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Boundary exact controllability for a porous elastic Timoshenko system. (English) Zbl 07250666
Summary: In this paper, we consider a one-dimensional system governed by two partial differential equations. Such a system models phenomena in engineering, such as vibrations in beams or deformation of elastic bodies with porosity. By using the HUM method, we prove that the system is boundary exactly controllable in the usual energy space. We will also determine the minimum time allowed by the method for the controllability to occur.
93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
Full Text: DOI
[1] Akil, M.; Chitour, Y.; Ghader, M.; Wehbe, A., Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary, (to appear) in Asymptotic Anal
[2] Júnior, D. S. Almeida; Ramos, A. J. A.; Santos, M. L., Observability inequality for the \hbox{finite}-difference semi-discretization of the 1-d coupled wave equations, Adv. Comput. Math. 41 (2015), 105-130
[3] Júnior, D. S. Almeida; Santos, M. L.; Rivera, J. E. Muñoz, Stability to weakly dissipative Timoshenko systems, Math. Methods Appl. Sci. 36 (2013), 1965-1976
[4] Aouadi, M.; Campo, M.; Copetti, M. I. M.; Fernández, J. R., Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects, Z. Angew. Math. Phys. 70 (2019), Article ID 117, 26 pages
[5] Araruna, F. D.; Zuazua, E., Controllability of the Kirchhoff system for beams as a limit of the Mindlin-Timoshenko system, SIAM J. Control Optim. 47 (2008), 1909-1938
[6] Cavalcanti, M. M.; Cavalcanti, V. N. Domingos; Nascimento, F. A. Falcão; Lasiecka, I.; Rodrigues, J. H., Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (2014), 1189-1206
[7] Cowin, S. C.; Nunziato, J. W., Linear elastic materials with voids, J. Elasticity 13 (1983), 125-147
[8] Dell’Oro, F.; Pata, V., On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equations 257 (2014), 523-548
[9] Dridi, H.; Djebabla, A., On the stabilization of linear porous elastic materials by microtemperature effect and porous damping, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 66 (2020), 13-25
[10] Infante, J. A.; Zuazua, E., Boundary observability for the space semi-discretizations of the 1-d wave equation, M2AN, Math. Model. Numer. Anal. 33 (1999), 407-438
[11] Lagnese, J. E.; Leugering, G.; Schmidt, E. J. P. G., Control of planar networks of Timoshenko beams, SIAM J. Control Optim. 31 (1993), 780-811
[12] Lagnese, J. E.; Lions, J.-L., Modelling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées 6. Masson, Paris (1988)
[13] Lions, J.-L., Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées 9. Masson, Paris (1988), French
[14] Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), 1-68
[15] Lions, J.-L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications. Volume III, Die Grundlehren der mathematischen Wissenschaften 183. Springer, Berlin (1973)
[16] Magaña, A.; Quintanilla, R., On the time decay of solutions in one-dimensional theories of porous materials, Int. J. Solids Struct. 43 (2006), 3414-3427
[17] Medeiros, L. A., Exact controllability for a Timoshenko model of vibrations of beams, Adv. Math. Sci. Appl. 2 (1993), 47-61
[18] Medeiros, L. A.; Miranda, M. M.; Lourêdo, A. T., Introduction Exact Control Theory: Method HUM, EDUEPB, Campina Grande (2013)
[19] Mercier, D.; Régnier, V., Decay rate of the Timoshenko system with one boundary damping, Evol. Equ. Control Theory 8 (2019), 423
[20] Rivera, J. E. Muñoz; Quintanilla, R., On the polynomial decay in elastic solids with voids, J. Math. Anal. Appl. 338 (2008), 1296-1309
[21] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44. Springer, New York (1983)
[22] Ramos, A. J. A.; Júnior, D. S. Almeida; Freitas, M. M.; Santos, M. J. dos; Santos, A. R., Exponential stabilization for porous elastic system with one boundary dissipation, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 66 (2020), 113-134
[23] Raposo, C. A.; Apalara, T. A.; Ribeiro, J. O., Analyticity to transmission problem with delay in porous-elasticity, J. Math. Anal. Appl. 466 (2018), 819-834
[24] Raposo, C. A.; Bastos, W. D.; Santos, M. L., A transmission problem for the Timoshenko system, Comput. Appl. Math. 26 (2007), 215-234
[25] Said-Houari, B.; Laskri, Y., A stability result of a Timoshenko system with a delay term in the internal feedback, Appl. Math. Comput. 217 (2010), 2857-2869
[26] Santos, M. L.; Júnior, D. S. Almeida, On porous-elastic system with localized damping, Z. Angew. Math. Phys. 67 (2016), Article ID 63, 18 pages
[27] Shubov, M. A., Exact controllability of damped Timoshenko beam, IMA J. Math. Control Inf. 17 (2000), 375-395
[28] Soufyane, A., Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 731-734 French
[29] Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Lond. Edinb. Dubl. Phil. Mag., Ser. VI. 41 (1921), 744-746
[30] Zhang, C.; Hu, X., Exact controllability of a Timoshenko beam with dynamical boundary, J. Math. Kyoto Univ. 47 (2007), 643-655
[31] Zuazua, E., Exact controllability for the semilinear wave equation, J. Math. Pures Appl., IX. Sér. 69 (1990), 1-31
[32] Zuazua, E., Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev. 47 (2005), 197-243
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