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    This study introduces a novel Kullback-Leibler (KL) control for Boolean networks, extending cost functions to include control inputs. This approach approximates conventional dynamic programming, offering new insights for control system optimization.

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    Area of Science:

    • Control Theory
    • Computer Science
    • Applied Mathematics

    Background:

    • Boolean control networks (BCNs) are widely used to model complex systems.
    • Conventional Kullback-Leibler (KL) control problems in Markov decision processes do not account for control inputs in their cost functions.
    • There is a need for control strategies that incorporate control inputs into the cost function for BCNs.

    Purpose of the Study:

    • To introduce an extended stage cost function for KL control in BCNs that incorporates control inputs.
    • To develop an associated Bellman equation and a matrix-based iteration algorithm for this extended KL control problem.
    • To analyze the theoretical properties and convergence of the proposed method and compare it with conventional dynamic programming (DP).

    Main Methods:

    • Introduction of an extended stage cost function that depends on control inputs.
    • Formulation of a Bellman equation specific to the extended KL control problem.
    • Development and application of a matrix-based iteration algorithm for solving the control problem.
    • Theoretical analysis of the proposed KL control's relationship to conventional DP.
    • Convergence analysis of the weight parameter.

    Main Results:

    • The proposed KL control method approximates conventional dynamic programming (DP).
    • A matrix-based iteration algorithm is presented for practical implementation.
    • Convergence analysis demonstrates the behavior of the weight parameter under different conditions.
    • Illustrative examples show a comparison between the proposed KL control and conventional DP.

    Conclusions:

    • The extended KL control provides a viable approach for BCNs by integrating control inputs into the cost function.
    • The developed algorithm and theoretical analysis offer a foundation for applying this method.
    • The findings suggest that this approach can lead to improved control strategies for complex systems modeled by BCNs.