Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

Iman Malmir

doi:10.3934/naco.2021013

Article Contents

2022, Volume 12, Issue 2: 395-426. Doi: 10.3934/naco.2021013

This issue Previous Article A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem Next Article $ V $-$ E $-invexity in $ E $-differentiable multiobjective programming

Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

Iman Malmir

Department of Mechanical Engineering, Buali Sina University, Hamedan, Iran

Received: August 2020

Revised: March 2021

Early access: April 2021

Published: June 2022

Abstract / Introduction Full Text(HTML) Figure(4) / Table(7) Related Papers Cited by

Abstract

Caputo derivative operational matrices of the arbitrary scaled Legendre and Chebyshev wavelets are introduced by deriving directly from these wavelets. The Caputo derivative operational matrices are used in quadratic optimization of systems having fractional or integer orders differential equations. Using these operational matrices, a new quadratic programming wavelet-based method without doing any integration operation for finding solutions of quadratic optimal control of traditional linear/nonlinear fractional time-delay constrained/unconstrained systems is introduced. General strategies for handling different types of the optimal control problems are proposed.

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Mathematics Subject Classification: Primary: 26A33, 49M99; Secondary: 90C20.

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Figure 1. Numerical solutions for Example 1

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Figure 2. $ {x}^*(t) $ and $ u^*(t) $ for Case 2 of Example 4

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Figure 3. $ x^*(t) $ and $ u^*(t) $ for Example 5, $ \alpha = 1 $

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Figure 4. Optimal control for Example 6

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Table 1. $ u^* $ for some $ t $ in Example 1, $ k = 2 $

$ t $	Exact	CW, Type III, Method 1, $ \xi=2 $, $ M=7 $	CW, Type III, Method 2, $ \xi=2 $, $ M=7 $	CW, Type I, $ \xi=2 $, $ M=7 $	LW, Type I, $ \xi=4 $, $ M=7 $
0	$ -0.0870988 $	$ -0.0865881 $	$ -0.0870909 $	$ -0.0870909 $	$ -0.0870982 $
0.2	$ -0.0336738 $	$ -0.0335485 $	$ -0.0336745 $	$ -0.0336745 $	$ -0.0336737 $
0.4	$ \phantom{+}0.0218134 $	$ \phantom{+}0.0219177 $	$ \phantom{+}0.0218149 $	$ \phantom{+}0.0218149 $	$ \phantom{+}0.0218134 $
0.6	$ \phantom{+}0.0774030 $	$ \phantom{+}0.0773811 $	$ \phantom{+}0.0774012 $	$ \phantom{+}0.0774012 $	$ \phantom{+}0.0774030 $
0.8	$ \phantom{+}0.1301664 $	$ \phantom{+}0.1298957 $	$ \phantom{+}0.1301676 $	$ \phantom{+}0.1301676 $	$ \phantom{+}0.1301664 $
1	$ \phantom{+}0.1758728 $	$ \phantom{+}0.1757065 $	$ \phantom{+}0.1758696 $	$ \phantom{+}0.1758696 $	$ \phantom{+}0.1758731 $
1.2	$ \phantom{+}0.2205516 $	$ \phantom{+}0.2079875 $	$ \phantom{+}0.2205631 $	$ \phantom{+}0.2205631 $	$ \phantom{+}0.2205505 $
1.4	$ \phantom{+}0.2751223 $	$ \phantom{+}0.2785798 $	$ \phantom{+}0.2751004 $	$ \phantom{+}0.2751004 $	$ \phantom{+}0.2751218 $
1.6	$ \phantom{+}0.3417751 $	$ \phantom{+}0.3464972 $	$ \phantom{+}0.3417959 $	$ \phantom{+}0.3417959 $	$ \phantom{+}0.3417753 $
1.8	$ \phantom{+}0.4231851 $	$ \phantom{+}0.4093533 $	$ \phantom{+}0.4231753 $	$ \phantom{+}0.4231753 $	$ \phantom{+}0.4231855 $
2	$ \phantom{+}0.5226194 $	$ \phantom{+}0.5872923 $	$ \phantom{+}0.5226835 $	$ \phantom{+}0.5226835 $	$ \phantom{+}0.5226206 $

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Table 2. $ J^* $ for some $ \alpha $ in (94), Example 1, $ k = 2 $

$ \alpha $	CW, Type I, $ \xi=2 $, $ M=7 $	LW, Type I, $ \xi=2 $, $ M=7 $	LW, Type I, $ \xi=4 $, $ M=8 $
2	0.1974785	0.1974785	0.1974785
1.99	0.1934399	0.1934481	0.1934459
1.98	0.1894355	0.1894379	0.1894288
1.97	0.1854667	0.1854495	0.1854282
1.96	0.1815345	0.1814848	0.1814454
1.95	0.1776403	0.1775456	0.1774818
1.94	0.1737852	0.1736341	0.1735389
1.93	0.1699705	0.1697526	0.1696184
1.92	0.1661976	0.1659034	0.1657223
1.91	0.1624677	0.1620891	0.1618525
1.9	0.1587825	0.1583125	0.1580112

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Table 3. Comparison of $ J^* $ for Example 2, $ k = 2 $

$ \mathfrak{a} $	$ \alpha $	CW², $ \xi=4, M=8 $	CW³, $ \xi=2, M=7 $	[29], $ \xi=2, M=7 $	[27], $ \xi=2, M=7 $
0	1		4.79679791916	4.79679791916	4.79679791913
0	0.999		4.79684648427	4.79697117915
0	0.99		4.79728438115	4.79853220259
0	0.95		4.79931928713	4.80544625813
0	0.9		4.80232045821	4.81377758646
1	0.9		5.26128658218	5.27371052428
1	0.95		5.24909761641	5.25573149155
1	0.99		5.23983310060	5.24118253395
1	0.999		5.23778132890	5.23791614199
1	1		5.23755370619	5.23755370619	5.23755370619
1	1.001	5.53898632105	5.49646031081	5.49646031081
1	1.01	5.53527054351	5.49452438269	5.49452438269
1	1.1	5.49587484403	5.46836520817	5.46836520817
1	1.2	5.44737670256	5.42892272878	5.42892272878
1	1.3	5.39396724356	5.38170076246	5.38170076246

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Table 4. $ J^* $ for Example 3

$ \mathfrak{a} $	$ \alpha_1 $	$ \alpha_2 $	This work, LW	This work, CW	[29]
0	1		1.56224137366	1.56224137355	1.56224137355
1	1	0.999	1.41013313297	1.41013507255	1.41013747158
1	1	0.99	1.41080207527	1.41094921928	1.41106062244
1	1	0.95	1.41374036510	1.41394156982	1.41378641071
1	1	0.91	1.41655889207	1.41695140257	1.41668866306
1	1	0.9	1.41729512016	1.41766964017	1.41740062973
1	1	0.8	1.42440244581	1.42467128854	1.42442691113

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Table 5. $ J^* $ for Case 1 of Example 4

$ \alpha $	This work, LW	This work, CW	[31]
1	2.54807	2.54807	2.54807
0.999	2.54887	2.54884	2.54887
0.99	2.55613	2.55586	2.55616
0.95	2.59051	2.58935	2.59081
0.91	2.62858	2.62681	2.62918
0.9	2.63869	2.63682	2.63937
0.8	2.75391	2.75189	2.75520

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Table 6. $ J^* $ for Example 5; for both wavelets, we set $ k = 2, \xi = 3 $, $ M = 8 $

	This work , CW		This work , LW		[32]
$ \alpha $	$ J_{QP}^* $	$ J^* $	$ J_{QP}^* $	$ J^* $	$ J^* $
1	0.592319871	0.758986537	0.592319871	0.758986537	0.833609761
0.999	0.592609998	0.759276665	0.592610281	0.759276948
0.99	0.595224540	0.761891207	0.595227150	0.761893817
0.98	0.598136364	0.764803031	0.598141037	0.764807703
0.97	0.601054736	0.767721403	0.601060817	0.767727484
0.96	0.603979032	0.770645699	0.603985765	0.770652432
0.95	0.606908605	0.773575272	0.606915143	0.773581809
0.94	0.609842787	0.776509454	0.609848197	0.776514863
0.93	0.612780884	0.779447551	0.612784163	0.779450830
0.92	0.615722181	0.782388847	0.615722265	0.782388932
0.91	0.618665935	0.785332602	0.618661716	0.785328382
0.9	0.621611381	0.788278047	0.621601717	0.788268384
0.8	0.650973598	0.817640265	0.650850308	0.817516974
0.7	0.679517928	0.846184595	0.679227593	0.845894260
0.6	0.706304327	0.872970994	0.705895247	0.872561914

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Table 7. Comparison of results for Example 6

Method	$ i_{max} $	$ J^{*} $	$ \mathbf{K}_{x} $	$ \mathbf{K}_{x_h} $
[31]	30	46.190119	$ [1.114775 \;\; 11.230139] $	$ [8.331317 \;\; 1.595857] $
This work, CW	29	46.190067	$ [1.144505 \;\; 11.220603] $	$ [8.297858 \;\; 1.589278] $
This work, LW	29	46.190014	$ [1.144826 \;\; 11.220515] $	$ [8.297552 \;\; 1.589329] $

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