Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality release_i4vj5gill5ci7ccl62zblwctdi

by Mariela Olguín, Domingo A. Tarzia

Published in International Journal of Differential Equations by Hindawi Limited.

2015   Volume 2015, p1-7

Abstract

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange's triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energy<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>. The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:math>(the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:math>goes to zero.
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