Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

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March 2012, 11(2): 785-807. doi: 10.3934/cpaa.2012.11.785

Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

Kaïs Ammari ^1, , Denis Mercier ^2, , Virginie Régnier ^3, and Julie Valein ^4,

1.	Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir
2.	LAMAV, FR CNRS 2956, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 VALENCIENNES Cedex 9
3.	Univ Lille Nord de France, F-59000 Lille, France, UVHC, LAMAV, FR CNRS 2956, F-59313 Valenciennes, France
4.	Institut Elie Cartan de Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex

Received March 2010 Revised May 2011 Published October 2011

Abstract
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We consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected to one another and forming a particular network which is a chain beginning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the solution of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions on the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second system, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

Keywords: Network, wave equation, spectrum, resolvent method, Euler-Bernoulli beam equation, feedback stabilization..

Mathematics Subject Classification: Primary: 35L05, 35M10, 35R02; Secondary: 47A10, 93D15, 93D2.

Citation: Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure and Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785

References:

[1]	F. Ali Mehmeti, J. von Below and S. Nicaise (eds.), "Partial Differential Equations on Multistructures," Lecture Notes in Pure and Appl. Math., vol. 219, Marcel Dekker, New York, 2001.
[2]	K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Appl. Anal., 86 (2007), 1529-1548. doi: 10.1080/00036810701734113.
[3]	K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Diff. Integral. Equations, 17 (2004), 1395-1410.
[4]	K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings, Appl. Maths., 52 (2007), 327-343. doi: 10.1007/s10492-007-0018-1.
[5]	K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Cont. Syst., 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.
[6]	K. Ammari, M. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system, Netw. Heterog. Media., 4 (2009), 19-34. doi: 10.3934/nhm.2009.4.19.
[7]	K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback, Mathematics of Control, Signals, and systems, 15 (2002), 229-255. doi: 10.1007/s004980200009.
[8]	K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727. doi: 10.1016/j.jde.2010.03.007.
[9]	K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM J. Control. Optim., 39 (2000), 1160-1181. doi: 10.1137/S0363012998349315.
[10]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.2307/2000826.
[11]	H. T. Banks, R. C. Smith and Y. Wang, "Smart Materials Structures," Wiley, 1996.
[12]	C. 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.
[13]	H. Brezis, "Analyse Fonctionnelle, Théorie et Applications," Masson, Paris, 1983.
[14]	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.
[15]	R. Dáger, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 43 (2004), 590-623. doi: 10.1137/S0363012903421844.
[16]	R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, in "Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000)", 1006-1010, SIAM, Philadelphia, PA, 2000.
[17]	R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1-d$ Flexible Multi-structures," volume 50 of Mathématiques & Applications (Berlin), Springer-Verlag, 2006.
[18]	T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 Edition, Springer-Verlag, Berlin, 1995.
[19]	J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures," Birkhäuer, Boston-Basel-Berlin, 1994.
[20]	Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.
[21]	D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problem, Netw. Heterog. Media., 4 (2009), 709-730. doi: 10.3934/nhm.2009.4.709.
[22]	D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams, J. Math. Anal. and Appl., 337 (2007), 174-196. doi: 10.1016/j.jmaa.2007.03.080.
[23]	D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams, J. Math. Anal. and Appl., 342 (2008), 874-894. doi: 10.1016/j.jmaa.2007.12.062.
[24]	D. Mercier and V. Régnier, Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses, Collect. Math, 60 (2009), 307-334. doi: 10.1007/BF03191374.
[25]	S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.
[26]	W. H. Paulsen, The exterior matrix method for sequentially coupled fourth-order equations, J. of Sound and Vibration, 308 (2007), 132-163.
[27]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983.
[28]	M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. II. Controllability and stability, SIAM J. Control Optim., 42 (2003), 907-935. doi: 10.1137/S0363012901399295.
[29]	M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.
[30]	J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797. doi: 10.1137/080733590.
[31]	G. Q. Xu and N. E. Mastorakis, Stability of a star shaped coupled networks of strings and beams, WSEAS, Proceedings of the 10th WSEAS International Conference on Technique and Computations, Technical University of Sofia (Bulgaria), 2008.
[32]	K. T. Zhang, G. Q. Xu and N. E. Mastorakis, Stability of a complex network of Euler-Bernoulli beams, WSEAS Trans. Syst., 8 (2009), 379-389.
[33]	E. Zuazua, Control and stabilization of waves on 1-d networks, in "Lecture Notes in Mathematics", CIME subseries, "Traffic Flow on Networks" (eds. B. Piccoli and M. Rascle), 2011.

show all references

References:

[1]	F. Ali Mehmeti, J. von Below and S. Nicaise (eds.), "Partial Differential Equations on Multistructures," Lecture Notes in Pure and Appl. Math., vol. 219, Marcel Dekker, New York, 2001.
[2]	K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Appl. Anal., 86 (2007), 1529-1548. doi: 10.1080/00036810701734113.
[3]	K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Diff. Integral. Equations, 17 (2004), 1395-1410.
[4]	K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings, Appl. Maths., 52 (2007), 327-343. doi: 10.1007/s10492-007-0018-1.
[5]	K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Cont. Syst., 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.
[6]	K. Ammari, M. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system, Netw. Heterog. Media., 4 (2009), 19-34. doi: 10.3934/nhm.2009.4.19.
[7]	K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback, Mathematics of Control, Signals, and systems, 15 (2002), 229-255. doi: 10.1007/s004980200009.
[8]	K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727. doi: 10.1016/j.jde.2010.03.007.
[9]	K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM J. Control. Optim., 39 (2000), 1160-1181. doi: 10.1137/S0363012998349315.
[10]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.2307/2000826.
[11]	H. T. Banks, R. C. Smith and Y. Wang, "Smart Materials Structures," Wiley, 1996.
[12]	C. 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.
[13]	H. Brezis, "Analyse Fonctionnelle, Théorie et Applications," Masson, Paris, 1983.
[14]	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.
[15]	R. Dáger, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 43 (2004), 590-623. doi: 10.1137/S0363012903421844.
[16]	R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, in "Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000)", 1006-1010, SIAM, Philadelphia, PA, 2000.
[17]	R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1-d$ Flexible Multi-structures," volume 50 of Mathématiques & Applications (Berlin), Springer-Verlag, 2006.
[18]	T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 Edition, Springer-Verlag, Berlin, 1995.
[19]	J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures," Birkhäuer, Boston-Basel-Berlin, 1994.
[20]	Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.
[21]	D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problem, Netw. Heterog. Media., 4 (2009), 709-730. doi: 10.3934/nhm.2009.4.709.
[22]	D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams, J. Math. Anal. and Appl., 337 (2007), 174-196. doi: 10.1016/j.jmaa.2007.03.080.
[23]	D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams, J. Math. Anal. and Appl., 342 (2008), 874-894. doi: 10.1016/j.jmaa.2007.12.062.
[24]	D. Mercier and V. Régnier, Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses, Collect. Math, 60 (2009), 307-334. doi: 10.1007/BF03191374.
[25]	S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.
[26]	W. H. Paulsen, The exterior matrix method for sequentially coupled fourth-order equations, J. of Sound and Vibration, 308 (2007), 132-163.
[27]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983.
[28]	M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. II. Controllability and stability, SIAM J. Control Optim., 42 (2003), 907-935. doi: 10.1137/S0363012901399295.
[29]	M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.
[30]	J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797. doi: 10.1137/080733590.
[31]	G. Q. Xu and N. E. Mastorakis, Stability of a star shaped coupled networks of strings and beams, WSEAS, Proceedings of the 10th WSEAS International Conference on Technique and Computations, Technical University of Sofia (Bulgaria), 2008.
[32]	K. T. Zhang, G. Q. Xu and N. E. Mastorakis, Stability of a complex network of Euler-Bernoulli beams, WSEAS Trans. Syst., 8 (2009), 379-389.
[33]	E. Zuazua, Control and stabilization of waves on 1-d networks, in "Lecture Notes in Mathematics", CIME subseries, "Traffic Flow on Networks" (eds. B. Piccoli and M. Rascle), 2011.

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