# American Institute of Mathematical Sciences

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. [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992). [3] V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control, 9 (2003), 197. doi: 10.1051/cocv:2003009. [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. [5] V. Barbu, S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations,, 2010, (). [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. [7] A. V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary control,, Sbornik: Mathematics, 192 (2001), 593. doi: 10.1070/SM2001v192n04ABEH000560. [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. [9] A. V. Fursikov, Feedback stabilization for the 2D Navier-Stokes equations,, in, 223 (2002), 179. [10] A. V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks,, in, 143 (2002), 169. [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. [12] A. V. Fursikov, Real process corresponding to 3D Navier-Stokes system, and its feedback stabilization from boundary,, in, 206 (2002), 95. [13] A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control,, in, 2 (2002), 137. [14] A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Transl. of Math. Mongraphs, 187 (2000). [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. [16] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Ser., 34 (1996). [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. [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. [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. [20] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). [21] K. Iosida, "Functional Analysis,", Springer-Verlag, (1965). [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. [23] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Revised English edition, (1963). [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. [25] J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications,, Vol. 1, (1968). [26] J. E. Marsden and M. McCracken, "The Hopf Bifurcation and Its Applications,", Applied Mathematical Sciences, (1976). [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. [28] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Third editon, 2 (1984). [29] M. I. Vishik and A. V. Fursikov, "Mathematical Problems of Statistical Hydromechanics,", Kluwer Acad. Publ., (1988).

show all references

##### 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. [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992). [3] V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control, 9 (2003), 197. doi: 10.1051/cocv:2003009. [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. [5] V. Barbu, S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations,, 2010, (). [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. [7] A. V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary control,, Sbornik: Mathematics, 192 (2001), 593. doi: 10.1070/SM2001v192n04ABEH000560. [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. [9] A. V. Fursikov, Feedback stabilization for the 2D Navier-Stokes equations,, in, 223 (2002), 179. [10] A. V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks,, in, 143 (2002), 169. [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. [12] A. V. Fursikov, Real process corresponding to 3D Navier-Stokes system, and its feedback stabilization from boundary,, in, 206 (2002), 95. [13] A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control,, in, 2 (2002), 137. [14] A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Transl. of Math. Mongraphs, 187 (2000). [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. [16] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Ser., 34 (1996). [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. [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. [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. [20] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). [21] K. Iosida, "Functional Analysis,", Springer-Verlag, (1965). [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. [23] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Revised English edition, (1963). [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. [25] J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications,, Vol. 1, (1968). [26] J. E. Marsden and M. McCracken, "The Hopf Bifurcation and Its Applications,", Applied Mathematical Sciences, (1976). [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. [28] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Third editon, 2 (1984). [29] M. I. Vishik and A. V. Fursikov, "Mathematical Problems of Statistical Hydromechanics,", Kluwer Acad. Publ., (1988).
 [1] A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289 [2] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147 [3] Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159 [4] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89 [5] Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109 [6] Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169 [7] V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117 [8] Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761 [9] Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375 [10] Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021 [11] Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153 [12] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [13] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2017149 [14] P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785 [15] Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 [16] G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583 [17] Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495 [18] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [19] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [20] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

2016 Impact Factor: 0.826