Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension

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2012, 1(1): 1-16. doi: 10.3934/eect.2012.1.1

Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension

1.	Octav Mayer Institute of Mathematics (Romanian Academy), Bd. Carol I, no. 8, Iaşi 700505, Romania
2.	Octav Mayer Institute of Mathematics (Romanian Academy), and Alexandru Ioan Cuza University (Department of Mathematics), Bd. Carol I, no. 8, Iaşi 700505, Romania

Received August 2011 Revised October 2011 Published March 2012

Abstract
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One designs an internal stabilizing feedback controller, for the Navier-Stokes equations, which steers, in finite time, the initial value $X_o$ in $X_e+\mathcal{X}_s$, where $X_e$ is any equilibrium solution and $\mathcal{X}_s$ is a finite codimensional space, consisting of stable modes.

Keywords: eigenvalue, Navier-Stokes equation, feedback controller, steady state solution., Oseen-Stokes system.

Mathematics Subject Classification: Primary: 93B52, 93C20, 93D15; Secondary: 35Q30, 76D5.

Citation: Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1

References:

[1]	A. V. Balakrishanan, "Applied Functional Analysis,", Second editon, 3 (1981).
[2]	V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM COCV, 9 (2003), 197. doi: 10.1051/cocv:2003009.
[3]	V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite dimensional controllers,, Indiana Univ. Math. Journal, 53 (2004), 1443. doi: 10.1512/iumj.2004.53.2445.
[4]	V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Memoires AMS, 851 (2006).
[5]	V. Barbu, R. Triggiani and I. Lasiecka, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high and low-gain feedback controllers,, Nonlin. Anal., 64 (2006), 2704. doi: 10.1016/j.na.2005.09.012.
[6]	V. Barbu, Optimal stabilizable feedback controller for Navier-Stokes equations,, in, 513 (2010), 43.
[7]	V. Barbu and C. Lefter, Internal stabilizability of the Navier-Stokes equations. Optimization and control of distributed systems,, Systems and Control Letters, 48 (2003), 161. doi: 10.1016/S0167-6911(02)00261-X.
[8]	V. Barbu, "Stabilization of the Navier-Stokes Flows,", Springer, (2010).
[9]	V. Barbu, I. Lasiecka and R. Triggiani, The unique continuation property of eigenfunctions to Stokes-Oseen operator is generic with respect to the coefficients,, Nonlin. Anal. Ser. A: Theory Meth. and Appl., ().
[10]	M. Bedra, Feedback stabilization of the 2-D and 3-D Navier Stokes equations based on an extended system,, ESAIM COCV, 15 (2009), 934. doi: 10.1051/cocv:2008059.
[11]	M. Bedra, Lyapunov functions and local feedback stabilization of the Navier-Stokes equations,, SIAM J. Control Optimiz., 48 (2009), 1797. doi: 10.1137/070682630.
[12]	J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007).
[13]	A. Fursikov, Stabilization for the 3D Navier-Stokes systems by feedback boundary control,, Discrete and Contin. Dyn. Syst., 10 (2004), 289. doi: 10.3934/dcds.2004.10.289.
[14]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theory,", Cambridge Univ. Press, (2000).
[15]	J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures et Appl. (9), 87 (2007), 627. doi: 10.1016/j.matpur.2007.04.002.
[16]	S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD,, J. Scientific Computing, 15 (2000), 457.
[17]	A. Shirikyan, Exact controllability in projections for three-dimensional Navier-Stokes equations,, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 521.
[18]	P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).

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References:

[1]	A. V. Balakrishanan, "Applied Functional Analysis,", Second editon, 3 (1981).
[2]	V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM COCV, 9 (2003), 197. doi: 10.1051/cocv:2003009.
[3]	V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite dimensional controllers,, Indiana Univ. Math. Journal, 53 (2004), 1443. doi: 10.1512/iumj.2004.53.2445.
[4]	V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Memoires AMS, 851 (2006).
[5]	V. Barbu, R. Triggiani and I. Lasiecka, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high and low-gain feedback controllers,, Nonlin. Anal., 64 (2006), 2704. doi: 10.1016/j.na.2005.09.012.
[6]	V. Barbu, Optimal stabilizable feedback controller for Navier-Stokes equations,, in, 513 (2010), 43.
[7]	V. Barbu and C. Lefter, Internal stabilizability of the Navier-Stokes equations. Optimization and control of distributed systems,, Systems and Control Letters, 48 (2003), 161. doi: 10.1016/S0167-6911(02)00261-X.
[8]	V. Barbu, "Stabilization of the Navier-Stokes Flows,", Springer, (2010).
[9]	V. Barbu, I. Lasiecka and R. Triggiani, The unique continuation property of eigenfunctions to Stokes-Oseen operator is generic with respect to the coefficients,, Nonlin. Anal. Ser. A: Theory Meth. and Appl., ().
[10]	M. Bedra, Feedback stabilization of the 2-D and 3-D Navier Stokes equations based on an extended system,, ESAIM COCV, 15 (2009), 934. doi: 10.1051/cocv:2008059.
[11]	M. Bedra, Lyapunov functions and local feedback stabilization of the Navier-Stokes equations,, SIAM J. Control Optimiz., 48 (2009), 1797. doi: 10.1137/070682630.
[12]	J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007).
[13]	A. Fursikov, Stabilization for the 3D Navier-Stokes systems by feedback boundary control,, Discrete and Contin. Dyn. Syst., 10 (2004), 289. doi: 10.3934/dcds.2004.10.289.
[14]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theory,", Cambridge Univ. Press, (2000).
[15]	J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures et Appl. (9), 87 (2007), 627. doi: 10.1016/j.matpur.2007.04.002.
[16]	S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD,, J. Scientific Computing, 15 (2000), 457.
[17]	A. Shirikyan, Exact controllability in projections for three-dimensional Navier-Stokes equations,, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 521.
[18]	P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).

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