An adaptative antidissipative method for optimal control problems

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(1)An adaptative antidissipative method for optimal control problems Olivier Bokanowski, Nadia Megdich, Hasnaa Zidani. To cite this version: Olivier Bokanowski, Nadia Megdich, Hasnaa Zidani. An adaptative antidissipative method for optimal control problems. [Research Report] RR-5770, INRIA. 2005, pp.21. �inria-00070250�. HAL Id: inria-00070250 https://hal.inria.fr/inria-00070250 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 adaptative antidissipative method for optimal control problems Olivier Bokanowski — Nadia Megdich — Hasnaa Zidani. N° 5770 Novembre 2005. ISSN 0249-6399. ISRN INRIA/RR--5770--FR+ENG. Thème NUM. apport de recherche.

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(84) npgpniU@tYl<j<4fnpYCor*Ødc©Y6tYC@¨gire^}vo j@j tvXgyU<}Ác }Ádo¨t4j@j4B9 ggp}vU@o j Y6epepor}ehX6giYYnX6YYC } e j«U<"}Á}vd o l@j@Yore\gpU@niYBY;Seij@}vYCX6®Y6ª X6YC¡ gi}Yj ny"gpU@}Y¨l@eYY 8tn " :}egiU@Yo nko X8X6Yfo }vgpY j@YCo¨4UL>tnio j@8Y e }8j@Yo Ud>tnio¨j@8YC¨Dorek}8Y epU<}vnio j@'}j YfY tn } tjLYCgin}gioŸj=«}kjLc6o¨gpUforepgpU@tYjLgpo tjdj<l@oŸghc'nij@}vYCjS6U<Y }Á2¬®YVoggitU<}4SeyY^j@tvt4g }nij@epYYCj<YCYCeicep7}npgpo U@ c orePgiorU@eQYgiU@epY^}X6t4Y6}vnepepo Y?j@"Y :o j@ª' TVehU@giYY~<eY6eCª YCö rj<e }f¨ tc¬Lj@ØtvYg 4YgkgiU@Ynior G kU@YCnpY}Y Dtv¡(X6o¨j<o¨X}veo ?Y YCogiU@Yn^tjLg}vo j<ek}foreptjLgpo jdl@oghc tnkore+@ }6j@YCo¨4UL>tn t¡}6YC¨

(85) tjLg}vo j@o¨j<6}6foreitjLgio¨jdl@o¨ghcBtnVore+}6eoregpYnkt¡epl<U*}Y 2ª D ² ² ¶ C

(86)

(87) ¶ ¶

(88) ² /· G ª aLgpYC~ ª 7(T

(89) }°Y G = K 9 t4j@YyY :¬<}vjS f/Y ;<j@Y G }4eØgpU@Y ft4X6}o¨j K ep~@ ogpgpYCo jLgpt'¡£tl@n YC¨reªafYg l = 1 ª aLgpYC~ ªÆZ 7 @ tn 1 ≤ l ≤ L − 1 ª @ tn}v PYC¨re M ∈ G \ G , tX6~<}npY'gpU<Y8"}¨l@Y ttjfgM}vo j©k}'o¨j<gpUY og4eVnpj@orYo fUdYj@>ttgpniYo¨j< '"}v l@Yeª 6õ¡giU@Y "}v l@YCeV}vniY f\o a:YniYjLg¬dgpU@YCj niY ;<j<Y Y M ª Y ªadYg }vj<t8gitadgpY~ 4ª @ª z\gpU@YCnpkorepY¬@epYg l = L }vj<G t8gptBaLgiY~ l =ª 1 lª + 1 ª aLgpYC~ ª 1 7PadYg Ge =G aLgpYC~I4ª ^ 7 @ tn l = L , · · · , 2 ¬S¡£tn\YCYCnpc*YC¨ M ∈ Ge ∩ G ¬So¨¡PgpU<Y'epo egpYCnie\tv¡ M U<}Á4Y gpU@Y'ep}X8Y X6YC}j "}¨l@Y 9 6tn :Ø}j<o¡gpU<Y j@Yo Ud>tnio¨j@Y eVt¡

(90) giU@Y ¡£tl@nêorehgiYneU<}ÁY }vrept gpadU@YY+g ei}vX6YV"}v l@¬fY}¬j<gpU@YCjt8gpt4tB}aLniepgiYYj~ Mª ^ ª ko¨gpz\U6gioÜ@giYePniepkorehorgieY4Yn¬@eeª Yg Ykt4fgi¬@}}o¨j6j< }yj@t8Y gptBaLnio gi8Y~ fYCj@ªÆ ftvgiª YC Ge ª l =l−1 l=1 a LgpYC~ ªÆZ 7PadYg G := Ge }vjS f/Y ;<j@Y V tj G ª tnØY =@}vX6~@ Y¬4gpU@Yyt4j<ehginplSgpo tjt¡®gpU@Yy}@}~fgpYnior ¡£t ¨t"+ePgpU@Y^np/Y ;<j@YCX8YCjLgegpY~Se/Y =d~<öro¨gpY o @ j ;<l<npY 1 ªÆ¡£tn L = 3. TVU@Y"}v l@Y6¡£l@j<gpo tj&U@YCGnpY8g}v°Ye "}v l@Y*SYC¨t"gpU@Y@o eit4j4gio¨jdl@o¨ghc or}e+j<o j Á]0,}¨l<1[.YB {Ø>Y Y c4eyt@j<¬Dª ¬ 1 g }npgpY U@YBniY ep;<YCj<YCt*j<gitL¨tYCSYCYQtv}v¡ØlSnieY Y ;Sgij@U@YYX6o nyYjLÁ}g¬

(91) ¨l<Y'YC¨Bf0o a:ore YnniekY ;<¡£nij<tYCX }gpe U@Y'o¨gi"e }vX6 l@YCY}tvj¡Q"gi}U@Ÿl@o Yn . . 0. 0. 0,0. 0,1. max. 0,l. j. 0,l−1. j. j. 0,l+1. max. 0,Lmax. 0,Lmax. max. j. 0,1. 0. 0. 0. max. Õ

(92) Õ Ç

(93) e<ÔÓÓÙ. l. j. 0,l−1. j. 0. 0,l.

(94) . (U " /% [T! ! I VHK )'

(95) S K K

(96) . j<Yo UdSt4nPYC¨:fª(Z+tvgio YVgpU<}vgC¬d}"gP Y4Y 1 ¬ G o eØel<U8giU<}"gØ}¨SY rePt4j4g}vo j@o j@ gpU@Y^fo eit4jLgpo jLl<oghc }npYytv¡(X6o¨j<o¨X}v

(97) eo ?Y }eVY D}egiU@Yo n+j@YCo¨4UdSt4npo j@6Y reª t4¡ }gpYCniepnYnpj/Y ;<el@j@:YCX8YCYC¨rjLeVgCtv¬D¡PfY4¬ }v1 ni@ni¬ c©1 t8lf}vgj<}*gpU@tL}vYCj©neYCej@l@o¨:j<YCërekgpYCt~D¡ª @@ ¬t }t"j<k o¨j@1 ª gpU<Y6gpYCegtv¡ØgpU@YB}¨4tniogiU@X*¬®ØY 0,3. . . @ C 4 1 1. . 1. 1. ```. 11. ``` ```. . . f 1. ```. ```. ```. o¨4l@niGY 7Ø{Øtj<egpnil<gpo tjt¡ G ,GgpU@Y@o eit4j4gio¨jdl@o¨ghcBorek~@ tvgpgpGYC ko¨gpU«ftgieCª Z\t" ¬®¡£tn n ≥ 0 ¬DØY6U<}Á4YgiU@YB}<}v~fgiYCnio G 9 }"g gpo X6Y t :y}vj< gpU@Yjdl@X6YCnpor}Pet4¨lfgio¨t4j tj ª'TVU@Y6foreptjLgpo jdl@o¨ghc«}"g oŸe o¨j gpU@Y6niYo tj tv¡ kU@YniYgiU@YYC¨re }vniYtv¡X6o¨j@o X}v V enipYo ?4Yo¨t4ª jI¦}"GYg }vlSeYtv9 ¡ygpU@Y{ @ tj<fo¨gpto tj 9 1 :¬ØY*°Lj<t" gpUS}"GggpU<Y«fo eit4jLgpo jLl<oghc¢oreBegpo ¨\o¨j gpU<o e }e}¨niYC}4fc Y =f~@r}vo j@YCIgpU@Y©foreitjLgio¨jdl@o¨ghc¢ftdYCe8j@tg6YCt4¨4Y t¡\X6tniYgiU<}vj tj@Y Y

(98) t¡Øep0o ?CY t (∆X , ∆X ) @l@npo j@}Bgio¨X6Y6ehgiY~ :ª Yt4j< l<fYgpU<}vg V = V 9 Btn8" : k@tU<j@YYBj@YCtYnin npYMe~>tj<∈ Ggpt«orgpeÛ@j<Ytvg Y tv ¡eX6tv¡ko j@X6o X6o¨j@}o QXe}vo ?Y4e¬Do ?}vY j<«gpU@Y6tj<¨c«tX6~@lfª©g}"TVgpU@o tYjSeYBe^kC}vU<ro l@Ur}"npgiYCo¨t4X6j<}e o¨j«kgpo ¨t*ØSSYY , ∆X ) @tj@Y e'ko¨gpYUj@gpU@YYY¢[^j@¦ YCõo¨¤^4UL¥4>¦ tnieio j@U@«YCX8"}vY4 ª l@YCe'gitft } (∆X l@ }vgpo tj<etj } 4o¨4YjIY tv¡ ¬Pt4j@Y X}Ác tjSfYnØkU<}"gVo egpt8>Yyft4j@Y+¡£t4nY et¡DX81 o j@o X}v®eo ?Y^kogiUj@YCo¨4UdSt4nieØj@tvgtvM¡DgiU@Y ep}GX8Yyeo ?Y4ª @ tn /Y =@}vX6~@ YkkU<}"gQgitft ¡£tnØY >@ko¨D tj<¨lSfY gpU<}vg+gp9U<Y ¡£l@j<gpo tj v g}vj °4YC;<ek4YCl@niYY nicdkª U@ª Y6Ïnij6Y ¡¯}9gt4}4neP"} :Vt4tLj}vngpeU@Ykore\Y Y< /orªVeQj@¤+tvYCgØj<tvY4¡:¬<X8gio U@j@Yo XX6}vYCS}epjo ?"Y}¬4¨Øl@YY t4jgpU<YyY SoreV t4n^ :P}j<BftdYCepj gV@Y~>Yj<tjgiU@YyY ®eo ?KY 7

(99) Y\l<epY+giU@YyX8Y}vj Á}¨l<Y\tjBgpU@Y YC¨ }j<ft}vrl< }vgpo tj<ek}4eVo¡(og+V}et¡X6o j@o¨X}eo ?Y4ª D ² ´ ¶ f

(100) ¶ · ¶ C v

(101) ¶ ¶ G aLgpYC~fª 7Dx^t+}vj o¨gpYn}"gio¨t4j tv¡f[\¦ õ¤^¥4¦epU<YX6YQtjygpU<YPY re®tv¡fX6o¨j<o¨X}v4eo ?Y (∆X , ∆X ) tv¡ G . Y t4fgi}o¨j V tj G . ¬<}j< V := V tj G ª aLgpYC~«fª 7Qx\/Y ;<j@Y G := G tX6~<}npY6giU@YB"}v l@Y aLgpYC~fªÆZ 7 @ tn 1 ≤ l ≤ L − 1 ¬¡£t4n}v YC¨re M ∈ G ∩G, 4 t j k ¨ o p g U ¨ o i g ^ e @ j C Y ¨ o 4 L U > t i n o @ j B " v } @ l C Y C e ª T õ 6 ¡ p g @ U 8 Y " v } @ l C Y ^ e v } i n ' Y f o 0 > a C Y p n YCjLgC¬>gpU@YCj«niY ;<j@Y8YC¨ V M }"gginpo @lfgiY gptgpU@Y'@}vl@gpU@YCn^Y e\tv¡ gpU<Y'ei}vX6Y"}v l@Y V ª\TVU@ore\f/Y ;<j@YCey}j@YC M , nio G ªadYg l = l + 1 ¬@}j< t8gptBMaLgiY1 ~ fªÆfª z\gpU@YCnpkorepY¬@epYg l = L }vj< t8gptegpYC~©@ª ª ÕDÀh½p¹×Æ×Ì2ÖÁ¹iÌ

(102) â¨ÄËDÀhÊÀhË£ò È

(103) Å Ì/Ö ÈCÌ2ÖÀ Ý Àp¹Ç+Ê¹×Æ¿À ÅÆÂDÀ¾¿"¹×dÌ/Ä ÄË » `. `. 0,3. 0,1. @. `. 0. 0. n. n. n. n. n. n. n+1. j. 1 min n. n+1 j. 2 min. 1 min. 2 min. . j. . . n. n. n+1 . n+1. n. n+1,0. n. 1 min. n+1,0. max. n+1,l j. . n j. n+1. j. 2 min. n. n+1,l.

(104). l. j. j. j. n+1,l j. n+1,l+1. max. . Mj ∈ G n+1,l ∩ Gl . l < Lmax. Vjn+1,l. 0. 1. ð ã ÕDð æ.

(105) 4.

(106)

(107)

(108) !"$#!%&!'

(109) (*)'! %+!

(110) (,-%+!./(0/&. }j< Ve aLgpYC~«fª 1 7PadYg Ge . := G := V a Lgp^ YC~fª ^ 7 @ tn l = L , · · · , 2 ¬>ftB}BtneYCj@o¨j<6egpYC~*¡£t ¨t"ko j@8gpU<Y'ei}vX6Y o @YC}}e+o¨j aLgiY~ ª ª 6õ¡gpU@YCnpY oreVj@ttL}vneYCj@o¨j<git6@t<¬fgiU@Yj©epYg l = 1 ¬S}vj< t8gptBaLgiY~«fªÆ fª ªØTVU@o e\tniniYCep~St4j<@eØgpt6giU@Y}v~@~<npt=fo aLgpYC~ fªÆZ 7adYg G := Ge , }vjS V := Ve X}"gpo tj*t4j G t¡

(111) giU@Yept lfgpo tj vˆ tv¡ 9 K :}"g t = t ª ¦c tj<egpnil<gpo tjD¬dYyUS}ÁYygpU@Y ¡£t4¨ t"ko j@'YqLl@o¨"}ŸCj<Y niYCepl@"g 7 . ,& VHK/% K% )!

(112) -

(113) !

(114)

(115) -%+!K

(116) !

(117) ²d¶ ² !A#

(118) H L [ W X

(119) [ /)'!

(120)

(121)

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(123) I /

(124) I Z% )(M/!

(125) ( Z !

(126) . n+1,Lmax. n+1,Lmax. n+1,Lmax. n+1,Lmax. max. n+1. n+1,1. n+1. n+1,1. n+1. n+1. S

(127) %+/!

(128) ( Z !

(129) ,H . . 6Ïj. . . max. . [$W S X[ )/D!< %+VH Z(0%$H% . *4 V#(%Ø$¢ 4 $

(130) . GLmax. . giU@Y4ni}~@U@ore giU@nitl@4U }¨gpU@oreepYCgpo tjD¬(YBl<epYgpU<Y@r}° t tn¡£tnY ekogiU¢X6YC}vj&"}¨l@Y egpniorgp c*SYghØYCYj }j<&¬®kU<ogiY¡£t4nyY eyko¨gpU "}¨l@Y6}vj<© o ULgy4ni}Ác ¡£tn Y e^kogiU"}¨l@Y ª Y}vrept'l<epY gpU<Y j@tvg}"gpo tj B(c , r) ¡£tnVgpU<Y <}v DYCj4giYniYC o¨j c }vj< kogiU*ni}4fo¨lSe r. Y ;Sniegêgi }vnp² g+k o¨gpU }~@ni~<tnp~St4}v~<4}v}vLgp}"o j@gpo 'j@8¡£npt4¡£nij4tgkjL~@gie\nit~@<np¨t4YC@X YX*ª+TVU@Yo j@o¨gpor}v(tjSfogio¨t4j©o e\U@YniY ghtBet4l@nYCe ¡£nitX§kU@orUBF} ;<niY\ep~@npY}@eCª DYg ϕ SY\}¡£l@j<gpo tjgiU<}"gX6tffYCö ?YeQgpU<Y\@l@nijLgØniY4o¨t4j6}vg t = 0, }vj< @Y ;<j@Y*}Fe 7 o¡ x ∈ B(c , 0.1) ∪ B(c , 0.1), 0 tvgpU<Yniko epY , ϕ(x) = 1 k}4epoëpgptfU cor}"gi=YØgi(0.4, ª v X6YY 2.5] × [−1.5, 1.5] t gpU<o 0.4) e~@npt4}v@j< Y X c gpU@=Yk(0.6, ¡£l<j<gi0.6). o¨t4j vˆ DkYg U@KorU6@giY}j@°tYCgpe(YV"gp}U<Ÿ+l@fY^t Xo¨j}vo (t,j [−0.5, ¨ o ® ¡ p g @ U Y >} ¡£nitjLgkU<}4ek}v npY}fcniYC}4U@YC x }"g t, }vjS tgpU@YCnpkoreY4ª 6Ïj ¡¯}4g¬ vˆ ep}vgporYe x);<Ye∈gpU@[0,Y To]¨°4×tj<K}®YCqLl<}"gio¨t4jD¬ 9^ : vˆ (t, x) + ||∇ˆ v (t, x)|| + (−x , x ) .∇ˆ v (t, x) = 0, ∀t ∈ [0, T ], ∀x = (x , x ) ∈ K, vˆ(0, x) = ϕ(x), ∀x ∈ K. TVU<Y'foreptjLgpo jdl@o¨ghc t¡ npvYCˆ~@}vnpg\YegpYCo jLX6gieØY gptU@orY e+<giU@l@Y8npjL~Sg$tL?Ceto¨gpj@o Y tj«}"gVtgi¡Qo¨gpX6U@Y Y S}vX6Y¡£npt4j4gy}"g\gio¨X6Y t ª^¤\Yj<YgiU@Y'epYg {x ∈ K, vˆ(t, x) = 0} giU@tl<U&giU@o e'~@nit@ YX tX6YCe¡£npt4X ¡£9 nit^ jLg~<npt4~<}vL}"gpo tjt.¬

(131) o¨g'g}v°4YCe~< }4YBo j¢giU@YB¡£t4npX}v oreXØY egplSfc*ª 6Ïj<@YYCD¬fgpU@Y o¨°4tj<}>YqLl<}"gio¨t4j :VC}vjSY knio¨ggpYCj©}eV}vj©¤\¥L¦ Yq4lS}"gpo tj 7 0. . 0. 1. 1. 2. 2. 2. t. 1. t. 1. . . vˆt (t, x) − mina∈A f (x, a).∇ˆ v (t, x) = 0, ∀t ∈ [0, T ], ∀x ∈ K, vˆ(0, x) = ϕ(x), ∀x ∈ K,. kU<YniY^giU@YepYgktv¡Qt4j4ginpt4 eore A = [0, 2π], }j< gpU@Yfcdj<}X6o CeVo eV4o¨4YjLc-7. f (x, a) = (x2 − cos(a), −x1 − sin(a))t , ∀x ∈ R I 2 , ∀a ∈ A.. Õ

(132) Õ Ç

(133) e<ÔÓÓÙ. 2.

(134) Á. (U " /% [T! ! I VHK )'

(135) S K K

(136) . 6ÏjgpU@Y8jdl@X6Ynio C}vgpYehge¬®Y6fo einiYgio0?CY o¨jLgpt tjLginpt4 eC¬>}j<«Y6U@tdt4epY }4ekX}

(137) =fo X6} Y4YDt¡(niY;Sj@YX6YjLgª Ydorel<}Aö ?YgpU@YNtX6=~@8lfgiYC©ept lfgpo tj«}j<gpU<YYninpt4LnkkU@or=U©o 6e Y/@=@Y };<j@g+Yeptt lfj*gpo YCt}4j Uvˆ tYj* MYC¨ ¬fM¡£tn }vjgV∈gpo X8J ¬<Y Ltc .ε = |V − Ve | ªØ¤+YniY Ve o ekgpU@Y}Á4Yn}vY^"}v l@Y tv¡gpU@Y a. j. n. j. n j. n j. +. 1.500. max. n j. 1.071. 0.643. 0.643. 0.214. 0.214. -0.214. -0.214. -0.643. -0.643. -1.071. -1.071. @. -0.071. 9 }K:VtX6~@lfgiYCet4¨lfgio¨t4j 0.357. 0.786. 1.214. 1.643. 2.071. +. 1.500. 1.071. + -1.500 -0.500. n j. + -1.500 -0.500. 2.500. -0.071. 9 :VYninpt4n. 0.357. 0.786. 1.214. (εnj )j∈J. o¨4l@niY 1 7Ø{ØtX6~@lfgiYCet4¨l@gpo tj*}vj<YninitnV}"g\T @ª 4¬ YC¨re ^^ ¬ L +. 1.500. 1.071. 0.643. 0.643. 0.214. 0.214. -0.214. -0.214. -0.643. -0.643. -1.071. max. =6. 2.071. 2.500. ª. +. 1.500. 1.071. 1.643. -1.071. +. + -1.500 -0.500. 9 },:VtX6~@l@gpYCept lfgpo tj 9 :Yninpt4n (εn ) j j∈J o@ ¨4l@niY ^ Ø7 {ØtX6~@lfgiYCet4¨l@gpo tj*}vj<YninitnV}"g\T @ª QLf¬ YC¨re Q<Áf¬ Lmax = 6 ª Yforep9 ~@ }Ác ^ n}v~<U@o Ce}vg T = 0.11 9 <; l<npY 1 : kU@YCj gpU<Y«ght&¡£npt4jLgieBX6YCYgC¬^}vj< giU@YjJ}"g S; l@niY P: kU<YjØYyYgt4j@¨ct4j@Y^¡£npt4j4gVkU@orU oreV}v npY}fc8¡¯}nØ¡£nitX giU@Y et4l@nYCeØtv¡ ;<npY4ª T = 0.87 Z\tvgio Y gpU<}vg+gpU@YYninpt4norek tf}v o ? YCo j«}8gpU@o j©niYo tj*}vnitl<j<BgiU@YforeitjLgio¨jdl@o¨ghct¡<}j<fkordgpU©t¡ j<tX6t4npY6giU<}vj ghkorY8gpU@Y eo ? Y6tv¡k}*X6o¨j<o¨X}vØY 2ªTVU@ore o e gpU@YB}j4gio @o eieo ~<}vgpo Y6SYCU<}Ádo¨t4n tv¡gpU@Y eiU@YCX8Y4ªZ+tgporY6}vretgpU<}vg^giU@Y}v~@~@nitf= o¨X}vgpo tj«qLl<}v o¨ghc©o epj g foregptnpgpY«kU@YCj«giU@Yfo eit4jLgpo jLl<oghc -1.500 -0.500. -0.071. 0.357. 0.786. 1.214. 1.643. 2.071. 2.500. -0.071. 0.357. 0.786. 1.214. 1.643. 2.071. 2.500. YCt4¨4YCeo¨j gpo X6YªTVU<o eVore+}vj<tvgpU<Yn¡£YC}vgpl@niY tv¡gpU@Y}j4gio @o eieo ~<}vgpo YySYCU<}Ádo¨t4nCª. ð ã ÕDð æ.

(138) 1.

(139)

(140)

(141) !"$#!%&!'

(142) (*)'! %+!

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(155) . Yt4j<epo fYCngpU@YC}v~fgil@npY <}4eo jtv¡ C SY¡£tniY^gio¨X6Y T 7 {}~fg (T, C) := {x ∈ K, ∃t ∈ [0, T ], ∃(a, θ) ∈ L ([0, T ]; A), y (.) ∈ K, y (t) ∈ C}. 6õgVoreŸ}vnØgpU<}vg T 7→ {}~fg (T, C) oreØo j<npY}epo¨j@¡£tnVo¨jS l<eo tjª(]«tniYt"YCnC¬ØY }j~@nit"Y O `

(156) R:giU<}"g {}~fg (T, C) = {}v~fg (C). Yg+l<e+epYg lim o¡ }j< 1 tgpU@YCnpkoreY4¬ ϕ(x) = 0 x ∈ C, }j<tjSeorfYnVgiU@YepYg õ"}v l@YC X}v~*fY ;Sj@YCdc  o¨¡ x ∈ C,  0 o¨¡ Λ(x) =  [0, 1] o¨¡ x ∈ ∂C, {1} x ∈ K \ C. Yg v >Y gpU@Y "}¨l@Yy¡£l@jSgpo tj*t¡

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