Finite-Time Synchronization Control for Generalized Reaction–Diffusion Systems
Resumo
This paper establishes a framework for achieving finite-time synchronization (FTSY) in coupled
reaction-diffusion systems (RDs) configured in a master-slave arrangement. The systems are governed by
partial differential equations (PDEs) with nonlinear reaction terms and homogeneous Neumann boundary
conditions. Control laws are designed for the slave system to synchronize with the master system within
a predetermined finite time. Using Lyapunov stability (LS) theory, we prove finite-time stability (FTS) of
the synchronization error dynamics and derive an explicit settling time formula. Numerical simulations
validate the theoretical results using the Degen-Harrison and Lengyel-Epstein models, demonstrating
synchronization within approximately five seconds and three seconds, respectively. The approach ac
commodates spatial coupling and boundary conditions, providing a foundation for applications requiring
precise spatiotemporal coordination in chemical, biological, and physical systems.
Downloads
Não há dados estatísticos.
Referências
[1] H. Nakao, T. Yanagita, and Y. Kawamura, “Phase-reduction approach to synchronization of spatiotem
poral rhythms in reaction-diffusion systems,” Physical Review X, vol. 4, no. 2, p. 021032, Apr. 2014. doi:
10.1103/PhysRevX.4.021032
[2] W. Yang and S. Zheng, “Impulsive Synchronization for Reaction-Diffusion System,” in 2010 In
ternational Workshop on Chaos-Fractal Theories and Applications, Oct. 2010, pp. 29–33. doi:
0.1109/IWCFTA.2010.17
[3] H. M. Almimi, M. M. Abu Hammad, G. Farraj, I. Bendib, and A. Ouannas, “A New Investigation on
Dynamics of the Fractional Lengyel-Epstein Model: Finite Time Stability and Finite Time Synchroniza
tion,” Computation, vol. 12, no. 10, p. 197, Sep. 2024. doi: 10.3390/computation12100197
[4] A. Qazza, I. Bendib, R. Hatamleh, R. Saadeh, and A. Ouannas, “Dynamics of the Gierer–Meinhardt
reaction–diffusion system: Insights into finite-time stability and control strategies,” Partial Differential
Equations in Applied Mathematics, vol. 14, p. 101142, Jun. 2025. doi: 10.1016/j.padiff.2025.101142
[5] H. Nakao, T. Yanagita, and Y. Kawamura, “Phase-reduction approach to synchronization of spatiotem
poral rhythms in reaction-diffusion systems,” Physical Review X, vol. 4, no. 2, p. 021032, Apr. 2014. doi:
10.1103/PhysRevX.4.021032
[6] O. Abdullah Almatroud, I. Bendib, A. Hioual, and A. Ouannas, “On stability of a reaction diffusion
system described by difference equations,” Journal of Difference Equations and Applications, vol. 30, no.
6, pp. 706–720, Jun. 2024. doi: 10.1080/10236198.2024.2322728
[7] J. X. Chen, H. Zhang, and Y. Q. Li, “Synchronization of a spiral by a circularly polarized electric
f
ield in reaction-diffusion systems,” The Journal of Chemical Physics, vol. 130, no. 12, Mar. 2009. doi:
10.1063/1.3098543
[8] X. Y. Lou and B. T. Cui, “Asymptotic synchronization of a class of neural networks with reaction
diffusion terms and time-varying delays,” Computers & Mathematics with Applications, vol. 52, no. 6–7,
pp. 897–904, Sep. 2006. doi: 10.1016/j.camwa.2006.05.013
[9] P. He and Y. Li, “Synchronization of coupled reaction-diffusion neural networks with mixed delays,”
Complexity, vol. 21, no. S2, pp. 42–53, Nov. 2016. doi: 10.1002/CPLX.21782
[10] B. Ambrosio and M. A. Aziz-Alaoui, “Synchronization and control of a network of coupled reaction
diffusion systems of generalized FitzHugh-Nagumo type,” in ESAIM: Proceedings, vol. 39, pp. 15–24,
Mar. 2013. doi: 10.1051/PROC/201339003
[11] Z. Aminzare and E. D. Sontag, “Some remarks on spatial uniformity of solutions of reaction-diffusion
PDE’s and a related synchronization problem for ODE’s,” arXiv preprint arXiv:1312.7145, Dec. 2013.
[12] X. X. Han, H. Y. Yu, and K. N. Wu, “Finite-time synchronisation control of coupled reaction-diffusion
systems,” International Journal of Systems Science, vol. 50, no. 15, pp. 2838–2852, Nov. 2019. doi:
10.1080/00207721.2019.1690718
[13] Y. Xu, Y. Wu, and W. Li, “Finite-time lag synchronization of coupled reaction–diffusion systems
with time-varying delay via periodically intermittent control,” Applicable Analysis, vol. 98, no. 9, pp.
1660–1676, Jul. 2019. doi: 10.1080/00036811.2018.1437419
[14] S. Chen, G. Song, B. C. Zheng, and T. Li, “Finite-time synchronization of coupled reac
tion–diffusion neural systems via intermittent control,” Automatica, vol. 109, p. 108564, Nov. 2019.
doi: 10.1016/j.automatica.2019.108564
[15] R. Tang, S. Yuan, X. Yang, P. Shi, and Z. Xiang, “Finite-time synchronization of intermittently controlled
reaction–diffusion systems with delays: A weighted LKF method,” Communications in Nonlinear Science
and Numerical Simulation, vol. 127, p. 107571, Dec. 2023. doi: 10.1016/j.cnsns.2023.107571
[16] A. Aarab, N. E. Alaa, and H. Khalfi, “Generic reaction-diffusion model with application to image
restoration and enhancement,” Electronic Journal of Differential Equations, vol. 2018, no. 125, pp.
1–12, 2018.
[17] J. Wu and H. Chen, “Generalized master equation for non-extensive reaction-diffusion systems,” Modern
Physics Letters B, vol. 21, no. 02n03, pp. 103–108, Jan. 2007. doi: 10.1142/S0217984907012451
[18] A. Abbad, S. Abdelmalek, S. Bendoukha, and G. Gambino, “A generalized Degn–Harrison reac
tion–diffusion system: Asymptotic stability and non-existence results,” Nonlinear Analysis: Real World
Applications, vol. 57, p. 103191, Feb. 2021. doi: 10.1016/j.nonrwa.2020.103191
[19] R. Aldana-López, D. Gómez-Gutiérrez, E. Jiménez-Rodríguez, J. D. Sánchez-Torres, and M. Defoort,
“Enhancing the settling time estimation of a class of fixed-time stable systems,” International Journal
of Robust and Nonlinear Control, vol. 29, no. 12, Aug. 2019. doi: 10.1002/rnc.4604
[20] X. Yang, Q. Song, Y. Liu, and Z. Zhao, “Finite-time stability analysis of fractional-order neural networks
with delay,” Neurocomputing, vol. 152, pp. 19–26, 2015. doi: 10.1016/j.neucom.2014.11.023
[21] R. Rao, Z. Lin, X. Ai, and J. Wu, “Synchronization of Epidemic Systems with Neumann Boundary
Value under Delayed Impulse,” Mathematics, vol. 10, p. 2064, Dec. 2022. doi: 10.3390/math10122064
[22] R. Hatamleh, I. Bendib, A. Qazza, R. Saadeh, A. Ouannas, and M. Dalah, “Finite-Time Stability and
Synchronization of the Glycolysis Reaction-Diffusion Model,” in International Journal of Neutrosophic
Science, vol. 2025, no.–, pp. 371–386, 2025. doi: 10.54216/IJNS.250431
[23] D. E. Harrison and S. J. Pirt, “The influence of dissolved oxygen concentration on the respiration and
glucose metabolism of Klebsiella aerogenes during growth,” Microbiology, vol. 46, no. 2, pp. 193–211,
Feb. 1967. doi: 10.1099/00221287-46-2-193
[24] G. M. Dunn, R. A. Herbert, and C. M. Brown, “Influence of oxygen tension on nitrate reduction by a
Klebsiella sp. growing in chemostat culture,” Microbiology, vol. 112, no. 2, pp. 379–383, Jun. 1979. doi:
10.1099/00221287-112-2-379
[25] D. E. Harrison and J. E. Loveless, “The effect of growth conditions on respiratory activity and growth
efficiency in facultative anaerobes grown in chemostat culture,” Microbiology, vol. 68, no. 1, pp. 35–43,
Sep. 1971. doi: 10.1099/00221287-68-1-35
[26] S. Abdelmalek and S. Bendoukha, “The Lengyel–Epstein Reaction Diffusion System,” in Applied Mathe
matical Analysis: Theory, Methods, and Applications, H. Dutta and J. Peters, Eds., Studies in Systems,
Decision and Control, vol. 177. Cham: Springer, 2020. doi: 10.1007/978-3-319-99918-0-10
poral rhythms in reaction-diffusion systems,” Physical Review X, vol. 4, no. 2, p. 021032, Apr. 2014. doi:
10.1103/PhysRevX.4.021032
[2] W. Yang and S. Zheng, “Impulsive Synchronization for Reaction-Diffusion System,” in 2010 In
ternational Workshop on Chaos-Fractal Theories and Applications, Oct. 2010, pp. 29–33. doi:
0.1109/IWCFTA.2010.17
[3] H. M. Almimi, M. M. Abu Hammad, G. Farraj, I. Bendib, and A. Ouannas, “A New Investigation on
Dynamics of the Fractional Lengyel-Epstein Model: Finite Time Stability and Finite Time Synchroniza
tion,” Computation, vol. 12, no. 10, p. 197, Sep. 2024. doi: 10.3390/computation12100197
[4] A. Qazza, I. Bendib, R. Hatamleh, R. Saadeh, and A. Ouannas, “Dynamics of the Gierer–Meinhardt
reaction–diffusion system: Insights into finite-time stability and control strategies,” Partial Differential
Equations in Applied Mathematics, vol. 14, p. 101142, Jun. 2025. doi: 10.1016/j.padiff.2025.101142
[5] H. Nakao, T. Yanagita, and Y. Kawamura, “Phase-reduction approach to synchronization of spatiotem
poral rhythms in reaction-diffusion systems,” Physical Review X, vol. 4, no. 2, p. 021032, Apr. 2014. doi:
10.1103/PhysRevX.4.021032
[6] O. Abdullah Almatroud, I. Bendib, A. Hioual, and A. Ouannas, “On stability of a reaction diffusion
system described by difference equations,” Journal of Difference Equations and Applications, vol. 30, no.
6, pp. 706–720, Jun. 2024. doi: 10.1080/10236198.2024.2322728
[7] J. X. Chen, H. Zhang, and Y. Q. Li, “Synchronization of a spiral by a circularly polarized electric
f
ield in reaction-diffusion systems,” The Journal of Chemical Physics, vol. 130, no. 12, Mar. 2009. doi:
10.1063/1.3098543
[8] X. Y. Lou and B. T. Cui, “Asymptotic synchronization of a class of neural networks with reaction
diffusion terms and time-varying delays,” Computers & Mathematics with Applications, vol. 52, no. 6–7,
pp. 897–904, Sep. 2006. doi: 10.1016/j.camwa.2006.05.013
[9] P. He and Y. Li, “Synchronization of coupled reaction-diffusion neural networks with mixed delays,”
Complexity, vol. 21, no. S2, pp. 42–53, Nov. 2016. doi: 10.1002/CPLX.21782
[10] B. Ambrosio and M. A. Aziz-Alaoui, “Synchronization and control of a network of coupled reaction
diffusion systems of generalized FitzHugh-Nagumo type,” in ESAIM: Proceedings, vol. 39, pp. 15–24,
Mar. 2013. doi: 10.1051/PROC/201339003
[11] Z. Aminzare and E. D. Sontag, “Some remarks on spatial uniformity of solutions of reaction-diffusion
PDE’s and a related synchronization problem for ODE’s,” arXiv preprint arXiv:1312.7145, Dec. 2013.
[12] X. X. Han, H. Y. Yu, and K. N. Wu, “Finite-time synchronisation control of coupled reaction-diffusion
systems,” International Journal of Systems Science, vol. 50, no. 15, pp. 2838–2852, Nov. 2019. doi:
10.1080/00207721.2019.1690718
[13] Y. Xu, Y. Wu, and W. Li, “Finite-time lag synchronization of coupled reaction–diffusion systems
with time-varying delay via periodically intermittent control,” Applicable Analysis, vol. 98, no. 9, pp.
1660–1676, Jul. 2019. doi: 10.1080/00036811.2018.1437419
[14] S. Chen, G. Song, B. C. Zheng, and T. Li, “Finite-time synchronization of coupled reac
tion–diffusion neural systems via intermittent control,” Automatica, vol. 109, p. 108564, Nov. 2019.
doi: 10.1016/j.automatica.2019.108564
[15] R. Tang, S. Yuan, X. Yang, P. Shi, and Z. Xiang, “Finite-time synchronization of intermittently controlled
reaction–diffusion systems with delays: A weighted LKF method,” Communications in Nonlinear Science
and Numerical Simulation, vol. 127, p. 107571, Dec. 2023. doi: 10.1016/j.cnsns.2023.107571
[16] A. Aarab, N. E. Alaa, and H. Khalfi, “Generic reaction-diffusion model with application to image
restoration and enhancement,” Electronic Journal of Differential Equations, vol. 2018, no. 125, pp.
1–12, 2018.
[17] J. Wu and H. Chen, “Generalized master equation for non-extensive reaction-diffusion systems,” Modern
Physics Letters B, vol. 21, no. 02n03, pp. 103–108, Jan. 2007. doi: 10.1142/S0217984907012451
[18] A. Abbad, S. Abdelmalek, S. Bendoukha, and G. Gambino, “A generalized Degn–Harrison reac
tion–diffusion system: Asymptotic stability and non-existence results,” Nonlinear Analysis: Real World
Applications, vol. 57, p. 103191, Feb. 2021. doi: 10.1016/j.nonrwa.2020.103191
[19] R. Aldana-López, D. Gómez-Gutiérrez, E. Jiménez-Rodríguez, J. D. Sánchez-Torres, and M. Defoort,
“Enhancing the settling time estimation of a class of fixed-time stable systems,” International Journal
of Robust and Nonlinear Control, vol. 29, no. 12, Aug. 2019. doi: 10.1002/rnc.4604
[20] X. Yang, Q. Song, Y. Liu, and Z. Zhao, “Finite-time stability analysis of fractional-order neural networks
with delay,” Neurocomputing, vol. 152, pp. 19–26, 2015. doi: 10.1016/j.neucom.2014.11.023
[21] R. Rao, Z. Lin, X. Ai, and J. Wu, “Synchronization of Epidemic Systems with Neumann Boundary
Value under Delayed Impulse,” Mathematics, vol. 10, p. 2064, Dec. 2022. doi: 10.3390/math10122064
[22] R. Hatamleh, I. Bendib, A. Qazza, R. Saadeh, A. Ouannas, and M. Dalah, “Finite-Time Stability and
Synchronization of the Glycolysis Reaction-Diffusion Model,” in International Journal of Neutrosophic
Science, vol. 2025, no.–, pp. 371–386, 2025. doi: 10.54216/IJNS.250431
[23] D. E. Harrison and S. J. Pirt, “The influence of dissolved oxygen concentration on the respiration and
glucose metabolism of Klebsiella aerogenes during growth,” Microbiology, vol. 46, no. 2, pp. 193–211,
Feb. 1967. doi: 10.1099/00221287-46-2-193
[24] G. M. Dunn, R. A. Herbert, and C. M. Brown, “Influence of oxygen tension on nitrate reduction by a
Klebsiella sp. growing in chemostat culture,” Microbiology, vol. 112, no. 2, pp. 379–383, Jun. 1979. doi:
10.1099/00221287-112-2-379
[25] D. E. Harrison and J. E. Loveless, “The effect of growth conditions on respiratory activity and growth
efficiency in facultative anaerobes grown in chemostat culture,” Microbiology, vol. 68, no. 1, pp. 35–43,
Sep. 1971. doi: 10.1099/00221287-68-1-35
[26] S. Abdelmalek and S. Bendoukha, “The Lengyel–Epstein Reaction Diffusion System,” in Applied Mathe
matical Analysis: Theory, Methods, and Applications, H. Dutta and J. Peters, Eds., Studies in Systems,
Decision and Control, vol. 177. Cham: Springer, 2020. doi: 10.1007/978-3-319-99918-0-10
Publicado
2026-02-04
Seção
Artigos
Copyright (c) 2026 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
