Stability of neutral delay differential equations modeling wave propagation in cracked media

Previous Article
Linear model of traffic flow in an isolated network
PROC Home
This Issue
Next Article
Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space

2015, 2015(special): 678-685. doi: 10.3934/proc.2015.0678

Stability of neutral delay differential equations modeling wave propagation in cracked media

Stéphane Junca ^1, and Bruno Lombard ^2,

1.	Laboratoire J.A. Dieudonné, UMR 7351 CNRS, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
2.	Laboratoire de Mécanique et dAcoustique, UPR 7051 CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France

Received September 2014 Revised August 2015 Published November 2015

Abstract
Related Papers

Propagation of elastic waves is studied in a 1D medium containing $N$ cracks modeled by nonlinear jump conditions. The case $N=1$ is fully understood. When $N>1$, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case $N=2$, some mathematical results about the existence, uniqueness and attractivity of periodic solutions have been obtained in 2012 by the authors, under the assumption of small sources. The difficulty of analysis follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. Here we propose a new result of stability in the homogeneous case, based on an energy method. One deduces the asymptotic stability of the zero steady-state. Extension to $N=3$ cracks is also considered, leading to new results in particular configurations.

Keywords: linear elasticity, stability, nonlinear cracks, Neutral delay differential equations, energy method..

Mathematics Subject Classification: Primary: 34K40, 34K20; Secondary: 74J20, 74K1.

Citation: Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678

References:

[1]	R. Bellman and K. Cooke, Differential Difference Equations,, Academic Press, (1963).
[2]	J. Cermák and J. Hrabalová, Delay-dependent stability criteria for neutral delay differential and difference equations,, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 4577.
[3]	H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations,, Funkcial. Ekvac., 34 (1991), 187.
[4]	P. S. Gromova, Stability of solutions of nonlinear equations in the asymptotically critical case,, Mathematical Notes, 1-6 (1967), 1.
[5]	J. K. Hale, S. M. Verduyn Lunel, and M. Sjoerd, Introduction to functional-differential equations,, Applied Mathematical Sciences, (1993).
[6]	S. Junca, Oscillating Waves for Nonlinear Conservation Laws,, Habilitation dissertation, (2013).
[7]	S. Junca and B. Lombard, Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave,, SIAM J. Appl. Math., 70 (2009), 735.
[8]	S. Junca and B. Lombard, Interaction between periodic elastic waves and two contact nonlinearities,, Math. Models Methods Appl. Sci., 22 (2012).
[9]	S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation,, J. Differential Equations, 256 (2014), 2368.
[10]	B. Lombard, Modélisation numérique de la propagation et de la diffraction d'ondes mécaniques, [in French],, Habilitation dissertation, (2010).
[11]	R. Rabah, G. M. Sklyar, and A. V. Rezounenko, Stability analysis of neutral type systems in Hilbert space,, J. Differential Equations, 214 (2005), 391.
[12]	X. F. Zhou, J. Liang, and T. J. Xiao, Asymptotic stability of the zero solution for degenerate retarded differential equations,, Nonlinear Analysis, 70 (2009), 1415.

show all references

References:

[1]	R. Bellman and K. Cooke, Differential Difference Equations,, Academic Press, (1963).
[2]	J. Cermák and J. Hrabalová, Delay-dependent stability criteria for neutral delay differential and difference equations,, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 4577.
[3]	H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations,, Funkcial. Ekvac., 34 (1991), 187.
[4]	P. S. Gromova, Stability of solutions of nonlinear equations in the asymptotically critical case,, Mathematical Notes, 1-6 (1967), 1.
[5]	J. K. Hale, S. M. Verduyn Lunel, and M. Sjoerd, Introduction to functional-differential equations,, Applied Mathematical Sciences, (1993).
[6]	S. Junca, Oscillating Waves for Nonlinear Conservation Laws,, Habilitation dissertation, (2013).
[7]	S. Junca and B. Lombard, Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave,, SIAM J. Appl. Math., 70 (2009), 735.
[8]	S. Junca and B. Lombard, Interaction between periodic elastic waves and two contact nonlinearities,, Math. Models Methods Appl. Sci., 22 (2012).
[9]	S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation,, J. Differential Equations, 256 (2014), 2368.
[10]	B. Lombard, Modélisation numérique de la propagation et de la diffraction d'ondes mécaniques, [in French],, Habilitation dissertation, (2010).
[11]	R. Rabah, G. M. Sklyar, and A. V. Rezounenko, Stability analysis of neutral type systems in Hilbert space,, J. Differential Equations, 214 (2005), 391.
[12]	X. F. Zhou, J. Liang, and T. J. Xiao, Asymptotic stability of the zero solution for degenerate retarded differential equations,, Nonlinear Analysis, 70 (2009), 1415.

[1]	Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[2]	Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105
[3]	Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[4]	Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906
[5]	Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[6]	Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[7]	Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[8]	Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[9]	Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685
[10]	Mustafa Hasanbulli, Yuri V. Rogovchenko. Classification of nonoscillatory solutions of nonlinear neutral differential equations. Conference Publications, 2009, 2009 (Special) : 340-348. doi: 10.3934/proc.2009.2009.340
[11]	Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[12]	Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[13]	Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[14]	Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[15]	John R. Graef, R. Savithri, E. Thandapani. Oscillatory properties of third order neutral delay differential equations. Conference Publications, 2003, 2003 (Special) : 342-350. doi: 10.3934/proc.2003.2003.342
[16]	Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124-137. doi: 10.3934/proc.1998.1998.124
[17]	Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 67-84. doi: 10.3934/dcdss.2020004
[18]	A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373
[19]	Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[20]	C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002

Impact Factor:

American Institute of Mathematical Sciences