A Fully Discrete Approximation for Control Problems governed by Parabolic Variational Inequalities
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Fully Discrete Approximation for Control Problems governed by Parabolic Variational Inequalities Maïtine Bergounioux — Housnaa Zidani. N° 3940 Mai 2000. ISSN 0249-6399. ISRN INRIA/RR--3940--FR+ENG. THÈME 4. apport de recherche.
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(152) ~ SeÌÌe SÀA§hl Jl oz}|jv{ l ~Al | $z| v e eJe ¼ ek e e e N$ZU¨e¤´{n ´ P µRQ À w| µ ³g6hrt2 zk9~jvtlyJl_yhl!~jzsz«³. "! #$ "% $ (
(153) ( Ë^pN e =U 1{
(154) & c;Z-e µ g%adcZ-ekWhg 7
(155) ~ A9£¤N\ U 1{] ¨->X->f!# ´
(156) ´ £ s » ¼ MN>-= !,Ñ& R 2
(157) { P @n !!#& { #&-n ¨ !,1{&- ? >- #^ P
(158) % , &fN$J Çe H' e *#"B,7?.3e /0!##&e#´{n *#"B´ 7$ / e13*#" 7e 3 / !#U
(159) ;&
(160) %!,f X& &=
(161) "7 k' efe
(162) ¨-> ¼ µ - >f!, L ´{n on´ P ¸s ¹
(163) P µRQ-Q q S ´ Ä N%n N%© UÐonN%© U UÅO©w´ P µRQ O © » N S U{Ì uwv .0 8 k ' 1 ´<l$ly^jnh oF n 'Àfv n ^jnN%A© CU Àfn ^j n ¤N%A© CU À ¢^KcA© v ekA© %ýc «¿zK
(164) QT}S ³ ´µl2wvxz?ly ^ o À ¥ ^ S Ä -N%© UO© «¿z
(165) +Q SÀ}w| ^ «¿z
(166) d^ Zj³ ¥ |9~wyrxujvxw §l6hw©¯l ^ n obn ³ ¥ |o§hHw'y`«¿zvtvtz'§2 À rt»w¦ z|yJw|Syrt| l_~Hl_| l_|}yz« e 9§hjrxChok9w© l ~Al | z}| ¼ e} e v ³»pslyyrt|jHÑ ^ rt| òE j³ ,H srtl v !¹ n e µ N%n e U N%n U{e µ n e ^
(167) e µ Ì Eòj³ H Á|yhl!zyhl hw| À«¿z}2w|s ~ v NJmCU Àj§l!hw©¯l}¹ O nNP© U{e µ NPn/NP© U{eU NPnN%© UkU¨e µ nNP© U{e ^
(168) N%© U{e µ Ì O © . ÁÇn ~ v{s s C N$\ ¼U À"yJhjl |n ~` v NJm!U w| §l kªw© µChjzsz}l F^ r{|¾yhjl ~l_¯SrtzuH$l_±Suwv{r{y ³ g6hl |1ujk9k9rt|jªz'¯l_ rt¯lL ¹. n e µ N n e U e µ n e ^
(169) e µ Ì XE j³_ H . P. J. . L. L. J. P. . . . . . . 1P. . . J. . . . 1P. . . P. . . . . . 5. K. ?. N. @M. M. M. N. N. M. M. N. N. M. N. @M. I. N. M. I. N. . @M. Ý"ÝÒ#Ò ¨Ó{Ô. N. M. N. M. N. I. @M. . N.
(170) #&-
(171) % , w
(172) !#&=
(173). ©. . . ´µl$uj¬yCw¼yÇEXj³ H»«¿z}k Eò³2L Hw| ujk u ~«¿Jzk
(174) d^ S¦yJzH
(175) d^
(176) ES »
(177) » BH¹. . . q q E ¥ H E ¥¥ H ¥ ^ N%n oÑn Ue µ $y/N n e UÐoÑy/N%n e U u¢ v u¢v ^ q n e8N Ço fU µ q n e µ o n e µ u¢v uwv. @M. N. @M. N. . M. N. . M. N. q wu v NPn U¨e µ o e µ q wu v
(178) e8N Ço U µ ^ ¥ E ¥¥¥ H E ¥ ËH E H E ¥ HÌ ´µl?|jz'§ lLyJr{kªwyl?l_w}ChyJl Jk z«"yhjl$w¬Hz'¯}lÁlL±}uHwvtr²y }³ rtJy_À§l?§r{yl"E ¥ Hr{|yhjl?«¿z}v{vtz'§rt|j,«¿z}k1¹. N ¥ U+^ q NPn oFn U¨e µ uw v. ^ q e µ q NPn o n U¨e µ ^ ¥ N ¥ U v N ¥ U Ì uwv uwv rt|joyhl?«¿zvtv{z'§rt|jor l_|Syr{y FûE «¿z}2w|s XÍ ¬jrtv{rt|jl_w{H ¹. q Í(N%z oFz v e zR U+^jÍ(NPzR ekzR ¤UÐoÍ(NPz ekz U q Í(NPzR woFz v e zR ozR v U{e EXj³¤H. uwv. uwv M. N. A. A. M. N. @M. N. A. &. @M. N. @M. . N. @M. N. . §l?z¬yCwrt|ý¹. N ¥ U v ^ q. . . u¢v. M. . o. . v e µ ^ S ´ ´¨ s µ o S ´ ¸ ´¨ s µ S q ´ o. uwv N. . . z}k Jz~Az}r²yJr{z}|³2}Às§l!}ly!¹. . . . v ´ s µ Ì. &. . N ¥ U v Q R´ ´ s µ o q ´ o v ´¨ s µ Ì uwv l_wv{rt|j9§r{yhyhjl!lLyJr{kªwyl?z«0N ¥ U À§l!z¬l J¯l ½ CyyJhw'yyJhjl?«¿zvtv{z'§rt|jor{|l_±Suwvtr{y hjzv _¹ ´{nN%© UÐo n ´ s µ » ¢´ n © ´ P µd¶ Q Ï
(179) ~ S eÌÌ2e*w| © ~ Ì EXj³L H . . . . . . . þ¿ÿÝýþûâ.
(180)
(181)
(182) "!#$
(183) % #&(' # !*)+ ,&-$ #/.0 12 n ^[n . ´ r{yh. L. w| XE j³L H¼À§l¦hHw©¯l. . N ¥ U ^³ %n o n e µ o q %n v o n v e o v µ u. n ´ P µÐ¶ Q ´ o n. ´ q P d ´ ´ ´ Ð ´. s µ. » d µ ¶ Q u © © . » ´ ´ s µ q u¢v ´ o v ´ s µ ¢´ n © ´ P µÐ¶ Q. . q » ´ ´¨ s µ u¢v ´ o v ´ s µ ¢´ n © ´¨ P Q e. ´¨. q ´ » s µ u¢v ´ o v ´ s µ fe M. . N. M. . . . . . N. . . v ´ sµ. . . . . . . J. . . §hl Jl. YZo§rtv{vý¬Al$ChjzSl_|wyyJhjl!l | E ¥¥ +H ^ q. . z«"lLyJr{kªw'yJl__³ jJzk yJhjl l ½ |jr{yrtz|1z« HÀr²y«¿z}v{vtz'§2 ¹ &. $y=N n e uwv Q q ¢´ ´¨ P µ Q Ì uwv. e UÐoÑy=NPn . . P. U . . . ^ q fy/N e UD^ q f y/N e U uwv uwv. . g6hlÁyJl Jk E ¥ ¥¥ Hrt6¬Azuj| l ¬s k9l_w|6z«}Eò³ ¬=Hw| z«dIýl_kokªwª³2?w«¿z}v{vtz'§2 ¹. q ¨´ n on v ´ s µ ´ ´ P µ Q n eN& Ëo µ » . uwv. u¢v. » q u¢v ´ ´¨ P µRQ N U . q » uwv ´ ´¨ P µRQ Ì. 2E ¥¥¥ H;^ q. @M. U N. . . P. P. Ý"ÝÒ#Ò ¨Ó{Ô. . . . . ". . 1P. .
(184) #&-
(185) % , w
(186) !#&=
(187). _m. . . ´µl!¬Azuj| E ¥ Hur{|+yJhjl$Jwk9l!wJujk9l |SyCw}«¿zÇE ¥¥¥ H¹ A. . . E ¥ H{U » q n on e µ 0 q n e8N Ço U µ uw v. uwv » q uwv ¢´ ´¨ s µ q uwv ¢´¨n ´ s µ ´¤N o ÅU ´ s µ. ´ q q ¢ ´ » uwv s µ u¢v ¢´¨n ´ P µ Q ´ ´ P µ Q. . . . q q q » uwv ¢´ ´ s µ uwv ¢´ ´ P µ Q uwv ¢´{n ´ P Rµ Q. . . ´ q q ¢ ´ » uwv s µ uwv ¢´ ´ P µ Q Ì A. . @M. N. . . @M. N. . . . 1P. . . . P. . P. . P. . . . P. . §l!¬Hz}uj| yhl!yl_k r{|s¯z}v{¯srt|jo|jz|1vtr{|jlLwJr²yJr{lL ³`´µl$Jl §Jr²yJl"E Hr{|¶w9ur²yCw¬jvtl!kªw|j|jl__³ ¥ | w j r{yrtz|ýÀ§l?ul?yhjl?«òw}¼y6yhwy rxwov{rt~JChjr{y·?«¿uj|yrtz|¶w| I l k9kªw9j³2}À§hjrxChvtl_w yJz¹ 5z'§. A. . . E H; » q NPn UÐo e µ q N%n U¨eN ho%U µ uwv. uwv. q» Ä N NPn U¢o NPnN%© UkU UÅO ©¨e µ q w N%n U{e8N ho% U µ . uwv. uwv . . . q q » uwv ´ Ä n oFnN%© UO ©8´ s µ ´ ´ s µ uwv ´{n ´ s µ NPZÅU ´¤N& o U ´ s µ . q q » uwv ¢´ ´ s µ Ä ¸ ´ N%© U8´ s µ O© u¢v N¨´{n ´ s µ S#U ´ ´ P µRQ. N # S U. ´ P q q ¢ ´ Ä ´ P N © U ¨ ´ ¢´ ´¨ P µRQ » uwv µQ ¸ sµ uwv. . » Ä ¸ ´ NP© U´¨ s µ O© q u¢v ¢´ ´¨ P Rµ Q Ì A. @M. N. M. @M. N. N. . M. N. . . . . P. . . . 1P. . B. . . . P. P. . . . þ¿ÿÝýþûâ.
(188)
(189)
(190) "!#$
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(192) w´ £ s. [» ¼ Àj§l?hw©¯}l¹. q ¥ 2E H;^
(193) eN o U . uwv » q u¢v ´
(194) ´ s ¶µ J. A. @M. . v q µ » wu v Ä ´
(195) /NP© U´ s µ O© д ´ P Rµ Q . q ¢´ ´¨ P Rµ Q » q ¢´ ´¨ P µ Q Ì. u¢v. uwv . N. . . . P. 1P. . . P. . . . g"zªukokªwrt· l}À§l?hw©¯}lÁ~jJz'¯l Jy hjl?«¿zvtv{z'§rt|j9l_yrtkªw'yJl¹. q q v ´ ´ s µ ´ o ´ s µ ¢´ ´ P µ Q » uwv uwv. . P. N ¤U Ä ´ N%© U8´¨ s µ O © q ¢´ ´¨ s µ ¸ uw v. . . EXj³_H ´ ´¨ s µ q wu v ´ o v ´¨ s µ q uwv ¢´ ´ P µ Q Ì ^ v zªyhHw'y2yJhjl$vxwy2yhjJl l?yJl Jkª2z«yJhjl$Jr{}h}yNhHw| r l$rt|¶rt|jl_±Suwv{r{y EXj³_H ¬s ªyhl?yl Jkª6z«yhjl?vtl«ûyNhw| r l?z«}òE ³2L H ³g6hsu_Às«¿z2w|s H
(196) ~ S eÌÌ2Ìe }Às§l . . . 5z'§. . . §l,ly w l?z}|SyJzvtv{l hHw©¯l¹ . q ´ ´ s µ ¢ ´ ´ P µ Q . . . . . P. . . q N U Ä ´ N%© U8´ s µ O© ¢´ ´ s µ Ì » ¸ uwv uwv EXj³ mÅH I lyuz}¬l J¯lÁyhwy«¿zK
(197) ~ S eÌÌ2Ìe ?w| © ~ cA© v e A© ec À§l!hHw©¯l¹ ´ NP© U´ s µ » ´{nNP© Uo n ´ s µ ´ n on ´ s µ » ¢´ n © ´ P Q ´ ´ s µ EXj³ H » ´ ´ s µ Ì g6hrt6z}¬l_¯'wyrtz|§r{yh EXj³ mH srtl v ¹. ´ ´ s µ q ¢´ ´¨ P µRQ » N ;U q ¢´ ´ s µ Ì EXj³ H u¢v uwv rt|jªw rx l yl ¦z}|s§wv{vWrt|jl_±Suwv{r{y BòE l lÇ I l k9kªw9j³Ø¼H À§l?z¬yCwrt|ý¹. q EXj³ H ´ ´¨ s µ ¢´ ´¨ P µ Q » ®N ¤U uwv P. . . . . . . . . B. . P. . J. . . . +>. . P. Ý"ÝÒ#Ò ¨Ó{Ô. . . :?.
(198) #&-
(199) % , w
(200) !#&=
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(202) $(N Ç
(203) #"& C !,w
(204) & @-!#
(205) P U 7 NPy =U !,& N -U 1m C Ë 0 @-&{¨ m '¢- ¤ ¨
(206) $?
(207) 9¤K&-"1{{ = 2 N =U 1{K!,&X
(208) & {!# {
(209) & Ë 0@-&/{ k'9f # ¨
(210) P
(211) ?9#h&-"1{ = !#& "2 Y[Z ¨- >*? >-!# v y » q fy t Ï Q Se tu¸ CD
(212) - > y » 7HMN>f& = - >f9' #26 CD
(213) &
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