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

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Arab Firoozjaee Mohammad; Hashemi Borzabadi Akbar; Seyahchereh Khelardi Zahra

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Journal Paper

Paper Information

Journal: Journal of Advanced Mathematical Modeling Year:2024 | Volume:14 | Issue:4 Page(s): 128-139

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Information Journal Paper

Title

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

Author(s)

Arab Firoozjaee Mohammad | Hashemi Borzabadi Akbar | Seyahchereh Khelardi Zahra | Issue Writer Certificate

Keywords

Optimization problem

calculus of variations

Ritz method

parabolic partial equation

Abstract

In this article, we solve control problems with the conditions of parabolic partial differentialequations. 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.

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