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

  • * Corresponding author: Mani Mehra

    * Corresponding author: Mani Mehra 
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  • 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.

    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

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

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

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

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
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