×

Stability and instability results for coupled waves with delay term. (English) Zbl 1454.35026

Summary: In this paper, we consider a coupled system of two wave equations. One of these equations is conservative, and the other has damping and delay terms. If the damping acts with more force than the delay term, we show polynomial stability for strong solutions of the system. Explicit decay rates are found, and the optimality of those rates is discussed. On the other hand, if the damping acts with the same or less force than the delay term, then we obtain a result of instability by constructing a sequence of time delays and initial data such that the solutions are not asymptotically stable.
©2020 American Institute of Physics

MSC:

35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L53 Initial-boundary value problems for second-order hyperbolic systems
35R10 Partial functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional-Differential Equations, x+447 (1993), Springer-Verlag: Springer-Verlag, New York · Zbl 0787.34002
[2] Nicaise, S.; Pignotti, C., Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45, 1561-1585 (2006) · Zbl 1180.35095 · doi:10.1137/060648891
[3] Alabau-Boussouira, F.; Nicaise, S.; Pignotti, C., Exponential stability of the wave equation with memory and time delay, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, 1-22 (2014), Springer: Springer, Cham · Zbl 1394.35037
[4] Guesmia, A., Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inf., 30, 507-526 (2013) · Zbl 1279.93090 · doi:10.1093/imamci/dns039
[5] Ammari, K.; Nicaise, S.; Pignotti, C., Feedback boundary stabilization of wave equations with interior delay, Syst. Control Lett., 59, 623-628 (2010) · Zbl 1205.93126 · doi:10.1016/j.sysconle.2010.07.007
[6] Kirane, M.; Said-Houari, B., Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62, 1065-1082 (2011) · Zbl 1242.35163 · doi:10.1007/s00033-011-0145-0
[7] Liu, G.; Zhang, H., Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67, 6 (2016) · Zbl 1347.35144 · doi:10.1007/s00033-015-0593-z
[8] Nicaise, S.; Pignotti, C., Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integr. Equations, 21, 935-958 (2008) · Zbl 1224.35247
[9] Nicaise, S.; Pignotti, C., Stabilization of second-order evolution equations with time delay, Math. Control Signals Syst., 26, 563-588 (2014) · Zbl 1307.35173 · doi:10.1007/s00498-014-0130-1
[10] Wang, D.; Li, G.; Zhu, B., Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl., 09, 1202-1215 (2016) · Zbl 1330.35044 · doi:10.22436/jnsa.009.03.46
[11] Wu, S.-T., Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwan. J. Math., 17, 765-784 (2013) · Zbl 1297.35044 · doi:10.11650/tjm.17.2013.2517
[12] Zitouni, S.; Ardjouni, A.; Zennir, K.; Amiar, R., Existence and exponential stability of solutions for transmission system with varying delay in \(\mathbb{R} \), Math. Moravica, 20, 143-161 (2016) · Zbl 1474.35462 · doi:10.5937/matmor1602143z
[13] Xu, G. Q.; Yung, S. P.; Li, L. K., Stabilization of wave systems with input delay in the boundary control, ESAIM: Control Optim. Calculus Var., 12, 770-785 (2006) · Zbl 1105.35016 · doi:10.1051/cocv:2006021
[14] Datko, R.; Lagnese, J.; Polis, M. P., An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24, 152-156 (1986) · Zbl 0592.93047 · doi:10.1137/0324007
[15] Datko, R., Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26, 697-713 (1988) · Zbl 0643.93050 · doi:10.1137/0326040
[16] Gerbi, S.; Said-Houari, B., Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Appl. Math. Comput., 218, 11900-11910 (2012) · Zbl 1278.35147 · doi:10.1016/j.amc.2012.05.055
[17] Nicaise, S.; Pignotti, C., Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. - Ser. S, 9, 791-813 (2016) · Zbl 1346.35121 · doi:10.3934/dcdss.2016029
[18] Rebiai, S. E.; Sidi Ali, F. Z., Uniform exponential stability of the transmission wave equation with a delay term in the boundary feedback, IMA J. Math. Control Inf., 33, 1-20 (2016) · Zbl 1335.93114 · doi:10.1093/imamci/dnu021
[19] Russell, D. L., A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173, 339-358 (1993) · Zbl 0771.73045 · doi:10.1006/jmaa.1993.1071
[20] Alabau, F.; Cannarsa, P.; Komornik, V., Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equations, 2, 127-150 (2002) · Zbl 1011.35018 · doi:10.1007/s00028-002-8083-0
[21] Oquendo, H. P.; Raya, R. P., Best rates of decay for coupled waves with different propagation speeds, Z. Angew. Math. Phys., 68, 77 (2017) · Zbl 1382.35045 · doi:10.1007/s00033-017-0821-9
[22] Alabau-Boussouira, F.; Léautaud, M., Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calculus Var., 18, 548-582 (2012) · Zbl 1259.35034 · doi:10.1051/cocv/2011106
[23] Benhassi, E. M. A.; Ammari, K.; Boulite, S.; Maniar, L., Exponential energy decay of some coupled second order systems, Semigroup Forum, 86, 362-382 (2013) · Zbl 1270.35096 · doi:10.1007/s00233-012-9440-0
[24] Ferhat, M.; Hakem, A., Global existence and asymptotic behavior for a coupled system of viscoelastic wave equations with a delay term, J. Partial Differ. Equations, 27, 293-317 (2014) · Zbl 1340.35210 · doi:10.4208/jpde.v27.n4.2
[25] Oquendo, H. P.; Pacheco, P. S., Optimal decay for coupled waves with Kelvin-Voigt damping, Appl. Math. Lett., 67, 16-20 (2017) · Zbl 1365.35089 · doi:10.1016/j.aml.2016.11.010
[26] Liu, Z.; Zheng, S., Semigroups Associated with Dissipative Systems, x+206 (1999), Chapman & Hall/CRC: Chapman & Hall/CRC, Boca Raton, FL · Zbl 0924.73003
[27] Borichev, A.; Tomilov, Y., Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347, 455-478 (2010) · Zbl 1185.47044 · doi:10.1007/s00208-009-0439-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.