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

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

Serge Nicaise 1, , Cristina Pignotti 2, and  Julie Valein 3,

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.
Keywords: stabilization., Wave equation, delay feedbacks.
Mathematics Subject Classification: 35L05, 93D1.
Citation: Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and 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, Akademie-Verlag, Berlin, 1994.

[2]

G. Chen, Control and stabilization for the wave equation in a bounded domain I, SIAM J. Control Optim., 17 (1979), 66-81.

[3]

G. Chen, Control and stabilization for the wave equation in a bounded domain II, SIAM J. Control Optim., 19 (1981), 114-122.

[4]

M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94.

[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), 697-713.

[6]

R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Automatic Control, 42 (1997), 511-515.

[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), 152-156.

[8]

L. C. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math., 26 (1977), 1-42.

[9]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 21, Pitman, Boston-London-Melbourne, 1985.

[10]

T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520.

[11]

T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, C.I.M.E., II ciclo (1976), 125-191.

[12]

T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems," Lezioni Fermiane, [Fermi Lectures]. Scuola Normale Superiore, Pisa, 1985.

[13]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.

[14]

V. Komornik, Exact controllability and stabilization, the multiplier method, RAM, 36, Masson, Paris, 1994.

[15]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1980), 33-54.

[16]

J. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.

[17]

J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.

[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), 340-390.

[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), 13-57.

[20]

C. Y. Lin, Time-dependent nonlinear evolution equations, Differential Integral Equations, 15 (2002), 257-270.

[21]

J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications. Vol. 1," Travaux et Recherches Mathématiques, 17, Dunod, Paris, 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), 563-578.

[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), 1561-1585.

[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), 425-479.

[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, Springer-Verlag, New York, 1983.

[27]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Math. Surveys Monographs, Vol. 49, Amer. Math. Soc., 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), 770-785.

[29]

E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.

show all references

References:
[1]

F. Ali Mehmeti, "Nonlinear Waves in Networks," Mathematical Research, 80, Akademie-Verlag, Berlin, 1994.

[2]

G. Chen, Control and stabilization for the wave equation in a bounded domain I, SIAM J. Control Optim., 17 (1979), 66-81.

[3]

G. Chen, Control and stabilization for the wave equation in a bounded domain II, SIAM J. Control Optim., 19 (1981), 114-122.

[4]

M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94.

[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), 697-713.

[6]

R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Automatic Control, 42 (1997), 511-515.

[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), 152-156.

[8]

L. C. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math., 26 (1977), 1-42.

[9]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 21, Pitman, Boston-London-Melbourne, 1985.

[10]

T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520.

[11]

T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, C.I.M.E., II ciclo (1976), 125-191.

[12]

T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems," Lezioni Fermiane, [Fermi Lectures]. Scuola Normale Superiore, Pisa, 1985.

[13]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.

[14]

V. Komornik, Exact controllability and stabilization, the multiplier method, RAM, 36, Masson, Paris, 1994.

[15]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1980), 33-54.

[16]

J. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.

[17]

J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.

[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), 340-390.

[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), 13-57.

[20]

C. Y. Lin, Time-dependent nonlinear evolution equations, Differential Integral Equations, 15 (2002), 257-270.

[21]

J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications. Vol. 1," Travaux et Recherches Mathématiques, 17, Dunod, Paris, 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), 563-578.

[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), 1561-1585.

[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), 425-479.

[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, Springer-Verlag, New York, 1983.

[27]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Math. Surveys Monographs, Vol. 49, Amer. Math. Soc., 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), 770-785.

[29]

E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.

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