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Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control

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(1)Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control J. Frederic Bonnans, Audrey Hermant. To cite this version: J. Frederic Bonnans, Audrey Hermant. Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control. [Research Report] RR-5889, INRIA. 2006. �inria-00071379�. HAL Id: inria-00071379 https://hal.inria.fr/inria-00071379 Submitted on 23 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. Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control Frédéric Bonnans — Audrey Hermant. N° 5889 April 2006. N 0249-6399. ISRN INRIA/RR--5889--FR+ENG. Thème NUM. apport de recherche.

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(194)   &  1*+.". ‘. . a fvd€(§ ŠdyfXr{fvHt¦t{adf‰‚•–t{fXy{v‰‚t{z©µ‹fi«­€‹ye@wd•}‰(tz–€ZvÃt{a¥‰(tb§z–•©•?®¥fwr{fXŒ4z–v.t{afr{ad€q€‚tz–v›6‰‚•©›‹€Zy{z–t{ae¯ ¶ A,z©yr¨tkz©vHt{y€sŒswfFŒMadfXwdyz©r¨t{z}‰‹•–•©p.z©vXF ™I´±Pzªtz}rk®¥‰‹r{fXŒ²€Zv²t{adf*wr{f6€‹«t{af6ez qfFŒ²f sŠd•©z}z–t€‹vr¨t{y%‰‚z©vHt €Zv.®R€‹wvŒd‰‚ypɂy%rX¯ /! ftt{adfQ."K=$"+ -    .+ #_/  # ." H˜ : R × Rn × Rn∗ × g (q) (u(t), y(t)) =fXŒ 0 ®qp

(195) ¥ ® # f s Œ f   d v R 7→ R »ì—FH ½ ˜ H(u, y, pq , ηq ) = H(u, y, pq ) + ηq g (q) (u, y), §afyf q Œsfv€‚t{fFrbt{af#€‹y%Œsfyb€‚«t{adf

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(204) . '. *4." A# = q ([0, T ]; Rn∗ ) ηq ∈ P CTq [0, T ] K+7U7 .*3.+ /*

(205) ν j j = 1, . . . , q .p q ν∈ P

(206) CK TLG  .  "> b#%$G*:>.9# 

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(209) V 3 T 4Z + # ) 7 4=4TL1#U9#+  6  . to. en. y˙. =. −p˙ q. =. ˜ p (u, y, pq , ηq ) = f (u, y) €‹v [0, T ] ; y(0) = y0 H ˜ y (u, y, pq , ηq ) = Hy (u, y, pq ) + ηq g (q) (u, y) €‹v [0, T ] \ T H y. pq (T ) = φy (y(T )) ˜ u (u, y, pq , ηq ) = Hu (u, y, pq ) + ηq gu(q) (u, y) 0 = H. «­€Zy j = 0, 1, . . . , q − 1 ; €‹v I «­€Zy:‰‚b•©• τ ∈ T €‹v [0, T ] \ I to. (j). g (y(τ )) = 0 g (q) (u, y) = 0 g(y(τ )) = 0 ηq (t) = 0 = −. [pq (τ )]. τ ∈ Ten. b. q X. ντj gy(j−1) (y(τ )). «­€‹y:‰‹•–•. τ ∈ Ten. €‹v. [0, T ] \ T. »ì—‹—‹½ »K—‹šZ½ »K"— [H½ »K—Z‹½ »ì—‚™Z½ »ì)— R‚½ »ì—‚‘Z½ »ì—‚’Z½ »Kš‹˜Z½. »KšHF½ «­€Zy:‰‚•©• τ ∈ T . »KšZ—‹½ [pq (τ )] = −ντ gy (y(τ )) to YÚv tadfMadfXwdy{z}r¨t{z}‰‚•:«­€‹ye@wd•}‰(tz–€Zv^€‹«F ™I´±bfX|Hw‰‚t{z©€‹vrM»ì—‹—‚½Ú³%»ì—‚‘‹½Ã‰‚yf²z–vHt{fXy{Šy{ft{fXŒ ‰ZrvdfXfXrr{‰‹y{p €ZŠst{z©e‰‚•©zªtupM€‹vŒdzªtz–€Zvrk«­€‹y t{adfŠdy€‹®•–fXe €‹«ºez©vdz–ezPdz©vd› » HX½ r{wd®sufF¾tt{€ »K—Z½k‰‹vŒ»ì—‚™‹½Ú³%»ì—‚‘H½«­€Zy‰ V %3kr{ft€‚«duwv¾tz–€Zvr¦t{z©efXr T ¯“i•ªtfyv‰(tz–µZf¤uwdeŠŠ‰‚y%‰‚eft{fy%r ν 1:q ‰‚ŠdŠRfX‰‹y{z©vd›*z–v¶»¿šZ˜Z½¦‰‚yfkr{ffXv ‰Zre6wd•ªtz–Š•–z©fy%rº‰Zr{r{€sz}‰(tfXŒ*§zªtaÃtadf q z©vHt{fyz©€‹y¤ŠR€‹z©vHt¦€‹vr¨t{y%‰‚z©vHtrz©τv»엂™‹½º‰(tº‰@y{fX›‹wd•}‰‚y¤fvHt{yp@tz–ef ¯ τen `badf6‰‹rr¨wdeŠstz–€Zv²fX|Hwdz©µ(‰‚•©fvHt:t{€¹»¿“k—Z½¦«­€‹yit{adf6‰‚•–t{fyv‰‚t{z©µ‹f«­€‹ye@w•©‰‚t{z©€‹v/±dz}rtadf

(210) «­€Z•–•©€(§z©vd›±r{ff yfe‰‹:y E4—d¯ šd»¿z–zÁ½

(211) œ  «­€Zy:‰‚•©• t ∈ [0, T ] ‰‹vŒ uˆ ∈ R. ˜ (ˆ ∃ α > 0, H u, y(t), p (t± ), η (t± )) ≥ α, j=1. [pq (τ )]. = 0. «­€‹y:‰‚•©•. τ ∈ Tex. en. q. uu. q. q.   --0 .' *,.0-  ./ ˆ:f•}‰(t{z©€‹v¥r²»K—‹—‹½´³¾»¿š‹˜Z½Œswdf't{€ F ™Ii‰‹y{f'vdfFfFr{r‰‚yp‹±º®dwstÃvd€‹t4r¨wsÀÍz–fXvZt4€‹vŒdzªtz–€Zvr*«­€‹y~$€‹vHty{pH‰‚›‹z©v f qt{yfe‰‚•}r¯7`baz©r§b‰‹r6wdvŒsfXy{•©z©vdfXŒ^z–vBFPHXš I§adfXy{f'r¨€Zef‰ZŒdŒsz–t{z©€‹v‰‹•¦vfXfFr{r‰‚yp €ZvŒsz–t{z©€‹vr*§¦fyf. Ý­Þ,ß/ÝÁà.

(212)  

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(215)   ! " #%$'&( $) "*#+, *-/./102 "

(216)  *3.#4658749#+."802 " 9*: ;*: =<>+1

(217) u’ Šy{€(µqz}ŒsfXŒ?±bta‰(t'‰‚•©•–€(§¦fXŒ t€^r{ad€(§ t{a‰‚t'‰ t{y%‰(ufXt{€Zy{p8»­§z–t{a\‰7«­€‹wdy{t{a\€‹y%Œsfyrut%‰(t{f¶€Zvruty‰‹z–vHt¾½ §b‰‹r vd€‹t

(218) ‰4~$€‹vHty{pH‰‚›‹z©vMf Hty{fXe‰‚•ì¯1a¶fr¨t‰‚t{fz©v¹~y€‹Š ¯—d¯ š4t{adf%a‰‹y‰Z¾tfyz>dF‰(t{z©€‹v²€‹«º~$€‹vHty{pH‰‚›‹z©v f qt{yfe‰‚•}r¦®‰‹r{fXŒ.€‹vt{af

(219) ‰‚•–t{fyv‰‚t{z©µ‹fk«­€‹ye@wd•}‰(tz–€Zv/¯ a¹f#vdfXfXŒr¨€Zef Šy{fX•–z©ez–v‰‹y{p4•–fXe*e‰Zr¯ ºÅ  7¡  082  (u, y) .  *:.  L/ * Z. (p , η ) ∈ P C 1 ([0, T ], Rn∗) × P C 0 [0, T ]

(220) ." #

(221)   #%$ 4 &35 6 84 &9:6 . 4 5)5 6 84 5 6 4 5 76 4 5  6  ;  q q(u, y, p ,Tη ) <3> " $"

(222) J/ G  T

(223)  P C q [0, T ] × q q q T  P CTq ([0, T ]; Rn ) × P CTq ([0, T ], Rn∗ ) × P CTq [0, T ]  *: !  i`baz©r#z}r

(224) ‰²€‹vr{fX|HwdfXvf€‚«¦t{af YÚe*Š•–z}z–t Adwv¾tz–€Zv7`badf€Zy{fXem‰‚ŠdŠ•–z©fXŒ¶•©€qX‰‚•©•–pMt{€7»K—Z‹½ €Zv z©vHt{fXy{z©€‹y‰‚y%rX±s‰‚vŒ.t€'»ì—‹Z½º‰‹vŒ »K— R(½¦€‹v®¥€ZwdvŒd‰‹y{p.‰‚y%rX¯ ºÅ  7¡

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(228) .> " $ .  *:.  L/ * (u, y) .,#  u ∈   "* . = 3L/ *

(229) (gy (y(t)), · · · , gy(q−1)(y(t))) ."*: #4.*  #=/7  P CTq [0, T ] )-4 .  L3 q ≤ n6  t ∈ [0, T ]  *: !  knsz–vf u ∈ P C q [0, T ] ±‹t{adfke‰‹ŠdŠdz©vd›Zr (A ) f vdfFŒz©vŒswt{z©µ‹f•©p n Œd l 0≤l≤q−1 : [0, T ] \ T 7→ R T ®qp

(230)  »Kš‹šZ½ A0 (t) := fu (u(t), y(t)) . . Al (t) := fy (u(t), y(t))Al−1 (t) − A˙ l−1 (t). l = 1, . . . , q − 1,. ‹‰ y{f §¦f•©•ª³ÚŒsf ¥vdfXŒ?±¥‰‚vŒ ¯bY´t:a‰‹rb®RffXv²r¨ad€(§¦fXŒ.z©vGF H R!I q−l n «­€‹y ta‰(ttadf«­€‹•©•©€(§z–vd›*yf•}‰(At{lz©€‹∈v¥rºPaC€‹T•}Œdr±d([0, «­€‹y:T‰‚];•©• Rt ∈) [0, Tl ]=

(231) 0, . . . , q − 1 ( «­€‹y l = 0, . . . , q − 2 ; j = 0, . . . , q − 2 − l, (j) »Kš"[H½ gy (y(t))Al (t± ) = 0 «­€Zy l = 0, . . . , q − 1. (q) (q−l−1) gu (u(t± ), y(t)) = gy (y(t))Al (t± ) nsft C := (g (y(t))∗ , · · · , g(q−1) (y(t))∗ ) ¯ `badf'‰‚®R€(µ‹f.y{fX•©‰‚t{z©€‹vr@z–eŠd•©p t{a‰‚t6tadf q × q e‰(ty{z y y z}r

(232) •©€(§ºfXyt{yz©‰‹vd›‹w•©‰‹y#§zªta vd€‹v dXfy€MŒsz}‰‚›‹€Zv‰‚•DfX•–fXefvHtr#fF|Hw‰‚• D := C > (Aq−1 (t± ), . . . , A0 (t± )) t€ (q) ± ), y(t)) ±¤afvfa¥‰‹r*y‰‹v E q ¯^`badfXy{f«­€‹yf C a‰‹r*y%‰‚v E7‰‚t*•©fX‰Zrut q ¯^`baf'€‹v•–w¥r¨z©€‹v «­€Z•–•©g€(u§:rX(u(t ¯ 1   É É ž  Ÿ ¿É   82  (u, y) .'9*:. L "*' 

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(234)   # $ 4 &35 6 84 &9:6 .1S."_ *4." #A=   *+ K>." 9#  4 5)5 6 84 5 86    V 'OK4L9# 

(235) η 0 ≤ j ≤ q − 1 OqL

(236)  ."  p ."4  K+7J7 .*:."+ *

(237) ν j T ."4 ν <  . en. Tex.  *. dq−j j = 0, . . . , q − 1 ; t ∈ [0, T ] \ T , ηq (t) dtq−j q X ηj (t)gy(j−1) (y(t)) t ∈ [0, T ] \ T , p(t) = pq (t) +. ηj (t) = (−1)q−j. j=1. ντ1en. + − η1 (τen ),. ;  #

(238) .  - *=.!$"#\=9*:+." ."

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(244) V   ντen. =. ∀ τen ∈ Ten. g(y(t)) < 0 dq η0 (t) = (−1)q q ηq (t) ≥ 0 dt. ßß Öeå¾ç¾çf. ;. − ), ντex = η1 (τex 4 :5 6 84  76

(245). »¿š‹™H½ »Kš R‚½. Q# ."   "> b#%$ .=)#9# .. . €Zv Z€ v. »KšZ‹½. ∀ τex ∈ Tex .. [0, T ] \ (Ib ∪ Tto ) Ib. »Kš‹‘Z½ »Kš‹’Z½.

(246) HF˜. ( -4."

(247)   &  1*+.". “¦ti‰‹•–•?fXvHt{ypt{z©ef . “¦ti‰‹•–•?f szªttz–ef.

(248). τen. + ντ1en = η1 (τen ) 1 + ντen ≥ η1 (τen ).

(249). τex  − η1 (τex )=0 − η1 (τex ) ≥ 0. “¦ti‰‹•–•Pt€‹w%a.t{z©ef.

(250). #  q #

(251) 1 `= #  q #

(252) 1 =  #  q #

(253) 1 !) #  q #

(254) 17=. + ντjen = ηj (τen ) ; j = 2, . . . , q. ;. ;. − ηj (τex ) = 0 ; j = 2, . . . , q.. . » [Z˜Z½ » [ HF½. » [H—‹½ .Å 7¡s   »¿z}½;Y´« z©r‰*yf›Zwd•©‰‹yº~$€ZvZty{pH‰‚›Zz–v.f qt{yfe‰‚•?r{€‹•©wst{z©€‹v€‚«»HF—‚½Ú³%»HX’‹½±st{adf«­wv¾³ tz–€Zvr η ± 1 ≤ j ≤ q ±b(u,€Hy) rut%‰(tf p ‰‚v¥ŒMuwdeŠ Š‰‹y‰‹eft{fXyr ν 1:q r{w%ata‰(t (u, y) r‰(tz©rfFr*t{adf ‰‹•ªtfyv‰(jtz–µZf:«­€Zy{e6wd•©‰‚t{z©€‹v¶»K—‹—‹½´³¾»K—‚‘H½¦q ‰‚vŒ.‰‹ŒdŒsz–t{z©€‹v¥‰‚•?€ZvŒsz–t{z©€‹vr#T »Kš‹‘‹½Ú³%» [ HX½¦‰‹v.®Rfy{fF€(µZfyfXŒ*«­y€‹e ± ‰‚v¥Œ ‰ZrD«­€Z•–•©€(§:r¯$`badfi«­wdv¾tz–€Zvr ‰‚yf y%rutº›‹z©µ‹fv«­y€‹e!»¿šH‹½®Hpr{wfXrr¨z©µ‹f:z©vHt{fX›‹y%‰(t{z©€‹v¥r p η0 €‹« η €(µZfykν®¥T €ZwdvŒd‰‹y{p'‰‚y%rX±§z–t{a¶€‹v¥rut%‰‚vHηtkj ‰(t f sz–tit{z©efXr τ ›‹z©µ‹fXv²®Hp »Kš R(½:«­€‹y j = 1 ‰‹vŒ7» [ HF½ «­€Zy j0 = 2, . . . , q ¯D‡º€Hrut%‰(tf p «­€Z•–•©€(§:r,t{afv«­y€‹e!»¿š‹™Z½¾±q‰‚vŒkuex wdeŠŠ‰‹y‰‹e*ft{fXyrD‰‚t¤fvHt{yp#tz–efXr j ‰‹y{f›Zz–µZfv®qp'»¿š R‚½$«­€‹y j = 1q ‰‚vŒ²» [Z˜Z½$«­€‹y j = 2, . . . , q ¯,¼ZwdeŠrºŠ‰‚y%‰‚eftfy%r ν ‰‹rr{€qz©‰‚t{fFŒ6§ντz–t{a T t€‹w%aŠR€‹z©vZt%r‰‚yf t{adf

(255) r‰‚efz–v'®R€‚ta.«­€Zy{e6wd•}‰(t{z©€‹v¥r¯ »¿z–zÁ½“irr{wdeŠst{z©€‹vr (A2) ‰‚vŒ (A2 ) ‰‚yf6fF|Zwz–µ(‰‚•©fvHtF±r{z–vf*t{adfÍ€‹vr¨t{y%‰‚z©vHt®Rfz©vd›'€‚«b€‹y%ŒsfXy q ±§ºf q a¥‰°µ‹f

(256) τto. ντto ≥ 0.. en. en. to. ˜ uu (u, y, pq , ηq ) = Huu (u, y, p) − Pq ηj (t)gy(j−1) (y)fuu (u, y) + ηq gu(q) (u, y) H Pj=1 (j) = Huu (u, y, p) − q−1 j=1 ηj (t)guu (y)(u, y) = Huu (u, y, p)..  *: ! 1  *: 37

(257) #9#  5  inqz©vf z©rDŠz–fFf§z}r{f q ®qp !/fee‰#—sP¯ H‚±‚«­wv¾tz–€Zvr ± ‰‹y{f §¦f•©•ª³ÚŒsf ¥vdfXŒ?¯ba¶f

(258) r{ad€(§ tadf#fXη|Hqwdz©µ(‰‚•©fvf ®Rftu§¦CffXv'fX|Hw‰(tz–€Zvr#» H°—‚½´³¾» HF’‹½b‰‚v¥Œ.ηjfF|H0w‰(≤tz–€Zjv≤r#»Kq—Z−—‚½´1³ »KšZ—‚½b‰‹wd›‹efXvZtfXŒ.§zªta »¿š‹‘Z½´³¾» [H—‹½¾¯ ¤|Hwdz©µ(‰‚•©fvfM®Rftu§¦ffv r¨t‰‚t{f²fX|Hw‰‚t{z©€‹vr» HF—‹½‰‚vŒ8»ì—‹—‹½± v‰‚• €Hrut%‰(t{f²€‹vŒsz–t{z©€‹v¥rM» H![Z½‰‚v¥Œ »ì— [Z½±srut%‰(t{f €Zvruty‰‹z–vHt¤fF|Hw‰(tz–€Zvri» HX™Z½¤‰‚vŒÂ»K—‚™Z½¾±/»K— R(½¾±?»ì—‚’‹½¤€‹vîR€‹wvŒd‰‚yp‰‚y%rX±¥» HF‘Z½¤‰‚vŒM»ì—‚‘H½‰(t t€‹w%aŠR€‹z©vHtrX±qz©r¤€‹®qµqz–€Zwr¯ ¤|Hwdz©µ(‰‚•©fvfi®Rftu§¦ffvÍ€Zr¨t‰‚t{f:fF|Zw¥‰(t{z©€‹v¥ri» HFšZ½¤‰‚vŒ²»ì—‚šZ½±H‰‹vŒ®Rftu§¦ffXv €‹vHt{y€‹•ºfX|Hw‰‚t{z©€‹vr» H°‚½@‰‹vŒ\»ì—‹‚½

(259) «­€‹•©•©€(§:r

(260) «­y€‹e¸‰‹•©wd•}‰(t{z©€‹v ±Dwr¨z©vd›Ât{adf'yf•}‰(t{z©€‹v¥r

(261) ®Rftu§¦ffvt{adf «­wv¾tz–€Zvr η ± p ‰‹vŒ p ‰‚vŒ.t{af«¿‰‹¾tta‰(tbtadf

(262) rut%‰(tf

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