Stabilization of linear systems by noise: Application to flow induced oscillations

Abstract

In a recent paper, Popp and Romberg reported on stabilization by grid generated turbulence of a smooth circular cylinder immersed in the wake from an identical cylinder in an array of aluminium tubes. Although these results were obtained experimentally, so far they have not been explained from a stochastic point of view on a rigorous theoretical basis. This paper provides analytical results which may explain this stabilization phenomenon by modelling the immersed cylinder as a two degree of freedom oscillator and the turbulence as a stochastic process. We obtain general asymptotic approximation for the moment Lyapunov exponent, g…p †, and the Lyapunov exponent, ¶, for a four-dimensional system with one critical mode and another asymptotically stable mode driven by a small intensity stochastic process. These results, pertaining to pth moment stability and almost-sure stability, explain how the stochastic components that couple the stable and the critical modes play an important role in determining whether a noisy excitation can stabilize or destabilize the oscillatory critical mode. They are then applied to a prototypical¯ow induced oscillation model to justify the experimental results.

Perturbations of linear delay differential equations at the verge of instability

“…7 note that the solution is similar to (33). 8 Note that sign(ς 1 ) = sign( c+c cc ) = sign( 1 c + 1 c ).…”

“…Delay equations with noise perturbations as considered in section 6 display interesting similarities with non-delay systems. For example, [33] considers coupled oscillators with one of the oscillators stable, in the following form. Let J be the symplectic matrix 0 1 −1 0 , I be the 2 × 2 identity matrix and O be the 2 × 2 zero matrix.…”

“…The oscillator with frequency ω 1 is coupled to the stable oscillator of frequency ω 2 . [33] shows that the Lyapunov exponent of the above system can be written in terms of quantities analogous to R 0 , R 2c , Ri defined in section 6.1. Further they show that both stabilization and destabilization are possible depending on the matrix coefficients K, M and N .…”

Simultaneous hollowisation, joint numerical range, and stabilization by noise

“…Since the pioneering work [2] these phenomena have received ongoing attention, with current interest e.g. in stochastic partial differential equations or Hamiltonian systems, [33,7,20]. Our new contribution concerns simultaneous stabilization by noise and features a new method of proof, which relies on an orthogonal transformation of matrices to hollow form.…”