American Institute of Mathematical Sciences

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

Boundary feedback stabilization of a chain of serially connected strings

Kaïs Ammari 1, and  Denis Mercier 2,

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.
Keywords: boundary feedback stabilization., resolvent method, transfer function, Network, wave equation.
Mathematics Subject Classification: Primary: 35L05, 35M10; Secondary: 35R02, 47A10, 93D15, 93D2.
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

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.  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

[1]

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

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004
[3]

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

Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091
[5]

Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control & Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013
[6]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
[7]

Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization. Networks & Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299
[8]

Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323
[9]

Roberto Guglielmi. Indirect stabilization of hyperbolic systems through resolvent estimates. Evolution Equations & Control Theory, 2017, 6 (1) : 59-75. doi: 10.3934/eect.2017004
[10]

Abdallah Benabdallah, Mohsen Dlala. Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021092
[11]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029
[12]

Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057
[13]

Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005
[14]

Muhammad I. Mustafa. Viscoelastic plate equation with boundary feedback. Evolution Equations & Control Theory, 2017, 6 (2) : 261-276. doi: 10.3934/eect.2017014
[15]

Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469
[16]

Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations & Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029
[17]

Ionuţ Munteanu. Boundary stabilization of non-diagonal systems by proportional feedback forms. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3113-3128. doi: 10.3934/cpaa.2021098
[18]

Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225
[19]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021
[20]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy