Global exponential stability in Lagrange sense for recurrent neural networks with time delays

Nonlinear Analysis: Real World Applications

Volumn 9, Issue 4, 2008, Pages 1535-1557

  Liao, Xiaoxin ^a   Luo, Qi ^b   Zeng, Zhigang ^c   Guo, Yunxia ^d  

^a HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b NANJING UNIVERSITY OF INFORMATION SCIENCE AND TECHNOLOGY   (China)

^c WUHAN UNIVERSITY OF TECHNOLOGY   (China)

^d Zhuhai Radio and TV University   (China)

Author keywords

Delays; Global exponential attractivity; Lagrange stability; Recurrent neural networks

Indexed keywords

ASYMPTOTIC STABILITY; GLOBAL OPTIMIZATION; LAGRANGE MULTIPLIERS; LYAPUNOV FUNCTIONS; TIME DELAY;

ACTIVATION FUNCTIONS; GLOBAL EXPONENTIAL ATTRACTIVITY; LAGRANGE STABILITY; MULTISTABLE NEURAL NETWORKS;

RECURRENT NEURAL NETWORKS;

EID: 43649099130     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2007.03.018     Document Type: Article

References (40)

1. C.-Y. Cheng, K.-H. Lin, C.-W. Shih, Multistability in recurrent neural networks, SIAM J. Appl. Math. 66(4), (2006) 1301-1320.
2. Cheng C.-Y., Lin K.-H., and Shih C.-W. Multistability and convergence in delayed neural networks. Physica D 225 (2007) 61-74
3. Chua L.O., and Yang L. Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35 (1988) 1257-1272
4. Forti M. On global asymptotic stability of a class of nonlinear systems arising in neural network theory. J. Differential Equations 113 (1994) 246-264
5. Forti M., and Tesi A. New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans. Circuits Syst. I 42 7 (1995) 354-366
6. Foss J., Longtin A., Mensour B., and Milton J. Multistability and delayed recurrent loops. Phys. Rev. Lett. 76 4 (1996) 708-711
7. Hahnloser R. On the piecewise analysis of networks of linear threshold neurons. Neural Networks 11 (1998) 691-697
8. Hahnloser R., Sarpeshkar R., Mahowald M., Douglas R., and Seung H.S. Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit. Nature 405 (2000) 947-951
9. A. Hassibi, S.P. Boyd, J.P. How, A class of Lyapunov functionals for analyzing hybrid dynamical systems, in: Proceedings of American Control Conference, San Diego, California, June, 1999.
10. Hopfield J.J. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81 (1984) 3088-3092
11. LaSalle J. Some extensions of Liapunov's second method. IRE Trans. Circuit Theory 7 4 (1960) 520-527
12. Q.D. Li, X.S. Yang, Complex dynamics in a simple Hopfield-type neural network, in: J. Wang, X.F. Liao, Y. Zhang (Eds.), Lecture Notes in Computer Science, vol. 3496, New York, Springer, Berlin, Heidelberg, 2005, pp. 357-362.
13. Liao X.X. Stability of Hopfield-type neural networks. Sci. China Ser. A 23 (1993) 1025-1035
14. Liao X.X. Mathematical theory of cellular neural networks (I). Sci. China Ser. A 24 (1994) 902-910
15. Liao X.X. Mathematical theory of cellular neural networks (II). Sci. China Ser. A 24 (1994) 1037-1046
16. Liao X.X. Theory and Applications of Stability for Dynamical Systems (2000), National Defense Publishing House, Beijing
17. Liao X.X., and Wang J. Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays. IEEE Trans. Circuits Syst. I 50 (2003) 268-275
18. Liao X.X., and Wang J. Global dissipativity of continuous-time recurrent neural networks with time delay. Phys. Rev. E 68 (2003) 016118
19. Liao X.X., Wang J., and Zeng Z.G. Global asymptotic stability and global exponential stability of delayed cellular neural networks. IEEE Trans. Circuits Syst. II Express Briefs 52 (2005) 403-409
20. Liao X.F., Wong K.-W., Leung C.-S., and Wu Z. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotone activation function. Chaos, Solitons and Fractals 12 (2001) 1535-1547
21. X.X. Liao, Z.G. Zeng, Global exponential stability in Lagrange sense of continuous-time recurrent neural networks, in: J. Wang, et al. (Eds.), Lecture Notes in Computer Science, vol. 3971, New York, Springer, Berlin, Heidelberg, 2006, pp. 115-121.
22. Mensour B., and Longtin A. Controlling chaos to store information in delay differential equations. Phys. Lett. A 205 (1995) 18-24
23. Michel A.N., and Liu D. Qualitative Analysis and Synthesis of Recurrent Neural Networks (2002), Marcel Dekker, New York
24. Morita M. Associative memory with nonmonotone dynamics. Neural Networks 6 (1993) 115-126
25. Passino K.M., and Burgess K.L. Lagrange stability and boundedness of discrete event systems. Discrete Event Dyn. Syst: Theory Appl. 5 (1995) 383-403
26. Rekasius Z.V. Lagrange stability of nonlinear feedback systems. IEEE Trans. Automat. Control 8 2 (1963) 160-163
27. Seiler G., and Nossek J.A. Winner-take-all cellular neural networks. IEEE Trans. Circuits Syst. II 40 3 (1993) 184-190
28. Thornton K.W., and Mulholland R.J. Lagrange stability and ecological systems. J. Theor. Biol. 45 (1974) 473-485
29. Townley S., Ilchmann A., Weiß M.G., Ruiz W.C., Owens D.H., and Prätzel-Wolters D. Existence and learning of oscillations in recurrent neural networks. IEEE Trans. Neural Networks 11 1 (2000) 205-214
30. Wang J., Duan Z., and Huang L. Control of a class of pendulum-like systems with Lagrange stability. Automatica 42 (2006) 145-150
31. Wersing H., Beyn W., and Ritter H. Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Comput. 13 (2001) 1811-1825
32. Y. Yang, L. Huang, Lagrange stability of a class of nonlinear discrete-time systems, in: First IEEE Conference on Industrial Electronics and Applications, May 2006, pp. 1-6.
33. Yi Z., and Tan K.K. Convergence Analysis of Recurrent Neural Networks (2004), Kluwer Academic Publishers, Dordrecht
34. Yoshizawa T. Stability Theory by Liapunov's Second Method (1966), Mathematical Society of Japan, Japan, Tokyo
35. Zeng Z.G., Wang J., and Liao X.X. Global exponential stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans. Circuits Syst. I 50 (2003) 1353-1358
36. Zeng Z.G., Wang J., and Liao X.X. Stability analysis of delayed cellular neural networks described using cloning templates. IEEE Trans. Circuits Syst. I 51 (2004) 2313-2324
37. X.H. Zhang, S.B. Zhou, Chaos synchronization for bi-directional coupled two-neuron systems with discrete delays, in: J. Wang, X.F. Liao, Y. Zhang (Eds.), Lecture Notes in Computer Science, vol. 3496, New York, Springer, Berlin, Heidelberg, 2005, pp. 351-356.
38. S.B. Zhou, H. Li, Z. Wu, Stability and chaos of a neural network with uncertain time delays, in: J. Wang, X.F. Liao, Y. Zhang (Eds.), Lecture Notes in Computer Science, vol. 3496, New York, Springer, Berlin, Heidelberg, 2005, pp. 340-345.
39. Zhou S.B., Liao X.F., Yu J.B., and Wong K.W. Chaos and its synchronization in two-neuron systems with discrete delays. Chaos Solitons and Fractals 21 (2004) 133-142
40. Zipser D., Kehoe B., Littlewort G., and Fuster J. A spiking network model of short-term active memory. J. Neurosci. 13 8 (1993) 3406-3420