## A stochastic version of Jurdjevic-Quinn theorem.(English)Zbl 0810.60051

Consider the following controlled stochastic differential equation: $x_ t = x_ 0 + \int^ t_ 0 \left( X_ 0 (x_ s) + \sum^ p_{i=1} u^ j Y_ j(x_ s) \right) ds + \sum^ m_{i=1} \int^ t_ 0 X_ i (x_ s) \circ dW^ i_ s, \tag $$*$$$ where $$X_ i$$ and $$Y_ i$$ are $$C^ 1_ b$$-vector fields on $$\mathbb{R}^ n$$ vanishing at the origin and $$u^ j$$ are bounded real-valued control laws. Let $$L$$ be the second-order differential operator given by $$L = X_ 0 + {1 \over 2} \sum^ m_{i=1} X^ 2_ t$$. The author proves that if there exists a $$C^ 2$$ function $$V$$: $$\mathbb{R}^ n \to \mathbb{R}$$ that is proper and positive definite such that $$LV(x) \leq 0$$ for all $$x \in \mathbb{R}^ n$$ and the set $$\{x \in \mathbb{R}^ n \mid L^{k+1} V(x) = L^ k Y_ j V(x) = 0$$, $$k \in \mathbb{N}$$, $$j=1,2, \dots, p\}$$ is reduced to $$\{0\}$$, then the control law $$u$$ defined on $$\mathbb{R}^ n$$ by $$u^ j(x) = - Y_ j V(x)$$, $$1 \leq j \leq p$$, is a stabilizing feedback law for the system $$(*)$$. This is an extension of the result by V. Jurdjevic and J. P. Quinn [J. Differ. Equations 28, 381-389 (1978)].
Reviewer: J.H.Kim (Pusan)

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Zbl 0417.93012
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### References:

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