A Jurdjevic-Quinn Theorem for Stochastic Nonlinear Systems

Abstract

In this paper, we provide a formula for a stabilizing feedback law for control stochastic differential equations. This result extends Jurdjevic-Quinn's theorem to nonlinear control systems corrupted by noise.

On a universal formula for the stabilization of control stochastic nonlinear systems

“…The goal is to provide explicit feedback control law that globally asymptotically stabilizes in probability a given nonlinear stochastic system, under the assumption that a 'control-Lyapunov function' is known. The result represents a continuation of a line of work started in [3], that concerns stabilization in probability of c.ontro1 stochastic systems which are affine in control, where every thing is corrupted by a noise. In Bilinear stochastic case, Chabour and Florchinger have study the stabilizability in probability of a class of this system in [4].…”

A Stochastic Jurdjevic--Quinn Theorem

“…In fact, it is proved in [2] that the stabilizer given in [3] for the deterministic part of the system remains valid in the stochastic context provided the system coefficients satisfy a rank condition, which can easily be deduced from that stated in [7]. An extension of this result has been obtained by Chabour and Oumoun in [1], where it is proved that under the stabilizability condition provided in [2] one can compute stabilizing state feedback laws for stochastic differential systems in the form…”

“…However, in the proofs of the main results in [2] and [1], 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 [2] and [1] in order to be able to design stabilizers for a wider class of stochastic differential systems than that considered in [2] and [1]. This paper is divided into four sections and is organized as follows.…”

“…The aim of this paper is to take this fact into account to improve the stabilizability conditions stated in [2] and [1] in order to be able to design stabilizers for a wider class of stochastic differential systems than that considered in [2] and [1]. In section four, we apply the result proved in section three to a working example which cannot be stabilized by using the results proved in [2] and [1]. In section one, we recall some basic facts on the asymptotic stability in probability for the equilibrium solution of a stochastic differential equation.…”

“…Then the stochastic process solution x t of the stochastic differential equation (1) tends to the largest invariant set whose support is contained in the locus LV (x t ) = 0 for any t ≥ 0 with probability 1. Then the stochastic process solution x t of the stochastic differential equation (1) tends to the largest invariant set whose support is contained in the locus LV (x t ) = 0 for any t ≥ 0 with probability 1.…”