Frequency domain approach for the polynomial stability of a system of partially damped wave equations. (English) Zbl 1152.35069
The authors study stability of a system of wave equations which are weakly coupled and partially damped. Using a frequency domain approach based on the growth of the resolvent on the imaginary axis, the polynomial energy decay rate for smooth initial data is established. It is shown that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations.
Reviewer: Anatoly Martynyuk (Kyïv)
References:
| [1] | Afilal, M.; Ammar Khodja, F., Stability of coupled second order equations, Comput. Appl. Math., 19, 91-107 (2000) · Zbl 1119.93046 |
| [2] | Agmon, S., Lectures on Elliptic Boundary Values Problems (1965), Van Nostrand · Zbl 0151.20203 |
| [3] | Ammar Khodja, F.; Benabdallah, A., Sufficient conditions for uniform stabilization of second order equations by dynamical controllers, Dyn. Contin. Discrete Impuls. Syst., 7, 207-222 (2000) · Zbl 0946.35013 |
| [4] | 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 |
| [5] | 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 |
| [6] | Alabau-Boussouira, F., Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41, 511-541 (2002) · Zbl 1031.35023 |
| [7] | Alabau-Boussouira, F., A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42, 871-906 (2003) · Zbl 1125.93311 |
| [8] | Bátkai, A.; Engel, K.-J.; Prüss, J.; Schnaubelt, R., Polynomial stability of operator semigroups, Math. Nachr., 279, 1425-1440 (2006) · Zbl 1118.47034 |
| [9] | Benchimol, C. D., A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16, 373-379 (1978) · Zbl 0384.93035 |
| [10] | Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Math., vol. 160 (2004), Cambridge Univ. Press · Zbl 1055.11002 |
| [11] | Garofalo, N.; Lin, F. H., Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40, 347-366 (1987) · Zbl 0674.35007 |
| [12] | Huang, F., Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Chinese Ann. Differential Equations, 1, 43-56 (1985) · Zbl 0593.34048 |
| [13] | Lasiecka, I., Mathematical Control Theory of Coupled PDE’s, CBMS-NSF Lecture Notes (2002), SIAM · Zbl 1032.93002 |
| [14] | Lasiecka, I.; Tataru, D., Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6, 3, 507-533 (1993) · Zbl 0803.35088 |
| [15] | Lasiecka, I.; Lions, J. L.; Triggiani, R., Non-homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9), 65, 2, 149-192 (1986) · Zbl 0631.35051 |
| [16] | Lasiecka, I.; Toundyakov, D., Energy decay rates for semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal., 64, 1757-1797 (2006) · Zbl 1096.35021 |
| [17] | Lebeau, G., Equation des ondes amorties, Math. Phys. Stud., 19, 73-109 (1996) · Zbl 0863.58068 |
| [18] | Lebeau, G.; Robbiano, L., Stabilisation de l’équation des ondes par le bord, Duke Math. J., 86, 465-491 (1997) · Zbl 0884.58093 |
| [19] | Lions, J.-L.; Magenes, E., Problèmes Non-Homogenes et Applications, vol. 1 (1968), Dumond: Dumond Paris · Zbl 0165.10801 |
| [20] | Lions, J.-L., Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, vol. 1 (1984), Masson: Masson Paris |
| [21] | W. Littman, L. Markus, Some recent results on control and stabilization of flexible structures, in: Proc. COMCON Workshop, Montpelier, 1987; W. Littman, L. Markus, Some recent results on control and stabilization of flexible structures, in: Proc. COMCON Workshop, Montpelier, 1987 · Zbl 0761.73080 |
| [22] | Littman, W.; Liu, B., On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59, 17-34 (1999) · Zbl 0921.35131 |
| [23] | Liu, Z.; Rao, B., Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56, 630-644 (2005) · Zbl 1100.47036 |
| [24] | Liu, Z.; Zheng, S., Semigroup Associated with Dissipative Systems, Res. Notes Math., vol. 394 (1999), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · Zbl 0924.73003 |
| [25] | Loreti, P.; Rao, B., Compensation spectrale et taux de décroissance optimal de l’energie de systèmes partiellement amortis, C. R. Math. Acad. Sci. Paris. C. R. Math. Acad. Sci. Paris, SIAM J. Control Optim., 45, 1612-1632 (2006), English transl.: Optimal energy decay rate for partially damped systems by spectral compensation · Zbl 1127.93054 |
| [26] | J. Muñoz Rivera, R. Quintanilla, On the time polynomial decay in elastic solids with voids, preprint, 2006; J. Muñoz Rivera, R. Quintanilla, On the time polynomial decay in elastic solids with voids, preprint, 2006 |
| [27] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1984), Springer: Springer New York · Zbl 0516.47023 |
| [28] | Prüss, J., On the spectrum of \(C_0\)-semigroups, Trans. Amer. Math. Soc., 284, 847-857 (1984) · Zbl 0572.47030 |
| [29] | Rao, B., Stabilization of a plate equation with dynamical boundary control, J. SIAM Control Optim., 36, 148-164 (1998) · Zbl 0911.35021 |
| [30] | Rao, B.; Wehbe, A., Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent, J. Evol. Equ., 5, 137-152 (2005) · Zbl 1091.93033 |
| [31] | Rauch, J.; Zhang, X.; Zuazua, E., Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84, 407-470 (2005) · Zbl 1077.35030 |
| [32] | Russell, D., A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173, 339-354 (1993) · Zbl 0771.73045 |
| [33] | Zhang, X.; Zuazua, E., Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differential Equations, 204, 380-438 (2004) · Zbl 1064.93008 |
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