  We propose a novel methodology to solve a key eigenvalue optimization problem
which arises in the contractivity analysis of neural ODEs. When looking at
contractivity properties of a one layer weight-tied neural ODE
$\dot{u}(t)=\sigma(Au(t)+b)$ (with $u,b \in {\mathbb R}^n$, $A$ is a given $n
\times n$ matrix, $\sigma : {\mathbb R} \to {\mathbb R}^+$ denotes an
activation function and for a vector $z \in {\mathbb R}^n$, $\sigma(z) \in
{\mathbb R}^n$ has to be interpreted entry-wise), we are led to study the
logarithmic norm of a set of products of type $D A$, where $D$ is a diagonal
matrix such that ${\mathrm{diag}}(D) \in \sigma'({\mathbb R}^n)$. Specifically,
given a real number $c$ (usually $c=0$), the problem consists in finding the
largest positive interval $\chi\subseteq \mathbb [0,\infty)$ such that the
logarithmic norm $\mu(DA) \le c$ for all diagonal matrices $D$ with $D_{ii}\in
\chi$. We propose a two-level nested methodology: an inner level where, for a
given $\chi$, we compute an optimizer $D^\star(\chi)$ by a gradient system
approach, and an outer level where we tune $\chi$ so that the value $c$ is
reached by $\mu(D^\star(\chi)A)$. We extend the proposed two-level approach to
the general multilayer, and possibly time-dependent, case $\dot{u}(t) = \sigma(
A_k(t) \ldots \sigma ( A_{1}(t) u(t) + b_{1}(t) ) \ldots + b_{k}(t) )$ and we
propose several numerical examples to illustrate its behaviour, including its
stabilizing performance on a one-layer neural ODE applied to the classification
of the MNIST handwritten digits dataset.
