Indirect stabilization of weakly coupled systems with hybrid boundary conditions

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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

Abstract
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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 and 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.
[2]	F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368.
[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.
[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.
[5]	F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems,, ESAIM COCV., ().
[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.
[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.
[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.
[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.
[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.
[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.
[12]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," 2^nd edition, Systems & Control: Foundations & Applications, Birkhäuser Bostonm, Inc., Boston, MA, 2007.
[13]	A. Beyrath, Indirect linear locally distributed damping of coupled systems, Bol. Soc. Parana. Mat. (3), 22 (2004), 17-34.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	B. Kapitonov, Stabilization and simultaneous boundary controllability for a pair of Maxwell's equations, Mat. Apl. Comput., 15 (1996), 213-225.
[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.
[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.
[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.
[25]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.
[26]	A. Lunardi, "Interpolation Theory," 2^nd edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009.
[27]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[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.
[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.
[30]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2^nd edition, Johann Ambrosius Barth, Heidelberg, 1995.
[31]	W. Youssef, "Contrôle et Stabilisation de Système Élastiques Couplés," Ph.D thesis, University Paul Verlaine-Metz, 2009.
[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.

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.
[2]	F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368.
[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.
[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.
[5]	F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems,, ESAIM COCV., ().
[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.
[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.
[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.
[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.
[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.
[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.
[12]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," 2^nd edition, Systems & Control: Foundations & Applications, Birkhäuser Bostonm, Inc., Boston, MA, 2007.
[13]	A. Beyrath, Indirect linear locally distributed damping of coupled systems, Bol. Soc. Parana. Mat. (3), 22 (2004), 17-34.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	B. Kapitonov, Stabilization and simultaneous boundary controllability for a pair of Maxwell's equations, Mat. Apl. Comput., 15 (1996), 213-225.
[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.
[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.
[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.
[25]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.
[26]	A. Lunardi, "Interpolation Theory," 2^nd edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009.
[27]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[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.
[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.
[30]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2^nd edition, Johann Ambrosius Barth, Heidelberg, 1995.
[31]	W. Youssef, "Contrôle et Stabilisation de Système Élastiques Couplés," Ph.D thesis, University Paul Verlaine-Metz, 2009.
[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.

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