International Journal of Nonlinear Analysis and Applications، جلد ۱۶، شماره ۸، صفحات ۱۰۳-۱۱۹
عنوان فارسی
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عنوان انگلیسی Generalizations of the Hilbert-Weierstrass theorem and Tonelli-Morrey theorem: The regularity of solutions of differential equations and optimal control problems
چکیده انگلیسی مقاله 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.
کلیدواژه‌های انگلیسی مقاله Boundary value problems,classical solution,regularity,weak solution
نویسندگان مقاله Saman Khoramian |
Faculty of Mathematics and Computer, Kharazmi University, Tehran, Iran

نشانی اینترنتی https://ijnaa.semnan.ac.ir/article_9084_d8256250998b3b6f070f55569ca2d0ad.pdf
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