A Closed-Form Feedback Controller for Stabilization of Linearized Navier-Stokes Equations: The 2D Poisseuille Flow

Abstract

We present a formula for a boundary control law which stabilizes the parabolic profile of an infinite channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poisseuille flow, this problem is frequently cited as a paradigm for transition to turbulence, whose stabilization for arbitrary Reynolds numbers, without using discretization, has so far been an open problem. Our result achieves exponential stability in the L 2 norm for the linearized Navier-Stokes equations, guaranteeing local stability for the fully nonlinear system. Explicit solutions are obtained for the closed loop system. This is the first time explicit formulae are produced for solutions of the Navier-Stokes equations. The result is presented for the 2D case for clarity of exposition. An extension to 3D is available and will be presented in a future publication.

Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

“…A 3D channel flow, periodic in two directions, is also tractable, adding some refinements which include actuation of the spanwise velocity at the wall (see [9] for the new techniques and difficulties involved). All cited references consider the steady problem of stabilizing a given Poiseuille profile, therefore some modifications to account for the unsteady coefficients have to be done, in the same way the present paper extends the results of [42].…”

Control of PDE–ODE cascades with Neumann interconnections

“…The controller for the system (15)- (18) is given explicitly in [8], however, we do not repeat it here because of its complexity. The difference between the systems (10)- (13) and (15)-(18) may appear subtle, because it is only a difference of one time derivative between the heat equation (11) and the wave equation (16).…”

“…We provide infinite-dimensional full-state feedback laws with explicit gain kernels that compensate the PDE dynamics and achieve stabilization of the PDE-ODE system. The key tool in this work is the continuum version of backstepping method [1,2,4,5,11,16,18] which employs infinite-dimensional transformations for the design of the controller and Lyapunov functions for the stability proof.…”

Stable Poiseuille flow transfer for a Navier-Stokes system

“…This is a problem of practical interest which, to the best of our knowledge, has not been solved or even been considered A possible solution for the problem would be to apply quasi-static deformation theory; this would require to modify the pressure gradient very slowly, and simultaneously gain-schedule a fixed Reynold number boundary controller like [17] for tracking a (slowly) time varying trajectory, which in general would not be an exact solution of the system. This idea has been already used for moving between equilibria of a nonlinear parabolic equation [4], or a wave equation [5].…”

“…The problem of locally stabilizing the equilibrium has been solved by means of optimal control [8], and backstepping [17]. Observers have been developed using dual methods [18].…”