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An Implementation of the Shooting Algorithm for Solving Optimal Control Problems

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(1)An Implementation of the Shooting Algorithm for Solving Optimal Control Problems J. Frederic Bonnans, S. Maurin. To cite this version: J. Frederic Bonnans, S. Maurin. An Implementation of the Shooting Algorithm for Solving Optimal Control Problems. [Research Report] RT-0240, INRIA. 2000, pp.24. �inria-00069932�. HAL Id: inria-00069932 https://hal.inria.fr/inria-00069932 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(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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(178) afXb˜_S^ ‚s egRKXb[]aYc][g_jR ^)i_h^)c]Xk_`afafRc c]i¢\ka)XZi`c]’R

(179) R˜fl ^ R R s s ^)Xk’¤˜?r\[g^‚c<¬_ho%Xkw ˜‚_hXkRKafic<Xkii ^fz¢s iRš }n”C[]RX \bsaY\]R []^‚l _hi*XZ‘Yc]_ mO Ri®R

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(181) R^)s si \ ¨< •v ÈrE ¥

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(184) _ O a l ™ < È 

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(187) *†o

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(190) Re]l _ ˜f¥ R ¨•ÈsBš ih AXkaf >c„†‚Ž&œ¤€gaY)™ c¹$o l*a)  ’

(191) ¦C *Vg†(¥yC™ Ÿ ¨ ¬ ¥ A z R!š  ih¨ a‚> œã’“ij_•œ¤o%aY”|Xkc\k^^f‘n RK–uiC¦€]K™^ y ¹ Ÿ Ÿ R ™ ^fc [gl cˆ”|”1R Xks_ R¢_h˜fRKR ih_`ihR Xkl afc aYc„_ OgR

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(204) \bRK˜ni XZ^)] _`XbaYcQ™ A®c]Xk_ ] s l µ ] i`eHR

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(209) afi`_ ] ‚œ ¹]K™ ¹ Ÿ ¬XZcg^)\ªo

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(233) _`R s i ] Õ ] o

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(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_•mˆ^‚_ 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\bRƒa‚’ l R˜YXZ^)_`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. ӓÔÕªÓ»È.

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(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

(248) ™. 10. 1.0. 1.5. 2.0. 2.5. 3.0. variation of rho 1.25. 1.00. ×. ×. ×. 0.75. 0.50. 0.25. number of iterations 0.00 1.0. ÕÖ-ÎØ×Ú Ù4P%Ú. s`O a’¤[gc<o%_hXkafca‚’*Xk_hR s ^‚_hXkafcgi™ 1.5. 2.0. 2.5. 3.0. u9.

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(255) œ  Ÿ µ  É u[q €)u  ´

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(257) afi`_ ] ‚œ ¹]™Í#€ ¼ Ÿ ¬XZcg^)\ªo

(258) afcgi`_ s ^)XZcY_`i ] ‚œ ¹]™ÍN€ ¾ Ÿ ‚œ ¹]™Í€  Ÿ. u  9E)5(5    µ. . Hu 3µ  qÁu.  €   qÁu  €Ø´  qÁu  €. € . . .  x ´Šx. ,  qÁu. . ӓÔÕªÓ»È.

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(265) o

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(272)     -¥\ NO 1O RƒR

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(274) ˜‚¦ af \k[gN _hXkafc a‚’ _Xk_hR Rs ^‚\baf_hXk afcga)ip’ª_^ s R!R!c]e]a\ka‚_`S _hR la)’§aYi cˆan_ a‚O _`RƒXkc]’“a‚ \k\b’¤a[g”1cgXZo

(275) c]_hXk „afcde]XZ^‚o%_jc [ s RK_ i ] RR

(276) ˜‚af\k[g_hXkafc„a‚’ )”1Xb_ Rije<RKo%_|_haš w®š› i . ÕÖ-ÎØ×Ú Ù4P%Ú.

(277)  1   ¥\)*. u.¼. speed function of time 0.38. 0.36. 0.34. 0.32. 0.30. 0.28 time in sec 0.26 0. 20. 40. 60. 80. ije<RR l ™ 100. 120. 140. 160. 180. 200. flight-path angle function of time 0. -0.02. -0.04. -0.06. -0.08. -0.10. -0.12. -0.14 time in sec -0.16 0. 20. 40. 60. . Xb O _u–•eg^)_ O ^fc] ‚\kR‚™ 80. 100. 120. 140. 160. 180. 200. altitude function of time 4. 3. 2. 1. 0. time in sec -1 0. 20. 40. 60. ^‚\b_`Xb_j[ l R‚™. 80. 100. 120. 140. 160. 180. 200. ӓÔÕªÓ»È.

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