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Global exponential estimates for uncertain Markovian jump neural networks with reaction-diffusion terms. (English) Zbl 1411.60114

Summary: The robust global exponential estimating problem is investigated for Markovian jumping reaction-diffusion delayed neural networks with polytopic uncertainties under Dirichlet boundary conditions. The information on transition rates of the Markov process is assumed to be partially known. By introducing a new inequality, some diffusion-dependent exponential stability criteria are derived in terms of relaxed linear matrix inequalities. Those criteria depend on decay rate, which may be freely selected in a range according to practical situations, rather than required to satisfy a transcendental equation. Estimates of the decay rate and the decay coefficient are presented by solving these established linear matrix inequalities. Numerical examples are provided to demonstrate the advantage and effectiveness of the proposed method.

MSC:

60J28 Applications of continuous-time Markov processes on discrete state spaces
35K51 Initial-boundary value problems for second-order parabolic systems
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[1] Agarwal, R.P., O’Regan, D.: Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems. Springer, New York (2009)
[2] Ahn, C.: Delay-dependent state estimation for T-S fuzzy delayed Hopfield neural networks. Nonlinear Dyn. 61, 483–489 (2010) · Zbl 1204.93047 · doi:10.1007/s11071-010-9664-z
[3] Arik, S.: Stability analysis of delayed neural networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47(7), 1089–1092 (2000) · Zbl 0992.93080 · doi:10.1109/81.855465
[4] Balasubramaniam, P., Syed Ali, M., Arik, S.: Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple time-varying delays. Expert Syst. Appl. 37, 7737–7744 (2010) · doi:10.1016/j.eswa.2010.04.067
[5] Balasubramaniam, P., Lakshmanan, S., Jeeva Sathya Theesar, S.: State estimation for Markovian jumping recurrent neural networks with interval time-varying delays. Nonlinear Dyn. 60, 661–675 (2010) · Zbl 1194.62109 · doi:10.1007/s11071-009-9623-8
[6] Cao, J., Wang, J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50(1), 34–44 (2003) · Zbl 1368.34084 · doi:10.1109/TCSI.2002.807494
[7] Chen, W., Zheng, W.X.: Global asymptotic stability of a class of neural networks with distributed delays. IEEE Trans. Circuits Syst. I 53(3), 644–652 (2006) · Zbl 1374.34280 · doi:10.1109/TCSI.2005.859051
[8] Cui, B., Lou, X.: Global asymptotic stability of BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solitons Fractals 27(5), 1347–1354 (2006) · Zbl 1084.68095 · doi:10.1016/j.chaos.2005.04.112
[9] He, Y., Wang, Q., Zheng, W.: Global robust stability for delayed neural networks with polytopic type uncertainties. Chaos Solitons Fractals 26(5), 1349–1354 (2005) · Zbl 1083.34535 · doi:10.1016/j.chaos.2005.04.005
[10] Hu, C., Jiang, H., Teng, Z.: Impulsive control and synchronization for delayed neural networks with reaction diffusion terms. IEEE Trans. Neural Netw. 21(1), 67–81 (2010) · doi:10.1109/TNN.2009.2034318
[11] Huang, X., Cao, J., Huang, D.: LMI-based approach for delay-dependent exponential stability analysis of BAM neural networks. Chaos Solitons Fractals 24(3), 885–898 (2005) · Zbl 1071.82538 · doi:10.1016/j.chaos.2004.09.037
[12] Li, H., Chen, B., Zhou, Q., Qian, W.: Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 39(1), 94–102 (2009) · doi:10.1109/TSMCB.2008.2002812
[13] Li, H., Gao, H., Shi, P.: New passivity analysis for neural networks with discrete and distributed delays. IEEE Trans. Neural Netw. 21(11), 1842–1847 (2010) · doi:10.1109/TNN.2010.2059039
[14] Liang, J., Cao, J.: Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays. Phys. Lett. A 314, 434–442 (2003) · Zbl 1052.82023 · doi:10.1016/S0375-9601(03)00945-9
[15] Liao, X., Chen, G., Sanchez, E.: LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(7), 1033–1039 (2002) · Zbl 1368.93598 · doi:10.1109/TCSI.2002.800842
[16] Liu, Y., Wang, Z., Liu, X.: Design of exponential state estimators for neural networks with mixed time delays. Phys. Lett. A 364, 401–412 (2007) · Zbl 05839158 · doi:10.1016/j.physleta.2006.12.018
[17] Li, P., Lam, J., Shu, Z.: On the transient and steady-state estimates of interval genetic regulatory networks. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(2), 336–349 (2010) · doi:10.1109/TSMCB.2009.2022402
[18] Li, X., Cao, J.: Delay-independent exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms. Nonlinear Dyn. 50, 363–371 (2007) · Zbl 1176.92004 · doi:10.1007/s11071-006-9164-3
[19] Lu, J.: Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos Solitons Fractals 35, 116–125 (2008) · Zbl 1134.35066 · doi:10.1016/j.chaos.2007.05.002
[20] Lu, J., Ho, D., Cao, J., Kurths, J.: Exponential synchronization of linearly coupled neural networks with impulsive disturbances. IEEE Trans. Neural Netw. 22(2), 329–355 (2011) · Zbl 1335.93058 · doi:10.1016/j.neunet.2008.11.003
[21] Ma, Q., Shi, G., Xu, S., Zou, Y.: Stability analysis for delayed genetic regulatory networks with reaction–diffusion terms. Neural Comput. Appl. 20, 507–516 (2011) · doi:10.1007/s00521-011-0575-9
[22] Ma, Q., Xu, S., Zou, Y., Shi, G.: Synchronization of stochastic chaotic neural networks with reaction-diffusion terms. Nonlinear Dyn. (2011, in press). doi: 10.1007/s11071-011-0138-8 · Zbl 1243.93106
[23] Shi, P., Mahmound, M., Nuang, S.K., Ismail, A.: Robust filtering for jumping systems with mode-dependent delays. Signal Process. 86, 140–152 (2006) · Zbl 1163.94387 · doi:10.1016/j.sigpro.2005.05.005
[24] Shu, Z., Lam, J.: Global exponential estimates of stochastic interval neural networks with discrete and distributed delays. Neurocomputing 71, 2950–2963 (2008) · Zbl 05718710 · doi:10.1016/j.neucom.2007.07.003
[25] Singh, V.: Global robust stability of delayed neural networks: An LMI approach. IEEE Trans. Circuits Syst. II, Express Briefs 52(1), 33–36 (2005) · doi:10.1109/TCSII.2004.840118
[26] Song, Q., Cao, J., Zhao, Z.: Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal., Real World Appl. 7, 65–80 (2006) · Zbl 1094.35128 · doi:10.1016/j.nonrwa.2005.01.004
[27] Sziranyi, T., Zerubia, J.: Markov random field image segmentation using cellular neural network. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 44(1), 86–89 (1997) · doi:10.1109/81.558448
[28] Tian, L., Liang, J., Cao, J.: Robust observer for discrete-time Markovian jumping neural networks with mixed mode-dependent delays. Nonlinear Dyn. 67, 47–61 (2011) · Zbl 1242.93072 · doi:10.1007/s11071-011-9956-y
[29] Tolstov, G.P., Silverman, R.A.: Fourier Series. Dover, New York (1976)
[30] Wang, L., Zhang, Z., Wang, Y.: Global exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 372, 3201–3209 (2008) · Zbl 1220.35090 · doi:10.1016/j.physleta.2007.07.090
[31] Wang, Z., Liu, Y., Liu, X.: Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 356, 346–352 (2006) · Zbl 1160.37439 · doi:10.1016/j.physleta.2006.03.078
[32] Wang, Z., Zhang, H., Li, P.: An LMI approach to stability analysis of reaction-diffusion Cohen-Grossberg neural networks concerning Dirichlet boundary conditions and distributed delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(6), 1596–1606 (2010) · doi:10.1109/TSMCB.2010.2043095
[33] Wu, Z., Su, H., Chu, J.: State estimation for discrete Markovian jumping neural networks with time delay. Neurocomputing 73, 2247–2254 (2010) · Zbl 05721421 · doi:10.1016/j.neucom.2010.01.010
[34] Xu, S., Zheng, W., Zou, Y.: Passivity analysis of neural networks with time-varying delays. IEEE Trans. Circuits Syst. II, Express Briefs 56(4), 325–329 (2009) · doi:10.1109/TCSII.2009.2015399
[35] Ye, H., Michel, A.N., Wang, K.: Global stability and local stability of Hopfield neural networks with delays. Phys. Rev. E 50, 4206–4213 (1994) · doi:10.1103/PhysRevE.50.4206
[36] Yu, F., Jiang, H.: Global exponential synchronization of fuzzy cellular neural networks with delays and reaction-diffusion terms. Neurocomputing 74, 509–515 (2011) · Zbl 05849772 · doi:10.1016/j.neucom.2010.08.017
[37] Zhang, B., Xu, S., Zou, Y.: Relaxed stability conditions for delayed recurrent neural networks with polytopic uncertainties. Int. J. Neural Syst. 16, 473–482 (2006) · Zbl 05153439 · doi:10.1142/S0129065706000871
[38] Zhang, B., Xu, S., Zong, G., Zou, Y.: Delay-dependent exponential stability for uncertain stochastic Hopfield neural networks with time-varying delays. IEEE Trans. Circuits Syst. I 56(6), 1241–1247 (2009) · doi:10.1109/TCSI.2008.2008499
[39] Zhang, L., Lam, J.: Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions. IEEE Trans. Autom. Control 55(7), 1695–1701 (2010) · Zbl 1368.93782 · doi:10.1109/TAC.2010.2046607
[40] Zhang, Y., He, Y., Wu, M., Zhang, J.: Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices. Automatica 47, 79–84 (2011) · Zbl 1209.93162 · doi:10.1016/j.automatica.2010.09.009
[41] Zhu, Q., Cao, J.: Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 41(2), 341–353 (2011)
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