Fredholm Backstepping with Heymann Pre-Feedback for Linearized Navier--Stokes 2D Channel

Abstract: We stabilize the linearized 2-D Poiseuille channel flow at arbitrary Reynolds numbers, and assign the decay of every wavenumber to any user-prescribed rate. The actuation is as in Vazquez--Krstic (IEEE TAC, 2007), VK'07 below: the two velocity components at only one wall. This is the first Fredholm backstepping design for a Navier--Stokes system. In VK'07 the wall-normal input renders the plant spatially causal, after which the tangential input closes a Volterra backstepping loop. Here the normal input carries a finite-rank feedback of arbitrarily small gain, whose sole role, in the sense of Heymann's lemma (TAC, 1968), is to give the plant simple spectrum and make every mode controllable from the tangential velocity. The tangential input then applies a stabilizing feedback, obtained through an invertible Fredholm transform to a Stokes target PDE, shifted left by an arbitrary decay. No simplicity of eigenvalues of the Orr--Sommerfeld operator is assumed and the exclusions of plant coefficient values that have accompanied Fredholm backstepping are removed. We state the comparison with the Volterra design without embellishment: the Fredholm architecture is powerful, but its gain is spectral where VK'07 is closed-form, the simpler of the two, and one from which the Fredholm approach takes its model, its actuation, and its target. What is demonstrated here is the reach of the Fredholm method on its most demanding plant to date. The feedback is tested at Reynolds number $30000$, with the dominant part of the closed-loop spectrum assigned so that it amounts to slowing the flow about twentyfold.