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    Area of Science:

    • Control Theory
    • Applied Mathematics
    • Optimization

    Background:

    • Nonlinear systems require advanced control strategies for optimal performance.
    • Solving infinite-horizon optimal control problems often involves complex Hamilton-Jacobi-Bellman (HJB) equations.

    Purpose of the Study:

    • To develop an iterative adaptive dynamic programming (ADP) algorithm for solving the HJB equation in continuous-time nonlinear systems.
    • To introduce a novel 'min-Hamiltonian' function for a unified framework.

    Main Methods:

    • Formulation of HJB equation and policy iteration (PI) using the min-Hamiltonian.
    • Development of an iterative ADP algorithm accounting for approximation errors.
    • Derivation of stability and convergence conditions based on iterative value gradient.
    • Inclusion of a model-free extension using off-policy reinforcement learning (RL).

    Main Results:

    • The min-Hamiltonian unifies the HJB equation and PI algorithm.
    • The proposed iterative ADP algorithm addresses approximation errors.
    • Sufficient conditions for closed-loop stability and optimal value convergence are established.
    • Numerical results validate the framework's effectiveness.

    Conclusions:

    • The Hamiltonian-driven framework with the min-Hamiltonian provides an effective approach for optimal control of nonlinear systems.
    • The iterative ADP algorithm offers robust solutions with guaranteed stability and convergence.
    • Model-free RL extension broadens the applicability of the proposed method.