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Stabilization of passive nonlinear stochastic differential systems by bounded feedback. (English) Zbl 1048.93094

The author considers the following input/output stochastic differential system \[ x_t = x_0 + \int_0^t f(x_s, u)\,ds + \int_0^t g(x_s)\,dw_s, \] where \((x_t, y_t) \in \mathbb{R}^n \times \mathbb{R}^n\), \(u\) is an \(\mathbb{R}^m\)-valued measurable control law, \(f\) and \(g\) are smooth Lipschitz functions, vanishing in the origin. The main results of this paper concern the asymptotic stabilization in probability by means of bounded state feedback laws for this system. For example, under some restrictions on the functions \(f\) and \(g\) the author proves the following result: Assume that the stochastic differential system can be written in the form \[ x_t = x_0 + \int_0^t \biggl( f(x_s, 0) + \sum_{i=1}^m \bar{f}_i (x_s) u_i\biggr)\,ds + \int_0^t g(x_s)\,dw_s \] and is Lyapunov stable in probability with some smooth Lyapunov function \(V\) defined on \(\mathbb{R}^n\). Assume further that the stochastic differential system is asymptotically stabilizable in probability by an arbitrarily small smooth state feedback law. In particular \[ u(x) = - \beta +(1 + \| \nabla V(x) \bar{f} (x) \|^2 (\nabla V(x) \bar{f} (x)), \] for any \(\beta >0\), is a possible choice of a stabilizing state feedback law. The results of this paper are applied in order to construct explicitly a dynamic compensator for a class of stochastic differential systems not necessarily affine in the control. The results of the paper are illustrated by a design example.

MSC:

93E15 Stochastic stability in control theory
93C10 Nonlinear systems in control theory
93D15 Stabilization of systems by feedback
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