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June
2022, 15(6): 1573-1597.
doi: 10.3934/dcdss.2022099
Local indirect stabilization of same coupled evolution systems through resolvent estimates
University of Sousse, ESSTHS, LAMMDA, Tunisia |
In this paper, we consider same systems of two coupled equations (wave-wave, Schrödinger-Schrödinger) in a bounded domain. Only one of the two equations is directly damped by a localized damping term (indirect stabilization). Under geometric control conditions on both coupling and damping regions (internal or boundary), we establish the energy decay rate by means of a suitable resolvent estimate. The numerical contribution is interpreted to confirm the theoretical result of a wave-wave system.
Keywords:
Coupled system,
indirect stabilization,
energy decay,
boundary damping,
internal damping,
resolvent estimate.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Ayechi Radhia, Khenissi Moez. Local indirect stabilization of same coupled evolution systems through resolvent estimates. Discrete and Continuous Dynamical Systems - S,
2022, 15
(6)
: 1573-1597.
doi: 10.3934/dcdss.2022099
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References:
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F. Alabau,
Stabilisation frontière indirecte de systéme, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020.
doi: 10.1016/S0764-4442(99)80316-4. |
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F. Alabau, P. Cannarsa and V. Komornik,
Indirect internal stabilisation of weakly coupled evolution equations, Journal of Evolution Equation, 2 (2002), 127-150.
doi: 10.1007/s00028-002-8083-0. |
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F. Alabau-Boussouira, R. Brockett, O. Glass, J. le Rousseau and E. Zuazua, Control of Partial Differential Equations, Lecture Notes in Mathematics, 2048. Fondazione CIME/CIME Foundation Subseries, Springer, Heidelberg, Fondazione C.I.M.E., Florence, 2012.
doi: 10.1007/978-3-642-27893-8. |
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F. Alabau-Boussouira, Z. Wang and L. Yu,
A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.
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L. Aloui and M. Daoulatli,
Stabilization of two coupled wave equations on a compact manifold with boundary, J. Math. Anal. Appl., 436 (2016), 944-969.
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F. Ammar-Khodja and A. Bader,
Stability of systems of one dimensional wave equations by internal or boundary control force, SIAM J. Control Optim., 39 (2001), 127.
|
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C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
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M. Bassam, D. Mercier, S. Nicaise and A. Wehbe,
Stabilisation frontière indirecte du système de Timoshenko, C. R. Math. Acad. Sci. Paris, 349 (2011), 379-384.
doi: 10.1016/j.crma.2011.03.011. |
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A. Borichev and Y. Tomilov,
Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, 347 (2010), 455-478.
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L. Gearhart,
Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.
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T.-E. Ghoul, M. Khenissi and B. Said-Houari,
On the stability of the Bresse system with frictional damping, J. Math. Anal. Appl., 455 (2017), 1870-1898.
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R. Guglielmi,
Indirect stabilization of hyperbolic systems through resolvent estimates, Evol. Equ. Control Theory, 6 (2017), 59-75.
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F. L. Huang,
Strong asymptotic stability of linear dynamical systems in Banach spaces, Journal of Differential Equations, 104 (1993), 307-324.
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B. Jacop and H. Zwart,
On the Hautus test for exponentially stable ${\rm{C}}_{0}$-groups, SIAM J. Control Optim., 48 (2009), 1275-1288.
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B. Kapitonov,
Stabilization and simultaneous boundary controllability for a class of evolution systems, Comput. Appl. Math., 17 (1998), 149-160.
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B. V. Kapitonov,
Uniform stabilization and simultaneous exact boundary controllability for a pair of hyperbolic systems, Siberian Math. J., 35 (1994), 722-734.
doi: 10.1007/BF02106615. |
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B. V. Kapitonov,
Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., 15 (1996), 199-212.
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C. Kassema, A. Mortada, L. Toufayli and A. Wehbe,
Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions, Comptes Rendus Mathematique, 357 (2019), 494-512.
|
| [19] |
G. Lebeau,
Équation des ondes amorties, Algebraic and Geometric Methods in Mathematical Physics, Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht, 19 (1996), 73-109.
|
| [20] |
G. Lebeau and L. Robbiano.,
Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.
|
| [21] |
K. Liu,
Locally distributed control and damping for the conservative systems, Journal on Control and Optimization, 35 (1997), 1574-1590.
|
| [22] |
W.-J. Liu and E. Zuazua,
Uniform stabilization of the higher-dimensional system of thermoelasticity with a nonlinear boundary feedback, Quarterly of Applied Mathematics, 59 (2001), 269-314.
doi: 10.1090/qam/1828455. |
| [23] |
Z. Liu and B. Rao,
Energy decay rate of the thermoelastic Bresse system, Zeitschrift für Angewandte Mathematik und Physik, 60 (2009), 54-69.
|
| [24] |
M. Negreanu and E. Zuazua,
Uniform boundary controlability of a discrete 1-D wave equation, Systems Control Lett., 48 (2003), 261-279.
doi: 10.1016/S0167-6911(02)00271-2. |
| [25] |
A. Pazy, Semigroups of Linear Operator and applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [26] |
J. Prüss,
On the spectrum of ${\rm{C}}_{0}$ semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857.
|
| [27] |
D. L. Russell,
A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-354.
doi: 10.1006/jmaa.1993.1071. |
| [28] |
L. Tebou,
Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60.
doi: 10.3934/mcrf.2012.2.45. |
show all references
References:
| [1] |
F. Alabau,
Stabilisation frontière indirecte de systéme, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020.
doi: 10.1016/S0764-4442(99)80316-4. |
| [2] |
F. Alabau, P. Cannarsa and V. Komornik,
Indirect internal stabilisation of weakly coupled evolution equations, Journal of Evolution Equation, 2 (2002), 127-150.
doi: 10.1007/s00028-002-8083-0. |
| [3] |
F. Alabau-Boussouira, R. Brockett, O. Glass, J. le Rousseau and E. Zuazua, Control of Partial Differential Equations, Lecture Notes in Mathematics, 2048. Fondazione CIME/CIME Foundation Subseries, Springer, Heidelberg, Fondazione C.I.M.E., Florence, 2012.
doi: 10.1007/978-3-642-27893-8. |
| [4] |
F. Alabau-Boussouira, Z. Wang and L. Yu,
A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.
doi: 10.1051/cocv/2016011. |
| [5] |
L. Aloui and M. Daoulatli,
Stabilization of two coupled wave equations on a compact manifold with boundary, J. Math. Anal. Appl., 436 (2016), 944-969.
doi: 10.1016/j.jmaa.2015.12.014. |
| [6] |
F. Ammar-Khodja and A. Bader,
Stability of systems of one dimensional wave equations by internal or boundary control force, SIAM J. Control Optim., 39 (2001), 127.
|
| [7] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
| [8] |
M. Bassam, D. Mercier, S. Nicaise and A. Wehbe,
Stabilisation frontière indirecte du système de Timoshenko, C. R. Math. Acad. Sci. Paris, 349 (2011), 379-384.
doi: 10.1016/j.crma.2011.03.011. |
| [9] |
A. Borichev and Y. Tomilov,
Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, 347 (2010), 455-478.
doi: 10.1007/s00208-009-0439-0. |
| [10] |
L. Gearhart,
Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.
doi: 10.1090/S0002-9947-1978-0461206-1. |
| [11] |
T.-E. Ghoul, M. Khenissi and B. Said-Houari,
On the stability of the Bresse system with frictional damping, J. Math. Anal. Appl., 455 (2017), 1870-1898.
doi: 10.1016/j.jmaa.2017.04.027. |
| [12] |
R. Guglielmi,
Indirect stabilization of hyperbolic systems through resolvent estimates, Evol. Equ. Control Theory, 6 (2017), 59-75.
doi: 10.3934/eect.2017004. |
| [13] |
F. L. Huang,
Strong asymptotic stability of linear dynamical systems in Banach spaces, Journal of Differential Equations, 104 (1993), 307-324.
doi: 10.1006/jdeq.1993.1074. |
| [14] |
B. Jacop and H. Zwart,
On the Hautus test for exponentially stable ${\rm{C}}_{0}$-groups, SIAM J. Control Optim., 48 (2009), 1275-1288.
doi: 10.1137/080724733. |
| [15] |
B. Kapitonov,
Stabilization and simultaneous boundary controllability for a class of evolution systems, Comput. Appl. Math., 17 (1998), 149-160.
|
| [16] |
B. V. Kapitonov,
Uniform stabilization and simultaneous exact boundary controllability for a pair of hyperbolic systems, Siberian Math. J., 35 (1994), 722-734.
doi: 10.1007/BF02106615. |
| [17] |
B. V. Kapitonov,
Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., 15 (1996), 199-212.
|
| [18] |
C. Kassema, A. Mortada, L. Toufayli and A. Wehbe,
Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions, Comptes Rendus Mathematique, 357 (2019), 494-512.
|
| [19] |
G. Lebeau,
Équation des ondes amorties, Algebraic and Geometric Methods in Mathematical Physics, Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht, 19 (1996), 73-109.
|
| [20] |
G. Lebeau and L. Robbiano.,
Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.
|
| [21] |
K. Liu,
Locally distributed control and damping for the conservative systems, Journal on Control and Optimization, 35 (1997), 1574-1590.
|
| [22] |
W.-J. Liu and E. Zuazua,
Uniform stabilization of the higher-dimensional system of thermoelasticity with a nonlinear boundary feedback, Quarterly of Applied Mathematics, 59 (2001), 269-314.
doi: 10.1090/qam/1828455. |
| [23] |
Z. Liu and B. Rao,
Energy decay rate of the thermoelastic Bresse system, Zeitschrift für Angewandte Mathematik und Physik, 60 (2009), 54-69.
|
| [24] |
M. Negreanu and E. Zuazua,
Uniform boundary controlability of a discrete 1-D wave equation, Systems Control Lett., 48 (2003), 261-279.
doi: 10.1016/S0167-6911(02)00271-2. |
| [25] |
A. Pazy, Semigroups of Linear Operator and applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [26] |
J. Prüss,
On the spectrum of ${\rm{C}}_{0}$ semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857.
|
| [27] |
D. L. Russell,
A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-354.
doi: 10.1006/jmaa.1993.1071. |
| [28] |
L. Tebou,
Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60.
doi: 10.3934/mcrf.2012.2.45. |
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