is positive definite and makes the $m$ forms $p(x)-p(A_jx)$ 