Constructive design method of stochastic continuous feedback laws for stabilization of deterministic nonlinear systems

The purpose of this paper is to present a constructive design method of a stochastic feedback law for a deterministic nonlinear affine system. This design method gives a feedback law in the case that a randomized system is given as a Stratonovich-type stochastic differential equation, based on a result of Wong and Zakai. It is proved that a designed feedback law can globally asymptotically stabilize a randomized system with probability one. The design method employs a notion of a stochastic control Lyapunov function and the so-called Sontag's formula. The method in this paper aims the stabilization of nonholomic systems and non-Euclidean systems, which cannot be stabilized by any continuously differentiable time-invariant feedback laws. We can obtain a continuous feedback law for a state variable in the stabilization of such systems.