Generalizations of the Hilbert-Weierstrass theorem and Tonelli-Morrey theorem: The regularity of solutions of differential equations and optimal control problems

Abstract

One of the basic problems in the “Calculus of Variations" is the minimization of the following functional:$$F(x)=\int_a^b f(t,x(t),x'(t)) dt,$$over a class of functions $x$ defined on the interval $[a,b]$. According to a regularity theorem, solutions to this fundamental problem are found in a smaller class of more regular functions. However, they were originally considered to belong to a larger class. In this context, two theorems attributed to “Hilbert-Weierstrass" and “Tonelli-Morrey" are two classical studies of the regularity of discussion for the solutions to this problem. As higher-order differential equations and higher-order optimal control problems become more prevalent in the literature, regularity issues for these problems should receive more attention. Therefore, a generalization of the above regularity theorems is presented here, namely the regularity of solutions to the following functional$$F(x)=\int_a^b f(t,x(t),x'(t),\dots,x^{(n-1)}(t)) dt$$where $n \geq 2$. It is expected that this extension will be helpful in discussing the regularity of higher-order differential equations and optimal control problems.