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 and 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), 43-161. doi: 10.1070/RM1964v019n03ABEH001149.

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[3]

V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM Control, Optimization and Calculus of Variation, 9 (2003), 197-206. 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), 2704-2746. 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), 1443-1494. doi: 10.1512/iumj.2004.53.2445.

[7]

A. V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary control, Sbornik: Mathematics, 192 (2001), 593-639. 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), 259-301. doi: 10.1007/PL00000972.

[9]

A. V. Fursikov, Feedback stabilization for the 2D Navier-Stokes equations, in "The Navier-Stokes Equations: Theory and Numerical Methods" (Varenna, 2000), Lecture Notes in Pure and Appl. Math., 223, Dekker, New-York, (2002), 179-196.

[10]

A. V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks, in "Proceedings of the 8th Conference on Control of Distributed Parameter Systems," International Series of Numerical Mathematics, 143, Birkhäser Verlag, (2002), 169-187.

[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), 289-314.

[12]

A. V. Fursikov, Real process corresponding to 3D Navier-Stokes system, and its feedback stabilization from boundary, in "Patial Differential Equations," Amer. Math. Soc. Transl. Series 2, 206, Amer. Math. Soc., Providence, RI, (2002), 95-123.

[13]

A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control, in "Nonlinear Problems in Mathematical Physics and Related Topics," II, In Honor of Professor O. A. Ladyzhenskaya, Int. Math. Ser. (N. Y.), 2, Kluwer/Plenum, New-York, (2002), 137-177.

[14]

A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications," Transl. of Math. Mongraphs, 187, AMS, Providence, Rhode Island, 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, Series S, 3 (2010), 269-289. doi: 10.3934/dcdss.2010.3.269.

[16]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Ser., 34, Seoul National University, Research Institute of Mathematics, Global Anal. Res. Center, Seoul, 1996.

[17]

A. V. Fursikov and O. Yu. Imanuvilov, Exact controllability of Navier-Stokes and Boussinesq equations, Russian Math. Surveys, 54 (1999), 565-618. 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), 209-258. doi: 10.1007/s002050200200.

[19]

A. V. Gorshkov, Stabilization of the one-dimensional heat equation on a semibounded rod, Uspekhi Mat. Nauk, 56 (2001), 213-214; English transl. in Russ. Math. Surv., 56 (2001), 409-410. doi: 10.1070/RM2001v056n02ABEH000388.

[20]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[21]

K. Iosida, "Functional Analysis," Springer-Verlag, Berlin, 1965.

[22]

A. A. Ivanchikov, On numerical stabilization of unstable Couette flow by the boundary conditions, Russ. J. Numer. Anal. Math. Modelling, 21 (2006), 519-537. doi: 10.1515/rnam.2006.21.6.519.

[23]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Revised English edition, Gordon and Breach Science Publishers, New York-London, 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, 7, Zap. Naučn. Sem. LOMI, 38 (1973), 46-93.

[25]

J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, (French), Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.

[26]

J. E. Marsden and M. McCracken, "The Hopf Bifurcation and Its Applications," Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New-York, 1976.

[27]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87 (2007), 627-669. doi: 10.1016/j.matpur.2007.04.002.

[28]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Third editon, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.

[29]

M. I. Vishik and A. V. Fursikov, "Mathematical Problems of Statistical Hydromechanics," Kluwer Acad. Publ., Dordrecht, Boston, London, 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), 43-161. doi: 10.1070/RM1964v019n03ABEH001149.

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[3]

V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM Control, Optimization and Calculus of Variation, 9 (2003), 197-206. 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), 2704-2746. 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), 1443-1494. doi: 10.1512/iumj.2004.53.2445.

[7]

A. V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary control, Sbornik: Mathematics, 192 (2001), 593-639. 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), 259-301. doi: 10.1007/PL00000972.

[9]

A. V. Fursikov, Feedback stabilization for the 2D Navier-Stokes equations, in "The Navier-Stokes Equations: Theory and Numerical Methods" (Varenna, 2000), Lecture Notes in Pure and Appl. Math., 223, Dekker, New-York, (2002), 179-196.

[10]

A. V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks, in "Proceedings of the 8th Conference on Control of Distributed Parameter Systems," International Series of Numerical Mathematics, 143, Birkhäser Verlag, (2002), 169-187.

[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), 289-314.

[12]

A. V. Fursikov, Real process corresponding to 3D Navier-Stokes system, and its feedback stabilization from boundary, in "Patial Differential Equations," Amer. Math. Soc. Transl. Series 2, 206, Amer. Math. Soc., Providence, RI, (2002), 95-123.

[13]

A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control, in "Nonlinear Problems in Mathematical Physics and Related Topics," II, In Honor of Professor O. A. Ladyzhenskaya, Int. Math. Ser. (N. Y.), 2, Kluwer/Plenum, New-York, (2002), 137-177.

[14]

A. V. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications," Transl. of Math. Mongraphs, 187, AMS, Providence, Rhode Island, 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, Series S, 3 (2010), 269-289. doi: 10.3934/dcdss.2010.3.269.

[16]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Ser., 34, Seoul National University, Research Institute of Mathematics, Global Anal. Res. Center, Seoul, 1996.

[17]

A. V. Fursikov and O. Yu. Imanuvilov, Exact controllability of Navier-Stokes and Boussinesq equations, Russian Math. Surveys, 54 (1999), 565-618. 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), 209-258. doi: 10.1007/s002050200200.

[19]

A. V. Gorshkov, Stabilization of the one-dimensional heat equation on a semibounded rod, Uspekhi Mat. Nauk, 56 (2001), 213-214; English transl. in Russ. Math. Surv., 56 (2001), 409-410. doi: 10.1070/RM2001v056n02ABEH000388.

[20]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[21]

K. Iosida, "Functional Analysis," Springer-Verlag, Berlin, 1965.

[22]

A. A. Ivanchikov, On numerical stabilization of unstable Couette flow by the boundary conditions, Russ. J. Numer. Anal. Math. Modelling, 21 (2006), 519-537. doi: 10.1515/rnam.2006.21.6.519.

[23]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Revised English edition, Gordon and Breach Science Publishers, New York-London, 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, 7, Zap. Naučn. Sem. LOMI, 38 (1973), 46-93.

[25]

J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, (French), Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.

[26]

J. E. Marsden and M. McCracken, "The Hopf Bifurcation and Its Applications," Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New-York, 1976.

[27]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87 (2007), 627-669. doi: 10.1016/j.matpur.2007.04.002.

[28]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Third editon, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.

[29]

M. I. Vishik and A. V. Fursikov, "Mathematical Problems of Statistical Hydromechanics," Kluwer Acad. Publ., Dordrecht, Boston, London, 1988.

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