# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2020033

## 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

Received  December 2019 Revised  May 2020 Published  June 2020

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.

Citation: Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020033
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##### References:
A sketch of the star graph with $k$ edges along with boundary control
Convergence of $y_i(x)$, $i=1,2,3$ for the optimality system $(50)$ for $\alpha=3/2$
State variables $y_i(x)$, $i=1,2,3$, for different fractional order $\alpha$ for the optimality system $(50)$ with $N=64$
Convergence of $y_i(x)$, $i=1,2,3$ for the optimality system $(54)$ for $\alpha=3/2$
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
 $N$ $u_1$ $u_2$ $u_3$ 32 .1867 .1792 .1749 64 .1834 .1762 .1718 128 .1817 .1746 .1702 256 .1808 .1738 .1694 512 .1804 .1734 .1690 1024 .1802 .1732 .1688
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
 $\alpha$ $u_1$ $u_2$ $u_3$ 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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