# American Institute of Mathematical Sciences

April  2016, 12(2): 505-514. doi: 10.3934/jimo.2016.12.505

## A weak condition for global stability of delayed neural networks

 1 School of Computer and Information Engineering, Hechi University, Guangxi, Yizhou 546300, China 2 Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845 3 School of Mathematics and Statistics, Hechi University, Guangxi, Yizhou 546300, China 4 Department of Mathematics and Statistics, Curtin University, Perth,WA 6845, Australia 5 School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China

Received  September 2014 Revised  February 2015 Published  June 2015

The classical analysis of asymptotical and exponential stability of neural networks needs assumptions on the existence of a positive Lyapunov function $V$ and on the strict negativity of the function $dV/dt$, which often come as a result of boundedness or uniformly almost periodicity of the activation functions. In this paper, we investigate the asymptotical stability problem of Hopfield neural networks with time delays under weaker conditions. By constructing a suitable Lyapunov function, sufficient conditions are derived to guarantee global asymptotical stability and exponential stability of the equilibrium of the system. These conditions do not require the strict negativity of $dV/dt$, nor do they require that the activation functions to be bounded or uniformly almost periodic.
Citation: Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial and Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505
##### References:
 [1] S. Abe, J. Kawakami and K. Hirasawa, Solving inequality constrained combinatorial optimization problems by the hopfield neural networks, Neural Networks, 5 (1992), 663-670. doi: 10.1016/S0893-6080(05)80043-7. [2] P. Baldi and A. F. Atiya, How delays affect neural dynamics and learning, IEEE Transactions on Neural Networks, 5 (1994), 612-621. doi: 10.1109/72.298231. [3] D. Calabuig, J. F. Monserrat, D. Gmez-Barquero and O. Lzaro, An efficient dynamic resource allocation algorithm for packet-switched communication networks based on Hopfield neural excitation method, Neurocomputing, 71 (2008), 3439-3446. [4] Y. Chen and H. Xu, Exponential stability analysis and impulsive tracking control of uncertain time-delayed systems, Journal of Global Optimization, 52 (2012), 323-334. doi: 10.1007/s10898-011-9669-2. [5] M. Forti, On global asymptotic stability of a class of nonlinear systems arising in neural network theory, J. Differential Equations, 113 (1994), 246-264. doi: 10.1006/jdeq.1994.1123. [6] M. Forti, S. Maneti and M. Marini, Necessary and sufficient conditions for absolute stability of neural networks, IEEE Trans. Circuits Syst. I, 41 (1994), 491-494. doi: 10.1109/81.298364. [7] M. Forti and A. Tesi, New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans. Circuits Syst. I, 42 (1995) 354-366. doi: 10.1109/81.401145. [8] A. Gasull, J. Llibre and J. Sotomayor, Global asymptotic stability of differential equations in the plane, J. Differential Equations, 91 (1991) 327-336. doi: 10.1016/0022-0396(91)90143-W. [9] K. Gopalsamy and X. He, Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76 (1994), 344-358. doi: 10.1016/0167-2789(94)90043-4. [10] M. W. Hirsch, Convergent activation dynamics in continuous time networks, Neural Netw., 2 (1989), 331-349. doi: 10.1016/0893-6080(89)90018-X. [11] J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Aead. Sci. USA, 81 (1984), 3088-3092. doi: 10.1073/pnas.81.10.3088. [12] A. O. Ignatyev, On the stability of equilibrium for almost periodic systems, Nonlinear Anal., 29 (1997), 957-962. doi: 10.1016/S0362-546X(96)00078-8. [13] E. Kaszkurewicz and A. Bhaya, On a class of globally stable neural circuits, IEEE Trans. Circuits Syst. I, 41 (1994), 171-174. doi: 10.1109/81.269055. [14] D. G. Kelly, Stability in contractive nonlinear neural networks, IEEE Trans. Biomed. Eng., 3 (1990), 231-242. doi: 10.1109/10.52325. [15] N. Laskaris, S. Fotopoulos, P. Papathanasopoulos and A. Bezerianos, Robust moving averages, with Hopfield neural network implementation, for monitoring evoked potential signals, Electroencephalography and Clinical Neurophysiology/ Evoked Potentials Section, 104 (1997), 151-156. doi: 10.1016/S0168-5597(97)96681-8. [16] X. Liao, G. Chen and E. N. Sanchez, Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach, Neural Networks, 15 (2002), 855-866 [17] X. Liu and T. Chen, A new result on the global convergence of Hopfield neural networks, IEEE Trans. Circuits Syst. I, 49 (2002), 1514-1516. doi: 10.1109/TCSI.2002.803358. [18] C. Marcus and R. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A, 39 (1989), 347-359. doi: 10.1103/PhysRevA.39.347. [19] A. N. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J., 12 (1960), 305-317. [20] Z. Orman, New sufficient conditions for global stability of neutral-type neural networks with time delays, Neurocomputing, 97 (2012), 141-148. doi: 10.1016/j.neucom.2012.05.016. [21] S. Rout, Seethalakshmy, P. Srivastava and J. Majumdar, Multi-modal image segmentation using a modified Hopfield neural network, Pattern Recognition, 31 (1998), 743-750. doi: 10.1016/S0031-3203(97)00089-7. [22] R. Sammouda, N. Adgaba, A. Touir and A. Al-Ghamdi, Agriculture satellite image segmentation using a modified artificial Hopfield neural network, Computers in Human Behavior, 30 (2014), 436-441. doi: 10.1016/j.chb.2013.06.025. [23] P. Suganthan, E. Teoh and D. Mital, Pattern recognition by homomorphic graph matching using Hopfield neural networks, Image and Vision Computing, 13 (1995), 45-60. doi: 10.1016/0262-8856(95)91467-R. [24] H. Tamura, Z. Zhang, X. S. Xu, M. Ishii and Z. Tang, Lagrangian object relaxation neural network for combinatorial optimization problems, Neurocomputing, 68 (2005), 297-305. doi: 10.1016/j.neucom.2005.03.003. [25] W. Walter, Analysis I, second ed., Springer-Verlag, Berlin, 1990. [26] R. L. Wang, Z. Tang and Q. P. Cao, A lear ning method in Hopfield neural network for combinatorial optimization problem, Neurocomputing, 48 (2002), 1021-1024. [27] H. Wersing, W. J. Beyn and H. Ritter, Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions, Neural Comput., 13 (2001), 1811-1825. doi: 10.1162/08997660152469350. [28] J. Wu, Symmetric functional-differential equations and neural networks with memory, Trans. Am. Math. Soc., 350 (1999), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2. [29] J. Wu and X. Zou, Patterns of sustained oscillations in neural networks with time delayed interactions, Appl. Math. Comput., 73 (1995), 55-75. doi: 10.1016/0096-3003(94)00203-G. [30] H. Xu and Y. Chen and K. L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Applied Mathematics and Computations, 217 (2010), 537-544. doi: 10.1016/j.amc.2010.05.087. [31] H. Xu and K. L. Teo, $H_\infty$ optimal stabilization of a class of uncertain impulsive systems: an LMI approach, Journal of Industrial and management optimization, 5 (2009), 153-159. doi: 10.3934/jimo.2009.5.153. [32] F. Zhang, The Schur complement and its applications. Numerical Methods and Algorithms, New York: Springer-Verlag, 4 (2005), 34. doi: 10.1007/b105056. [33] W. Zhang, A weak condition of globally asymptotic stability for neural networks, Applied Mathematics Letters, 19 (2006), 1210-1215. doi: 10.1016/j.aml.2006.01.009. [34]

show all references

##### References:
 [1] S. Abe, J. Kawakami and K. Hirasawa, Solving inequality constrained combinatorial optimization problems by the hopfield neural networks, Neural Networks, 5 (1992), 663-670. doi: 10.1016/S0893-6080(05)80043-7. [2] P. Baldi and A. F. Atiya, How delays affect neural dynamics and learning, IEEE Transactions on Neural Networks, 5 (1994), 612-621. doi: 10.1109/72.298231. [3] D. Calabuig, J. F. Monserrat, D. Gmez-Barquero and O. Lzaro, An efficient dynamic resource allocation algorithm for packet-switched communication networks based on Hopfield neural excitation method, Neurocomputing, 71 (2008), 3439-3446. [4] Y. Chen and H. Xu, Exponential stability analysis and impulsive tracking control of uncertain time-delayed systems, Journal of Global Optimization, 52 (2012), 323-334. doi: 10.1007/s10898-011-9669-2. [5] M. Forti, On global asymptotic stability of a class of nonlinear systems arising in neural network theory, J. Differential Equations, 113 (1994), 246-264. doi: 10.1006/jdeq.1994.1123. [6] M. Forti, S. Maneti and M. Marini, Necessary and sufficient conditions for absolute stability of neural networks, IEEE Trans. Circuits Syst. I, 41 (1994), 491-494. doi: 10.1109/81.298364. [7] M. Forti and A. Tesi, New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans. Circuits Syst. I, 42 (1995) 354-366. doi: 10.1109/81.401145. [8] A. Gasull, J. Llibre and J. Sotomayor, Global asymptotic stability of differential equations in the plane, J. Differential Equations, 91 (1991) 327-336. doi: 10.1016/0022-0396(91)90143-W. [9] K. Gopalsamy and X. He, Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76 (1994), 344-358. doi: 10.1016/0167-2789(94)90043-4. [10] M. W. Hirsch, Convergent activation dynamics in continuous time networks, Neural Netw., 2 (1989), 331-349. doi: 10.1016/0893-6080(89)90018-X. [11] J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Aead. Sci. USA, 81 (1984), 3088-3092. doi: 10.1073/pnas.81.10.3088. [12] A. O. Ignatyev, On the stability of equilibrium for almost periodic systems, Nonlinear Anal., 29 (1997), 957-962. doi: 10.1016/S0362-546X(96)00078-8. [13] E. Kaszkurewicz and A. Bhaya, On a class of globally stable neural circuits, IEEE Trans. Circuits Syst. I, 41 (1994), 171-174. doi: 10.1109/81.269055. [14] D. G. Kelly, Stability in contractive nonlinear neural networks, IEEE Trans. Biomed. Eng., 3 (1990), 231-242. doi: 10.1109/10.52325. [15] N. Laskaris, S. Fotopoulos, P. Papathanasopoulos and A. Bezerianos, Robust moving averages, with Hopfield neural network implementation, for monitoring evoked potential signals, Electroencephalography and Clinical Neurophysiology/ Evoked Potentials Section, 104 (1997), 151-156. doi: 10.1016/S0168-5597(97)96681-8. [16] X. Liao, G. Chen and E. N. Sanchez, Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach, Neural Networks, 15 (2002), 855-866 [17] X. Liu and T. Chen, A new result on the global convergence of Hopfield neural networks, IEEE Trans. Circuits Syst. I, 49 (2002), 1514-1516. doi: 10.1109/TCSI.2002.803358. [18] C. Marcus and R. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A, 39 (1989), 347-359. doi: 10.1103/PhysRevA.39.347. [19] A. N. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J., 12 (1960), 305-317. [20] Z. Orman, New sufficient conditions for global stability of neutral-type neural networks with time delays, Neurocomputing, 97 (2012), 141-148. doi: 10.1016/j.neucom.2012.05.016. [21] S. Rout, Seethalakshmy, P. Srivastava and J. Majumdar, Multi-modal image segmentation using a modified Hopfield neural network, Pattern Recognition, 31 (1998), 743-750. doi: 10.1016/S0031-3203(97)00089-7. [22] R. Sammouda, N. Adgaba, A. Touir and A. Al-Ghamdi, Agriculture satellite image segmentation using a modified artificial Hopfield neural network, Computers in Human Behavior, 30 (2014), 436-441. doi: 10.1016/j.chb.2013.06.025. [23] P. Suganthan, E. Teoh and D. Mital, Pattern recognition by homomorphic graph matching using Hopfield neural networks, Image and Vision Computing, 13 (1995), 45-60. doi: 10.1016/0262-8856(95)91467-R. [24] H. Tamura, Z. Zhang, X. S. Xu, M. Ishii and Z. Tang, Lagrangian object relaxation neural network for combinatorial optimization problems, Neurocomputing, 68 (2005), 297-305. doi: 10.1016/j.neucom.2005.03.003. [25] W. Walter, Analysis I, second ed., Springer-Verlag, Berlin, 1990. [26] R. L. Wang, Z. Tang and Q. P. Cao, A lear ning method in Hopfield neural network for combinatorial optimization problem, Neurocomputing, 48 (2002), 1021-1024. [27] H. Wersing, W. J. Beyn and H. Ritter, Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions, Neural Comput., 13 (2001), 1811-1825. doi: 10.1162/08997660152469350. [28] J. Wu, Symmetric functional-differential equations and neural networks with memory, Trans. Am. Math. Soc., 350 (1999), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2. [29] J. Wu and X. Zou, Patterns of sustained oscillations in neural networks with time delayed interactions, Appl. Math. Comput., 73 (1995), 55-75. doi: 10.1016/0096-3003(94)00203-G. [30] H. Xu and Y. Chen and K. L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Applied Mathematics and Computations, 217 (2010), 537-544. doi: 10.1016/j.amc.2010.05.087. [31] H. Xu and K. L. Teo, $H_\infty$ optimal stabilization of a class of uncertain impulsive systems: an LMI approach, Journal of Industrial and management optimization, 5 (2009), 153-159. doi: 10.3934/jimo.2009.5.153. [32] F. Zhang, The Schur complement and its applications. Numerical Methods and Algorithms, New York: Springer-Verlag, 4 (2005), 34. doi: 10.1007/b105056. [33] W. Zhang, A weak condition of globally asymptotic stability for neural networks, Applied Mathematics Letters, 19 (2006), 1210-1215. doi: 10.1016/j.aml.2006.01.009. [34]
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