# American Institute of Mathematical Sciences

March  2019, 9(1): 97-116. doi: 10.3934/mcrf.2019005

## Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions

 Lebanese University-Faculty of Sciences, Khawarezmi laboratory for mathematics and applications-(KALMA), Beirut, Lebanon

Received  November 2017 Revised  April 2018 Published  August 2018

Fund Project: This research is supported by the Lebanese University.

In this paper, we consider a multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system in the absence of the compactness of the resolvent and without any additional geometric conditions. Next, we show that our system is not uniformly stable in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions and by using the exponential decay of the wave equation with a standard damping, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative.

Citation: Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control and Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005
##### References:
 [1] Z. Abbas and S. Nicaise, The multidimensional wave equation with generalized acoustic boundary conditions, SIAM J. Control Optim., 53 (2015), 2582-2607.  doi: 10.1137/140971348. [2] Z. Achouri, N.-E. Amroun and A. Benaissa, The Euler-Bernouilli beam equation with boundary dissipation of fractional derivative type, Mathematical Methods in the Applied Sciences, 40 (2017), 3837-3854.  doi: 10.1002/mma.4267. [3] F. Alabau-Boussouira, J. Prüss and R. Zacher, Exponential and Polynomial stability of a wave equation for boundary memory damping with singular kernels, Comptes Rendues Mathématiques, 347 (2009), 277-282.  doi: 10.1016/j.crma.2009.01.005. [4] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3. [5] R. L. Bagley and P. J. Torvik, A different approach to the analysis of viscoelasticity damped structures, The American Institue of Aeronautics and Astronautics, 21 (1983), 741-748. [6] R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mechn., 51 (1983), 294-298. [7] R. L. Bagley and P. J. Torvik, A Theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheonology, 27 (1983), 201-210.  doi: 10.1122/1.549724. [8] 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. [9] A. Bátkai, K. J. Engel, J. Prüss and R. Shnaubelt, Polynomial stability of operator semigroup, Mth. Nashr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429. [10] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semigroup on Banach spaces, Journal of Evolution Equations, 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1. [11] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379.  doi: 10.1137/0316023. [12] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann, 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0. [13] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [14] M. Caputo, Vibrations of an infinite plate with a frequency independant, Q. J. Acoustic Soc. Am., 60 (1976), 634-639. [15] H. Dai and H. Zhang, Exponential growth for wave equation with fractional boundary dissipation and boundary source term, Boundary Value Problems, 2014 (2014), 1-8.  doi: 10.1186/s13661-014-0138-y. [16] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical system in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56. [17] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6. [18] I. Lasiecka and R. Triggiani, Uniform Stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.  doi: 10.1007/BF01182480. [19] J. L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1 (1968), ⅹⅹ+372 pp. [20] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4. [21] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC Research Notes in Mathematics, Boca Raton, FL, 1999. [22] P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632.  doi: 10.1137/S0363012903437319. [23] F. Mainardi and E. Bonetti, The application of real order derivatives in linear viscoelasticity, Rheol. Acta., 26 (1988), 64-67.  doi: 10.1007/978-3-642-49337-9_11. [24] M. Matignon, J. Audounet and G. Montesny, Energy decay rate for wave equations with damping of fractional order, Fourth interantional conference on Mathematical and Numerical Aspects of Wave propagation Phenomena, (1998), 638-640. [25] B. Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematics Control and information, 23 (2006), 237-257.  doi: 10.1093/imamci/dni056. [26] B. Mbodje and G. Montesny, Boundary fractional derivative control of the wave equation, IEEE Transactions onAutomatic Control, 40 (1995), 378-382.  doi: 10.1109/9.341815. [27] N. Najdi, Étude de la Stabilisation Exponentielle et Polynomiale de Certains Systèmes D'équations Couplées par des Contrôles Indirects Bornés ou non Bornés, Thèse université de Valenciennes, http://ged.univ-valenciennes.fr/nuxeo/site/esupversions/aaac617d-95a5-4b80-8240-0dd043f20ee5 (2016). [28] S. Nicaise, M. A. Sammoury, D. Mercier and A. Wehbe, Indirect Stability of the wave equation with a dynamic boundary control, Mathematische Naschrichten, 291 (2018), 1114-1146. [29] H. J. Park and J. R. Kang, Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation, IMA J. Appl. Math, 76 (2011), 340-350.  doi: 10.1093/imamat/hxq040. [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sciences, 24, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-5561-1. [31] J. Prüss, On the spectrum of $C_0-$semigroup, Trans. Amer. Math. Soc, 284 (1984), 847-857.  doi: 10.2307/1999112. [32] B. Rao and A. Wehbe, Polynomial energy decay rate and strong stability of Kirchoff plates with non-compact resolvent, J. Evol. Equ, 5 (2005), 137-152.  doi: 10.1007/s00028-005-0171-5. [33] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach [Engl. Trans. from the Russian], 1993. [34] M. Slemrod, Feedbacks stabilization of a linear system in a Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.

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##### References:
 [1] Z. Abbas and S. Nicaise, The multidimensional wave equation with generalized acoustic boundary conditions, SIAM J. Control Optim., 53 (2015), 2582-2607.  doi: 10.1137/140971348. [2] Z. Achouri, N.-E. Amroun and A. Benaissa, The Euler-Bernouilli beam equation with boundary dissipation of fractional derivative type, Mathematical Methods in the Applied Sciences, 40 (2017), 3837-3854.  doi: 10.1002/mma.4267. [3] F. Alabau-Boussouira, J. Prüss and R. Zacher, Exponential and Polynomial stability of a wave equation for boundary memory damping with singular kernels, Comptes Rendues Mathématiques, 347 (2009), 277-282.  doi: 10.1016/j.crma.2009.01.005. [4] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3. [5] R. L. Bagley and P. J. Torvik, A different approach to the analysis of viscoelasticity damped structures, The American Institue of Aeronautics and Astronautics, 21 (1983), 741-748. [6] R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mechn., 51 (1983), 294-298. [7] R. L. Bagley and P. J. Torvik, A Theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheonology, 27 (1983), 201-210.  doi: 10.1122/1.549724. [8] 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. [9] A. Bátkai, K. J. Engel, J. Prüss and R. Shnaubelt, Polynomial stability of operator semigroup, Mth. Nashr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429. [10] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semigroup on Banach spaces, Journal of Evolution Equations, 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1. [11] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379.  doi: 10.1137/0316023. [12] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann, 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0. [13] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [14] M. Caputo, Vibrations of an infinite plate with a frequency independant, Q. J. Acoustic Soc. Am., 60 (1976), 634-639. [15] H. Dai and H. Zhang, Exponential growth for wave equation with fractional boundary dissipation and boundary source term, Boundary Value Problems, 2014 (2014), 1-8.  doi: 10.1186/s13661-014-0138-y. [16] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical system in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56. [17] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6. [18] I. Lasiecka and R. Triggiani, Uniform Stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.  doi: 10.1007/BF01182480. [19] J. L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1 (1968), ⅹⅹ+372 pp. [20] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4. [21] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC Research Notes in Mathematics, Boca Raton, FL, 1999. [22] P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632.  doi: 10.1137/S0363012903437319. [23] F. Mainardi and E. Bonetti, The application of real order derivatives in linear viscoelasticity, Rheol. Acta., 26 (1988), 64-67.  doi: 10.1007/978-3-642-49337-9_11. [24] M. Matignon, J. Audounet and G. Montesny, Energy decay rate for wave equations with damping of fractional order, Fourth interantional conference on Mathematical and Numerical Aspects of Wave propagation Phenomena, (1998), 638-640. [25] B. Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematics Control and information, 23 (2006), 237-257.  doi: 10.1093/imamci/dni056. [26] B. Mbodje and G. Montesny, Boundary fractional derivative control of the wave equation, IEEE Transactions onAutomatic Control, 40 (1995), 378-382.  doi: 10.1109/9.341815. [27] N. Najdi, Étude de la Stabilisation Exponentielle et Polynomiale de Certains Systèmes D'équations Couplées par des Contrôles Indirects Bornés ou non Bornés, Thèse université de Valenciennes, http://ged.univ-valenciennes.fr/nuxeo/site/esupversions/aaac617d-95a5-4b80-8240-0dd043f20ee5 (2016). [28] S. Nicaise, M. A. Sammoury, D. Mercier and A. Wehbe, Indirect Stability of the wave equation with a dynamic boundary control, Mathematische Naschrichten, 291 (2018), 1114-1146. [29] H. J. Park and J. R. Kang, Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation, IMA J. Appl. Math, 76 (2011), 340-350.  doi: 10.1093/imamat/hxq040. [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sciences, 24, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-5561-1. [31] J. Prüss, On the spectrum of $C_0-$semigroup, Trans. Amer. Math. Soc, 284 (1984), 847-857.  doi: 10.2307/1999112. [32] B. Rao and A. Wehbe, Polynomial energy decay rate and strong stability of Kirchoff plates with non-compact resolvent, J. Evol. Equ, 5 (2005), 137-152.  doi: 10.1007/s00028-005-0171-5. [33] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach [Engl. Trans. from the Russian], 1993. [34] M. Slemrod, Feedbacks stabilization of a linear system in a Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.
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