BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (6): 1881-1902.

BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS

刘汉兵, 肖海军

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

收稿日期:2017-03-31 修回日期:2018-01-20 出版日期:2018-12-25 发布日期:2018-12-28
通讯作者: Hanbing LIU E-mail:hanbing272003@aliyun.com
作者简介:Haijun XIAO,E-mail:xiaohj@cug.edu.cn
基金资助:
This work was supported by NSFC (11401544).

BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS

Hanbing LIU, Haijun XIAO

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Received:2017-03-31 Revised:2018-01-20 Online:2018-12-25 Published:2018-12-28
Contact: Hanbing LIU E-mail:hanbing272003@aliyun.com
Supported by:
This work was supported by NSFC (11401544).

摘要/Abstract

摘要： The aim of this work is to design oblique boundary feedback controller for stabilizing the equilibrium solutions to Boussinesq equations on a bounded and open domain in R². Two kinds of such feedback controller are provided, one is the proportional stabilizable feedback control, which is obtained by spectrum decomposition method, while another one is constructed via the Ricatti operator for an infinite time horizon optimal control problem. An example of periodic Boussinesq flow in 2-D channel is also given.

关键词: Boussinesq equations, boundary feedback controller, eigenvalue

Abstract: The aim of this work is to design oblique boundary feedback controller for stabilizing the equilibrium solutions to Boussinesq equations on a bounded and open domain in R². Two kinds of such feedback controller are provided, one is the proportional stabilizable feedback control, which is obtained by spectrum decomposition method, while another one is constructed via the Ricatti operator for an infinite time horizon optimal control problem. An example of periodic Boussinesq flow in 2-D channel is also given.

Key words: Boussinesq equations, boundary feedback controller, eigenvalue

刘汉兵, 肖海军. BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS[J]. 数学物理学报(英文版), 2018, 38(6): 1881-1902.

Hanbing LIU, Haijun XIAO. BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS[J]. Acta mathematica scientia,Series B, 2018, 38(6): 1881-1902.

参考文献

[1] Barbu V, Triggiani R. Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ Math J, 2004, 53:1443-1494
[2] Fursikov A V. Stabilization for the 3D Navier-Stokes systems by feedback boundary control. Discrete Contin Dyn Syst, 2004, 10:289-314
[3] Barbu V, Lasiecka I, Triggiani R. Tangential boundary stabilization of Navier-Stokes equations. Mem Amer Math Soc, 2006, 181(852):107-116
[4] Raymond J P. Feedback boundary stabilization of the two-dimensional Navier-Stokes equations. SIAM J Control Optim, 2006, 45:790-828
[5] Raymond J P. Feedback boundary stabilization of the three-dimensional incomplessible Navier-Stokes equations. J Math Pures Appl, 2007, 87(9):627-669
[6] Raymond J P, Thevenet L. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete Contin Dyn Syst, 2010, 27:1159-1187
[7] Badra M. Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J Control Optim, 2009, 48:1797-1830
[8] Badra M, Takahashi T. Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers:application to the Navier-Stokes system. SIAM J Control Optim, 2011, 49:420-463
[9] Barbu V, Wang G. Feedback stabilization of semilinear heat equations. Abstr Appl Anal, 2003, 12:697-71
[10] Barbu V, Wang G. Internal stabilization of semilinear parabolic systems. J Math Anal Appl, 2003, 285(2):387-40
[11] Barbu V, Wang G. Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems. Indiana Univ Math J, 2005, 54(6):1521-1546
[12] Barbu V. Stabilization of a plane channel flow by wall normalcontrollers. Nonlinear Anal, Theory Methods Appl, 2007, 67(9):2573-2588
[13] Munteanu I. Normal feedback stabilization of periodic flows in a two dimensional channel. J Optim Theory Appl, 2012, 152:413-438
[14] Lefter C G. Feedback stabilization of magnetohydrodynamic equations. SIAM J Control Optim, 2011, 49:963-983
[15] Wang Y, Li S. Stabilization of vibrating beam by velocity feedback control. Acta Math Sci, 2000, 20B(2):235-244
[16] Chai S. Boundary feedback stabilization of Naghdi's model. Acta Math Sin, Eng Ser, 2005, 21(1):169-184
[17] Barbu V. Stabilization of Navier-Stokes equations by oblique boundary feedback controllers. SIAM J Control Optim, 2012, 50:2288-2307
[18] Barbu V. Boundary stabilization of equilibrium solutions to parabolic equations. IEEE Trans Automat Control, 2013, 58(9):2416-2420
[19] Munteanu I. Boundary stabilization of the phase field system by finite-dimensional feedback controllers. J Math Anal Appl, 2014, 412:964-975
[20] Wang G. Stabilization of the Boussinesq equation via internal feedback controls. Nonlinear Anal, TMA, 2003, 52:485-506
[21] Lefter A I. On the feedback stabilization of Bossinesq euqations. Annals of Alexandru Ioan Cuza UniversityMathematics, 2011, 57(2):285-310
[22] Badra M. 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 Cont Dyn Syst-Series A, 2012, 32(4):1169-1208
[23] Barbu V. Stabilization of Navier-Stokes Flows. New York:Springer-Verlag, 2011
[24] Munteanu I. Stabilization of parabolic semilinear equations. Int J Control, 2016:1-20
[25] Lasiecka I, Triggiani R. Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers. Nonlinear Anal, TMA, 2015, 121:424-446
[26] Lasiecka I, Triggiani R. Uniform stabilization with arbitrary decay rates of the Oseen equation by finitedimensional tangential localized interior and boundary controls. J Differential Equations, 2015, 79(2):340-381
[27] Raymond J P. Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Annales De Linstitut Henri Poincare Non Linear Analysis, 2007, 24(6):921-951
[28] Balakrishanan A V. Appplied Functional Analysis. 2nd ed. Spring-Verlag, 1981
[29] Temam R. Navier-Stokes Equations. Amsterdam:North-Holland, 1979

BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS

BOUNDARY FEEDBACK STABILIZATION OF BOUSSINESQ EQUATIONS

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