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Stabilisation of Functional Differential Equations by Noise
 

Summary: Stabilisation of Functional Differential
Equations by Noise
John A. D. Appleby 1
School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland.
Abstract
This paper extends, by an alternative method, a result of Mao (Systems and Con-
trol Letters, 1994) which shows that solutions of nonlinear differential equations
can be stabilised by noise. Here, we show that a linear multiplicative noise can
always stabilise a general finite-dimensional functional differential equation, when-
ever the delay is sufficiently small. The result is also extended to a scalar functional
differential equation with nonlinear multiplicative noise.
Key words: stochastic stabilisation, stochastic functional differential equation,
almost sure exponential stability, Liapunov exponent, top Liapunov exponent,
nonlinear system, Volterra equations
1991 MSC: 60H10, 35K20
1 Introduction
This paper is inspired by Mao's paper in this journal [1]. In it, he proves that
general finite dimensional stochastic differential equations can be stabilised or
destabilised by Brownian motion. As a consequence, one would expect that
a similar result should be true for delay differential equations, or functional

  

Source: Appleby, John - School of Mathematical Sciences, Dublin City University

 

Collections: Mathematics