Exponential stabilization of magnetoelastic waves in a Mindlin-Timoshenko plate by localized internal damping. (English) Zbl 1320.74068
Summary: This article is a continuation of our earlier work in [the author, ibid. 63, No. 6, 1047–1065 (2012; Zbl 1254.74069)] on the polynomial stabilization of a linear model for the magnetoelastic interactions in a two-dimensional electrically conducting Mindlin-Timoshenko plate. We introduce nonlinear damping that is effective only in a small portion of the interior of the plate. It turns out that the model is uniformly exponentially stable when the function \(\mathbf{p}(\mathbf{x},\mathbf{U}_t)\), that represents the locally distributed damping, behaves linearly near the origin. However, the use of Mindlin-Timoshenko plate theory in the model enforces a restriction on the region occupied by the plate.
MSC:
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 93B15 | Realizations from input-output data |
Keywords:
Mindlin-Timoshenko plate; magnetic field; dissipation; localized viscous damping; exponential stabilityCitations:
Zbl 1254.74069References:
| [1] | Avalos G., Toundykov D.: Boundary stabilization of structural acoustic interactions with interface on a Reissner-Mindlin plate. Nonlinear Anal.- Real 12, 2985-3013 (2011) · Zbl 1231.35253 |
| [2] | Borichev A., Tomilov Y.: Optimal polynomial decay rate of functions and operator semigroups. Math. Ann. 347, 455-478 (2010) · Zbl 1185.47044 · doi:10.1007/s00208-009-0439-0 |
| [3] | Cavalcanti M.M., Domingos Cavalcanti V.N., Falcão Nascimento F.A., 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, 1189-1206 (2014) · Zbl 1316.35034 · doi:10.1007/s00033-013-0380-7 |
| [4] | Charão R.C., Oliveira J.C., Perla Menzala G.: Energy decay rates of magnetoelastic waves in a bounded conductive medium. Discrete Cont. Dyn. Syst. Ser. A 25(3), 797-821 (2009) · Zbl 1176.35030 · doi:10.3934/dcds.2009.25.797 |
| [5] | Charão R.C., Oliveira J.C., Perla Menzala G.: Decay rates of magnetoelastic waves in an unbounded conductive medium. Electron. J. Differ. Equ. 127, 1-14 (2011) · Zbl 1233.35033 |
| [6] | Dautray R., Lions J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2: Functional and Variational Methods. Springer, Berlin (2000) · Zbl 0942.35001 · doi:10.1007/978-3-642-58090-1 |
| [7] | Dunkin J.W., Eringen A.C.: On the propagation of waves in an electromagnetic elastic solid. Int. J. Eng. Sci. 1, 461-495 (1963) · Zbl 0128.42704 · doi:10.1016/0020-7225(63)90004-1 |
| [8] | Duyckaerts T.: Stabilisation Haute Fréquence d’équationes aux Derivées Partielles Linéaires, Thèse de Doctorat. Université Paris XI, Orsay (2004) |
| [9] | Fernandes A., Pouget J.: An accurate modelling of piezoelectric multi-layer plates. Eur. J. Mech. 21, 629-651 (2002) · Zbl 1062.74013 · doi:10.1016/S0997-7538(02)01224-X |
| [10] | Fernández Sare H.D.: On the stability of Mindlin-Timoshenko plates. Q. Appl. Math. LXVII(2), 249-263 (2009) · Zbl 1175.35020 · doi:10.1090/S0033-569X-09-01110-2 |
| [11] | Grobbelaar-Van Dalsen M.: On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam. J. Math. Anal. Appl. 320, 121-144 (2006) · Zbl 1130.35367 · doi:10.1016/j.jmaa.2005.06.034 |
| [12] | Grobbelaar-Van Dalsen M.: Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface. Math. Meth. Appl. Sci. 29, 1749-1766 (2006) · Zbl 1110.35010 · doi:10.1002/mma.737 |
| [13] | Grobbelaar-Van Dalsen M.: On a structural acoustic model which incorporates shear and thermal effects in the structural component. J. Math. Anal. Appl. 341, 1253-1270 (2008) · Zbl 1137.35436 · doi:10.1016/j.jmaa.2007.10.073 |
| [14] | Grobbelaar-Van Dalsen M.: Strong stabilization of a structural acoustic model which incorporates shear and thermal effects in the structural component. Math. Meth. Appl. Sci. 33, 1433-1445 (2010) · Zbl 1204.35043 |
| [15] | Grobbelaar-Van Dalsen M.: Strong stabilization of models incorporating the thermoelastic Reissner-Mindlin plate equations with second sound. Appl. Anal. 90, 1419-1449 (2011) · Zbl 1228.35052 · doi:10.1080/00036811.2010.530259 |
| [16] | Grobbelaar-Van Dalsen M.: On the dissipative effect of a magnetic field in a Mindlin-Timoshenko plate model. Z. Angew. Math. Phys. 63, 1047-1065 (2012) · Zbl 1254.74069 · doi:10.1007/s00033-012-0206-z |
| [17] | Grobbelaar-Van Dalsen M.: Stabilization of a thermoelastic Mindlin-Timoshenko plate model revisited. Z. Angew. Math. Phys. 64, 1305-1325 (2013) · Zbl 1310.74031 · doi:10.1007/s00033-012-0289-6 |
| [18] | Grobbelaar-Van Dalsen M.: Polynomial decay rates of a thermoelastic Mindlin-Timoshenko plate model with Dirichlet boundary conditions. Z. Angew. Math. Phys. 66, 113-128 (2015) · Zbl 1317.74052 · doi:10.1007/s00033-013-0391-4 |
| [19] | Kim J.U., Renardy Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25, 1417-1429 (1987) · Zbl 0632.93057 · doi:10.1137/0325078 |
| [20] | Krommer M.: Piezoelectric vibrations of composite Reissner-Mindlin type plates. J. Sound Vib. 263, 871-891 (2003) · doi:10.1016/S0022-460X(02)01169-0 |
| [21] | Lagnese, J.: Boundary Stabilization of Thin Plates. SIAM Studies in Applied Mathematics SIAM, Philadelphia (1989) · Zbl 0696.73034 |
| [22] | Lasiecka I., Tataru D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507-533 (1993) · Zbl 0803.35088 |
| [23] | Lions, J.L.: Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson, RMA 8, Paris (1988) · Zbl 0653.93002 |
| [24] | Mindlin R.D.: Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18, 31-38 (1951) · Zbl 0044.40101 |
| [25] | Mindlin R.D.: Thickness-shear and flexural vibrations of crystal plates. J. Appl. Phys. 22, 316-323 (1951) · Zbl 0042.18603 · doi:10.1063/1.1699948 |
| [26] | Muñoz Rivera J.E., Racke R.: Polynomial stability in two-dimensional magneto-elasticity. IMA J. Appl. Math. 66, 359-384 (2001) · Zbl 1029.74020 · doi:10.1093/imamat/66.3.269 |
| [27] | Muñoz Rivera J.E., Racke R.: Mildly dissipative nonlinear Timoshenko systems. J. Math. Anal. Appl. 276, 248-278 (2002) · Zbl 1106.35333 · doi:10.1016/S0022-247X(02)00436-5 |
| [28] | Muñoz Rivera J.E., Racke R.: Global stability for damped Timoshenko sytems. Discrete Contin. Dyn. Syst. Ser. B 9(6), 1625-1639 (2003) · Zbl 1047.35023 · doi:10.3934/dcds.2003.9.1625 |
| [29] | Muñoz Rivera J.E., Portillo Oquendo H.: Asymptotic behavior on a Mindlin-Timoshenko plate with viscoelastic dissipation of memory type on the boundary. Funkcial. Ekvac. 46, 363-382 (2003) · Zbl 1259.74015 · doi:10.1619/fesi.46.363 |
| [30] | Muñoz Rivera J.E., Lima Santos M.de : Polynomial stability to three-dimensional magnetoelastic waves. Acta Appl. Math. 76, 265-281 (2003) · Zbl 1023.74022 · doi:10.1023/A:1023223517930 |
| [31] | Nakao M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305, 403-417 (1996) · Zbl 0856.35084 · doi:10.1007/BF01444231 |
| [32] | Oliveira J.C., Charào R.C.: Stabilization of a locally damped thermoelastic system. Comput. Appl. Math. 27, 319-357 (2008) · Zbl 1171.35019 |
| [33] | Paria G.: On magneto-thermo-elastic plane waves. Proc. Cambr. Phil. Soc. 58, 527-531 (1962) · doi:10.1017/S030500410003680X |
| [34] | Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Springer, New York (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [35] | Perla Menzala G., Zuazua E.: Energy decay of magnetoelastic waves in a bounded conductive medium. Asymptot. Anal. 18, 349-362 (1998) · Zbl 0931.35017 |
| [36] | Pokojovy, M.: On stability of hyperbolic thermoelastic Reissner-Mindlin Timoshenko plates. MMA (to appear). doi:10.1002/mma.3140 (2014) · Zbl 1320.35059 |
| [37] | Purushotham C.M.: Magneto-elastic coupling effects on the propagation of harmonic waves in an electrically conducting elastic plate. Proc. Cambr. Phil. Soc. 63, 503-511 (1967) · Zbl 0161.44902 · doi:10.1017/S0305004100041451 |
| [38] | Racke R., Muñoz Rivera J.: Magneto-thermoelasticity, large time behaviour for linear systems. Adv. Differ. Equ. 6, 359-384 (2001) · Zbl 1023.74017 |
| [39] | Raposo C.A., Ferreira J., Santos M.L., Castro N.N.O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 7, 535-541 (2005) · Zbl 1072.74033 · doi:10.1016/j.aml.2004.03.017 |
| [40] | Reissner E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184-191 (1944) · Zbl 0061.42501 |
| [41] | Santos M.L., Almeida Júnior D.S., Muñoz Rivera J.E.: The stability number of the Timoshenko system with second sound. J. Differ. Equ. 253, 2715-2733 (2013) · Zbl 1250.74013 · doi:10.1016/j.jde.2012.07.012 |
| [42] | Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635-664 (1983) · Zbl 0524.76099 · doi:10.1002/cpa.3160360506 |
| [43] | Tébou L.R.T.: Well-posedness and energy decay estimates for the damped wave equation with Lr localizing coefficient. Comm. Partial Differ. Eqs. 23, 1839-1855 (1998) · Zbl 0918.35078 |
| [44] | Tébou L.R.T.: Stabilization of the wave equation with localized nonlinear damping. J. Differ. Equ. 145, 502-524 (1998) · Zbl 0916.35069 · doi:10.1006/jdeq.1998.3416 |
| [45] | Timoshenko S.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. 41, 744-746 (1921) · doi:10.1080/14786442108636264 |
| [46] | Wehbe A., Youssef W.: Stabilization of the uniform Timoshenko beam by one locally distributed feedback. Appl. Anal. 88, 1067-1078 (2009) · Zbl 1172.74024 · doi:10.1080/00036810903156149 |
| [47] | Willson A.J.: The propagation of magneto-thermo-elastic plane waves. Proc. Cambr. Phil. Soc. 59, 483-488 (1963) · Zbl 0131.45101 · doi:10.1017/S0305004100037087 |
| [48] | Yang J.S.: Equations for elastic plates with partially electroded piezoelectric actuators in flexure with shear deformation and rotary inertia. J. Intell. Mat. Syst. Struct. 8, 444-451 (1997) · doi:10.1177/1045389X9700800507 |
| [49] | Yang J.S.: Equations for elastic plates with partially electroded piezoelectric actuators and higher order electric fields. Smart Mat. Struct. 8, 73-82 (1999) · doi:10.1088/0964-1726/8/1/008 |
| [50] | Yang J.S., Yang X., Turner A., Kopinski J.A., Pastore R.A.: Two-dimensional equations for electrostatic plates with relatively large shear deformations. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 765-771 (2003) · doi:10.1109/TUFFC.2003.1214496 |
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