This paper investigates the prescribed-time projective synchronization (PTPS) for complex networks (CNs) with different dimension. To solve this problem, a projective synchronization error is constructed and a novel performance value function integrated with the prescribed time and desired accuracy is proposed. Subsequently, a fuzzy controller is introduced to address the prescribed-time projective synchronization issue. The controller is realized through a fuzzy adaptive dynamic programming (ADP)-based framework. Additionally, the convergence analysis of the proposed methodology is provided, demonstrating that the projective synchronization error can converge to a predefined residual set within the prescribed time, which means the synchronization of CNs is solved. Finally, a numerical example is presented to verify the obtained results.
- Open Access
- Article
Prescribed-Time Projective Synchronization for Different Dimensional Complex Networks via Fuzzy Reinforcement Learning
- Xin Qu,
- Tao Dong *
Author Information
Received: 14 Sep 2025 | Revised: 21 Nov 2025 | Accepted: 12 Jan 2026 | Published: 26 Jan 2026
Abstract
Keywords
prescribed-time | complex networks | projective synchronization | fuzzy fuzzy reinforcement learning
References
- 1.
Wu, J.; Xia, Y. Complex-Network-Inspired Design of Traffic Generation Patterns in Communication Networks. IEEE Trans. Circuits Syst. II Express Briefs 2016, 64, 590–594.
- 2.
Xuan, Q.; Zhang, Y.Z.; Fu, C. Social Synchrony on Complex Networks. IEEE Trans. Cybern. 2017, 48, 1420–1431.
- 3.
Chen, Z.; Wu, J.; Xia, Y. Robustness of Interdependent Power Grids and Communication Networks: A Complex Network Perspective. IEEE Trans. Circuits Syst. II Express Briefs 2017, 65, 115–119.
- 4.
Rubinov, M.; Sporns, O. Complex Network Measures of Brain Connectivity: Uses and Interpretations. Neuroimage 2010, 52, 1059–1069.
- 5.
Guo, W.; Toader, B.; Feier, R. Global Air Transport Complex Network: Multi-Scale Analysis. SN Appl. Sci. 2019, 1, 680.
- 6.
Li, B.; Wang, Z.; Ma, L. An Event-Triggered Pinning Control Approach to Synchronization of Discrete-Time Stochastic Complex Dynamical Networks. IEEE Trans. Neural Netw. Learn. Syst. 2018, 29, 5812–5822.
- 7.
Wang, L.; Wang, Z.; Huang, T. An Event-Triggered Approach to State Estimation for a Class of Complex Networks with Mixed Time Delays and Nonlinearities. IEEE Trans. Cybern. 2015, 46, 2497–2508.
- 8.
Dong, T.; Wang, A.J.; Zhu, H.Y.; et al. Event-Triggered Synchronization for Reaction–Diffusion Complex Networks via Random Sampling. Phys. Stat. Mech. Its Appl. 2018, 495, 454–462.
- 9.
Dong, H.L.; Zhou, J.M.; Wang, B.C. Synchronization of Nonlinearly and Stochastically Coupled Markovian Switching Networks via Event-Triggered Sampling. IEEE Trans. Neural Netw. Learn. Syst. 2018, 29, 5691–5700.
- 10.
Tang, Z.; Park, J.H.; Shen, H. Finite-Time Cluster Synchronization of Lur’e Networks: A Nonsmooth Approach. IEEE Trans. Syst. Man, Cybern. Syst. 2017, 48, 1213–1224.
- 11.
Liu, X.; Chen, T. Finite-Time and Fixed-Time Cluster Synchronization with or Without Pinning Control. IEEE Trans. Cybern. 2016, 48, 240–252.
- 12.
Tan, M.; Tian, W. Finite-Time Stabilization and Synchronization of Complex Dynamical Networks with Nonidentical Nodes of Different Dimensions. Nonlinear Dyn. 2015, 79, 731–741.
- 13.
Zhang, C.; Wang, X.Y. Robust Outer Synchronization Between Two Nonlinear Complex Networks with Parametric Disturbances and Mixed Time-Varying Delays. Phys. Stat. Mech. Its Appl. 2018, 494, 251–264.
- 14.
Lin, D.; Liu, J.M.; Zhang, F. Adaptive Outer Synchronization of Delay-Coupled Nonidentical Complex Networks in the Presence of Intrinsic Time Delay and Circumstance Noise. Nonlinear Dyn. 2015, 80, 117–128.
- 15.
Fan, A.; Li, J. Adaptive Neural Network Prescribed Performance Matrix Projection Synchronization for Unknown Complex Dynamical Networks with Different Dimensions. Neurocomputing 2018, 281, 55–66.
- 16.
Jian, L.X.; Guang, Y.H. Adaptive Fault-Tolerant Synchronization Control of a Class of Complex Dynamical Networks with General Input Distribution Matrices and Actuator Faults. IEEE Trans. Neural Netw. Learn. Syst. 2015, 28, 559–569.
- 17.
Wu, Y.B.; Fu, S.X.; Li, W.X. Exponential Synchronization for Coupled Complex Networks with Time-Varying Delays and Stochastic Perturbations via Impulsive Control. J. Frankl. Inst. 2019, 356, 492–513.
- 18.
Wen, Z. Synchronization of Stochastic Dynamical Networks Under Impulsive Control with Time Delays. IEEE Trans. Neural Netw. Learn. Syst. 2013, 25, 1758–1768.
- 19.
Wang, X.; Liu, X.; She, K. Delay-Dependent Impulsive Distributed Synchronization of Stochastic Complex Dynamical Networks with Time-Varying Delays. IEEE Trans. Syst. Man, Cybern. Syst. 2018, 49, 1496–1504.
- 20.
Li, X.; Wang, N.; Lu, J. Pinning Outer Synchronization of Partially Coupled Dynamical Networks with Complex Inner Coupling Matrices. Phys. Stat. Mech. Its Appl. 2019, 515, 497–509.
- 21.
Huang, H.; Xu, J.; Wang, J.; et al. Reinforcement Learning-Based Secure Synchronization for Two-Time-Scale Complex Dynamical Networks with Malicious Attacks. Appl. Math. Comput. 2024, 479, 128840.
- 22.
Zhang, H.; Yan, D.; Zhang, Y.; et al. Distributed Synchronization Based on Model-Free Reinforcement Learning in Wireless ad Hoc Networks. Comput. Netw. 2023, 227, 109670.
- 23.
Yu, X.; Khani, A.; Chen, J.; et al. Real-Time Holding Control for Transfer Synchronization via Robust Multiagent Reinforcement Learning. IEEE Trans. Intell. Transp. Syst. 2022, 23, 23993–24007.
- 24.
Li, M.; Qin, J.; Ma, Q.; et al. Hierarchical Optimal Synchronization for Linear Systems via Reinforcement Learning: A Stackelberg-Nash Game Perspective. IEEE Trans. Neural Netw. Learn. Syst. 2021, 32, 1600–1611.
- 25.
Zhou, Z.; Liu, Y.; Lu, J.; et al. Cluster Synchronization of Boolean Networks Under State-Flipped Control with Reinforcement Learning. IEEE Trans. Circuits Syst. II Express Briefs 2022, 69, 5044–5048.
- 26.
Qin, J.; Li, M.; Shi, Y.; et al. Optimal Synchronization Control of Multiagent Systems with Input Saturation via Off-Policy Reinforcement Learning. IEEE Trans. Neural Netw. Learn. Syst. 2019, 30, 85–96.
- 27.
Li, K.; Yang, L.; Guan, C.; et al. Reinforcement Learning-Based Pinning Control for Synchronization Suppression in Complex Networks. Heliyon 2024, 10, e34065.
- 28.
Zhong, M.; Huang, C.; Cao, J.; et al. Adaptive Fuzzy Echo State Network Optimal Synchronization Control of Hybrid-Order Chaotic Systems via Reinforcement Learning. Chaos Solitons Fractals 2024, 181, 114665.
- 29.
Zhou, Z.; Liu, Y.; Cao, J.; et al. Cluster Synchronization of Boolean Control Networks with Reinforcement Learning. IEEE Trans. Circuits Syst. II Express Briefs 2023, 70, 4434–4438.
- 30.
Guleva, V. Node Correlation Effects on Learning Dynamics in Networked Multiagent Reinforcement Learning. In Proceedings of the 6th Scientific School Dynamics of Complex Networks and their Applications, Kaliningrad, Russia, 14–16 September 2022; pp. 110–113.
- 31.
Wang, W.; Feng, C.; Quan, W. Data-Driven Control of a Class of Discrete-Time Linear Complex Dynamical Networks. In Proceedings of the IEEE International Conference on Big Data, Los Angeles, CA, USA, 9–12 December 2019; pp. 6243–6245.
- 32.
Destro, A.; Giorgi, G. Reinforcement Learning Applied to Network Synchronization Systems. In Proceedings of the IEEE International Symposium on Measurements & Networking, Padua, Italy, 18–20 July 2022; pp. 1–6.
- 33.
Dong, T.; Gong, X.; Li, H.; et al. Data-Driven Tracking Control for Multi-Agent Systems with Unknown Dynamics via Multithreading Iterative Q-Learning. IEEE Trans. Syst. Man, Cybern. Syst. 2023, 53, 2533–2542.
- 34.
Dong, T.; Li, K.; Huang, T. Event-Based ADP Tracking Control of Complex-Valued Nonlinear Systems. IEEE Trans. Emerg. Top. Comput. Intell. 2024, 8, 1086–1096.
- 35.
Shen, Z.; Dong, T.; Huang, T. Asynchronous Iterative Q-Learning Based Tracking Control for Nonlinear Discrete-Time Multi-Agent Systems. Neural Netw. 2024, 180, 106667.
- 36.
Wang, L.X.; Mendel, J.M. Fuzzy Basis Functions, Universal Approximation, and Orthogonal Least Squares Learning. IEEE Trans. Neural Netw. 1992, 3, 807–814.
- 37.
Hu,W.; Gao, L.; Dong, T. Event-Based Projective Synchronization for Different Dimensional Complex Dynamical Networks with Unknown Dynamics by Using Data-Driven Scheme. Neural Process. Lett. 2021, 53, 3031–3048.
- 38.
Sheng, L.; Yang, H. Exponential Synchronization of a Class of Neural Networks with Mixed Time-Varying Delays and Impulsive Effects. Neurocomputing 2008, 71, 3666–3674.
- 39.
Li, J.; Pei, X.; Ji, L. Data-Based Optimal Couple-Group Consensus Control for Heterogeneous Multi-Agent Systems via Policy Gradient Reinforcement Learning. J. Mach. Learn. Inf. Secur. 2026, 2, 1.
- 40.
Tang, H.; Wang, Y.; Li, B.; Deng, J.; Zhang, Z. Bayesian Local Differential Privacy for Implicit Feedback Recommendation. J. Mach. Learn. Inf. Secur. 2025, 1, 6.
- 41.
Shao, Y.; Li, Y.; Wu, B. A Directional Attention Fusion and Multi-Head Spatial-Channel Attention Network for Facial Expression Recognition. J. Mach. Learn. Inf. Secur. 2025, 1, 7.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Suite 4002 Level 4, 447 Collins Street, Melbourne, Victoria 3000, Australia
General Inquiries: info@sciltp.com