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Higher-order numerical scheme for linear quadratic problems with bang-bang controls.

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This study enhances optimal control methods for problems with hypercube constraints. A new discretization scheme significantly improves convergence rates for bang-bang controls.

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Area of Science:

  • Control Theory
  • Applied Mathematics

Background:

  • Linear-quadratic optimal control problems are fundamental in engineering and economics.
  • Hypercube constraints on control functions present unique challenges.
  • Bang-bang control strategies are common but require careful analysis.

Purpose of the Study:

  • To develop an efficient numerical method for a class of linear-quadratic optimal control problems.
  • To address problems where control functions are constrained within a hypercube.
  • To improve the convergence rate of existing discretization schemes.

Main Methods:

  • Consideration of a linear-quadratic optimal control problem with hypercube control constraints.
  • Assumption of purely bang-bang optimal controls and specific properties of the switching function.
  • Introduction of a novel discretization scheme.
  • Utilizing recent findings on the stability of optimal solutions.

Main Results:

  • A discretization scheme is proposed that doubles the convergence rate compared to Euler's scheme.
  • The accuracy of the method is proven using stability analysis of optimal solutions.
  • The scheme is effective for optimal control problems with bang-bang solutions.

Conclusions:

  • The developed discretization scheme offers a significant improvement for solving optimal control problems with hypercube constraints.
  • The findings contribute to more efficient and accurate numerical solutions in control theory.
  • The stability analysis provides a robust theoretical foundation for the proposed method.