# Global Asymptotic Stabilization of Cellular Neural Networks with Proportional Delay via Impulsive Control

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## Abstract

This paper considers the global asymptotic stabilization of a class of cellular neural networks with proportional delay via impulsive control. First, from impulsive control point of view, some delay-dependent criteria are established to guarantee the asymptotic stability and global asymptotic stability of the zero solution of a general impulsive differential system with proportional delay by applying the Lyapunov–Razumikhin method. Second, based on the obtained criteria and LMI approach, several delay-dependent conditions are derived to ensure the uniqueness and global asymptotic stability of the equilibrium point of a class of cellular neural networks with impulses and proportional delay. It is shown that impulses can be used to globally asymptotically stabilize some unstable and even chaotic cellular neural networks with proportional delay. Moreover, the proposed stability conditions expressed in terms of linear matrix inequalities (LMIs) can be checked by the Matlab LMI toolbox, and so it is effective to implement in real problems. Finally, three numerical examples are provided to illustrate the effectiveness of the theoretical results.

## Keywords

Global asymptotic stability Cellular neural network Proportional delay Impulsive control Lyapunov–Razumikhin method Linear matrix inequality (LMI)## Mathematics Subject Classification

34D23 93D20## Notes

### Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for a number of valuable comments and constructive suggestions that have improved the presentation and quality of this paper. This work was supported by the Natural Science Foundation of Guangdong Province, China (Nos. 2016A030313005 and 2015A030313643) and the Innovation Project of Department of Education of Guangdong Province, China (No. 2015KTSCX147).

## References

- 1.Ahmada S, Stamova IM (2008) Global exponential stability for impulsive cellular neural networks with time-varying delays. Nonlinear Anal Theory Methods Appl 69(3):786–795MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Akca H, Benbourenane J, Covachev V (2014) Global exponential stability of impulsive Cohen–Grossberg-Type BAM neural networks with time-varying and distributed delays. Int J Appl Phys Math 4(3):196–200CrossRefGoogle Scholar
- 3.Arik S, Tavanoglu V (2000) On the global asymptotic stability of delayed cellular neural networks. IEEE Trans Circuits Syst I 47:571–574MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Balasubramaniam P, Vaitheeswaran V, Rakkiyappan R (2012) Global robust asymptotic stability analysis of uncertain switched Hopfield neural networks with time delay in the leakage term. Neural Comput Appl 21(7):1593–1616CrossRefGoogle Scholar
- 5.Bemporad A (1998) Predictive control of teleoperated constrained systems with unbounded communication delays. In: Proceedings of the 37th IEEE conference on decision and control, Tampa, Florida, USAGoogle Scholar
- 6.Chen T, Wang L (2007) Power-rate global stability of dynamical systems with unbounded time-varying delays. IEEE Trans Circuits Syst II 54(8):705–709CrossRefGoogle Scholar
- 7.Dovrolis C, Stiliadisd D, Ramanathan P (1999) Proportional differential services: delay differentiation and packet scheduling. ACM SIGCOMM Comput Commun Rev 29(4):109–120CrossRefGoogle Scholar
- 8.Eastham J, Hastings K (1988) Optimal impulse control of portfolios. Math Oper Res 4:588–605MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Fox L, Mayers DF, Ockendon JR, Taylor AB (1971) On a functional differential equation. J Inst Math Appl 8:271–307MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Guan K, Luo Z (2013) Stability results for impulsive pantograph equations. Appl Math Lett 26:1169–1174MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Guan K, Wang Q (2018) Impulsive control for a class of cellular neural networks with proportional delay. Neural Process Lett. https://doi.org/10.1007/s11063-017-9776-2
- 12.Hardy GH, Littlewood JE, Polya G (1952) Inequalities, 2nd edn. Cambridge University Press, LondonzbMATHGoogle Scholar
- 13.He H, Yan L, Tu J (2012) Guaranteed cost stabilization of time-varying delay cellular neural networks via Riccati inequality approach. Neural Process Lett 35(2):151–158CrossRefGoogle Scholar
- 14.Hien LV, Son DT (2015) Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl Math Comput 251:14–23MathSciNetzbMATHGoogle Scholar
- 15.Huang Z (2017) Almost periodic solutions for fuzzy cellular neural networks with multi-proportional delays. Int J Mach Learn Cybern 8(4):1323–1331CrossRefGoogle Scholar
- 16.Kato T, Mcleod JB (1971) The functional-differential equation \(y^{\prime }(x)=ay(\lambda x)+by(x)\). Bull Am Math Soc 77(6):891–937MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Kinh CT, Hien LV, Ke TD (2018) Power-rate synchronization of fractional-order nonautonomous neural networks with heterogeneous proportional delays. Neural Process Lett 47(1):139–151CrossRefGoogle Scholar
- 18.Koo M, Choi H, Lim J (2012) Output feedback regulation of a chain of integrators with an unbounded time-varying delay in the input. IEEE Trans Autom Control 57(10):2662–2667MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Kulkarm S, Sharma R, Mishra I (2012) New QoS routing algorithm for MPLS networks using delay and bandwidth constrainst. Int J Inf Commun Technol Res 2(3):285–293Google Scholar
- 20.Li X (2009) Uniform asymptotic stability and global stability of impulsive infinite delay differential equations. Nonlinear Anal 70:1975–1983MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Li L, Huang L (2010) Equilibrium analysis for improved signal range model of delayed cellular neural networks. Neural Process Lett 31(3):177–194CrossRefGoogle Scholar
- 22.Li X, Song S (2013) Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans Neural Netw Learn Syst 24(6):868–877CrossRefGoogle Scholar
- 23.Li X, Wu J (2016) Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64:63–69MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Li X, Song S (2017) Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans Autom Control 62(1):406–411MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Li X, Cao J (2017) An impulsive delay inequality involving unbounded time-varying delay and applications. IEEE Trans Autom Control 62(7):3618–3625MathSciNetCrossRefzbMATHGoogle Scholar
- 26.Li X, Ding Y (2017) Razumikhin-type theorems for time-delay systems with persistent impulses. Syst Control Lett 107:22–27MathSciNetCrossRefzbMATHGoogle Scholar
- 27.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 Math Comput 330:152–169MathSciNetGoogle Scholar
- 28.Liu B (2017) Finite-time stability of a class of CNNs with heterogeneous proportional delays and oscillating leakage coefficients. Neural Process Lett 45:109–119CrossRefGoogle Scholar
- 29.Liu X, Teo K, Xu B (2005) Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays. IEEE Trans Neural Netw 16:1329–1339CrossRefGoogle Scholar
- 30.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
- 31.Liu Y, Zhang D, Lu J (2017) Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dyn 87(1):553–565CrossRefzbMATHGoogle Scholar
- 32.Liu Y, Xu P, Lu J, Liang J (2016) Global stability of Clifford-valued recurrent neural networks with time delays. Nonlinear Dyn 84(2):767–777MathSciNetCrossRefzbMATHGoogle Scholar
- 33.Lu J, Ho DWC, Cao J (2010) A unified synchronization criterion for impulsive dynamical networks. Automatica 46:1215–1221MathSciNetCrossRefzbMATHGoogle Scholar
- 34.Ockendon JR, Taylor AB (1971) The dynamics of a current collection system for an electric locomotive. Proc R Soc Lond Ser A 332:447–468CrossRefGoogle Scholar
- 35.Prussing JE, Wellnitz LJ (1989) Optimal impulsive time-fixed directascent interaction. J Guid Control Dyn 12:487–494CrossRefGoogle Scholar
- 36.Raja R, Sakthivel R, Marshal Anthoni S, Kim H (2011) Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays. Int J Appl Math Comput Sci 21(1):127–135MathSciNetCrossRefzbMATHGoogle Scholar
- 37.Roska T, Chua LO (1992) Cellular neural networks with delay type template elements and nonuniform grids. Int J Circuit Theory Appl 20(4):469–481CrossRefzbMATHGoogle Scholar
- 38.Song X, Zhao P, Xing Z, Peng J (2016) Global asymptotic stability of CNNs with impulses and multi-proportional delays. Math Methods Appl Sci 39:722–733MathSciNetCrossRefzbMATHGoogle Scholar
- 39.Stamova IM, Stamov GT (2011) Impulsive control on global asymptotic stability for a class of impulsive bidirectional associative memory neural networks with distributed delays. Math Comput Model 53:824–831MathSciNetCrossRefzbMATHGoogle Scholar
- 40.Tao W, Liu Y, Lu J (2017) Stability and \(L_{2}\)-gain analysis for switched singular linear systems with jumps. Math Methods Appl Sci 40(3):589–599MathSciNetCrossRefzbMATHGoogle Scholar
- 41.Tan J, Li C (2017) Finite-time stability of neural networks with impulse effect and time-varying delay. Neural Process Lett 46:29–39CrossRefGoogle Scholar
- 42.Tank D, Hopfield J (1987) Neural computation by concentrating information in time. Proc Natl Acad Sci U S 84:1896–1900MathSciNetCrossRefGoogle Scholar
- 43.Wang G, Guo L, Wang H, Duan H, Liu L, Li J (2014) Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Comput Appl 24:853–871CrossRefGoogle Scholar
- 44.Wang W, Li L, Peng H, Kurths J, Xiao J, Yang Y (2016) Anti-synchronization control of memristive neural networks with multiple proportional delays. Neural Process Lett 43:269–283CrossRefGoogle Scholar
- 45.Wang Z, Wang X, Li Y, Huang X (2017) Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay. Int J Bifurc Chaos 27(13):1750209MathSciNetCrossRefzbMATHGoogle Scholar
- 46.Yang T, Chua LO (1997) Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans Circuits Syst I Fundam Theory Appl 44:976–988MathSciNetCrossRefGoogle Scholar
- 47.Yang X, Lu J (2016) Finite-time synchronization of coupled networks with Markovian topology and impulsive effects. IEEE Trans Autom Control 61(8):2256–2261MathSciNetCrossRefzbMATHGoogle Scholar
- 48.Yang X, Cao J, Lu J (2011) Synchronization of delayed complex dynamical networks with impulsive and stochastic effects. Nonlinear Anal Real World Appl 12:2252–2266MathSciNetCrossRefzbMATHGoogle Scholar
- 49.Yang Z, Xu D (2007) Stability analysis and design of impulsive control systems with time delay. IEEE Trans Autom Control 52(8):1448–1454MathSciNetCrossRefzbMATHGoogle Scholar
- 50.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 265:696–707MathSciNetzbMATHGoogle Scholar
- 51.Zhang A (2015) New results on exponential convergence for cellular neural networks with continuously distributed leakage delays. Neural Process Lett 41:421–433CrossRefGoogle Scholar
- 52.Zhou L (2013) Dissipativity of a class of cellular neural networks with proportional delays. Nonlinear Dyn 73:1895–1903MathSciNetCrossRefzbMATHGoogle Scholar
- 53.Zhou L (2013) Delay-dependent exponential stability of cellular neural networks with multi-proportional delays. Neural Process Lett 38(3):347–359CrossRefGoogle Scholar
- 54.Zhou L, Chen X, Yang Y (2014) Asymptotic stability of cellular neural networks with multi-proportional delays. Appl Math Comput 229:457–466MathSciNetzbMATHGoogle Scholar
- 55.Zhou L (2014) Global asymptotic stability of cellular neural networks with proportional delays. Nonlinear Dyn 77(1):41–47MathSciNetCrossRefzbMATHGoogle Scholar
- 56.Zhou L, Zhang Y (2015) Global exponential stability of cellular neural networks with multi-proportional delays. Int J Biomath 8(6):1–17MathSciNetCrossRefGoogle Scholar
- 57.Zhou L (2015) Delay-dependent exponential synchronization of recurrent neural networks with multi-proportional delays. Neural Process Lett 42:619–632CrossRefGoogle Scholar
- 58.Zhou L (2015) Novel global exponential stability criteria for hybrid BAM neural networks with proportional delays. Neurocomputing 161:99–106CrossRefGoogle Scholar
- 59.Zhou L, Zhao Z (2016) Exponential stability of a class of competitive neural networks with multiproportional delays. Neural Process Lett 44:651–663CrossRefGoogle Scholar