generalizations of the hilbert-weierstrass theorem and tonelli-morrey theorem: the regularity of solutions of differential equations and optimal control problems

One of the basic problems in the “calculus of variations” is the minimization of the following functional:f(x) =z b a 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) = z b a f(t, x(t), x′(t), . . . , x(n−1)(t))dt where n ≥ 2. it is expected that this extension will be helpful in discussing the regularity of higher-order differential equations and optimal control problems.