Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control
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(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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(215) ! " #%$'&( $) "*#+, *-/./102 "
(216) *3.#4658749#+."802 " 9*: ;*: =<>+1
(217) u y{(µqz}sfX?±bta(t'©(§¦fX t^r{ad(§ t{at' t{y%(ufXt{Zy{p8»§zt{a\7«wdy{t{a\y%sfyrut%(t{f¶ZvrutyzvHt¾½ §br vdt
(218) 4~$vHty{pHz©vMf Hty{fXeì¯1a¶fr¨tt{fz©v¹~y ¯d¯ 4t{adf%ayZ¾tfyz>dF(t{z©v²«º~$vHty{pHz©v f qt{yfe}r¦®r{fX.vt{af
(219) t{fyvt{z©µfk«ye@wd}(tzZv/¯ a¹f#vdfXfXr¨Zef y{fXz©ezvy{p4fXe*eZr¯ ºÅ 7¡ 082 (u, y) . *:. L/ * Z. (p , η ) ∈ P C 1 ([0, T ], Rn∗) × P C 0 [0, T ]
(220) ." #
(221) #%$ 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> " $"
(222) J/ G T
(223) P C q [0, T ] × q q q T P CTq ([0, T ]; Rn ) × P CTq ([0, T ], Rn∗ ) × P CTq [0, T ] *: ! i`baz©r#z}r
(224) ²vr{fX|HwdfXvf«¦t{af YÚe*z}zt Adwv¾tzZv7`badfZy{fXemdz©fX¶©qX©pMt{7»KZ½ Zv z©vHt{fXy{z©yy%rX±sv.t'»ìZ½ºv »K R(½¦v®¥Zwdvdy{p.y%rX¯ ºÅ 7¡
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(227) 9*:.#-6*39$=K>."*# 4 & ^ 6 ">
(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 ] *: ! knszvf u ∈ P C q [0, T ] ±t{adfkeddz©vdZr (A ) f vdfFz©vswt{z©µf©p n d l 0≤l≤q−1 : [0, T ] \ T 7→ R T ®qp
(230) »KZ½ 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©ª³Úsf ¥vdfX?±¥v ¯bY´t:arb®RffXv²r¨ad(§¦fX.z©vGF H R!I q−l n «y ta(ttadf«©©(§zvd*yf}(At{lz©∈v¥rºPaCT}dr±d([0, «y:T];© Rt ∈) [0, Tl ]=
(231) 0, . . . , q − 1 ( «y l = 0, . . . , q − 2 ; j = 0, . . . , q − 2 − l, (j) »K"[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(µf.y{fX©t{z©vr@zed©p t{at6tadf q × q e(ty{z y y z}r
(232) ©(§ºfXyt{yz©vdw©y#§zªta vdv dXfyMsz}ZvDfXfXefvHtr#fF|Hw D := C > (Aq−1 (t± ), . . . , A0 (t± )) t (q) ± ), y(t)) ±¤afvfa¥r*yv E q ¯^`badfXy{f«yf C ar*y%v E7t*©fXZrut q ¯^`baf'vw¥r¨z©v «Z©g(u§:rX(u(t ¯ 1 É É ¿É 82 (u, y) .'9*:. L "*'
(233) .9#
(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) . - *=.!$"#\=9*:+." ."
(239) ." #
(240) V
(241) L "4#9(u, # y)
(242) 1.*:'
(243) .9#
(244) V ντen. =. ∀ τen ∈ Ten. g(y(t)) < 0 dq η0 (t) = (−1)q q ηq (t) ≥ 0 dt. ßß Öeå¾ç¾çf. ;. − ), ντex = η1 (τex 4 :5 6 84 76
(245). »¿H½ »K R½. Q# ." "> b#%$ .=)#9# .. . Zv Z v. »KZ½. ∀ τex ∈ Tex .. [0, T ] \ (Ib ∪ Tto ) Ib. »KZ½ »KZ½.
(246) HF. ( -4."
(247) & 1*+.". ¦ti?fXvHt{ypt{z©ef . ¦ti?f szªttzef.
(248). τen. + ντ1en = η1 (τen ) 1 + ντen ≥ η1 (τen ).
(249). τex − η1 (τex )=0 − η1 (τex ) ≥ 0. ¦tiPtw%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.. . » [ZZ½ » [ HF½. » [H½ .Å 7¡s »¿z}½;Y´« z©r*yfZwd©yº~$ZvZty{pHZzv.f qt{yfe?r{©wst{z©v«»HF½Ú³%»HX½±st{adf«wv¾³ tzZvr η ± 1 ≤ j ≤ q ±b(u,Hy) rut%(tf p v¥Muwde yeft{fXyr ν 1:q r{w%ata(t (u, y) r(tz©rfFr*t{adf ªtfyv(jtzµZf:«Zy{e6wd©t{z©v¶»K½´³¾»KH½¦q v.dszt{z©v¥?Zvszt{z©vr#T »K½Ú³%» [ HX½¦v.®Rfy{fF(µZfyfX*«ye ± v¥ ZrD«Z©(§:r¯$`badfi«wdv¾tzZvr yf y%rutºz©µfv«ye!»¿H½®Hpr{wfXrr¨z©µf:z©vHt{fXy%(t{z©v¥r p η0 « η (µZfykν®¥T Zwdvdy{p'y%rX±§zt{a¶v¥rut%vHηtkj (t f sztit{z©efXr τ z©µfXv²®Hp »K R(½:«y j = 1 v7» [ HF½ «Zy j0 = 2, . . . , q ¯DºHrut%(tf p «Z©(§:r,t{afv«ye!»¿Z½¾±qvkuex wdeye*ft{fXyrDt¤fvHt{yp#tzefXr j y{fZzµZfv®qp'»¿ R½$«y j = 1q v²» [ZZ½$«y j = 2, . . . , q ¯,¼Zwderºy%eftfy%r ν rr{qz©t{fF6§ντzt{a T tw%aRz©vZt%ryf t{adf
(255) refzv'®Rta.«Zy{e6wd}(t{z©v¥r¯ »¿zzÁ½irr{wdest{z©vr (A2) v (A2 ) yf6fF|Zwzµ(©fvHtF±r{zvf*t{adfÃvr¨t{y%z©vHt®Rfz©vd'«by%sfXy q ±§ºf q a¥°µf
(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©vf z©rDzfFf§z}r{f q ®qp !/fee#sP¯ H±«wv¾tzZvr ± y{f §¦f©ª³Úsf ¥vdfX?¯ba¶f
(258) r{ad(§ tadf#fXη|Hqwdz©µ(©fvf ®Rftu§¦CffXv'fX|Hw(tzZvr#» H°½´³¾» HF½bv¥.ηjfF|H0w(≤tzZjv≤r#»KqZ−½´1³ »KZ½bwdefXvZtfX.§zªta »¿Z½´³¾» [H½¾¯ ¤|Hwdz©µ(©fvfM®Rftu§¦ffv r¨tt{f²fX|Hwt{z©vr» HF½v8»ì½± v Hrut%(t{f²vszt{z©v¥rM» H![Z½v¥ »ì [Z½±srut%(t{f ZvrutyzvHt¤fF|Hw(tzZvri» HXZ½¤v»KZ½¾±/»K R(½¾±?»ì½¤vîRwvdypy%rX±¥» HFZ½¤vM»ìH½(t tw%aRz©vHtrX±qz©r¤®qµqzZwr¯ ¤|Hwdz©µ(©fvfi®Rftu§¦ffvÃZr¨tt{f:fF|Zw¥(t{z©v¥ri» HFZ½¤v²»ìZ½±Hv®Rftu§¦ffXv vHt{yºfX|Hwt{z©vr» H°½@v\»ì½
(259) «©©(§:r
(260) «ye¸©wd}(t{z©v ±Dwr¨z©vdÂt{adf'yf}(t{z©v¥r
(261) ®Rftu§¦ffvt{adf «wv¾tzZvr η ± p v p v.t{af«¿¾tta(tbtadf
(262) rut%(tf
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