  We analyze Neural Ordinary Differential Equations (NODEs) from a control
theoretical perspective to address some of the main properties and paradigms of
Deep Learning (DL), in particular, data classification and universal
approximation. These objectives are tackled and achieved from the perspective
of the simultaneous control of systems of NODEs. For instance, in the context
of classification, each item to be classified corresponds to a different
initial datum for the control problem of the NODE, to be classified, all of
them by the same common control, to the location (a subdomain of the euclidean
space) associated to each label. Our proofs are genuinely nonlinear and
constructive, allowing us to estimate the complexity of the control strategies
we develop. The nonlinear nature of the activation functions governing the
dynamics of NODEs under consideration plays a key role in our proofs, since it
allows deforming half of the phase space while the other half remains
invariant, a property that classical models in mechanics do not fulfill. This
very property allows to build elementary controls inducing specific dynamics
and transformations whose concatenation, along with properly chosen
hyperplanes, allows achieving our goals in finitely many steps. The
nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results
apply also in the particular case of the ReLU activation function. We also
present the counterparts in the context of the control of neural transport
equations, establishing a link between optimal transport and deep neural
networks.
