A Appendix

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.