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Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control
1. | Indian Institute of Science Education and Research, Kolkata, West Bengal, India |
2. | Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India |
3. | Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA |
In this paper, we consider the Stokes equations in a two-dimensional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Müntz-Szász Theorem.
References:
[1] |
O. M. Aamo, M. Krstic and T. R. Bewley,
Control of mixing by boundary feedback in 2D-channel, Automatica J. IFAC, 39 (2003), 1597-1606.
doi: 10.1016/S0005-1098(03)00140-7. |
[2] |
A. Balogh, W.-J. Liu and M. Krstic,
Stability enhancement by boundary control in 2D channel flow, IEEE Trans. Automat. Control, 46 (2001), 1696-1711.
doi: 10.1109/9.964681. |
[3] |
V. Barbu,
Stabilization of a plane periodic channel flow by noise wall normal controllers, Systems Control Lett., 59 (2010), 608-614.
doi: 10.1016/j.sysconle.2010.07.005. |
[4] |
V. Barbu,
Stabilization of a plane channel flow by wall normal controllers, Nonlinear Anal., 67 (2007), 2573-2588.
doi: 10.1016/j.na.2006.09.024. |
[5] |
V. Barbu,
Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series. Springer, London, 2011.
doi: 10.1007/978-0-85729-043-4. |
[6] |
S. Chowdhury and S. Ervedoza, Open loop stabilization of incompressible Navier-Stokes equations in a 2d channel using power series expansion,
https://www.math.univ-toulouse.fr/~ervedoza/Publis/Chowdhury-Erv.pdf. |
[7] |
J.-M. Coron,
Control and Nonlinearity, Math. Surveys Monogr. 136, American Mathematical Society, Providence, RI, 2007.
doi: 10.1090/surv/136. |
[8] |
J.-M. Coron and E. Crépeau,
Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc., 6 (2004), 367-398.
doi: 10.4171/JEMS/13. |
[9] |
J.-M. Coron and S. Guerrero,
Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl.(9), 92 (2009), 528-545.
doi: 10.1016/j.matpur.2009.05.015. |
[10] |
J.-M. Coron and P. Lissy,
Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.
doi: 10.1007/s00222-014-0512-5. |
[11] |
A. Lopez and E. Zuazua,
Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 543-580.
doi: 10.1016/S0294-1449(01)00092-0. |
[12] |
I. Munteanu,
Normal feedback stabilization of periodic flows in a two-dimensional channel, J Optim. Theory Appl., 152 (2012), 413-438.
doi: 10.1007/s10957-011-9910-7. |
[13] |
I. Munteanu,
Tangential feedback stabilization of periodic flows in a 2-D channel, Differ. Integral Equ., 24 (2011), 469-494.
|
[14] |
J.-P. Raymond,
Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
doi: 10.1016/j.anihpc.2006.06.008. |
[15] |
J.-P. Raymond,
Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.
doi: 10.1137/050628726. |
[16] |
R. Triggiani,
Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D wall-normal boundary controller, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 279-314.
doi: 10.3934/dcdsb.2007.8.279. |
[17] |
R. Vázquez, E. Trélat and J.-M. Coron,
Control for fast and stable laminar-to-high-Reynolds-numbers transfer in a 2D Navier-Stokes channel flow, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 925-956.
doi: 10.3934/dcdsb.2008.10.925. |
[18] |
R. Vázquez and M. Krstic,
A closed-form feedback controller for stabilization of the linearized 2-D Navier-Stokes Poiseuille system, IEEE Trans. Automat. Control, 52 (2007), 2298-2312.
doi: 10.1109/TAC.2007.910686. |
show all references
References:
[1] |
O. M. Aamo, M. Krstic and T. R. Bewley,
Control of mixing by boundary feedback in 2D-channel, Automatica J. IFAC, 39 (2003), 1597-1606.
doi: 10.1016/S0005-1098(03)00140-7. |
[2] |
A. Balogh, W.-J. Liu and M. Krstic,
Stability enhancement by boundary control in 2D channel flow, IEEE Trans. Automat. Control, 46 (2001), 1696-1711.
doi: 10.1109/9.964681. |
[3] |
V. Barbu,
Stabilization of a plane periodic channel flow by noise wall normal controllers, Systems Control Lett., 59 (2010), 608-614.
doi: 10.1016/j.sysconle.2010.07.005. |
[4] |
V. Barbu,
Stabilization of a plane channel flow by wall normal controllers, Nonlinear Anal., 67 (2007), 2573-2588.
doi: 10.1016/j.na.2006.09.024. |
[5] |
V. Barbu,
Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series. Springer, London, 2011.
doi: 10.1007/978-0-85729-043-4. |
[6] |
S. Chowdhury and S. Ervedoza, Open loop stabilization of incompressible Navier-Stokes equations in a 2d channel using power series expansion,
https://www.math.univ-toulouse.fr/~ervedoza/Publis/Chowdhury-Erv.pdf. |
[7] |
J.-M. Coron,
Control and Nonlinearity, Math. Surveys Monogr. 136, American Mathematical Society, Providence, RI, 2007.
doi: 10.1090/surv/136. |
[8] |
J.-M. Coron and E. Crépeau,
Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc., 6 (2004), 367-398.
doi: 10.4171/JEMS/13. |
[9] |
J.-M. Coron and S. Guerrero,
Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl.(9), 92 (2009), 528-545.
doi: 10.1016/j.matpur.2009.05.015. |
[10] |
J.-M. Coron and P. Lissy,
Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.
doi: 10.1007/s00222-014-0512-5. |
[11] |
A. Lopez and E. Zuazua,
Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 543-580.
doi: 10.1016/S0294-1449(01)00092-0. |
[12] |
I. Munteanu,
Normal feedback stabilization of periodic flows in a two-dimensional channel, J Optim. Theory Appl., 152 (2012), 413-438.
doi: 10.1007/s10957-011-9910-7. |
[13] |
I. Munteanu,
Tangential feedback stabilization of periodic flows in a 2-D channel, Differ. Integral Equ., 24 (2011), 469-494.
|
[14] |
J.-P. Raymond,
Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
doi: 10.1016/j.anihpc.2006.06.008. |
[15] |
J.-P. Raymond,
Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.
doi: 10.1137/050628726. |
[16] |
R. Triggiani,
Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D wall-normal boundary controller, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 279-314.
doi: 10.3934/dcdsb.2007.8.279. |
[17] |
R. Vázquez, E. Trélat and J.-M. Coron,
Control for fast and stable laminar-to-high-Reynolds-numbers transfer in a 2D Navier-Stokes channel flow, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 925-956.
doi: 10.3934/dcdsb.2008.10.925. |
[18] |
R. Vázquez and M. Krstic,
A closed-form feedback controller for stabilization of the linearized 2-D Navier-Stokes Poiseuille system, IEEE Trans. Automat. Control, 52 (2007), 2298-2312.
doi: 10.1109/TAC.2007.910686. |
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