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Generalizations of the Hilbert-Weierstrass theorem and Tonelli-Morrey theorem: The regularity of solutions of differential equations and optimal control problems | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 10، دوره 16، شماره 8، آبان 2025، صفحه 103-119 اصل مقاله (430.27 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.27413.3731 | ||
| نویسنده | ||
| Saman Khoramian* | ||
| Faculty of Mathematics and Computer, Kharazmi University, Tehran, Iran | ||
| تاریخ دریافت: 25 تیر 1401، تاریخ بازنگری: 25 مهر 1401، تاریخ پذیرش: 12 آبان 1401 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Boundary value problems؛ Classical solution؛ Regularity؛ Weak solution | ||
| مراجع | ||
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