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

Shirshendu Chowdhury; Debanjana Mitra; Michael Renardy

doi:10.3934/eect.2018022

Article Contents

2018, Volume 7, Issue 3: 447-463. Doi: 10.3934/eect.2018022

This issue Previous Article Control problems and invariant subspaces for sabra shell model of turbulence Next Article Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications

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
^* Corresponding author: Michael Renardy

Received: December 31, 2017

Revised: February 28, 2018

Published: July 2018

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.

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 93C20, 93B05; Secondary: 76D05.

Citation:

Full Text(HTML)

Related Papers

Cited by

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., 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.