# American Institute of Mathematical Sciences

June  2012, 1(1): 109-140. doi: 10.3934/eect.2012.1.109

## Certain questions of feedback stabilization for Navier-Stokes equations

 1 Department of Mechanics & Mathematics, Moscow State University, Moscow 119991 2 Department of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russian Federation

Received  November 2011 Revised  February 2012 Published  March 2012

The authors study the stabilization problem for Navier-Stokes and Oseen equations near steady-state solution by feedback control. The cases of control in initial condition (start control) as well as impulse and distributed controls in right side supported in a fixed subdomain of the domain $G$ filled with a fluid are investigated. The cases of bounded and unbounded domain $G$ are considered.
Citation: Andrei Fursikov, Alexey V. Gorshkov. Certain questions of feedback stabilization for Navier-Stokes equations. Evolution Equations & Control Theory, 2012, 1 (1) : 109-140. doi: 10.3934/eect.2012.1.109
##### References:
 [1] M. S. Agranovich and M. I. Vishik, Elliptic boundary problems with parameter and parabolic problems of general type, (Russian),, Russian Math. Surveys, 19 (1964), 43.  doi: 10.1070/RM1964v019n03ABEH001149.  Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992).   Google Scholar [3] V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control, 9 (2003), 197.  doi: 10.1051/cocv:2003009.  Google Scholar [4] V. Barbu, I. Lasiecka and R. Triggiani, Abstract setting of tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers,, Nonlinear Analysis, 64 (2006), 2704.  doi: 10.1016/j.na.2005.09.012.  Google Scholar [5] V. Barbu, S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations,, 2010, ().   Google Scholar [6] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimentional controllers,, Indiana Univ. Math. J., 53 (2004), 1443.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar [7] A. V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary control,, Sbornik: Mathematics, 192 (2001), 593.  doi: 10.1070/SM2001v192n04ABEH000560.  Google Scholar [8] A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control,, J. of Math. Fluid Mechanics, 3 (2001), 259.  doi: 10.1007/PL00000972.  Google Scholar [9] A. V. Fursikov, Feedback stabilization for the 2D Navier-Stokes equations,, in, 223 (2002), 179.   Google Scholar [10] A. V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks,, in, 143 (2002), 169.   Google Scholar [11] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial Differential Equations and Applications,, Discrete and Cont. Dyn. Syst., 10 (2004), 289.   Google Scholar [12] A. V. Fursikov, Real process corresponding to 3D Navier-Stokes system, and its feedback stabilization from boundary,, in, 206 (2002), 95.   Google Scholar [13] A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control,, in, 2 (2002), 137.   Google Scholar [14] A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Transl. of Math. Mongraphs, 187 (2000).   Google Scholar [15] A. V. Fursikov, Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations,, Discrete and Continuous Dynamical System, 3 (2010), 269.  doi: 10.3934/dcdss.2010.3.269.  Google Scholar [16] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Ser., 34 (1996).   Google Scholar [17] A. V. Fursikov and O. Yu. Imanuvilov, Exact controllability of Navier-Stokes and Boussinesq equations,, Russian Math. Surveys, 54 (1999), 565.  doi: 10.1070/RM1999v054n03ABEH000153.  Google Scholar [18] Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbbR^2$,, Arch. Ration. Mech. Anal., 163 (2002), 209.  doi: 10.1007/s002050200200.  Google Scholar [19] A. V. Gorshkov, Stabilization of the one-dimensional heat equation on a semibounded rod,, Uspekhi Mat. Nauk, 56 (2001), 213.  doi: 10.1070/RM2001v056n02ABEH000388.  Google Scholar [20] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar [21] K. Iosida, "Functional Analysis,", Springer-Verlag, (1965).   Google Scholar [22] A. A. Ivanchikov, On numerical stabilization of unstable Couette flow by the boundary conditions,, Russ. J. Numer. Anal. Math. Modelling, 21 (2006), 519.  doi: 10.1515/rnam.2006.21.6.519.  Google Scholar [23] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Revised English edition, (1963).   Google Scholar [24] O. A. Ladyžhenskaya and V. A. Solonnikov, On linearization principle and invariant manifolds for problems of magnetichydromechanics, (Russian), Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 38 (1973), 46.   Google Scholar [25] J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications,, Vol. 1, (1968).   Google Scholar [26] J. E. Marsden and M. McCracken, "The Hopf Bifurcation and Its Applications,", Applied Mathematical Sciences, (1976).   Google Scholar [27] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures Appl. (9), 87 (2007), 627.  doi: 10.1016/j.matpur.2007.04.002.  Google Scholar [28] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Third editon, 2 (1984).   Google Scholar [29] M. I. Vishik and A. V. Fursikov, "Mathematical Problems of Statistical Hydromechanics,", Kluwer Acad. Publ., (1988).   Google Scholar

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##### References:
 [1] M. S. Agranovich and M. I. Vishik, Elliptic boundary problems with parameter and parabolic problems of general type, (Russian),, Russian Math. Surveys, 19 (1964), 43.  doi: 10.1070/RM1964v019n03ABEH001149.  Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992).   Google Scholar [3] V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control, 9 (2003), 197.  doi: 10.1051/cocv:2003009.  Google Scholar [4] V. Barbu, I. Lasiecka and R. Triggiani, Abstract setting of tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers,, Nonlinear Analysis, 64 (2006), 2704.  doi: 10.1016/j.na.2005.09.012.  Google Scholar [5] V. Barbu, S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations,, 2010, ().   Google Scholar [6] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimentional controllers,, Indiana Univ. Math. J., 53 (2004), 1443.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar [7] A. V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary control,, Sbornik: Mathematics, 192 (2001), 593.  doi: 10.1070/SM2001v192n04ABEH000560.  Google Scholar [8] A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control,, J. of Math. Fluid Mechanics, 3 (2001), 259.  doi: 10.1007/PL00000972.  Google Scholar [9] A. V. Fursikov, Feedback stabilization for the 2D Navier-Stokes equations,, in, 223 (2002), 179.   Google Scholar [10] A. V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks,, in, 143 (2002), 169.   Google Scholar [11] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial Differential Equations and Applications,, Discrete and Cont. Dyn. Syst., 10 (2004), 289.   Google Scholar [12] A. V. Fursikov, Real process corresponding to 3D Navier-Stokes system, and its feedback stabilization from boundary,, in, 206 (2002), 95.   Google Scholar [13] A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control,, in, 2 (2002), 137.   Google Scholar [14] A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Transl. of Math. Mongraphs, 187 (2000).   Google Scholar [15] A. V. Fursikov, Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations,, Discrete and Continuous Dynamical System, 3 (2010), 269.  doi: 10.3934/dcdss.2010.3.269.  Google Scholar [16] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Ser., 34 (1996).   Google Scholar [17] A. V. Fursikov and O. Yu. Imanuvilov, Exact controllability of Navier-Stokes and Boussinesq equations,, Russian Math. Surveys, 54 (1999), 565.  doi: 10.1070/RM1999v054n03ABEH000153.  Google Scholar [18] Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbbR^2$,, Arch. Ration. Mech. Anal., 163 (2002), 209.  doi: 10.1007/s002050200200.  Google Scholar [19] A. V. Gorshkov, Stabilization of the one-dimensional heat equation on a semibounded rod,, Uspekhi Mat. Nauk, 56 (2001), 213.  doi: 10.1070/RM2001v056n02ABEH000388.  Google Scholar [20] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar [21] K. Iosida, "Functional Analysis,", Springer-Verlag, (1965).   Google Scholar [22] A. A. Ivanchikov, On numerical stabilization of unstable Couette flow by the boundary conditions,, Russ. J. Numer. Anal. Math. Modelling, 21 (2006), 519.  doi: 10.1515/rnam.2006.21.6.519.  Google Scholar [23] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Revised English edition, (1963).   Google Scholar [24] O. A. Ladyžhenskaya and V. A. Solonnikov, On linearization principle and invariant manifolds for problems of magnetichydromechanics, (Russian), Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 38 (1973), 46.   Google Scholar [25] J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications,, Vol. 1, (1968).   Google Scholar [26] J. E. Marsden and M. McCracken, "The Hopf Bifurcation and Its Applications,", Applied Mathematical Sciences, (1976).   Google Scholar [27] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures Appl. (9), 87 (2007), 627.  doi: 10.1016/j.matpur.2007.04.002.  Google Scholar [28] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Third editon, 2 (1984).   Google Scholar [29] M. I. Vishik and A. V. Fursikov, "Mathematical Problems of Statistical Hydromechanics,", Kluwer Acad. Publ., (1988).   Google Scholar
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