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

Tuğba Akman Yıldız; Amin Jajarmi; Burak Yıldız; Dumitru Baleanu

doi:10.3934/dcdss.2020023

Article Contents

2020, Volume 13, Issue 3: 407-428. Doi: 10.3934/dcdss.2020023

This issue Previous Article Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method Next Article A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel

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

1.
Department of Logistics Management, University of Turkish Aeronautical Association, 06790 Ankara, Turkey
2.
Department of Electrical Engineering, University of Bojnord, Bojnord, Iran
3.
Hurma Mah., 252. Sokak, 2/5, Konyaaltı, Antalya, Turkey
4.
Department of Mathematics, Çankaya University, 06530, Ankara, Turkey
5.
Institute of Soft Matter Mechanics, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu 210098, China
6.
Institute of Space Sciences, Magurele-Bucharest 077125, Romania

^* Corresponding author: Tuğba Akman Yıldız
^* Corresponding author: Tuğba Akman Yıldız

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

Received: May 31, 2018

Revised: August 31, 2018

Published: December 2019

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Abstract

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.

Keywords:

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

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

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

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

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

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

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

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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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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