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December  2011, 1(4): 413-436. doi: 10.3934/mcrf.2011.1.413

Indirect stabilization of weakly coupled systems with hybrid boundary conditions

Fatiha Alabau-Boussouira 1, , Piermarco Cannarsa 2, and  Roberto Guglielmi 3,

1. 

Present position Délégation CNRS at MAPMO, UMR 6628, Current position Université Paul Verlaine-Metz, Ile du Saulcy, 57045 Metz Cedex 1, France
2. 

Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma
3. 

Dipartimento di Matematica - Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1 - 00133 Roma, Italy

Received  March 2011 Revised  July 2011 Published  November 2011

We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with hybrid boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.
Keywords: Indirect stabilization, hyperbolic systems., interpolation spaces, evolution equations, energy estimates.
Mathematics Subject Classification: 93D15, 35L53, 47D06, 46B7.
Citation: Fatiha Alabau-Boussouira, Piermarco Cannarsa, Roberto Guglielmi. Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Mathematical Control & Related Fields, 2011, 1 (4) : 413-436. doi: 10.3934/mcrf.2011.1.413
References:
[1]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020. doi: 10.1016/S0764-4442(99)80316-4.  Google Scholar

[2]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368.  Google Scholar

[3]

F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.  Google Scholar

[4]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0.  Google Scholar

[5]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems,, ESAIM COCV., ().   Google Scholar

[6]

F. Ammar Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851. doi: 10.1137/S0363012900366613.  Google Scholar

[7]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[8]

G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., 22 (2008), 817-833. doi: 10.3934/dcds.2008.22.817.  Google Scholar

[9]

G. Avalos, I. Lasiecka and R. Triggiani, Beyond lack of compactness and lack of stability of a coupled parabolic-hyperbolic fluid-structure system, in "Optimal Control of Coupled Systems of Partial Differential Equations," 1-33, Internat. Ser. Numer. Math., 158, Birkhäuser Verlag, Basel, 2009.  Google Scholar

[10]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. doi: 10.1002/mana.200410429.  Google Scholar

[11]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.  Google Scholar

[12]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," 2nd edition, Systems & Control: Foundations & Applications, Birkhäuser Bostonm, Inc., Boston, MA, 2007.  Google Scholar

[13]

A. Beyrath, Indirect linear locally distributed damping of coupled systems, Bol. Soc. Parana. Mat. (3), 22 (2004), 17-34.  Google Scholar

[14]

A. Beyrath, Indirect internal observability stabilization of coupled systems with locally distributed damping, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 451-456. doi: 10.1016/S0764-4442(01)01974-7.  Google Scholar

[15]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.  Google Scholar

[16]

M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., 14 (2008), 1-42. doi: 10.1051/cocv:2007031.  Google Scholar

[17]

N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math., 180 (1998), 1-29. doi: 10.1007/BF02392877.  Google Scholar

[18]

J.-M. Coron and S. Guerrero, Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl. (9), 92 (2009), 528-545. doi: 10.1016/j.matpur.2009.05.015.  Google Scholar

[19]

R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1-d$ flexible Multi-Structures," Mathématiques & Applications (Berlin) [Mathematics & Applications], 50, Springer-Verlag, Berlin, 2006.  Google Scholar

[20]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[21]

B. Kapitonov, Stabilization and simultaneous boundary controllability for a pair of Maxwell's equations, Mat. Apl. Comput., 15 (1996), 213-225.  Google Scholar

[22]

O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87 (2007), 408-437. doi: 10.1016/j.matpur.2007.01.005.  Google Scholar

[23]

G. Lebeau, Équation des ondes amorties, in "Algebraic and Geometric Methods in Mathematical Physics" (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, (1996), 73-109.  Google Scholar

[24]

P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632. doi: 10.1137/S0363012903437319.  Google Scholar

[25]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[26]

A. Lunardi, "Interpolation Theory," 2nd edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009.  Google Scholar

[27]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[28]

J.-P. Raymond and M. Vanninathan, Null controllability in a fluid-solid structure model, J. Differential Equations, 248 (2010), 1826-1865. doi: 10.1016/j.jde.2009.09.015.  Google Scholar

[29]

D. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358. doi: 10.1006/jmaa.1993.1071.  Google Scholar

[30]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2nd edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[31]

W. Youssef, "Contrôle et Stabilisation de Système Élastiques Couplés," Ph.D thesis, University Paul Verlaine-Metz, 2009. Google Scholar

[32]

X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction, in "Free Boundary Problems," Internat. Ser. Numer. Math., 154, Birkhäuser, Basel, (2007), 445-455.  Google Scholar

show all references

References:
[1]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020. doi: 10.1016/S0764-4442(99)80316-4.  Google Scholar

[2]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368.  Google Scholar

[3]

F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.  Google Scholar

[4]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0.  Google Scholar

[5]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems,, ESAIM COCV., ().   Google Scholar

[6]

F. Ammar Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851. doi: 10.1137/S0363012900366613.  Google Scholar

[7]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[8]

G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., 22 (2008), 817-833. doi: 10.3934/dcds.2008.22.817.  Google Scholar

[9]

G. Avalos, I. Lasiecka and R. Triggiani, Beyond lack of compactness and lack of stability of a coupled parabolic-hyperbolic fluid-structure system, in "Optimal Control of Coupled Systems of Partial Differential Equations," 1-33, Internat. Ser. Numer. Math., 158, Birkhäuser Verlag, Basel, 2009.  Google Scholar

[10]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. doi: 10.1002/mana.200410429.  Google Scholar

[11]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.  Google Scholar

[12]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," 2nd edition, Systems & Control: Foundations & Applications, Birkhäuser Bostonm, Inc., Boston, MA, 2007.  Google Scholar

[13]

A. Beyrath, Indirect linear locally distributed damping of coupled systems, Bol. Soc. Parana. Mat. (3), 22 (2004), 17-34.  Google Scholar

[14]

A. Beyrath, Indirect internal observability stabilization of coupled systems with locally distributed damping, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 451-456. doi: 10.1016/S0764-4442(01)01974-7.  Google Scholar

[15]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.  Google Scholar

[16]

M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., 14 (2008), 1-42. doi: 10.1051/cocv:2007031.  Google Scholar

[17]

N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math., 180 (1998), 1-29. doi: 10.1007/BF02392877.  Google Scholar

[18]

J.-M. Coron and S. Guerrero, Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl. (9), 92 (2009), 528-545. doi: 10.1016/j.matpur.2009.05.015.  Google Scholar

[19]

R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1-d$ flexible Multi-Structures," Mathématiques & Applications (Berlin) [Mathematics & Applications], 50, Springer-Verlag, Berlin, 2006.  Google Scholar

[20]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[21]

B. Kapitonov, Stabilization and simultaneous boundary controllability for a pair of Maxwell's equations, Mat. Apl. Comput., 15 (1996), 213-225.  Google Scholar

[22]

O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87 (2007), 408-437. doi: 10.1016/j.matpur.2007.01.005.  Google Scholar

[23]

G. Lebeau, Équation des ondes amorties, in "Algebraic and Geometric Methods in Mathematical Physics" (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, (1996), 73-109.  Google Scholar

[24]

P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632. doi: 10.1137/S0363012903437319.  Google Scholar

[25]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[26]

A. Lunardi, "Interpolation Theory," 2nd edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009.  Google Scholar

[27]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[28]

J.-P. Raymond and M. Vanninathan, Null controllability in a fluid-solid structure model, J. Differential Equations, 248 (2010), 1826-1865. doi: 10.1016/j.jde.2009.09.015.  Google Scholar

[29]

D. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358. doi: 10.1006/jmaa.1993.1071.  Google Scholar

[30]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2nd edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[31]

W. Youssef, "Contrôle et Stabilisation de Système Élastiques Couplés," Ph.D thesis, University Paul Verlaine-Metz, 2009. Google Scholar

[32]

X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction, in "Free Boundary Problems," Internat. Ser. Numer. Math., 154, Birkhäuser, Basel, (2007), 445-455.  Google Scholar

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