# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Global stability of stochastic high-order neural networks with discrete and distributed delays. (English) Zbl 1141.93416
Summary: High-order neural networks can be considered as an expansion of Hopfield neural networks, and have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks. In this paper, the global asymptotic stability analysis problem is considered for a class of stochastic high-order neural networks with discrete and distributed time-delays. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are derived, which guarantee the global asymptotic convergence of the equilibrium point in the mean square. It is shown that the stochastic high-order delayed neural networks under consideration are globally asymptotically stable in the mean square if two linear matrix inequalities (LMIs) are feasible, where the feasibility of LMIs can be readily checked by the Matlab LMI toolbox. It is also shown that the main results in this paper cover some recently published works. A numerical example is given to demonstrate the usefulness of the proposed global stability criteria.

##### MSC:
 93E15 Stochastic stability 93D20 Asymptotic stability of control systems 92B20 General theory of neural networks (mathematical biology)
##### Software:
LMI toolbox; Matlab
Full Text:
##### References:
 [1] Arik, S.: Global robust stability analysis of neural networks with discrete time delays, Chaos, solitons & fractals 26, No. 5, 1407-1414 (2005) · Zbl 1122.93397 [2] Artyomov, E.; Yadid-Pecht, O.: Modified high-order neural network for invariant pattern recognition, Pattern recognit lett 26, No. 6, 843-851 (2005) [3] Blythe, S.; Mao, X.; Liao, X.: Stability of stochastic delay neural networks, J franklin inst 338, 481-495 (2001) · Zbl 0991.93120 · doi:10.1016/S0016-0032(01)00016-3 [4] Boyd, S.; Ghaoui, L. Ei; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, (1994) · Zbl 0816.93004 [5] Cao, J.; Chen, T.: Globally exponentially robust stability and periodicity of delayed neural networks, Chaos, solitons & fractals 22, No. 4, 957-963 (2004) · Zbl 1061.94552 · doi:10.1016/j.chaos.2004.03.019 [6] Cao, J.; Huang, D. -S.; Qu, Y.: Global robust stability of delayed recurrent neural networks, Chaos, solitons & fractals 23, 221-229 (2005) · Zbl 1075.68070 · doi:10.1016/j.chaos.2004.04.002 [7] Cao, J.; Liang, J.; Lam, J.: Exponential stability of high-order bidirectional associative memory neural networks with time delays, Physica D: Nonlinear phenom 199, No. 3 -- 4, 425-436 (2004) · Zbl 1071.93048 · doi:10.1016/j.physd.2004.09.012 [8] Dembo, A.; Farotimi, O.; Kailath, T.: High-order absolutely stable neural networks, IEEE trans circ syst 38, No. 1, 57-65 (1991) · Zbl 0712.92002 · doi:10.1109/31.101303 [9] Friedman, A.: Stochastic differential equations and their applications, (1976) · Zbl 0323.60057 [10] Gahinet, P.; Nemirovsky, A.; Laub, A. J.; Chilali, M.: LMI control toolbox: for use with Matlab, (1995) [11] Gao, H.; Lam, J.; Xie, L.; Wang, C.: New approach to mixed H2/H$\infty$ filtering for polytopic discrete-time systems, IEEE trans signal process 53, No. 8, 3183-3192 (2005) [12] Gao, H.; Lam, J.; Wang, C.; Wang, Y.: Delay-dependent output-feedback stabilization of discrete-time systems with time-varying state delay, IEE proc control theory appl 151, No. 6, 691-698 (2004) [13] Gao, H.; Lam, J.; Wang, C.: Robust energy-to-peak filter design for stochastic time-delay systems, Syst control lett 55, No. 2, 101-111 (2006) · Zbl 1129.93538 · doi:10.1016/j.sysconle.2005.05.005 [14] Gu K. An integral inequality in the stability problem of time-delay systems. In: Proceedings of 39th IEEE conference on decision and control, December 2000, Sydney, Australia, 2000. p. 2805 -- 10. [15] Hale, J. K.: Theory of functional differential equations, (1977) · Zbl 0352.34001 [16] Huang, H.; Ho, D. W. C.; Lam, J.: Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays, IEEE trans circ syst: part II 52, No. 5, 251-255 (2005) [17] Karayiannis, N. B.; Venetsanopoulos, A. N.: On the training and performance of high-order neural networks, Math biosci 129, No. 2, 143-168 (1995) · Zbl 0830.92005 · doi:10.1016/0025-5564(94)00057-7 [18] Liu, Y.; Wang, Z.; Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays, Neural networks 19, No. 5, 667-675 (2006) · Zbl 1102.68569 · doi:10.1016/j.neunet.2005.03.015 [19] Psaltis, D.; Park, C. H.; Hong, J.: Higher order associative memories and their optical implementations, Neural networks 1, 143-163 (1988) [20] Ren, F.; Cao, J.: LMI-based criteria for stability of high-order neural networks with time-varying delay, Nonlinear anal ser B: real world appl 7, No. 5, 967-979 (2006) · Zbl 1121.34078 · doi:10.1016/j.nonrwa.2005.09.001 [21] Rong, L.: LMI-based criteria for robust stability of Cohen -- Grossberg neural networks with delay, Phys lett A 339, No. 1 -- 2, 63-73 (2005) · Zbl 1137.93401 · doi:10.1016/j.physleta.2005.03.023 [22] Ruan, S.; Filfil, R. S.: Dynamics of a two-neuron system with discrete and distributed delays, Physica D 191, 323-342 (2004) · Zbl 1049.92004 · doi:10.1016/j.physd.2003.12.004 [23] Sun, J.; Wan, L.: Global exponential stability and periodic solutions of Cohen -- Grossberg neural networks with continuously distributed delays, Physica D: Nonlinear phenom 208, No. 1 -- 2, 1-20 (2005) · Zbl 1086.34061 · doi:10.1016/j.physd.2005.05.009 [24] Wan, L.; Sun, J.: Mean square exponential stability of stochastic delayed Hopfield neural networks, Phys lett A 343, No. 4, 306-318 (2005) · Zbl 1194.37186 · doi:10.1016/j.physleta.2005.06.024 [25] Wang, Z.; Liu, Y.; Liu, X.: On global asymptotic stability of neural networks with discrete and distributed delays, Phys lett A 345, No. 4 -- 6, 299-308 (2005) · Zbl 05314210 [26] Wang, Z.; Shu, H.; Liu, Y.; Ho, D. W. C.; Liu, X.: Robust stability analysis of generalized neural networks with discrete and distributed time delays, Chaos, solitons & fractals 30, No. 4, 886-896 (2006) · Zbl 1142.93401 · doi:10.1016/j.chaos.2005.08.166 [27] Xie, L.: Output feedback H$\infty$ control of systems with parameter uncertainty, Int J control 63, 741-750 (1996) · Zbl 0841.93014 · doi:10.1080/00207179608921866 [28] Xu, B.; Liu, X.; Liao, X.: Global asymptotic stability of high-order Hopfield type neural networks with time delays, Comput math appl 45, No. 10 -- 11, 1729-1737 (2005) · Zbl 1045.37056 · doi:10.1016/S0898-1221(03)00151-2 [29] Zhao, H.: Global asymptotic stability of Hopfield neural network involving distributed delays, Neural networks 17, 47-53 (2004) · Zbl 1082.68100 · doi:10.1016/S0893-6080(03)00077-7 [30] Zhao, H.: Existence and global attractivity of almost periodic solution for cellular neural network with distributed delays, Appl math comput 154, 683-695 (2004) · Zbl 1057.34099 · doi:10.1016/S0096-3003(03)00743-4