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Lyapunov-like techniques for stochastic stability. (English) Zbl 0845.93085

The author considers the dynamical system

$x\left(t\right)={x}_{0}+{\int }_{0}^{t}f\left(x\left(s\right),u\right)ds+{\int }_{0}^{t}g\left(x\left(s\right)\right)dw\left(s\right)$

where $x\in {ℝ}^{n}$, ${x}_{0}$ is an initial state, $u\in {ℝ}^{p}$ is a control law, $f:{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{n}$, $g:{ℝ}^{n}×{ℝ}^{p}\to {ℝ}^{n}$ are Lipschitz functionals mapping, $f\left(0,0\right)=0$, $g\left(0\right)=0$, $w\left(t\right)$ 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