Local Synchronization for Delayed Complex Dynamical Networks via Self-Triggered Impulsive Control Involving Delays.

Area of Science:

• Complex Systems
• Control Theory
• Network Science

Background:

• Complex dynamical networks (CDNs) exhibit intricate behaviors crucial in various scientific fields.
• Synchronization in CDNs is vital for understanding collective dynamics but challenging due to inherent time delays.
• Existing self-triggered impulsive control (STIC) methods often overlook delays in system dynamics or impulse application.

Purpose of the Study:

• To develop novel design criteria for STIC strategies to achieve local synchronization in delayed CDNs.
• To address and incorporate time delays present in both the continuous and discrete dynamics of the networks.
• To ensure the avoidance of Zeno behavior in the proposed control strategy.

Main Methods:

• Lyapunov-Razumikhin methods are employed to analyze system stability and synchronization.
• The comparison principle is utilized to derive the design criteria for the STIC strategy.
• A novel self-triggered mechanism (STM) with an explicit expression is proposed for direct calculation of triggering instants.

Main Results:

• Design criteria for STIC are established, guaranteeing local synchronization for delayed CDNs with delayed impulses.
• The proposed STM provides a direct, simple, and easily implementable approach to control triggering.
• Zeno behavior is successfully avoided, a common issue in impulsive control systems.
• Time delays in both continuous and discrete system dynamics are comprehensively considered.

Conclusions:

• The developed STIC approach effectively achieves local synchronization in delayed CDNs.
• The proposed explicit self-triggered mechanism simplifies control implementation and avoids Zeno behavior.
• The findings offer a significant advancement in the control of complex dynamical networks with time delays.