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 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.

[1]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

[2]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[3]

Ammar Khemmoudj, Imane Djaidja. General decay for a viscoelastic rotating Euler-Bernoulli beam. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3531-3557. doi: 10.3934/cpaa.2020154

[4]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029

[5]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks and Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709

[6]

Mohammad Akil, Ibtissam Issa, Ali Wehbe. Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021059

[7]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675

[8]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks and Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[9]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control and Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45

[10]

Zhiling Guo, Shugen Chai. Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022001

[11]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425

[12]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control and Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015

[13]

Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems and Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051

[14]

Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks and Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19

[15]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations and Control Theory, 2022, 11 (2) : 373-397. doi: 10.3934/eect.2021004

[16]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152

[17]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057

[18]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021

[19]

Kevin Zumbrun. L resolvent bounds for steady Boltzmann's Equation. Kinetic and Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048

[20]

Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (17)

[Back to Top]