# American Institute of Mathematical Sciences

2011, 4(3): 693-722. doi: 10.3934/dcdss.2011.4.693

## Exponential stability of the wave equation with boundary time-varying delay

 1 Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9 2 Dipartimento di Matematica Pura e Applicata, Università di L'Aquila, Via Vetoio, Loc. Coppito, 67010 L'Aquila 3 Institut Elie Cartan de Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

Received  March 2009 Revised  November 2009 Published  November 2010

We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain $\Omega\subset\RR^n.$ Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version of the model.
Citation: Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
##### References:
 [1] F. Ali Mehmeti, "Nonlinear Waves in Networks,", Mathematical Research, 80 (1994). [2] G. Chen, Control and stabilization for the wave equation in a bounded domain I,, SIAM J. Control Optim., 17 (1979), 66. [3] G. Chen, Control and stabilization for the wave equation in a bounded domain II,, SIAM J. Control Optim., 19 (1981), 114. [4] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Israel J. Math., 11 (1972), 57. [5] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,, SIAM J. Control Optim., 26 (1988), 697. [6] R. Datko, Two examples of ill-posedness with respect to time delays revisited,, IEEE Trans. Automatic Control, 42 (1997), 511. [7] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations,, SIAM J. Control Optim., 24 (1986), 152. [8] L. C. Evans, Nonlinear evolution equations in an arbitrary Banach space,, Israel J. Math., 26 (1977), 1. [9] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 21 (1985). [10] T. Kato, Nonlinear semigroups and evolution equations,, J. Math. Soc. Japan, 19 (1967), 508. [11] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type,, C.I.M.E., (1976), 125. [12] T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,", Lezioni Fermiane, (1985). [13] V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197. [14] V. Komornik, Exact controllability and stabilization, the multiplier method,, RAM, 36 (1994). [15] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1980), 33. [16] J. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. [17] J. Lagnese, Note on boundary stabilization of wave equations,, SIAM J. Control Optim., 26 (1988), 1250. [18] I. Lasiecka and R. Triggiani, Uniform exponential decay of wave equations in a bounded region with $L_2(0,T; L_2(\Sigma))$-feedback control in the Dirichlet boundary conditions,, J. Differential Equations, 66 (1987), 340. [19] I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13. [20] C. Y. Lin, Time-dependent nonlinear evolution equations,, Differential Integral Equations, 15 (2002), 257. [21] J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications. Vol. 1,", Travaux et Recherches Mathématiques, 17 (1968). [22] S. Nicaise and C. Pignotti, Boundary stabilization of Maxwell's equations with space-time variable coefficients,, ESAIM Control Optim. Calc. Var., 9 (2003), 563. [23] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedback,, SIAM J. Control Optim., 45 (2006), 1561. [24] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425. [25] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays,, to appear in Discrete Contin. Dyn. Syst. Ser. S., (). [26] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Math. Sciences, 44 (1983). [27] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Math. Surveys Monographs, 49 (1997). [28] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optim. Calc. Var., 12 (2006), 770. [29] E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.

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##### References:
 [1] F. Ali Mehmeti, "Nonlinear Waves in Networks,", Mathematical Research, 80 (1994). [2] G. Chen, Control and stabilization for the wave equation in a bounded domain I,, SIAM J. Control Optim., 17 (1979), 66. [3] G. Chen, Control and stabilization for the wave equation in a bounded domain II,, SIAM J. Control Optim., 19 (1981), 114. [4] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Israel J. Math., 11 (1972), 57. [5] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,, SIAM J. Control Optim., 26 (1988), 697. [6] R. Datko, Two examples of ill-posedness with respect to time delays revisited,, IEEE Trans. Automatic Control, 42 (1997), 511. [7] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations,, SIAM J. Control Optim., 24 (1986), 152. [8] L. C. Evans, Nonlinear evolution equations in an arbitrary Banach space,, Israel J. Math., 26 (1977), 1. [9] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 21 (1985). [10] T. Kato, Nonlinear semigroups and evolution equations,, J. Math. Soc. Japan, 19 (1967), 508. [11] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type,, C.I.M.E., (1976), 125. [12] T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,", Lezioni Fermiane, (1985). [13] V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197. [14] V. Komornik, Exact controllability and stabilization, the multiplier method,, RAM, 36 (1994). [15] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1980), 33. [16] J. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. [17] J. Lagnese, Note on boundary stabilization of wave equations,, SIAM J. Control Optim., 26 (1988), 1250. [18] I. Lasiecka and R. Triggiani, Uniform exponential decay of wave equations in a bounded region with $L_2(0,T; L_2(\Sigma))$-feedback control in the Dirichlet boundary conditions,, J. Differential Equations, 66 (1987), 340. [19] I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13. [20] C. Y. Lin, Time-dependent nonlinear evolution equations,, Differential Integral Equations, 15 (2002), 257. [21] J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications. Vol. 1,", Travaux et Recherches Mathématiques, 17 (1968). [22] S. Nicaise and C. Pignotti, Boundary stabilization of Maxwell's equations with space-time variable coefficients,, ESAIM Control Optim. Calc. Var., 9 (2003), 563. [23] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedback,, SIAM J. Control Optim., 45 (2006), 1561. [24] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425. [25] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays,, to appear in Discrete Contin. Dyn. Syst. Ser. S., (). [26] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Math. Sciences, 44 (1983). [27] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Math. Surveys Monographs, 49 (1997). [28] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optim. Calc. Var., 12 (2006), 770. [29] E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.
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