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

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 L2norm 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.

[1]  K. W. Cattermole The Fourier Transform and its Applications , 1965 .

[2]  Ronald N. Bracewell,et al.  The Fourier Transform and Its Applications , 1966 .

[3]  W. Rudin Real and complex analysis , 1968 .

[4]  W. Rudin Real and complex analysis, 3rd ed. , 1987 .

[5]  Dan S. Henningson,et al.  Pseudospectra of the Orr-Sommerfeld Operator , 1993, SIAM J. Appl. Math..

[6]  Miroslav Krstic,et al.  Nonlinear and adaptive control de-sign , 1995 .

[7]  C. Fabre,et al.  Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems , 1996 .

[8]  J. Coron,et al.  On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions , 1996 .

[9]  O. Imanuvilov,et al.  On exact controllability for the Navier-Stokes equations , 1998 .

[10]  Miroslav Krstic,et al.  Stability enhancement by boundary control in 2D channel flow. I. Regularity of solutions , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[11]  P. Christofides,et al.  Nonlinear Control of Incompressible Fluid Flow: Application to Burgers' Equation and 2D Channel Flow☆ , 2000 .

[12]  P. Schmid,et al.  Stability and Transition in Shear Flows. By P. J. SCHMID & D. S. HENNINGSON. Springer, 2001. 556 pp. ISBN 0-387-98985-4. £ 59.50 or $79.95 , 2000, Journal of Fluid Mechanics.

[13]  Bartosz Protas,et al.  Optimal rotary control of the cylinder wake in the laminar regime , 2002 .

[14]  Fernando Paganini,et al.  Distributed control of spatially invariant systems , 2002, IEEE Trans. Autom. Control..

[15]  Viorel Barbu,et al.  Feedback stabilization of Navier–Stokes equations , 2003 .

[16]  Dan S. Henningson,et al.  Linear feedback control and estimation of transition in plane channel flow , 2003, Journal of Fluid Mechanics.

[17]  M. Krstić,et al.  Title Closed-Form Boundary State Feedbacks for a Class of 1-D Partial Integro-Differential Equations Permalink , 2004 .

[18]  M. Krstic,et al.  A Closed-Form Observer for the Channel Flow Navier-Stokes System , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[19]  M. Krstić,et al.  Backstepping observers for a class of parabolic PDEs , 2005, Syst. Control. Lett..

[20]  Miroslav Krstic,et al.  Boundary Control of the Linearized Ginzburg--Landau Model of Vortex Shedding , 2005, SIAM J. Control. Optim..

[21]  Bassam Bamieh,et al.  Componentwise energy amplification in channel flows , 2005, Journal of Fluid Mechanics.

[22]  George Haller,et al.  Closed-loop separation control: An analytic approach , 2006 .

[23]  Jean-Pierre Raymond,et al.  Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations , 2006, SIAM J. Control. Optim..

[24]  Miroslav Krstic,et al.  Explicit integral operator feedback for local stabilization of nonlinear thermal convection loop PDEs , 2006, Syst. Control. Lett..