  Control problems are always challenging since they arise from the real-world
systems where stochasticity and randomness are of ubiquitous presence. This
naturally and urgently calls for developing efficient neural control policies
for stabilizing not only the deterministic equations but the stochastic systems
as well. Here, in order to meet this paramount call, we propose two types of
controllers, viz., the exponential stabilizer (ES) based on the stochastic
Lyapunov theory and the asymptotic stabilizer (AS) based on the stochastic
asymptotic stability theory. The ES can render the controlled systems
exponentially convergent but it requires a long computational time; conversely,
the AS makes the training much faster but it can only assure the asymptotic
(not the exponential) attractiveness of the control targets. These two
stochastic controllers thus are complementary in applications. We also
investigate rigorously the linear controller and the proposed neural stochastic
controllers in both convergence time and energy cost and numerically compare
them in these two indexes. More significantly, we use several representative
physical systems to illustrate the usefulness of the proposed controllers in
stabilization of dynamical systems.
