Fractional optimal control problems on a star graph: Optimality system and numerical solution

Mehandiratta Vaibhav; Mehra Mani; Leugering Günter

doi:10.3934/mcrf.2020033

Article Contents

2021, Volume 11, Issue 1: 189-209. Doi: 10.3934/mcrf.2020033

This issue Previous Article On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems Next Article Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Fractional optimal control problems on a star graph: Optimality system and numerical solution

1.
Department of Mathematics, Indian Institute of Technology Delhi, 110016, Delhi, India
2.
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Lehrstuhl Angewandte Mathematik Ⅱ, Cauerstr. 11, 91058 Erlangen, Germany

^* Corresponding author: Mani Mehra
^* Corresponding author: Mani Mehra

Received: November 30, 2019

Revised: April 30, 2020

Early access: June 2020

Published: February 2021

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Abstract

In this paper, we study optimal control problems for nonlinear fractional order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the $ L2 $ scheme and the Grünwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are provided to demonstrate the feasibility of the numerical method.

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Mathematics Subject Classification: Primary:34A08, 49J15, 26A33;Secondary:49K15, 93C15.

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Figure 1. A sketch of the star graph with $ k $ edges along with boundary control

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Figure 2. Convergence of $ y_i(x) $, $ i=1,2,3 $ for the optimality system $ (50) $ for $ \alpha=3/2 $

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Figure 3. State variables $ y_i(x) $, $ i=1,2,3 $, for different fractional order $ \alpha $ for the optimality system $ (50) $ with $ N=64 $

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Figure 4. Convergence of $ y_i(x) $, $ i=1,2,3 $ for the optimality system $ (54) $ for $ \alpha=3/2 $

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Table 1. Control variable $ u=(u_1,u_2,u_3) $ for different values of $ N $

$ N $	$ u_1 $	$ u_2 $	$ u_3 $
32	.1867	.1792	.1749
64	.1834	.1762	.1718
128	.1817	.1746	.1702
256	.1808	.1738	.1694
512	.1804	.1734	.1690
1024	.1802	.1732	.1688

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Table 2. Control variable $ u=(u_1,u_2,u_3) $ for different fractional order $ \alpha $ with $ N=64 $

$ \alpha $	$ u_1 $	$ u_2 $	$ u_3 $
1.2	.2017	.1959	.1910
1.4	.1894	.1824	.1778
1.6	.1775	.1703	.1662
1.8	.1666	.1598	.1563
2	.1572	.1511	.1482

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