Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow
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We consider the problem of generating and tracking a trajectory between two arbitrary parabolic profiles of a periodic 2D channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our procedure consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. Using a backstepping method, we then design boundary control laws guaranteeing that the error between the state and the trajectory decays exponentially in $L^2$, $H^1$, and $H^2$ norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear Navier-Stokes system.
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Cites
Partial Differential Equations
Chern, S.-S. (1988) · Lecture notes in mathematics · doi:10.1007/bfb0082920 · Cited by 23,546 · see citing works
Garreau, G. A., & Rudin, W. (1987) · Journal of the Royal Statistical Society Series D (The Statistician) · doi:10.2307/2348852 · Cited by 10,941 · see citing works
Nonlinear and adaptive control design
Krstić, M., Kokotović, P. V., & Kanellakopoulos, I. (1995) · Cited by 10,429 · see citing works
Navier Stokes Equations: Theory and Numerical Analysis
Témam, R., & CHORIN, A. J. (1978) · Journal of Applied Mechanics · doi:10.1115/1.3424338 · Cited by 4,028 · see citing works
(1992) · Acta Applicandae Mathematicae · doi:10.1007/bf00047090 · Cited by 3,072 · see citing works
Stability and Transition in Shear Flows
Schmid, P. J., & Henningson, D. S. (2001) · Applied Mathematical Sciences · doi:10.1007/978-1-4613-0185-1 · Cited by 2,610 · see citing works
Partial Differential Equations
Abbott, S., & Evans, L. C. (1999) · The Mathematical Gazette · doi:10.2307/3618751 · Cited by 1,429 · see citing works
An Introduction to Fluid Mechanics
Batchelor, G. K., & Young, A. D. (1968) · Journal of Applied Mechanics · doi:10.1115/1.3601282 · Cited by 1,012 · see citing works
Closed-Form Boundary State Feedbacks for a Class of 1-D Partial Integro-Differential Equations
Smyshlyaev, A., & Krstić, M. (2004) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2004.838495 · Cited by 595 · see citing works
Backstepping observers for a class of parabolic PDEs
Smyshlyaev, A., & Krstić, M. (2004) · Systems & Control Letters · doi:10.1016/j.sysconle.2004.11.001 · Cited by 469 · see citing works
Exact controllability theorems for linear parabolic equations in one space dimension
Fattorini, H. O., & Russell, D. L. (1971) · Archive for Rational Mechanics and Analysis · doi:10.1007/bf00250466 · Cited by 446 · see citing works
Pseudospectra of the Orr–Sommerfeld Operator
Reddy, S. C., Schmid, P. J., & Henningson, D. S. (1993) · SIAM Journal on Applied Mathematics · doi:10.1137/0153002 · Cited by 434 · see citing works
Introduction to Fluid Mechanics
(1998) · Elsevier eBooks · doi:10.1016/b978-0-340-67649-3.x5000-3 · Cited by 317 · see citing works
Motion planning for the heat equation
Laroche, B., Martin, P., & Rouchon, P. (2000) · International Journal of Robust and Nonlinear Control · doi:10.1002/1099-1239(20000715)10:8 · Cited by 222 · see citing works
On control design for PDEs with space-dependent diffusivity or time-dependent reactivity
Smyshlyaev, A., & Krstić, M. (2005) · Automatica · doi:10.1016/j.automatica.2005.04.006 · Cited by 177 · see citing works
Linear feedback control and estimation of transition in plane channel flow
HGBERG, M., Bewley, T., & Henningson, D. S. (2003) · Journal of Fluid Mechanics · doi:10.1017/s0022112003003823 · Cited by 174 · see citing works
Global Steady-State Controllability of One-Dimensional Semilinear Heat Equations
Coron, J., & Trélat, E. (2004) · SIAM Journal on Control and Optimization · doi:10.1137/s036301290342471x · Cited by 169 · see citing works
Self-citationControllability of a quantum particle in a moving potential well
Beauchard, K., & Coron, J. (2005) · Journal of Functional Analysis · doi:10.1016/j.jfa.2005.03.021 · Cited by 162 · see citing works
Self-citationLocal controllability of a 1-D Schrödinger equation
Beauchard, K. (2005) · Journal de Mathématiques Pures et Appliquées · doi:10.1016/j.matpur.2005.02.005 · Cited by 161 · see citing works
Fursikov, A. V., Gunzburger, M., & Hou, L. (1998) · SIAM Journal on Control and Optimization · doi:10.1137/s0363012994273374 · Cited by 161 · see citing works
Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations
Raymond, J. (2006) · SIAM Journal on Control and Optimization · doi:10.1137/050628726 · Cited by 156 · see citing works
Tangential boundary stabilization of Navier-Stokes equations
Barbu, V., Lasiecka, I., & Triggiani, R. (2006) · Memoirs of the American Mathematical Society · doi:10.1090/memo/0852 · Cited by 123 · see citing works
$H^\infty$ Feedback Boundary Stabilization of the Two-Dimensional Navier–Stokes Equations
Dharmatti, S., Raymond, J., & Thevenet, L. (2011) · SIAM Journal on Control and Optimization · doi:10.1137/100782607 · Cited by 119 · see citing works
The Velocity Tracking Problem for Navier--Stokes Flows With Boundary Control
Gunzburger, M., & Manservisi, S. (2000) · SIAM Journal on Control and Optimization · doi:10.1137/s0363012999353771 · Cited by 107 · see citing works
Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations
Raymond, J.-P. (2007) · Journal de Mathématiques Pures et Appliquées · doi:10.1016/j.matpur.2007.04.002 · Cited by 103 · see citing works
State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows
Hœpffner, J., Chevalier, M., Bewley, T., & Henningson, D. S. (2005) · Journal of Fluid Mechanics · doi:10.1017/s0022112005004210 · Cited by 98 · see citing works
Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
Coron, J. (2002) · ESAIM Control Optimisation and Calculus of Variations · doi:10.1051/cocv:2002050 · Cited by 96 · see citing works
Self-citationStabilization for the 3D Navier-Stokes system by feedback boundary control
Fursikov, A. V. (2004) · Discrete and Continuous Dynamical Systems · doi:10.3934/dcds.2004.10.289 · Cited by 94 · see citing works
Coron, J. (1999) · SIAM Journal on Control and Optimization · doi:10.1137/s036301299834140x · Cited by 92 · see citing works
Self-citationStabilizability of Two-Dimensional Navier—Stokes Equations with Help of a Boundary Feedback Control
Fursikov, A. V. (2001) · Journal of Mathematical Fluid Mechanics · doi:10.1007/pl00000972 · Cited by 92 · see citing works
GLOBAL STEADY-STATE STABILIZATION AND CONTROLLABILITY OF 1D SEMILINEAR WAVE EQUATIONS
Coron, J., & Trélat, E. (2006) · Communications in Contemporary Mathematics · doi:10.1142/s0219199706002209 · Cited by 90 · see citing works
Self-citationTwo results on exact boundary control of parabolic equations
Seidman, T. I. (1984) · Applied Mathematics & Optimization · doi:10.1007/bf01442174 · Cited by 82 · see citing works
Flow Control by Feedback: Stabilization and Mixing
Aamo, O., Krstić, M., & W Shyy (2003) · Applied Mechanics Reviews · doi:10.1115/1.1584420 · Cited by 80 · see citing works
The instability of oscillatory plane Poiseuille flow
von Kerczek, C. H. (1982) · Journal of Fluid Mechanics · doi:10.1017/s002211208200038x · Cited by 79 · see citing works
Vázquez, R., & Krstić, M. (2007) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2007.910686 · Cited by 77 · see citing works
Self-citationStability enhancement by boundary control in 2-D channel flow
Balogh, A., Liu, W., & Krstić, M. (2001) · IEEE Transactions on Automatic Control · doi:10.1109/9.964681 · Cited by 77 · see citing works
COLTON, D. L. (1977) · Journal of Differential Equations · doi:10.1016/0022-0396(77)90189-9 · Cited by 54 · see citing works
Vázquez, R., & Krstić, M. (2006) · Systems & Control Letters · doi:10.1016/j.sysconle.2005.09.019 · Cited by 43 · see citing works
Self-citationStabilization of a plane channel flow by wall normal controllers
Barbu, V. (2006) · Nonlinear Analysis · doi:10.1016/j.na.2006.09.024 · Cited by 33 · see citing works
Vázquez, R., & Krstić, M. (2006) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2005.1583349 · Cited by 31 · see citing works
Self-citationBackstepping boundary control of Navier-Stokes channel flow: a 3D extension
Cochran, J., Vázquez, R., & Krstić, M. (2006) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2006.1655449 · Cited by 27 · see citing works
Self-citationA Closed-Form Observer for the Channel Flow Navier-Stokes System
Vázquez, R., & Krstić, M. (2006) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc.2005.1583115 · Cited by 22 · see citing works
Self-citationGlobal steady-state controllability of 1-D semilinear heat equations
Coron, J., & Trélat, E. (2004) · HAL (Le Centre pour la Communication Scientifique Directe) · Cited by 22 · see citing works
Self-citationTriggiani, R. (2007) · Discrete and Continuous Dynamical Systems - B · doi:10.3934/dcdsb.2007.8.279 · Cited by 19 · see citing works
Glass, O. (2005) · SIAM Journal on Control and Optimization · doi:10.1137/s0363012903431153 · Cited by 19 · see citing works
Controllability of couette flows
Schmidt, M., & Trélat, E. (2006) · Communications on Pure & Applied Analysis · doi:10.3934/cpaa.2006.5.201 · Cited by 17 · see citing works
Self-citationNonexistence for a boundary value problem arising in parabolic theory
Kannai, Y. (1990) · Israel Journal of Mathematics · doi:10.1007/bf02773753 · Cited by 4 · see citing works
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- Authors
- Vázquez, R., Trélat, E., & Coron, J.-M.
- Affiliations
- University of California System (US)
- Laboratoire de Mathématiques Analyse, Probabilités, Modélisation Orléans (FR)
- Laboratoire de Mathématiques d'Orsay (FR)
- Venue
- Discrete and Continuous Dynamical Systems - B
- Published
- 2008
- Volume
- 10
- Issue
- 4
- Pages
- 925-956
- ISSN
- 1531-3492 (ISSN-L)
- 1553-524X
- OpenAlex
- W1979079822
- Cited by
- 58
- References
- 47
- Citations by year
- 2026: 1 · 2025: 6 · 2024: 2 · 2023: 3 · 2022: 3 · 2021: 2 · 2020: 4 · 2019: 7 · 2018: 4 · 2017: 2 · 2016: 10 · 2015: 5 · 2014: 1 · 2013: 1 · 2012: 2
- FWCI
- 3.08
- Type
- journal article
- Language
- en
- Open access
- green
- hal.science
- Licence
- other-oa
- Locations
- Discrete and Continuous Dynamical Systems - B (publishedVersion)
- HAL (submittedVersion, other-oa)
- idUS (Universidad de Sevilla) (submittedVersion, cc-by-nc-nd)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Stability and Controllability of Differential Equations
- Physical Sciences › Mathematics › Applied Mathematics › Navier-Stokes equation solutions
- Physical Sciences › Engineering › Computational Mechanics › Fluid Dynamics and Turbulent Flows
- Keywords
- laminar-turbulent transition, Poiseuille flow, channel flow, backstepping control, boundary control, trajectory tracking, linear instability, high Reynolds number, Stokes equations, Navier-Stokes equations, exponential stability
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