Generalizations of the Hilbert-Weierstrass Theorem and Tonelli-Morrey Theorem: the Regularity of Solutions of Differential Equations and Optimal Control Problems

TL;DR

This paper extends classical regularity theorems by Hilbert-Weierstrass and Tonelli-Morrey to high-order differential equations and optimal control problems, aiding researchers in establishing solution regularity.

Contribution

It provides generalized regularity theorems applicable to high-order differential equations and optimal control problems, expanding the scope of classical results.

Findings

Generalized regularity theorems for high-order differential equations

Application to optimal control problems with higher complexity

Enhanced tools for proving solution regularity

Abstract

Two Theorems attributed to Hilbert-Weierstrass and Tonelli-Morrey respectively are two classical studies for the regularity discussion around the solutions of some problems in the realm of Calculus of Variations. Now, since differential equations and optimal control problems with high-order have been growing in the literature, addressing the regularity issues for these problems should be paid more attention. In this regard, here, a generalization for the regularity theorems will be presented. It is desired that these theorems will be useful for researchers to prove the regularity properties of differential equations or optimal control problems.