*(English)*Zbl 0845.93085

The author considers the dynamical system

where $x\in {\mathbb{R}}^{n}$, ${x}_{0}$ is an initial state, $u\in {\mathbb{R}}^{p}$ is a control law, $f:{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$, $g:{\mathbb{R}}^{n}\times {\mathbb{R}}^{p}\to {\mathbb{R}}^{n}$ are Lipschitz functionals mapping, $f(0,0)=0$, $g\left(0\right)=0$, $w\left(t\right)$ is a standard ${R}^{n}$-valued Wiener process. He obtains sufficient conditions for the existence of stabilizing feedback laws that are smooth, except possibly at the equilibrium point of the system using stochastic Lyapunov techniques. The main results are obtained for the class of nonlinear stochastic systems that are affine in the control. Moreover, a stabilization result is also proved for a class of stochastic bilinear systems and for some nonlinear stochastic differential systems by means of a slight extension of the stochastic Jurdjevic-Quinn theorem obtained previously. In the last section, the necessary and sufficient conditions for the dynamic stabilization of control nonlinear stochastic systems are given.

##### MSC:

93E15 | Stochastic stability |

93D30 | Scalar and vector Lyapunov functions |

60H10 | Stochastic ordinary differential equations |

93D15 | Stabilization of systems by feedback |

93D20 | Asymptotic stability of control systems |

93C10 | Nonlinear control systems |