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

• 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.

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

 Citation: • • Figure 1.  Numerical solutions for Example 1

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

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

Figure 4.  Optimal control for Example 6

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 $\phantom{+}0.5226194$ $\phantom{+}0.5872923$ $\phantom{+}0.5226835$ $\phantom{+}0.5226835$ $\phantom{+}0.5226206$

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

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

 $\mathfrak{a}$ $\alpha$ CW2, $\xi=4, M=8$ CW3, $\xi=2, M=7$ , $\xi=2, M=7$ , $\xi=2, M=7$ 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

Table 4.  $J^*$ for Example 3

 $\mathfrak{a}$ $\alpha_1$ $\alpha_2$ This work, LW This work, CW  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

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

 $\alpha$ This work, LW This work, CW  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

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

 This work , CW This work , LW  $\alpha$ $J_{QP}^*$ $J^*$ $J_{QP}^*$ $J^*$ $J^*$ 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

Table 7.  Comparison of results for Example 6

 Method $i_{max}$ $J^{*}$ $\mathbf{K}_{x}$ $\mathbf{K}_{x_h}$  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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