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Feedback optimal control of distributed parameter systems by using finite-dimensional approximation schemes.

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    This study introduces a novel method for designing approximate feedback controllers for systems governed by partial differential equations. The approach simplifies complex infinite-dimensional problems into finite-dimensional optimization tasks, offering a versatile framework for various control strategies.

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    Area of Science:

    • Control Theory
    • Applied Mathematics
    • Computational Science

    Background:

    • Optimal control problems governed by partial differential equations (PDEs) are often infinite-dimensional and computationally challenging.
    • Existing methods like dynamic programming can be limited in scope or efficiency for complex systems.

    Purpose of the Study:

    • To develop a general methodology for designing approximate feedback controllers for systems described by PDEs.
    • To reduce infinite-dimensional optimal control problems to finite-dimensional mathematical programming problems.

    Main Methods:

    • Constraining the control law to a fixed structure with a finite number of parameters.
    • Reducing the optimization problem to a finite-dimensional mathematical programming problem.
    • Employing sequential quadratic programming for parameter optimization.
    • Utilizing linear combinations of fixed and parameterized basis functions for control law approximation.

    Main Results:

    • The proposed method successfully reduces complex optimal control problems to solvable finite-dimensional optimization problems.
    • Two distinct finite-dimensional approximation schemes were developed using different basis function combinations.
    • The methodology accommodates both distributed and boundary control problems within a unified framework.
    • Simulations demonstrated the approach's potential compared to traditional dynamic programming.

    Conclusions:

    • The developed methodology offers a powerful and generalizable framework for approximate optimal control of PDE systems.
    • The approach is applicable to a wide range of systems, including linear/nonlinear and elliptic/parabolic/hyperbolic equations.
    • This method provides a computationally tractable alternative to existing optimal control techniques for complex systems.