Equivalent Formulations of Optimal Control Problems with Maximum Cost and Applications.

Area of Science:

• Optimal control theory
• Numerical analysis
• Mathematical optimization

Background:

• The optimal control problem with maximum cost presents challenges for direct numerical solution.
• Existing methods may lack efficiency or applicability for certain problem structures.

Purpose of the Study:

• To reformulate the optimal control problem with maximum cost into forms amenable to numerical methods.
• To introduce novel mathematical frameworks for analyzing and solving these problems.
• To develop approximation schemes for optimal values.

Main Methods:

• Reformulation as extended Mayer problems with state constraints.
• Reformulation as a differential inclusion with an upper-semi-continuous right member.
• Development of a numerical approximation scheme for the optimal value.

Main Results:

• Two equivalent reformulations as Mayer problems were derived.
• A third reformulation as a differential inclusion was proposed.
• A scheme for approximating the optimal value from below was presented.
• The effectiveness of the approaches was demonstrated through examples.

Conclusions:

• The proposed reformulations provide versatile tools for numerical optimal control.
• The differential inclusion approach offers a state-constraint-free alternative.
• The approximation scheme enhances the practical applicability of the methods.