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Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks
Laboratory ACEDP, Djillali Liabes university, 22000 Sidi Bel Abbes, Algeria |
$\left(|u_{t}|^{\gamma-2}u_{t}\right)_{t}-Lu-\int_{0}^{t}g(t-s)L u(s)ds+ \mu_{1} \psi(u_{t}(x, t))+ \mu_{2} \psi(u_{t}(x, t-\tau(t)))=0.$ |
References:
[1] |
C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed Positive Feedback Can Stabilize Oscillatory System, ACC, San Francisco, (1993), 3106-3107. |
[2] |
V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1.![]() ![]() ![]() |
[3] |
A. Benaissa and A. Guesmia,
Energy decay for wave equations of ϕ-Laplacian type with weakly nonlinear dissipation, Electron. J. Differ. Equations, 2008 (2008), 1-22.
|
[4] |
M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka,
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, Jour. Diff. Equa., 236 (2007), 407-459.
doi: 10.1016/j.jde.2007.02.004. |
[5] |
G. Chen,
Control and stabilization for the wave equation in a bounded domain, Part Ⅰ, SIAM J. Control Optim., 17 (1979), 66-81.
doi: 10.1137/0317007. |
[6] |
G. Chen,
Control and stabilization for the wave equation in a bounded domain, Part Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.
doi: 10.1137/0319009. |
[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.
doi: 10.1137/0324007. |
[8] |
M. Eller, J. E. Lagnese and S. Nicaise,
Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Computational. Appl. Math., 21 (2002), 135-165.
|
[9] |
A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161-179. |
[10] |
T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, Hyperbolicity, C. I. M. E. Summer Sch. , Springer, Heidelberg, 72 (2011), 125-191. |
[11] |
T. Kato,
Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures]. Scuola Normale Superiore, Pisa 1985. |
[12] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.
![]() ![]() |
[13] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of semilinear wave equations with nonlinear boundary dampin, Diff. Integr. Equa., 6 (1993), 507-533.
|
[14] |
J. L. Lions,
Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969. |
[15] |
W. J. Liu and E. Zuazua,
Decay rates for dissipative wave equations, Ricerche di Matematica, 48 (1999), 61-75.
|
[16] |
P. Martinez and J. Vancostenoble,
Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim, 39 (2000), 776-797.
doi: 10.1137/S0363012999354211. |
[17] |
M. Nakao,
Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl, 60 (1977), 542-549.
|
[18] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim, 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[19] |
S. Nicaise and C. Pignotti,
Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integr. Equat, 21 (2008), 935-958.
|
[20] |
S. Nicaise and J. Valein,
Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var, 16 (2010), 420-456.
doi: 10.1051/cocv/2009007. |
[21] |
S. Nicaise, C. Pignotti and J. Valein,
Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.
doi: 10.3934/dcdss.2011.4.693. |
[22] |
S. Nicaise, J. Valein and E. Fridman,
Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.
doi: 10.3934/dcdss.2009.2.559. |
[23] |
J. Y. Park and S. H. Park, General decay for a nonlinear beam equation with weak dissipation, J. Math. Phys. , 51 (2010), 073508, 8pp. |
[24] |
W. Rudin, Real and Complex Analysis, second edition, McGraw-Hill, Inc, New York, 1974.
![]() ![]() |
[25] |
F. G. Shinskey, Process Control Systems, McGraw-Hill Book Company, 1967.
![]() |
[26] |
I. H. Suh and Z. Bien,
Use of time delay action in the controller design, IEEE Trans. Autom. Control, 25 (1980), 600-603.
|
[27] |
C. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var, 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[28] |
Q. C. Zhong, Robust Control of Time-delay Systems, Springer, London, 2006.
![]() |
show all references
References:
[1] |
C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed Positive Feedback Can Stabilize Oscillatory System, ACC, San Francisco, (1993), 3106-3107. |
[2] |
V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1.![]() ![]() ![]() |
[3] |
A. Benaissa and A. Guesmia,
Energy decay for wave equations of ϕ-Laplacian type with weakly nonlinear dissipation, Electron. J. Differ. Equations, 2008 (2008), 1-22.
|
[4] |
M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka,
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, Jour. Diff. Equa., 236 (2007), 407-459.
doi: 10.1016/j.jde.2007.02.004. |
[5] |
G. Chen,
Control and stabilization for the wave equation in a bounded domain, Part Ⅰ, SIAM J. Control Optim., 17 (1979), 66-81.
doi: 10.1137/0317007. |
[6] |
G. Chen,
Control and stabilization for the wave equation in a bounded domain, Part Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.
doi: 10.1137/0319009. |
[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.
doi: 10.1137/0324007. |
[8] |
M. Eller, J. E. Lagnese and S. Nicaise,
Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Computational. Appl. Math., 21 (2002), 135-165.
|
[9] |
A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161-179. |
[10] |
T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, Hyperbolicity, C. I. M. E. Summer Sch. , Springer, Heidelberg, 72 (2011), 125-191. |
[11] |
T. Kato,
Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures]. Scuola Normale Superiore, Pisa 1985. |
[12] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.
![]() ![]() |
[13] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of semilinear wave equations with nonlinear boundary dampin, Diff. Integr. Equa., 6 (1993), 507-533.
|
[14] |
J. L. Lions,
Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969. |
[15] |
W. J. Liu and E. Zuazua,
Decay rates for dissipative wave equations, Ricerche di Matematica, 48 (1999), 61-75.
|
[16] |
P. Martinez and J. Vancostenoble,
Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim, 39 (2000), 776-797.
doi: 10.1137/S0363012999354211. |
[17] |
M. Nakao,
Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl, 60 (1977), 542-549.
|
[18] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim, 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[19] |
S. Nicaise and C. Pignotti,
Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integr. Equat, 21 (2008), 935-958.
|
[20] |
S. Nicaise and J. Valein,
Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var, 16 (2010), 420-456.
doi: 10.1051/cocv/2009007. |
[21] |
S. Nicaise, C. Pignotti and J. Valein,
Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.
doi: 10.3934/dcdss.2011.4.693. |
[22] |
S. Nicaise, J. Valein and E. Fridman,
Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.
doi: 10.3934/dcdss.2009.2.559. |
[23] |
J. Y. Park and S. H. Park, General decay for a nonlinear beam equation with weak dissipation, J. Math. Phys. , 51 (2010), 073508, 8pp. |
[24] |
W. Rudin, Real and Complex Analysis, second edition, McGraw-Hill, Inc, New York, 1974.
![]() ![]() |
[25] |
F. G. Shinskey, Process Control Systems, McGraw-Hill Book Company, 1967.
![]() |
[26] |
I. H. Suh and Z. Bien,
Use of time delay action in the controller design, IEEE Trans. Autom. Control, 25 (1980), 600-603.
|
[27] |
C. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var, 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[28] |
Q. C. Zhong, Robust Control of Time-delay Systems, Springer, London, 2006.
![]() |
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