  This paper proposes a novel approach to improve the performance of
distributed nonlinear control systems while preserving stability by leveraging
Deep Neural Networks (DNNs). We build upon the Neural System Level Synthesis
(Neur-SLS) framework and introduce a method to parameterize stabilizing control
policies that are distributed across a network topology. A distinctive feature
is that we iteratively minimize an arbitrary control cost function through an
unconstrained optimization algorithm, all while preserving the stability of the
overall network architecture by design. This is achieved through two key steps.
First, we establish a method to parameterize interconnected Recurrent
Equilibrium Networks (RENs) that guarantees a bounded $\mathcal{L}_2$ gain at
the network level. This ensures stability. Second, we demonstrate how the
information flow within the network is preserved, enabling a fully distributed
implementation where each subsystem only communicates with its neighbors. To
showcase the effectiveness of our approach, we present a simulation of a
distributed formation control problem for a fleet of vehicles. The simulation
demonstrates how the proposed neural controller enables the vehicles to
maintain a desired formation while navigating obstacles and avoiding
collisions, all while guaranteeing network stability.
