An Implementation of the Shooting Algorithm for Solving Optimal Control Problems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An implementation of the shooting algorithm for solving optimal control problems F. Bonnans , S. Maurin. N ˚ 0240 May 2000 ` THEME 4. ISSN 0249-0803. apport technique.
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(123) s _ R!O a o ls Xb _`R N1s XbOaYRc [gi`[<^)\|^?m-a s Ä]Í\ u ³ ] x ' ³ []R!_`adÄ]v u% ¬ |R O ^?R_ O ^_ l Ä]Í l ) ³ q) p x $ ' ³ x $ ³ s []Xk\k²R RKc 7_ C] _a¢O e<R^ s ^s i`S'_1R
(124) _`^)R \Zs []
(125) i yR u ^¬ c l ¬ XZ c¬ L %m u¬ i`¬ [gXbo_*O Xki§_ _ O O ^)R _ s R
(126) a s R;o
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(128) \baY³ ihRXki RKc]^ afs []R
(129) YO []\k_h^ a s ³
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(131) 1_haYcQ i1SR_ O a l O O l O ËrWaËrW Ñ Í ± ¤" Z ± Í ¶ª¶ " Z YQÎ ± Í ¶ · W N1O l ^ s XZ^cYm5l _a)¬ E_a O s R&ãxi`XkS e]m\kR " mi O $anau _`Xk¬ c]1 Xk_ S'O R
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(133) O ^)s _^_h1XZc]Xk_ 2O 5 ³ u K ¹ L ^!j5 o
(134) u^- ¾ c l X al ^)R _hs Rp l a s m!_l O R1? ^¬ \k[g1R¢Xb_ a)O ³XZc]$ Xkl _h XZ^)h \ o
(135) afc l Xb_`XbaYc ³ l x ³ N1O Rcd_ O R NprEp r !uf ¹ (!u-¾ XZi1afYnXkaf[gi`\bmRK}n[]Xb^\b RKcf_p#"_`a É ³ l x ³ m &x m"" m $ uNm É ( " Ä]Í7¹ h ½ ³ L m ³ $ x R OO ^?l R|c]N1aO 2^i`mni`_hRKs S a) Q  R}n[g^_hXkafcgi^fcs l [<c Ênc]a¢cgi ¬ s Xkcgi`_hRKl^ l a)1 l a O s _ O R1O ihXZSe]\kRpi O ana_`Xkcgl XZS'cRih_ XZSa e]\k Ri O anXkia_`]XkcgXk RS'Ri`_ mnO i`a _hlRKSi a_hRKc£^egeHR^ i _haHRHR
(136) _`_hR o
(137) afc Xk_hXkafcgR _ ^c _ RaYc]R$afg_`^)XZc]R Ñ8Ò Ó Ô'Ò W O s O O ¨W Ñ Í ±¶ *. Z | ¶ Y ¶ * ¶ ± #g- aYe]R%_h 9XZSc]R!^)\p_ O o%R aYcfO _ ^s S'af\|Xb\ke_haYs c]aY]XZ^\bcRKS ¤[gcg! uo%v_`u XbaY j"c 5 u! T ^fco
(138) ^l c0_ O <$R R#`o
(139) Xk^fc]ihXk_h_'XZ^)Xkc0\bL H_ cgO ^)R-\ ¥ a ^)¨us c£S s ^)_^f_c]XZ i a)XZ ^i`R&c _ o
(140) O%(_hR Xk' afNpc ^frC|i|pR a\krR\bao
(141) ¢^5 \biu\ ]K ¹ aj 5 u_- ¾ R 9 vu ! l*m5n¦m!oQm*) x,+¢$l*m!n¦m!o q-) g". z$l*m!n¦m!o 9 Í 'A$ n i m5n g m0/ xs n i m!n g q1/ . n i m!n g R O ^?fR®_ O Rcd_ O R!_ O Ra s RS LihRR -u* ] ÓÔÕªÓ»È.
(142)
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(147) s ^)_`XbaYcYmeg^ s _ji ¬ |Raf]_j^)XZc w h f g Õ ` n¦m!oQm*)¨m /¨m x $ Õ ` ! $ l*m!n´ l m!o l m*)ª$ l h q )ª$ l h . $ l & p l i 9 v u 9 q Õ ` ' n´ã m!n´$ m0/ q )ªL h L q hL ! q Õ ` 'A$ n´ã m!n´ m / $ )ª h" m'o%\Z^ijihXZo
(148) ^\g^ s f[gS'RcY_ji_ O XZiE^fSaY[gcY_XkØi
(149) R s aX¸©x 9 T ¬ 9 ¬ ^c l 9 O a\ l N1O R s RKi`[g\b_af\b\ka¢i
(150) ÕÖ-ÎØ×Ú Ù4P%Ú.
(151) 1 ¥\)*. ¦ È È È È È È . 0¥\ 2$#. l Ro
(152) \k^ s ^_hXkafc RcgR s ^)_hXkafc Xkihc]XZSXk_hXZe]^)\b\kR$Xvi ^)O _`anXbaYac _`Xkcg S'R_ O a l % S[]\k_hXZe]\kR$i O anaf_hXZc] SR_ O a l Xkc]Xk_hXZ^)\kvX ^)_`XbaYc !#"%$ ½'&)(+ *K-½ ,/. 0*%½ 1¤%½ 2Â93¾ &4(6 5 ! 1¤½`¾. &)o l* ) ) (u & )o l*
(153) *.
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(156) * \
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(159) i`^o
(160) _hX§XkafXZc c ¬ XZ_o
(161) O^)R_`R[<i ih R s R
(162) e _ s aR nX l _ RKi1R_ <|\baR < È &)o l* ) l R% HcgRi¢^\b\§cgafccn[gS'R s Xko l ^_`^a_ O R^fege]\kXko^)_hXkafc ] cg^S'RKi1a)o%aYcgiu _¤j[<^cgcY_-o%_`Xbe<aY^ cgs i ^] S'ihm]R
(163) _`i`R_hRs iS ¬ li`_`m]^)cg_`^fRASXZ^ o
(164) si XZ¬^l ]XZ\kihR_ i s ¬ XZg^[]c _`lR l o%o
(165) afafcYi`_ _ s ¬ afe<\¢R s ^ a s s Xk^fS]^\bRKc<i o%¬ RE^fo si Xk_h|R sR
(166) Xk\kaf\c ^f¬ i^c _ lO Ro%afcgc<afihc]_ s\kXk^cgXkRcY^ _`si < HR
(167) È_| R
(168) RKc&)XZc]o Xkl*_hXZ
(169) ^) *\§i`_` ^Xk_hR$Ri-^fc ^)l\k\® 9c]cgR^)R \§l Ri`_`l ^_hcnRf[g S'R s Xko^)\ l ^)_`^ ] ^\k[]RKi a!o
(170) afcgi`_`^fcY_eg^ s ^S'R_hR s i ¬ o
(171) af^cg\k[]i`RK_ i s ^)a XZcYs _`RKipihS_`XkS[g\b^_`Xk_he]R\ki XkR sa s /§i` _`^_hR ¬ o
(172) afi`_`^)_`R^fc l o%aYcY_ s a\?^ s XZ^g\bRKi ^)_XZc]Xb_`Xk^\_hXZSR ¬ ^c l _ O R = R s R!^ s R_ O R! RKc]R s XZo¢ <\kRi ] < È ) (u ¬ e s ano
(173) RijihRKiE_ O R[gihR s H\bR È &)o1l* ) ¬ ^fc l fRc]R s ^_hRiE_ O R^ l tuaXZcf_ l mnc<{^N S'Xkoi^i|s R
(174) \k\¬ ^fi!ihaYS'R's e]XkRo
(175) RiaCs _ O R ¥ z {NpNp¦?{ > s ReHs a s _
(176) z \ki`a ¬ [gihXZc] d_ OO Rvz y s{ w ¬w o
(177) l afR Ss Xbeg^)[]_`_jXbi`^)fa)_`RXbi_aYpCc$^ N1a)HR O _ R O Rpo
(178) afXb_S^ s egRKXb[]aYc][g_jR ^)i_h^)c]Xk_`afafRc c]i¢\ka)XZi`c]R
(179) Rfl ^ R R s s ^)Xk¤?r\[g^c<¬_ho%Xkw _hXkRKafic<Xkii ^fz¢s iR }nC[]RX \bsaY\]R []^l _hi*XZYc]_ mO Ri®R
(180) _ X O s^)R\k \basVg1i`yC_XZ¨c]^¥ c z l _ ihRRKXko%cYcnaf_h[gc RS'l s Ra ^)s _`XZl ao
(181) R^)s si \ ¨< v ÈrE ¥
(182) ^ *c l cuNz
(183) Vg *V ¥ i`a \kRi|_ O R!afe]_`XkS^\ªo%afcY_ s af\ªe s aY]\kRS Ym _ O R$i O aYaf_hXZc] S'R
(184) _ O a l < È
(185) * %
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(187) *o
(188) afSeg[]_`Ri^\k[]RKia)_ O RAo%aYcf_ s af\ki^fc l Xb_ji_hXZS'R l R s Xk^)_hXkRKi^)_ XZc]Xk_hXZ^)\ ` _ k X ' S f R N1O R-i`a)_p^ s RN1i!O s Xb_`_hRKc Xkc s VgyCs ¨ ¥ zs ^ s R-O ^)\k\EXZs cgo%\Z[ l R l s Xkc0afc]R <\bR ÈrE ¥* ¦ µ @ ®R
(189) q] o
(190) Re]l _ f¥ R ¨ÈsB ih AXkaf >c&¤gaY) c¹$o l*a)
(191) ¦C *Vg(¥yC ¨ ¬ ¥ A z R! ih¨ a> ãij_¤o%aY|Xkc\k^^fn RKuiC¦]K^ y ¹ R ^fc [gl c|1R Xks_ R¢_hfRKR ih_`ihR Xkl afc aYc_ OgR
(192) s \kR
(193) RR!^fihwR e<¹R s a)^*_hvXZc] 'z VYm]i`_hRKS'ãSi ^] e]Vg\kwR ¥ ¹ z { ^¨`c V ¬ CrW ± Zã C
(194) *· ² W N1O R$S^)_`R s Xk^\Qa)_ O R!_|ac]R%q]_¢ij[ggi`Ro
(195) _hXkafcgipXZi1 RKc]R s ^)_`R l ^[]_`afS^ _`c<Xk^oS'^)\kR \bm aY´m-_ O w!Ry1ih_jV9^)]_h®R®a_`^ Rs Xk_^fO ]^)\bRf_| _ O RR i` fm]XbSfR¢Ha_|\Qaa$*R%o%qgaY^ihS_j^)e]_h\kRRiE^ a)s Xk^fs R]i`\kaR\Zi|[]^ _hsXkafRco
(196) afacgªoaf^)e]_h_`RKXkcgS^)^_`\9XbaYo
(197) cafcYa)_ s a^ \9)-e s1aYXk]_ O\kRS_ O i
(198) R Ñ8Ò Ó Ô'Ò µ Z¤ Í ± Z ¶ Q· Z¤N Z ² FK7 Z ¤] ¶ E± D ± W CrWV W Y E± D ± Ð ¶ ¯ ÓÔÕªÓ»È.
(199)
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(201) 8. ±£¶ "Z 0 E ¶ Y ¶ * ¶ ± Y · Í ± ·ÏZ ± ± Y . Z¤ ± *g" Z ¶ Y W 1|yCafcgi`_`^fcY_pe<^ s ^S'R
(202) _`R s i ] ] o%anR : o%XkRcY_u ( p ] o%anR#:o
(203) XbRKcY_1 ] l |´ Vn] _`^)^f^f_`c]SR! e]\kR\b^ Xka)_`s [Xk ^fl]R R
(204) \bRKni XZ^)] _`XbaYcQ A®c]Xk_ ] s l µ ] i`eHR
(205) R l A®c]Xk_ ] s l eHR s i`Ro ll ] _hXZS'R A®c]Xk_ ] i`Ro ¬ l s O l l l O sl s |oz®yCip] aY_ o
(206) cYafR_ cYs _ as \´^)a_`\L^ ^ s Xk^fR]eH\bRRKc i ] afc-_hXZS' R l afcgRo
(207) afcgi`X R ~i l|^fi1^_ X ^ Xk^f]\kRai`_`^_hR ll Xb©9R s RKcf_`Xk^\QR}n[g^_hXkafcgipa_ O R!afe]_`XkS^\ªo%afcY_ s af\´e s aY]\bRKS ! u ] w ¹]vu ´xµ CrWV WV W Ñ Í. . (%! ^*1y# Q201$)4 \ )576
(208) . w µ x . ¹]Í . . £i`XZcQ¾ ¼ . . £i`XkcQ¹ ¼. )5 ¤ q h . ¹] 9 XZih_ s XZg[]_`R l o%aYih_ ] ¹] T ¬XZcg^)\ªo
(209) afi`_ ] ¹]K ¹ ¬XZcg^)\ªo
(210) afcgi`_ s ^)XZcY_`i ] ¹] ¼ ¹]- ¾ 0 ª 4®¥ yCafi`_`^_hR^ s Xk^f]\bRKi ] )1´ ¬ )µ ¬ )ll yCaYih_j^)_`RRK}n[g^)_`XbaYcgi ] ¹] . )5# ¤ )4 ¤ $ ´ É qo . . w. ll3xÇu u °o qÁu Ø´ . . . . ´ $ x µ $ , x.
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(212) K. 1 ¥\)*. u. ¹] . w ) µyx $ \ w. ¹]vu) l l3x. $ 1,. ¾ ؼ . Ao%afi(¾ ¼. . )5# ¤ . . . )5 ¤ q£¹ ؼ . . ¿ )5 * 0 )8 -ª ^24#)5 s afS _ O R r afcY_ s ©mY^) f[gXkc]Rf iCe s Xkcgo
(213) Xkeg\bR!aYc]R l R l [<o%Ri ] ¹]vu7u o/x $ , { Rij[]\b_`Xkcg i`_`^_hR
(214) Äo
(215) afi`_`^)_`R!ihm]i`_hRS L ] w ¹]v uK ´xµ. £o
(216) afi,¹ ¼. )5 ¤ $ . . )4 ¤ . É . ¸¬. ¹]vu 9 . w µ x y. . £i`Xkcª¾ ¼ . . £ihXZcQ¹ ¼. )4 ¤ q h . ¹]vu
(217) T . w. ) ´²x $ ´^q. . É $ ,. ,¸)5 ¤ . w ) µyx $ \. ¹]vu"¼ w. ¹]vuc)¾ l l3x. . )5# ¤ $ )5# ¤ $ ´ . ll3xÇu w. ¹]vu.¹ . . Ao%afi(¾ ¼ )5 # ¤ )5 ¤ q£¹ ¼Ø£o
(218) afi,¹ ¼ )5 ¤ $ )4 ¤ CrWV WaËrW ± Z¤f E· ² W N 1O R l -¥Xb©9 R s !RK cf_¢o
(219) ªafcg24i`#_`^f*cY_`ip$_`N^ Ê)R_ O R!?^\k[gRi ] ¹]vu% yx u ¹]vu% N p x u ¹]Í) x K¹ N1O Ri`a\Z[]_`XbaYc-XZi1o%aYSeg[]_hR l 1Xb_ O _ O R$S[g\b_`Xke]\kR$i O ana_hXZc] ^)\k a s Xk_ O S `_ ^7ÊYXZc] xÇu< $ 1,. . ¾ ؼ . . . . . . É. . ÓÔÕªÓ»È.
(220)
(221) !"#$&%' %()*&+ ) # -,.(%/01 324#)5 & )576
(222) 8. uNu. . 0,.)y ª %N#)*\. ¢¨uc]Xk_hXZ^)\ª^)\Z[]Ra)o%aYih_j^)_`R^_1Xkc]Xk_hXZ^)\§_hXZSR¤af[gc l 1Xk_ O ij[go
(223) o
(224) RijihXkR_hRKih_ji ] ) ´ i x $ ¼u%Y)fff#Õq0 )µ i x $ f)Y 9 ¼(¹)ff# Õq0 )ll i x ffff)N Õ»q0 . ¨ucgXb_`Xk^\ª^)\Z[]Ra)i O anaf_hXZc] ¤[<cgo%_`XbaYcdu x gÍ)T&¾&u? 9N9 u"¼N¼(¾\u# q \u ¬XZcg^)\Q^\k[]R!ai O ana_`Xkc] '¤[gc<o%_hXkafc-u^x g ¼ 9 (¾ N¹N¾ 7¼ ¬XZ[gcgS^)\QHR ^s \k[]a)*R! a¦ o
(225) afi`N _`^)w®_`R! ^)i`_1_hRKXZc]egXbip_`Xk8^\ªu^_hxXZS'9R ] ) ´ i x $ (¾\uK TN¼fN¹#¼N¼(¾cÕq0 )µ i x Í )TT(¹fT 9 (¾#¼ \u.Õ $ 7¼. )ll i x -¾ &¾ 9 f)TfTYN¾T 9 7Õ $ N ¹. N O 1O RaR
(226) 1fa
(227) \Z[]© O _hXkaf cs1 Xb_ O- -s¥\RO ije<RKo%_p_ha'_hXZS'R!a)§_ O R$l i`_`^)O _`R^fc l o%afcY_ s af\9^ s XZ^O]\ksRi|^c l _ O RR
(228) ¦ af \k[gN _hXkafc a _Xk_hR Rs ^\baf_hXk afcga)ipª_^ s R!R!c]e]a\ka_`S _hR la)§aYi can_ aO _`RXkc]a \k\b¤a[g1cgXZo
(229) c]_hXk afcde]XZ^o%_jc [ s RK_ i ] RR
(230) af\k[g_hXkafca )1Xb_ Rije<RKo%_|_ha w® i CrWaËrW Y E± D ± ¶ Ð ¶ *. Z ¶ Y
(231) ¶ * ¶ ± - Í ± Z Î1Í ¶ Ð$ ¶ ¶ ² ±9±³ Í Z¤ ± W CrWaËrWV W Ñ Í ±£¶ " Z 0 E ¶ Y ¶ * ¶ ± Y · Í ± Ï· Z ± ± Y . Z¤ ± *g"Z ¶ Y W (%! ^*1y# Q201$)4 \ )576
(232) 1|yCafcgi`_`^fcY_pe<^ s ^S'R
(233) _`R s i ] Õ ] o
(234) aYR#: o%XkRcY_1_ha^)R s a l m]cg^fS'Xko^)\ l s ^) o%anR : o%XkRcY_ + ] o
(235) anR : o%XkRcY_p_ha ^R s a l mnc<^S'Xko^)\9\bXb_ o
(236) aY#R : o%XkRcY_ º ] s ^?nXk_`^_hXkafcg^\9^fo
(237) o%R\bR s ^_hXkafc #´¨u ] o%anR : o
(238) XbRKcf_pXZco%afi`_|¤[gcgo
(239) _hXkafc #´H ] ^)_jSaYi`e O R s Xko l RKcgihXk_m^_ O x ] o
(240) aYR
(241) ©A<R_`^ #u ] o
(242) anR : o%XkRcY_u . ÕÖ-ÎØ×Ú Ù4P%Ú. .
(243) ?. 1 ¥\)*. u. angle of deviation 0.3. 0.2. 0.1. 0. -0.1. -0.2 time in sec -0.3 0. 4. 8. ^c] f\bRa l RYXZ^)_`XbaYcQ 12. 16. 20. 24. 28. 32. 36. 40. speed 0.3. 0.2. 0.1. 0. -0.1. -0.2 time in sec -0.3 0. 4. 8. 12. 16. ije<RR l 20. 24. 28. 32. 36. 40. time 40 36 32 28 24 20 16 12 8 4. time in sec. 0 0. 4. 8. 12. 16. 20. _`XkS'R. 24. 28. 32. 36. 40. ÓÔÕªÓ»È.
(244)
(245) !"#$&%' %()*&+ ) # -,.(%/01 324#)5 & )576
(246) 8. control 0.9. 0.7. 0.5. 0.3. 0.1. -0.1. -0.3. -0.5. -0.7 time in sec -0.9 0. 4. 8. 12. o%afcY_ s af\ã. 16. 20. 24. 28. 32. 36. 40. norm of shooting function 1. 10 0. 10 -1. 10 -2. 10 -3. 10 -4. 10 -5. 10 -6. 10. number of iterations -7. cga s S a)i O aYaf_hXZc] '¤[gcgo
(247) _hXkafcd! s _1Xb_`R s ^)_hXkafc<i
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