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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 and 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.

[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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