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 control in both convergence time and energy cost and numerically compare it with the proposed controllers in these terms. More significantly, we use several representative physical systems to illustrate the usefulness of the proposed controllers in stabilization of dynamical systems.

A.1 Proof of Proposition 3.2 First, we consider the solution of Eq. (9) for u( x) = kx . Thus, from the property of the martingale and It ˆ o's isometry formula, it follows that Eη (t) = Eη (0) = 0, Eη (t) So, to satisfy Condition (iii) in Theorem 2.2, we have to set Therefore, the exponential stability of the zero solution is assured. Now, applying Gronwall's inequality, we get E[ x (t) This therefore completes the proof of the whole theorem. A.3.2 Proof of Theorem 4.2 First we prove the estimation for E[ τ Applying It ˆ o's formula to log V (x) yields: log V ( x( t)) = log V (x Then, similar to the procedure for the energy cost in A.3.1, we can get that E[ x (t) Here we explain this term in more detail. The training for ES framework is not as efficient as AS.