Abstract
This paper is concerned with strict \((\mathcal {Q}, \mathcal {S}, \mathcal {R})-\gamma \) - dissipativity and passivity analysis for discrete-time Markovian jump neural networks involving both leakage and discrete delays expressed in terms of two additive time-varying delay components. The discretized Wirtinger inequality is utilized to bound the forward difference of finite-sum term in the Lyapunov functional. By constructing a suitable Lyapunov–Krasovskii functional, sufficient conditions are derived to guarantee the dissipativity and passivity criteria of the proposed neural networks. These conditions are presented in terms of linear matrix inequalities (LMIs), which can be efficiently solved via LMI MATLAB Toolbox. Finally, numerical examples are given to illustrate the effectiveness of the proposed results.
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References
Arunkumar A, Sakthivel R, Mathiyalagan K, Park JH (2014) Robust stochastic stability of discrete-time fuzzy Markovian jump neural networks. ISA Trans 53(4):1006–1014
Banu LJ, Balasubramaniam P, Ratnavelu K (2015) Robust stability analysis for discrete-time uncertain neural networks with leakage time-varying delay. Neurocomputing 151:808–816
S. Boyd, L.EI. Ghaoui, E. Feron, and V. Balakrishnan. Linear matrix inequalities in system and control theory. SIAM, Philadelphia, 1994
Feng Z, Lam J (2011) Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans Neural Netw 22(6):976–981
Feng Z, Zheng WX (2015) On extended dissipativity of discrete-time neural networks with time delay. Neural Networks and Learning Systems, IEEE Transactions on 26(12):3293–3300
Hill DJ, Moylan PJ (1980) Dissipative dynamical systems: basic input-output and state properties. J Franklin Inst 309(5):327–357
Hou L, Cheng J, Wang H (2016) Finite-time stochastic boundedness of discrete-time Markovian jump neural networks with boundary transition probabilities and randomly varying nonlinearities. Neurocomputing 174:773–779
Hu M, Cao J, Hu A (2014) Exponential stability of discrete-time recurrent neural networks with time-varying delays in the leakage terms and linear fractional uncertainties. IMA J Math Control Inf 31(3):345–362
Li J, Hu M, Guo L, Yang Y, Jin Y (2015) Stability of uncertain impulsive stochastic fuzzy neural networks with two additive time delays in the leakage term. Neural Comput Appl 26(2):417–427
Lin DH, Wu J, Li JN (2016) Less conservative stability condition for uncertain discrete-time recurrent neural networks with time-varying delays. Neurocomputing 173:1578–1588
Liu B (2013) Global exponential stability for BAM neural networks with time-varying delays in the leakage terms. Nonlinear Anal Real World Appl 14(1):559–566
Liu Y, Lee S, Lee HG (2015) Robust delay-depent stability criteria for uncertain neural networks with two additive time-varying delay components. Neurocomputing 151:770–775
Liu X-G, Wang F-X, Shu Y-J (2016) A novel summation inequality for stability analysis of discrete-time neural networks. J Comput Appl Math 304:160–171
Mohamad S, Gopalsamy K (2003) Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl Math Comput 135(1):17–38
Nam PT, Pathirana PN, Trinh H (2015) Discrete wirtinger-based inequality and its application. J Franklin Inst 352(5):1893–1905
Park P, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1):235–238
Park MJ, Kwon OM, Park JH, Lee SM, Cha EJ (2012) Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay. J Franklin Inst 349(5):1699–1720
Raja R, Zhu Q, Senthilraj S, Samidurai R (2015) Improved stability analysis of uncertain neutral type neural networks with leakage delays and impulsive effects. Appl Math Comput 266:1050–1069
Sakthivel R, Rathika M, Santra S, Zhu Q (2015) Dissipative reliable controller design for uncertain systems and its application. Appl Math Comput 263:107–121
Shao H, Han Q-L (2011) New delay-dependent stability criteria for neural networks with two additive time-varying delay components. IEEE Trans Neural Netw 22(5):812–818
Shi P, Zhang Y, Agarwal RK (2015) Stochastic finite-time state estimation for discrete time-delay neural networks with Markovian jumps. Neurocomputing 151:168–174
Shu Y, Liu X, Liu Y (2015) Stability and passivity analysis for uncertain discrete-time neural networks with time-varying delay. Neurocomputing 173:1706–1714
Song C, Gao H, Zheng WX (2009) A new approach to stability analysis of discrete-time recurrent neural networks with time-varying delay. Neurocomputing 72(10):2563–2568
Song Q, Wang Z (2007) A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays. Phys Lett A 368(1):134–145
Tian J, Zhong S (2012) Improved delay-dependent stability criteria for neural networks with two additive time-varying delay components. Neurocomputing 77(1):114–119
Willems JC (1972) Dissipative dynamical systems part I: General theory. Arch Ration Mech Anal 45(5):321–351
Wu M, Liu F, Shi P, He Y, Yokoyama R (2008) Improved free-weighting matrix approach for stability analysis of discrete-time recurrent neural networks with time-varying delay. IEEE Trans Circuits Syst II 55(7):690–694
Wu Z-G, Shi P, Su H, Chu J (2011) Passivity analysis for discrete-time stochastic Markovian jump neural networks with mixed time delays. IEEE Trans Neural Netw 22(10):1566–1575
Wu Z-G, Park JH, Su H, Chu J (2012) Admissibility and dissipativity analysis for discrete-time singular systems with mixed time-varying delays. Appl Math Comput 218(13):7128–7138
Xiao J, Zeng Z, Shen W (2015) Passivity analysis of delayed neural networks with discontinuous activations. Neural Process Lett 42(1):215–232
Xiao N, Jia Y (2013) New approaches on stability criteria for neural networks with two additive time-varying delay components. Neurocomputing 118:150–156
Xu Z, Su H, Xu H, Wu Z-G (2015) Asynchronous \(H_\infty \) filtering for discrete-time Markov jump neural networks. Neurocomputing 157:33–40
Yang R, Wu B, Liu Y (2015) A Halanay-type inequality approach to the stability analysis of discrete-time neural networks with delays. Appl Math Comput 265:696–707
Yu J, Zhang K, Fei S (2010) Exponential stability criteria for discrete-time recurrent neural networks with time-varying delay. Nonlinear Anal Real World Appl 11(1):207–216
Zeng H-B, Park JH, Zhang C-F, Wang W (2015) Stability and dissipativity analysis of static neural networks with interval time-varying delay. J Franklin Inst 352(3):1284–1295
Zhang B, Xu S, Zou Y (2008) Improved delay-dependent exponential stability criteria for discrete-time recurrent neural networks with time-varying delays. Neurocomputing 72(1):321–330
Zhang X-M, Han Q-L (2015) Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems. Automatica 57:199–202
Zhao Y, Gao H, Mou S (2008) Asymptotic stability analysis of neural networks with successive time delay components. Neurocomputing 71(13):2848–2856
Zhao H, Li L, Peng H, Kurths J, Xiao J, Yang Y (2015) Anti-synchronization for stochastic memristor-based neural networks with non-modeled dynamics via adaptive control approach. Eur Phys J B 88(5):1–10
Zhao H, Li L, Peng H, Xiao J, Yang Y, Zheng M (2016) Impulsive control for synchronization and parameters identification of uncertain multi-links complex network. Nonlinear Dyn 83(3):1437–1451
Zhao H, Li L, Peng H, Xiao J, Yang Y (2015) Finite-time boundedness analysis of memristive neural network with time-varying delay. Neural Process Lett. doi:10.1007/s11063-015-9487-5
Zheng C-D, Zhang X, Wang Z (2016) Mode and delay-dependent stochastic stability conditions of fuzzy neural networks with Markovian jump parameters. Neural Process Lett 43(1):195–217
Zhu Q, Cao J, Hayat T, Alsaadi F (2015) Robust stability of Markovian jump stochastic neural networks with time delays in the leakage terms. Neural Process Lett 41(1):1–27
Acknowledgments
This work is supported by the University Grants Commission - Basic Science Research (UGC - BSR) - Research fellowship in Mathematical Sciences—2013–2014, Govt. of India, New Delhi. The authors are very much thankful to the editors and anonymous reviewers for their careful reading, constructive comments and fruitful suggestions to improve this manuscript.
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Ramasamy, S., Nagamani, G. & Radhika, T. Further Results on Dissipativity Criterion for Markovian Jump Discrete-Time Neural Networks with Two Delay Components Via Discrete Wirtinger Inequality Approach. Neural Process Lett 45, 939–965 (2017). https://doi.org/10.1007/s11063-016-9559-1
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DOI: https://doi.org/10.1007/s11063-016-9559-1