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    This study develops a state observer for delayed genetic regulatory networks with reaction-diffusion terms. The proposed method ensures accurate estimation of mRNA and protein concentrations using linear matrix inequalities.

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    Area of Science:

    • Systems Biology
    • Control Theory
    • Mathematical Biology

    Background:

    • Genetic regulatory networks (GRNs) are fundamental to cellular processes.
    • Incorporating delays and spatial diffusion is crucial for realistic GRN modeling.
    • State estimation in such complex systems presents significant challenges.

    Purpose of the Study:

    • To design a state observer for delayed genetic regulatory networks (DGRNs) with reaction-diffusion terms.
    • To estimate the concentrations of mRNAs and proteins in DGRNs.
    • To ensure the stability and feasibility of the state estimation process.

    Main Methods:

    • Utilizing the Hill function for nonlinear regulation.
    • Developing a Lyapunov-Krasovskii functional with novel integral terms.
    • Applying Wirtinger-type integral inequality, convex analysis, Green's identity, and Wirtinger's inequality.
    • Establishing stability criteria via linear matrix inequalities (LMIs).

    Main Results:

    • An asymptotic stability criterion for the error system was derived.
    • The criterion is dependent on delay bounds and their derivatives.
    • Feasibility of the LMIs guarantees successful state estimation for DGRNs.

    Conclusions:

    • The proposed observer design enables accurate state estimation for DGRNs with delays and reaction-diffusion terms.
    • The method is validated through numerical examples, demonstrating its effectiveness.
    • This work contributes to the robust analysis and control of biological systems.