  In order to stabilize nonlinear systems modeled by stochastic differential
equations, we design a Fast Exponentially Stable and Safe Neural Controller
(FESSNC) for fast learning controllers. Our framework is parameterized by
neural networks, and realizing both rigorous exponential stability and safety
guarantees. Concretely, we design heuristic methods to learn the exponentially
stable and the safe controllers, respectively, in light of the classic
stochastic exponential stability theory and our established theorem on
guaranteeing the almost-sure safety for stochastic dynamics. More
significantly, to rigorously ensure the stability and the safety guarantees for
the learned controllers, we develop a projection operator, projecting to the
space of exponentially-stable and safe controllers. To reduce the high
computation cost of solving the projection operation, approximate projection
operators are delicately proposed with closed forms that map the learned
controllers to the target controller space. Furthermore, we employ Hutchinson's
trace estimator for a scalable unbiased estimate of the Hessian matrix that is
used in the projection operator, which thus allows for computation cost
reduction and therefore can accelerate the training and testing processes. More
importantly, our approximate projection operations can be applied to the
nonparametric control methods to improve their stability and safety
performance. We empirically demonstrate the superiority of the FESSNC over the
existing methods.
