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

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  • 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:

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  • 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 $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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] $
     | Show Table
    DownLoad: CSV
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