Article Contents
Article Contents

# Stability of nonlinear differential systems with delay

• 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.
Mathematics Subject Classification: Primary: 34K20, 93D20.

 Citation:

•  [1] R. Bellman and K. L. Cooke, Differential Difference Equations, The Rand Corporation USA, 1963. [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. [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. [4] J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1969. [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. [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. [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. [8] W. M. Haddad, V. Chellaboina and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, 2010.doi: 10.1515/9781400832248. [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. [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. [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. [12] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986. [13] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, 1993. [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. [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. [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. [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. [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. [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. [20] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science, 1976. [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. [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. [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. [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. [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. [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.