American Institute of Mathematical Sciences

Advanced
December  2011, 15(3): 491-511. doi: 10.3934/dcdsb.2011.15.491

Boundary stabilizability of the linearized viscous Saint-Venant system

Hassen Arfaoui 1, , Faker Ben Belgacem 2, , Henda El Fekih 1, and  Jean-Pierre Raymond 3,

1. 

LAMSIN, Ecole Nationale d'Ingénieurs de Tunis, B.P. 37, 1002 Tunis Le Belvédère, Tunisia, Tunisia
2. 

Université de Technologie de Compiègne, BP 20529, 60205 Compiègne cedex, France
3. 

Université de Toulouse & CNRS, Institut de Mathématiques, UMR 5219, 31062 Toulouse Cedex 9

Received  March 2009 Revised  May 2010 Published  February 2011

We consider a shallow water flow in a channel modeled by the Saint-Venant equations with a viscous term. We are interested in the stabilization of the flow at a steady state. We establish that the semi-group of the linearized system is exponentially stable. However when the convection coefficient is dominant, the natural stabilization turns out to be very slow. One way to enhance the stabilization of the system is to use boundary controls by means of a moving device located at the extremities of the channel. We determine, by an extension method due to Fursikov, boundary Dirichlet controls able to accelerate the stabilization of the flow. Numerical experiments illustrate the efficiency of the control.
Keywords: Saint-Venant equations, linear stability, stabilizability., Viscous shallow water model, boundary control.
Mathematics Subject Classification: 76D55, 93C2.
Citation: Hassen Arfaoui, Faker Ben Belgacem, Henda El Fekih, Jean-Pierre Raymond. Boundary stabilizability of the linearized viscous Saint-Venant system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 491-511. doi: 10.3934/dcdsb.2011.15.491
References:
[1]

M. B. Abbott and A. W. Minns, "Computational Hydraulics,", 2nd edition, (1998).   Google Scholar

[2]

S. N. Antontsev, A. V. Kazhikov and V. N. Monakhov, "Boundary Values Problems In Mechanics Of Nonhomogeneous Fluids,", North-Holland, (1990).   Google Scholar

[3]

H. Arfaoui, "Contrôle et Stabilisation des Équations de Saint-Venant,", Ph.D thesis, (2006).   Google Scholar

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," Vol. 2,, Birkhäuser, (1993).   Google Scholar

[5]

J. A. Cunge, F. M. Holly and Jr. A. Verwey, "Practical Aspects of Computational River Hydraulics,", Pitman Advanced Publishing Program, (1980).   Google Scholar

[6]

A. V. Fursikov, Stabilizability of quasilinear parabolic equation by by means of feedback boundary control,, Sbornik Mathematics, 192 (2001), 593.  doi: 10.1070/SM2001v192n04ABEH000560.  Google Scholar

[7]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control,, J. Math. Fluid Mech., 3 (2001), 259.  doi: 10.1007/PL00000972.  Google Scholar

[8]

A. V. Fursikov, Real Processes and Realizability of a Stabilization Method for Navier-Stokes Equations by Boundary Feedback Control,, in, (2002), 137.   Google Scholar

[9]

J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation,, Discrete Continuous Dynam. Systems - B, 1 (2001), 89.   Google Scholar

[10]

J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin, Boundary feedback control in networks of open channels,, Automatica, 39:8 (2003), 1365.  doi: 10.1016/S0005-1098(03)00109-2.  Google Scholar

[11]

Z. Huang, S. Jin, P. A. Markowich, C. Sparber and C. Zheng, A Time-Splitting spectral Scheme for the Dirac-Maxwell System,, J. Comp. Physics, 208:2 (2005), 761.  doi: 10.1016/j.jcp.2005.02.026.  Google Scholar

[12]

L. A. Khan and P. L.-F. Liu, Numerical analyses of operator-splitting algorithms for the two-dimensional advection-diffusion equation,, Comput. Meth. Appl. Mech. Engng., 152 (1998), 337.  doi: 10.1016/S0045-7825(97)00127-8.  Google Scholar

[13]

D. A. Lyn and P. Goodwin, Stability of a general Preissmann scheme,, Journal of Hydraulic engineering, 113 (1987), 16.  doi: 10.1061/(ASCE)0733-9429(1987)113:1(16).  Google Scholar

[14]

G. Mathieu, "Étude et Contrôle des Équations de la Théorie 'Shallow Water' en Dimension un,", Ph.D thesis, (1998).   Google Scholar

[15]

M. Renardy, Are viscoelastic flows under control or out of control?,, Systems & Control Letters, 54 (2005), 1183.  doi: 10.1016/j.sysconle.2005.04.006.  Google Scholar

[16]

J. J. Stoker, "Water Waves, the Mathematical Theory with Applications," Pure and Applied Mathematics,, Vol. \textbf{IV}, IV ().   Google Scholar

[17]

D. H. Wagner, Equivalence of the Euler and Lagrangian equations of a gas dynamics for weak solutions,, Journal of Differential Equations, 68 (1987), 118.  doi: 10.1016/0022-0396(87)90188-4.  Google Scholar

show all references

References:
[1]

M. B. Abbott and A. W. Minns, "Computational Hydraulics,", 2nd edition, (1998).   Google Scholar

[2]

S. N. Antontsev, A. V. Kazhikov and V. N. Monakhov, "Boundary Values Problems In Mechanics Of Nonhomogeneous Fluids,", North-Holland, (1990).   Google Scholar

[3]

H. Arfaoui, "Contrôle et Stabilisation des Équations de Saint-Venant,", Ph.D thesis, (2006).   Google Scholar

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," Vol. 2,, Birkhäuser, (1993).   Google Scholar

[5]

J. A. Cunge, F. M. Holly and Jr. A. Verwey, "Practical Aspects of Computational River Hydraulics,", Pitman Advanced Publishing Program, (1980).   Google Scholar

[6]

A. V. Fursikov, Stabilizability of quasilinear parabolic equation by by means of feedback boundary control,, Sbornik Mathematics, 192 (2001), 593.  doi: 10.1070/SM2001v192n04ABEH000560.  Google Scholar

[7]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control,, J. Math. Fluid Mech., 3 (2001), 259.  doi: 10.1007/PL00000972.  Google Scholar

[8]

A. V. Fursikov, Real Processes and Realizability of a Stabilization Method for Navier-Stokes Equations by Boundary Feedback Control,, in, (2002), 137.   Google Scholar

[9]

J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation,, Discrete Continuous Dynam. Systems - B, 1 (2001), 89.   Google Scholar

[10]

J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin, Boundary feedback control in networks of open channels,, Automatica, 39:8 (2003), 1365.  doi: 10.1016/S0005-1098(03)00109-2.  Google Scholar

[11]

Z. Huang, S. Jin, P. A. Markowich, C. Sparber and C. Zheng, A Time-Splitting spectral Scheme for the Dirac-Maxwell System,, J. Comp. Physics, 208:2 (2005), 761.  doi: 10.1016/j.jcp.2005.02.026.  Google Scholar

[12]

L. A. Khan and P. L.-F. Liu, Numerical analyses of operator-splitting algorithms for the two-dimensional advection-diffusion equation,, Comput. Meth. Appl. Mech. Engng., 152 (1998), 337.  doi: 10.1016/S0045-7825(97)00127-8.  Google Scholar

[13]

D. A. Lyn and P. Goodwin, Stability of a general Preissmann scheme,, Journal of Hydraulic engineering, 113 (1987), 16.  doi: 10.1061/(ASCE)0733-9429(1987)113:1(16).  Google Scholar

[14]

G. Mathieu, "Étude et Contrôle des Équations de la Théorie 'Shallow Water' en Dimension un,", Ph.D thesis, (1998).   Google Scholar

[15]

M. Renardy, Are viscoelastic flows under control or out of control?,, Systems & Control Letters, 54 (2005), 1183.  doi: 10.1016/j.sysconle.2005.04.006.  Google Scholar

[16]

J. J. Stoker, "Water Waves, the Mathematical Theory with Applications," Pure and Applied Mathematics,, Vol. \textbf{IV}, IV ().   Google Scholar

[17]

D. H. Wagner, Equivalence of the Euler and Lagrangian equations of a gas dynamics for weak solutions,, Journal of Differential Equations, 68 (1987), 118.  doi: 10.1016/0022-0396(87)90188-4.  Google Scholar

[1]

Jean-Frédéric Gerbeau, Benoit Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 89-102. doi: 10.3934/dcdsb.2001.1.89
[2]

Marie-Odile Bristeau, Jacques Sainte-Marie. Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 733-759. doi: 10.3934/dcdsb.2008.10.733
[3]

Georges Bastin, Jean-Michel Coron, Brigitte d'Andréa-Novel. On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks & Heterogeneous Media, 2009, 4 (2) : 177-187. doi: 10.3934/nhm.2009.4.177
[4]

E. Audusse. A multilayer Saint-Venant model: Derivation and numerical validation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 189-214. doi: 10.3934/dcdsb.2005.5.189
[5]

Emmanuel Audusse, Fayssal Benkhaldoun, Jacques Sainte-Marie, Mohammed Seaid. Multilayer Saint-Venant equations over movable beds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 917-934. doi: 10.3934/dcdsb.2011.15.917
[6]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[7]

Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020097
[8]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[9]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375
[10]

Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks & Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145
[11]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020230
[12]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355
[13]

Marta Strani. Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1653-1667. doi: 10.3934/cpaa.2014.13.1653
[14]

Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69
[15]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[16]

Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799
[17]

Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, 2019, 6 (1) : 1-37. doi: 10.3934/jcd.2019001
[18]

Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
[19]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[20]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences