# American Institute of Mathematical Sciences

March  2015, 4(1): 1-19. doi: 10.3934/eect.2015.4.1

## 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

Received  August 2014 Revised  January 2015 Published  February 2015

We consider $N$ strings connected one to another and forming a particular network which is a chain of strings. We study a stabilization problem and precisely we prove that the energy of the solutions of the dissipative system decays exponentially to zero when the time tends to infinity, independently of the densities of the strings. Our technique is based on a frequency domain method and a special analysis for the resolvent. Moreover, by the same approach, we study the transfer function associated to the chain of strings and the stability of the Schrödinger system.
Citation: Kaïs Ammari, Denis Mercier. Boundary feedback stabilization of a chain of serially connected strings. Evolution Equations & Control Theory, 2015, 4 (1) : 1-19. doi: 10.3934/eect.2015.4.1
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Diff. Integral. Equations, 17 (2004), 1395-1410.  Google Scholar [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.  Google Scholar [9] H. T. Banks, R. C. Smith and Y. Wang, Smart Materials Structures, Wiley, 1996. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 248 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Diff. Integral. Equations, 17 (2004), 1395-1410.  Google Scholar [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.  Google Scholar [9] H. T. Banks, R. C. Smith and Y. Wang, Smart Materials Structures, Wiley, 1996. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 248 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar [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.  Google Scholar
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