Adaptive Exponential Synchronization of Complex Networks With Nondifferentiable Time-Varying Delay

Area of Science:

• Control Theory
• Complex Networks
• Nonlinear Systems

Background:

• Adaptive exponential synchronization (AES) is crucial for complex networks.
• Existing methods often assume differentiable time-varying delays, limiting applicability.
• Handling nondifferentiable delays in AES remains a significant challenge.

Purpose of the Study:

• To investigate the AES problem in complex networks with general time-varying delays, particularly nondifferentiable ones.
• To develop novel theoretical tools for analyzing systems with such delays.
• To establish new criteria for achieving robust AES.

Main Methods:

• A novel delay differential inequality was proposed, generalizing the Halanay inequality.
• The boundedness of the adaptive control gain was rigorously proven.
• New AES criteria were derived using the proposed inequality and control gain analysis.

Main Results:

• The proposed delay differential inequality effectively addresses exponential stability for delayed nonlinear systems.
• Theoretical proof confirms the boundedness of the adaptive control gain.
• Novel AES criteria were established for complex networks with general time-varying delays, including nondifferentiable cases.

Conclusions:

• The study successfully addresses the AES problem for complex networks with general time-varying delays, overcoming the differentiability assumption.
• The developed methods and criteria offer a more general and practical approach to synchronization in delayed complex networks.
• The findings are validated through a practical example, demonstrating their effectiveness.