Asymptotic expansion of the optimal control under logarithmic penalty: worked example and open problems
17
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Asymptotic expansion of the optimal control under logarithmic penalty: worked example and open problems Felipe Alvarez — J. Frédéric Bonnans — Julien Laurent-Varin. N° 6170 April 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6170--FR+ENG. Thème NUM.
(3)
(4)
(5)
(6)
(7)
(8)
(9) !#"$% & (') "+*,,- .' " /# '01
(10) ,!234 "1 576 8 '
(11) 94-:#,";# -< =4>,'0 ?,"@! ACBEDGFIHJBK,DMLON5PQBER SJT0U ACPQVEW0VEPQFGXZY\[^])]CN5])_ SJTO` DIFGBE]ba@N ` PcBE]edcfhg#NOPQFG] i.jlkmonprqtJs2}uw n|hzrvy%x{z}mo|hkmomoncz4~l~yz lmoc Elncz @lO|4ln nQjlnjln~O)7u@lh0ytuQ \¡ncz ∗. †. ‡. ¢¤£0¥c¦§©¨lª¦« ¬ nt{ zzhz|hjlnrlh®ncm¯°n±{~z²³~°{|hm5~E|hI´Ez²|h|nt~z²|h©|hn@µ4jlnc~ ³¡Eh¶|hjlmo³\On~¶|}x=³z#ll³³nc3|7~z²|³~E|hzc· ¬ nz²jl©µ¸|j5©|c´0³~+,z³m\n¤czn´(|j||jln ¹hl n© |h~Z° ´{ºMµ4jlhncn¡n l 1 z.»C|}jln~515|nc³~~$|}Ox³~E|ch´0m\~n3|ncc°%· |jln¤{|hm~E|hI´(z²|h|n~5=z²|h©|hno³z ° ¼¾½E¿^ÀhÁ#ε Âlog§©Ã0ε¥«Ä l|³moε (~E|hI´E³¡Eh¶|hjlmo³tOn~¶|}x´~|he~5z}|hh~E|hzc´{z²nc~z¶|h ¹ |}x·. Ù0Ð ß¾Å0Ý^à1Æ}Î1áÇ©Ì4âcÈã1Ì@ÉËÊIØÌ4ÈÙ(ÍÆ}ÍÜQÍÒ ÛrÊeÚÆ}Îñ{ÉGÏEÛGƲÒÔÐÈÜ©Ì ÈÈhÜäÊIØ(ÑÜòËÆ}ç0ÆJÌ@ßÝ^ÈhòôÊGÑó
(12) ÒÔÒÔÓ²ÕÔå ÈÆæõ©ÕåMïQÝeÖÊIç
(13) ïQÍÉG×1ß0Û²ÒÔÍØCèÆ}é©ÈÆ}ÝeÍÉGÒÔÜZÈÍQÛI×ÒÔÝ^ÕÔØ©ÕêÆ}È4Ù
(14) ÍãÍQÊGâÉMÒ ÚÆ&áhƲåËë4ÉMÜÛIÆ#ÒÔèQÜÈÐ ÈÍÜ4ÊIÈhÒêÊGÜÈÑÆC×1ö}Î@Ì@Ý^ëQÑÈhØÒÔÊIÕÔÒÔÝeÆ(÷ÑQïÞZÒÔÆ}ÕÔÛäÆÝ^ì{Ʋó
(15) ϳÍÈÇÊGÕ ÉGÇÚ1Æ0ÕÔÈÒÔÉMÏ÷Î1Æ}ïíÉ^ö}î0ÐÆ}ÛÜØ0ÈhÒÔÌ ÊGÖ5Ñé Æ}Ó}ïÌ@Î1ÓðÕÔÑÆ#ÈhÒÔÊGÕÔÒÔølÆ1Óé Î1ÈÓ}ÕÕ Õùé Ð ÊGƲÎÓIÑÜÍÆ}ÕÔÒÔÕÔ÷ÒÔÍï×Æ1ØØ ú ã1ã²à1û4øÈÕêÈÒÔÛGƲÈïEØõ©ÉGÈÍÓ}Æ1ì^õ©ÉMƲÜQƲÉMÒÔÓ1é ü^Î1ÍÍ©ÈÍÛðî0ÒÔÍÉMÒêÈQé ϳÉ}é õ©ÉIÈÍÓ}ý{Æ1Èì ïMïÍQÓðÕÔÑÒÔÆ}Æ}ÍEÉ(é ÕêÅ0ÈïÒÔÉGÉMÆ}Æ}ÓæÍÊGÊMÎ1å ÉIÚÈhÈÊGÉGÆ1ÒÔÍØî0Ý^ƲÓ}ÍÍÊGÆ}ÉGÛ²ÆCé ϳÉ}ç
(16) é ÈhÊIÒÔÎ1ÍÈÕÜEþ ÖlÊIïQÜÆ}Û(ècÇ©ÈhÊGÒêÈÕÔƲ۲ØßÎ1ÍÜQåôølÎ1ÒÔÍÊÜÆ(Õ þ Ö5ÛGÇ©ÈÓ}ÆØ ú ã²á1à1ëÖlÚQÉËùrÝ^Æ}ÜÆæÿØ ∗. † ‡. Unité de recherche INRIA Futurs Parc Club Orsay Université, ZAC des Vignes, 4, rue Jacques Monod, 91893 ORSAY Cedex (France) Téléphone : +33 1 72 92 59 00 — Télécopie : +33 1 60 19 66 08.
(17) 4" 3@!4# -, " % & ! -,; ¤ 0!" 4 ,"
(18)
(19) # " /# '01
(20) ,!\*, 6 @"4 ,'0 ?,"@!4 #* .'
(21)
(22) ln3 ¥
(23) |h |7 n« |7pr|h©|7zo{ }z³{~E||Z~z&zhz²{n,³lchzc´t®lE³kmo~n { ~ { ¹ lnc lE5lncn mo³n~E|oOl~n Zz|³mo~ m¡~{n,|jllmo|³ moEln3n´{n± | ~E|³~E|nQz·+prz\mo~E|h~5z&l~z\l~ zozmol³nn ¹ ©|h~ l~ 5~E|{n\}³~1|h~ lncz#~E|³~E|nQz´0momon¤{n¤³¾mom~{n¤{|hm³nn|&{n |h|#n|\|h|&©}³~|#zz{³cz nQz}| h{hn ε log ε ´{ ε nQz}|4³n mok|hn {n O~ zh©|³~,³¡|jm\ Eln· ¦Q¥ ª ¥ « m  ~{nÀ ´lz n~5z²³®l³|%· momo~{nl|³mon´%O~5³³zh©|h~ ³¡|jm\ Eln´%~E|hh~E|nQzoz²l m.
(24) .
(25) !" $#%'&()! * "+,! -.
(26) /) 01. & (') -,* J
(27) i.j³z zjl| ~l|hn& z ~E|hn~{nQ7|h¾{³zh5zz zmon\ElnQz}|h~zänc³|nQ¾|,|hjlno5nc²|hl®5©|³~3~x{z³z °{|hmO~E|h5l®l³nmzJµ4jn~|hjlnr³¡Eh¶|hjlmo³r5nc~|}xo³z%znc°ËJ³~lnQ5³¶|}x~z²|³~|z· 4æ~h{nc|. 5l± {nQzc´{n|@z4~z² {nc@~,{|hm~E|h^lh®ncm °0|jln °Ëm · n· t ∈ [0, T ]; Z ˙ = f (y(t), u(t)), y(t) l · n· t ∈ [0, T ]; º}Q» Min `(y(t), u(t))dt + Φ(y(T )); 0 ≤ g(y(t), u(t)), y(0) = y , µ4jnhn f : R ×R → R ´ g : R ×R → R ´ ` : R ×R → R ´~5 Φ : R → R hn.zhz²lmonQ |h|ho¡®5nn|jznmo.Eµ4¶||hj,j,°Ë|ljl~n O|³n~~zc·7 6i.nc4jln ³~l¡~l³~l¡|jm\Ez} | ä5nc~|}x¤~z³z²|hz.³~~E|h{{~¡o\hmon|nc ε > 0 P ° g(y, u) > 0, ºG» `(y, u) − ε log[g (y, u)] |hjlnhµ4 z²n . ` (y, u) := +∞, ®Oprnj|h³~ ~l¡¤|hjl|n jmo| n`|j(y,y,u) z.|↓`(y, z ¹ u)nt|jlz n εOn↓~50³°Ë* 6cncä,n ¹nc{|hx m³ (y,~E|hu)Ozj®l|ncjm | g(y, u) > 0 ´5|jln&³lnc Z l· n· t ∈ [0, T ]; ºM » y(t) ˙ = f (y(t), u(t)), Min ` (y(t), u(t))dt + Φ(y(T )); y(0) = y , °Ë=zm³ ε > 0 ´%~$| 8n |j³zOn|®Onc z²l|³~ zo~ llh±{m©|h~+°t|jln{|³m zt z 9~ :;!
(28) !<7 /=; >&&~=|jn\zn~zn&|j| |hjln~E|°Ëhnc
(29) z²°Ë³®lt³n¾|j5n\~E|³z¡³~°7(ºM El»hzh®l©³|n m z²°Ëx$º²Q|»j·#| i.jllog[g ³@z ?(n®Oncz¡ln³~E|n¡h®l³n~ jln~5n (y(t), u(t))] l · n . · ³ ~ Ë ° r · B A%x~E|³~Ezx n|²|h~l¡ |hn~,|h¤l´5|jl z g (y(t), u(t)) > 0 °0{lnQl{hlh±yn4³m³zJ©n|h±{näOznc1³|h{nc|\³|h~5z[0,¡|ncj~lTn|r]©|hzn.xEmoC~ {|h Di'|h=E³ c=1,³.x F..,r~ y´nc|j¡n@|%| ©zµ.´E ¤~ monn±l|n1h|@7 6znc#³{ε°M|mo³~³x ° ε 7→º²Q»u· (·) vy³mo %~E|hn¬h ð5~E|Cmon|hjl{lzhn4µ4 {n³x&5z²nQ&¹ µ4|j¤¡hnc©|znczhzC³~³~lncJ~5&~~l³~nc 5©|j5¡zCjmo mon4~l®O¡5n·nc~~E¶|h|hjln³~~\zG5n³~lx|³n~ ð{nc³z²mo|³n¡~z|³nc~·0i.°Ëjlncmo%ncµ%~"xE8^|´c |4jln%~5|&©¡}nnQ1m\|n|³nch³z(.|nc5 nc²®y|hx nQ|hz)jljn% {n4|®O³mnnc~ n±{l¶|hnc3¹ |hZ{n ¹ z²=n %¹ <0; * HI¹ ,+³¡|jmoz |j|&j ¹ n;n J³n~E|#On°Ëhm~ncz ³~+l¹ | n· )xy~lm\
(30) ®5l~lzä~¾|jlnomol{|h|³~0mol³n±{|}x °|jlnQz²n\³¡|jlmzrj ¹ n&®5ncn~ZnQz}|® ³M z²jlnQeL· Ko{n|³nQ+n±{Oz¶|h~z\°@|hjln |jlnchx+~ nQn~E|o{n ¹ n³lmon~E|z\znn´C°Ë\³~z}|~n´ ´K) Nl¤%´ N({O
(31) |³~5mä|hjln ~En|h°Ënh@n~lhncz%®l³|njmnhzn³|h~jl· nZ³~E|nc³ IO³~E|¤lEj z²ncnmz|h ®OnZEl¶|hnZn)POnQ1| n z³~n.|hjln@zhz²{³|nc&³~lnQJ¡n®ltmox#|h( 8n4 ¹ ~|¡n°5|hjln@®~oz}|h51|n4°5|hjln@hnczl¶|h~¹ ¡ m|h³nczc· Ä {|³mo*6Q©|³~¾¡h¬¶|hjlmz%®5M znc ~~E|hnhM ð5~E|@³¡|jmozj ¹ n M ®5ncn~¾l³nc | {|³m~E|W¬ hVOlh®l³nM mz.®yx h³¡jE| QQ OI´ Rr n~E|hn QQ OI´ ?(n³®{°Ë"| 6 ~,vljz S Oð´ T{ 8n~lj U ¹ nI´ A³n¡nc.~ r¶|hjll¡j |jlncjE|nnc z OI· ¹ z}|³¶|hn©|n,~ |jln¾m|jlncmo| .°Ël~©|³~5z&°ä³~E|nc³ IO³~E| mon|jl{lzä°ËB 55~l¶|hnæ{³m\nc~z~)m©|hjlnm©|h³c)lh¡hm\mo³~l¡´5|jnoz||³~3³z El|n{* POncnc~| °Ël|³mo~E|hllh®ncmozc·
(32) i.jn@~l³x#hn°Ëncnc~n4µXn 8y~l©µ ³z M > Oð´Eµ4jlnhn.|jlnt{|hjlz0|hnc||jln 2. 3. T. u. 0. 0. n. m. ε. n. n. m. r. n. r i=1. m. n. i. ε. T. u. ε. 0. 0. i. i. ε. ß
(33) ß ÍYZQãâá.
(34) N. B !" . B . . . #%!")< !
(35) . M°³¡E1|Q´5h¶l|h~5jl{mon³ tz²O5n~jZ¶|}x¤~°Ëz}|h.h|j~En|z@z²³mo~¾lnc°Ë@tz²|z²hn ~°)¡³®Ox,l~~ ¹ n± ~³z²~l|nQ³ä~E|hz5°0{|jl©|hn ³ |}xyOn ®lanc≤mzu(t) ´5|jlnc≤xbl· ¹4æ~ n i.j~ n¹ x=ncz¡jln©~µ n#|j°|#{z²³|mo³m hnc~Ez|lh|hzCjl~ ³=z²°Ë|h|ncz³ºG~¡ n ³z ,zhz z²°.|hl|hncz®l»t³n~ mLz µ4jl~ncnoL|hjln ´
(36) rnQz²m\Onc³|| ¹~n³³x~ · °Ë~1|h~3 z~ ¹ n±Zµ4¶|hj nQz²Onc| ||jn~E|h ¹ ®n·@ z²5´|jnxZnQz}|®l³ z²j=|j|&~E|h z hznmm³e~n~l~ll¶¡°Ëj mo· ³x©|4&l³z²|h~nä°
(37) |³ncz}| cε °Ëhm |hz®5l~lzc´yµ4jlnhn c > 0 z.l~l°Ëhm °Ë.³ %~ ¹ nh¡nc~εnnQz²¶|zJzC|jlEz²n4®{|h~lnQ\~ M Q O5n4³mo5²|~E|®Onczn4|jlncx#n±y³~o|hjln4|}xy5n mo°Jnc~|\~ ¹ ~nh ¡ncz²~|n {x3~n#°4z²~ljncµ¯l ¾~n ±{O;n Jnc|rnc°Ë~h|\m ³|¡jln&h|5jlncm~zc´e|}x|jlmon n|~jyx{z·t³z p@m&n ¹ ncz²²||hjl®5nn,³nczhz²zl´°Ëltncmo|jlnn\~E|{nQn =¹ nµ4³| j ¡,nmo~lnchhno(hnczmoll|h³zän|hno~Z~ |jl¹ nQn\z}|hh¡E|©n\|h°%~7°%~ ¹ |ncjln¡n~z²xynmoµ%{|l| 7\®O®5n ncj8 n¹ x|ty³°%zr°Ël|ä³mo|jn\C©~|hz²j|zch · 1K|h ~Z³~z}°J|¡~y{n ´ Ol|nhl³ncz#~ |jn¤On~¶|}x3mon|nQ´
(38) ~$zjll $l x=Zn~E|hhn¤~+~Ex moln±{¶|}x ~®l³x{ncz²m zz.°n ¹ |jnc~¤nä°Ëz4z{nc ³©||nQ ¹oncx©|hzj moI°Ël³³n©µ4zl³~l®O¡#³z¡znczh¶|hjl°)mz~·)z²i(|&³~llnc¤®Onc{z²||h8ym~l©µ4³nc~E{|h¡n´O|jlnQz²®lnäncmhnrz· 5nc~ 4æ~$|hjl z#~l|n´0µn³³z²|©|hn|jlh¡j$ ncx3z²³mol³n¤n±lm\n|hjln mol{|©|h~$°@ 5hz²| hln äzxym\l||h³än±y5~z~¤°
(39) |hjln {|hm¹ e©|hjz.~,|hnhmoz.°
(40) |hjln On~¶|}x5mon|hnQ· /#4 .')" ')4e
(41) ')
(42) >
(43) ,4 ')@ 4æ~o|hjlnrn±lmol³n4|j|%µn@µ4³Oz}|h{xo~|jln@~n±y|znc|³~´Eµ%nrµ4³5{nQ5µ4¶|hj¤~³mol |.ncE©|h~ |hj©|\³z\~l|\{* P^nhn~E| ®l³n¤µ4|j nQz²Onc|#|h7|jn¤On|lh®©|h~+hm\n|ncc·3rz\Z~zncEln~5n´ |& z~|#Ozhz²³®l³n|=ll³x=|jln,z}|~lh 4æmol³³¶ | Kll~51|³~ i.jlnncm,· 4æ~z²|nQe´)µnµ4³%z²n |hz²|hjln~°Ël³h©+µ4z~l¡7G}nci.|jl nn¤ncmm ll´C³~l~¡Ze|h´0jlm\nncnm nQ° ä z²nc x´(nQ|hz¤jlºMnzn%n MOI´ h~xZ {·¾ z²i. jlM nQ z²Oôn¤»·+@n |¹ jll ¡~Ej |hz°Ëo°.|jlln lOznczt| z% JnQ¹ z@|h j ¹ n 5~¶|hnæ{monc~z²³~5 ¹ ¹ nz~z@°|jlnQz²n\hnczl¶|z´^µn&z²|h|no~Zl ¹ n |h°Ëjlnmom ncµ%°Ë."8^®´yz²|~,®O| ncA.~zntj¤|hjlznx mncz%x|o®5z}n |hnQ°0z³zJ~|{jncnt5°Mnc~1{|n|~Ej|@³|.~E||njlhncncxz²|c· hn ¹ ³³ µ4|jl³~,zj,¡nc~ln ½y § ½ ½ ¥¦  §©¨{¦  ¦ ½y § ½ ; X & Y BF D ( E )E!
(44) ( i. 2. i. ∞. . . . . . . . . & # LF: U. . "! $#. &%. . (. ). E # #- @=DF ,+L WW D)9 )) ) ⊂ X ×E → Y #%= )!" DF
(45) #%! ) )E 0 'D U! (ˆ#x , #-ε0 @)=∈DFU -+ F (ˆ x,& ε0/)#%=' ;0
(46) HI H ) # D A:X →Y c : R + → R+ c(β) ↓ 0 β↓0 x ∈ B(ˆ x, β) '& ; 0 x ∈ B(ˆ x, β). '. ,+. *. ,. '. *. ε ∈ B(ε0 , β). -). ). º
(47) NE». kF (x0 , ε) − F (x, ε) − A(x0 − x)k ≤ c(β)kx0 − xk.. '; )!"="F E # C '; ! 'η & > 0 .. -). (x, ε). ) * Q )@) #(+>C:. (ˆ x, ε0 ). ). )!"=DF E. x ¯. # D. º». ½ + ®l¨{³§n jl% l~zc{´l|l³~l~°Ëhº
(48) NEmo»oxmo³nc~ ~zo· |hj©| A = D F (ˆx, ε ) ~ |j| z}|h 1|¤l P^nhn~E| ®³¶|}x ³~ ¹ x ε F (¯ x, ε) = 0 0/1 . k¯ x − xk ≤ ηkF (x, ε)k. x. . 0. òËç0ßòôó. ).
(49)
(50) !" $#%'&()! * "+,! -.
(51) /) 01 . § hzn#° A ´
(52) °Ëµ4jl@ j µn Â{Ân~|n A%®yx7x |Bjlno´lI· On·%n~ mºËOzhlz®l~³¡¤x¤|h~jln~l³nc~m,nc´1|»%jlmncnolnl±{³~l z}¡ |z B,: ®5Yl→~{XnQZz²h¡j,jE| |hj³~ ©¹ | ncABy =y ´l~ y∈Y ºME» kBk := sup{kByk/kyk | y ∈ Y, y 6= 0} z 5~l|n· K0¶± β > 0 z²5j¤|hj©| ºI» L := c(β)kBk < 1. ?(n| ~E|ρ³~yl>50z´yµ~nä¾m| 8x¤n xzzl∈moB(ˆ nä|xj,ρ| ) ´ ε ∈ B(ε , ρ ) .· Ax¾|h( 8E³~l¡ ρ > 0 zmo)nc~ll¡j´z F z º E» ρ + (1 − L ) kB[F (x, ε)]k ≤ β. ?(n| ´ ´®5n |hjlnz²nQEln~n {;n 5~nc,®yx x = x ~ |hjlnºËmo{{* 5ncp@ncµ.|~,³7 8nQ»z²|nc {x } n ∈ N º » x = x − BF (x , ε). i.jn~ º}Q» kx − x k = kB[F (x , ε)]k ≤ kBk kF (x , ε)k. @n ©|³~ º E»mol³³ncz º}Q» F (x , ε) + A(x − x ) = 0. ¬ nrzjl©µ ®yx&~l1|h~o|j| {x } ncmo~5zC~ B(ˆx, β) I· Ax,º »´|hjl³zJ³zC|hln4¶° n = 0 · K n = 1 ´ µn j ¹ näµ4|j$º}Q».~5=º » . β. 0. 0. 0. 0. β. 0. 0. −1. 0. n. n+1. n+1. n. n. n. n. n. n. n+1. n. n. i.jn~,¶° x ∈ B(ˆx, β) ´{°Ë@³ 1 ≤ i ≤ n ´(º
(53) NE»~ º²Q»%³molx. kx1 − x ˆk ≤ kx1 − x0 k + kx0 − xˆk ≤ kB[F (x0 , ε)]k + ρ0 ≤ β.. º²Q». i. %m#®l³~l³~l¡oµ4¶|hj$º}cE»´lµn ¡n|. kF (xn , ε)k ≤ c(β)kxn − xn−1 k.. º²c ». kxn+1 − xn k ≤ Lβ kxn − xn−1 k ≤ · · · ≤ (Lβ )n kx1 − x0 k,. ~ jln~n´lµ4|j$º»1´. kxn+1 − x0 k ≤ (1 − Lβ )−1 kx1 − x0 k ≤ (1 − Lβ )−1 kB[F (x0 , ε)]k ≤ β − ρ0 .. v{~n ´yµnt{nQ{nr|j| ´l~5jln~5n´E|hjlnäzncElnc~n hnm~z ³~ B(ˆxkx, β)−· ¬ xˆk¶|h<j7ρº}Q»0~,º}Q »1´µ%n.®{x|h~&|∈jB(ˆ | xx, β)~ ¹ nh¡nQz|hz²mon x¯ zj&|j{x| F}(¯x, ε) = 0 ~ k¯x − x k ≤ (1 − L ) kBkkF (x , ε)k ´{µ4jl jlh ¹ nczäº »µ4¶|hj¾~5z}|~E| η ¡ ¹ nc~¤®yx º}S NE» η = (1 − L ) kBk. 0. 0. n+1. n. n. 0. β. −1. . 0. β. −1. . ß
(54) ß ÍYZQãâá.
(55) . B !" . B . . #%!")< ! . .
(56) . ½³~++º ¨{§»´µ4jlp@nc|n n%|j zr|c´o³~®O|hjln4~{nc®O ¹ hn%³¡ljE|@y~ °}´©nµ%n%z²n j¹° n%®{|~ ³~lnc|jln.ncz²|³m©|h·4nä@º²) NEz²5»´{°Ë|
(57) jn#|jjyn.xy5~|z²jl|hnQz²~E z| η |hzj©j | (x,|jε)|#º³Ez »jl³B zlnrzc·nc~ll¡j r zzh©|h³z5nQµ4¹ jlnc~ln ¹ n Ax ∈ B(ˆLx, ρ= )c(β)kBk ~ ε ∈ B(ε , ρ ) µ4jlnhn ρ ³z  §  M'¨{§ )¿ ; ! )$ ) * ; -1D: ; ) G ' !D; @ ) *+>& '!"& >&( S)& 1 BS) !" &(#%)&F D:& &! +Q-1 ;#! D ) ε ε F (·, ε) xˆ x A º² » x =x ˆ − BF (ˆ x, ε) + r(ε), 0/1 . . . β. . 0. . (.. .. ). *. ; ! ) =. r(ε). ). . '. '. ε. . ε. kr(ε)k ≤ c(β)(1 − c(β)kBk)−1 kBk2 kF (ˆ x, ε)k.. ;' #(+Q. 0. . '. '. ) E ;. 0.
(58) ). ). 0. H ;!D !") &()!. 0. º}Q». §  ?n| xˆ(ε) := xˆ − BF (ˆx, ε) · ¬ n@j ¹ n|hj©| F (ˆx, ε) + A(ˆx(ε) − xˆ) = 0 ~ kˆx(ε) − xˆk ≤ ¬ kBkkF |hjlnncm(ˆx· , ε)k ·W4æ~ ¹ ncµ °¾º
(59) NE»´ kF (ˆx(ε), ε)k ≤ c(β)kBkkF (ˆx, ε)k · n7~³{n¾µ4|j |hjln. ½ +¨{§ 4hl~l¡¤|$º}Q»´°Ëä|jlnll³³c©|³~5zr°4%h³ hxl´e®5n|²|nc ncz²|³m©|nQzä~ xy³n zjOn4®Ol~5lz.~,|jn ncmo~5{n.³~ º² »·i.jln ®Oncz²|r~5z}|~E| c(β) z4¡ ¹ nc~¤®yx c(β) º²» kF (x , ε) − F (x, ε) − A(x − x)k , c(β) := sup kx − xk µ4jnhnt|hjlnzllhnm#m ³z| 8n~, ¹ n4 x 6= x ´l®O|hj,³~ B(ˆx, β) · - 4" #" %~z³{nc|jln °Ë³©µ4~¡\ncncmon~E|hxzh³{|³m~E|helh®ncm Z º² » Min (u(t) − 2t) dt + y(1) , y(t) ˙ = u(t); u(t) ≥ 0, t ∈ [0, 1]; y(0) = . 4ch{³~l¡¤|,~E|hx¡l³~ zrmo~l³m#m lh~n´e|jlnozhzy³|nQ 5z}| Ih{nct{|³m³¶|}x7~ l¶|h~, z.¡ ¹ nc~ ®yx|hjln °Ë³³©µ4~l¡ozxyz²|nc m β>0. *. . 0/1 . . 0. 0. 0. x,x0. 0. 1. 1 2. 0. 2. 1 2. 2. 3 4. y(t) ˙ = Hp (u(t), y(t), p(t), t), t ∈ [0, 1], y(0) = 43 , p(t) ˙ = −Hy (u(t), y(t), p(t), t), t ∈ [0, 1], p(1) = y(1), u(t) ∈ Argmin H(v, y(t), p(t), t), t ∈ [0, 1].. º² ». v≥0. òËç0ßòôó.
(60) .
(61) !" $#%'&()! * "+,! -.
(62) /) 01. x nc~E|~E³|xh{µ4{|²|hn~l~¾¡oz ~l~lOz¶|h ¹ n ?
(63) ¡h~l¡näm#l|³l³ncc´|jln z²|r~{|³~³~+º} E»%mox ®5n#ncEl ¹ H (u(t), y(t), p(t), t) + λ(t) = 0, ºI» u(t) ≥ 0, λ(t) ≤ 0, λ(t)u(t) = 0. 4æ~,|jnz²Onc 55 cznt°0®lncm2º} »J|jnhhncz5~{³~l¡ tmo³¶|h~l ~ °Ël~|³~³z.¡ ¹ n~,®yx A. u. i.jyz°Ë.|jln}³~E|@z}|©|n v{n|²|h~¡. H(u, y, p, t) = 12 (u − 2t)2 + pu. p(t). µnä¡n|. −p(t) ˙ = 0, t ∈ [0, 1];. p0. p(1) = y(1) =. 3 4. +. Z. 1. u(t)dt.. ´ ´{µ%n ®{|h~,|j5©|4|jnä{|³m(~E|h^ z.¡ ¹ nc~ ®yx := p(1) = p(t) t ∈ [0, 1] 0. µ4jnhn p z² ¹ ncz|jn ncE©|h~. u0 (t) = max{2t − p0 , 0} = [2t − p0 ]+ , t ∈ [0, 1],. 0. p0 = y(1) =. µ4jznä~l³Eln z³{|³~, z4¡ ¹ nc~ ®yx i.jnhn°Ëhn´E|hjln {|hm~E|he³z. 3 4. +. Z. ºG» ºI ». 1. [2t − p0 ]+ dt,. 0. ºGlQ». p0 = 1.. ¶° ºINE» ° tt ≥≤ ,. i.jno~z²|³~E|t~Z|hjln~E|h
(64) jzä~l³x7~nr}l~51|³~3O³~||ä|hmon ·#i.jl z }~1|h~3 z hn¡³c´e³~Z|hjlnzn~zn\|j| u˙ (t) jz ,~l~6nc,³mo¶|µ4jnτ~ t=↓p1/2/2·\=p@1/2 |h³no|j| ~ln jz|hjlnhnczO~{³~l¡&m#l|³l³nc ° t ≤ , 2t − 1 ° t ≥ . λ (t) = −[1 − 2t] = 0 |¤·molncmon~E|hx´J|hj©|¤ zc´ λ (t) < 0 µ4jn~ u (t) = 0 ´n±lnl|°Ë|hjln }µ4j~³1|hj zh~ ©||h zmo5n nczτ z²|=h 11/2 Kt|jl zän±lmol³n|hjln#°Ëm&l ©|³~Z°C|jn&³¡Eh¶|hjlmo³c³x,5nc~³7 6nc7l®l³nm³zc´^°Ëä¡ ¹ nc~ ´ u0 (t) = [2t − 1]+ =. . 0 2t − 1. 1 2 1 2. 0. 0. 0. 1 2 1 2. +. 0. 0. 0. ε>0. . ß
(65) ß ÍYZQãâá. Min. 1 2. Z. 1 0. . (u(t) − 2t)2 − ε log(u(t)) dt + 12 y(1)2 ,. y(t) ˙ = u(t), t ∈ [0, 1];. y(0) = 43 ,. 0.
(66) B !" . . B . . . #%!")< !
(67) . 4µz²|hjn|n4hn&zh©|| z³z 5nc³mozCl¡ ³~ |³xZºI{hnc»1E´l~³h&nc¾µ%n4|hj{©nc| ~lu(t) |hn4®E>x p0 °Ë¶|z · n~·z}|t~E∈| ¹[0,³1]ln·\·)i.4jln{~hnc¡rz|h5t~|hjl{n4³~lmo¡ ~m&}l³~Em | ³~³l³n´y|hjln {|hm(~|hG´yµ4jl j,µ%n{nc~l|hn ®Ex u (t) ´5zh©|h³z5ncz ºG » u (t) > 0; u (t) − 2t + p − ε/u (t) = 0. pr|h³n |hj©|®yx+~5³¡x µ4|j ºG»1´%µ%n~ ³~E|h{{n~ |* 55 m&l¶|h³n ¹ h ®l³n¤®yx zn||³~l¡ ºG» λ (t) = −ε/u (t), µ4j³j,³z@~ln¡E©| ¹ n ~5,zh©|h³z 5nQzJ|hjln °Ë³³©µ4~l¡oOn|lh®Onc,mol³nmonc~|hx¤~{|³~ ε. ε. . ε. ε. ε. ε. . . ε. ε. i.jnä~l³Elnz³{|h~ ° ºI »J z4¡ ¹ n~,®Ex. λε (t)uε (t) = −ε.. . °Ë v{~n. uε (t) = φε (2t − pε ), φε (x) := 21 (x +. p. x2 + 4ε) =. ´{µn {nc{5n |j5©| φ (−x)φ (x) = ε ε. . ε. ¶° x = 0, ¶° x 6= 0.. √ ε. [x]+ + ε/|x| + O(ε2 ). ºI» ºI ». °Ë³©4ðµ4|4³h~ln¡&mnQE³~z%©|h|o~¤°Ëm\ p{|nä |jn ¹ ln p °|hjln }~E|4z}|©|hn·qrz³~l¡¾ºG{»J~3ºG»´{µnt¡n|.|jln Z ºI » φ (2t − p )dt = + [Φ (2 − p ) − Φ (−p )] , p = + µ4jnhn Φ z4ol³mo¶|h ¹ n ° φ ´~mon³x ºM » Φ (x) = Φ (x) + Φ (x). µ4jnhn p ºM Q» Φ (x) = x + x x + 4ε ~ p ºG » Φ (x) = ε log x + x + 4ε . 4æ~hlne|h@®{|³~@³zncä°Ëm °Ë µ%n%z²jll 5~ ®yxäz ~l¡&ºG E»1·0q@~{°Ë²|hl~©|hn³x´ |hjl~, zz²|hxyhmo~{z|hn~| länn~E±{|%nQ~5z²©³|³~,~¤° ³zl°Ë5 hn~E|uE³x\z²nä(t)l|h J ·C¶i(|\|\|jz²³z4 ¹ ncnr~{ep³´lnQµ1n |³{xn;· 55¹ 4æ~l~nz²|nQe´Eµnrµ4³emol{|n p ε 0 ° ε > 0, x − − Φ (2 − x) + Φ (−x) ºG » F (x, ε) = ° x − − [2 − x] + [−x] ε = 0, λε (t) = −ε/uε(t) = −φε (pε − 2t). ε. ε. 1. ε. ε. 3 4. ε. 3 4. ε. 0. 1 2. ε. ε. ε. ε. ε. ε. 1,ε. 1,ε. 1 2 4. 2,ε. 1 4. 2. 2. 2,ε. ε. ε. ε. 3 4. 1 2. 3 4. 1 4. 1 2. ε. 2 +. 1 4. ε. 2 +. òËç0ßòôó.
(68)
(69) !" $#%'&()! * "+,! -.
(70) /) 01 . µ4° j(p³j, |h0)l=~5zr(1,{0)|r|h· ¬ ®5n nrj} ³¹ ~Enä|³|hxj©|4~E°Ë|³4~yln ¹ nc5z@x ε~≥Zz²0mo´ py|hzhj¾©|µ4 z¶5|hj7ncz nQz²Onc|r|h x ~7~lnc¡jE®Ohjly{ ºM( NE» F (p , ε) = 0. Ä ®5z²nc ¹ n¾|j|,µn=c~l~|¾ll³x | ºM( NE»o|hjln zhz² B 4æmol³³¶| Kl~1|h~ i.jlnchnm®5nQz²n ³z¤~l| ~E|³~yl5z²³x l P^nhn~E| ®nZ| (p , 0) = (1, 0) · 4æ~ °M1|Q´r|jll¡j (x, ε) → F (x, ε) . z ³ ~ { nnQz²moy|hj,l~ p = 1 ~ mohn ¹ nc x → F (x, ε) ºG » D F (p , 0) = , ~\|jln4|jlncCj~´|jn@| llnh ¹ ©|h ¹ n D F (p , 0) zC~l|µ%ncl;n 5~lnQo®5nQz²n.µn4j ¹ n|hj©| ºM » 1 1 [F (p , ε) − F (p , 0)] = F (p , ε) = log ε − + O(ε). ε ε K ¹ nmo~l¡ |hjl z¤|hncjl~³ctl J¶|}x´rµnZµ4³ lx %h³ hx $°&vync|³~ $| ºG E»1´ oncEl ¹ nc~E|³x´C|h ºG Ny»19· K0³hz²|c´%®yx ¹ ³|ln°t|jln¾{nQm\Oz|³~ ºG E»ºG »° Φ ´%µ%n,µ4|n µ4jlncn F (x, ε) = F (x, ε) + F (x, ε) ° ε > 0, x − − Φ (2 − x) + Φ (−x) ºG E» F (x, ε) = ¶ ° ε = 0, x − − [2 − x] + [−x] ~ ° ε > 0, − Φ (2 − x) + Φ (−x) ºG » F (x, ε) = ¶ ° 0 ε = 0. A|j\°Ël~|³~zn@~E|³~yl5zC~ozm\y|joµ4¶|hjnQz²Onc|C| ~ ~ln³¡jy®Ohjlyy&° (p , ε ) = ·vyn|²|³~l¡ A = D F (1, 0) ~ A = D F (1, 0) ´ i = 1, 2x´yµ%n j ¹ n (1, 0) ºG » A = 0, A = A = . v{~n ³ztz²moy|hj ºË°Ë@µ4j5©|4°Ë³³©µ@z.¶|tz% JnQz °C zhz »4~¾o~ln³¡jy®Ohjly{¤° ´ µ5z²n#³~lj5¡7 ¹Fº}n©|h»j©~5|o3º NyºG»@ Ejl»´{³|jzrn °Ë®5 nQz}F|rµ4~¶|hz²j|hc~|4(β) z = c βF°Ëäzmon&Cllhlh ©|n&~5z}|~E| c · Kl(1, F0) ´ 0. ε. ε. 0. 0. 0. ε. 0. . 3 2. 0. x. 0. 1 2. 0. 1 2. ε. 1. 2. 1. 3 4. 1 2. 3 4. 1 4. 1 2. 1 2. 1,ε. 2 +. 1,ε. 2 +. 1 4. 1 2. 2,ε. 2,ε. 2. 0. x. i. 2. 1. 1. 1. 1. 0. x i. 1. 3 2. 2. 0 1. 0 1. 2. º
(71) N» µ4~j nhnµ%|jn n\jz lln hnm&lm³zä ¹ nc x 6= x ´e®O|j=~Z|jln\³~|hn ¹ [1 − β, 1 + β] · A{|ä°Ëä5Ez²| ¹ n y ¹ c2 (β) := sup x,x0. |F2 (x0 , ε) − F2 (x, ε)| , |x0 − x|. 0. z. ß
(72) ß ÍYZQãâá. p −y + y 2 + 4ε √ Φ2,ε (−y) − Φ2,ε (−z) := ε log −z + √ z 2 + 4ε z + z 2 + 4ε p = O(ε|y − z|). = ε log y + y 2 + 4ε. º
(73) NQ».
(74) Q. B !" . B . . . #%!")< !
(75) . h|i.jj©n|#hnº
(76) NE°Ë»%hjlnµ³ln&z4m°Ë x,F|h(µ48n|jc (β) = c ε ~ 5~³³x´ez²³~n ε zäjzn~Zz|hj©| ε ≤ β ´µn#®{|h~ º
(77) NE» c(β) = (c + c )β. i.jnhn°Ëhn i.jlnncm ~7% hx¤olx· ¬ n mx,jlyzn ρ = ε ´5~z~n |F (p , ε)| = ´(º »³mol³ncz β = O(ε log ε) ·i.jyz´{5z²³~l¡ZºG »%µn ®{|³~¤|hjln °Ë³©µ4³~l¡ohnczl¶| O(ε log ε) §  J ¥ M¦  '. D ; ! * D+F ! %
(78) 1 ;' 0 >&!" *) ) 0 2. 2. 0 1. 0 2. 0. !. ! $#. . .. 0. . '. = p0 − A−1 F (p0 , ε) + O (ε log ε)2. pε. . º
(79) N ». = 1 − 31 ε log ε + 13 ε + O(ε2 ) + O (ε log ε)2 ,. = 1 − 31 ε log ε + 13 ε + O((ε log ε)2 ). i.jl z&³z|Z®On monQ=µ4|j+|jln,z}|~lh$cznZºË°r7zm\y|j$On|lh®©|h~5» µ4jnhn|jln n±{~z³~¤µ³ ®5n °
(80) |hjln |}xy5n p + dε + O(ε ) °Ë@~llhlh ©|nä~z}|~E| d · 4nQ^|hj©|Q´l®yx ¹ |nä°ºG»~ ºG »´µ%n j ¹ nr°Ë ε > 0 √ ° t = p , ε º
(81) N(NE» u (t) = φ (2t − p ) = h | l j n h 4 µ ³ z n , [2t − p ] + ε/|2t − p | + O(ε ) µ4j³n u (t) = [2t − 1] . v{n|²|h~¡ |hjln~,°Ë@³ ε > 0 zm³en~ll¡j t |=zh©p| z²=°Ëx t −> tε log= ε´{+µ%O(ε), n ¡n| 2. 0. 1 2 ε. ε. ε. ε. ε +. ε. 0. ε. 1 2 ε. +. 1 2. 1 6. ε. √. 2. 0. √. 1 2. √ ε + 31 ε log ε + O(ε) = O( ε).. h|v{j~©| n °Ë t 6= t µnj ¹ n |u (t) − u (t)| ≤ O(ε log ε) √~l¶°Ëmo³x µ4|j¾hnczOnc1|4|h t ´µn#{nQ{n Ä ~ |jn |jn4j~e´l|@³z.~|r{*Jlku|4|− ¹ unck°Ëx|hj=©| O( ε). uε (tε ) − u0 (tε ) = ε. ε. ε − (2tε − 1) = 0. ε. æ~,°M1|c´. kuε − u0 kL1 = O(ε log ε).. 4. kuε − u0 kL1 ≥. 0 L∞. Z. 1 2 0. φε (2t − pε )dt = 21 [Φε (1 − pε ) − Φε (−pε )] = − 21 ε log ε + O(ε).. òËç0ßòôó.
(82) .
(83) !" $#%'&()! * "+,! -.
(84) /) 01. Ä ~ |jn |jn4j~ kuε − u0 k. L1. ≤ =. x3ºG E»1´{µn j ¹ nt|j5©|. =. A. Z. 1 0. [φε (2t − pε ) − [2t − pε ]+ ]dt +. 1 2 [Φε (2 1 2 [Φε (2 1 2 [Φε (2. Z. 1. [[2t − 1]+ − [2t − pε ]+ ]dt. 0. − pε ) − Φε (−pε )] − 14 (2 − pε )2 + − pε ) − Φε (−pε )] −. 1 4. 1 4. − 41 (2 − pε )2. − 13 ε log ε + o(ε log ε).. − pε ) − Φε (−pε )] = pε −. 3 4. =. 1 4. − 13 ε log ε + O(ε). · @n~5n. ≤ − 32 ε log ε + o(ε log ε).. vy³mo @mol{|©|h~z4~,nQz}|hm©|hncz.jl³¤°Ë λ (·) · 8 '0-, = #((?," /4,.')"
(85) # s7| ©|hnc ®Exo|hjln lhn ¹ zn±lmol³n´y³n|@z.~z³ln|jln mohnä¡n~n^³~lnQ ðE{h| 4lh® ³nm ¹ kuε − u0 kL1. ε. . . µ4jnhn. r : [0, T ] → Rm. Z 1 1 T 2 min ky(T )k + ku(t) − r(t)k2 dt u 2 2 0 y(t) ˙ = Ay(t) + Bu(t), t ∈ [0, T ], y(0) = y0, u ∈ L2 (0, T ; U ),. º
(86) N » . z@z²monä¡ ¹ n~ n°Ënhn~n ~E|hI´ A ∈ R ´ B ∈ R ´~ n×n. n×m. m U := Rm + = {u ∈ R | u ≥ 0}.. rnhn´y|hjln rm\³|~³~¤°Ël~51|³~ z4¡ ¹ nc~ ®yx. H(u, y, p, t) =. 1 ku − r(t)k2 + p> (Ay + Bu), 2. ~¤|jlnzz{ ©|nQ)~E|hxE¡~ mo±{m&lm lh~5³ln z. y(t) ˙ = Hp (u(t), y(t), p(t), t) = Ay(t) + Bu(t), t ∈ [0, T ], p(t) ˙ = −Hy (u(t), y(t), p(t), t) = −A> p(t), t ∈ [0, T ], y(0) = y 0 , p(T ) = y(T ), H(v, y(t), p(t), t), t ∈ [0, T ], u(t) ∈ Argmin m v∈R+. ß
(87) ß ÍYZQãâá.
(88) . B !" . B . . . #%!")< !
(89) . i.jnhn°Ëhn´E|hjlnEz}|©|n ~¤{|³m(~E|henänQz²Onc| ¹ ncx¡ ¹ n~,®yx >. >. p(t) = e−tA p(0) = e(T −t)A p(T ). ~ µ4jnhn p(T ) ³z%³m\u |(t)³x=j[r(t) |−nh7B6nc\p(t)] ®yx\|jlnt=nc[r(t) E|³−~ Bp(Te ) = y(Tp(T ·) Ä )]~|, jlnt|jnJj5~e´E|hjln ³~E|h{{1|h~ °(|hjln ¡|jlmo ä5nc~|}xxync³z|jln {|³m(~E|h >. 0. > (T −t)A>. +. +. >. uε (t) = φε (r(t) − B > pε (t)) = φε (r(t) − B > e(T −t)A pε (T )),. ³z ¡ n~7®yx ºI »ä~7 z ll³³nc=jlnhn\moO~lnc~|}µ4 zn·rz ®On°Ën´e¶| hnm³~zr| 5µ4~j nhpn (Tφ (x) ®) yx z ¹ ¹ ~¡#|hjln ncE|³~ º NE» p (T ) = y (T ) Z º N{» e Bu (s)ds = e y + Z º N » = e y + e Bφ (r(s) − B e p (T ))ds. qr~{°Ë|~©|hn³x´|³z ~|³ncäj©µ |¾{nQCµ4|j=|jln ©||nc ~3{nä|h,®On®n\|¾ll³x¾|jln ¡n~lnch^nQz}|h©|³~|jlnchnm z.~,|jn ln ¹ z.n±lmol³n· ,%"*,-, / '0@! '09 K4³~ncr@Elh| l¡momo~¡&lh¡hmzJ|r z45Ezz³®ln |hm\{|n#~¾zxEmo{|h|h³än± 5~z~#M |hjln.|jn@n~E|hh{©|hj´znn4s¾~|hn³ ~5\i0zjl³x M QQ OI´Es¾7 6l~l5F´ TEhn.~5\vE|hyn M {´ OI´ °zJ|hµjlnn.³ ©|h´j %G°Ëj³{©µ4|hn³ ~l¡ä OI·)¡i.hjl¶|h zjlmn±{z· 4æ~5~&z²³~¤²|h³³l© µ@Qz´|jlln4hnc®O³z nä¹ n~hn°Ë³nx{hz²n ~5znc°z)|{j n zhz²zhxyz)mo{h|hhnc|1 @|h®O~nj5| nc¹m³z ³~\{n
(90) |ä®{|³~\zlOnh³~lnQ)~ ¹ nh¡n~5n³~#|hjln4®zn~5n°5z²|h³|mol³nmonc~|h¶|}x· 4ð|Cµ%l ®On{i.nQjlz²³nthl®lhnc³nrzn|h~E|J{nh5 ¹ ncnäJz zJ³m\³ 55h4z²|%lhz²|nO¤n|~³nc|hz.j³©~,||{jlhn nc|³~E|~n· ±y|@Ä ® °
(91) l|z³mox#|jlntn~±l|hmoell³hn@ ®lz ³nmnczcxo· zmol³n· vy|³C| jlOn~5zr|hj©| |hjln&y°%³zä~l| zz²³moln·#@~3³~|hn¹ hncz²|³~l¡ j5³nc~l¡n\³zt|h ¹ n±E|hn~Z|j³z 8y³~,°0nQz²l|hz|ho¡n~n~ ¹ n±³³~lnc.E{©| älh®l³nmzc· 4 ©.')4 4 M ) OT5· K%· A~~~zc ´ T5·©%j· ä³®On|c´ytQ· ?(nmhcj5I´Q~\t·v{¡z²|7 6 ®I· #% )!
(92) > <
(93) !")E
(94) G& #% )!
(95) /Q ;E 1·+q@~l ¹ nchz¶|hn±y|vynhnQz·vy³~l¡nc RCnh ¡´ Anh³~´ l·Cvync~ J{|³~(· 5. ε. ε. ε. ε. T. (T −s)A. TA 0. ε. 0. T. TA 0. (T −s)A. ε. > (T −s)A>. ε. 0. . . . . . . . . òËç0ßòôó.
(96) Q .
(97) !" $#%'&()! * "+,! -.
(98) /) 01. °Ë5· .%·{|hm~l~5e~z~E~|hOi.ljh· ä®l³ln³m®5zc· e·qt z²³ ~l ¡ ³ ¡Eh!"¶ |h jlmo³(.'O 0
(99) n ~5
(100) |³nc z% ~&|hjlCnr)zjly S&|h 1´5~l¡&N ³¡hl|jl m l´ l· M OT5· s$·%A%µn³~·(vE|®l³³¶|}x~hn¡³JO³~|z°(³~lncE|}xz²x{z²|ncmozc·% #%!' > <
(101) ' !1 & F 0
(102)
(103) 1´ N {´O l· M N(O #· Kz²¡nc~´y0· @· ä³G´O~s$· · ¬ h¡jE|c· 4æ~E|nc³4mon|jl{lz.°Ë4~l~³~lnc4{|hmo7 6c|³~·
(104) B >
(105) )HC´N(N y\ºËn³nc|h~l c» ºI »´lE{· M O?J· s+· äh ¹ nczc·(vymonämoll³~l¡|hjlnncmz· #FC) E
(106) - #%! ê´e ^S N´{ · M O i %· T{ 8n~j U ¹ ncG(´ ?J·{i ,· A³n¡³nQ´y~#· W ¬ V jE|ncc· txy~mo @{|³mo* 6Q©|³~¤°e|jnr|hn~l~lnQzznn nQz²|m~ {nQzzz²³~l¡ |hjlnZ{|h~E|h nc~|hn· G#%:)! & )
(107) / -+ 'D)! ,+´ E ³ c ^ l´E l· M Q OKQ· ?nc®{°Ëh"| 6J~ @· ¬ ·v{jzc· 4æ~ln±l|
(108) v t7³~E|nh³eO³~|
(109) m\n|jl{lz0~5t h¡nzhnC{|³m ~E|h{®lncmz·
(110) - #%! !" '& 'E . >
(111) ´y E{ ºËncnQ1|h~l³Q»1´ l· M Ove·(s¾7 6l~l5·¯zlOnh³~lnQhx7~ ¹ nc¡n~E|t³~{°Ëncz²³®l³nð~E|hnh ð5~E| ³¡h|jlm°Ë ¡nmon| h / ?)%Jzoµ4|jl{|¤ z}|h 1|hx mol³nmon~E|hx+~5{¶|h~ · C) E
(112) ; C )!" E
(113) B )!D D{´5{ -Nl´ l· M Ove·s¾7 6l~l5´ K· TEhn´y~ T·vy|ynQ· l~l* 5nc¾llhj|³~{°Ëncz²³®l³nð~E|hnh ð5~E|4³¡ h|'jlEm .z >¹
(114) ¡n´O mo n |h ³c{ N5³~´lnc ä N5· m\ncm\nc~E|hh|}x¤lh®l³nmzc · F 0
(115) & C) E
(116) ; & M c O · &·Ôt·@s7~E|nch ~ i ·@i0z²5jlxEl· ?³mo¶|h~¡ ®Onj5 ¹ ³ °&|jln {nh ¹ | ¹ nQz¤°\nc²|³~ |hh}nc|h³nczzhzy³|nQ µ4|jb+m\~l|h~ln¾jh* 6c~E|h4³~lnc¤molncmon~E|h|}x+lh®l³nm· C ;=
(117) )!" E
(118) G; )D !" y´5l ê ) N´ l· M ) O!\·(p@nQz}|hnh ¹ ~3·
(119) p@nmo³ ¹ "z 8yI· :;!
(120) !<7 01
(121) +F ! % = "=!" < +(!" . ,+´ ³lmonc o° " E#&
(122) F 0
(123) &#C) E
(124) ; 1·rvy{³n|}x°Ë/4æ~{5z}|h ~@l³nc¹ s7|jlncmo| z ºI-v 4}rsZ»´j {ncjl³´le´ (N· M Q O ?J· p#· Rr nc~|hn· Ä ~&~E|nc³ ð5~E|0prnµ.|h~o³¡h|jlmz(°ËC{ zn|7 6nc#{|hmy~E|hElh® ³nmz4µ4|j¾z²|h|n ~z}|hh~E|z· > E
(125) $C; >&Q & H !D´ Ô N {E ´ l· M c O ve· T5· ¬ h¡j|Q(· 4æ~E|nc³(5~E|
(126) mon|jl{lz(°Ë
(127) {|hmE~|h°{l³zhhn|hnC|³monzx{z}|hnmz%· - #%! > E
(128) ' !1 '& F 0
(129) E
(130) 1´^ ³cl& O Ey´5 l· M ) N(O ve· T5· ¬ h¡jE|c( · ' ! 0<&# )!
(131) !<7 ) >&Q 1· v{yn|}x3°Ë. 4æ~{z²|h³%~+@l³nc sZ©|jnm©|h³czºG-v 4}tsZ»1´jl³³{n³ljl l´le#´( E{· M OT K -A. . . . ). . .. . . . . . . . . . . . . . . . . . *. . . . . . . ß
(132) ß ÍYZQãâá. . .. . .
(133) Unité de recherche INRIA Futurs Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(134). . ISSN 0249-6399.
(135)
Documents relatifs
La loi du 29 juillet 1998 relative à la lutte contre les exclu- sions comporte un chapitre concernant les « Mesures d’urgence contre le saturnisme infantile dans le code de
Finally, the proposed migration-cost aware ACO-based algorithm outperforms FFD and Sercon in the number of released PMs and requires less migrations than FFD and V-MAN when used in
L’objectif de cette étude visait (1) la reconstitution de l’alimentation de 33 sujets du site de la nécropole de l’Ilot de la Boucherie à Amiens (III-Vème siècles ap JC)
One of the main improvement that must be made is the ability to speak with the companions in natural language. The current system features a set of vocal commands, for
Developing an extremely high impe dance, the condenser mike needs amplification close to the mike , (studio versions f rom the early thirties on actually included
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
The present study aimed to compare couples based on their dyadic AEE and examine whether they differ on the intensity of pain reported by the women, and
Les crustae sont taillées dans un calcaire marmoréen blanc, à gros grains, identique à celui des crustae du pourtour de la salle S; approximativement carrés ou
