Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method

Kareem T. Elgindy

doi:10.3934/jimo.2017056

Article Contents

2018, Volume 14, Issue 2: 473-496. Doi: 10.3934/jimo.2017056

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Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method

Kareem T. Elgindy^,

Mathematics Department, Faculty of Science, Assiut University, Assiut, 71516, Egypt

* Corresponding author: Kareem T. Elgindy
* Corresponding author: Kareem T. Elgindy

Received: May 31, 2016

Revised: September 30, 2016

Early access: May 2017

Published: March 2018

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Abstract

In this paper, we introduce a novel fully exponentially convergent direct integral pseudospectral method for the numerical solution of optimal control problems governed by a parabolic distributed parameter system. The proposed method combines the superior advantages possessed by the family of pseudospectral methods with the well-conditioning of integral operators through the use of the integral formulation of the distributed parameter system equations, and the spectral accuracy provided by the latest technology of Gegenbauer barycentric quadratures in a fashion that allows us to take advantage of the strengths of these three methodologies. A rigorous error analysis of the method is presented, and a numerical test example is given to show the accuracy and efficiency of the proposed integral pseudospectral method.

Keywords:

Mathematics Subject Classification: Primary: 65D30, 65D32, 90-08, 49J20.

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Figure 1. The figure shows the plots of the approximate optimal cost functional $J_{N,N}^*$ (upper left), the feasibility of the optimal solution $\boldsymbol{Z}^*$ as reported by the solver (upper right), the maximum error in the initial condition (3), ${\psi}_1$, at the $101$ linearly spaced nodes $(x_i, y_i)$ in the $y$-and $t$-directions from $0$ to $4$, and $0$ to $1$, respectively in a semi-logarithmic scale (lower left), and the maximum error in the boundary condition (11), ${\psi}_2$ (lower right). All of the plots were generated using the same $101$ points $(x_i, y_i)$

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Figure 2. The figure shows the state and control profiles on $D_{4,1}^2$ using $N = 4, 12$ and $\alpha = -0.2$

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Figure 3. The figure shows the state and control profiles at the midpoint $y = 2$ using $N = 4, 12$ and $\alpha = -0.2$

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