Finite-time stabilization in probability for stochastic reaction–diffusion Cohen–Grossberg neural networks with mixed delays

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Abstract

We studied finite-time stabilization in probability for stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed delays and multiplicative noise under Neumann boundary conditions. By developing a Lyapunov-Krasovskii functional and using stochastic theory with the Neumann-Poincaré inequality, we obtained several verifiable stabilization criteria. A componentwise controller with a nonlinear finite-time term was further designed to induce a constant drift effect in the Lyapunov evolution, leading to an explicit stochastic settling time bound that quantifies the roles of diffusion, delays, couplings, noise, and control gains. Simulations confirmed the effectiveness of the proposed method.

Keywords

finite-time stabilization / stochastic reaction-diffusion / mixed delays / generalized Halanay inequality / Lyapunov-Krasovskii functional

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References

[1] H. Xu, Q. Zhu, Finite-time stability of stochastic systems with proportional delay involving hybrid impulses, Commun. Nonlinear Sci. Numer. Simul., 143 (2025), 108600. https://doi.org/10.1016/j.cnsns.2025.108600

[2] Y. Bai, L. Wang, W. Xu, Z. Zhang, D. Li, Stochastic dynamics analysis of nonlinear systems with time delay based on path integration method, J. Sound Vib., 615 (2025), 119169. https://doi.org/10.1016/j.jsv.2025.119169

[3] Y. Huang, A. Wu, J. E. Zhang, Exponential stability of stochastic time-delay neural networks with random delayed impulses, Neural Process. Lett., 56 (2024), 38. https://doi.org/10.1007/s11063-024-11521-3

[4] Z. Zhang, X. Zhang, T. Yu, Global exponential stability of neutral-type Cohen-Grossberg neural networks with multiple time-varying neutral and discrete delays, Neurocomputing, 490 (2022), 124-131. https://doi.org/10.1016/j.neucom.2022.03.068

[5] Z. Zhang, T. Yu, X. Zhang, Algebra criteria for global exponential stability of multiple time-varying delay Cohen-Grossberg neural networks, Appl. Math. Comput., 435 (2022), 127461. https://doi.org/10.1016/j.amc.2022.127461

[6] D. Maity, M. H. Mamduhi, S. Hirche, K. H. Johansson, Optimal LQG control of networked systems under traffic-correlated delay and dropout, IEEE Control Syst. Lett., 6 (2022), 1280-1285. https://doi.org/10.1109/LCSYS.2021.3091492

[7] F. Huang, S. Gao, Stochastic integral input-to-state stability for stochastic delayed networked control systems and its applications, Commun. Nonlinear Sci. Numer. Simul., 138 (2024), 108177. https://doi.org/10.1016/j.cnsns.2024.108177

[8] Z. Zhu, S. Liao, F. Jia, Nonovershooting prescribed finite-time control for nonlinear pure-feedback systems, Complex Syst. Stab. Control., 1 (2025), 4. https://doi.org/10.53941/cssc.2025.100004

[9] C. He, A. Martinez, B. Niu, D. K. Jain, Y. Jiang, D. Zubov, et al., Predefined-time fault-tolerant consensus tracking control for multiple flexible robotic manipulator systems with prescribed performance, Complex Syst. Stab. Control, 1 (2025), 7. https://doi.org/10.53941/cssc.2025.100007

[10] J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. U.S.A., 79 (1982), 2554-2558. https://doi.org/10.1073/pnas.79.8.2554

[11] B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Man Cybern., 18 (1988), 49-60. https://doi.org/10.1109/21.87054

[12] M. A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern., SMC-13 (1983), 815-826. https://doi.org/10.1109/TSMC.1983.6313075

[13] Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lévy processes, IEEE Trans. Autom. Control, 70 (2025), 1176-1183. https://doi.org/10.1109/TAC.2024.3448128

[14] H. Yuan, Q. Zhu, Stabilities of delay stochastic McKean-Vlasov equations in the G-framework, Sci. China Inf. Sci., 68 (2025), 112203. https://doi.org/10.1007/s11432-024-4075-2

[15] H. Xu, Q. Zhu, Stability of discrete-time impulsive stochastic systems with hybrid non-deterministic delays, IEEE Trans. Autom. Control, 2026 (2026), 1-8. https://doi.org/10.1109/TAC.2026.3666702

[16] M. S. Ali, S. Saravanan, L. Palanisamy, Stochastic finite-time stability of reaction-diffusion Cohen-Grossberg neural networks with time-varying delays, Chin. J. Phys., 57 (2019), 314-328. https://doi.org/10.1016/j.cjph.2018.09.039

[17] Z. Li, M. Zhao, A. Abdurahman, Fixed-time synchronization of memristor-based stochastic BAM neural networks with time-varying delays, Complex Syst. Stab. Control, 1 (2025), 8. https://doi.org/10.53941/cssc.2025.100008

[18] Q. Zhu, J. Cao, Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays, IEEE Trans. Cybern., 41 (2011), 341-353. https://doi.org/10.1109/TSMCB.2010.2053354

[19] C. Zhao, Y. Song, Q. Zhu, K. Shi, Input-to-state stability analysis for stochastic mixed time-delayed neural networks with hybrid impulses, Math. Probl. Eng., 2022 (2022), 6135390. https://doi.org/10.1155/2022/6135390

[20] X. Han, K. Wu, X. Ding, Finite-time stabilization for stochastic reaction-diffusion systems with Markovian switching via boundary control, Appl. Math. Comput., 385 (2020), 125422. https://doi.org/10.1016/j.amc.2020.125422

[21] C. Aouiti, H. Jallouli, Q. Zhu, T. Huang, K. Shi, New results on finite/fixed-time stabilization of stochastic second-order neutral-type neural networks with mixed delays, Neural Process. Lett., 54 (2022), 5415-5437. https://doi.org/10.1007/s11063-022-10868-9

[22] F. Kong, Q. Zhu, Fixed-time stabilization of discontinuous neutral neural networks with proportional delays via new fixed-time stability lemmas, IEEE Trans. Neural Netw. Learn. Syst., 34 (2023), 775-785. https://doi.org/10.1109/TNNLS.2021.3101252

[23] W. Fang, T. Xie, Q. Zhu, T. Huang, Fixed time stabilization of stochastic neutral-type fuzzy inertial neural network with mixed delays, Fuzzy Sets Syst., 520 (2025), 109566. https://doi.org/10.1016/j.fss.2025.109566

[24] R. Chitre, W. M. Haddad, Finite time stability and optimal control for stochastic dynamical systems, Axioms, 14 (2025), 767. https://doi.org/10.3390/axioms14100767

[25] H. Peng, Q. Zhu, Finite-time stability and stabilization of highly nonlinear stochastic systems via the noise control, IEEE Trans. Autom. Control, 2026 (2026), 1-8. https://doi.org/10.1109/TAC.2026.3660601

[26] Y. Li, F. Deng, Finite time stability analysis of the coupled stochastic reaction-diffusion systems on networks, Commun. Nonlinear Sci. Numer. Simul., 131 (2024), 107882. https://doi.org/10.1016/j.cnsns.2024.107882

[27] K. Wu, M. Ren, X. Liu, Exponential input-to-state stability of stochastic delay reaction-diffusion neural networks, Neurocomputing, 412 (2020), 399-405. https://doi.org/10.1016/j.neucom.2019.09.118

[28] F. Zheng, P. M. Frank, Robust control of uncertain distributed delay systems with application to the stabilization of combustion in rocket motor chambers, Automatica, 38 (2002), 487-497. https://doi.org/10.1016/S0005-1098(01)00232-1

[29] H. Yu, S. Zhu, S. Wen, C. Mu, Finite-time stability in probability of stochastic delay systems via generalized Halanay inequality, Syst. Control Lett., 195 (2025), 105969. https://doi.org/10.1016/j.sysconle.2024.105969

[30] J. Yin, S. Khoo, Z. Man, X. Yu, Finite-time stability and instability of stochastic nonlinear systems, Automatica, 47 (2011), 2671-2677. https://doi.org/10.1016/j.automatica.2011.08.050

[31] X. Zong, T. Li, G. Yin, J. Zhang, Delay tolerance for stable stochastic systems and extensions, IEEE Trans. Autom. Control, 66 (2021), 2604-2619. https://doi.org/10.1109/TAC.2020.3012525

[32] S. P. Bhat, D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766. https://doi.org/10.1137/S0363012997321358

[33] A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Autom. Control, 57 (2012), 2106-2110. https://doi.org/10.1109/TAC.2011.2179869

[34] A. Polyakov, L. Hetel, Relay control design for robust stabilization in a finite-time, IEEE Trans. Autom. Control, 62 (2017), 1985-1991. https://doi.org/10.1109/TAC.2016.2591725

[35] G. Zuo, Y. Wang, Adaptive prescribed finite time control for strict-feedback systems, IEEE Trans. Autom. Control, 68 (2023), 5729-5736. https://doi.org/10.1109/TAC.2022.3225465

[36] B. Liu, H. J. Marquez, Razumikhin-type stability theorems for discrete delay systems, Automatica, 43 (2007), 1219-1225. https://doi.org/10.1016/j.automatica.2006.12.032

[37] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007.