# Event Triggered Finite Time \(H_{\infty }\) Boundedness of Uncertain Markov Jump Neural Networks with Distributed Time Varying Delays

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## Abstract

This paper is concerned with the problem of event triggered finite time \(H_{\infty }\) boundedness of uncertain Markov jump neural networks with distributed time varying delays. To reduce the limited network bandwidth, an event triggered scheme is proposed in this paper. The main objective of this paper is to design the event triggered scheme, such that the proposed neural networks have finite time boundedness with admissible uncertainties. The integral terms in the derivative of the Lyapunov–Krasovskii functional are handled by the Wirtinger and Auxillary function based inequality techniques. The proposed conditions are represented by linear matrix inequalities to determine the finite time stability. The advantage of the method in this paper over some existing ones is shown by comparing the results with the existing results. At the end, numerical examples are given to verify the efficiency of the proposed method among them one example was supported by real-life application of the benchmark problem.

## Keywords

Event triggered Markovian jump parameters Lyapunov–Krasovskii functional Linear matrix inequality Finite time boundedness## Notes

## References

- 1.He Y, Liu GP, Rees D, Wu M (2007) Stability analysis for neural networks with time-varying interval delay. IEEE Trans Neural Netw 18:1850–1854CrossRefGoogle Scholar
- 2.Cichocki A, Unbehauen R (1993) Neural networks for optimization and signal processing. Wiley, ChichesterzbMATHGoogle Scholar
- 3.Liu Y, Zhang D, Lou J, Lu J, Cao J (2017) Stability analysis of quaternion-valued neural networks: decomposition and direct approaches. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2017.2755697 Google Scholar
- 4.Liu Y, Zhang D, Lu J, Cao J (2016) Global \(\mu \)-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Inf Sci 360:273–288CrossRefGoogle Scholar
- 5.Zeng HB, He Y, Wu M, Zhang C (2011) Complete delay-decomposing approach to asymptotic stability for neural networks with time-varying delays. IEEE Trans Neural Netw 22:806–812CrossRefGoogle Scholar
- 6.Liu Y, Zhang D, Lu J (2017) Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dyn 87:553–565CrossRefzbMATHGoogle Scholar
- 7.Wang Z (2013) A numerical method for delayed fractional order differential equations. J Appl Math https://doi.org/10.1155/2013/256071
- 8.Sun J, Liu GP, Chen J, Rees D (2009) Improved stability criteria for neural networks with time-varying delay. Phys Lett A 373:342–348CrossRefzbMATHGoogle Scholar
- 9.Wang Z, Ding S, Shan Q, Zhang H (2017) Stability of recurrent neural networks with time-varying delay via flexible terminal method. IEEE Trans Neural Netw Learn Syst 28:2456–2463MathSciNetCrossRefGoogle Scholar
- 10.Shi K, Zhong S, Zhu H, Liu X, Zeng Y (2015) New delay-dependent stability criteria for neutral-type neural networks with mixed random time-varying delays. Neurocomputing 168:896–907CrossRefGoogle Scholar
- 11.Ge C, Hua C, Guan X (2014) New delay-dependent stability criteria for neural networks with time-varying delay using delay-decomposition approach. IEEE Trans Neural Netw 25:1378–1383CrossRefGoogle Scholar
- 12.Seuret A, Gouaisbaut F (2013) Wirtinger-based integral inequality: application to time-delay systems. Automatica 49:2860–2866MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Park P, Lee W, Lee SY (2015) Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J Franklin Inst 352:1378–1396MathSciNetCrossRefGoogle Scholar
- 14.Zhou X, Tian J, Ma H, Zhong S (2014) Improved delay-dependent stability criteria for recurrent neural networks with time-varying delays. Neurocomputing 129:401–408CrossRefGoogle Scholar
- 15.Tian J, Zhong S (2011) Improved delay-dependent stability criterion for neural networks with time-varying delay. Appl Math Comput 217:10278–10288MathSciNetzbMATHGoogle Scholar
- 16.Tian JK, Xiong WJ, Xu F (2014) Improved delay-partitioning method to stability analysis for neural networks with discrete and distributed time-varying delays. Appl Math Comput 223:152–164MathSciNetzbMATHGoogle Scholar
- 17.Syed Ali M (2015) Stability of Markovian jumping recurrent neural networks with discrete and distributed time-varying delays. Neurocomputing 149:1280–1285CrossRefGoogle Scholar
- 18.Zhang D, Kou K, Liu Y, Cao J (2017) Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field. Neural Netw 94:55–66CrossRefGoogle Scholar
- 19.Li L, Wang Z, Li Y, Shen H, Lu J (2018) Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays. Appl 330:152–169MathSciNetGoogle Scholar
- 20.Wu B, Liu Y, Lu J (2012) New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays. Math Comput Model 55:837–843MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Arbi A, Cherif F, Aouiti C, Touati A (2016) Dynamics of new class of hopfield neural networks with time-varying and distributed delays. Acta Math Sci 36:891–912MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Wu Z, Dong S, Su H, Li C (2017) Asynchronous dissipative control for fuzzy Markov jump systems. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2017.2739754 Google Scholar
- 23.Chen W, Ma Q, Miao G, Zhang Y (2013) Stability analysis of stochastic neural networks with Markovian jump parameters using delay-partitioning approach. Neurocomputing 103:22–28CrossRefGoogle Scholar
- 24.Liang K, Dai M, Shen H, Wang J, Chen B (2018) \(L_2-L_{\infty }\) synchronization for singularly perturbed complex networks with semi-Markov jump topology. Appl Math Comput 321:450–462MathSciNetGoogle Scholar
- 25.Kovacic M (1991) Markovian neural networks. Biol Cybern 64:337–342MathSciNetCrossRefzbMATHGoogle Scholar
- 26.Hou N, Dong H, Wang Z, Ren W, Alsaadi F (2016) Non-fragile state estimation for discrete Markovian jumping neural networks. Neurocomputing 179:238–245CrossRefGoogle Scholar
- 27.Syed Ali M, Saravanakumar R, Arik S (2015) Delay-dependent stability criteria of uncertain Markovian jump neural networks with discrete interval and distributed time-varying delays. Neurocomputing 158:167–173CrossRefGoogle Scholar
- 28.Xiong J, Lam J (2007) Stabilization of linear systems over networks with bounded packet loss. Automatica 43:80–87MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Chen B, Niu Y, Zou Y (2014) Sliding mode control for stochastic Markovian jumping systems subject to successive packet losses. J Frankl Inst 351:2169–2184MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Zhu Q, Cao J (2010) Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. Neurocomputing 73:2671–2680CrossRefGoogle Scholar
- 31.Xie J, Kao Y (2015) Stability of Markovian jump neural networks with mode-dependent delays and generally incomplete transition probability. Neural Comput Appl 26:1537–1553CrossRefGoogle Scholar
- 32.Shao L, Huang H, Zhao H, Huang T (2015) Filter design of delayed static neural networks with Markovian jumping parameters. Neurocomputing 153:126–132CrossRefGoogle Scholar
- 33.Tan F, Zhou B, Duan G (2016) Finite-time stabilization of linear time-varying systems by piecewise constant feedback. Automatica 68:277–285MathSciNetCrossRefzbMATHGoogle Scholar
- 34.Weiss L, Infante EF (1967) Finite time stability under perturbing forces and on product spaces. IEEE Trans Autom Control 12:54–59MathSciNetCrossRefzbMATHGoogle Scholar
- 35.Dorato P (1961) Short time stability in linear time-varying systems. In: Proceeding of IRE international convention record, pp 83–87Google Scholar
- 36.Shen H, Park JuH, Wu Z (2014) Finite-time synchronization control for uncertain Markov jump neural networks with input constraints. Nonlinear Dyn 77:1709–1720MathSciNetCrossRefzbMATHGoogle Scholar
- 37.Wang L, Shen Y, Ding Z (2015) Finite time stabilization of delayed neural networks. Neural Netw 70:74–80CrossRefGoogle Scholar
- 38.Syed Ali M, Saravanan S (2018) Finite-time \(L_2\)-gain analysis for switched neural networks with time-varying delay. Neural Comput Appl 29:975–984CrossRefGoogle Scholar
- 39.Wang L, Shen Y, Sheng Y (2016) Finite-time robust stabilization of uncertain delayed neural networks with discontinuous activations via delayed feedback control. Neural Netw 76:46–54CrossRefGoogle Scholar
- 40.Zhang Y, Shi P, Nguang SK, Zhang J, Karimi HR (2014) Finite-time boundedness for uncertain discrete neural networks with time-delays and Markovian jumps. Neurocomputing 140:1–7CrossRefGoogle Scholar
- 41.Syed Ali M, Saravanan S (2016) Robust finite-time \(H_{\infty }\) control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays. Neurocomputing 177:454–468CrossRefGoogle Scholar
- 42.Shi P, Zhang Y, Agarwal RK (2016) Stochastic finite-time state estimation for discrete time-delay neural networks with Markovian jumps. Neurocomputing 151:168–174CrossRefGoogle Scholar
- 43.Wu Z, Shen Y, Su H, Lu R, Huang T (2017) \(H_2\) performance analysis and applications of 2-D hidden Bernoulli jump system. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2017.2745679 Google Scholar
- 44.Syed Ali M, Saravanakumar R (2014) Novel delay-dependent robust \(H_\infty \) control of uncertain systems with distributed time-varying delays. Appl Math Comput 249:510–520MathSciNetzbMATHGoogle Scholar
- 45.Syed Ali M, Saravanakumar R, Zhu Q (2015) Less conservative delaydependent \(H_\infty \) control of uncertain neural networks with discrete interval and distributed time-varying delays. Neurocomputing 166:84–95CrossRefGoogle Scholar
- 46.Du Y, Liu X, Zhong S (2016) Robust reliable \(H_\infty \) control for neural networks with mixed time delays. Chaos Solitons Fractals 91:1–8MathSciNetCrossRefzbMATHGoogle Scholar
- 47.Sakthivel R, Mathiyalagan K, Marshal S (2012) Anthoni, Robust \(H_\infty \) control for uncertain discrete-time stochastic neural networks with time-varying delays. IET Control Theory Appl 6:1220–1228MathSciNetCrossRefGoogle Scholar
- 48.Fujinami T, Saito Y, Morishita M, Koike Y, Tanida K (2001) A hybrid mass damper system controlled by \(H_\infty \) control theory for reducing bending-torsion vibration of an actual building. Earthq Eng Struct Dyn 30:1639–1653CrossRefGoogle Scholar
- 49.Syed Ali M, Saravanan S, Arik S (2016) Finite-time \(H_{\infty }\) state estimation for switched neural networks with time-varying delays. Neurocomputing 207:580–589CrossRefGoogle Scholar
- 50.Luan X, Liu F, Shi P (2010) Robust finite-time \(H_\infty \) control for nonlinear jump systems via neural networks. Circuits Syst Signal Process 29:481–498MathSciNetCrossRefzbMATHGoogle Scholar
- 51.Phat VN, Trinh H (2013) Design of \(H_\infty \) control of neural networks with time-varying delays. Neural Comput Appl 22:323–331CrossRefGoogle Scholar
- 52.Ma Y, Jia X, Liu D (2016) Robust finite-time \(H_\infty \) control for discrete-time singular Markovian jump systems with time-varying delay and actuator saturation. Appl Math Comput 286:213–227MathSciNetGoogle Scholar
- 53.Xiang Z, Sun YN, Mahmoud MS (2012) Robust finite-time \(H_{\infty }\) control for a class for a class of uncertain switched neutral systems. Commun Nonlinear Sci Numer Simul 17:1766–1778MathSciNetCrossRefzbMATHGoogle Scholar
- 54.Ma Y, Jia X, Liu D (2016) Robust finite-time \(H_{\infty }\) control for discrete-time singular Markovian jump systems with time-varying delay and actuator saturation. Appl Math Comput 286:213–227MathSciNetGoogle Scholar
- 55.Zhang XM, Han QL (2014) Event-triggered dynamic output feedback control for networked control systems. IET Control Theory Appl 8:226–234MathSciNetCrossRefGoogle Scholar
- 56.Wang H, Shi P, Lim C, Xue Q (2015) Event-triggered control for networked Markovian jump systems. Int J Robust Nonlinear 25:3422–3438MathSciNetCrossRefzbMATHGoogle Scholar
- 57.Wu Z, Xu Y, Pan Y, Shi P, Wang Q (2017) Event-triggered pinning control for consensus of multiagent systems with quantized information. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2017.2773634 Google Scholar
- 58.Li H, Zuo Z, Wang Y (2016) Event triggered control for Markovian jump systems with partially unknown transition probabilities and actuator saturation. J Frankl Inst 353:1848–1861MathSciNetCrossRefzbMATHGoogle Scholar
- 59.Wu Z, Wu Y, Wu Z, Lu J (2017) Event-based synchronization of heterogeneous complex networks subject to transmission delays. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2017.2723760 Google Scholar
- 60.Wu Z, Xu Y, Lu R, Wu Y, Huang T (2017) Event-triggered control for consensus of multiagent systems with fixed/switching topologies. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2017.2744671 Google Scholar
- 61.Wang H, Shi P, Agarwal RK (2016) Network-based event-triggered filtering for Markovian jump systems. Int J Control 89:1096–1110MathSciNetCrossRefzbMATHGoogle Scholar
- 62.Tan Y, Du D, Qi Q (2016) State estimation for Markovian jump systems with an event-triggered communication scheme. Circuits Syst Signal Process 36:2–24MathSciNetCrossRefzbMATHGoogle Scholar
- 63.Wang H, Ying Y, Lu R, Xue A (2016) Network-based \(H_{\infty }\) control for singular systems with event-triggered sampling scheme. Inf Sci 329:540–551CrossRefzbMATHGoogle Scholar
- 64.Xue A, Wang H, Lu R (2016) Event-based \(H_\infty \) control for discrete Markov jump systems. Neurocomputing 190:165–171CrossRefGoogle Scholar
- 65.Zhang H, Cheng J, Wang H, Chen Y, Xiang H (2016) Robust finite-time event-triggered \(H_\infty \) boundedness for network-based Markovian jump nonlinear systems. ISA Trans 63:32–38CrossRefGoogle Scholar
- 66.Ma G, Liu X, Qin L, Wu G (2016) Finite-time event-triggered \(H_\infty \) control for switched systems with time-varying delay. Neurocomputing 207:828–842CrossRefGoogle Scholar
- 67.Wang H, Ying Y, Lu R, Xue A (2017) Event-triggered control for the disturbance decoupling problem of Boolean control networks. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2017.2746102 Google Scholar
- 68.Huang T, Li C, Duan S, Starzyk JA (2012) Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects. IEEE Trans Neural Netw Learn Syst 23:866–875CrossRefGoogle Scholar
- 69.Johansson KH (2000) The quadruple-tank process: a multivariable laboratory process with an adjustable zero. IEEE Trans Control Syst Technol 8:456–465CrossRefGoogle Scholar
- 70.Lee TH, Park Ju H, Kwon OM, Lee SM (2013) Stochastic sampled-data control for state estimation of time-varying delayed neural networks. Neural Netw 46:99–108CrossRefzbMATHGoogle Scholar