No-gap Second-order Optimality Conditions for Optimal Control Problems with a Single State Constraint and Control
34
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. No-gap Second-order Optimality Conditions for Optimal Control Problems with a Single State Constraint and Control Frédéric Bonnans — Audrey Hermant. N° 5837 February 2006. N 0249-6399. ISRN INRIA/RR--5837--FR+ENG. Thème NUM. apport de recherche.
(3)
(4)
(5)
(6) !"#$ % '&)(*+ % , !+-/.01 + , !"# (*2 3 4#657 8#$
(7) 9"-;:< = ,>?@ $#$
(8) A B ,C ,
(9) D(*-E 3 $2 F (*44 , # GIHKJMLJNHPORQTSBUWVVYXZV[ \]+^ LHP_a`cb+_NHPdeX'VEf ghji!k lmonqIp1{ rt zl!yqsvsMuxwzy{i!k lKw |v}jk ~KM}jlKw 9aj' ayjllP%hjl!%%hjl|'YaavTr@jl!}auaNCr1aBaalPw ∗ †. ‡†. a| BK a !|j¡0¢jlq£awz ay%¤0ygl9hj! alB|w®y%'l!©|MyKx¯lP a°±¥wo{¦j§ªyy{h¨{a auxy{ N§{kjal!¥P²,! a³¨|Mloy{$ a N¥,|jw{©x ajlK¥©l!ykhjlqwq!¦aªwy{lh« N°W|j´¥§u¬|ªy{ Nl!|j¥©u2l2k! a|M|vyu2{ aZ¥ N0}j|{©jaaj¥©u l a%hZ{Kw%a=|y{«l!{§y{¼K alq}w%lKh¨! ay|©k v¸»lK awK%²+xlK³¨lµM} NNxxy{a{§|¶y·a|j{ a 0¸¹¦Myh±«² yhjlK a{uº °9wlK! a|v¸» a%xlKC! a|jªy{§ N|w!¯ ¥©¥§ 0¦©|jy ½¬j{¾M¿WyÀ{qÁ4N ¡0 0à ¦ay{h±¤ ¯vqlKax}jy¥©k z¥Y}j|Z$ N=|Ny©y{ a|Z Nw!¥R¯E² wlK! a|v¸» a%xlKo ajy©k a¥§§yzuT$ N|x§y© a|wK¯±w®y{0y{lB! a|w®y%©|MyK¯EM}0¸. ∗ † ‡. Ä©Õ×ÅÇÖ ÆzرÈÉÆzÕÚÅRÙÜÊÌË=ÛÍ ÎÝEÏ=Þ·ÐÉÐ0ß®Ñ%à=ÐÉáPÒ»ÛaÓ â%ãPÊÌÐÉß®ÅÇá=ÊÔäÑPÍ Ä©å Å ÆçæWèÉÆzÒRÐ0Ñ®é Û0ê ÅRÑ%ÐÉËzÆ Ñ%ëÉÈÉÅÇÆzé$Í èÉÆzÅÇì9Ñ%Ð!í»Ó ÊÌÐÉÅÇÊÔÑPÍ Ä©Å. Unité de recherche INRIA Rocquencourt.
(10) *( % , $-<z + , !"4#$ % Ü
(11) -,-3 =
(12) -<
(13) /- -,2 ,
(14) - -,
(15) Ü +
(16) « #$
(17) - 8·# 9"
(18) -<
(19) Ü4 "4+
(20) , K"#$
(21)
(22)
(23) 2 3#$
(24) -3#$4 =
(25)
(26)
(27) 2 3 $2 ,
(28) - # , ¤ |Zw2!l$yBa®y{©!¥§lN¯|j a}w4~$y{}x© a|w2}j|6j{ a¥§iKkBl+xl+! ak k|xl+ axy{§ka¥§l ÉNlK}| !w a a|M|Myyw®a}¥©l4jZw{! Maw®~P¥©aw§{l!l4|ºl!|y a}jk4|lBl2! a´|M|jy%¹²2©|Mmoy al2}ww}j aj¥ y ~$l!y%|0 ay| wC x alP%xw{$l4 aM|Z}jxl!ªy{¥§$ N N||wCM}j la a² x 3y{§lKkw§¥©|Z§ywz~y%x|M}«y{wwxlKlq! az| aº|$ ay%© ax|{lw |~K$lPww{©{lKwÜl$yow"} !µwa|Ny{lKwK¯jM}j±'l!{kBl!y®y{l!|Myoxl!a{N=y~Kwl!¥© ${ aw{wa|$lCM}ax%0y{©M}jlN² ¥%#* ~$y{ PyK ¯À !£ $ N © w{w{¤ |!Ülq aMk }kaj{|Zxyl9N a}jlay¯P©zk aa|¥§$lNy¯x© a$| a|Zw9xª~Ky{a§ N}j|¥©§wiK lP w! N² xy©k¥©ªy{~qx}w®lP$ N| a%x{la¯v$ a|My{{a§|Myl9w}j.
(29)
(30) !"#$ ! %'&( ! ) +*-,./ ) !"#(&(0 #12 435#6 *. . 2 3+ç , $ |¦ÜlK§ Ny$|hBlPww©w®w{x alK¸»{K{u¨aj¥© a¥§¥©lP2lwl5$}"9' N!µ N|j®$lÜy%©wl!!|M ahy |Éw®N aylx%y{'©§|Mkl!lKy=a|T<×¥§¹§²xyzlNu¶ a²3|y{$lhj al|Z§|·x$ª ay{y{|Zhj§ Nwzl|y{{w4aN§°×wz|M ayy, °×k N a9xay{j{§lKjk x©|j}©¼Ko!§k2y|© a}|6wzyy hjjZl{l aM©|2¥§·lKk '$l$w2 ayz|v¦©|;Nl!ll!:>|¬$w| alKN|!%la ah6¯É| avw® ¸»Z ak a%$x alP{lKw!l ¯ y{y{Nhjhjl!l!lq|j NlK!{}ju+a{¥§É¥© u·°Iy§w}j| lK{$lo N| a$°Ev a¸»y|v ahjN%lCl x>T$l! Nw®|vl!|ay lKl?$$> ²2lPww®gw{l!hjyK{¯xu wwl!°× N{lBxaAyk ©kÉl!¦¦a¥© aªw{ yzu@+@v$©D| NCF$EK|¥©x}G¹§x¯ylP©Ü awC a|kk wo©§|j|v|val$u¬y Ny¥§ aHNxlKC y{w§G¹k²I :ay{¥l!u{!k/$ a N|M|Ny%yy{{{ a@vN¥©wz|jjyPµ¯N{ aw©|NlK!y{¥§lK a k |avwKK¸»²4$ a N%g}jxhj|MlKly ³¨ww"}lK!lq!µ aw{$|É©xl!uB¸¹|M Nyy3{h NxxlKyy©4 akx|jy{ ¥©§¸»ªkyzNuaµ!¥©§ ayz$u2| ax|Z$§ Nxy|©ª ay{x§|Z N§ywE|µ©yz auvhj|j NwI¥©!j©wçwK¥©Z¯M¥©uo¦l!yz©hj¦Ü|Ml!NlK|l! a|¥©y{ahjlYl|jwz ay{ |jw®¥©=u}Zy%%hCh|yBl!|j{N|jKk l9 NJ|µ'w®l$lKwzyzy{lI¦lal!=² l!yI| ² ©|jpT|jlKlKM!}}lKaw{lK¥§w§a3yzuauB²H|PM a| L'yw® 0hZ"} ¦Ü0!µNlyÜ!CÌK§O lKaE|NwG¹ly ² §ay,lw±lK}Zw{w®w}l!|May¥§¥©uo¥©'¥§uC Nw{yzw¦§j C¥©lY!ay{w olK aw,j¦y{hjlK©||B|% h¸»Na{2N=$y alK|Z©x¼Kªy{y§© N a||w ¦Ü°vlKyhllÜ awjlKy{$ N©||jvlP¸»R aQ=%x<ÇS l!?,Ey{Nhjl 0¦Z Ny{h¥§uN$lK ax|Z{©xçªy{°×§% N|±k ²,l!g¦h al! {@Wl ¯ ©Tq|·y%hl4O} >K!NC Uw®lGWa¦|hjµlK|·p¬y©ahj|l T9By lKCÌaw{wU ©G¹a¯x|¬ j° ¥§ 3©lKNy{ 4a|j Naxy©|kw9¥' $ a ±|MlKy{a NlK|¥xj{{lq aj°× N¥©l!kk·wç¯Z§ a|+{§laN² §|²Yysv©4 |j@N a§¥©|T 0¦9ywhj@vlD ¦CÌN aV GE@µ|Zvu lP:aw| Nw\ |jl!|M|<×y{©©! |Za?¥§Ww y¥©u hCFEXl4$G¹¯x 0w®aNlP|l!$ N% wÜ|yvyhhj¸»lo al4%l Kx>val!y{wl!lC|{ l!xN°IlP}j$¥© Na'|N ay{§yz¥©uMu Nh¥±°×lK%$x N{k |©!wzl!y{ª¦yz{u2a a§ °×|M%@Wy%¯xw[k §|M<Çl!y{¦|Z x ax ´Z@4}|j!©lKªY| y{·l!C ¥©Z©u+\|¯xsvkC lK a$G |Mya©u a|| ´] |C a² j¥3G¹¯Z² wz ¦y%G)0JNªy{y{ylhºhj$wç a°×|Z{jwza¥§y{kB{!alK§y¦Ü|M© ay{ Nw| ?X@ w J y{w ¨lK! awl!|k x¸¹ Nx{xl!lK´ç|j§$y Nl·|x ax§yy{©§ ak |wÜ©¼K¦yl!©{ al9|± a²exy%³«©l|jlKµl!°×l!MBuyL'^ lK©C ZG|µ°× a© |_CÌ| UGW 0°×a NlKçvk ©l!$>x¦;lP °o$ Ny|MhjylP{w® al¥§¸»yw®hy{l! ayl9{©$lK NwK²6|w®ym9% ¸»©|NNy%aw! ¯ }Zw®©|j` lK$|j Nl!|0%z}j¥©¥§Nu yw®lC''lK4 a@v©|N§|jyZy{¯Ühjl! N Nuµj¥§alK|k·w ¦ª!y{0h yy|jE alP|N}Z'0 Ny©w aªy{|Z§w!v² §yzue$ N|w®y%©|Ny%wB§| wa!lKw a°C! a|My©|v}j a}w °×K}a|wl·=y{§¦ NN|w2w2 Njx Ty{a|j§ |y2lK ´jy vau§|MyA TÉy¦hjNlPww®4 l @v°×B {aCFEPkBvlKb¯ ¦ÜE5 NZGo@x©wK| ²Tyghjhjlºlµ al5|j>vl¬xlP©k wwl!§ N||¶w§ N |°y{¥9hjl+Ka!w}jla¯0|y}je{lalKy|jlKl!%k ¥©©©¼!|¶lKeyh©©| w Ü Nk §|jl!y®y{3|l!| cy CÔd G»-² T9 0¦l!Nl!P¯M N|j¥©u+w"} !µ$©l!|Myo$ N|x§y© a|w¦§yh a}xyo!}j0y}j{lyl!{kw¦l!{l @v|j 0¦|3²«g,¦Ü «l>j$lKxy© a|w2{l+TM}j§yl·wZlP$§´Z wªy{}0y{§ N| w®y}x©lK §|eC G[<Ǧªy{heajj¥©©K0y© a|Zwy wK ak |¬l'll!©a{lKlKx|M}0a!lK¥§}j·l¥© xj!{ aj¥©¥©¥©u l!ky{ w ?=´¯j|ja§|ylK¥§uy{hjkl!|vNu+w®l©|j lP°±N´Z}Z|jª¥©y{ªl!yz¥©u·u $k N|wz|vy{uµ{a!§ a|M|My%w!y{¯N=wyl!l4Zla N²§|MN²y{wKT9¯x¦l!y®hjylK%| hºy{ahj|l' jfa{ a Nj|j¥©al!lKk | C gUG»² y{y{l!hj{l!k1 Nq}ju+©|M y º°Wkz}j©|| ´$|y{©ªlK ay{wlµ|º}j¥§|v!y}j a©k4|w2j'yªl!y{h§l+ N| a°×°9w9 N¥§$©¥©|Y 0 a¦|MfNy{©N|j$©j Nh²}xy{w®:§ N N|·u6|wl$«yq ¥§°a x¥R'!²ia aCF¥§}jEP©¼K|dGYy© a{|u«| ¬pº%{!aw}j}jk a{l!|l!c¨|MCFyPEVy¯Y Ng¦ÜG¹}¯'%l·¦Üh w®lZa¥§ N§y §l2|My%yw!hjj²·l¬¥©lno$}yw §|0j¬0y{ 0y}jhjN{lll y{y{hl!{0k yP¯ç©}jw4|Z¼!xl!l!{ ²«M}jj§y al·2¦Üy NlP} %@6h¶N'w aw}j©|Nk y%w!x¯Yy©¦ alµ|Zw!}¯Ywy{l+hjl¬º$ NlPx|N}y{$©yj©}x a|6y{§ N| {a a}°okBZlK N|M}jy|°×j aa4u6yhja M{w®Kl wyy{h ¨yByhjl¬{lµ$}lPww0l!0|My{y}j{l ¥ lP<ÇRw²°×wla}jl!²o||Myy{=h©y{a§ N¥jyq|µ a'|j©|l!lP¥©w! a²|jg hjyl ²I Ny|jsvhj¥§©lBu |$w®xlq}jlKª¥§yZ!© Nwçy®|jly ' y°Y ax©y|Nh!µyÜl4©!wIk4}jy¥ª} yÜ¥ªy{y$§ B Njk ¥©wz©y%l!j0X}xy?9lqyl9w®|Z"} y¬hj!µl9¦Ü!5l §l4lK>x|Nk y4$|@a awl2|Z© ax|j| ª µy{ § N°Whv|uvywYhZl9y% a yk hj@v©lP§|j|jw®lP©k4w©|N}°× ay{k 4qy0ahKl4$¥©}j N|jl9}j N|M a| y° lP%hZwwl!%|May{=©y{al!¥,{{§¼KlKlqx}ZMW$}©Njx¥§l4{yy N}o%Nhº 0'¦ ay{©h+|My{©wK|¬¯'¦ lB$ a a|vxNy%l!|j©|T©l!|M§|«yyzy¦Üh© aw¸¹|j¦ NÉu+k|jw aw¸¹l$My®y{º§|! a² |jªy{§ N|w!¯'y{h0yC§|¨ajjªy{§ N| y{'§ a N©||MgwYy{hjwE alB a°Z°xy{hjMal9©Zl!¦lKa aZ°xwlKaç a{w®N{yla%|ja§|µ=§¼KyEwlK NlKTx$yyN©©w9 ak |°×§ NwI¼P¥§00¥© 0y¸ ©¦9 awK|¦[² hjaP|lK|«Ülwzy{lK}jhj=|y{l9$§ Ny|j©|¨ a a¸¹|vMw ¯E$ ¦ NlB|w®lPx$§lP ay!©|Z av|¥©¥,¸¹w± Nya{hjj|l l!Yk¥©ux$0w® Ny{|wKl!²±x{©g§ay©hj¥, alI|j| klK©lKwIxlK©lP|w®Ey{$¯E N©|N|«¥§y{w®'h© jlKy{}xEh ¸ ² sxlK=y{§ N| w®y{ylKwy{hjlwlK$ N|v¸» a%xl!|jlP$lKw{w{{uµ$ a|Zxªy{§ N
(31) | <R$ NkB}xy{y© a|· °,yhjl!}j0y}j{lyl!{.k ?=² 7. 8. 2,∞. Ø,Ø Ðj á=ã=ä$â.
(32) Z. B0 * 5X + . sxwlKj=y{{§lK Nw|Yl!|MZµylPh a|©|·x¥© alK%woxyl!hjlBy{ µw®lPx$lK Na|¥Ev¦¸» aª%y{xh+l!y{9hjwl}"!µ|j N$|x©l!¸¹¼K|Ml!y{ 2! aZ|jªyy{§ a N°3|±y{²WhjP l|«$w}jlK{$0y0© ay{}j|ºlv¯Ey{ l!{klPx² }=y{§ N|Tj Ma%h
(33) 4"
(34) :* ³¨! al|My${ N a|¥)Qw®xlK3y{hjl°× a¥©¥§ 0¦©|jo Nxy©k¥x$ N|My{ a¥Mj Nj¥©l!k¦§yh qw{!a¥©a w®y{yl$ a|Zwzy{{a§|My a|Bqw{!a¥©a (P). min. Z. < Ed? <R4? <Ç?. T. `(u(t), y(t))dt + φ(y(T )). !w ² yK² y(t) j² la² t ∈ [0, T ] ; y(0) = y ˙ = f (u(t), y(t)) gy{hjhllqxjuv|y{akB a°±!yw hlqj{ a¥§lKk aloyhjlC¯jg(y(t)) yxhwzly{wz©y%j0}xy≤y{llK0! a!| Nw®w®yy%∀t`©|M:∈y R[0,×TR]. → R ¯M¯jyyhhlolC´´|Z|¥W¥±$y N©kBw®y l φ : R →¯Z|R ¯ y{hjl©|jªy{©a¥3$ N|xf§y:©R a| ×yR∈ →R R²Y³«lk@NlyhjlC°× N¥§¥© 0¦§|BgNw:wR}jk x→y© aR|Zw a|+yhjljy{+Q T > 0 g|h+l4hkÉNloj¥© v§K|jM¥©w¥§u ` 3¯ φ§Z¯ wf%hj§y|¼C $ Ng|My©{|Ml }k a¸¹}yw©kBw®lPlP$wq N!| av|M¸»y a©%|vx}jl! a}Zxw®¥©l!u+{§0x6 9Wy©l!a{lKl!wK|M¯xy|jµ¥©l yhj(Clx)uv|¦aªk y{h ©Kkw ≥ ©2w f ±©w{%hjªy{¼$ N|Ny{§|v}j N}wK² ghl§|j§y¥,$ a|Zxªy{§ N|w{0y{©w®´lPw g(y ) < 0 ² ghj{ a}ahj N}xyyhjlaZlKK¯x§y9waw{w®}jk lPµy{h0yoNww}jk xy© a]| <q ?hj N¥©w!² u,y. 0. 0. n. n. 0. n. n. n. n. k. 0. . "!$#&%('*)+#-,/.0#1 231546'#15798. a 1 ≤ p ≤ ∞ ¯ L (0, T ) xl!| ylPwyhjlc:a|a%hw®N$lC °,°×}j|$y© a|w9w}%h+yhy ! Z °× N p < ∞; kuk := supess |u(t)| < ∞, <∞ |u(t)| dt kuk := aw|N $l¬W °4(0,! a|MTy©)|v}jj al!}Z|jw ay°×lK}jw|y{$hjyl© asv| aw+' 0 aa¥©l!lKµ wN$l w °±°×j}jl!|Z|j= ayy© alK|Z wvhu Év§|j 2¦Ü¯lP¦ @µªy{hxlKyh©0lº0y|j©a Nlqk §| L ²Ighjl [0, T ] C[0, T ] kxk = ² sup |x(t)| ql!|j aylvu |ºlKL¥§lKkB(0,lK|My T ; R) <×{lKw±² Y :=w{W0yw®°×uM©(0,|j+Tyhj; Rlw®)y{? yyl hjl9lKM!} a|Myy©{ a a|;D¥ <Ç<¹lP?=w®² 3² ` w®§y{Nl!y| dl ?wN$la¯² );:5 5 X%¬©wUa:= 3jl!§Z|jw a%yhjlµ§yv¼µu gyhl! a∈{l!Yk·¯3yhyhjl w<×}k|j©(u,M}jjUl y)§?|¬w∈ a©¥©w4}xUy¦Ü©× alK| ¥§¥§Y¸»j c°l$´<R|j lP?=²º no||¨xl! °9a!w{¥©Nw}jww k xy©² a| ³¨<<lµok?$ɯYuºvuº¦{y{ªhjy{l·lµj{ua a}j∈%¥©hvl!UuNk ¸ C awVQ p. 1/p. T. p. p. ∞. 0. 1,p. p. ∞. ∞. 1,∞. n. u. k. (P). min J(u) ; u∈U. G(u) ∈ K. < Z?. Õ×Ö Ø±ÕÚÙ.
(35) .
(36) !"#$ ! %'&( ! ) +*-,./ ) !"#(&(0 #12 435#6 *. ¦hl!{l. J :U →R. a|. G : U → C[0, T ]. J(u) =. Z. alCxl!´|jlK±¯jlPw®'lK$y©alK¥§uN¯xMu. T. `(u(t), yu (t))dt + φ(yu (T )). ;. G(u) = g(yu ).. ¦g§hylKh·³¨wl20lBk¥©w{}jÉlPu+wjy§§h| |NRy w² {l C wq² T9l!<צÜ{l lPK@ ?=¥© xC!a¥,[0, NxTy©]k©wq¥yw®h Nl ¥§}xwy{l$§yC N|T ° $°b a<|MZy{?9§|vy}jhZ N0}yw9w°×}y|wz=´y{lP§ Nw|y{wChjl 0alK+ 4[0,T ) ] ¯ 4I _ ! u¯§° ∈yUhjlKlCl>xwzy%w α > 0 a| ρ > 0 w}%h y{h0ydQ °× a9a¥§¥ u˜ ∈ B (u, ρ), <R4 ? J(˜ u) ≥ J(u) + α k˜ u − uk ¦©|vhal! N{¥§Nl lKBwçyz¦Ü(u, |jρ) NxklKwK|j¯ y{lKw,yhjl a'°×l! a|2y{hj¥©l¥M§|j| l!L©ahv'(0, a{hjT v) x¦Eª¯xy{hB|! l!|MylK u |Z°× a%axy{hj©}lw aρ{², 0¦gyhh©w$$ a N|Z|xxªy{§y§© N a|±| ² ghjl9¯vwwNx$l!l|j a°WylP{ 0¦ vuLalP=(0,y{ a%w,T)wIxalK||jµ y{xlKBl!|Mvy{uª´RlP¦²Yªy{gh hj°×l9}jw|Z=ay!© alL|Z aw°W(0, o°,aT'x a) N}|B|k xlKlP+aw0}j{lP©w!¯Ny©x a}|µ0¥Zaw®|j©Nw$hjl©|jy C[0, T ] y{hjy2l ¼Kl!'{ a ¥²ºBg$hj Nl|jlµ$ N |j° lµ ²«°9M[0, |g ahj|jl|jlKxTN}]ay¥§©§ayzlµu«k lK xNxw®}}$lPy2w 0©NwBl!xlK|j y{lK¶vu M [0, T]w4ax|lK¶|j y{w4lK¶lPMv}u ¥çy K ¯ K M[0, T ] × C[0, T ] hη, xi = R É z a © M | y a ' ! l % 0 y N { w Ç < a | y % | w Z M ® w l © |. ? { l j ! l j | a y K l e v u z w % y ² ©j|l!w®{y{§x(t)dη(t). a0|0y{$§l NlKw2 ° f ¯Il$y%² ¦² P² yP²e¯ {a}kBlK|My{w u ∈ R ¯ y ∈R¯xRl$y{a¯Ü² al·xl!| ylP vu6¨w®}w!©xyP~P¯Y%hj°× al$y ql$´|lf y(u,hjly)!¥©N=wwD!¥ f (u,|ZM y)!# f+ (u,«y)|Z =/D54f4(u, MTy)uQ °×}|=y{§ N|w9 a°,j{ aj¥©l!k (P) ¯lPw®'lK$y©al!¥©u H :R×R ×R →R L : U × M[0, T ] → R <Ç ? H(u, y, p) := `(u, y) + pf (u, y) ; L(u, η) := J(u) + hη, G(u)i . ql!| yl+vu ¯v¥©l$BVyy{hj(0,l$T M)wzy%y{0hjyl+l wN$l 'lC a°yhj°×}jl|}j$|y©©M a}j|lw2w a°9¥©}x'y a©}j a||Zx©lK| Éa0y{§ N|±² ` ©a lK° |Q u ∈ U | η ∈ M [0, T ] p BV (0, T ; R ) <¹ ? − dp = (` (u, y ) + p f (u, y ))dt + g (y )dη ; p (T ) = φ (y (T )). ` ©alK| v ∈ U ¯j¥©l$yy{hjl¥©§|jlP{§¼KlK+wzy%0yl z ∈ Y 'lw® N¥§}jy© a| ° Q <Ç ? z˙ = f (u, y )z + f (u, y )v ; z (0) = 0. ghlgkhjl2aj|jjl ©>v|jy ¥§lKUkB→k Ya¯ ©valK7→wyzhjl2l >x©wj{y{lKhjw{lw®©j a{|~Kw9%hj l!°çy9xxlKl!{©0§00y©ya©alPlCw °,° y ,hjlN{ka|jaj§|±|¯x¦u§y7→h«ylPw®'0lKy9$'y9 ay© |My yhjul ² ©k j¥©©!ªyzu¬ a°Ü|j ay{0y{§ N|±¯E² ¦lB¦{§ylB§|«yhjlwlKM}jlK¥ D H (u, y, p)(v, z) ©|wzy{lKN a! a° |MDy{ a¥¹²4 aw®H(u, y, p)((v, z), (v, z)) ¢ 5 . ܾ 6
(37)
(38) + u 7→ L(u, η) S* ,5#6* * C 5 U I ! (*X B *5 4 5X ) U *Bη ∈ 5]M35% [0,,T ]#!# M ! ) ! !" M O 5+ 0. k. −. −. 2 2. ∞. ∞. ∞. ∞. 2. n∗. −. +. T 0. n. ∗. n. u. u. n. 2 uu. uu. n∗. +. n∗. u,η. u,η. y. u. u,η y. u. y. u. u,η. y. u. u,v. u,v. y. u. u,v. u. u. u,v. u,v. u. 2. 2 (u,y),(u,y). . +. v∈U. Du L(u, η)v =. Ø,Ø Ðj á=ã=ä$â. 2. 2. . Z. (u,y)2. T. Hu (u, yu , pu,η )vdt, 0. < g?.
(39) . B0 * 5X + . 2 Duu L(u, η)(v, v). Z. T. D2 H(u,y)2 (u, yu , pu,η )(v, zu,v )2 dt Z T ∗ ∗ + zu,v (T ) φyy (yu (T ))zu,v (T ) + zu,v gyy (yu )zu,v dη,. + . =. 0. <EK?. S*b4 U 5
(40) 35% z p *5# O Y * + ) # % , H u,v u,η 1 V, qsx§|!l u 7→ y w C 2 ¯yhjl¨a}%hvuM¸ 3§w{%hj§y¼¬ghjlK a{l!kDlK|w}jlPw yhlTl>xwzy{l!|!l¬ °Cyhjl wlK! a|x¸¹ N{xlKÜl5x> |w©u a| a°,y{hjlw®y{0y{l <E4Ed? 1 yu+v = yu + zu,v + zu,vv + o kvk2∞ . 2 P»yq©wlPaw§¥©u wl!lK|±¯Zw}jw®y§y}xy{§|_< EEV? ©|Ny{ yhjl2wzy%0yllPM}0y{§ N|·a| a@ lKl!j©|j yhlCyl!{kw9 a°wlK! a|v¸ N{jl!P¯Ny{h0y zu,vv ww a¥©}xy© a| ° Q <PE 4? z˙u,vv = fy (u, yu )zu,vv + D2 f(u,y) (u, yu )(v, zu,v )2 ; zu,vv (0) = 0. nqaw®N©}j|jk o!l! N|Mw®y{y{w?XyQ lYlKM}y© a|.»< E? a|C¥§©|jlP{§¼KlKw®y{0y{lIlKM}y© a|w-Ç< 3? |"<PE $? ¯É¦ÜlINl$y3lPaw§¥©u ×< ak §y®y{§|j 0. 2. Du L(u, η)v. −. =. 2 Duu L(u, η)(v, v). =. Z. Z. T 0. (dpu,η zu,v + pu,η z˙u,v dt) + φy (yu (T ))zu,v (T ) Z T Hu vdt; + 0. T. 2. 2. D H(u,y)2 (v, zu,v ) dt + zu,v (T )∗ φyy (yu (T ))zu,v (T ) Z T ∗ + zu,v gyy (yu )zu,v dη 0 Z T − (dpu,η zu,vv + pu,η z˙u,vv dt) + φy (yu (T ))zu,vv (T ). 0. l5g,>x Bj alPxwy%w©© a|
(41) |w < g4?|¦ª<y{EPh 4?ªy9wa}"|!µ$lP¦w!ª¯xy{h©|+v©l!¦ ¯j a{°lKw 3Zl!lPk =y{k§Nl!¥©4ua² ² j¯vy §|My{l!a%0y{lqvua®y%w§|+yhjlaZ 0Nl 0. pu,η. zu,v. zu,vv. ¡0! ¡Éà ¾ ¡
(42) ¾ £ ¾ PP¢x¡ ¿ 2Â Ã ÇÂ a ¯0xl!´|jlÜyhjl (*X /4 5W+ 45 *5 5 I(x) := {t ∈ [0, T ] ; x(t) = 0} ²Ighjll5>xjxlPw∈w© aK| a=°,Cy{hj−l"(0, T) 5+ |
(43) X # ! a|jlPw[<ש|·yhjl wR< w®l!lK|ZlCw®lBla² °²N$C Z aG!|v?Nal|>T|a©aa¥§lKux|±w©¯Mw°×? ay{ K yC'<Úy ah©lK|Mwy lxw®¯3l!y{{wlKw'Zl!lP©|j=y{ §Nl!l!k ¥©u xyzTu+Kª(x)° | ?=N¯xvKu(x)Q ¯,al2¦ÜlK¥§¥§¸ @M| 0¦| x∈K x∈ /K . a | I(x)}, N (x) = {η ∈ M [0, T ] ; supp(η) ⊂ I(x)}. Tol!{lvu supp(η) ¦lxl!| ylCy{hjl"* d 4 +X a° η ∈ M[0, T ] ¯xR² la²çyhjl! ak j¥©l!k l!|My9©| [0, T ] °y{hjl ¥0a{|ja©lPwwzhjy©|j N ZlK a|6| w®l!y W ⊂ [0,² T ] y{h0y2w{0y{©w®´lPw5Q R x(t)dη(t) = 0 ¯,°× a4a¥§¥I°×}j|=y{§ N|w x ∈ C[0, T ] TK (x) K. [0, T ] \ W. = {h ∈ C[0, T ] ; h(t) ≤ 0 +. T 0. Õ×Ö Ø±ÕÚÙ.
(44) .
(45) !"#$ ! %'&( ! ) +*-,./ ) !"#(&(0 #12 435#6 *. °× N¥§¥© 0 3¦l$y©|ju4∈´Z{Uw®y9²o a³«%xlBl!wÉ|ju lP$ylPhww{y {ηu ∈ axMy{§k[0,¥©§yzTu+]$ Nwo|x §y/© a54|hj a¥j MwVQ +. O# ) # 5µaw{w v!©ylP+¦§yh. ghlCw Nl$jy9©| a°w® N ,| wa%$ a||Zawzly{{k4a§}j|M¥§yy©Nj}Z¥©§lK¥©{ªw´ZKN0wy{w§ x N$|
(46) 0<Ryw®lPlKµlM¦ÇZZyh ¯audG ?Üw9°×x al!| y NlPj ¥§lKvk u Λ(u) ² < Z ?©wNw°× a¥©¥§ 0¦9wVQ Du L(u, η) = DJ(u) + DG(u)∗ η = 0 ;. Tol!{l. η ∈ NK (G(u)).. u. §°Yyhjl <EK ? <E5Z?. xl>vl!|y yyhjlPlKw ay{hl!lk }j|jw §y ¦Ü<×lK a¥§'¥§¸ l!@M||? 0¦|e¥©¥E<Ç w°l!l·C[0,lN² ZT²e] ²C ZG»¯ 3l!k k j² gN«a| ghjl! NlKkDj² g?$² mo yl B g j h ¬ l | y{h0y°× a v² ∈ U ¯3¦ÜlBhÉal DG(u)v = g (y )z ¯E¹² lN²§¯ (DG(u)v)(t) = g (y (t))z (t) ¯W°× a¥©¥ t ∈ [0, T ] ¾v ¡ ¾ <ÇS ?" 45 5X d ) h , " S* 5 M S*X * v ∈ U; g (y (t))z (t) < 0 ,#!# t ∈ I(g(y )). <EP4 ? <ǧ! ? 5 u 3 .#6d#/*5# O ) ] , * S* ,% !
(47) + I ! u S*U**5V5 K.\ b )% 3 O "*5 , /54 O# # 5* . ∃ ε > 0,. εBC ⊂ G(u) + DG(u)U − K.. C. y. u. y. u. u,v. y. u. u,v. u,v. u. ¾ £  à ¡0à ¾ ¡µ ¢ $ ¿ ± l!yyhjl5X ! ) #H 'lxl$´Z|jlKvuQ . . a h ∈ T (x) ¯xyhjl K. . *5 4 5+ 45 (*5 5 Üw9xl!´|jlP+vuQ. <EK?. C(u) = {v ∈ U ; DG(u)v ∈ TK (G(u)) ; DJ(u)v ≤ 0}.. <EÉ? $9 DJ(u)v = 0. P»°N< EP?hj N¥©w!¯xy{hjl!| DJ(u)v ≥ 0 °× No¥©¥ v w}%hyhy DG(u)v ∈ T (G(u)) a| K o ² v s © | $ l Z h a 9 w w j } j ' a o y © | E ¯ Z | N| I(G(u)) ¯W¦l axy%©| η ⊥ DG(u)v η≥0 I(G(u)) DG(u)v ≤ 0 y{hjlC°× a¥©¥© 0¦§|jKÇ< !¥©Nww!¥!?wzy%0y{l!k l!|MyVQ. ܾ 6¢ / (u, η) *5 ) S* ,%. + -*X 2 4 5[ X**5X% ! ) 5 <KE ? C(u) = {v ∈ U; DG(u)v ∈ TK (G(u)); supp(η) ⊂ I 2 (G(u), DG(u)v)}. jl$´g|jlPhjlMM !+uQ 2 |
(48) O M*5 [ 5+ *5 5 =* ¯x{lKwZlP=y©alK¥§u TK2,i(x, h) | TK2 (x, h) ¯jal I 2 (x, h) = {t ∈ I(x) ; h(t) = 0}.. 2,i TK (x, h) := {w ∈ C[0, T ]; dist(x + εh + 12 ε2 w, K) = o(ε2 ), ε ≥ 0}, 2 TK (x, h) := {w ∈ C[0, T ]; ∃εn ↓ 0, dist(x + εn h + 12 ε2n w, K) = o(ε2n )}.. ³¨w{@vloI{lKCFEPKv¯R¥©¥EVy{ZhjG=lq<Rw®%lKhZl%a¥w®= y{l!CÌ{U§G!¼P?0QYy{§ª° N|B a°Eyhjl©||j |jl!wlK$ N|v¸» a¯%yxhjl!lKI| y{|al!|MyÜwl$y T (x, h) x}loy{ A¦0¸ x∈K h ∈ T (x) N| [0, T ]}, <EVg? T (x, h) = {w ∈ C[0, T ] ; w(t) ≤ ς (t) 2,i K. K. 2,i K. Ø,Ø Ðj á=ã=ä$â. x,h.
(49) . B0 * 5X + . ¦hl!{l. wa©alK|+vuQ. ª° t ∈ (int I(x)) ∩ I (x, h) <¹? ª° t ∈ ∂I(x) ∩ I (x, h) (h(t) ) ς (t) = liminf 2x(t) yhl!{¦©wl . +∞ ¯aa| int S a| ³¨∂SlhxlKÉ|jal y{llPw®'lK$y©alK°×¥§u a y{hjl9©|Myl!{© aça|B|·Z N§yo}j|w9j|ja au y 0} aTo°Il!w®{l!l y h(t) ²S svl$:=y Tmax{h(t), ∩ I (x, h). ς (τ ) ≤ 0 τ ∈ T (x, h) 6j9 !µ$}¥ªy2y «%hlK(x,@«°×y ah)hy :=¥©¥ t ∂I(x) 4 w © ¥ 0 ¦ ! l 4 w ! l. k § » ¸ ! a M | y © v | j } a } K w « ² Ü a Z | ® w lPN©}|l!$|M{lKy¥©Nuaw®¯ ©|jT·w®lP(x,M}jh)l!|!6=l2 a∅° 7→ ς (t) ² P º | { y h 0 y ! N ® w N l ¯ © w { y j h. l j } j ' ! l © ¥ § . k § y ° ς ! a|Mςy©|v}j(t) a}Zwç>°×}−∞ |=y{§ N|w t² ` ©al!| ¯v¦lkÉu jl$´|jl.<Rw®lKlCla² ²NCFEPdG ?XQ ςx,h : [0, T ] → R. 2. 0 . x,h. +. 2. 2. t0 →t ; x(t)<0. +. 2. x,h. 2,i K. x,h. x,h. x,h. η ∈ M+ [0, T ] (Z. (ςn ). Z. ghl!|DQ. T. T. ςx,h (t)dη(t) := sup 0. 0. ¦hwl!{ylwz´Zσ(η, lKw S) = sup. ς(t)dη(t); ς ≤ ςx,h. 2,i σ(η, TK (x, h)) =. Z. ). T. ςx,h (t)dη(t),. ∈ R ∪ {+∞}.. <¹OEd?. xhjl!l!|j| aylPwyhl2w}jj' ay9°×}j|$y© a|¬ °y{hjl2wl$y S ²NP»° y{hjl2w}jj' ayo a° hη, wi j ¯ { y η supp(η) ⊂ I (x, h) <¹a4 ? σ(η, T (x, h)) ≤ 0. wlK! a|v¸» a%xlKÜ|jlP$lPww{{uµ$ N|x§y© a|·x}jlCy. Aɦaw{ @v/ C6EPU G35w Q ¾v ¡ ¾ / u 3 K#6d#= ) ! #I*5# O ) ;, *5 S* ,% ! 5 ,"#!# v ∈ + -,#!#6I ! #FU* C(u) n o <¹ ? sup D L(u, η)(v, v) − σ(η, T (G(u), DG(u)v)) ≥ 0. +¾ 6¢x¡ ghjl' 0alCwlK$ N|v¸» a%xl!|jlP$lPww{{uµ$ N|x§y© a|·¦Nw§k j{ 0alP vuÜ Nk §|jl!y®y{3©| C UG»¯jvu w®y{0y{§| yhy°× N9¥©¥3$ N|val> wl$y S ⊂ T (G(u), DG(u)v) ¯ . <¹ Z? sup D L(u, η)(v, v) − σ(η, S ) ≥ 0. g! ah3|²Ywjx² l!{©lKwܺ a§x|¨y%y©h|jlBlKB°×lP Nw®IlKy{|NhjylqZaZlKyK$¯3}j¦Ü¥l çN%ah§ a| $wl9}" a!µ° $S©l!|My=§|x°×T Nk(G(u), ²Ij NIyhlqj{ a¥§lKk DG(u)v) y © a « | × ° N k <R?M<Çwl!lI{ a' Nwªy{§ N| 6² Ed?=² 0. w∈S 2. 2,i K. η∈Λ(u). 2,i K. 2 uu. . u,v. η∈Λ(u). 2 K. 2 uu. u,v. u,v. 2,i K. Õ×Ö Ø±ÕÚÙ.
(50)
(51) !"#$ ! %'&( ! ) +*-,./ ) !"#(&(0 #12 435#6 *. g. 08 )+#15798 798015# 1 798 80' !&1 ! 3O4X% <. <. ¨³©|MyllK´0%wz¥Zy {°±lK'! Naw¥§¥9ªy{w§N al9k k l·lK$N¥w®a}jw{w®{l !a¥xl$´|ªy{§ Nw|}w!%² h y{h0y ×{lKw<×±{²lKw± !+² X ?'·?Yw°× a¨k ¥©¥ >x§k¥² =0 P»° [τ , τ ] ©w ' a}|j{u aI{a⊂¯ τ[0, Ta]| τ al·g(y(t)) ! a § ¥ © ¥ K l 5+ )X%6| g(y(t)) 4 ! B' a<©|N0yP¯lPw®'lK$yt©a∈l!¥©Iua² ç|Ny{uµa|+l>xªy' a©|Ny%w{lw{y{ 2'l O#F§°3y{hjl!u alql!|jZ N§|My °| ©|MylK© a%² + y{'¦ a ah}©l!|M%|h¬y τy'hj a§l9| ©|Mz(0,y{}jw|=Ty{{§)l4 N|!wÜwaa¥§'¥©|µlK a ©©w|M: ay{O¥w90y{a5lK µl !{ l! a* N|M}j +y{N¥©a= !K+yI² Z* P N|<ǧ a|My{y hj©<Ç wol! !" |jXZ*U'?= Nl!²o§P|M¯j³¨y N l2|j°±¥§w{uµyzɦu+yhj yl4h©|NKy{ayol!w{ylC§h N loÜ°za´}j{|j|ZKw §=y?$ylK²
(52) ©¥§ au+ ç|Z|Nk wqy{aauN|vl¯vu l{>x{l!l!ªNya}j}¥¥©|aP ¯ z}|=gy{hj§ Nlq|´w%wzwy¸¹x NlK{aj¥ªl!y9Y¦y{§§yk h±lC² xl!{§0y©al9 a°±yhjlwzy%0y{lq$ N|w®y%©|Ny¦hjl!| w{0y{©w®´lPwIy{hjlqw®y{yllKM}0y{§ N| <¹4 ?$¯a¹² lN²§¯ g (u, y) = g(y(t)) = g (y)f (u, y) ¯awIxlK|j y{lKBvu g y(y) §°Wyhjl°×}j|Z=y© a| R × R → J x vlPwC|j yxlKZlK|¨ a| <×yhy©wK¯Ey{hjlB°×}j|=y{§ N| w R (u, y) 7→ g (y)f (u, y) xlK|Ny{©K¥©¥§u ¼!l!{ ?$²Y³«lkÉu jl$´|jlw©kB©¥{¥§u g u, . . . , g §° g, f {l C (u,|y)§° g7→ g≡ 0(u,¯v°× ay)9¥©¥ ¯j|¦lhZÉal 'l!9y) a° =y{§gk lPwox(y)f ¯j °× a° yhjjl2=wzy%1,0y.l2. .$, aq|Z² wzy{{a§|MyK¯jw yhZ0yq j = 1, . . . , q − 1 (u, y) 3l$y q ≥ 1 'lyhjl2w®k¥©¥©lKw®y9|Mg}k4(u, K l © 0 0 y © a Z | w j<Rw®l!lK'lCl!la|² xlKN² |CÌ$d lG!?$¦² ² K² yK² u ZlP%w!²HP»° q wÜ´|j§ylN¯x¦ÜlCwÉuBy{h0y q wçy{hjl. 4 °±y{hjlwzy%0ylC$ N|w®y%©|Ny khZ2É}ja¥§ 3ly©l$0jy yY¥§©k l!u Nη∈w®yYaU!| a}jZ$|Ml¶ My%wzy%j0¥©y{wul ak¥©p}xy{§|v Nu|cw®x N ¥§}jw°!y a©y{ ahj|M| yl«© |v´(°}j{ª<¹w®yz4 y·u?=²Iy a©%svkBx©|l!lP+w!$l¯N|jηlK|!BalK|w{w{alÜpulKalK$ Nuv|{¦lxhj§ y°ElK© a'lÜ| a a}j<| |ZEKx lK?=¯ ¦0ª¥§y{{l!h °Ú0yçy 3© a|a|32a¯%{y§h|jNl!NhNu ly [0, T ] !j a©w{|M$y N©|v|N}jy{§ a|v}Z}jw!§yz²µuµ³¨ a° l xlK|jy yy{©kBllvu [η(τ )] =²,η(τ ¦ j h ! l { l y{hjlz}jk ) − η(τ ) η(τ ) = lim η(t) ¨ ³ l k N @ q l { y j h C l × ° a © ¥ © ¥ 0 ¦ § j |. a { w w j }. k x y © a | w V Q η τ ∈ [0, T ] + ghM l Q Tqk §¥§y N|j|¬wqw®y{ a|jN¥§u·$ N|val >¦² P² yK²qyhjlB$ a|My{ N¥30{©aj¥©la¯}|jª°× Nk ¥©u·¦² P² yP² t ∈ [0, T ] <¹a4 ? ∃ γ > 0, H (ˆ u, y (t), p (t )) ≥ γ ∀ˆ u ∈ R, ∀t ∈ [0, T ]. wzy%<»Ü0y{ al|$w®y a%|Zwz©y{|M{y4a§{|Ml!ya}©w¥©a a° §yz Nu {?CxlKg hjl aj|µy{¬y{hj al°$y Nhj|l+xj§y© N aj|·¥©l!'k l!¥© 0¦c{l hjC N¥©jVw ¯YQ R² la² k ≥ 2q ©| <q ?=¯y{hjl q <¹ ? ∃ β > 0, |g (ˆ u, y (t))| > β, ∀ˆ u ∈ R, ∀t ∈ [0, T ]. 6 ghl·y%0zlK$y Nu (u,¯Ny¦ª)y{h hNw ¯ (0 ! | *5 5 M, yh: l O S* ):5 + !0* W )<Ç !a" |XB*=¯ÜZ Myhww§yµ¥§¦u2§l!¥©¥9k 'xlTyzu x?l!w®|j} ayw®lPl! y{wYv au ° T =: T ∪ T ∪ T lPw®'lK$y©alK¥§uµ{l!N}j¥l!|xy{uN¯xTl>x§yoT| y N}T%h+' a©|My{wK¯|Z ¦lw®}jZ Mw®ly{h0y g(y (T )) < 0 ² ©+|¬¾ w®6lP=¢xy{§¡ N| Z |Zoj| ¯jN©w ww}jk xy{§ N|2¦lK4 @al!,yhZK| <q ?=¯ayhyI©wlK|j a}jNh4°× N yhjl9w"} !µ$©l!|Myç$ N|x§y© a|w * ) 54 + 5X& #6 d3X*5 _ ! ) ² lN² t ∈ [0, T ]. <¹N ? ∃ γ > 0, H (u(t), y (t), p (t)) ≥ γ en. ex. en. d dt. (1). ex. (1). y. n. (1) u. y. (2). (q). (j) u. q. (j−1) y. (j). u,η. u,η. +. uu. −. u. u,η. ±. t→τ ±. ±. 2q. (q) u. u. u. en. ex. to. en. ex. to. u. . uu. Ø,Ø Ðj á=ã=ä$â. u. u,η.
(53) EP. B0 * 5X + . 1y N}%²ºh g'h a©l |Ny w®l!τy2 a∈°lPTwwl!|M©y{w·©aw{¥Iy N}y% eh¶Z'l a©|N X*y%*5w 5 a+° ) yhj#Ô¯ql ªy°%y{0hjzlKl$y 3 NNu {a|jal¬k2¦}j¥§©y¥©©¥Üj'¥§©l+l! xηlK|j w{0y{lKy{© w®´vlKu w [η(τ )] > 0 (u, y ) ² T ¦jl!a{w§g0¥00hjy{y{l·§l!N9lK'$wÜ¥ 00ay3{l §z´}hMlK|uv+='y{§ |¬§ Ny{hj| pTlK'walK a}jw2©{|Nl!§y%ik w!²YC6jEdg¥©gUu G»hj² ylqh|jll >v! ay|Mjy{© a|v'}j Nªyzwu¨ªy{§ N°o|y©hjwl·x$}j Nlq|Nyy{M NfN¥çN0$ a{Zw® Nj|B¥©l+l!ya|a¶¥R² CFEP°odwG» a² k P»y{l+w j°o{ v§y{ w ° ¡  ç Ç 5 *5 S* ,% ! I ! /54 O# ) # 5 η
(54) U** O + #6 + u ∈ U <ÇS ? M0 # u S*c+ ) ! + *iU 5 [0, T ] ! +X ) O#6 (: O5 ) *= + !+ * τ ∈ T . C [0, T ] \ T O# # 5 η S* 0 ! + *X# % B 5+ ) 435#6 [0, T ] \ T <ǧ! ? , τ ∈ T ∪ T S*M O#FM 5+ )X% M 4 ! / + !0 + .<R ? , q S*B xj η q − 1 (*X b ) !" X ) U * , + ) ! + *. <×0? , S* l!Nl!| + (*X B !" . to. u. ess to. . . q. q−2 τ q 4 5X U *M , u M0 !u + *c τ <ǧ©!? , τ ∈ T S*M + " + !0 + c<R? + q − 2 (*X W4 5X ) U * , u " + ) ! + * ×< 0?M to, 5 # *5'0 ! + * τ + N S* , q = 1 5 (u, y ) τ u 4d X*c q = 1 X **5 5+ ) η#D + u˙ + !0 +¾ 6¢x¡ no|xl!qyhl4aw{w®}kBjy© a|w a°YI N±²ox²FE ¯Z¦ÜlhZÉalCyhl°× a¥©¥© 0¦§|jµjlK$ Nk Z Mw®§y© a|DQ en. ex. ¦ hjlKl δ xlK|j y{lKw9yhjl4q©{Nk lKaw}j{l0y9y{§k l τ ¯'y{hjl4xlK|w®§yzu xj|Z w9BwÜlKlPaM}j}¥¥Wy{l! |My{u Él5¥©kB>xª MywzZy Nl!§N|Ml!yP{¯uv¦|hj+l!§{° lqqa=| 1ν|:= τ[η(τ©wB)] y{≥ a}0%²Yh³«' alC©|NhyPɲ Nl ν = 0 ª° q w }j|©M}j² lK|jlKw{w °y{hjlk4}¥ªy{ª¸ ¥§©l!³¨P²Ylj al!|Zy{hjyhjwwo¦ÜlCwlK|j$ylK©lK a+|·yvhu+l l >x{jlK{wlK}jw{¥ªw®yq© a a|+|¬ $° N|yhjwzy{l{ay©§k |My9l4Mx}lK¥©©§0´Z0!y©ay©lP aw|¬ ° |DG(u)v. ܾ 6¢ ** O B f, g C + g ≡ 0 , j = 1, . . . , q − 1 + / #!# (,#!#6I ! K 5#6 ) +* #F . dη(t) = η0 (t)dt + η0 ∈ L1 (0, T ) τ. v∈U. P. τ ∈T. ντ δτ (t). τ. dη dt. τ. q. . dj gy (yu )zu,v dtj q d gy (yu )zu,v dtq. τ. (j) u. = gy(j) (u, yu )zu,v ,. j = 1, . . . , q − 1,. <R?. <R4g? !
(55) ,M !a4 ! ) S** S*2 5 DG(u) S*"h S*5") O S*X 3 5 - 5 L∞(0, T ) c*) W 4 ( 35% R< a? W := {ϕ ∈ W q,∞ (0, T ) ; ϕ(j) (0) = 0 ; j = 0, . . . , q − 1}. 1 V, <ÇS? :u_R< $? ¯j¦ÜlChÉNl4Q = gy(q) (u, yu )zu,v + gu(q) (u, yu )v.. d gy (yu )zu,v = gyy (yu )f (u, yu )zu,v + gy (yu )fy (u, yu )zu,v + gy (yu )fu (u, yu )v dt (1) (1) = gy (u, yu )zu,v + gu (u, yu )v.. Õ×Ö Ø±ÕÚÙ.
(56)
(57) !"#$ ! %'&( ! ) +*-,./ ) !"#(&(0 #12 435#6 *. EE. sx<Çw§§!|?b©|!P»lx°çl!g§'|¨l!|Za≡xjl!j|M0ªy{yo§°× N N N|a | j<Rva=¯4?91|Zµwy wy{hqy0−ywz´1y{lPhj¯MEl¦Ü¯WlqxªyCl! a{x©woÉy%lP©ya|µ©wa§vlC¥©u¬u ° w®§|lK al!j%|T}x=l!vy{ u\§ Nq|µ<R4hygahZ?9w0yy{y hhjlyolg°×>x ajC(y{lKw{¥©)z¥w®© a|+§=|\<¹g¯'g4y{?$hj(u,² l!{l4y l5)z>vw®y{w y{+hj}jl! N|jMlKk·}jl ² v ∈ U w}%hºyhy g (y )z = ϕ ²4ghjl! a|!¥§}w© a|¬°× N¥§¥© 0¦9wq°× Nk ϕ y∈hjl W a'l!|«kjj©|j ¡  ç Ç i** O" = + +#6* #F u +∈ /U54* S * ,% b O # # 5+ U* *5Vb5 3 !+ *5I ! * *X !+ +# = #FU * ! [ , dj dtj y. (j) u. y. . u. u. u,v. (j) y. u. u,v. u,v. . Λ(u) 6= ∅ η S * O+ + <שS?aa||· ±w®l!l!y k k x²FE4<×!?Üyhyc< E5Z?hj a¥²çjwsx§$|9a!l <<cEU?x vlKwK²gh©w ¯ 1 0VNlK, wiWP»×< !y=? ²wo aw{vw®v}j§k N}lCwÜy{Mhu0y ±ηl!k , kη ∈x²Λ(u) µ := η − η ∈ M[0, T ] DG(u)∗ µ = 0. u. §°× y9 NÜ°× Nw¥§ a¥© 0k ¦9l wyhy ²Rg,4ϕ(t)dµ(t) ¯j=°×y a©9 a|Ba¥§y{¥ ϕ ∈ W ²°Wsv°×}j©||Z$=l y©g(y a|ZwI)©| < 0 ¯j¦lhɯva¦l l9supp(µ) =0 ⊂ [2ε, T ] v @ § j | C { y j h o l { K l ® w y { a x % y © . | y j h lo¦hj a¥©l ¯ ε>0 [ε, T ] DG(u)U wN$l W (ε, T ) ² :Üu¨xl!|Zw®§yzuº a°y{hjl¥0y®y{l!©| C[ε, T ] ¦ÜlµjlKx}!lyhZ0y°× a2¥©¥ ϕ ∈ C[0, T] R R ² ( 9 T K l | $ l x ¯ ¦ j h © % ¬ h N % j h § K l a P l ç w { y j h l j { v , ° a 2 ° Ç < § ! = ? ² ϕ(t)dµ(t) = ϕ(t)dµ(t) = 0 dµ ≡ 0.
(58)
(59) F
(60) Ü
(61) -,-3 & (*4+ % , $4+ T 0. 1. 2. 2. 1. 0. q,∞. T 0. . T ε. ' !&1 ) ) 78 7 %
(62) % ) ! ! '*% 798 1;#15780! < 4? <<WZ? 4@ u∈U η ∈ Λ(u). {w 0y{©w®°×u4ª° aw{w}jw kBd4jy© a||Zw ¸ a|ª° w U 5² ² ³«lk alÜyhjl°× a¥©¥© 0¦§|jCaw{w}jk xy© a|wK² qˆ := 2q − 1 q qˆ := 2q − 2 q + <Rm9 N|·g |al!|My{©a¥§§yzu+Ü a|jªy{§ N|0? a9a¥§¥ElK|Ny{uµy©kBl a|+a¥§¥3l>xªyy{§k l τ ∈ T Q τ ∈T <R Ed? d d g(y (t))| g(y (t))| (−1) <0; < 0. dt dt j a9a¥§-¥ **5 0 #vy a}Z%h+' a©|My Q τ ∈T <RN4 ? d g(y (t))| < 0. dt <¹sMy{=yÜ ak j¥©l!k lK|Ny%{ªyz
(63) u
(64) 3O4X%*d?Q int I(G(u)) ⊂ supp(η). +¾ 6¢x¡ Ed?:Üu I NZ Mw®§y© a| xF² E¯Iy{hjl·l >xj{lKw{w®© a|Zw2ajZlP{§|«©| Nww}jk xy{§ Ne | <4 ?5<×! ?»¸ <ǧ! ?al¦l!¥©¥ª¸ xl!´|jlK±¯a| qˆ + 1 ©wyhjl wk¥©¥§lPwzy' Nw{w§j¥©lµ N{xlKC°× a2¦hj©%h y{hjl ! a{lPw®' a|j§|j j ?l!{q§0|j0¥©y{uT§Nlqyh al ° ag(yw{w®}kB) jkyÉ© auµ| '<lq|j a ?C|¼!Zl!lK{¥§ 0¦0y9¯,¦Ü|·lP l!@N|Ml!y{CuµyhZ N| l>x<ªqy9 '?= a¯©|MwyK²I}mowlK ¨ylC©|¶y{h|j0lKy !lKqˆw{=waquº°× a! a |qj=ªy{§ N1,|¨2 a² ° lKgk hxl! ayzua{l!¯xk °× N9jF²E¥©¯a¥ ©| a%xl!y{ Q l!|w}j{lyhyIyhjl9wlK! a|x¸¹ N{xlK y{|al!|MyIwl$y T (G(u), DG(u)v) wY|j ay 3l$y 3l$y. . en. qˆ+1. en. ex. qˆ+1. qˆ+1. qˆ+1. ex. − t=τen. u. . qˆ+1. u. ess to. to. 2. 2. u. t=τto. . u. v ∈ C(u). Ø,Ø Ðj á=ã=ä$â. 2,i K. + t=τex.
(65) EÉ. . B0 * 5X + . RsMy{|=+yÜlÜ>x N§kBy9Z N¥§lK§|MkBy lK|My{{§yzu x¯ y{Whjl! {0lC Xl% >xR © w® y{w ! Y °'Z N}jw}|%jh ay{uh0ayd{KQ w ?Q aY¥©¥xl!|My{u' a©|My τ T τ ∈T ε>0. <. . en. ex. ex. (τen , τen + ε) ⊂ supp(η). (τex − ε, τex ) ⊂ supp(η).. ;. en. ∈. <Ra?. mo y{lqy{h0y9¦lx |j ayNww}jk lwzy{=y9! ak j¥©l!k l!|My{a§yzu yy{ a}%h' a©|Ny%w!² ¾v ¡ ¾ < i* * O" c +#6* 5 u ∈ U 3 h !"#H*5# O \, I ! Y ! * /54 " O# # 5 *5 S* ,% ! U K / " -+ ! * 5 B, **5 0 # + + !+ η* ,c + c );:5 5 X% (u, yu ) #!# v ∈ C(u) . . Ttoess ντ = [η(τ )] > 0. ,. + , . τ ∈ Ttoess. <RZ?. (1). 2 Duu L(u, η)(v, v) −. X. ντ. (gy (yu (τ ))zu,v (τ ))2 ≥ 0. d2 dt2 g(yu (t))|t=τ. 2 * *5 ¡ 0 $ $Ç¢x#N¡ ¿ + Y + !0 4 5' !] + _X )* * OO#6b ) , + *_ K ,
(66) *X Y +* ) !+ c ,K S*Y + Y, )´Z{;:5w®y 5 a %Xx% l! (u, y ) +U*Y + . . , M#!#. . ess τ ∈Tto. u. . q = 1. . P |ºyhjlBw®lPN}l!¥¹¯Z¦l2xlK|j y{l I 2(G(u), DG(u)v) vu I 2 ²q aC¥©¥ v ∈ C(u) ¯Wvu\< EP4?=¯W¦l4hÉNl ² ±l!y}Zwjl!|j aylqyhjlCw®}w®l!y a°,${§yu,v ess 2 T ⊂ (T to ∩ Iu,v ) to y{ a}%h' a©|Ny Ç< R² la²©¯w}%h+yhy g(y (τ ))z (τ ) < 0 ¯j°× a9!¥©¥ ¥Eτx©∈{lKT=y{§ N\|Twçyessh?yvÉuaQ N©9|j N| lPwwl!|My¥ 2 Duu L(u, η)(v, v) ≥ 0. v ∈ C(u). u. u,v. to. to. 2 C0 (u) := {v ∈ C(u) ; Tto ∩ Iu,v = Ttoess }.. g!h§yl !´a%¥±w®xyB©w®lPy=l!ye© a |Z°9wÜy{©hj| lj v a°9² °qghjlK a{l!k j²FE+! a|wwzy%w4©|e! ak j}xy{§|j«yhlw§Nk0¸¹yl!{k°× aByhjl C (u) ¡  ç Ç < 5 v ∈ C (u) [ MU** OW * , + 5 - + X <RN4 ? (g (y (τ ))z (τ )) σ(η, T (G(u), DG(u)v)) = . ν g(y (t))| '1 a©V|My{, lKwKok ¯vgy§hj|hlKl |jyyh{hj vl y °9 Nv|juwBlCj <¹° §v y?$ ax¯}ZlK% ah+|j§¥©|M'uey a ¨©y|Mhjºy{lTwK¯w®'y alK|©|M w!y{² ´w |³¨§a| ¥§l ¥©u ´Z!{ aw®yB|!a¥§|}j¥©laux² w®l yhjl·hZÉ$ Na|NlTy{6©j!}x ay{|M§ Ny{|¶§ }x°oy©lK a|M| y{y{u ¶ 0l y>xhj§ly ∂I(G(u)) ∩ I w©ak+yl!{k·²²(mo: u
(67) y{l<Ryh?=¯jy ¦∂I(G(u)) · ² v s ! l y | ¥©l$y =T ς := ς =ς lChÉalQ 0. . . . 0. 2,i K. (1) y u d2 u dt2. τ. ess τ ∈Tto. u,v. 2. t=τ. 2 u,v. u,v. 2 τ ∈ T ∩ Iu,v. ςu,v (τ ) =. g(yu ),gy (yu )zu,v. ({gy (yu )zu,v (t)}+ )2 . 2g(yu (t)) g(yu (t))<0. liminf. G(u),DG(u)v. <Ra?. jlK al!lK{|±§EU0¯j?K0y{< §| IN lK|MwÜy{ u °± Éy{5l hj>vlC§§° yµ$ a'|M© aw9y{©|N v NXy ¥'?=E²²ÜyoÜ{l!w{ aNw®|}j}jwk ¥©lKaMçl·}jlKlKy{|N|Mhy{y0¥©y u ua¯x 0τlv>xu+∈ªyxTZl!´ N|j§|M∪T§yy%©w aa|·² l °,$y N!hj|N!ly{ a§%|v Nx{}jj© N|jl!}ºw y{}j ¶°,|MyyIhj©¥Wl2{ a awz±%y%x²0l!y ljq6² !E a−<×|©S 2w®?$y¯Ü§%° y{§q©k |MyKlw ¯ q−1 q t→τ ;. en. ex. Õ×Ö Ø±ÕÚÙ.
(68) EP.
(69) !"#$ ! %'&( ! ) +*-,./ ) !"#(&(0 #12 435#6 *. y{§° hjlyw©k xlKjw9E²IxlKT9lK©0|0$y{lq§NylKhjwlKu ° g(y¥©¥'Éa)|jaw®hlC0$y N|Nl!y{|M§y|v{}ju N É}lw>x§0yIy y©τk }l |Ny{§¥3 °3 a%2xZl! N}j2q|j−a2u©aw {qa²/wP»yl!N°× al!¥©|±¥§ 0¯j¦9a|wç yhZ2q0y−°× a1 0qy|jlK§NhM' a{hj v xµ a° τ a|+yhjlC©|Myl!{§ N%wxla¯j g ÉuMτ¥© al>xa|w§ N|+N§NlKwK¯Mvu jl$´|j§y© a| ° qˆQ t <RM ? (t − τ ) d g(y )| + o((t − τ ) ), g(y (t)) = dt (ˆ q + 1)! ¦hl!Ü{la N¯vk4°× aj©|jy{©hj|jl w{ ± l!@ak lk °Y¬w§xk ²Ìj¥©©!|ª^yzuN¯x<q¦Ü l ?=j¯3l!¦|j alµylCwl!vlBu yhτ yCl!°×§ ayhjlK ¥©¥ τ ª° τ ∈ T¯,yhj NlB °×τ}j|Z=§y° © aτ| ∈<Ç T°y©² kBlU? ¯xa|0 |w© 2whjxlK 2wYzªy%}wÜwzyC´{0w®°Úyy{l!olK|Myl!y{©kB©|jlCx alK9'©0l$0°×y N©allPwÜ¥©lKwÉ©v|©$|jloµyhjlv'¥© a∈}jy®y{|ZC(u) j{{u+loa${ NC|M Ny©|·|M} aw®}kwÜv¥©u ¥3© 3|Ml!yk lKk0 ¥ g (y )z ! l [τ, τ ± ε] ©j|M² +ylK<×!?=© a²çghj%l Cqw®¸¹xyh¬l Q xlK©00y{§Nl aq° −g 1(y )z ZlK§|µ Z N}j|xlP °×}|=y{§ N|vu
(70) <Rg?=¯j¦lChÉalN¯x a|yhjl <Ra ? |g (y (t))z (t)| ≤ C|t − τ | . w d4¯ ! ak2j§|§|jh <RM ?¦§yh |^ <Ra ?C|¨vuTy%|jNl!|My¥©ªyzu«aw{w®}jk xy{§ N| P»° © <<q ?<×! q?=¯¦ÜlxlPx}!lq°×{ ak <Ça ?ÜyhVy Q qˆ = 2q − 1 u. qˆ+1. qˆ+1. u. u. qˆ+1. qˆ+1. t=τ ±. ±. y. u. u. ex. u,v. y. ≥. ςu,v (τ ). q. +. en. u,v. y. P»°. −. w U 5¯/<ÇM?¦ªy{h ςu,v (τ ). lim. 2q t→τ ± d 2q dt. qˆ = 2q − 2 ≥. u. q. u,v. C 2 (t − τ )2q. > −∞.. 2q. ) 2q g(yu )|t=τ ± (t−τ (2q)! + o((t − τ ) ). ¯D<Ra?|]<4?5<×!?§|\<ÇN4?a©alQ C 2 (t − τ )2q. lim±. (t−τ )2q−1 d2q−1 ± dt2q−1 g(yu )|t=τ (2q−1)!. = 0.. xs §|!l ?Nς<Çg, a(τ}%)h≤' a0©|MvyuM?$²*<¹aqw?±w}jy k al9||jl! 0|M¦ y{uCy{h a03y l5>xªy Z N§|MyK¯0§y3°× N² ¥§¥©P» 0°W¦9yhwEy{yÜh0!yNw®<×l9¦hZhjlKj| 'ql!|w3wKlK¯vaw®l!©||$? l ς (τ ) = 0 ²¯ N}j hv²çuM'sv ©|y{hj$l lKwlKw ©k jhZ¥©au¶w9yBhw®y N¥©τylKw¥© x| !lK¥Ew{wkl!τ|M∈y>v©Tk4¥}y∩ NkF}I%0h y Z N¯ §|My w{0ya|w®°× uM©|j <ÇM4?=¯Éa|Z|jw®h·hjvlK0|y ∈$lNC¯ç¦y{(u) h©0¥§ly j h q≥2 g(y ) τ g(y ) g (y ) τ ç w | a j | ' N w ª { y § N o l Z | µ $ N N | { y § v | j } N } ç w y w § | ! l © w $ N M | y © M | } a } ç w M B u I { a ± I ² j²6E<×!?=² g = g (u, y ) τ u ³¨lCyhv}whZÉa4l Q <R4 g? (t − τ ) d + o((t − τ ) ). g(y (t)) = g (y )| dt 2 sx§|!l τ ∈¦ªIy{h a¯'¥§k ¦l MwzyB¥w®l! alKhÉuvN¦l hjglK(yl+º(τ'))z a}j|ZxlK(τe)w®=lP$ a0|Z²6gxhjl!l{©°×É}j|Zy=©ya©lN a¯Y| ¦Ügl(yal!yB)zvu <RZalK§?$|¯Y y{4C@M©|j<R«w®©|yhj$ll q ≥ 2) | a|j|jlKNy©alVy Q <ZN ? (g (y (t))z (t)) = (g (y (τ ))z (τ )(t − τ )) + o(t − τ ). gh Nl!k {l$°×<R a4 {g4la?$¯aH¯ y{<4 ZN@M4 ©?o|j4|yhj\ l <<q ?<ש! ?=¯¦(ghjlK| (y )z )$ NkB/g(ylPwYy{ 4) y%©wo @a¥§l!l9°Úy®y¸hjl |·{©a hM°3yq'$ Ny{|Mhµy©¥§|M©k } aªy%}ww¦¦hjhl!l!| | t → τ ² min t→τ a| ¯xyhv}w¦l alimjy{inf©|DQ t → τ t→τ. u,v. (1). u. u. u. (1). 2 u,v. y. y. u. u,v. 2. u. u. u,v. +. (1) y. u. u. 2 u,v +. 2. t=τ. y. u. u,v. u. u,v. 1. +. u. +. −. ςu,v (τ ). Ø,Ø Ðj á=ã=ä$â. 0. u. (2). u. t→τ. u,v. 2 u,v. to. d (1) dt. + o((t − τ )2q−1 ). = min. (. (1). (gy (yu (τ ))zu,v (τ ))2 ; 0 g (2) (u(τ ), yu (τ )). ). (1). =. (gy (¯ y (τ ))zu¯,v (τ ))2 > −∞. (2) g (u(τ ), yu (τ )). <Z+Ed?.
Documents relatifs
À cet effet, un nombre considérable de recherches démontre cliniquement que les enfants de parents souffrant de problème de santé mentale sont plus à risque d’un retard dans leur
usual route to school and provide us with some details about their travel routine (e.g., schedule, mode of transportation, people accompanying them, etc.); 3) recent interventions
La question de l’extension des limites de Paris jusqu’à cette ligne avait été discutée mais ne fut tranchée que par un décret impérial du 9 janvier 1859 qui décidait de
C'est aussi le résidant pas chez leurs parents fréquentent deux fois parativement à la province, les étudiants de Nanterre ne lement ce moindre attachement à la ville
In this paper, we study the performance of an enhanced channel estimation technique combining estimation using an autocorrelation based method and the Expectation-
Selon Laeven et Valencia (2012) , le coût budgétaire de la crise bancaire s’est respectivement élevé à 44 et 41 points de PIB en Islande et en Irlande, ce qui place ces deux
Dans ce travail, nous comparerons les entreprises coopératives, détenues par leurs membres, dont le produit risqué est la ressource critique apportée et dont l’objectif est
