Stability of linear differential equations with a distributed delay

Previous Article
Duffing--van der Pol--type oscillator system and its first integrals
CPAA Home
This Issue
Next Article
Exterior differential systems and prolongations for three important nonlinear partial differential equations

September 2011, 10(5): 1361-1375. doi: 10.3934/cpaa.2011.10.1361

Stability of linear differential equations with a distributed delay

Leonid Berezansky ^1, and Elena Braverman ^2,

1.	Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2.	Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4

Received August 2009 Revised August 2010 Published April 2011

Abstract
Related Papers

We present some new stability results for the scalar linear equation with a distributed delay

$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s) d_s R_k(t,s) =0, h_k(t)\leq t,$ su$p_{t\geq 0}(t-h_k(t))<\infty,$

where the functions involved in the equation are not required to be continuous.
The results are applied to integro-differential equations, equations with several concentrated delays and equations of a mixed type.

Keywords: exponential stability, explicit stability conditions., Distributed delay.

Mathematics Subject Classification: Primary: 34K20; Secondary: 34K2.

Citation: Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361

References:

[1]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191. Academic Press, Boston, MA, 1993.
[2]	V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558-560.
[3]	G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246.
[4]	A. D. Myshkis, "Linear Differential Equations with Retarded Argument," Nauka, Moscow, 1972.
[5]	L. Berezansky and E. Braverman, On oscillation of equations with distributed delay, Zeitschrift für Analysis und ihre Anwendungen, 20 (2001), 489-504.
[6]	L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Mathematical and Computer Modelling, 48 (2008), 287-304. doi: 10.1016/j.mcm.2007.10.003.
[7]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
[8]	N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential equation of evolution type, Differential Equations, 13 (1977), 1915--1925, 2106.
[9]	N. V. Azbelev, L. Berezansky, P. M. Simonov and A. V. Chistyakov, The stability of linear systems with aftereffect. I, Differential Equations, 23 (1987), 745-754, 914, II. Differential Equations, 27 (1991), 383-388, III. Differential Equations, 27 (1991), 1165-1172, IV. Differential Equations, 29 (1993), 153-160.
[10]	N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect," Stability and Control: Theory, Methods and Applications, 20. Taylor & Francis, London, 2003.
[11]	L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl., 314 (2006), 391-411. doi: 10.1016/j.jmaa.2005.03.103.
[12]	S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 1 (2001), 233-256. doi: 10.3934/dcdsb.2001.1.233.
[13]	S. A. Gusarenko, Criteria for the stability of a linear functional-differential equation, Boundary value problems, 41-45, Perm. Politekh. Inst., Perm, 1987 (Russian).
[14]	S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations, Functional-differential equations, 30-40, Perm. Politekh. Inst., Perm, 1987 (Russian).
[15]	I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl., 220 (1998), 290-312. doi: 10.1006/jmaa.1997.5883.

show all references

References:

[1]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191. Academic Press, Boston, MA, 1993.
[2]	V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558-560.
[3]	G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246.
[4]	A. D. Myshkis, "Linear Differential Equations with Retarded Argument," Nauka, Moscow, 1972.
[5]	L. Berezansky and E. Braverman, On oscillation of equations with distributed delay, Zeitschrift für Analysis und ihre Anwendungen, 20 (2001), 489-504.
[6]	L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Mathematical and Computer Modelling, 48 (2008), 287-304. doi: 10.1016/j.mcm.2007.10.003.
[7]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
[8]	N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential equation of evolution type, Differential Equations, 13 (1977), 1915--1925, 2106.
[9]	N. V. Azbelev, L. Berezansky, P. M. Simonov and A. V. Chistyakov, The stability of linear systems with aftereffect. I, Differential Equations, 23 (1987), 745-754, 914, II. Differential Equations, 27 (1991), 383-388, III. Differential Equations, 27 (1991), 1165-1172, IV. Differential Equations, 29 (1993), 153-160.
[10]	N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect," Stability and Control: Theory, Methods and Applications, 20. Taylor & Francis, London, 2003.
[11]	L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl., 314 (2006), 391-411. doi: 10.1016/j.jmaa.2005.03.103.
[12]	S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 1 (2001), 233-256. doi: 10.3934/dcdsb.2001.1.233.
[13]	S. A. Gusarenko, Criteria for the stability of a linear functional-differential equation, Boundary value problems, 41-45, Perm. Politekh. Inst., Perm, 1987 (Russian).
[14]	S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations, Functional-differential equations, 30-40, Perm. Politekh. Inst., Perm, 1987 (Russian).
[15]	I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl., 220 (1998), 290-312. doi: 10.1006/jmaa.1997.5883.

[1]	Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[2]	Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[3]	Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
[4]	Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[5]	István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[6]	Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations and Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[7]	Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[8]	Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[9]	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
[10]	Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[11]	Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
[12]	Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[13]	Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations and Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041
[14]	Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[15]	Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19
[16]	Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[17]	Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure and Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457
[18]	Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048
[19]	Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
[20]	Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Leonid Berezansky Elena Braverman
on Google Scholar
Leonid Berezansky Elena Braverman

[Back to Top]