December  2015, 4(4): 493-505. doi: 10.3934/eect.2015.4.493

Stability of nonlinear differential systems with delay

1. 

Vietnam National University-HCMC, International University, Department of Mathematics, Saigon, Vietnam

Received  December 2014 Revised  October 2015 Published  November 2015

General nonlinear time-varying differential systems with delay are considered. Several new explicit criteria for exponential stability are given. A discussion of the obtained results and two illustrative examples are presented.
Citation: 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
References:
[1]

R. Bellman and K. L. Cooke, Differential Difference Equations, The Rand Corporation USA, 1963.  Google Scholar

[2]

J. Cao and L. Wang, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Transactions on Neural Networks, 13 (2002), 457-463. Google Scholar

[3]

S. Dashkovskiy and L. Naujok, Lyapunov-Razumikhin and Lyapunov-Krasovskii theorems for interconnected ISS time-delay systems, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, (MTNS) 5-9 July, 2010, Budapest, Hungary, 1180-1184. Google Scholar

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J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1969.  Google Scholar

[5]

R. D. Driver, Existence and stability of solutions of a delay differential system, Archive for Rational Mechanics and Analysis, 10 (1962), 401-426. doi: 10.1007/BF00281203.  Google Scholar

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E. Fridman, New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems, Systems & Control Letters, 43 (2001), 309-319. doi: 10.1016/S0167-6911(01)00114-1.  Google Scholar

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A. Goubet Bartholoms, M. Dambrine and J. P. Richard, Stability of perturbed systems with time-varying delays, Systems & Control Letters, 31 (1997), 155-163. doi: 10.1016/S0167-6911(97)00032-7.  Google Scholar

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W. M. Haddad, V. Chellaboina and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, 2010. doi: 10.1515/9781400832248.  Google Scholar

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J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[10]

L. Huang, C. Huang and B. Liu, Dynamics of a class of cellular neural networks with time-varying delays, Physics Letters A, 345 (2005), 330-344. doi: 10.1016/j.physleta.2005.07.039.  Google Scholar

[11]

L. Idels and M. Kipnis, Stability criteria for a nonlinear nonautonomous system with delays, Applied Mathematical Modelling, 33 (2009), 2293-2297. doi: 10.1016/j.apm.2008.06.005.  Google Scholar

[12]

V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.  Google Scholar

[13]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, 1993.  Google Scholar

[14]

C. H. Li and S. Yang, Global attractivity in delayed Cohen-Grossberg neural network models, Chaos, Solitons and Fractals, 39 (2009), 1975-1987. doi: 10.1016/j.chaos.2007.06.064.  Google Scholar

[15]

X. Liu, W. Yu and L. Wang, Stability analysis for continuous-time positive systems with time-varying delays, IEEE Transactions on Automatic Control, 55 (2010), 1024-1028. doi: 10.1109/TAC.2010.2041982.  Google Scholar

[16]

W. Ma, Y. Saito and Y. Takeuchi, M-matrix structure and harmless delays in a Hopfield-type neural network, Applied Mathematics Letters, 22 (2009), 1066-1070. doi: 10.1016/j.aml.2009.01.025.  Google Scholar

[17]

P. H. A. Ngoc, On positivity and stability of linear Volterra systems with delay, SIAM Journal on Control and Optimization, 48 (2009), 1939-1960. doi: 10.1137/080740040.  Google Scholar

[18]

P. H. A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters, 25 (2012), 1208-1213. doi: 10.1016/j.aml.2012.02.041.  Google Scholar

[19]

P. H. A. Ngoc and L. T. Hieu, New criteria for exponential stability of nonlinear difference systems with time-varying delay, International Journal of Control, 86 (2013), 1646-1651. doi: 10.1080/00207179.2013.792004.  Google Scholar

[20]

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science, 1976.  Google Scholar

[21]

H. Smith, An Introduction to Delay Differential Equations with Sciences Applications to the Life, Texts in Applied Mathematics, vol. 57, Springer, New York, Dordrecht, Heidelberg, London, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[22]

N. K. Son and D. Hinrichsen, Robust stability of positive continuous-time systems, Numer. Funct. Anal. Optim., 17 (1996), 649-659. doi: 10.1080/01630569608816716.  Google Scholar

[23]

S. Xueli and P. Jigen, A novel approach to exponential stability of nonlinear systems with time-varying delays, Journal of Computational and Applied Mathematics, 235 (2011), 1700-1705. doi: 10.1016/j.cam.2010.09.011.  Google Scholar

[24]

F. Wang, Exponential asymptotic stability for nonlinear neutral systems with multiple delays, Nonlinear Analysis: Real World Applications, 8 (2007), 312-322. doi: 10.1016/j.nonrwa.2005.07.006.  Google Scholar

[25]

J. Zhang, Globally exponential stability of neural networks with variable delays, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 50 (2003), 288-291. doi: 10.1109/TCSI.2002.808208.  Google Scholar

[26]

B. Zhang, J. Lam, S. Xu and Z. Shu, Absolute exponential stability criteria for a class of nonlinear time-delay systems, Nonlinear Analysis: Real World Applications, 11 (2010), 1963-1976. doi: 10.1016/j.nonrwa.2009.04.018.  Google Scholar

show all references

References:
[1]

R. Bellman and K. L. Cooke, Differential Difference Equations, The Rand Corporation USA, 1963.  Google Scholar

[2]

J. Cao and L. Wang, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Transactions on Neural Networks, 13 (2002), 457-463. Google Scholar

[3]

S. Dashkovskiy and L. Naujok, Lyapunov-Razumikhin and Lyapunov-Krasovskii theorems for interconnected ISS time-delay systems, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, (MTNS) 5-9 July, 2010, Budapest, Hungary, 1180-1184. Google Scholar

[4]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1969.  Google Scholar

[5]

R. D. Driver, Existence and stability of solutions of a delay differential system, Archive for Rational Mechanics and Analysis, 10 (1962), 401-426. doi: 10.1007/BF00281203.  Google Scholar

[6]

E. Fridman, New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems, Systems & Control Letters, 43 (2001), 309-319. doi: 10.1016/S0167-6911(01)00114-1.  Google Scholar

[7]

A. Goubet Bartholoms, M. Dambrine and J. P. Richard, Stability of perturbed systems with time-varying delays, Systems & Control Letters, 31 (1997), 155-163. doi: 10.1016/S0167-6911(97)00032-7.  Google Scholar

[8]

W. M. Haddad, V. Chellaboina and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, 2010. doi: 10.1515/9781400832248.  Google Scholar

[9]

J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[10]

L. Huang, C. Huang and B. Liu, Dynamics of a class of cellular neural networks with time-varying delays, Physics Letters A, 345 (2005), 330-344. doi: 10.1016/j.physleta.2005.07.039.  Google Scholar

[11]

L. Idels and M. Kipnis, Stability criteria for a nonlinear nonautonomous system with delays, Applied Mathematical Modelling, 33 (2009), 2293-2297. doi: 10.1016/j.apm.2008.06.005.  Google Scholar

[12]

V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.  Google Scholar

[13]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, 1993.  Google Scholar

[14]

C. H. Li and S. Yang, Global attractivity in delayed Cohen-Grossberg neural network models, Chaos, Solitons and Fractals, 39 (2009), 1975-1987. doi: 10.1016/j.chaos.2007.06.064.  Google Scholar

[15]

X. Liu, W. Yu and L. Wang, Stability analysis for continuous-time positive systems with time-varying delays, IEEE Transactions on Automatic Control, 55 (2010), 1024-1028. doi: 10.1109/TAC.2010.2041982.  Google Scholar

[16]

W. Ma, Y. Saito and Y. Takeuchi, M-matrix structure and harmless delays in a Hopfield-type neural network, Applied Mathematics Letters, 22 (2009), 1066-1070. doi: 10.1016/j.aml.2009.01.025.  Google Scholar

[17]

P. H. A. Ngoc, On positivity and stability of linear Volterra systems with delay, SIAM Journal on Control and Optimization, 48 (2009), 1939-1960. doi: 10.1137/080740040.  Google Scholar

[18]

P. H. A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters, 25 (2012), 1208-1213. doi: 10.1016/j.aml.2012.02.041.  Google Scholar

[19]

P. H. A. Ngoc and L. T. Hieu, New criteria for exponential stability of nonlinear difference systems with time-varying delay, International Journal of Control, 86 (2013), 1646-1651. doi: 10.1080/00207179.2013.792004.  Google Scholar

[20]

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science, 1976.  Google Scholar

[21]

H. Smith, An Introduction to Delay Differential Equations with Sciences Applications to the Life, Texts in Applied Mathematics, vol. 57, Springer, New York, Dordrecht, Heidelberg, London, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[22]

N. K. Son and D. Hinrichsen, Robust stability of positive continuous-time systems, Numer. Funct. Anal. Optim., 17 (1996), 649-659. doi: 10.1080/01630569608816716.  Google Scholar

[23]

S. Xueli and P. Jigen, A novel approach to exponential stability of nonlinear systems with time-varying delays, Journal of Computational and Applied Mathematics, 235 (2011), 1700-1705. doi: 10.1016/j.cam.2010.09.011.  Google Scholar

[24]

F. Wang, Exponential asymptotic stability for nonlinear neutral systems with multiple delays, Nonlinear Analysis: Real World Applications, 8 (2007), 312-322. doi: 10.1016/j.nonrwa.2005.07.006.  Google Scholar

[25]

J. Zhang, Globally exponential stability of neural networks with variable delays, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 50 (2003), 288-291. doi: 10.1109/TCSI.2002.808208.  Google Scholar

[26]

B. Zhang, J. Lam, S. Xu and Z. Shu, Absolute exponential stability criteria for a class of nonlinear time-delay systems, Nonlinear Analysis: Real World Applications, 11 (2010), 1963-1976. doi: 10.1016/j.nonrwa.2009.04.018.  Google Scholar

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