Stabilisation and destabilisation of nonlinear differential equations by noise

John A. D. Appleby, Xuerong Mao, Alexandra Rodkina

Research output: Contribution to journalArticlepeer-review

123 Citations (Scopus)
93 Downloads (Pure)


This paper considers the stabilisation and destabilisa- tion by a Brownian noise perturbation which preserves the equilibrium of the ordinary dierential equation x0(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. .
Original languageEnglish
Pages (from-to)683-691
Number of pages9
JournalIEEE Transactions on Automatic Control
Issue number3
Publication statusPublished - Apr 2008


  • brownian motion
  • almost sure asymptotic stability
  • It^o's formula
  • stabilisation
  • destabilisation
  • control systems

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