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Boundary feedback stabilization of a chain of serially connected strings
1. | UR Analysis and Control of Pde, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia |
2. | Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Le Mont Houy, 59313 Valenciennes Cedex 9, France |
References:
[1] |
K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, Vol. 2124, Springer-Verlag, Berlin, 2015.
doi: 10.1007/978-3-319-10900-8. |
[2] |
K. Ammari, D. Mercier, V. Régnier and J. Valein, Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings, Commun. Pure Appl. Anal., 11 (2012), 785-807.
doi: 10.3934/cpaa.2012.11.785. |
[3] |
K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM Journal on Control and Optimization, 39 (2000), 1160-1181.
doi: 10.1137/S0363012998349315. |
[4] |
K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240. |
[5] |
K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Appl. Maths., 52 (2007), 327-343.
doi: 10.1007/s10492-007-0018-1. |
[6] |
K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386.
doi: 10.1051/cocv:2001114. |
[7] |
K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Diff. Integral. Equations, 17 (2004), 1395-1410. |
[8] |
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. |
[9] |
H. T. Banks, R. C. Smith and Y. Wang, Smart Materials Structures, Wiley, 1996. |
[10] |
J. von Below, Classical solvability of linear parabolic equations on networks, J. Diff. Eq., 72 (1988), 316-337.
doi: 10.1016/0022-0396(88)90158-1. |
[11] |
W. L. Chan and B. Z. Guo, Pointwise stabilization for a chain of vibrating strings, IMA J. Math. and Information, 7 (1990), 307-315.
doi: 10.1093/imamci/7.4.307. |
[12] |
G. Chen, M. Coleman and H. H. West, Pointwise stabilization in the middle of the span for second order systems, nonuniform exponential decay of solutions, SIAM J. Appl. Math., 47 (1987), 751-780.
doi: 10.1137/0147052. |
[13] |
G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, Stabilization and control of serially connected beams, SIAM J. Control Optim., 25 (1987), 526-546.
doi: 10.1137/0325029. |
[14] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in $1-d$ Flexible Multi-structures, Mathématiques & Applications, 50, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
[15] |
F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56. |
[16] |
J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston-Basel-Berlin, 1994.
doi: 10.1007/978-1-4612-0273-8. |
[17] |
K.-S. Liu, F.-L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM Journal on Applied Mathematics, 49 (1989), 1694-1707.
doi: 10.1137/0149102. |
[18] |
D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams, International Journal of Control, 87 (2014), 1266-1281.
doi: 10.1080/00207179.2013.874597. |
[19] |
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. |
[20] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[21] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 248 (1984), 847-857.
doi: 10.2307/1999112. |
[22] |
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. |
show all references
References:
[1] |
K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, Vol. 2124, Springer-Verlag, Berlin, 2015.
doi: 10.1007/978-3-319-10900-8. |
[2] |
K. Ammari, D. Mercier, V. Régnier and J. Valein, Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings, Commun. Pure Appl. Anal., 11 (2012), 785-807.
doi: 10.3934/cpaa.2012.11.785. |
[3] |
K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM Journal on Control and Optimization, 39 (2000), 1160-1181.
doi: 10.1137/S0363012998349315. |
[4] |
K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240. |
[5] |
K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Appl. Maths., 52 (2007), 327-343.
doi: 10.1007/s10492-007-0018-1. |
[6] |
K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386.
doi: 10.1051/cocv:2001114. |
[7] |
K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Diff. Integral. Equations, 17 (2004), 1395-1410. |
[8] |
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. |
[9] |
H. T. Banks, R. C. Smith and Y. Wang, Smart Materials Structures, Wiley, 1996. |
[10] |
J. von Below, Classical solvability of linear parabolic equations on networks, J. Diff. Eq., 72 (1988), 316-337.
doi: 10.1016/0022-0396(88)90158-1. |
[11] |
W. L. Chan and B. Z. Guo, Pointwise stabilization for a chain of vibrating strings, IMA J. Math. and Information, 7 (1990), 307-315.
doi: 10.1093/imamci/7.4.307. |
[12] |
G. Chen, M. Coleman and H. H. West, Pointwise stabilization in the middle of the span for second order systems, nonuniform exponential decay of solutions, SIAM J. Appl. Math., 47 (1987), 751-780.
doi: 10.1137/0147052. |
[13] |
G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, Stabilization and control of serially connected beams, SIAM J. Control Optim., 25 (1987), 526-546.
doi: 10.1137/0325029. |
[14] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in $1-d$ Flexible Multi-structures, Mathématiques & Applications, 50, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
[15] |
F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56. |
[16] |
J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston-Basel-Berlin, 1994.
doi: 10.1007/978-1-4612-0273-8. |
[17] |
K.-S. Liu, F.-L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM Journal on Applied Mathematics, 49 (1989), 1694-1707.
doi: 10.1137/0149102. |
[18] |
D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams, International Journal of Control, 87 (2014), 1266-1281.
doi: 10.1080/00207179.2013.874597. |
[19] |
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. |
[20] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[21] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 248 (1984), 847-857.
doi: 10.2307/1999112. |
[22] |
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. |
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