  Deep neural networks (DNN) are increasingly being used to learn controllers
due to their excellent approximation capabilities. However, their black-box
nature poses significant challenges to closed-loop stability guarantees and
performance analysis. In this paper, we introduce a structured DNN-based
controller for the trajectory tracking control of Lagrangian systems using
backing techniques. By properly designing neural network structures, the
proposed controller can ensure closed-loop stability for any compatible neural
network parameters. In addition, improved control performance can be achieved
by further optimizing neural network parameters. Besides, we provide explicit
upper bounds on tracking errors in terms of controller parameters, which allows
us to achieve the desired tracking performance by properly selecting the
controller parameters. Furthermore, when system models are unknown, we propose
an improved Lagrangian neural network (LNN) structure to learn the system
dynamics and design the controller. We show that in the presence of model
approximation errors and external disturbances, the closed-loop stability and
tracking control performance can still be guaranteed. The effectiveness of the
proposed approach is demonstrated through simulations.
