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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, Ashgate Publishing Company, Brookfield, USA, 1998. Google Scholar

[2]

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

[3]

H. Arfaoui, "Contrôle et Stabilisation des Équations de Saint-Venant," Ph.D thesis, Université Tunis El Manar, Université Paul Sabatier Toulouse, 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, Boston. London. Melbourne, 1980. Google Scholar

[6]

A. V. Fursikov, Stabilizability of quasilinear parabolic equation by by means of feedback boundary control, Sbornik Mathematics, 192 (2001), 593-639. 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), 259-301. 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 "Nonlinear Problems in Mathematical Physics and Related Topics II, In Honor of Professor O. A. Ladyzhenskaya," Kluwer/Plenum Publishers, New-York, Boston, Dordrecht, London, Moscow, (2002), 137-177.  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), 89-102.  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), 1365-1376. 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), 761-789. 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), 337-359. 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), 16-28. 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, Université Paul Sabatier Toulouse, 1998. Google Scholar

[15]

M. Renardy, Are viscoelastic flows under control or out of control?, Systems & Control Letters, 54 (2005), 1183-1193. 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), 118-136. 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, Ashgate Publishing Company, Brookfield, USA, 1998. Google Scholar

[2]

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

[3]

H. Arfaoui, "Contrôle et Stabilisation des Équations de Saint-Venant," Ph.D thesis, Université Tunis El Manar, Université Paul Sabatier Toulouse, 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, Boston. London. Melbourne, 1980. Google Scholar

[6]

A. V. Fursikov, Stabilizability of quasilinear parabolic equation by by means of feedback boundary control, Sbornik Mathematics, 192 (2001), 593-639. 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), 259-301. 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 "Nonlinear Problems in Mathematical Physics and Related Topics II, In Honor of Professor O. A. Ladyzhenskaya," Kluwer/Plenum Publishers, New-York, Boston, Dordrecht, London, Moscow, (2002), 137-177.  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), 89-102.  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), 1365-1376. 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), 761-789. 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), 337-359. 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), 16-28. 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, Université Paul Sabatier Toulouse, 1998. Google Scholar

[15]

M. Renardy, Are viscoelastic flows under control or out of control?, Systems & Control Letters, 54 (2005), 1183-1193. 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), 118-136. doi: 10.1016/0022-0396(87)90188-4.  Google Scholar

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