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October  2021, 14(10): 3641-3657. doi: 10.3934/dcdss.2020434

Feedback stabilization of bilinear coupled hyperbolic systems

Ilyasse Lamrani 1, , Imad El Harraki 2,3,, , Ali Boutoulout 4, and  Fatima-Zahrae El Alaoui 4,

1. 

TSI Team, Department of Mathematics, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco
2. 

Department of Industrial Engineering, National Superior School of Mines, Rabat, Morocco
3. 

LERMA, Mohammadia Engineering School, Mohamed V University in Rabat, Morocco
4. 

TSI Team, Department of Mathematics, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco

* Corresponding author: Imad El Harraki

Received  November 2019 Revised  July 2020 Published  October 2021 Early access  November 2020

This paper studies the problem of stabilization of some coupled hyperbolic systems using nonlinear feedback. We give a sufficient condition for exponential stabilization by bilinear feedback control. The specificity of the control used is that it acts on only one equation. The results obtained are illustrated by some examples where a theorem of Mehrenberger has been used for the observability of compactly perturbed systems [18].

Keywords: Coupled systems, bilinear control, observability, wave equation, exponential stabilization.
Mathematics Subject Classification: Primary: 93D15, 93D22; Secondary: 93B07.
Citation: Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3641-3657. doi: 10.3934/dcdss.2020434
References:
[1]

F. Alabeau-Boussouira, P. Cannarsa and V. Kormonik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ, 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.

[2]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim, 41 (2002), 511-541.  doi: 10.1137/S0363012901385368.

[3]

K. Ammari and S. Nicaise, Polynomial and analytic stabilization of a wave equation coupled with an Euler-Bernoulli beam, Math. Methods Appl. Sci, 32 (2009), 556-576.  doi: 10.1002/mma.1052.

[4]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar, 123 (2009), 1-10.  doi: 10.1007/s10474-009-8011-7.

[5]

K. Ammari, M. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system, Networks & Heterogeneous Media, 4 (2009), 19-34.  doi: 10.3934/nhm.2009.4.19.

[6]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM COCV, 6 (2001), 361-386.  doi: 10.1051/cocv:2001114.

[7]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, 2nd edition, SIAM, 2005.

[8]

J. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, J. Appl. Math. Optim, 5 (1979), 169-179.  doi: 10.1007/BF01442552.

[9] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional System, Academic Press, 1993. 
[10]

E. A. Benhassi, K. Ammari, S. Boulite and L. Maniar, Exponential energy decay of some coupled second order systems, Semigroup Forum, 86 (2013), 362-382.  doi: 10.1007/s00233-012-9440-0.

[11]

L. Berrahmoune, Stabilization and decay estimate for distributed bilinear systems, Systems & Control Letters, 36 (1999), 167–171. doi: 10.1016/S0167-6911(98)00065-6.

[12]

I. Bochicchio, C. Giorgi and E. Vuk, On the viscoelastic coupled suspension bridge, Evolution Equations & Control Theory, 3 (2014), 373-397.  doi: 10.3934/eect.2014.3.373.

[13]

J. Charles, M. Mbekhta and H. Queffélec, Analyse Fonctionnelle et Théorie des Opérateurs, Dunod, 2010.

[14]

I. El Harraki and A. Boutoulout, Controllability of the wave equation via multiplicative controls, IMA Journal of Mathematical Control and Information, 35 (2018), 393-409.  doi: 10.1093/imamci/dnw055.

[15]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal Math, 46 (1989), 245-258. 

[16]

S. Hongying and W. Xiang-Sheng Wang, Global dynamics of a coupled epidemic model, Discrete & Continuous Dynamical Systems - B, 22 (2017), 1575-1585.  doi: 10.3934/dcdsb.2017076.

[17]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, 1st edition, Masson, Paris, 1998.

[18]

M. Mehrenberger, Observability of coupled systems, Acta Math. Hungar, 4 (2004), 321-348.  doi: 10.1023/B:AMHU.0000028832.47891.09.

[19]

M. Ouzahra, Exponential and weak stabilization of constrained bilinear systems, SIAM J. Control Optim, 48 (2010), 3962-3974.  doi: 10.1137/080739161.

[20]

A. Pazy, Semi-Groups of Linear Operators and Applications to Partial Differential Equations, 1st edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Annali di Matematica Pura ed Applicata (Ⅳ), CXLVI (1987), 65-96.  doi: 10.1007/BF01762360.

[22]

A. Soufyane, Uniform stability of coupled second order equations, Electron. J. Diff. Equ, 25 (2001), 1-10. 

[23]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, 1st edition, Birkhauser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.

[24]

L. Yu, Exact Controllability of the Lazer-McKenna Suspension Bridge Equation, Ph.D thesis, Nevada University in Las Vegas, 2014.

[25]

A. Wehbe and W. Youssef, Observabilité et contrôlabilité exacte indirecte interne par un contrôle localement distribué de systèmes d'équations couplées, Comptes Rendus Mathématique, 348 (2010), 1169-1173.  doi: 10.1016/j.crma.2010.10.013.

[26]

E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm. in Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.

show all references

References:
[1]

F. Alabeau-Boussouira, P. Cannarsa and V. Kormonik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ, 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.

[2]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim, 41 (2002), 511-541.  doi: 10.1137/S0363012901385368.

[3]

K. Ammari and S. Nicaise, Polynomial and analytic stabilization of a wave equation coupled with an Euler-Bernoulli beam, Math. Methods Appl. Sci, 32 (2009), 556-576.  doi: 10.1002/mma.1052.

[4]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar, 123 (2009), 1-10.  doi: 10.1007/s10474-009-8011-7.

[5]

K. Ammari, M. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system, Networks & Heterogeneous Media, 4 (2009), 19-34.  doi: 10.3934/nhm.2009.4.19.

[6]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM COCV, 6 (2001), 361-386.  doi: 10.1051/cocv:2001114.

[7]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, 2nd edition, SIAM, 2005.

[8]

J. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, J. Appl. Math. Optim, 5 (1979), 169-179.  doi: 10.1007/BF01442552.

[9] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional System, Academic Press, 1993. 
[10]

E. A. Benhassi, K. Ammari, S. Boulite and L. Maniar, Exponential energy decay of some coupled second order systems, Semigroup Forum, 86 (2013), 362-382.  doi: 10.1007/s00233-012-9440-0.

[11]

L. Berrahmoune, Stabilization and decay estimate for distributed bilinear systems, Systems & Control Letters, 36 (1999), 167–171. doi: 10.1016/S0167-6911(98)00065-6.

[12]

I. Bochicchio, C. Giorgi and E. Vuk, On the viscoelastic coupled suspension bridge, Evolution Equations & Control Theory, 3 (2014), 373-397.  doi: 10.3934/eect.2014.3.373.

[13]

J. Charles, M. Mbekhta and H. Queffélec, Analyse Fonctionnelle et Théorie des Opérateurs, Dunod, 2010.

[14]

I. El Harraki and A. Boutoulout, Controllability of the wave equation via multiplicative controls, IMA Journal of Mathematical Control and Information, 35 (2018), 393-409.  doi: 10.1093/imamci/dnw055.

[15]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal Math, 46 (1989), 245-258. 

[16]

S. Hongying and W. Xiang-Sheng Wang, Global dynamics of a coupled epidemic model, Discrete & Continuous Dynamical Systems - B, 22 (2017), 1575-1585.  doi: 10.3934/dcdsb.2017076.

[17]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, 1st edition, Masson, Paris, 1998.

[18]

M. Mehrenberger, Observability of coupled systems, Acta Math. Hungar, 4 (2004), 321-348.  doi: 10.1023/B:AMHU.0000028832.47891.09.

[19]

M. Ouzahra, Exponential and weak stabilization of constrained bilinear systems, SIAM J. Control Optim, 48 (2010), 3962-3974.  doi: 10.1137/080739161.

[20]

A. Pazy, Semi-Groups of Linear Operators and Applications to Partial Differential Equations, 1st edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Annali di Matematica Pura ed Applicata (Ⅳ), CXLVI (1987), 65-96.  doi: 10.1007/BF01762360.

[22]

A. Soufyane, Uniform stability of coupled second order equations, Electron. J. Diff. Equ, 25 (2001), 1-10. 

[23]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, 1st edition, Birkhauser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.

[24]

L. Yu, Exact Controllability of the Lazer-McKenna Suspension Bridge Equation, Ph.D thesis, Nevada University in Las Vegas, 2014.

[25]

A. Wehbe and W. Youssef, Observabilité et contrôlabilité exacte indirecte interne par un contrôle localement distribué de systèmes d'équations couplées, Comptes Rendus Mathématique, 348 (2010), 1169-1173.  doi: 10.1016/j.crma.2010.10.013.

[26]

E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm. in Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.

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