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September  2018, 7(3): 447-463. doi: 10.3934/eect.2018022

Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control

Shirshendu Chowdhury 1, , Debanjana Mitra 2, and  Michael Renardy 3,,

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

* Corresponding author: Michael Renardy

Received  January 2018 Revised  March 2018 Published  July 2018

Fund Project: Shirshendu Chowdhury acknowledges financial support from an INSPIRE Fellowship. Michael Renardy and Debanjana Mitra acknowledge support from the National Science Foundation under Grant DMS-1514576.

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.

Keywords: Incompressible Stokes equations, null controllability, Dirichlet boundary control.
Mathematics Subject Classification: Primary: 93C20, 93B05; Secondary: 76D05.
Citation: Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations & Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series. Springer, London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[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. Google Scholar

[7]

J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Munteanu, Tangential feedback stabilization of periodic flows in a 2-D channel, Differ. Integral Equ., 24 (2011), 469-494.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series. Springer, London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[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. Google Scholar

[7]

J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Munteanu, Tangential feedback stabilization of periodic flows in a 2-D channel, Differ. Integral Equ., 24 (2011), 469-494.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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