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

Florchinger, Patrick

Stochastic Anal. Appl. 12, No. 4, 473-480 (1994).

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)

Keywords:

asymptotically stable in probability; controlled stochastic differential equation; stabilizing feedback law

Citations:

Zbl 0417.93012

References:

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