  In this investigation we use a simple model of the dynamics of an inviscid
vortex sheet given by the Birkhoff-Rott equation to obtain fundamental insights
about the potential for stabilization of shear layers using feedback control.
As actuation we consider two arrays of point sinks/sources located a certain
distance above and below the vortex sheet and subject to the constraint that
their mass fluxes separately add up to zero. First, we demonstrate using
analytical computations that the Birkhoff-Rott equation linearized around the
flat-sheet configuration is in fact controllable when the number of actuator
pairs is sufficiently large relative to the number of discrete degrees of
freedom present in the system, a result valid for generic actuator locations.
Next we design a state-based LQR stabilization strategy where the key
difficulty is the numerical solution of the Riccati equation in the presence of
severe ill-conditioning resulting from the properties of the Birkhoff-Rott
equation and the chosen form of actuation, an issue which is overcome by
performing computations with a suitably increased arithmetic precision.
Analysis of the linear closed-loop system reveals exponential decay of the
perturbation energy and of the corresponding actuation energy in all cases.
Computations performed for the nonlinear closed-loop system demonstrate that
initial perturbations of nonnegligible amplitude can be effectively stabilized
when a sufficient number of actuators is used. We also thoroughly analyze the
sensitivity of the closed-loop stabilization strategies to the variation of a
number of key parameters. Subject to the known limitations of inviscid vortex
models, our findings indicate that, in principle, it may be possible to
stabilize shear layers for relatively large initial perturbations, provided the
actuation has sufficiently many degrees of freedom.
