# American Institute of Mathematical Sciences

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

 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

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.
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 & 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., (2001). Google Scholar [2] K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams,, Appl. Anal., 86 (2007), 1529. doi: 10.1080/00036810701734113. Google Scholar [3] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Diff. Integral. Equations, 17 (2004), 1395. Google Scholar [4] K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings,, Appl. Maths., 52 (2007), 327. doi: 10.1007/s10492-007-0018-1. Google Scholar [5] K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, J. Dyn. Cont. Syst., 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar [6] K. Ammari, M. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system,, Netw. Heterog. Media., 4 (2009), 19. doi: 10.3934/nhm.2009.4.19. Google Scholar [7] K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback,, Mathematics of Control, 15 (2002), 229. doi: 10.1007/s004980200009. Google Scholar [8] K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation,, J. Differential Equations, 249 (2010), 707. doi: 10.1016/j.jde.2010.03.007. Google Scholar [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. doi: 10.1137/S0363012998349315. Google Scholar [10] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837. doi: 10.2307/2000826. Google Scholar [11] H. T. Banks, R. C. Smith and Y. Wang, "Smart Materials Structures,", Wiley, (1996). Google Scholar [12] C. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. Google Scholar [13] H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983). Google Scholar [14] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. Google Scholar [15] R. Dáger, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control Optim., 43 (2004), 590. doi: 10.1137/S0363012903421844. Google Scholar [16] R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings,, in, (2000), 1006. Google Scholar [17] R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1-d$ Flexible Multi-structures,", volume 50 of Math\'ematiques & Applications (Berlin), (2006). Google Scholar [18] T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 Edition, (1980). Google Scholar [19] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures,", Birkh\, (1994). Google Scholar [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. doi: 10.1007/s00033-004-3073-4. Google Scholar [21] D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problem,, Netw. Heterog. Media., 4 (2009), 709. doi: 10.3934/nhm.2009.4.709. Google Scholar [22] D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams,, J. Math. Anal. and Appl., 337 (2007), 174. doi: 10.1016/j.jmaa.2007.03.080. Google Scholar [23] D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams,, J. Math. Anal. and Appl., 342 (2008), 874. doi: 10.1016/j.jmaa.2007.12.062. Google Scholar [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. doi: 10.1007/BF03191374. Google Scholar [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. doi: 10.3934/nhm.2007.2.425. Google Scholar [26] W. H. Paulsen, The exterior matrix method for sequentially coupled fourth-order equations,, J. of Sound and Vibration, 308 (2007), 132. Google Scholar [27] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983). Google Scholar [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. doi: 10.1137/S0363012901399295. Google Scholar [29] M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups,", Birkh\, (2009). doi: 10.1007/978-3-7643-8994-9. Google Scholar [30] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590. Google Scholar [31] G. Q. Xu and N. E. Mastorakis, Stability of a star shaped coupled networks of strings and beams,, WSEAS, (2008). Google Scholar [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. Google Scholar [33] E. Zuazua, Control and stabilization of waves on 1-d networks,, in, (2011). Google Scholar

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##### References:
 [1] F. Ali Mehmeti, J. von Below and S. Nicaise (eds.), "Partial Differential Equations on Multistructures,", Lecture Notes in Pure and Appl. Math., (2001). Google Scholar [2] K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams,, Appl. Anal., 86 (2007), 1529. doi: 10.1080/00036810701734113. Google Scholar [3] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Diff. Integral. Equations, 17 (2004), 1395. Google Scholar [4] K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings,, Appl. Maths., 52 (2007), 327. doi: 10.1007/s10492-007-0018-1. Google Scholar [5] K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings,, J. Dyn. Cont. Syst., 11 (2005), 177. doi: 10.1007/s10883-005-4169-7. Google Scholar [6] K. Ammari, M. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system,, Netw. Heterog. Media., 4 (2009), 19. doi: 10.3934/nhm.2009.4.19. Google Scholar [7] K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback,, Mathematics of Control, 15 (2002), 229. doi: 10.1007/s004980200009. Google Scholar [8] K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation,, J. Differential Equations, 249 (2010), 707. doi: 10.1016/j.jde.2010.03.007. Google Scholar [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. doi: 10.1137/S0363012998349315. Google Scholar [10] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837. doi: 10.2307/2000826. Google Scholar [11] H. T. Banks, R. C. Smith and Y. Wang, "Smart Materials Structures,", Wiley, (1996). Google Scholar [12] C. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. Google Scholar [13] H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983). Google Scholar [14] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. Google Scholar [15] R. Dáger, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control Optim., 43 (2004), 590. doi: 10.1137/S0363012903421844. Google Scholar [16] R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings,, in, (2000), 1006. Google Scholar [17] R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1-d$ Flexible Multi-structures,", volume 50 of Math\'ematiques & Applications (Berlin), (2006). Google Scholar [18] T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 Edition, (1980). Google Scholar [19] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures,", Birkh\, (1994). Google Scholar [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. doi: 10.1007/s00033-004-3073-4. Google Scholar [21] D. Mercier, Spectrum analysis of a serially connected Euler-Bernoulli beams problem,, Netw. Heterog. Media., 4 (2009), 709. doi: 10.3934/nhm.2009.4.709. Google Scholar [22] D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams,, J. Math. Anal. and Appl., 337 (2007), 174. doi: 10.1016/j.jmaa.2007.03.080. Google Scholar [23] D. Mercier and V. Régnier, Control of a network of Euler-Bernoulli beams,, J. Math. Anal. and Appl., 342 (2008), 874. doi: 10.1016/j.jmaa.2007.12.062. Google Scholar [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. doi: 10.1007/BF03191374. Google Scholar [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. doi: 10.3934/nhm.2007.2.425. Google Scholar [26] W. H. Paulsen, The exterior matrix method for sequentially coupled fourth-order equations,, J. of Sound and Vibration, 308 (2007), 132. Google Scholar [27] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983). Google Scholar [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. doi: 10.1137/S0363012901399295. Google Scholar [29] M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups,", Birkh\, (2009). doi: 10.1007/978-3-7643-8994-9. Google Scholar [30] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590. Google Scholar [31] G. Q. Xu and N. E. Mastorakis, Stability of a star shaped coupled networks of strings and beams,, WSEAS, (2008). Google Scholar [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. Google Scholar [33] E. Zuazua, Control and stabilization of waves on 1-d networks,, in, (2011). Google Scholar
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