Indirect boundary stabilization of strongly coupled degenerate hyperbolic systems. (English) Zbl 07871647
Summary: In this paper, we consider the energy decay of a damped hyperbolic system of two degenerate wave equations coupled by velocities when only one equation is directly damped by a linear boundary feedback. To this aim, we first prove that the proposed system is well-posed using the semigroup theory. Then, under the hypothesis that the coupling coefficient is positive and small, we show that the total energy of the whole system decays exponentially. The explicit energy decay rate is established by using the energy multiplier method.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L53 | Initial-boundary value problems for second-order hyperbolic systems |
| 35L80 | Degenerate hyperbolic equations |
| 93D15 | Stabilization of systems by feedback |
| 93D23 | Exponential stability |
References:
| [1] | Ammari, K.; Tucsnak, M., Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6, 361-386, 2001 · Zbl 0992.93039 · doi:10.1051/cocv:2001114 |
| [2] | Bardos, C.; Lebeau, G.; Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 1024-1065, 1992 · Zbl 0786.93009 · doi:10.1137/0330055 |
| [3] | Bardos, C.; Lebeau, G.; Rauch, J., Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problemes hyperboliques, Rend. Sem. Mat. Univ. Politec. Torino, 46, 11-31, 1988 · Zbl 0673.93037 |
| [4] | Chen, G., A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19, 106-113, 1981 · Zbl 0461.93036 · doi:10.1137/0319008 |
| [5] | Cox, S.; Zuazua, E., The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44, 545-573, 1995 · Zbl 0847.35078 · doi:10.1512/iumj.1995.44.2001 |
| [6] | Gugat, M., Sigalotti, M., Tucsnak, M.: Robustness analysis for the boundary control of the string equation. In: Proceedings of the 9th European Control Conference, Kos, Greece (2007) |
| [7] | Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46, 245-258, 1989 · Zbl 0679.93063 |
| [8] | Haraux, A.; Zuazua, E., Decay estimates for some semilinear damped hyperbolic problems, Arch. Ration. Mech. Anal., 100, 191-206, 1988 · Zbl 0654.35070 · doi:10.1007/BF00282203 |
| [9] | Komornik, V., Exact Controllability and Stabilization (the Multiplier Method), 1995, Paris: Wiley, Paris |
| [10] | Lebeau, G.; Robbiano, L., Stabilisation de l’équation des ondes par le bord, Duke Math. J., 86, 465-491, 1997 · Zbl 0884.58093 · doi:10.1215/S0012-7094-97-08614-2 |
| [11] | Lions, JL, Contrôlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués, Tome 1, 1988, Paris: Masson, Paris · Zbl 0653.93002 |
| [12] | Lions, JL, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30, 1-68, 1988 · Zbl 0644.49028 · doi:10.1137/1030001 |
| [13] | Nicaise, S., Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications, Rend. Mat. Appl., 23, 83-116, 2003 · Zbl 1050.35053 |
| [14] | Rauch, J.; Taylor, M.; Phillips, R., Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24, 79-86, 1974 · Zbl 0281.35012 · doi:10.1512/iumj.1975.24.24004 |
| [15] | Tébou, LRT, 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 |
| [16] | Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping, Commun. Partial Differ. Equ., 15, 205-235, 1990 · Zbl 0716.35010 · doi:10.1080/03605309908820684 |
| [17] | Russell, DL, 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 |
| [18] | Alabau-Boussouira, F., Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41, 511-541, 2002 · Zbl 1031.35023 · doi:10.1137/S0363012901385368 |
| [19] | Alabau-Boussouira, F.; Léautaud, M., Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18, 548-582, 2012 · Zbl 1259.35034 · doi:10.1051/cocv/2011106 |
| [20] | Alabau-Boussouira, F.; Cannarsa, P.; Komornik, V., Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2, 127-150, 2002 · Zbl 1011.35018 · doi:10.1007/s00028-002-8083-0 |
| [21] | Khodja, FA; Bader, A., Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39, 1833-1851, 2001 · Zbl 0991.93053 · doi:10.1137/S0363012900366613 |
| [22] | Akil, M.; Ghader, M.; Wehbe, A., The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization, SeMA J., 78, 287-333, 2021 · Zbl 1476.34130 · doi:10.1007/s40324-020-00233-y |
| [23] | Akil, M.; Wehbe, A., Indirect stability of a multidimensional coupled wave equations with one locally boundary fractional damping, Math. Nachr., 295, 2272-2300, 2022 · Zbl 1523.35034 · doi:10.1002/mana.202100185 |
| [24] | Akil, M.; Badawi, H.; Nicaise, S.; Régnier, V., Stabilization of coupled wave equations with viscous damping on cylindrical and non-regular domains: cases without the geometric control condition, Mediterr. J. Math., 19, 271, 2022 · Zbl 1504.35595 · doi:10.1007/s00009-022-02164-6 |
| [25] | Akil, M.; Badawi, H.; Nicaise, S., Stability results of locally coupled wave equations with local Kelvin-Voigt damping: cases when the supports of damping and coupling coefficients are disjoint, Comput. Appl. Math., 41, 240, 2022 · Zbl 1513.35346 · doi:10.1007/s40314-022-01956-6 |
| [26] | Akil, M.; Badawi, H.; Nicaise, S.; Wehbe, A., Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface, Math. Methods Appl. Sci., 44, 6950-6981, 2021 · Zbl 1471.35038 · doi:10.1002/mma.7235 |
| [27] | Wehbe, A.; Issa, I.; Akil, M., Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients, Acta Appl. Math., 171, 1-46, 2021 · Zbl 1465.76008 · doi:10.1007/s10440-021-00384-8 |
| [28] | Akil, M.; Chitour, Y.; Ghader, M.; Wehbe, A., Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary, Asymptot. Anal., 119, 221-280, 2020 · Zbl 1452.35205 |
| [29] | Ammari, K.; Mehrenberger, M., Stabilization of coupled systems, Acta Math. Hungar., 123, 1-10, 2009 · Zbl 1224.93017 · doi:10.1007/s10474-009-8011-7 |
| [30] | Salah, MBH, Stabilization of weakly coupled wave equations through a density term, Eur. J. Control, 58, 315-326, 2021 · Zbl 1458.93212 · doi:10.1016/j.ejcon.2020.07.010 |
| [31] | Allouni, H.; Kesri, M.; Benaissa, A., On the asymptotic behaviour of two coupled strings through a fractional joint damper, Rendiconti del Circolo Matematico di Palermo Series, 2, 69, 613-640, 2020 · Zbl 1447.93277 · doi:10.1007/s12215-019-00423-2 |
| [32] | Gerbi, S.; Kassem, C.; Mortada, A.; Wehbe, A., Exact controllability and stabilization of locally coupled wave equations: theoretical results, Z. Anal. Anwend., 40, 67-96, 2021 · Zbl 1467.35216 · doi:10.4171/zaa/1673 |
| [33] | Kassem, C.; Mortada, A.; Toufayli, L.; Wehbe, A., Local indirect stabilization of n-d system of two coupled wave equations under geometric conditions, C. R. Math., 357, 494-512, 2019 · Zbl 1416.35157 · doi:10.1016/j.crma.2019.06.002 |
| [34] | Liu, Z.; Rao, B., Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335, 860-881, 2007 · Zbl 1152.35069 · doi:10.1016/j.jmaa.2007.02.021 |
| [35] | Alabau-Boussouira, F.: On some recent advances on stabilization for hyperbolic equations. In: Control of Partial Differential Equations, pp. 1-100 (2012) |
| [36] | Bastin, G.; Coron, JM, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, 2016, Cham: Birkhäuser, Cham · Zbl 1377.35001 · doi:10.1007/978-3-319-32062-5 |
| [37] | Alabau-Boussouira, F.; Cannarsa, P.; Leugering, G., Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55, 2052-2087, 2017 · Zbl 1373.35185 · doi:10.1137/15M1020538 |
| [38] | Alabau-Boussouira, F.; Wang, Z.; Yu, L., A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23, 721-749, 2017 · Zbl 1362.35176 · doi:10.1051/cocv/2016011 |
| [39] | Barbu, V., Partial Differential Equations and Boundary Value Problems, 1998, Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0898.35002 · doi:10.1007/978-94-015-9117-1 |
| [40] | Coron, JM, Control and Nonlinearity, 2007, Providence: American Mathematical Society, Providence · Zbl 1140.93002 |
| [41] | Lagnese, JE, Boundary Stabilization of Thin Plates, 1989, Philadelphia: SIAM Studies in Applied Mathematics, Philadelphia · Zbl 0696.73034 · doi:10.1137/1.9781611970821 |
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