Almost sure asymptotic stability of parabolic SPDEs with small multiplicative noise: with application to the perturbed Moore-Greitzer model

Abstract

In this paper, we investigate the almost-sure exponential asymptotic stability of the trivial solution of a parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise near the deterministic Hopf bifurcation point. We show the existence and uniqueness of the invariant measure under appropriate assumptions, and approximate the exponential growth rate via asymptotic expansion, given that the strength of the noise is small. This approximate quantity can readily serve as a robust indicator of the change of almost-sure stability. We apply the results to a simplified stochastic Moore-Greitzer PDE model in detecting the stall instabilities of modern jet-engine under the impact of multiplicative noise. A better understanding of the instability margin will eventually optimize the jet-engine operating range and thus lead to lighter and more efficient jet-engine design.