Synchronization of time-delay systems with impulsive delay via an average impulsive estimation approach.

Area of Science:

• Dynamical Systems and Control Theory
• Nonlinear Dynamics
• Control Systems Engineering

Background:

• Synchronization is crucial in various complex systems.
• Systems with mixed delays and delayed impulses present significant analytical challenges.
• Existing methods often have limitations in handling varying impulse thresholds.

Purpose of the Study:

• To investigate and establish sufficient conditions for the synchronization of dynamic systems with mixed delays and delayed impulses.
• To develop novel methods for analyzing the impact of delays and impulses on system synchronization.
• To improve upon existing synchronization criteria by addressing limitations in threshold dependency.

Main Methods:

• Impulsive control method applied to analyze system dynamics.
• Average impulsive interval approach for deriving stability conditions.
• Development of average positive impulsive estimation and average impulsive estimation concepts.
• Integration of impulsive delay information into the rate coefficient.

Main Results:

• Lyapunov sufficient conditions derived for synchronization under impulsive perturbation and control.
• Demonstrated flexibility of delays in continuous systems under impulsive perturbation.
• Showcased that synchronization is not strictly dependent on the size of impulsive delays.
• Established that impulsive delay can maintain synchronization effects even with integrated rate coefficients.

Conclusions:

• The derived conditions offer a more flexible framework for achieving synchronization in complex systems with delays and impulses.
• The proposed estimation concepts overcome limitations of fixed thresholds, enhancing synchronization analysis.
• The findings represent an improvement over previous synchronization results for such systems.
• Numerical examples validate the effectiveness and applicability of the proposed theoretical results.