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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

 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.

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.

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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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