Existence and Finite-Time Stability Analysis of Fractional Neural Networks Using Gronwall's Inequality

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Authors

  • K. Kaliraj Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600005, Tamil Nadu, India Author
  • S. John Lourdhu Antony Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600005, Tamil Nadu, India Author
  • R. Ruthra Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600005, Tamil Nadu, India Author

DOI:

https://doi.org/10.5890/DNC.2027.03.007

Abstract

This article delves into investigating the existence of solutions for fractional-order neural networks (FNN), alongside analyzing the finite-time stability (FTS) of fractional differential equations. FTS, which focuses on trajectories converging to equilibrium within a short time frame, is a central aspect of this exploration. Employing Gronwall's inequality as the primary analytical tool, the study derives conditions for FTS. The theoretical findings are then substantiated through rigorous numerical simulations.

References

[1] Atangana, A. and Bildik, N. (2013), The use of fractional order derivative to predict the groundwater flow, Mathematical Problems in Engineering, 2013(1), 1-9.

[2] Atangana, A. and Vermeulen, P.D. (2014), Analytical solutions of a space-time fractional derivative of groundwater flow equation, Abstract and Applied Analysis, 2014, 1-11.

[3] Metzler, R. and Klafter, J. (2000), The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339(1), 1-77.

[4] Dung, N.T. (2013), Fractional stochastic differential equations with applications to finance, Journal of Mathematical Analysis and Applications, 397(1), 334-348.

[5] Farhadi, A., Erjaee, G.H., and Salehi, M. (2017), Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Computers and Mathematics with Applications, 73(9), 2066-2075.

[6] Ladde, G. and Wu, L. (2010), Development of nonlinear stochastic models by using stock price data and basic statistics, Parallel and Scientific Computations, 18, 269-282.

[7] Debbouche, A. and Nieto, J.J. (2015), Relaxation in controlled systems described by fractional integro-differential equations with nonlocal control conditions, Electronic Journal of Differential Equations, 2015(89), 1-18.

[8] Debbouche, A. and Nieto, J.J. (2014), Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls, Applied Mathematics and Computation, 245, 74-85.

[9] Magin, R. (2004), Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering, 32(1), 1-104.

[10] Kavitha, K., Vijayakumar, V., Udhayakumar, R., and Ravichandran, C. (2021), Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian Journal of Control, 24(3), 1406-1415.

[11] Lendek, A. and Tan, L. (2021), Mitigation of derivative kick using time-varying fractional-order PID control, IEEE Access, 9, 55974-55987.

[12] Tarasov, V.E. (2011), Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer.

[13] Nisar, K.S., Jothimani, K., Kaliraj, K., and Ravichandran, C. (2021), An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos, Solitons and Fractals, 146(2), 110915.

[14] Kuang, Z., Sun, L., Gao, H., and Tomizuka, M. (2020), Precise motion control of wafer stages via adaptive neural network and fractional-order super-twisting algorithm, IFAC-PapersOnLine, 53(2), 8315-8320.

[15] Kamenkov, G. (1953), On stability of motion over a finite interval of time, Journal of Applied Mathematics and Mechanics, 17(2), 529-540.

[16] Chang, X., Xiao, Q., Zhu, Y., and Xiao, J. (2021), Stability analysis of two kinds of fractional-order neural networks based on Lyapunov method, IEEE Access, 9, 124132-124141.

[17] Jafarian, A., Mokhtarpour, M., and Baleanu, D. (2017), Artificial neural network approach for a class of fractional ordinary differential equations, Neural Computing and Applications, 28, 765-773.

[18] Sang, H., Nie, H., and Zhao, J. (2022), Dissipativity-based synchronization for switched discrete-time-delayed neural networks with combined switching paradigm, IEEE Transactions on Cybernetics, 52(8), 7995-8005.

[19] Ding, S., Wang, Z., and Rong, N. (2021), Intermittent control for quasisynchronization of delayed discrete-time neural networks, IEEE Transactions on Cybernetics, 51(2), 862-873.

[20] Zhao, D., Wang, Z., Chen, Y., and Wei, G. (2020), Proportional-integral observer design for multidelayed sensor-saturated recurrent neural networks: A dynamic event-triggered protocol, IEEE Transactions on Cybernetics, 50(11), 4619-4632.

[21] Li, M. and Wang, J.R. (2017), Finite time stability of fractional delay differential equations, Applied Mathematics Letters, 64, 170-176.

[22] Chen, L., Huang, T., Machado, J.A.T., Lopes, A.M., Chai, Y., and Wu, R. (2019), Delay-dependent criterion for asymptotic stability of a class of fractional-order memristive neural networks with time-varying delays, Neural Networks, 118, 289-299.

[23] Lazarević, M.P. and Spasić, A.M. (2009), Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach, Mathematical and Computer Modelling, 49(3-4), 475-481.

[24] Ding, X., Cao, J., Zhao, X., and Alsaadi, F.E. (2017), Finite-time stability of fractional-order complex-valued neural networks with time delays, Neural Processing Letters, 46(2), 561-580.

[25] Wang, H., Yu, Y., Wen, G., Zhang, S., and Yu, J. (2015), Global stability analysis of fractional-order Hopfield neural networks with time delay, Neurocomputing, 154, 15-23.

[26] Yang, F. and Wang, X. (2021), Dynamic characteristic of a new fractional-order chaotic system based on the Hopfield neural network and its digital circuit implementation, Physica Scripta, 96(3).

[27] Li, X., Liu, H., Liu, K., Bo, Q., and Wang, Y. (2020), An improved-gain analysis method of nonlinear delay systems and its application to delayed neural networks, International Journal of Control, 95, 1021-1031.

[28] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier: Amsterdam.

[29] Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley: New York.

[30] Podlubny, I. (1999), Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of their Solution and Some of Their Applications, Academic Press.

[31] Amato, F., Ambrosino, R., Ariola, M., Cosentino, C., and Tommasi, G.D. (2013), Finite-time Stability and Control, Lecture Notes in Control and Information Sciences, Springer: London.

[32] Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.

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How to Cite

Kaliraj, K., Antony, S. J. L., & Ruthra, R. (2026). Existence and Finite-Time Stability Analysis of Fractional Neural Networks Using Gronwall’s Inequality. Discontinuity, Nonlinearity, and Complexity, 16(1), 97-109. https://doi.org/10.5890/DNC.2027.03.007