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Global solutions to the incompressible magnetohydrodynamic equations
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 |
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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