  Recent research has shown that supervised learning can be an effective tool
for designing optimal feedback controllers for high-dimensional nonlinear
dynamic systems. But the behavior of these neural network (NN) controllers is
still not well understood. In this paper we use numerical simulations to
demonstrate that typical test accuracy metrics do not effectively capture the
ability of an NN controller to stabilize a system. In particular, some NNs with
high test accuracy can fail to stabilize the dynamics. To address this we
propose two NN architectures which locally approximate a linear quadratic
regulator (LQR). Numerical simulations confirm our intuition that the proposed
architectures reliably produce stabilizing feedback controllers without
sacrificing optimality. In addition, we introduce a preliminary theoretical
result describing some stability properties of such NN-controlled systems.
