# Reachable sets bounding for generalized neural networks with interval time-varying delay and bounded disturbances

- 358 Downloads
- 2 Citations

## Abstract

This paper deals with the problem of finding outer bound of forwards reachable sets and interbound of backwards reachable sets of generalized neural network systems with interval nondifferentiable time-varying delay and bounded disturbances. Based on constructing a suitable Lyapunov–Krasovskii functional and utilizing some improved Jensen integral-based inequalities, two sufficient conditions are derived for the existence of: (1) the smallest possible outer bound of forwards reachable sets and (2) the largest possible interbound of backwards reachable sets. These conditions are delay dependent and in the form of matrix inequalities, which therefore can be efficiently solved by using existing convex algorithms. Three numerical examples with simulation results are provided to demonstrate the effectiveness of our results.

## Keywords

Generalized neural networks Forwards reachable sets Backwards reachable sets Lyapunov–Krasovskii functional Linear matrix inequalities## Notes

### Acknowledgments

The author would like to thank the editor(s) and anonymous reviewers for their constructive comments which helped to improve the present paper. This work was partially supported by the Australian Research Council (Grant DP130101532) and the Ministry of Education and Training of Vietnam.

## References

- 1.Dongsheng Y, Liu X, Xu Y, Wang Y, Liu Z (2013) State estimation of recurrent neural networks with interval time-varying delay: an improved delay-dependent approach. Neural Comput Appl 23:1149–1158CrossRefGoogle Scholar
- 2.Song X, Gao H, Ding L, Liu D, Hao M (2013) The globally asymptotic stability analysis for a class of recurrent neural networks with delays. Neural Comput Appl 22:587–595CrossRefGoogle Scholar
- 3.Phat VN, Trinh H (2013) Design of \(H_{\infty }\) control of neural networks with time-varying delays. Neural Comput Appl 22:323–331CrossRefGoogle Scholar
- 4.Rajchakit M, Niamsup P, Rajchakit G (2013) A switching rule for exponential stability of switched recurrent neural networks with interval time-varying delay. Adv Differ Equ 44:10MathSciNetzbMATHGoogle Scholar
- 5.Rajchakit G (2013) Delay-dependent asymptotical stabilization criterion of recurrent neural networks. Appl Mech Mater 330:1045–1048CrossRefGoogle Scholar
- 6.Phat VN, Fernando T, Trinh H (2014) Observer-based control for time-varying delay neural networks with nonlinear observation. Neural Comput Appl 24:1639–1645CrossRefGoogle Scholar
- 7.Hua M, Tan H, Chen J (2014) Delay-dependent \(H_{\infty }\) and generalized \(H_2\) filtering for stochastic neural networks with time-varying delay and noise disturbances. Neural Comput Appl 25:613–624CrossRefGoogle Scholar
- 8.Xiao J, Zeng Z (2014) Global robust stability of uncertain delayed neural networks with discontinuous neuron activation. Neural Comput Appl 24:1191–1198CrossRefGoogle Scholar
- 9.Chen L, Liu C, Wu R, He Y, Chai Y (2015) Finite-time stability criteria for a class of fractional-order neural networks with delay. Neural Comput Appl 27:549–556CrossRefGoogle Scholar
- 10.Lee TH, Park JH, Park MJ, Kwon OM, Jung HY (2015) On stability criteria for neural networks with time-varying delay using Wirtinger-based multiple integral inequality. J Franklin Inst 352:5627–5645MathSciNetCrossRefGoogle Scholar
- 11.Rajchakit M, Niamsup P, Rojsiraphisal T, Rajchakit G (2012) Delay-dependent guaranteed cost controller design for uncertain neural networks with interval time-varying delay. Abstr Appl Anal 587426:16MathSciNetzbMATHGoogle Scholar
- 12.Niamsup P, Ratchagit K, Phat VN (2015) Novel criteria for finite-time stabilization and guaranteed cost control of delayed neural networks. Neurocomputing 160:281–286CrossRefGoogle Scholar
- 13.Feng J, Tang Z, Zhao Y, Xu C (2013) Cluster synchronisation of non-linearly coupled Lure networks with identical and non-identical nodes and an asymmetrical coupling matrix. IET Control Theory Appl 7:2117–2127MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Tang Z, Park JH, Lee TH (2016) Dynamic output-feedback-based \(H_{\infty }\) design for networked control systems with multipath packet dropouts. Appl Math Comput 275:121–133MathSciNetGoogle Scholar
- 15.Tang Z, Park JH, Lee TH, Feng J (2016) Mean square exponential synchronization for impulsive coupled neural networks with time-varying delays and stochastic disturbances. Complexity 21:190–202MathSciNetCrossRefGoogle Scholar
- 16.Tang Z, Park JH, Lee TH, Feng J (2016) Random adaptive control for cluster synchronization of complex networks with distinct communities. Int J Adapt Control Signal Process 30:534–549MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Tang Z, Park JH, Lee TH (2016) Distributed adaptive pinning control for cluster synchronization of nonlinearly coupled Lure networks. Commun Nonlinear Sci Numer Simul 39:7–20MathSciNetCrossRefGoogle Scholar
- 18.Xu C, Liao M, Zhang Q (2015) On the mean square exponential stability for a stochastic fuzzy cellular neural network with distributed delays and time-varying delays. Int J Innov Comput Inf Control 11:247–256Google Scholar
- 19.Shi P, Zhang Y, Chadli M, Agarwal RK (2016) Mixed \(H_\infty\) and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Trans Neural Netw Learn Syst 27:903–909MathSciNetCrossRefGoogle Scholar
- 20.Shi P, Li F, Wu L, Lim CC (2016) Neural network-based passive filtering for delayed neutral-type semi-markovian jump systems. IEEE Trans Neural Netw Learn Syst. doi: 10.1109/TNNLS.2016.2573853 Google Scholar
- 21.Chen H, Shi P, Lim CC (2016) Exponential synchronization for markovian stochastic coupled neural networks of neutral-type via adaptive feedback control. IEEE Trans Neural Netw Learn Syst. doi: 10.1109/TNNLS.2016.2546962 Google Scholar
- 22.Xu ZB, Qiao H, Peng J, Zhang B (2004) A comparative study of two modeling approaches in neural networks. Neural Netw 17:73–85CrossRefzbMATHGoogle Scholar
- 23.Zhang XM, Han QL (2011) Global asymptotic stability for a class of generalized neural networks with interval time-varying delays. IEEE Trans Neural Netw 22:1180–1192CrossRefGoogle Scholar
- 24.Liu Y, Wang Z, Liu X (2006) Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw 19:667–675CrossRefzbMATHGoogle Scholar
- 25.Wu M, Liu F, Shi P, He Y, Yokoyama R (2008) Exponential stability analysis for neural networks with time-varying delay. IEEE Trans Syst Man Cybern B Cybern 38:1152–1156CrossRefGoogle Scholar
- 26.Mahmoud MS, Ismail A (2010) Improved results on robust exponential stability criteria for neutral-type delayed neural networks. Appl Math Comput 217:3011–3019MathSciNetzbMATHGoogle Scholar
- 27.Liu Y, Ma Y, Mahmoud M (2012) New results for global exponential stability of neural networks with varying delays. Neurocomputing 97:357–363CrossRefGoogle Scholar
- 28.Gao H, Song X, Ding L, Liu D, Hao M (2013) New conditions for global exponential stability of continuous-time neural networks with delays. Neural Comput Appl 22:41–48CrossRefGoogle Scholar
- 29.Zhang W, Li C, Huang T, Tan J (2015) Exponential stability of inertial BAM neural networks with time-varying delay via periodically intermittent control. Neural Comput Appl 26:1781–1787CrossRefGoogle Scholar
- 30.Ji MD, He Y, Wu M, Zhang CK (2015) Further results on exponential stability of neural networks with time-varying delay. Appl Math Comput 256:175–182MathSciNetzbMATHGoogle Scholar
- 31.Weera W, Niamsup P (2016) Novel delay-dependent exponential stability criteria for neutral-type neural networks with non-differentiable time-varying discrete and neutral delays. Neurocomputing 173:886–898CrossRefGoogle Scholar
- 32.Shao HY (2008) Delay-dependent stability for recurrent neural networks with time-varying delays. IEEE Trans Neural Netw 19:1647–1651CrossRefGoogle Scholar
- 33.Zuo Z, Yang C, Wang Y (2010) A new method for stability analysis of recurrent neural networks with interval time-varying delay. IEEE Trans Neural Netw 21:339–344CrossRefGoogle Scholar
- 34.Wu ZG, Lam J, Su H, Chu J (2012) Stability and dissipativity analysis of static neural networks with time delay. IEEE Trans Neural Netw Learn Syst 23:199–210CrossRefGoogle Scholar
- 35.Bai YQ, Chen J (2013) New stability criteria for recurrent neural networks with interval time-varying delay. Neurocomputing 121:179–184CrossRefGoogle Scholar
- 36.Sun J, Chen J (2013) Stability analysis of static recurrent neural networks with interval time-varying delay. Appl Math Comput. 221:111–120MathSciNetzbMATHGoogle Scholar
- 37.Zhang XM, Han QL (2014) Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Netw 54:57–69CrossRefzbMATHGoogle Scholar
- 38.Zeng HB, Park JH, Zhang CF, Wang W (2015) Stability and dissipativity analysis of static neural networks with interval time-varying delay. J Franklin Inst 352:1284–1295MathSciNetCrossRefzbMATHGoogle Scholar
- 39.Zhang CK, He Y, Jiang L, Wu QH, Wu M (2014) Delay-dependent stability criteria for generalized neural networks with two delay components. IEEE Trans Neural Netw Learn Syst 25:1263–1276CrossRefGoogle Scholar
- 40.Zeng HB, He Y, Wu M, Xiao SP (2015) Stability analysis of generalized neural networks with time-varying delays via a new integral inequality. Neurocomputing 161:148–154CrossRefGoogle Scholar
- 41.Liu Y, Lee SM, Kwon OM, Park JH (2015) New approach to stability criteria for generalized neural networks with interval time-varying delays. Neurocomputing 149:1544–1551CrossRefGoogle Scholar
- 42.Chen ZW, Yang J, Zhong SM (2016) Delay-partitioning approach to stability analysis of generalized neural networks with time-varying delay via new integral inequality. Neurocomputing 191:380–387CrossRefGoogle Scholar
- 43.Fridman E, Shaked U (2003) On reachable sets for linear systems with delay and bounded peak inputs. Automatica 39:2005–2010MathSciNetCrossRefzbMATHGoogle Scholar
- 44.Kim JH (2008) Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances. Automatica 44:2940–2943MathSciNetCrossRefzbMATHGoogle Scholar
- 45.Kwon OM, Lee SM, Park JH (2011) On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances. Inform Sci 181:3735–3748MathSciNetCrossRefzbMATHGoogle Scholar
- 46.Nam PT, Pathirana PN (2011) Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays. Automatica 47:1838–1841MathSciNetCrossRefzbMATHGoogle Scholar
- 47.Nam PT, Pathirana PN, Trinh H (2013) Exponential convergence of time-delay systems in the presence of bounded disturbances. J Optim Theory Appl 157:843–852MathSciNetCrossRefzbMATHGoogle Scholar
- 48.Zuo Z, Chen Y, Wang W, Ho DWC, Chen MZQ, Li H (2013) A note on reachable set bounding for delayed systems with polytopic uncertain. J Franklin Inst 350:1827–1835MathSciNetCrossRefGoogle Scholar
- 49.Feng Z, Lam J (2014) An improved result on reachable set estimation and synthesis of time-delay systems. Appl Math Comput 249:89–97MathSciNetzbMATHGoogle Scholar
- 50.Nam PT, Pathirana PN, Trinh H (2015) Convergence within a polyhedron: controller design for time-delay systems with bounded disturbances. IET Control Theory Appl 9:905–914MathSciNetCrossRefGoogle Scholar
- 51.Shen C, Zhong S (2011) The ellipsoidal bound of reachable sets for linear neutral systems with disturbances. J Franklin Inst 348:2570–2585MathSciNetCrossRefzbMATHGoogle Scholar
- 52.Zuo Z, Fu Y, Wang Y (2012) Results on reachable set estimation for linear systems with both discrete and distributed delays. IET Control Theory Appl 6:2346–2350MathSciNetCrossRefGoogle Scholar
- 53.Zhang B, Lam J, Xu S (2014) Reachable set estimation and controller design for distributed delay systems with bounded disturbances. J Franklin Inst 351:3068–3088MathSciNetCrossRefzbMATHGoogle Scholar
- 54.That ND, Nam PT, Ha QP (2013) Reachable set bounding for linear discrete-time systems with delays and bounded disturbances. J Optim Theory Appl 157:96–107MathSciNetCrossRefzbMATHGoogle Scholar
- 55.Lam J, Zhang B, Chen Y, Xu S (2015) Reachable set estimation for discrete-time linear systems with time-delays. Int J Robust Nonlinear Control 25:269–281MathSciNetCrossRefzbMATHGoogle Scholar
- 56.Hien LV, Trinh H (2014) A new approach to state bounding for linear time-varying systems with delay and bounded disturbances. Automatica 50:1735–1738MathSciNetCrossRefzbMATHGoogle Scholar
- 57.Nam PT, Pathirana PN, Trinh H (2015) Reachable set bounding for nonlinear perturbed time-delay systems: the smallest bound. Appl Math Lett 43:68–71MathSciNetCrossRefzbMATHGoogle Scholar
- 58.Chen Y, Lam J, Zhang B (2015) Estimation and synthesis of reachable set for switched linear systems. Automatica 63:122–132MathSciNetCrossRefzbMATHGoogle Scholar
- 59.Feng Z, Lam J (2015) On reachable set estimation of singular systems. Automatica 52:146–153MathSciNetCrossRefzbMATHGoogle Scholar
- 60.Zuo Z, Wang Z, Chen Y, Wang W (2014) A non-ellipsoidal reachable set estimation for uncertain neural networks with time-varying delay. Commun Nonlinear Sci Numer Simul 19:1097–1106MathSciNetCrossRefGoogle Scholar
- 61.Trinh H, Nam PT, Pathirana PN, Le HP (2015) On backwards and forwards reachable sets bounding for perturbed time-delay systems. Appl Math Comput 269:664–673MathSciNetGoogle Scholar
- 62.Hien LV, Trinh H (2015) Refined Jensen-based inequality approach to stability analysis of time-delays systems. IET Control Theory Appl 9:2188–2194MathSciNetCrossRefGoogle Scholar
- 63.Park PG, Lee WI, 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
- 64.He Y, Ji MD, Zhang CK, Wu M (2016) Global exponential stability of neural networks with time-varying delay based on free-matrix-based integral inequality. Neural Netw 77:80–86CrossRefGoogle Scholar
- 65.Park PG, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47:235–238MathSciNetCrossRefzbMATHGoogle Scholar
- 66.Seuret A, Gouaisbaut F, Fridman E (2013) Stability of systems with fast-varying delay using improved Wirtinger’s inequality. IEEE conference on decision and control. Florence, Italy, pp 946–951Google Scholar