# Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 4, page 934-968
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBadra, Mehdi. "Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2009): 934-968. <http://eudml.org/doc/245958>.

@article{Badra2009,

abstract = {We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.},

author = {Badra, Mehdi},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Navier-Stokes equation; feedback stabilization; Dirichlet control; Riccati equation},

language = {eng},

number = {4},

pages = {934-968},

publisher = {EDP-Sciences},

title = {Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system},

url = {http://eudml.org/doc/245958},

volume = {15},

year = {2009},

}

TY - JOUR

AU - Badra, Mehdi

TI - Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2009

PB - EDP-Sciences

VL - 15

IS - 4

SP - 934

EP - 968

AB - We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.

LA - eng

KW - Navier-Stokes equation; feedback stabilization; Dirichlet control; Riccati equation

UR - http://eudml.org/doc/245958

ER -

## References

top- [1] M. Badra, Feedback stabilization of 3-D Navier-Stokes equations based on an extended system, in Proceedings of the 22nd IFIP TC7 Conference (2005). Zbl1214.93079MR2241694
- [2] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Contr. Opt. (to appear). MR2516189
- [3] V. Barbu, Feedback stabilization of Navier-Stokes equations. ESAIM: COCV 9 (2003) 197–206 (electronic). Zbl1076.93037MR1957098
- [4] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoirs of the American Mathematical Society 181. AMS (2006). Zbl1098.35026MR2215059
- [5] V. Barbu and R.L Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443–1494. Zbl1073.76017MR2104285
- [6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems 1, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, USA (1992). Zbl0781.93002MR2273323
- [7] P. Constantin and C. Foias, Navier-Stokes equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, USA (1988) Zbl0687.35071MR972259
- [8] J.-M. Coron and A.V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429–448. Zbl0938.93030MR1470445
- [9] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. Zbl1267.93020MR2103189
- [10] H. Fujita and H. Morimoto, On fractional powers of the Stokes operator. Proc. Japan Acad. 46 (1970) 1141–1143. Zbl0235.35067MR296755
- [11] A.V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control. J. Math. Fluid Mech. 3 (2001) 259–301. Zbl0983.93021MR1860125
- [12] A.V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete Contin. Dyn. Syst. 10 (2004) 289–314. Zbl1174.93675MR2026196
- [13] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I, Linearized steady problems, Springer Tracts in Natural Philosophy, Vol. 38. Springer-Verlag, New York (1994). Zbl0949.35004MR1284205
- [14] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. II, Nonlinear steady problems, Springer Tracts in Natural Philosophy, Vol. 39. Springer-Verlag, New York (1994). Zbl0949.35005MR1284206
- [15] P. Grisvard, Caractérisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal. 25 (1967) 40–63. Zbl0187.05901MR213864
- [16] P. Grisvard, Elliptic problems in nonsmooth domains, in Monographs and Studies in Mathematics, Vol. 24, Pitman (Advanced Publishing Program), Boston, MA, USA (1985). Zbl0695.35060MR775683
- [17] E. Hille and R.S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31. American Mathematical Society, Providence, RI, USA, revised edition (1957). Zbl0078.10004MR89373
- [18] I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. I. Abstract parabolic systems, in Encyclopedia of Mathematics and its Applications 74, Cambridge University Press, Cambridge (2000). Zbl0961.93003MR1745475
- [19] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I. Dunod, Paris (1968). Zbl0165.10801
- [20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, in Applied Mathematical Sciences 44, Springer-Verlag, New York (1983). Zbl0516.47023MR710486
- [21] J.-P. Raymond, Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt. 45 (2006) 790–828. Zbl1121.93064MR2247716
- [22] J.-P. Raymond, Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87 (2007) 627–669. Zbl1114.93040MR2335090
- [23] J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. H. Poincaré, Anal. Non Linéaire 24 (2007) 921–951. Zbl1136.35070MR2371113
- [24] M.E. Taylor, Partial differential equations. I. Basic theory, in Applied Mathematical Sciences 115, Springer-Verlag, New York (1996). Zbl0869.35002MR1395148
- [25] R. Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam, revised edition (1979). With an appendix by F. Thomasset. Zbl0426.35003MR603444
- [26] H. Triebel, Interpolation theory, function spaces, differential operators. Johann Ambrosius Barth, Heidelberg, second edition (1995). Zbl0830.46028MR1328645
- [27] A. Yagi, Coïncidence entre des espaces d’interpolation et des domaines de puissances fractionnaires d’opérateurs. C. R. Acad. Sci. Paris Sér. I Math. 299 (1984) 173–176. Zbl0563.46042MR759225

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.