به کارگیری روش ریتز برای حل مسائل کنترل بهینه معادلات سهموی با استفاده از کنترل کننده درونی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه آموزشی ریاضی ، دانشگاه علم و فناوری مازندران

2 گروه آموزشی ریاضی، دانشکده ریاضی، دانشگاه علم و فناوری مازندران

3 دانشگاه علم و فناوری مازندران

چکیده

در این مقاله یافتن جواب تقریبی برای مسایل کنترل بهینه تحت معادلات سهموی با در نظر گرفتن کنترل در منبع داخلی مورد بررسی قرار گرفته است. بدین منظور از روش تقریبی ریتز با در نظر گرفتن تابع برآورنده ای که در شرایط مرزی معادله حاکم به صورت دقیق صدق می کند استفاده شده است. با جایگذاری فرم های پارامتری در معادله دیفرانسیل و تابع هدف مساله به یک مساله بهینه سازی غیر خطی می رسیم که حل این روش به کمک روش های معمول بهینه سازی منجر به یافتن پارامترهای مجهول و در نهایت جواب تقریبی برای مساله خواهد شد. همگرایی این روش مورد بررسی قرار گرفته و به منظور بررسی توانمندی روش چند مثال عددی ارایه شده است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Applying the Ritz method to solve optimal control problems of parabolic equations using internal controller

نویسندگان [English]

  • Zahra Seyahchereh Khelardi 1
  • Akbar Hashemi Borzabadi 2
  • Mohammad Arab Firoozjaee 3
1 Department of Mathematics, University of science and Technology of Mazandaran
2 Department of Applied Mathematics University of Science and Technology of Mazandaran
3 Department of Mathematics, University of Science and Technology of Mazandaran, P.O. Box 48518-78195, Behshahr, Iran
چکیده [English]

In this article, we solve control problems with the conditions of parabolic partial differential
equations. In order to solve this type of mentioned problems, we use an iterative method based on basic polynomials and generating function. In this method, we first consider the solution of the problem based on the given partial differential equation, and then we approximate this solution based on the basic polynomials and the generating function (the function that applies to the initial conditions of the problem). which apply precisely in the given boundary conditions. After that, by placing this solution in the given sub-function and optimality conditions, we reach a system of nonlinear equations. By solving this device of unknown coefficients, we get the approximate answer. In fact, an approximation of the answer is made in a finitedimensional space. We have also shown the convergence of the proposed method for this type of problem.
In order to check the effectiveness of the method, we have solved some examples using it and presented the obtained numerical results.

کلیدواژه‌ها [English]

  • Optimization problem
  • calculus of variations
  • Ritz method
  • parabolic partial equation
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