Figure 1.
The network and the prescribed conditions
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September
2018, 7(3): 335-351.
doi: 10.3934/eect.2018017
Energy decay for the damped wave equation on an unbounded network
UR13ES64, Analysis and Control of PDEs, Faculty of Sciences of Monastir, University of Monastir, Monastir, 5019, Tunisia |
We study the wave equation on an unbounded network of $N, N∈\mathbb{N}^*$, finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant $α>0$ via the condition $\sum_{j = 0}^N\partial_xu_j(0, t) = α \partial_tu_0(0, t)$. We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for $α\not = N+1$ we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.
Mathematics Subject Classification:
Primary: 35B37, 35B40, 35L05; Secondary: 35B35.
Citation:
Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory,
2018, 7
(3)
: 335-351.
doi: 10.3934/eect.2018017
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References:
| [1] |
F. Alabau-Boussouira, V. Perrolaz and L. Rosier,
Finite-time stabilization of a network of strings, Mathematical Control and Related Fields, AIMS, 5 (2015), 721-742.
doi: 10.3934/mcrf.2015.5.721. |
| [2] |
L. Aloui and M. Khenissi,
Stabilisation de l'équation des ondes dans un domaine extérieur, Rev. Math. Iberoamericana, 18 (2002), 1-16.
doi: 10.4171/RMI/309. |
| [3] |
K. Ammari, M. Jellouli and M. Khenissi,
Stabilization of generic trees of strings, J. Dyn. Contol Syst., 11 (2005), 177-193.
doi: 10.1007/s10883-005-4169-7. |
| [4] |
K. Ammari and M. Jellouli,
Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2015), 327-343.
doi: 10.1007/s10492-007-0018-1. |
| [5] |
K. Ammari and M. Jellouli,
Méthode numérique pour la déroissance de l'éergie d'un réseau de cordes, Bulletin of the Belgian Mathematical Society, 4 (2010), 717-735.
|
| [6] |
R. Assel, M. Jellouli and M. Khenissi,
Optimal decay rate for the local energy of a unbounded network, J. Differential Equations, 261 (2016), 4030-4054.
doi: 10.1016/j.jde.2016.06.016. |
| [7] |
R. Assel, M. Khenissi and J. Royer, Energy Decay for a Unbounded Degenerate Network, work in progress. |
| [8] |
R. Dager and H. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications, 50 (2006), x+221 pp.
doi: 10.1007/3-540-37726-3. |
| [9] |
K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. |
| [10] |
P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-0741-2. |
| [11] |
M. Jellouli,
Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662.
doi: 10.1080/00207179.2015.1012652. |
| [12] |
M. Khenissi,
Equation des ondes amorties dans un domaine extérieur, Bull. Soc. math. France, 131 (2003), 211-228.
doi: 10.24033/bsmf.2440. |
| [13] |
P. Kuchment,
Quantum graphs: Ⅰ. Some basic structures, Ⅱ. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen, 14 (2004), S107-S128.
doi: 10.1088/0959-7174/14/1/014. |
| [14] |
J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Multi-Link Flexible Structures, Systems and Control: Foundations and Applications, Birkhauser, Basel, 1994.
doi: 10.1007/978-1-4612-0273-8. |
| [15] |
N. E. Mastorakis and G. Q. Xu,
Spectral distribution of star-shaped coupled network, WSEAS Transactions on Applied and Theoretical Mechanics, 4 (2008), 739-746.
|
| [16] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [17] |
H. Reinhard, Equations Différentielles, Fondements et Applications, Dunod, Paris, 1982. |
| [18] |
S.-H. Tang and M. Zworski,
Resonance expansions of scattered waves, Comm. Pure. Appl. Math., 53 (2000), 1305-1334.
|
| [19] |
B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989. |
| [20] |
B. Vainberg,
On the short wave asymptotic behavior of solutions of stationary problems and asymptotic behavior as t→+∞ of solutions of non-stationary problems, Uspehi Mat. Nauk, 30 (1975), 3-55.
|
| [21] |
G. Vodev,
On the uniform decay of the local energy, Serdica Math. J., 25 (1999), 191-206.
|
| [22] |
Y. N. Xu and G. Q. Xu,
Exponential stabilization of a tree- shaped network of strings with variable coefficients, Glasgow Math. J., 53 (2011), 481-499.
doi: 10.1017/S0017089511000085. |
show all references
References:
| [1] |
F. Alabau-Boussouira, V. Perrolaz and L. Rosier,
Finite-time stabilization of a network of strings, Mathematical Control and Related Fields, AIMS, 5 (2015), 721-742.
doi: 10.3934/mcrf.2015.5.721. |
| [2] |
L. Aloui and M. Khenissi,
Stabilisation de l'équation des ondes dans un domaine extérieur, Rev. Math. Iberoamericana, 18 (2002), 1-16.
doi: 10.4171/RMI/309. |
| [3] |
K. Ammari, M. Jellouli and M. Khenissi,
Stabilization of generic trees of strings, J. Dyn. Contol Syst., 11 (2005), 177-193.
doi: 10.1007/s10883-005-4169-7. |
| [4] |
K. Ammari and M. Jellouli,
Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2015), 327-343.
doi: 10.1007/s10492-007-0018-1. |
| [5] |
K. Ammari and M. Jellouli,
Méthode numérique pour la déroissance de l'éergie d'un réseau de cordes, Bulletin of the Belgian Mathematical Society, 4 (2010), 717-735.
|
| [6] |
R. Assel, M. Jellouli and M. Khenissi,
Optimal decay rate for the local energy of a unbounded network, J. Differential Equations, 261 (2016), 4030-4054.
doi: 10.1016/j.jde.2016.06.016. |
| [7] |
R. Assel, M. Khenissi and J. Royer, Energy Decay for a Unbounded Degenerate Network, work in progress. |
| [8] |
R. Dager and H. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications, 50 (2006), x+221 pp.
doi: 10.1007/3-540-37726-3. |
| [9] |
K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. |
| [10] |
P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-0741-2. |
| [11] |
M. Jellouli,
Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662.
doi: 10.1080/00207179.2015.1012652. |
| [12] |
M. Khenissi,
Equation des ondes amorties dans un domaine extérieur, Bull. Soc. math. France, 131 (2003), 211-228.
doi: 10.24033/bsmf.2440. |
| [13] |
P. Kuchment,
Quantum graphs: Ⅰ. Some basic structures, Ⅱ. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen, 14 (2004), S107-S128.
doi: 10.1088/0959-7174/14/1/014. |
| [14] |
J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Multi-Link Flexible Structures, Systems and Control: Foundations and Applications, Birkhauser, Basel, 1994.
doi: 10.1007/978-1-4612-0273-8. |
| [15] |
N. E. Mastorakis and G. Q. Xu,
Spectral distribution of star-shaped coupled network, WSEAS Transactions on Applied and Theoretical Mechanics, 4 (2008), 739-746.
|
| [16] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [17] |
H. Reinhard, Equations Différentielles, Fondements et Applications, Dunod, Paris, 1982. |
| [18] |
S.-H. Tang and M. Zworski,
Resonance expansions of scattered waves, Comm. Pure. Appl. Math., 53 (2000), 1305-1334.
|
| [19] |
B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989. |
| [20] |
B. Vainberg,
On the short wave asymptotic behavior of solutions of stationary problems and asymptotic behavior as t→+∞ of solutions of non-stationary problems, Uspehi Mat. Nauk, 30 (1975), 3-55.
|
| [21] |
G. Vodev,
On the uniform decay of the local energy, Serdica Math. J., 25 (1999), 191-206.
|
| [22] |
Y. N. Xu and G. Q. Xu,
Exponential stabilization of a tree- shaped network of strings with variable coefficients, Glasgow Math. J., 53 (2011), 481-499.
doi: 10.1017/S0017089511000085. |
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