A fractional Halanay inequality for neutral systems and its application to Cohen-Grossberg neural networks

Mohammed D. Kassim; Mohammed D. Kassim

doi:10.3934/math.2025115

AIMS Mathematics

2025, Volume 10, Issue 2: 2466-2491. doi: 10.3934/math.2025115

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Research article

A fractional Halanay inequality for neutral systems and its application to Cohen-Grossberg neural networks

Mohammed D. Kassim ^,

Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34151, Saudi Arabia

Received: 31 December 2024 Revised: 26 January 2025 Accepted: 08 February 2025 Published: 12 February 2025
MSC : 92B20, 26A33, 34D20

We expand the Halanay inequality to accommodate fractional-order systems incorporating both discrete and distributed neutral delays. By establishing specific conditions, we demonstrate that the solutions of these systems converge to zero at a Mittag-Leffler rate. Our analysis is versatile, accommodating a wide range of delay kernels. This versatility extends the applicability of our findings to fractional Cohen-Grossberg neural networks, offering valuable insights into their stability and dynamical behavior.
- Halanay inequality,
- Cohen-Grossberg neural network system,
- Caputo fractional derivative,
- neutral delay,
- Mittag-Leffler stability
Citation: Mohammed D. Kassim. A fractional Halanay inequality for neutral systems and its application to Cohen-Grossberg neural networks[J]. AIMS Mathematics, 2025, 10(2): 2466-2491. doi: 10.3934/math.2025115

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Abstract

We expand the Halanay inequality to accommodate fractional-order systems incorporating both discrete and distributed neutral delays. By establishing specific conditions, we demonstrate that the solutions of these systems converge to zero at a Mittag-Leffler rate. Our analysis is versatile, accommodating a wide range of delay kernels. This versatility extends the applicability of our findings to fractional Cohen-Grossberg neural networks, offering valuable insights into their stability and dynamical behavior.

References

[1]	C. T. H. Baker, A. Tang, Generalized Halanay inequalities for Volterra functional differential equations and discretized versions, Invited plenary talk, Volterra Centennial Meeting, UTA Arlington, June, 1996.
[2]	A. Boroomand, M. B. Menhaj, Fractional-order Hopfield neural networks, Int. Conf. Neural Inf. Process., 5506 (2008), 883–890.
[3]	R. Caponetto, L. Fortuna, D. Porto, Bifurcation and chaos in noninteger order cellular neural networks, Int. J. Bifurc. Chaos, 8 (1998), 1527–1539.
[4]	B. Chen, J. Chen, Razumikhin-type stability theorems for functional fractional-order differential systems and applications, Appl. Math. Comput., 254 (2015), 63–69. https://doi.org/10.1016/j.amc.2014.12.010 doi: 10.1016/j.amc.2014.12.010
[5]	C. J. Cheng, T. L. Liao, J. J. Yan, C. C. Hwang, Globally asymptotic stability of a class of neutral-type neural networks with delays, IEEE T. Syst. Man Cy. B, 36 (2006), 1191–1195. https://doi.org/10.1109/TSMCB.2006.874677 doi: 10.1109/TSMCB.2006.874677
[6]	M. Concezzi, R. Spigler, Some analytical and numerical properties of the Mittag-Leffler functions, Fract. Calc. Anal. Appl., 18 (2015), 64–94. https://doi.org/10.1515/fca-2015-0006 doi: 10.1515/fca-2015-0006
[7]	O. Faydasicok, S. Arik, An analysis of global stability of a class of neutral-type neural systems with time delays, Internat. J. Math. Comput. Sci., 6 (2012), 391–395.
[8]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. Y. Rogosin, Mittag-Leffler functions, related topics and applications, Berlin: Springer, 2014. https://doi.org/10.1007/978-3-662-43930-2
[9]	S. Grossberg, Nonlinear neural networks: Principles, mechanisms, and architectures, Neural Networks, 1 (1988), 17–61.
[10]	A. Halanay, Differential equations, New York: Academic Press, 1996.
[11]	B. B. He, H. C. Zhou, Y. Q. Chen, C. H. Kou, Asymptotical stability of fractional order systems with time delay via an integral inequality, IET Control Theory A., 12 (2018), 1748–1754. https://doi.org/10.1049/iet-cta.2017.1144 doi: 10.1049/iet-cta.2017.1144
[12]	L. V. Hien, V. Phat, H. Trinh, New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems, Nonlinear Dynam., 82 (2015), 563–575. https://doi.org/10.1007/s11071-015-2176-0 doi: 10.1007/s11071-015-2176-0
[13]	F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Cont. Dyn.-s, 13 (2020), 709–722.
[14]	Y. Ke, C. Miao, Stability analysis of fractional-order Cohen-Grossberg neural networks with time delay, Int. J. Comput. Math., 92 (2015), 1102–1113. http://dx.doi.org/10.1080/00207160.2014.935734 doi: 10.1080/00207160.2014.935734
[15]	R. P. Lippmann, D. M. Shahian, Coronary artery bypass risk prediction using neural networks, Ann. Thorac. Surg., 63 (1997), 1635–1643.
[16]	B. Liu, W. Lu, T. Chen, Generalized Halanay inequalities and their applications to neural networks with unbounded time-varying delays, IEEE T. Neural Networ., 22 (2011), 1508–1513. https://doi.org/10.1109/TNN.2011.2160987 doi: 10.1109/TNN.2011.2160987
[17]	X. Lou, Q. Ye, B. Cui, Exponential stability of genetic regulatory networks with random delays, Neurocomputing, 73 (2010), 759–769.
[18]	X. Luan, P. Shi, F. Liu, Robust adaptive control for greenhouse climate using neural networks, Int. J. Robust Nonlin., 21 (2011), 815–826. https://doi.org/10.1002/hipo.20797 doi: 10.1002/hipo.20797
[19]	F. Mainardi, On some properties of the Mittag-Leffler function $ E_{\alpha }(-t^{\alpha })$, completely monotone for $t>0$ with $0 < \alpha < 1$, Discrete Cont. Dyn.-B, 10 (2014), 2267–2278. https://doi.org/10.48550/arXiv.1305.0161 doi: 10.48550/arXiv.1305.0161
[20]	M. W. Michalski, Derivatives of noninteger order and their applications, Ph. D. Thesis, Polska Akademia Nauk, 1993.
[21]	S. Mohamed, K. Gopalsamy, Continuous and discrete Halanay-type inequalities, B. Aust. Math. Soc., 61 (2000), 371–385.
[22]	P. Niamsup, Stability of time-varying switched systems with time-varying delay, Nonlinear Anal.-Hybri., 3 (2009), 631–639.
[23]	T. Simon, Comparing Frechet and positive stable laws, Electron. J. Probab., 19 (2014), 1–25. https://doi.org/10.1214/EJP.v19-3058 doi: 10.1214/EJP.v19-3058
[24]	M. S. Ali, S. Saravanan, M. E. Rani, S. Elakkia, J. Cao, A. Alsaedi, et al., Asymptotic stability of Cohen-Grossberg BAM neutral type neural networks with distributed time varying delays, Neural Process. Lett., 46 (2017), 991–1007. https://doi.org/10.1007/s11063-017-9622-6 doi: 10.1007/s11063-017-9622-6
[25]	H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl., 270 (2002), 143–149. https://doi.org/10.1016/S0022-247X(02)00056-2 doi: 10.1016/S0022-247X(02)00056-2
[26]	L. Wen, Y. Yu, W. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169–178. https://doi.org/10.1016/j.jmaa.2008.05.007 doi: 10.1016/j.jmaa.2008.05.007
[27]	Z. Zeng, J. Wang, X. Liao, Global asymptotic stability and global exponential stability of neural networks with unbounded time-varying delays, IEEE T. Circuits-II, 52 (2005), 168–173. https://doi.org/10.1109/TCSII.2004.842047 doi: 10.1109/TCSII.2004.842047
[28]	B. Zheng, Z. Wang, Mittag-Leffler synchronization of fractional-order coupled neural networks with mixed delays, Appl. Math. Comput., 430 (2022), 127303. https://doi.org/10.1016/j.amc.2022.127303 doi: 10.1016/j.amc.2022.127303

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