# American Institute of Mathematical Sciences

June  2022, 12(2): 395-426. doi: 10.3934/naco.2021013

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

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

Received  August 2020 Revised  March 2021 Published  June 2022 Early access  April 2021

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.

Citation: Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 395-426. doi: 10.3934/naco.2021013
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Numerical solutions for Example 1
${x}^*(t)$ and $u^*(t)$ for Case 2 of Example 4
$x^*(t)$ and $u^*(t)$ for Example 5, $\alpha = 1$
Optimal control for Example 6
$u^*$ 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 $\phantom{+}0.5226194$ $\phantom{+}0.5872923$ $\phantom{+}0.5226835$ $\phantom{+}0.5226835$ $\phantom{+}0.5226206$
 $t$ 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 $\phantom{+}0.5226194$ $\phantom{+}0.5872923$ $\phantom{+}0.5226835$ $\phantom{+}0.5226835$ $\phantom{+}0.5226206$
$J^*$ 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
 $\alpha$ 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
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
 $\mathfrak{a}$ $\alpha$ 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
$J^*$ 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
 $\mathfrak{a}$ $\alpha_1$ $\alpha_2$ 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
$J^*$ 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
 $\alpha$ 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
$J^*$ 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
 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
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 $[1.144826 \;\; 11.220515]$ $[8.297552 \;\; 1.589329]$
 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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