On Hölder calmness and Hölder well-posedness for optimal control problems

Abstract

In this paper, we consider parametric optimal control problems with linear state equations and control constraints. Applying the KKM fixed point theorem and the Gronwall Lemma, conditions of the existence of solutions to such problems are gained. Under suitable tools and techniques, we formulate sufficient conditions of a Hölder property of solution maps to reference problems without assuming conditions related to differentiability properties. Also, we suggest and investigate a concept of Hölder well-posedness for these problems. Finally, as applications, employing the main results, we discuss the Hölder calmness property of the solution maps to two practical situations, namely linear quadratic regulator and fuel-optimal frictionless horizontal motion of a mass point problems.

Keywords:

• Optimal control problem
• linear quadratic regulator problem
• existence conditions
• Hölder calmness
• linear state equation

2010 Mathematics Subject Classifications:

• 49J15
• 49K40
• 90C31

Acknowledgments

The authors would like to thank the referees for their valuable remarks and suggestions that helped us significantly improve the paper. This is a result of the project under Grant number B2021-TCT-02 supported by Ministry of Education and Training of Viet Nam.

Disclosure statement

No potential conflict of interest was reported by the author(s).