Stabilization by Artificial Wiener Processes

In this technical note, we restage the positive effects of Gaussian white noises for continuous-time driftless input-affine nonholonomic systems. Compared with previous studies, we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi-dimensional Wiener processes. Furthermore, we found that the positive effects are artificially restaged by numerical calculations with use of Euler scheme for ordinary differential equations. Our results demonstrate that the positive effects act as “hidden independent control inputs” for nonholonomic systems. This implies that our results provide a constructive procedure for designing control inputs such that the origins become $p$-th moment exponentially or finite-time stable.