American Institute of Mathematical Sciences

Advanced
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

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 & 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, (1993).   Google Scholar

[2]

V. Volterra, Fluctuations in the abundance of species considered mathematically,, Nature, 118 (1926), 558.   Google Scholar

[3]

G. E. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221.   Google Scholar

[4]

A. D. Myshkis, "Linear Differential Equations with Retarded Argument,", Nauka, (1972).   Google Scholar

[5]

L. Berezansky and E. Braverman, On oscillation of equations with distributed delay,, Zeitschrift f\, 20 (2001), 489.   Google Scholar

[6]

L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay,, Mathematical and Computer Modelling, 48 (2008), 287.  doi: 10.1016/j.mcm.2007.10.003.  Google Scholar

[7]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[8]

N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential equation of evolution type,, Differential Equations, 13 (1977).   Google Scholar

[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.   Google Scholar

[10]

N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect,", Stability and Control: Theory, (2003).   Google Scholar

[11]

L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations,, J. Math. Anal. Appl., 314 (2006), 391.  doi: 10.1016/j.jmaa.2005.03.103.  Google Scholar

[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.  doi: 10.3934/dcdsb.2001.1.233.  Google Scholar

[13]

S. A. Gusarenko, Criteria for the stability of a linear functional-differential equation,, Boundary value problems, (1987), 41.   Google Scholar

[14]

S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations,, Functional-differential equations, (1987), 30.   Google Scholar

[15]

I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations,, J. Math. Anal. Appl., 220 (1998), 290.  doi: 10.1006/jmaa.1997.5883.  Google Scholar

show all references

References:
[1]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, (1993).   Google Scholar

[2]

V. Volterra, Fluctuations in the abundance of species considered mathematically,, Nature, 118 (1926), 558.   Google Scholar

[3]

G. E. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221.   Google Scholar

[4]

A. D. Myshkis, "Linear Differential Equations with Retarded Argument,", Nauka, (1972).   Google Scholar

[5]

L. Berezansky and E. Braverman, On oscillation of equations with distributed delay,, Zeitschrift f\, 20 (2001), 489.   Google Scholar

[6]

L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay,, Mathematical and Computer Modelling, 48 (2008), 287.  doi: 10.1016/j.mcm.2007.10.003.  Google Scholar

[7]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[8]

N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential equation of evolution type,, Differential Equations, 13 (1977).   Google Scholar

[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.   Google Scholar

[10]

N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect,", Stability and Control: Theory, (2003).   Google Scholar

[11]

L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations,, J. Math. Anal. Appl., 314 (2006), 391.  doi: 10.1016/j.jmaa.2005.03.103.  Google Scholar

[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.  doi: 10.3934/dcdsb.2001.1.233.  Google Scholar

[13]

S. A. Gusarenko, Criteria for the stability of a linear functional-differential equation,, Boundary value problems, (1987), 41.   Google Scholar

[14]

S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations,, Functional-differential equations, (1987), 30.   Google Scholar

[15]

I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations,, J. Math. Anal. Appl., 220 (1998), 290.  doi: 10.1006/jmaa.1997.5883.  Google Scholar

[1]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108
[2]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[3]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305
[4]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[5]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[6]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[8]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[9]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[10]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117
[11]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[12]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275
[13]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[14]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[15]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[16]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[17]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[18]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[19]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[20]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences