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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

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