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

* Corresponding author: Khenissi Moez

Received  January 2022 Revised  April 2022 Published  June 2022 Early access  April 2022

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.

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
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. AlabauP. 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-BoussouiraZ. 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. BardosG. 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. BassamD. MercierS. 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. GhoulM. 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. KassemaA. MortadaL. 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. AlabauP. 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-BoussouiraZ. 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. BardosG. 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. BassamD. MercierS. 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. GhoulM. 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. KassemaA. MortadaL. 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.

Figure 1.  $ u(x, t) $  Figure 2. $ v(x, t) $
Figure 3.  $ u(x, t) $  Figure 4. $ v(x, t) $
Figure 5.  $ u(x, t) $  Figure 6. $ v(x, t) $
Figure 7.  $ u(x, t) $  Figure 8. $ v(x, t) $
Figure 9.  $ u(x, t) $  Figure 10. $ v(x, t) $
Figure 11.  $ u(x, t) $  Figure 12. $ v(x, t) $
Figure 13.  $\log(E(t)/E(0))$
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