Spectral theory for the failure of linear control in a nonlinear stochastic system

Abstract

We consider the failure of localized control in a nonlinear spatially extended system caused by extremely small amounts of noise. It is shown that this failure occurs as a result of a nonlinear instability. Nonlinear instabilities can occur in systems described by linearly stable but strongly nonnormal evolution operators. In spatially extended systems the nonnormality manifests itself in two different but complementary ways: transient amplification and spectral focusing of disturbances. We show that temporal and spatial aspects of the nonnormality and the type of nonlinearity are all crucially important to understanding and describing the mechanism of nonlinear instability. Presented results are expected to apply equally to other physical systems where strong nonnormality is due to the presence of mean flow rather than the action of control.