No abstract

No 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.

We consider stochastic difference equationwhere functions f and g are nonlinear and bounded, random variables ξ i are independent and h > 0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x n ≡ 0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x n is approximately polynomial: we find α > 0 such that x n decays faster than n −α+ε but slower than n −α−ε for any ε > 0.It also turns out that if g(x) decays faster than f (x) as x → 0, the polynomial rate of decay can be established exactly, x n n α → const. On the other hand, if the coefficient by the noise does not decay fast enough, the approximate decay rate is the best possible result.1991 Mathematics Subject Classification. 39A10, 39A11, 37H10, 34F05, 93E15.

We consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to prediction-based control. These perturbations may be multiplicativeif they arise from stochastic variation of the control parameter, or additiveif they reflect the presence of systemic noise.We begin by relaxing the control parameter in the deterministic equation, and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval. Then, by allowing the variation to be stochastic, we derive sufficient conditions (less restrictive than known ones for the unperturbed equation) under which the positive equilibrium will be globally a.s. asymptotically stable: i.e. the presence of noise improves the known effectiveness of prediction-based control. Finally, we show that systemic noise has a "blurring" effect on the positive equilibrium, which can be made arbitrarily small by controlling the noise intensity. Numerical examples illustrate our results.