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Article Contents

# New aspects of time fractional optimal control problems within operators with nonsingular kernel

• * Corresponding author: Tuğba Akman Yıldız

PhD graduate from Department of Mathematics, Middle East Technical University, Ankara, Turkey

• This paper deals with a new formulation of time fractional optimal control problems governed by Caputo-Fabrizio (CF) fractional derivative. The optimality system for this problem is derived, which contains the forward and backward fractional differential equations in the sense of CF. These equations are then expressed in terms of Volterra integrals and also solved by a new numerical scheme based on approximating the Volterra integrals. The linear rate of convergence for this method is also justified theoretically. We present three illustrative examples to show the performance of this method. These examples also test the contribution of using CF derivative for dynamical constraints and we observe the efficiency of this new approach compared to the classical version of fractional operators.

Mathematics Subject Classification: Primary: 49K99, 34A08; Secondary: 34H05, 49M25.

 Citation:

• Figure 1.  Example 1: Comparative results of $u(t)$ and $u^*(t)$ for $M = 800$ and $\alpha = \{0.6, 0.7, 0.8, 0.9\}$

Figure 2.  Example 1: Comparative results of $x(t)$ and $x^*(t)$ for $M = 800$ and $\alpha = \{0.6, 0.7, 0.8, 0.9\}$

Figure 3.  Example 1: The absolute error plots for $x(t)$ (left) and $u(t)$ (right) with $M = 800$ and $\alpha = \{0.6, 0.7, 0.8, 0.9\}$

Figure 4.  Example 2: Numerical results of $x(t)$ (left) and $u(t)$ (right)

Figure 5.  Example 2: Numerical results of $x(t)$ for Caputo and CF derivatives

Figure 6.  Example 3: Numerical results of $x_1(t)$, $x_2(t)$ and $u(t)$

Figure 7.  Example 3: Numerical results of $x_1(t)$ for Caputo and CF derivatives

Figure 8.  Example 3: Numerical results of $x_2(t)$ for Caputo and CF derivatives

Table 1.  Example 1: The values of J, absolute error, order of convergence and computational time (CT) for α = {0.6, 0.7}

 $\alpha=0.6$ $\alpha=0.7$ $M$ $J$ $e_J$ $r_J$ CT $J$ $e_J$ $r_J$ CT 50 4.2795 0.0116 - 0.33 6.1951 0.0309 - 0.29 100 4.2818 0.0058 1.00 0.43 6.2030 0.0154 1.00 0.41 200 4.2838 0.0029 1.00 0.74 6.2088 0.0077 1.00 0.77 400 4.2850 0.0015 0.95 2.97 6.2122 0.0039 0.98 2.88 800 4.2857 0.00073 1.03 18.91 6.2140 0.0019 1.03 19.74

Table 2.  Example 1: The values of $J$, absolute error, order of convergence and computational time (CT) for $\alpha = \{0.8, 0.9\}$.

 $\alpha=0.8$ $\alpha=0.9$ $M$ $J$ $e_J$ $r_J$ CT $J$ $e_J$ $r_J$ CT 50 9.9900 0.0863 - 0.31 20.2108 0.2844 - 0.27 100 10.0091 0.0432 0.99 0.38 20.1666 0.1425 0.99 0.40 200 10.0247 0.0216 1.00 0.76 20.1912 0.0713 0.99 0.74 400 10.0339 0.0108 1.00 2.43 20.2152 0.0356 1.00 2.90 800 10.0390 0.0054 1.00 18.85 20.2301 0.0178 1.00 19.18

Table 3.  Example 1: The values of absolute error for $x(t)$ and the order of convergence for $\alpha = \{0.6, 0.7, 0.8, 0.9\}$

 $\alpha=0.6$ $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $M$ $e_x$ $r_x$ $e_x$ $r_x$ $e_x$ $r_x$ $e_x$ $r_x$ 50 0.0090 - 0.0085 - 0.0146 - 0.0531 - 100 0.0045 1.00 0.0043 0.98 0.0072 1.01 0.0260 1.03 200 0.0022 1.03 0.0021 1.03 0.0036 1.00 0.0129 1.01 400 0.0011 1.00 0.0011 0.93 0.0018 1.00 0.0064 1.01 800 5.57e-04 0.98 5.34e-04 1.04 8.91e-04 1.01 0.0032 1.00

Table 4.  Example 1: The values of absolute error for $u(t)$ and the order of convergence for $\alpha = \{0.6, 0.7, 0.8, 0.9\}$

 $\alpha=0.6$ $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $M$ $e_u$ $r_u$ $e_u$ $r_u$ $e_u$ $r_u$ $e_u$ $r_u$ 50 0.0240 - 0.0417 - 0.0770 - 0.1914 - 100 0.0120 1.00 0.0207 1.01 0.0381 1.01 0.0939 1.02 200 0.0060 1.00 0.0103 1.00 0.0190 1.00 0.0465 1.01 400 0.0030 1.00 0.0051 1.01 0.0095 1.00 0.0232 1.00 800 0.0015 1.00 0.0026 0.97 0.0047 1.01 0.0116 1.00

Table 5.  Example 2: The values of $J$ and computational time (CT)

 $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$ $M$ $J$ CT $J$ CT $J$ CT $J$ CT 50 0.2048 0.08 0.1912 0.08 0.1817 0.08 0.1754 0.08 100 0.2008 0.15 0.1909 0.15 0.1856 0.14 0.1842 0.14 200 0.2019 0.41 0.1945 0.40 0.1920 0.39 0.1940 0.41 400 0.2031 1.70 0.1971 1.67 0.1963 1.71 0.2002 1.61 800 0.2039 17.17 0.1986 17.55 0.1986 16.96 0.2035 17.18

Table 6.  Example 2: The comparative values of $J$ with $M = 800$

 $J$ FD $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$ Caputo 0.2301 0.2073 0.2002 0.2035 CF 0.2039 0.1986 0.1986 0.2035

Table 7.  Example 3: The values of $J$, rate of convergence and computational time (CT) for $\alpha = \{0.7, 0.8\}$

 $\alpha=0.7$ $\alpha=0.8$ $N$ $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT 50 12.1411 - - 0.63 13.8799 - - 0.64 100 7.9860 4.1551 - 1.32 8.6678 5.2121 - 1.25 200 6.6614 1.3246 1.65 4.31 6.9545 1.7133 1.61 4.33 400 6.1602 0.5012 1.40 17.18 6.3194 0.6351 1.43 16.97 800 5.9455 0.2147 1.22 146.48 6.0524 0.2670 1.25 144.69

Table 8.  Example 3: The values of $J$, rate of convergence and computational time (CT) for $\alpha = \{0.9, 1\}$

 $\alpha=0.9$ $\alpha=1$ $N$ $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT 50 15.6616 - - 0.73 18.3533 - - 0.69 100 9.9329 5.7287 - 1.39 12.3690 5.9843 - 1.73 200 7.7685 2.1644 1.40 4.30 9.7979 2.5711 1.22 3.81 400 6.9493 0.8192 1.40 18.26 8.7483 1.0496 1.29 18.53 800 6.6071 0.3422 1.26 148.13 8.3018 0.4465 1.23 167.40

Table 9.  Example 3: The comparative values of $J$ with $M = 800$

 $J$ FD $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$ Caputo 6.6976 6.6329 7.1959 8.3018 CF 5.9455 6.0524 6.6071 8.3018

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