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A stochastic Jurdjevic-Quinn theorem. (English) Zbl 1014.60062
Consider stochastic differential systems of the form \[ x_t=x_0+\int_0^t(b(x_s)+uf(x_s)) ds+\int_0^t\sigma(x_s) dw_s+ \int_0^tug(x_s) d\widetilde w_s,\tag{1} \] where \((w_t)_{t\geq 0}\) and \((\widetilde w_t)_{t\geq 0}\) are two independent Wiener processes. The aim of this paper is to design a state feedback law \(u\) such that the equilibrium solution of the closed-loop system deduced from (1) is asymptotically stable in probability. Sufficient conditions for the stabilizability of (1) when both the drift and diffusion terms are affine in the control, are obtained. This result extends to stochastic differential systems the well-known Jurdjevic-Quinn theorem and incorporates earlier stabilization results proved by P. Florchinger [Stochastic Anal. Appl. 12, No. 4, 473-480 (1994; Zbl 0810.60051)] and R. Chabour and M. Oumoun [ibid. 16, No. 1, 43-50 (1998; Zbl 0897.93052)]. The technique used in the cited papers is based on the stochastic Lyapunov analysis and the stochastic version of La Salle’s invariance principle. However, in the proofs of the main results in these papers, all of the information given by applying Itô’s formula, when using the stochastic La Salle theorem, has not been used. The aim of this paper is to take this fact into account to improve the stabilizability conditions stated in these papers in order to be able to design stabilizers for a wider class of stochastic differential systems. The obtained results are applied to a working example which cannot be stabilized by using the results proved in the mentioned two papers.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93C10 Nonlinear systems in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D15 Stabilization of systems by feedback
93E15 Stochastic stability in control theory
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