Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 16 8 2025 08 01 Generalizations of the Hilbert-Weierstrass theorem and Tonelli-Morrey theorem: The regularity of solutions of differential equations and optimal control problems 103 119 9084 10.22075/ijnaa.2022.27413.3731 EN Saman Khoramian Faculty of Mathematics and Computer, Kharazmi University, Tehran, Iran Journal Article 2022 07 16 One of the basic problems in the “Calculus of Variations” is the minimization of the following functional:<br />$$F(x)=\int_a^b f(t,x(t),x'(t)) dt,$$<br />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<br />$$F(x)=\int_a^b f(t,x(t),x'(t),\dots,x^{(n-1)}(t)) dt$$<br />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. https://ijnaa.semnan.ac.ir/article_9084_d8256250998b3b6f070f55569ca2d0ad.pdf