The decay of stabilizability with Reynolds number in a linear model of spatially developing flows

This article characterizes the gradual decay of stabilizability with Reynolds number in the linear complex Ginzburg-Landau model of spatially developing flow systems when a single spatially-localized actuator is used to apply the control forcing. It is shown that, technically, the system considered is linearly stabilizable for all actuator locations at any Reynolds number. However, as the Reynolds number is increased and an increasing number of modes of the open-loop system become unstable, the,control authority on some of these open-loop unstable modes is found to be exponentially small. Using finite-precision arithmetic and any given numerical method for computing the feedback gains, an effective upper bound on the Reynolds number is reached, above which it is not possible to compute a linearly stabilizing control algorithm. This 'effective upper bound', however, is not a fundamental characteristic of the system; rather, it is a persistent artefact of the numerical precision used in the controller calculation. The most suitable location for the actuator as the Reynolds number is increased is well predicted by analysis of the domain of support of the open-loop adjoint eigenfunctions. Further understanding is provided by analysis of the closed-loop system eigenfunctions, which are shown to become increasingly non-normal as the Reynolds number is increased.