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

Mehdi Badra

doi:10.3934/dcds.2012.32.1169

Article Contents

2012, Volume 32, Issue 4: 1169-1208. Doi: 10.3934/dcds.2012.32.1169

This issue Previous Article Second order approximations of quasistatic evolution problems in finite dimension Next Article Symbolic approach and induction in the Heisenberg group

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

Mehdi Badra^1,

1.
Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex

Received: October 2010

Revised: May 2011

Published: April 2012

Abstract / Introduction Related Papers Cited by

Abstract

Let $-A: \mathcal{D}(A)\to H$ be the generator of an analytic semigroup and $B : U \to [\mathcal{D}(A^*)]'$ a relatively bounded control operator such that $(A-\sigma,B)$ is stabilizable for some $\sigma>0$. In this paper, we consider the stabilization of the nonlinear system $y'+Ay+G(y,u)=Bu$ by means of a feedback or a dynamical control $u$. The control is obtained from the solution to a Riccati equation which is related to a low-gain optimal quadratic minimization problem. We provide a general abstract framework to define exponentially stable solutions which is based on the contruction of Lyapunov functions. We apply such a theory to stabilize, around an unstable stationary solution, the 2D or 3D Navier-Stokes equations with a Neumann control and the 2D or 3D Boussinesq equations with a Dirichlet control.

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Mathematics Subject Classification: Primary: 93D15, 93C20; Secondary: 76D05, 76D55, 35Q30, 35Q35.

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References

[1]	C. Amrouche and V. Girault, Problèmes généralisés de Stokes, Portugal. Math., 49 (1992), 463-503.
[2]	C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.
[3]	G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, In "Fluids and Waves," 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007.
[4]	G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Equ., 9 (2009), 341-370.doi: 10.1007/s00028-009-0015-9.
[5]	G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behaviour of the resolvent operator on the imaginary axis, Appl. Anal., 88 (2009), 1357-1396.doi: 10.1080/00036810903278513.
[6]	G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447.doi: 10.3934/dcdss.2009.2.417.
[7]	M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system, In "Systems, Control, Modeling and Optimization," 13-24, IFIP Int. Fed. Inf. Process., 202, Springer, New York, 2006.
[8]	M. Badra, Local stabilization of the Navier-Stokes equations with a feedback controller localized in an open subset of the domain, Numer. Funct. Anal. Optim., 28 (2007), 559-589.doi: 10.1080/01630560701348434.
[9]	M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.doi: 10.1051/cocv:2008059.
[10]	M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control Optim., 48 (2009), 1797-1830.doi: 10.1137/070682630.
[11]	M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463.doi: 10.1137/090778146.
[12]	V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 9 (2003), 197-206 (electronic).doi: 10.1051/cocv:2003009.
[13]	V. Barbu, "Stabilization of Navier-Stokes Flow," Communications and Control Engineering, Springer Verlag, Berlin-London, 2010.
[14]	V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, In "Fluids and Waves," 55-82, Contemp. Math, 440, Amer. Math. Soc., Providence, RI, 2007.
[15]	V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-1207.doi: 10.1512/iumj.2008.57.3284.
[16]	V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746.doi: 10.1016/j.na.2005.09.012.
[17]	V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), :x+128 pp.
[18]	V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d=2,3$, via feedback stabilization of its linearization, In "Control of Coupled Partial Differential Equations," 13-46, Internat. Ser. Numer. Math., 155, Birkhäuser, Basel, 2007.
[19]	V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization for Navier-Stokes equations by means of finite-dimensional distributed controls, 2010. Available from: http://hal.archives-ouvertes.fr/hal-00455250/fr/.
[20]	V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.doi: 10.1512/iumj.2004.53.2445.
[21]	F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006.
[22]	F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 (2010), 217-235.
[23]	F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505-521.doi: 10.3934/dcdss.2011.4.505.
[24]	P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[25]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp.
[26]	H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad., 46 (1970), 1141-1143.doi: 10.3792/pja/1195526510.
[27]	A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary, In "Partial Differential Equations," 95-123, Amer. Math. Soc. Transl. Ser. 2, 206, Amer. Math. Soc., Providence, RI, 2002.
[28]	A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 289-314.doi: 10.3934/dcds.2004.10.289.
[29]	P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63.
[30]	G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, Math. Scand., 69 (1991), 217-290.
[31]	S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61.
[32]	E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957.
[33]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478.doi: 10.1016/j.jde.2009.06.005.
[34]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254.
[35]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176.doi: 10.1088/0951-7715/24/1/008.
[36]	I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach, Appl. Anal., 42 (1991), 243-261.doi: 10.1080/00036819108840045.
[37]	I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati's feedback synthesis and regularity of optimal solution, Appl. Math. Optim., 16 (1987), 147-168.doi: 10.1007/BF01442189.
[38]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.
[39]	I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction, In "Optimal Control of Coupled Systems of Partial Differential Equations," 171-190, Internat. Ser. Numer. Math., 158, Birkhäuser Verlag, Basel, 2009.
[40]	I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems Control Lett., 58 (2009), 499-509.doi: 10.1016/j.sysconle.2009.02.010.
[41]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[42]	J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I, Dunod, Paris, 1968.
[43]	J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic).doi: 10.1137/050628726.
[44]	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.
[45]	J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
[46]	J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.
[47]	J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete and Continuous Dynamical Systems A, 27 (2010), 1159-1187.
[48]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," Second edition, Johann Ambrosius Barth, Heidelberg, 1995.
[49]	R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation, Nonlinear Anal., 71 (2009), 4967-4976.doi: 10.1016/j.na.2009.03.073.
[50]	R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645-677.doi: 10.3934/dcdss.2009.2.645.