Exponential stability of the  wave equation withboundary  time-varying delay

Serge Nicaise; Cristina Pignotti; Julie Valein

doi:10.3934/dcdss.2011.4.693

Article Contents

2011, Volume 4, Issue 3: 693-722. Doi: 10.3934/dcdss.2011.4.693

This issue Previous Article An identification problem for a linear evolution equation in a Banach space and applications Next Article Linear evolution equations with strongly measurable families and application to the Dirac equation

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

Received: February 28, 2009

Revised: October 31, 2009

Published: May 2011

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35L05, 93D15.

Citation:

Related Papers

Cited by

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.