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Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback. (English) Zbl 1191.35164

The authors consider the stabilization of the wave equation with variable coefficients and a delay in the dissipative boundary feedback. Using the Riemannian geometry methods, the energy-perturbed approach and the multiplier skills, they establish the uniform stability of the energy of the closed-loop system.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
35B35 Stability in context of PDEs
93D15 Stabilization of systems by feedback
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[1] Cavalcanti, M.M.; Khemmoudj, A.; Medjden, M., Uniform stabilization of the damped cauchy – ventcel problem with variable coefficients and dynamic boundary conditions, J. math. anal. appl., 328, 900-930, (2007) · Zbl 1107.35024
[2] Chai, S.; Liu, K., Observability inequalities for the transmission of shallow shells, Systems control lett., 55, 726-735, (2006) · Zbl 1100.93004
[3] Chai, Shugen; Guo, Bao-Zhu, Analyticity of a thermoelastic plate with variable coefficients, J. math. anal. appl., 354, 1, 330-338, (2009) · Zbl 1161.74037
[4] Chen, G., Control and stabilization for the wave equation in a bounded domain, part I, SIAM J. control optim., 17, 66-81, (1979) · Zbl 0402.93016
[5] Chen, G., Control and stabilization for the wave equation in a bounded domain, part II, SIAM J. control optim., 19, 114-122, (1981) · Zbl 0461.93037
[6] 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
[7] Datko, R., Two examples of ill-posedness with respect to time delays revisited, IEEE trans. automat. control, 42, 511-515, (1997) · Zbl 0878.73046
[8] 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
[9] Feng, S.J.; Feng, D.X., Boundary stabilization of wave equations with variable coefficients, Sci. China ser. A, 44, 3, 345-350, (2001) · Zbl 1054.35021
[10] Gulliver, R.; Lasiecka, I.; Littman, W.; Triggiani, R., The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, (), 73-181 · Zbl 1067.35014
[11] Hamchi, I., Uniform decay rates for second-order hyperbolic equations with variable coefficients, Asymptot. anal., 57, 1-2, 71-82, (2008) · Zbl 1148.35303
[12] Komornik, V., Rapid boundary stabilization of the wave equation, SIAM J. control optim., 29, 197-208, (1991) · Zbl 0749.35018
[13] Komornik, V., Exact controllability and stabilization. the multiplier method, Res. appl. math., vol. 36, (1994), Masson, John Wiley Paris, Chichester · Zbl 0937.93003
[14] Lagnese, J., Decay of solutions of the wave equations in a bounded region with boundary dissipation, J. differential equations, 50, 163-182, (1983) · Zbl 0536.35043
[15] Lagnese, J., Note on boundary stabilization of wave equations, SIAM J. control optim., 26, 1250-1256, (1988) · Zbl 0657.93052
[16] Lasiecka, I.; Triggiani, R., Uniform exponential energy decay of wave equations in a bounded region with L2(0,∞; L2(γ))-feedback control in the Dirichlet boundary conditions, J. differential equations, 66, 340-390, (1987) · Zbl 0629.93047
[17] Lasiecka, I.; Triggiani, R.; Yao, P.F., Inverse/observability estimates for second-order hyperbolic equations with variable systems, J. math. anal. appl., 235, 13-57, (1999) · Zbl 0931.35022
[18] 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, 5, 1561-1585, (2006) · Zbl 1180.35095
[19] Nicaise, S.; Pignotti, C., Internal and boundary observability estimates for the heterogeneous Maxwell’s system, Appl. math. optim., 54, 47-70, (2006) · Zbl 1100.93037
[20] Rebiai, S.E., Uniform energy decay of Schrödinger equations with variable coefficients, IMA J. math. control inform., 20, 335-345, (2003) · Zbl 1042.93037
[21] Xu, G.Q.; Yung, S.P.; Li, L.K., Stabilization of wave systems with input delay in the boundary control, ESAIM control optim. calc. var., 12, 4, 770-785, (2006) · Zbl 1105.35016
[22] Lasiecka, I.; Triggiani, R., Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. math. optim., 25, 189-224, (1992) · Zbl 0780.93082
[23] Yao, P.F., On the observability inequalities for the exact controllability of the wave equation with variable coefficients, SIAM J. control optim., 37, 6, 1568-1599, (1999) · Zbl 0951.35069
[24] Yao, P.F., Observability inequalities for the shallow shell, SIAM J. control optim., 38, 6, 1729-1756, (2000) · Zbl 0974.35013
[25] Yao, P.F., Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. differential equations, 241, 62-93, (2007) · Zbl 1214.35037
[26] P.F. Yao, Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., doi:10.1007/s00245-009-9088-7 · Zbl 1185.93018
[27] Yao, P.F., Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. ann. math. ser. B, 31, 1, 59-70, (2010) · Zbl 1190.35036
[28] Zhang, Z.F.; Yao, P.F., Global smooth solutions of the quasilinear wave equation with internal velocity feedbacks, SIAM J. control optim., 47, 4, 2044-2077, (2008) · Zbl 1357.35225
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