American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications. Vol. 1," Travaux et Recherches Mathématiques, 17, Dunod, Paris, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[26]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[27]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Math. Surveys Monographs, Vol. 49, Amer. Math. Soc., 1997.  Google Scholar

[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.  Google Scholar

[29]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

J. L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications. Vol. 1," Travaux et Recherches Mathématiques, 17, Dunod, Paris, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[26]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[27]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Math. Surveys Monographs, Vol. 49, Amer. Math. Soc., 1997.  Google Scholar

[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.  Google Scholar

[29]

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

[1]

Serge Nicaise, Julie Valein. Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks & Heterogeneous Media, 2007, 2 (3) : 425-479. doi: 10.3934/nhm.2007.2.425
[2]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024
[3]

Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057
[4]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057
[5]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021
[6]

Serge Nicaise. Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks. Mathematical Control & Related Fields, 2011, 1 (3) : 331-352. doi: 10.3934/mcrf.2011.1.331
[7]

Bei Gong, Zhen-Hu Ning, Fengyan Yang. Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $. Evolution Equations & Control Theory, 2021, 10 (2) : 321-331. doi: 10.3934/eect.2020068
[8]

Xiaorui Wang, Genqi Xu. Uniform stabilization of a wave equation with partial Dirichlet delayed control. Evolution Equations & Control Theory, 2020, 9 (2) : 509-533. doi: 10.3934/eect.2020022
[9]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[10]

Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005
[11]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure & Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
[12]

Imene Aicha Djebour, Takéo Takahashi, Julie Valein. Feedback stabilization of parabolic systems with input delay. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021027
[13]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004
[14]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029
[15]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
[16]

Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014
[17]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319
[18]

Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815
[19]

Alberto Bressan, Fabio S. Priuli. Nearly optimal patchy feedbacks. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 687-701. doi: 10.3934/dcds.2008.21.687
[20]

Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences