A Stochastic Jurdjevic--Quinn Theorem

Abstract

The purpose of this paper is to state sufficient conditions for the stabilizability of stochastic differential systems when both the drift and diffusion terms are affine in the control. This result extends to stochastic differential systems the well-known theorem of Jurdjevic-Quinn [J.

Time‐Varying Stabilizers for Stochastic Systems with no Unforced Dynamics

“…When for every t ≥0, one has, as noticed above, u 0 ( t , x t )= α ( t , x t ) for every t ≥0 and u j ( t , x t )=0 for every t ≥0 and j ∈{1,…, p } and therefore, the stochastic differential system reduces to the ordinary differential equation As a consequence, when using Itô's formula in the iterative procedure described in the above computations we do not get additional terms like those obtained in the proof of Theorem 3.1 in which could be permitted to assume a more general rank condition than H1 .…”

Stabilization of nonlinear stochastic systems without unforced dynamics via time--varying feedback

“…By using the techniques developed in [9] to obtain a stochastic version of the JurdjevicQuinn theorem with the approach used by Pomet [16] and Lin [15] to design time-varying stabilizers for deterministic driftless controllable systems, we propose a constructive method to design a stabilizing time-varying feedback law provided some rank condition involving the drift coefficients is satisfied. The main tools used in this paper are the stochastic Lyapunov stability theory introduced by Khasminskii in [13] combined with the stochastic La Salle invariance principle proved by Kushner [14] and the bounded feedback design technique for passive stochastic differential systems developed in [10].…”