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