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    Area of Science:

    • Complex Networks
    • Control Theory
    • Dynamical Systems

    Background:

    • Existing research on multi-weighted networks often assumes undirected and certain outer coupling matrices (OMs).
    • Parametric uncertainty and direct coupling in dynamical networks present significant challenges.
    • The role of inner coupling matrices (IMs) in directed networks has been underexplored.

    Purpose of the Study:

    • To address passivity and control issues in directed uncertain coupled dynamical networks.
    • To introduce and analyze the impact of inner coupling matrices (IMs).
    • To develop novel criteria for achieving passivity, synchronization, and pinning control.

    Main Methods:

    • Analysis of directed networks with multiple coupling matrices and parametric uncertainty.
    • Consideration of both diagonal and non-diagonal positive definite inner coupling matrices (IMs).
    • Utilizing weighted combinations of normalized left eigenvectors (NLEVec) and Chebyshev distance (Cheb-Dist) for analysis.
    • Development of adaptive coupling strength mechanisms.

    Main Results:

    • Established conditions for passivity and pinning control based on the strong connectivity of weighted OMs for diagonal IMs.
    • Derived passivity, synchronization, and pinning control criteria when the Cheb-Dist among NLEVecs is within a tolerance interval for non-diagonal IMs.
    • Successfully addressed adaptive coupling strengths in the control framework.

    Conclusions:

    • The proposed methods offer a significant advancement over existing results for directed uncertain coupled dynamical networks.
    • The findings provide robust criteria for ensuring network passivity, synchronization, and effective pinning control.
    • The developed techniques are validated through illustrative examples.