  We introduce a method for controlling systems with nonlinear dynamics and
full actuation by approximating the dynamics with polynomials and applying a
system level synthesis controller. We show how to optimize over this class of
controllers using a neural network while maintaining stability guarantees,
without requiring a Lyapunov function. We give bounds for the domain over which
the use of the class of controllers preserves stability and gives bounds on the
control costs incurred by optimized controllers. We then numerically validate
our approach and show improved performance compared with feedback linearization
-- suggesting that the SLS controllers are able to take advantage of
nonlinearities in the dynamics while guaranteeing stability.
