Stabilization and Destabilization of Nonlinear Differential Equations by Noise

Abstract

This paper considers the stabilisation and destabilisation by a Brownian noise perturbation which preserves the equilibrium of the ordinary differential equation x ′ (t) = f (x(t)). In an extension of earlier work, we lift the restriction that f obeys a global linear bound, and show that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g(X(t)) dB(t) either stabilises an unstable equilibrium, or destabilises a stable equilibrium. When the equilibrium of the deterministic equation is non-hyperbolic, we show that a non-hyperbolic perturbation suffices to change the stability properties of the solution.

The internal stabilization by noise of the linearized Navier-Stokes equation

“…However, the results obtained here are essentially stochastic not only because the stabilizable controller arises as multiplicative term of a Brownian N -dimensional motion but mainly because the asymptotic nature of stabilization results as well as the stochastic approach have no analogue in deterministic stabilization technique. As a matter of fact, it was known long time ago that one might use the multiplicative noise to stabilize differential systems (see [3]) and more recent results in this direction can be found in [1,2,[7][8][9]19]. (See also [11] for related results.)…”

“…Our purpose here is to stabilize the null solutions to (4.1) by a noise boundary controller u of the form 2) where N is, as above, the number of eigenvalues λ j of the operator A with Re λ j ≤ γ and φ i will be defined below. As in the previous case, ϕ * j are the eigenfunctions of A * corresponding to λ j (see (2.4)) and…”

Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

“…The structure of those test systems was motivated by well-established results demonstrating that multiplicative stochastic perturbations may act to stabilise or destabilise an equilibrium solution in the almost sure sense. A review of the extensive literature on the subject may be found in Appleby, Mao and Rodkina [3].…”

Stochastic suppression and stabilization of delay differential systems

“…For the nonlinear systeṁ x(t) = f 1 (x(t), t), if f 1 satisfies the local Lipschitz condition and the linear growth condition, this system may be stabilized by the Brownian noise (see [3,4]). Then Appleby and Mao [5] and Appleby et al [6] examined the stabilization of noise of the functional systemẋ(t) = f 2 (x t , t), where x t = x t ( ) := {x(t + ) :− 0} and f 2 : C([− , 0]; R n )×R + → R n and satisfies the onesided linear growth condition, which may cover more equations. For the detailed understanding of stabilization, [7] is a good reference.…”