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

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.