On universal classes of Lyapunov functions for linear switched systems^††thanks: This research was partially supported by the iCODE institute, research project of the Idex Paris-Saclay.

As $V$ is convex, $V$ is Lipschitz and the composition $V\circ\varphi$ is absolutely continuous. Hence for almost every $t\in I$ the derivatives of both $\varphi$ and $V\circ\varphi$ are well-defined. By definition of subdifferential, for every $t,s\in I$ and $l\in\partial V(\varphi(t))$ we have

We deduce that

holds true for almost every $t\in I$ and for every $l\in\partial V(\varphi(t))$. Similarly, taking the limit as $s\to t^{-}$, we obtain that $\frac{d}{dt}V(\varphi(t))\leq l^{\top}\dot{\varphi}(t)$ for almost every $t\in I$ and for every $l\in\partial V(\varphi(t))$. This concludes the proof of the lemma. ∎

The second part of the proposition and the if implication in the first part directly follow from Lemma 2.9. We are left to show that if $V$ is a nonstrict common Lyapunov function for $(\Sigma_{\mathscr{M}})$, then the inequality (4) holds true. By contradiction, suppose that there exist $x\in\mathbb{R}^{n},l\in\partial V(x)$, and $M\in\mathscr{M}$ such that $l^{\top}Mx>0$. By [21, Theorem 25.6] one may find a differentiability point $y$ of $V$ such that the pair $(y,\nabla V(y))$ is arbitrarily close to $(x,l)$. In particular we may assume $\nabla V(y)^{\top}My>0$, that is, $V$ is increasing at $t=0$ along the trajectory $t\mapsto e^{tM}y$, leading to a contradiction. ∎

Let $(\Sigma_{\mathscr{M}})$ be uniformly exponentially stable. Let $V$ be the absolutely homogeneous common Lyapunov function provided by Proposition 2.7. In particular, $V$ is smooth on $\mathbb{R}^{n}\setminus\{0\}$. In order to prove the proposition, it is enough to show that any convex absolutely homogeneous function close enough to $V$ on $S^{n-1}$ in uniform norm is itself a Lyapunov function for $(\Sigma_{\mathscr{M}})$.

We proceed by contradiction: we assume that there exists a sequence of convex absolutely homogeneous functions $(W_{k})_{k\in\mathbb{N}}$ converging uniformly to $V$ on $S^{n-1}$ as $k$ goes to infinity and such that each $W_{k}$ is not strictly decreasing along at least one trajectory of the system. In particular the derivative of $W_{k}$ along such a trajectory is nonnegative on a set of times of positive measure. By Lemma 2.9 and absolute homogeneity of $W_{k}$, we deduce that there exist $x_{k}\in S^{n-1}$ and $M_{k}\in\mathscr{M}$ such that, for every fixed $l_{k}\in\partial W_{k}(x_{k})$, one has $l_{k}^{\top}M_{k}x_{k}\geq 0$. By compactness, we may assume that $x_{k}$ tends to $\bar{x}\in S^{n-1}$ as $k$ goes to infinity. Then, by [21, Theorem 24.5], $l_{k}$ converges to $\nabla V(\bar{x})$, so that, by boundedness of $\mathscr{M}$, $\limsup_{k\to\infty}\nabla V(\bar{x})^{\top}M_{k}\bar{x}=\limsup_{k\to\infty}l_{k}^{\top}M_{k}x_{k}\geq 0$. However, it follows by the choice of $V$ and Proposition 2.7 that $\nabla V(\bar{x})^{\top}M_{k}\bar{x}\leq-1$, yielding a contradiction. ∎

Let $V$ be a convex absolutely homogeneous function. Let $(x_{i})_{i\in\mathbb{N}}$ be a dense sequence in $S^{n-1}$, and $l_{i}\in\partial V(x_{i})$. We consider the increasing sequence of absolutely homogeneous functions defined by

Observe that each $W_{i}$ is convex and $W_{i}(x)\leq V(x)$ for every $x\in\mathbb{R}^{n}$. Indeed

for every positive integer $j$, by definition of subgradient and since $V(x)=V(-x)$ and $V(x_{j})=l_{j}^{\top}x_{j}$. We deduce that $W_{i}(x_{k})=V(x_{k})$ for every $k\in\mathbb{N}$ and $i\geq k$, hence $\lim_{i\to\infty}W_{i}(x_{k})=V(x_{k})$ and we can apply [21, Theorem 10.8] to conclude that the sequence of functions $W_{i}$ converges to $V$ uniformly on $S^{n-1}$. By applying Proposition 3.1 we get that the family of polyhedral functions is a universal class of Lyapunov functions.

Let us now consider the absolutely homogeneous functions

The function $Z_{i}$ is convex since it is the composition of the $2i$-norm on $\mathbb{R}^{i}$, i.e., $\|y\|_{2i}=\left(\sum_{j=1}^{i}y_{j}^{2i}\right)^{\frac{1}{2i}}$ for $y\in\mathbb{R}^{i}$, with the linear function from $\mathbb{R}^{n}$ to $\mathbb{R}^{i}$ mapping $x$ to $(l_{1}^{\top}x,\dots,l_{i}^{\top}x)^{\top}$. Moreover it is immediate to see that $W_{i}(x)\leq Z_{i}(x)\leq i^{\frac{1}{2i}}W_{i}(x)$, and in particular

tends to zero uniformly on $S^{n-1}$ as $i$ goes to infinity. By applying again Proposition 3.1, it follows that the family of homogeneous sums of squares is a universal class of Lyapunov functions. ∎

Proposition 2.8 guarantees the existence of a convex absolutely homogeneous nonstrict Lyapunov function $V$ for $(\Sigma_{\mathscr{M}_{1}})$. Intuitively speaking, the lemma follows from the fact that, for $\lambda$ large enough, the vectors $Mx$, with $M\in\mathscr{M}_{2}^{\lambda}$ and $x\in\mathbb{R}^{n}$, point towards the interior of the sublevel set $V^{-1}([0,V(x)])$. Let us formalize this idea. Since $l^{\top}x=V(x)$ whenever $l\in\partial V(x)$, by boundedness of $\mathscr{M}_{2}$ and of $\cup_{x\in V^{-1}(1)}\partial V(x)$ [21, Theorem 24.7], for $\lambda>0$ large enough one has $l^{\top}(M-\lambda\mathrm{Id}_{n})x=l^{\top}Mx-\lambda l^{\top}x=l^{\top}Mx-\lambda<0$ for every $M\in\mathscr{M}_{2}$, $x\in V^{-1}(1)$, and $l\in\partial V(x)$. The result is then an immediate consequence of Proposition 2.10. ∎

We start by showing the theorem in the case $n=2$. We proceed by contradiction, assuming that every uniformly exponentially stable switched system in the case where $\mathscr{M}$ consists of two matrices in $M_{2}(\mathbb{R})$ admits a Lyapunov function in $\mathcal{P}$. We consider a switched system corresponding to $\mathscr{M}^{0}=\{M^{0}_{1},M^{0}_{2}\}\subset M_{2}(\mathbb{R})$, where $M^{0}_{1},M^{0}_{2}$ are Hurwitz, the corresponding trajectories rotate clockwise around the origin, the system is uniformly stable, but not attractive, and starting from every initial nonzero condition there exists a unique periodic trajectory, with four switches per period. The existence of such a system is obtained in [2, Theorem 1], where it corresponds to the case $\mathbf{S4}$, $\mathcal{R}=1$. In particular, there exist $t_{1},t_{2}>0$ such that $e^{t_{1}M^{0}_{1}}e^{t_{2}M^{0}_{2}}$ has an eigenvalue equal to $-1$, corresponding to an eigenvector $x_{0}$. Set $T=t_{1}+t_{2}$ and consider the switched systems associated with $\mathscr{M}^{\varepsilon}=\{M^{\varepsilon}_{1},M^{\varepsilon}_{2}\}\subset M_{2}(\mathbb{R})$, where $M^{\varepsilon}_{i}=M^{0}_{i}-\varepsilon\mathrm{Id}_{2}$ for $i=1,2$. For $\varepsilon\geq 0$ we consider the $T$-periodic switching sequence $A^{\varepsilon}(\cdot)$ which takes values $M^{\varepsilon}_{1}$ for $t\in[0,t_{1})$ and $M^{\varepsilon}_{2}$ for $t\in[t_{1},T)$.

Since every trajectory $x(\cdot)$ of $(\Sigma_{\mathscr{M}^{\varepsilon}})$ can be written as $t\mapsto e^{-\varepsilon t}y(t)$ where $y(\cdot)$ is a trajectory of $(\Sigma_{\mathscr{M}})$, then $(\Sigma_{\mathscr{M}^{\varepsilon}})$ is uniformly exponentially stable for $\varepsilon>0$. Hence, by assumption, it admits a Lyapunov function $V_{\varepsilon}(\cdot)$ in $\mathcal{P}$. Since the latter is compact, there exists a sequence $(\varepsilon_{k})_{k\in\mathbb{N}}$ converging to zero such that $(V_{\varepsilon_{k}})_{k\in\mathbb{N}}$ converges to some $\bar{V}\in\mathcal{P}$. Moreover, for every $t\geq 0$,

Since $\bar{V}(\Phi_{A^{0}}(2T,0)x_{0})=\bar{V}(x_{0})$ we deduce that $\bar{V}$ is constant along the trajectory $\Phi_{A^{0}}(\cdot,0)x_{0}$. The function $t\mapsto\bar{V}(e^{tM^{0}_{1}}x_{0})$ is analytic for $x_{0}\neq 0$, being the composition of analytic functions, and it is constantly equal to $\bar{V}(x_{0})$ for $t\in[0,t_{1}]$. By analyticity, $t\mapsto\bar{V}(e^{tM^{0}_{1}}x_{0})$ is constant for all $t>0$ and therefore it must be identically equal to $0$ since $\bar{V}(\lim_{t\to\infty}e^{tM^{0}_{1}}x_{0})=\bar{V}(0)=0$. Since every nonzero point of $\mathbb{R}^{2}$ may be written as $\mu e^{tM^{0}_{1}}x_{0}$ for some positive $\mu$ and $t$, we deduce that $\bar{V}$ must be identically zero, contradicting the assumptions on $\mathcal{P}$.

We are left to prove the result for $n>2$. For this purpose we consider $\mathscr{M}_{1}=\{\bar{M}^{0}_{1},\bar{M}^{0}_{2}\}$ with

where the matrices $M^{0}_{i}$ are defined as above. Let $\lambda>0$ and $\mathscr{M}_{2}^{\lambda}$ be given by Lemma 4.1 with $\mathscr{M}_{2}=\{M\in{\rm so}(n)\mid\|M\|\leq 1\}$, where ${\rm so}(n)$ denotes the space of skew-symmetric $n\times n$ matrices. Define $\bar{\mathscr{M}}^{0}=\mathscr{M}_{1}\cup\mathscr{M}_{2}^{\lambda}$ and, for $\varepsilon>0$, consider the switched system corresponding to $\bar{\mathscr{M}}^{\varepsilon}=\bar{\mathscr{M}}^{0}-\varepsilon\mathrm{Id}_{n}$. It is clear that $(\Sigma_{\bar{\mathscr{M}}^{\varepsilon}})$ is uniformly exponentially stable for every $\varepsilon>0$.

Letting $\Pi_{1,2}$ be the $(x_{1},x_{2})$ plane, i.e., $\Pi_{1,2}=\{x\in\mathbb{R}^{n}\mid x_{3}=\dots=x_{n}=0\}$, we notice that, starting from every $\bar{x}\in\Pi_{1,2}$, there exists a periodic trajectory of $(\Sigma_{\bar{\mathscr{M}}^{0}})$ lying on $\Pi_{1,2}$. The restrictions of functions in $\mathcal{P}$ to $\Pi_{1,2}$ form a compact set of functions on $\Pi_{1,2}$ that are analytic outside the origin. As in the case $n=2$, we prove by contradiction that $\mathcal{P}$ is not universal. Assume that there exists a sequence $(V_{\varepsilon_{k}}(\cdot))_{k\in\mathbb{N}}$ of Lyapunov functions in $\mathcal{P}$ for $(\Sigma_{\bar{\mathscr{M}}^{\varepsilon_{k}}})$ converging to $\bar{V}\in\mathcal{P}$. We can show as before that $\bar{V}$ is equal to $0$ on $\Pi_{1,2}$. Because of the choice of $\mathscr{M}_{2}$ and by construction of $\bar{\mathscr{M}}^{0}$, every $1$-dimensional linear subspace of $\mathbb{R}^{n}$ may be reached in finite time from $\Pi_{1,2}$ via a trajectory of $(\Sigma_{\bar{\mathscr{M}}^{0}})$. Since $\bar{V}$ is non-increasing along such a trajectory, we deduce that $\bar{V}\equiv 0$ on $\mathbb{R}^{n}$, obtaining a contradiction. ∎

Let $\{f_{1},\dots,f_{N}\}$ be a basis of $\mathcal{P}$. The linear map $\varphi:\mathbb{R}^{N}\to\mathcal{P}$ defined as $\varphi(\alpha)=\sum_{i=1}^{N}\alpha_{i}f_{i}$ is continuous when $\mathcal{P}$ is endowed with the topology of uniform convergence on compact sets. Indeed, on every compact set $K\subset\mathbb{R}^{n}$ and for every $\alpha^{1},\alpha^{2}\in\mathbb{R}^{N}$, one has

Hence the image of the unit sphere via the map $\varphi$ is a compact set $\bar{\mathcal{P}}\subset\mathcal{P}$. Furthermore, $\bar{\mathcal{P}}$ does not contain the zero function because $\{f_{1},\dots,f_{N}\}$ is a linearly independent subset of $\mathcal{P}$. Applying Theorem 4.2 we obtain that $\bar{\mathcal{P}}$ is not universal. As each element of $\mathcal{P}$ is a scalar multiple of an element of $\bar{\mathcal{P}}$ we deduce that $\mathcal{P}$ is not universal either, concluding the proof of the first part of the corollary. Concerning the second part, it is enough to observe that the set of polynomial functions of degree at most $m$ from $\mathbb{R}^{n}$ to $\mathbb{R}$ is a finite-dimensional vector space of analytic functions. ∎

We first claim that if $\mathcal{P}^{n}_{d,l}$ is universal, the same is true for $\mathcal{P}^{2}_{d,l}$. Indeed, for every $\mathscr{M}\subset M_{2}(\mathbb{R})$ such that $(\Sigma_{\mathscr{M}})$ is uniformly exponentially stable, consider $\hat{\mathscr{M}}\subset M_{n}(\mathbb{R})$ given by

If $\hat{V}\in\mathcal{P}^{n}_{d,l}$ is a common Lyapunov function for $(\Sigma_{\hat{\mathscr{M}}})$, then $V:\mathbb{R}^{2}\ni x\mapsto\hat{V}(x,0)$ is a common Lyapunov function for $(\Sigma_{\mathscr{M}})$ and $V\in\mathcal{P}^{2}_{d,l}$.

We are left to prove that $\mathcal{P}^{2}_{d,l}$ is not universal. Consider the switched systems $(\Sigma_{\mathscr{M}^{\varepsilon}})$ introduced in the proof of Theorem 4.2, which are uniformly exponentially stable for $\varepsilon>0$, and only uniformly stable for $\varepsilon=0$. Assume by contradiction that $\mathcal{P}^{2}_{d,l}$ is universal and, in particular, that for every $\varepsilon>0$ there exists a Lyapunov function $V^{\varepsilon}\in\mathcal{P}^{2}_{d,l}$ for $(\Sigma_{\mathscr{M}^{\varepsilon}})$. By definition of $\mathcal{P}^{2}_{d,l}$, for every $\varepsilon>0$ there exist $l$ polynomials $P_{1}^{\varepsilon},\dots,P_{l}^{\varepsilon}$ of degree at most $d$ such that $V^{\varepsilon}(x)\in\{P_{1}^{\varepsilon}(x),\dots,P_{l}^{\varepsilon}(x)\}$. Given $j,k\in\{1,\dots,l\}$, we investigate the set of zeroes of the polynomial $Q_{jk}^{\varepsilon}$ defined as the homogeneous polynomial corresponding to the terms of maximal degree of $P_{j}^{\varepsilon}-P_{k}^{\varepsilon}$. For this purpose, recall that, by the fundamental theorem of algebra, every homogeneous polynomial $Q$ of positive degree $m$ may be factorized as $Q=\prod_{k=1}^{m}(\alpha_{k}x_{1}+\beta_{k}x_{2})$, where $\alpha_{k},\beta_{k}\in\mathbb{C}$ for $k=1,\dots,m$, so that its zeroes correspond to the intersection of the unit circle $S^{1}$ with at most $m$ lines through the origin. Hence, it follows that either $Q_{jk}^{\varepsilon}\equiv 0$ (i.e., $P_{j}^{\varepsilon}\equiv P_{k}^{\varepsilon}$) or $Q_{jk}^{\varepsilon}$ vanishes at most $2d$ times on the unit circle. Moreover, for every $\varepsilon>0$, the integer $N=2d\binom{l-1}{2}+1$ is a strict upper bound for the total number of zeroes of $Q_{jk}^{\varepsilon}$ for $j,k\in\{1,\dots,l\}$. Partitioning the circle into $N$ arcs $\mathcal{C}_{1},\dots,\mathcal{C}_{N}$ of equal length, for every $\varepsilon>0$ there exists an arc $\mathcal{C}_{n_{\varepsilon}}$ which contains no zero of the nontrivial polynomials $Q_{jk}^{\varepsilon}$ in its interior. Denote by $\mathcal{A}_{n_{\varepsilon}}$ the closed middle third of $\mathcal{C}_{n_{\varepsilon}}$ (see Figure 1).

We next claim that for every $\varepsilon>0$ there exists $\nu_{\varepsilon}>0$ large enough such that the restriction of $V^{\varepsilon}$ to the dilated arc $\nu_{\varepsilon}\mathcal{A}_{n_{\varepsilon}}$ coincides with the restriction to the same arc of one of the polynomials $P_{1}^{\varepsilon},\dots,P_{l}^{\varepsilon}$. By definition of the function $V^{\varepsilon}$ and taking into account its continuity, it is enough to prove that, for every $\varepsilon>0$ there exists $\nu_{\varepsilon}>0$ large enough such that, in the interior of the arc $\nu_{\varepsilon}\mathcal{A}_{n_{\varepsilon}}$, one has for every $j,k\in\{1,\dots,l\}$ that either $P_{j}^{\varepsilon}\equiv P_{k}^{\varepsilon}$ or $P_{j}^{\varepsilon}-P_{k}^{\varepsilon}$ is never vanishing. To see that, it is enough to prove that if $Q_{jk}^{\varepsilon}$ does not vanish on $\mathcal{A}_{n_{\varepsilon}}$ then $P_{j}^{\varepsilon}-P_{k}^{\varepsilon}$ does not vanish on $\nu_{\varepsilon}\mathcal{A}_{n_{\varepsilon}}$ for $\nu_{\varepsilon}>0$ large enough independent of $j,k$. In that case, one has, for $\nu>0$ large enough, $x\in S^{1}$ and $j,k\in\{1,\dots,l\}$,

where $d^{\prime}$ is the positive degree of $Q_{jk}^{\varepsilon}$ and $o(1)$ is a function of $x$ and $\nu$ tending to $0$ as $\nu$ tends to infinity uniformly with respect to $x\in S^{1}$ and $j,k\in\{1,\dots,l\}$. This concludes the proof of the claim.