American Institute of Mathematical Sciences

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April  2012, 32(4): 1169-1208. doi: 10.3934/dcds.2012.32.1169

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  October 2011

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.
Keywords: Dirichlet control., Riccati equation, Navier-Stokes, Feedback stabilization, Neumann control, Boussinesq, Lyapunov function.
Mathematics Subject Classification: Primary: 93D15, 93C20; Secondary: 76D05, 76D55, 35Q30, 35Q3.
Citation: 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
References:
[1]

C. Amrouche and V. Girault, Problèmes généralisés de Stokes,, Portugal. Math., 49 (1992), 463.   Google Scholar

[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.   Google Scholar

[3]

G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties,, In, 440 (2007), 15.   Google Scholar

[4]

G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Equ., 9 (2009), 341.  doi: 10.1007/s00028-009-0015-9.  Google Scholar

[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.  doi: 10.1080/00036810903278513.  Google Scholar

[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.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[7]

M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system,, In, 202 (2006), 13.   Google Scholar

[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.  doi: 10.1080/01630560701348434.  Google Scholar

[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.  doi: 10.1051/cocv:2008059.  Google Scholar

[10]

M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control Optim., 48 (2009), 1797.  doi: 10.1137/070682630.  Google Scholar

[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.  doi: 10.1137/090778146.  Google Scholar

[12]

V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 9 (2003), 197.  doi: 10.1051/cocv:2003009.  Google Scholar

[13]

V. Barbu, "Stabilization of Navier-Stokes Flow,", Communications and Control Engineering, (2010).   Google Scholar

[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, 440 (2007), 55.   Google Scholar

[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.  doi: 10.1512/iumj.2008.57.3284.  Google Scholar

[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.  doi: 10.1016/j.na.2005.09.012.  Google Scholar

[17]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006).   Google Scholar

[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, 155 (2007), 13.   Google Scholar

[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: \url{http://hal.archives-ouvertes.fr/hal-00455250/fr/}., (2010).   Google Scholar

[20]

V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers,, Indiana Univ. Math. J., 53 (2004), 1443.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar

[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, (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcdss.2011.4.505.  Google Scholar

[24]

P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).   Google Scholar

[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).   Google Scholar

[26]

H. Fujita and H. Morimoto, On fractional powers of the Stokes operator,, Proc. Japan Acad., 46 (1970), 1141.  doi: 10.3792/pja/1195526510.  Google Scholar

[27]

A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary,, In, 206 (2002), 95.   Google Scholar

[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.  doi: 10.3934/dcds.2004.10.289.  Google Scholar

[29]

P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40.   Google Scholar

[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.   Google Scholar

[31]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29.   Google Scholar

[32]

E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups,", rev. ed., (1957).   Google Scholar

[33]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system,, J. Differential Equations, 247 (2009), 1452.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[34]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system,, Adv. Differential Equations, 15 (2010), 231.   Google Scholar

[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.  doi: 10.1088/0951-7715/24/1/008.  Google Scholar

[36]

I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach,, Appl. Anal., 42 (1991), 243.  doi: 10.1080/00036819108840045.  Google Scholar

[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.  doi: 10.1007/BF01442189.  Google Scholar

[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, (2000).   Google Scholar

[39]

I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction,, In, 158 (2009), 171.   Google Scholar

[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.  doi: 10.1016/j.sysconle.2009.02.010.  Google Scholar

[41]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[42]

J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I,, Dunod, (1968).   Google Scholar

[43]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790.  doi: 10.1137/050628726.  Google Scholar

[44]

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

[45]

J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921.   Google Scholar

[46]

J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398.   Google Scholar

[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.   Google Scholar

[48]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", Second edition, (1995).   Google Scholar

[49]

R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation,, Nonlinear Anal., 71 (2009), 4967.  doi: 10.1016/j.na.2009.03.073.  Google Scholar

[50]

R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645.  doi: 10.3934/dcdss.2009.2.645.  Google Scholar

show all references

References:
[1]

C. Amrouche and V. Girault, Problèmes généralisés de Stokes,, Portugal. Math., 49 (1992), 463.   Google Scholar

[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.   Google Scholar

[3]

G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties,, In, 440 (2007), 15.   Google Scholar

[4]

G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Equ., 9 (2009), 341.  doi: 10.1007/s00028-009-0015-9.  Google Scholar

[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.  doi: 10.1080/00036810903278513.  Google Scholar

[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.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[7]

M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system,, In, 202 (2006), 13.   Google Scholar

[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.  doi: 10.1080/01630560701348434.  Google Scholar

[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.  doi: 10.1051/cocv:2008059.  Google Scholar

[10]

M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control Optim., 48 (2009), 1797.  doi: 10.1137/070682630.  Google Scholar

[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.  doi: 10.1137/090778146.  Google Scholar

[12]

V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 9 (2003), 197.  doi: 10.1051/cocv:2003009.  Google Scholar

[13]

V. Barbu, "Stabilization of Navier-Stokes Flow,", Communications and Control Engineering, (2010).   Google Scholar

[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, 440 (2007), 55.   Google Scholar

[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.  doi: 10.1512/iumj.2008.57.3284.  Google Scholar

[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.  doi: 10.1016/j.na.2005.09.012.  Google Scholar

[17]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006).   Google Scholar

[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, 155 (2007), 13.   Google Scholar

[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: \url{http://hal.archives-ouvertes.fr/hal-00455250/fr/}., (2010).   Google Scholar

[20]

V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers,, Indiana Univ. Math. J., 53 (2004), 1443.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar

[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, (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcdss.2011.4.505.  Google Scholar

[24]

P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).   Google Scholar

[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).   Google Scholar

[26]

H. Fujita and H. Morimoto, On fractional powers of the Stokes operator,, Proc. Japan Acad., 46 (1970), 1141.  doi: 10.3792/pja/1195526510.  Google Scholar

[27]

A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary,, In, 206 (2002), 95.   Google Scholar

[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.  doi: 10.3934/dcds.2004.10.289.  Google Scholar

[29]

P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40.   Google Scholar

[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.   Google Scholar

[31]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29.   Google Scholar

[32]

E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups,", rev. ed., (1957).   Google Scholar

[33]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system,, J. Differential Equations, 247 (2009), 1452.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[34]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system,, Adv. Differential Equations, 15 (2010), 231.   Google Scholar

[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.  doi: 10.1088/0951-7715/24/1/008.  Google Scholar

[36]

I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach,, Appl. Anal., 42 (1991), 243.  doi: 10.1080/00036819108840045.  Google Scholar

[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.  doi: 10.1007/BF01442189.  Google Scholar

[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, (2000).   Google Scholar

[39]

I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction,, In, 158 (2009), 171.   Google Scholar

[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.  doi: 10.1016/j.sysconle.2009.02.010.  Google Scholar

[41]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[42]

J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I,, Dunod, (1968).   Google Scholar

[43]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790.  doi: 10.1137/050628726.  Google Scholar

[44]

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

[45]

J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921.   Google Scholar

[46]

J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398.   Google Scholar

[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.   Google Scholar

[48]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", Second edition, (1995).   Google Scholar

[49]

R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation,, Nonlinear Anal., 71 (2009), 4967.  doi: 10.1016/j.na.2009.03.073.  Google Scholar

[50]

R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645.  doi: 10.3934/dcdss.2009.2.645.  Google Scholar

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