Using Viscosity Solution for Approximations in Piecewise Deterministic Control Systems

(1)

HAL Id: inria-00072982

https://hal.inria.fr/inria-00072982

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Using Viscosity Solution for Approximations in Piecewise Deterministic Control Systems

Francesco Moresino, Odile Pourtallier, Mabel M. Tidball

To cite this version:

Francesco Moresino, Odile Pourtallier, Mabel M. Tidball. Using Viscosity Solution for Approximations

in Piecewise Deterministic Control Systems. RR-3687, INRIA. 1999. �inria-00072982�

(2)

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Using Viscosity Solution for Approximations in Piecewise Deterministic Control Systems

Francesco Moresino — Odile Pourtallier — Mabel Tidball

N° 3687

Mai 1999

(3)

(4)

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

012)23426522)"+789;:<"=>?@2A+-

BDCFEHGJILKNMFILOQPROSCFKLMFTUGJO=VXWZY\[JTU]^K`_aOHbJC8cdEH]^]UTUKNC*eWZPfEghKL])i?T^[JgJEH]^]Aj

kmlonpq.rs tvuwpxoy{z!|}u~Lq|A~|}upu{z|uw~

q.|n8p>qF)~Lp>yqvqF

D~q|a\u{z~Lx

Az~L| q.}q8loq8}lqHJLNs zu8LsdzLq8

* ¡8¢£!¤o¥¢L¦?§©¨wFªF«Jz?vxopq}uw8zyzoo}~dup?z!|}u~Lloq8pquw¬o}~H~Lq '~¬|loq>~Lyxo|uw~

~ ouqFq8®)uwq q|q8puwou{|}uw~LN|}~yJ|qp?¯UZ°²±mto³´kmlou{.zo~duwpz|uw~X}loqpq>uw

µ zLq ~Lz¶|}upq u{}}q|uw·8z|uw~>®)lou{l>}q ~Lpxoy{z!|qF|loq)~LuwuwzyD°.±mt>uwL|}~z.|}~lz¸

|u{>o}~zp\´ §`|lou{²zHq²®qo~!¹Lq.|lq~v¹q8Lqq~ |loq?zoo}~dvuwp?z!|}uo*¹q8|~L

¹!zywxoq xo|uw~X|~?|}loq¹Lq8|}~A¹!zyxoq xo|}u~L~ |}loq~}uwuwzyZ°²±mtzL)|}loq u{q|uw·8z¸

|uw~º)|q8|}q m|~·q8~´

»¼N½S¾¿À £!Á¡L¦ ouwq8q®)u{q q|q}puuw|u{?~LL|}~Ly|}qp?8Â|}~lz|u{axop@Â¹Lq8|}~}uwzy

¹!zywxoq xo|uw~@ÂHzoo}~dup?z|uw~@ÂN¹vuw}~Nu|~Lyxo|uw~

ÃAÄÅÆÇÈÄvÉJÊHËÍÌÎSÏÐÑÒFÓÔÕÖU×ØÓÙÈÚ¶ÛÜ9ÆÕÒdÕÝÔÞFÐoßàÓÙ^áÕÖUâãÞÒdäNåæ@çdÓè×ÞéFÙ^çdÛÖà@Þ×J×^édêdêÛÖÍÙØÕäëÚ²Þ)ìÖØÞÒÙ

ÜÖ^ÛíîÙ^çdÕ)ßàÓè×^×ïÞ}Ù^ÓèÛÒ!Þâ9ßðÓèÕÒdðÕmñ!ÛéFÒdä!Þ}Ù^ÓèÛÒNåæ@çdÓè×Ö^Õ×ØÕÞÖUðòçà9Þ×ðÛÒFädédðÕäàçdÓèâèÕZÙØçdÓè×JÞéFÙØçFÛÖDà9Þ×

Ô8Óè×ØÓÙ^ÓèÒdìÇÍïóÇÈÉZå

ô

ÇÍïóÇÈÉZÐ9ß8ÛêFçdÓãÞÉÒÙ^ÓèêÛâèÓè×ÐHñ!ÖØÞÒdðÕå\æ@çdÓè×mÞéFÙØçFÛÖAà9Þ×)×ØédêdêÛÖÍÙØÕäëÚÞìÖ^ÞÒÙAÜÖ^ÛíÙ^çdÕßàÓè×Ø×

ïÞ}Ù^ÓèÛÒ!ÞâoßðÓèÕÒdðÕDñ!ÛédÒdädÞ}ÙØÓèÛÒNå

õ

ÇÍïó@ÇÈÉhÐ!ß8ÛêdçdÓãÞÉÒÙ^ÓèêÛâèÓè×ÐFñ!ÖØÞÒdðÕå@Ï@ÅJïÇUÏ@ÎæÐ!ÑÒFÓÔÕÖU×ØÓÙÈÚ¬ÛÜó@Û×òÞÖUÓèÛdÐdÉÖUìÕÒÙ^ÓèÒ!ÞFå

(5)

,%%'")(*+-,. ',/2) @?@A+-2) h")

/%,"4+-"2),?(

¡ ¦ ºz|u{ywq'¨wFªd«Zo8qN|qxo©}lp?z ºzo~duwpz|uw~ Nxp8uNxoq~Lxo.y{z

~ywx|uw~ ºx o}~ µ yn8p>q q?~LL|}Lyq |}q}puou{|q>z²p~qFzx9´±hq||qzo~duwpz¸

|uw~Â µ zL8qxoxoq u{|u{z|uw~ x |}qp8ÂHq}pq| q?}q ~Lpxoywq²yq?~ µ yn8pq q

~LN|ywq)q?xo?o}~ µ yn8p>q qAo}~L}zp>p?z|uw~|}~vlz|uLxq´°¶zZq¬zouwqh~o}~xo¹Lq

y{z*~Lv¹q}q8q qy{z ~L|}u~L`¹Lq8|}~}uq8yywqzoo}~dvuwp?zN|q¹qaywz ~|uw~`¹!zyq8xoa¹Lq8¸

|~Luwqywywq x'o}~ µ ywnpq q²~N|}Lyqauou|u{zyNxz yq¶z q u{}}q|u{}z!|uw~~v¹Lq}qA¹Lq

·8~

1À ¢8¡ ¾ ¥! ¡ ¦ ±h~N|}Lyq |q}puuw|q?zp~qFzx9Â}zxo|}.|~lz|uNxoq88Â ~L|}u~L

¹!zywqxo)¹Lq8|~}uwqywyqLÂvzoo}~dup?z!|}u~L@Â~Lyxo|uw~ q²¹Nu{}~Lu|

(6)

% &

" 'DJ

uwq8q®)u{q q|}q}puou{|u{a~N|}~yH|qp?²¯UZ°²±mto³zq¶|}~vlz|u{¬~LN|}~ywyq o}~vq8}qF

|lz!|q8¹~Ly¹Lqz~ uwo\|~ z q|}q}puou{|u{ vzp>u{?qq|z!|~Lp>q|}~vlz|u{|uwpq

®)loq8q)z.|~lz|u{xop?~ |loqa|z!|qm¹!z}u{z µ ywqm~v8xo8´('~J|lq8q q88z qFÂZ°²±mt>lzd¹Lq

µ qq©|loq~ µ qF|~ ~u qz µ yquwv¹qF|}uNz!|uw~ uw=±h~N|}~LyZkmloq~LQ¯Uqq ~>qozpoywq

¨!«òÂ¨w*)d«òÂw¨Fd«ØÂJ¨,+)d«z ¨+o«Í³´akmlq\~-Sq¶z®)u q¹!z}uwq|'~ zoyu{z|uw~ ~¬|loqp>~ ¸

qywuwo ~ uw x|}u{zy)z qF~~puwo}~q8}q8*.h®q'8z Nxo~|}qÂhzp~oN|~|}loq>|luoNÂ

p?zvx zL|xuwo`¯^q8q0/21,31>¨èªF«òÂ¨,+F«z ¨8r!«È³ÂS~|}up?zyqoyw~z!|}u~Lz ~xop|}u~L~

}qoq8®z µ yqz o~L¸òq8oq®mz µ ywq}q8~x}q8¨4+F«ØÂq8p?zqN|.loq8zy|}l uwxo}zq'¨w«Zz

zzu|?qzuw~ ¨!8«Ø´

§ Q¨dªF«òÂ65azxo}uwqz \~LqFuwo~\o}~H~Lq z\vxopq}uw8zyZzoo}~dvuwp?z!|}u~L }loqpq ~L

|lq~ywx|uw~©~ zXqFu{zymy{z}~ Z°²±mtS´kmloq'|z!|}q*¹!z}uwz µ ywq?~ |lq'D°.±mt©|x uq

u{~Lp~Nq ~ zX~LL|}uvxo~Lx|}z|q¹!z}u{z µ ywq|lz|q¹L~yw¹qFz8~ uwo'|~`z~N|}~ywywq

u7-Hq8q8N|u{zyq Nxz!|}u~LXz z uw}}q|}q|}z!|}q¹!z}uwz µ ywq²|lz|aq¹L~yw¹qFAz8~ uwo>|~'zxo¸

~LN|}~ywyq xop~qFZ|}z98vuo>¹!zywxoq8huw'z;:ou|q.q|8´kmlou{mzo~duwpz|uw~}loqpq¶uw

µ zLq ~Lz¶|}upq u{}}q|uw·8z|uw~>®)lou{l>}q ~Lpxoy{z!|qF|loq)~LuwuwzyD°.±mt>uwL|}~z.|}~lz¸

|u{~Lzp\´¬kmloqpzu ~Hq~ |}louw¶zq8¬u{A|}~*o}~!¹qLÂxuwo*¹vu{~Lu|*|q8louNxoqFÂ

|lq>~v¹q8Lqq.~ |}louw u{}}q|uw·8z|uw~`}loq8p>qL´=<¬¬|loq>¹!zywxoq x|}u~L`~ |loqD°.±mt

u{)z¹q8|~L¹!zyxoq xo|}u~L@Âou|Auwmq8qF}z}|~qN|}q |}loqy{z}u{zyS¹Nu{}~Lu|*~Lyxo|uw~

:|¶uN|}~ xq \µ `±h}z zyyØÂ §louwuz uw~¯^q8q0/21,31>¨!«Í³´²tvxl`zXqv|q8u~L®zL

zyw}q8z ~q)u\¨ãF«z ¨Fd«uw?z.ywuLlN|yw u7-Hq8q8N|Z~N|}qv|8´Jkmloq¬}loqpqm~ o}~v~ xq

uw*|}louwAzHq)u{)z'qv|q8u~L~ ~pq¬|}q8louNxoqF>:}|)xq *~ q|q}puuw|u{²~N|}~Ly

v|q8p µ \±zoxo··8~¸Ý°¬~Lywq||}zu ¨«Ø´

kmloqzHq>uw~}Lzouw·q zL ~Lyyw~!®A8´ §=tq8|}u~L?+®q ~Lpxoy{z!|q*|loq\D°.±mt©z

|lqXz pu{uµ yqX|z!|}quwq8?z uw¹q'|lqlNvH~|}loq8q8?qq q |lo}~xlo~Lx||loqXzHqF´

@ qqFzyyJ|}loq vzp>u{o}~L}zp>puwo\o}uuyq|}z|q uwî¨Fd«z o}~!¹qq8xoy{z}u|

o}~Hq|uwq8}z!|}uwA:q µ .|lq)¹q8|~L¹dzyxq x|}u~L>~ |}loqAD°.±mtS´ §tvqF|}u~L>²®q q:q

|lqZ¹qF|}~¹vu{~Lu|¶~Lyxo|uw~z l~!® |}lz!|u|uw~Luw|q8L|®)u|}l.|loqy{z}u{zyL~ywx|uw~´

@ qao}~!¹qLÂxuwoB:oq H~uwL|mz}xopq8L|ÂL|lz!||loqa¹q8|~LZ¹!zywxoq xo|}u~L*~ |loq²D°.±mt

u{A|}loqxouNxoq ulou|·>~LL|}uvxo~LxA¹Lq8|~¬¹Nu{}~Lu|~ywx|uw~~ z|q8p ~ ~xoyq

zo~duwp?z!|uw~X}loq8p>q>|}x uwq u ¨dªF«Jz u|}¶uwN|q}o}q|z!|uw~uwX|}q}p?¬~ |}~lz|u{

o}~L}zp'´@ q?o}~!¹q|lz|.|loqq Nxoq8q~ ¹qF|}~.¹!zywxoq xo|uw~z}~u{z!|}q |~\|lq

zo~duwp?z!|uwo|~lzL|}uw)~Lzp?~LN¹Lq}qFJ|~.|}loq¶¹q8|~LD¹vuw}~Nu|>~ywx|}u~L?~ |lq

5HC3E |}qp\Âoz ~Lq NxoqN|}y|}~|loq²¹qF|}~m¹!zywxoq xo|}u~L\~ |}loq.uwou|}uwzy@D°.±mtSÂzL

|lq uw}}q|}u·Fz!|}u~L@ºm|q8|}q m|~·q8~´

I JLK

2 %\2)2342'22)"+ '9J&h"©@?92A+

MONAP QSRUTUV"WYXAZ\[]_^a`cbUdfegQfhi

@ qa~Lu q8hzouqFq8®)uwq q|q8puwouw|u{¬~N|}~y|qp\Âv®)u|llv µ }u |}z|qkjljnmpopqArjnmposoÂ

®)loq8qljnmpo=tuvxw q8o~|}q8¶|}loq~LL|}uvxo~Lx²z|.~ |loq|}z|q®)louwyqarjmpo q8o~|}q8¶|lq

u{q|q²z|F´Zkmloq

u{q|q|}z|qÂyrjnmpoÂ

µ

q8y~LoL|}~zk:ou|qq|{z}|~33qpcq*qAuLÂoz

q¹L~yw¹qFAz8~

uwo>|~*z*~N|}uvxo~Lxm|uwpq>zs8L~!¹.xop`~qFA®)u|}l\|zu|}u~L}z|qF

q:oq

'µ

rjm\}mpo(|}rjnmpo6|n| *,*mj3mpoqtazq|}

rjm\}mpo6|Srjnmpo6|n| {} m\j3mpoyq ¯d³

®)u|l

7 |

D¡

¢ ,=£S¤ ¯n+L³

(7)

z

j3mpo

3m

| ¤

¯UN³

kmloq~LL|}uvxo~Lx²|z!|}qq8¹~yw¹qF¬zL~L uo*|~\z u-Sq}qN|u{zyJq Lxz!|uw~`|lz| qHq a~L

|lq¹!zyxoq|z8q8 µ ²|}loq u{q|qm|z!|qL´J~}qho}q8uwqywÂ!u |loq u{q|q)|}z!|}q)z!|J|uwpqOmuw

rjnmpo"|SÂ|}loq|}loq¬~N|uwvxo~xZ|z!|q¬q¹~Ly¹Lq8zL~L uo¶|}~²|loq ~Lyyw~!®)uwo|z!|}qAq Nxz|uw~

}~p |}upqBmh~'xoN|uwy9|loq.qv|xop\~ |loq u{}}q|q.|}z|q²~x}

ljnmpo(|"!

jljmposq$#(jnmpososq%#(jnmpo_t'&

q ¯Írv³

®)loq8q¶|lq~N|}~Ly(#(jnmpo|z8Lq8¹!zywxoq.uzq|)&

´

*¡F¡+*¢-,À/."0213&

3 4/4 65)789{/(9:yFkzH1

*¡F¡+*¢-,À/.<;=1>@?/A9yB!

q"_tazC6s/ED"7yF?HG;yyyIKJ"yyy

jIqL#oM(p(3/4NJ{1/21

O

7!

jIq$#oP!

jnl qL#o7 QSRUT7I l\qWV _tazqLIqsl$t0uv w q$#tX&

q

O !

jIq$#oH;nyxjIq$#o$J

O

7!

jIq$#oYQ<Z T qWV(jIq$#o_t0uvFw\[X&

<¬}xop|uw~z +uxo}q²|lqqu{|}qq²z xoouNxoqoqFA~ |loq~ywx|}u~L~ ¯UrN³ ~L

q8zLlH~L}uµ ywq²uu|}uwzy@~LuN|=jIqApohz||}upqBm´

°aqo~|q>]

_^

uvxw6|loq ~pzL|q|'~ H~L}uµ ywqXuu|}uwzy¶~N|uwvxo~x*|z!|qL´ q|

` ^ uv w [ ¤ q$a µ q`|loq`q|~ ~LuN| jlqpmpo>xl=|lz|*|lq}qquw|}z |z!qF|}~}

|}z|uwoz|A|uwpq;m{| ¤ }~p4z~LuN|Igt>]

z }q8zLlouwo>|loq~N|uwvxo~xA|z!|}qBlz|

|uwpqam´ § |.uw²ywq8z ~Lp <¬}xop|}u~L>>z +|lz!|²|loq}q8zLlz µ ywqq|:`6uw²z'yw~Lq

µ ~xo q x µ q|)~ uvFw\[ ¤ qLaò´

°aq:oq:b

z ~Lyyw~!®A-c

b | de8f

gih4jklhFm

nioprq s!

jIqL#o+!

jlq$#osqLIa lut

Iglv ¯^ªL³

*¡F¡+*¢-,

À/._w@1Mb

QSRyxyqzRyx|{ ¤

}Z¼

¤

0213D/OjIj~9osqsrjnmpoAo|( jnl j~9osqArjmpoAop/ {$L//k6s/7/nH/

H5a/xm>|

¤

jI

qAr

o/$jnl

qsr

o32D/"? ?/k65a/ays2#(j$7o:(0?/

65a/3p/Gi 9=?/ 5)772p*/p1@?/'{/3?Y4/B96:(~3{

¤

Ij~9o l j~9oYQI l x

£ À@À( ¦ tvuwq|lqq¹L~ywx|uw~`~ |loq u{q|q|}z|q ~vq8ao~| qHq ~|}loq¹!zywxoq~

|lq>~N|uwvxo~x¶|}z!|}qÂS|}loqywqpp?z ~ywy~!®A uwqF|yw }~p¶~Lv®zy |}loq~Lq8p ¯Uqq /*1,1

¨w«Í³´

M

(8)

MONnM HWYXA[[XAd Z]{T `c] s[ V6THXAdZd?Xs[3d ]=dTA]>] [3`cV(`d{XAd[

'o~L.z*Lu¹Lq`uwou|u{zylv µ }u |z!|}q~ |lq|}qp jIqApo¬z|a|}upq9Âz z puw}uµ ywq~Lq8

yw~N~L'~N|}~yØÂv®)uywy µ qz>p>qFzxoz µ ywqap?zoouwo

#(j$7ocq$ak &

xl|lz!|¶|loq?~ywx|uw~X~ ¯Írv³Â9®)u|l`uwou|u{zy~L u|uw~gl joO| I qvu{|}²z uw¶xoouNxoq´

q|Ax q8o~|}q

|loqq|A~ xlp?zoouwoL8´

§ |}lou{)~N|}qv|Az|z!|}q~ |lq²~N|}~ywywquw qFuµ q uw|loq ~ywyw~!®)uo®zdL´"<|

q8zLl?xop |uwp>q~ |loq?|}qp\Â"1/*1>z|¶qFzl`uw|}zL|`®)lq zxop ~ |}loq uw}}q|}q

|}z!|}q~xoÂ|loq~LN|}~ywyq8¬~ µ q8¹Lq8m|}loqq®lv µ u |}z|q$jnljosqsrjosoU| jIqspomz

lo~v~Lq8¬zXz p>u{}uµ ywq.~HqXyw~v~X~LN|}~y9uw|loq>q|

´Akmlouw¬~Hqyw~v~X~LN|}~y@®)uwyy

µ q¶zooywuq ÂvxoN|uwy|}upqØÂN®)lq*q8u|}loqmzq® xop~ |lq¶|}qp ~xo}8ÂN~D|}loqH:zy

|uwpquw}q8zLloq ÂU1/*1x| a´=°¬xuwo |loquwN|q8¹!zym~ |}upqyq DãÂh|lq\zL~u{z!|}q

|z!qF|}~}al jLoD~ |}loq~LL|}uvxo~Lx)|}z!|}q²uwA~ywx|}u~L~

l jnmpo"| !

jljnmpopq$#(jnmpoAopq)®)u|}l l jo6| I

k~xop4xoz>|z!|}qL!uwm|}lvx)zp?zoouwo }~p cq$a l["uvxw [z |}~#" %$'& ´

(A~|u{q.|}lz!|²uwq®hqlzd¹qz q|q8puwou{|}uw~N|}~Ly@o}~ µ ywqp µ q|®hq8q\|®~*xq8¸

u¹Lq)xop8Â9~HqXyw~v~`z yw~Lq yw~v~`|z!|}qLuqF¬z}qq Lxu¹!zywqN| µ q|®hq8q|}loq8q|®~

xop8´Ztv~?uw*|}louw¬zq²ouwq8q8®)uwq²~Hq'y~v~L'z uqFq®)u{q²y~Nq yw~v~\|}}z|q8uwq8mz}q

q Nxouw¹!zywqN|8´

< ~poywq|}q?z oqFu{q qFuw|uw~ ~ z pu{uµ yq|}}z|q8uwq8auv¹~Ly¹Lq8¬|loqxq>~

z?~Lq8|¬~

~ywx|uw~\~

z\~

uz}

u7-Hq8q8L|}uwzyq Lxz!|uw~\®)u|l

u{}~N|}uvxo~LxmuwlN|¸

lz

u

qÂuwq>|lq~N|}~yA#(j$7oAz

~Lq NxoqN|yw\|lq

x|}u~L3!

jnljnmposqL#jmpoAoAz

µ q uw}~LN|uwNx~x8´kmlouw)8z µ q ~Lxo uw ¨w8!«uz>p~}q²qqzy9q|xo´

MON) *V+ ,d ^-,HTZ `cX ]{T V"T ] U`cXAW V+AX`R d/.,V(`XA]{T[

tvxo~Nq*|lz|8ÂDz!||uwpq0mÂJ|}loq'|}z|q~ |lqv|q8p uw jIqApo t ] [ z)Â®)loq}q]

|

~-I6jIqpmpo0t\`0NÂz |lz||loqX~N|}~ywyq8xqF?z uw¹q8 |z!|q80!´îkmloq\|z!qF|}~}

lj$7o.~ |}loqX~LN|uwNx~x|}z!|}q'|~Lq|}loq®)u|l |loq\|}}zq8|}~} ~ |}loqX~LN|}~yB#jLou{?z

|~lz|u{Jo}~q8}q8S®)u|l.pq8zLxo}q"=u xq ²µ !´@ q q:q|loqZq¹!zywxz!|}u~L xo|}u~L

z}~vuwz|q |~|}loq|z!|q8!Xz uwou|u{zy|}z|qajIqApoZzL

1

2

jIqsmpo"|34O56

87 9;:

<

jnl j~9osq$#(j~9oso3 ~k4ljnmpo6|<Iqsrjnmpo6| q ¯UN³

*¡F¡+*¢-,

À/.>=K1>@?/ c /4HD"7yF?HG pyyy I@Jxyyy

jIqL#oM(p(3/4NJ{1/

O 7 jIq$#o

7

jnlq$#oYQ_R@?6Iglq%V(IqAltX` q$#t>&

J

O 7

jIqL#oFByyyxjIqL#oLJ

O 7

jIq$#oYQ<Z?UV(I t0uvxw;qL# tX&

1

kmlo}~xoLlo~xo|h|}loq.zHq)®qz}xopq¬|}lz!|<axop|uw~.¶|~rz}q¶}z!|}uwA:q ´

@ q zq uN|q8qF|}q |~?: |loq ~|}up?zy¶¹Lq8|~'¹!zywxoq xo|}u~L jBA

jIqsmposo

¢DC;EGFGFGFGEH

®)loq8q

A

jIqsmpo"| JILK

2 1

2

jIqsmposq 6|f3q*qsu ¯Ø³

(A~!®<®hq|z!|}q|}loq vzpu{~Lzppuo*ouwuoywq¹q8u7:q Xµ MA

´kmlou{²qFxy|

z µ q ~.qzpoyq ~x u=¨Fd«òÂuw z\yuwlN|yw u7-Hq8q8N| ~Lpxoy{z!|uw~Â@~²uw z'p~}q

q8oqzy9q|Axo\uw ¨w8!«Ø´

(9)

£ À * À

¡i,È¢-,

À(.C0(1M6:yajIqsmpoF` (37nH/:{/3?Y4/

A

jIqsmpo6| ILK

:<$ k A

jljm\*osqpm*o

5 7

jlj~9opq$#(j~9oAo

$'&

A

jnlj~9osq4~o ~

¯UN³

A?/s/lj$7oHH/P/=6|9q**qAu\1

kmloqo}~N~ ~ |lou{Jo}~H~Lu|}u~Lz µ q ~Lxo u'¨Fd«o~uw¨w8d« ~LJz¶yuwlN|yw u7-Hq8q8N|

¹q8}uw~@´

¼vÀ

£ ¼

0(19F 5)7c D ? ?/S?/9 nc A

jIqsmpos/>p(3/4

/D"7yF?HGyyyxjIqpmpo1

£ À@À( ¦ kmloq µ ~Lxo q oq8}>uw|zuwlN|~L®mz ~Lp |loq q:ou|uw~ ¯UN³~ |lq\~L|

xo|uw~Q|}~q|loq8®)u|l <axop|uw~îr´ q|*x*~Lpox|qX|loq u-Sq}qq A j! Iq mpoO

A

jIqsmpo}´_@ q.lzd¹q

A

j" I(qmso A jIqpmpo

QfA

j" I(qmpo A

j! Iqpmpo* A

j" I(qsmpo A jIqsmpo

¯UN³

# |loq.~Loq²lz Â µ q:ou|}u~L ¯UL³h~ |}loq~N| xo|uw~@Â®hq²lzd¹Lq

A

j! Iqpmpo A jIqpmpo

|%$

$$$$IK

2

34'&506

7 9 :

<

j! lj~9osqL#j~9oAoY~rjmpo"| q( l jnmpo(|% I*)

JILK

2

34'&56

7 9;:

<

jnl j~9osq$#(j~9osoY~rjnmpo"|qAljmpo(|_I+),$

$$$$

| de8f

2 3 4 &5 6 7 9 :

<

j! lj~9osqL#j~9oAoY~

7 9 :

<

jnlj~9osqL#j~oso ~

rjmpo6|q- ljmpo6|. I(qAljnmpo(| I )

kmloq u}lou|·²o}~Hq|?~ 7

z yq8ppz\²uwpoy

A

j" I(qsmpo A

jIqsmpo Q / I I67R ?=R

C Q j" I(q mpo jIqsmpoR@?@R

C

¯*)N³

®)loq8q3R

C

|1032-45

6 4

uw)z~L|}zN|F´

# |loq.~|loq8)lz Â

A

j! Iq mpo A

j" I(qsmpo | $$$$$ILK

2 3 4'& 5 6

7 7 9 :

<

j8 lj~opq$#(j~9oAo ~k rjmpo(|q( l jmpo"|. I )

IK

2

349& 5 6

7 9 :

<

jlj~9osqL#j~9oAoY~rjmpo"|pqsljnmpo(|% I+):$

$$$$

(10)

@îu|lo~Lx|?y~N~

Lqoq8}zyu|ÂJ®qp?zd xooH~Lq|lz!| m { m´auwo`|loq

z||}lz!|>|lq

uw|zN|}zoq~Lxm~N|

7

u{

µ

~Lxo

q

u|¬~Lpq8|lz!|

A

j" Iq mpo A j" I(qsmpo

Q $$$$$ JILK

2 3 4'&56 :7 <

7 7 9 :

<

j! lj~9opq$#(j~9oAo ~ rjmpo"|Sq( l jmo(|% I ) 56

6

:

7

<Z ? ~

IK

2 3 49& 506

7 9 :

<

jlj~9osqL#j~9oAoY~rjmpo6| pqsljnmpo(|% I ) $$$$$

kmloq:|.z |loq?y{z|²|q}p?.uw`|}loq?o}q¹vu~Lx²qo}q8}u~L`uwN¹L~yw¹q|}~lz|u{}qyu{zL

z zq¶|lq}q ~}q²q Nxzy^´6@ q²|lvxm~ µ |}zuw

A

j" I(q mpo A

j" I(qsmpoYQ_Z ? m m8QSZ ? j! Iq mpo?jIqpmpo ¯d³

kz8vu|}~Lq|lqmuq Nxzywu|uwq8¯^L³Â@¯*)N³mz ¯LF³hyqFz |}~|loq ~Lyyw~!®)uwo}q8xoy|

A

j" I(qmso A

jIqsmpo Q jZ? R ?/R

C

oj! Iq mpo?jIqpmpoq ¯2+L³

®)lou{l'~yx q8|}loq.o}~N~ ´

M

2)Z"J, 8?

§ |lou{q8|}u~L`®qqv|q8 |}loq?o~|uw~`~ ywzLuw8zyJ¹vuw}~Nu|X~ywx|uw~ ~z'|qp6~

:|)~L q8mz|u{zy u7-Sq}qN|u{zy9q Lxz!|uw~´

)ONAP QdxTX`XA]{T V6Ty]=d`cXd[

@ q>~u

qA|}loq

~Lyyw~!®)uwo*v|q8p&~

~xooywq

:}|a~L

q

u7-Sq}qN|u{zyq Nxz|uw~a®)u|}l

µ

~xo

z}*~L

u|uw~

C

jIqpmq

C

jIqsmpopq**q

H

jIqsmposq

C

jIqpmpoAo6| ¤

v jIqpmq

C

jIqsmpopq**q

H

jIqsmposq

v

jIqpmpoAo6| ¤

´´´

H jIqpmq

C

jIqsmpopq*q

H

jIqsmposq

H

jIqsmpoAo_| ¤ q V(jIqsmpo_tX`

C

jIq$a=o(|

v

jIqLa=ou-|

H

jIqLa=o(| ¤ V/Iq

¯8N³

®)loq8q

zq.~LN|uwNx~x xo|uw~ ~Lp uvxw [0uv [0uv H [ uvxw C

|}~ uv>´

'¼ . ,È¢-,À/. 0(1 @?/pyyyA/D6(9 n$9`ajIqsmpo4|j C jIqsmposq**q H jIqpmpoAo

x$s/$yyy/pD6M2 9=?/B4/5 "9

O

96 gt'R C j`0oLJ 9Bjnlq~9oF$724K5"!3H5 5 9#

96=5/HOtzUJ {/

?Y6/

jlq4~9q C jlq~9osq**q H jnl q4~9osqjlq4~osoQ ¤ q ¯rv³

(

O

96 t'R

C

j` o$JO9Bjlq4~ox 725 yH5 5 9$

9B65a/HtzUJB{/

?Y6/

jnlq~3q C jlq~9opq*q H jlq~9osqjnlq~9oAo&% ¤ ¯FªL³

(11)

l9 jIqsmpo/

/ y ! s/7c/H4/ iJ|{/4 4/ *4 2

s/7\1UN7c/n(91

< upuwywz q:ou|uw~®mz~L~Nq uw¨ã8«Sz u`¨Fd«Ø´kmlou{ q:ou|uw~?u{Zz|}}zuLlL|¸

~}®mz qv|quw~X~ |loq~LN|uwNx~x¬¹Nu{}~Lu|~ywx|uw~ ~L:}|¬~ q8¬z|}uwzy u7-Hq8¸

qN|}uwzy9q Nxz!|}u~L$:}| q¹Lqyw~Hq 'µ '±h}z zyyØÂ §louwuz u~L.¯Uqq$/21,1A¨d«Í³´

kmloq ~ywy~!®)uwo?|}loq~Lq8p4ywuy8v¶y{z}uw8zy~ywx|}u~L`z ¹Nu{}~Lu|'~ywx|uw~~ ¯FL³ÂSuw

|lqzLq?®)lq}qX¯8N³.z p>u|}zXy{z}uw8zyZ~ywx|uw~î¯A1/*1 zX~N|uwvxo~xyw u7-Hq8q8L|}uwz µ ywq

~ywx|uw~³´

¼vÀ £ ¼ ;@1; N7r7c4/#

jIqsmpoER

C

`J89McU;zyJ6?/ ajIqpmpoMap4

9 a9:/$ya9= : /D6M2pnn 9 91

£ À@À(

¦

q|$

µ

qz¹vu{~Lu|~Lyx|}u~L~

¯8N³´{@ q²|}z98q |

uw\|loq

q:ou|uw~ ´

tvuwqq8zl\H~uwN|¬~ `&u{¬zp?z!uwpxop&z zp>uwouwpxop&~

`®hq~ µ |}zu\|}loq.|®~

uwoq Nxzyu|uwq8

jIqpmq C4jIqsmpopq*q H9jIqpmposq

jIqpmpoAoQ ¤

z

jIqpmq C4jIqsmpopq*q H9jIqpmposq

jIqpmpoAo&% ¤

~!¹q8E`\Âz ~yx qa|}lz!| u{)zy{z}u{zy@~Lyxo|uw~@´

±h~v¹Lqq8yLÂ9u u{zy{z}u{zy~Lyx|}u~L ~ ¯8N³²z u l uwz\y~zyp?z!upxop ¯$s/7\1

puupxop³~ Â®q©lzd¹Lq

jnl o| jl oz ~ jnl qjl opqjl oAo?|

jnl qajnlopq

jl oAo"| ¤ Â }~p$®)louwl'®q¶~yx qa|}lz!|# u{)z¹vu{~Lu|x µ ¸Ý~Lyxo|uw~

¯p/71hxoq8¸Ý~ywx|uw~³´

M

)ONnM bUdUXA[Z]{[XD`R [] ,`cX ]{T ]_^$`bd e Q hi

kmloq vzpuw²o}~L}zpp>uwo>o}uuyqzooywuq |~D°.±mt q:oq8Az?|}qp4~ ~xoyq

5¬zpuy|~L¸DCLz~ µ u¸DEhq8yywp?z@º¶¯5HC3E)³q Nxz|uw~8´

¼vÀ £ ¼ wK1X@?/34/Ny/ 9 9F?/_!#(uJA?|jA

C

q**qA H

oLJ9/ /4

|"? M|24nng9x?/k/un |A?/s/;?/65 yc:6s/

9/

/4

jIqpmq C jIqsmpopq*q H jIqpmposq jIqpmpoAo6|

I

$

k

~3

7

jIq$#o $'&

*,

jIqsmpo

m

jIqpmpo

jIqsmpo!

jIqL#os 6|9q*qAu

¯8N³

£ À@À( ¦ @ q$:}|~!¹Lq|lz||}loq*¹!zywxoq xo|}u~LA uw>x µ ¸ò~ywx|}u~L@´ ±h~u q z

R C

j`0o xo|uw~@´ tvxo~Nq'|}lz!|z y~8zympuouwpxp ~ A

uw?z||zuwoq ~ jI(q

mpo´

@îu|lo~Lx|hzvy~NJ~ Lqoq8}zyu|®qAzzLxopq|lz| A jIq

mpo"|>jI(q

mpo}´D±h~Lq NxoqN|}yLÂ

|lq}q*qu{|z}q8zyZH~Lu|}u¹Lqvxop µ qFÂvÂZxl |lz| ~>zywyHjnlq~9o¶xl |lz!| jI(q

mpo_

jlqo7YQ®q²lzd¹q

A

jlq4~o Q>jlq~9osq

z |}loq ~,ax! ?uq8N|yw*p?zyyØÂoxu>~L~Nu|uw~ ¶®qz®)}u|qÂ

>jI(q

mpo">jl\j

m\opq

mo Q>A

jIq

mpo A j#lj

m\*osq

m*o

Q 5$

$ 7

j#l j~9osq$#o3Y~> 5$

$

A

j#l j~9osq~9o3Y~q

¯d³

(12)

®)loq8q ljL oSuw|lq~N|uwvxo~x|z!|}qD|z!qF|}~}a~L}q8H~

uwom|}~¬uwou|}uwzy~

u|}u~L_jIqspo

z!|²|}upq

mÂ@®)loq8`|}loq?~|zN|²~N|}~y #(j$7oU|\# t &

uw²zooywuq

´>°auw¹Nu

uwo\q Nxz|uw~

¯!³ µ Az

yq||uwom|}q

|}~·q}~>®hq.~

µ

|}zu@c

m

>jIq

mpo >jI6q

mso!

jIqL#o

7

jI(q$#o

4¡

¢ A jIq

mpo A jIq

mpoQ ¤

kmlou{Auwoq Nxzyu|*u{)|}xq ~¬zyy#ÂHz uwXz|}uwxoywz ~)|}loq~N|}~y=#`|lz|¬puupuw·q8

|lqywq |lz u q~ |}loqmywzL|Juwoq Nxzyu|´@ q|lvx8z>~ywx qh|lz|+A |fjA C q**q A H o

u{)z?x µ ¸ò~ywx|uw~~ ¯FL³h®)u|}l ¯8N³Âuw'|loqqq²~ |loq q:ou|uw~ L´

@ qo~!®o}~!¹q|lz|Afu{azxoHq¸Ý~Lyx|}u~L@´ q| tPR C j`0oAz jI(q

mpo µ qzyw~zy

p?z!uwpxop ~ A ´"<ALzu®qAz}xp>qm|}lz!|A

jI(q

mpo(| jIq

mso´Dtv~Â|loq8q¬quw|} { ¤

xl*|}lz!|FÂ

~Azv jlq~9oZ}z!|}uw

vuo jnl q4~9o?jIq

mpoYQNÂ®hq.lzd¹Lqc

A

jlq4~o %>jlq~9os

'o}~p;¯UL³h®q.~

µ

|zuw*|}loq

~ywyw~!®)uo>q Nxzyu|/c

A jIq

mpo | JI

:<$ k~ A

j#lj

m *opq

m\ *o

5 $

$

7

j#lj~opq$#(j~9oAo\?

$'&

A

jlj~9osq4~9oY~

z ~q Nxoq8L|}y

jIq

mpo % JI

:<$ k~ j#lj

m *opq

m\ *o

5 $

$

7

j#l j~9osq$#(j~9oso

$'&

,

A

jlj~9opq~9o3Y~

¬uwo>|loq.q Lxzywu|

>jl\j

m\opq

mo(|>jI(q

mpo

5 $

$

m

>jlj~9osq4~9o >j#l j~9osq~9o!

j#l j~9osq$#(j~9osoY~9q

®q¶}q®)}u|q¶|loq.y{z|)uq Nxzywu|

¤ % JI

:< $ k~ 5 $

$

7

j#l j~9osq$#(j~9oso?

$'&

*, A

jlj~9opq~9o

m

>jlj~9osq4~9o >j#l\j~9osq~9o!

j#l j~9osq$#(j~9osoY~

®)lou{luwpoywuqF

¤ % I

:<$ k~ 5 $

$

I

$

k 7

j#l j~9osqBo\ , A

j#lj~opq~9o

m j#lj~9osq4~o\ >jl\j~opq~9o!

jlj~opq=onY~9

~AzLzu

¤ % JI

:<$ k ~ 5 $

$

jlj~9opq~3q AHjlj~9opq~9osq >j#l\j~9osq4~oso3Y~

(13)

°au¹vu

u

µ

,Az

ywq||}u)|q8

|~·q8~®q²~

µ

|zuw

¤ Q jIq

mq AHjIq

mpopq>jI6q

mpososq ¯8N³

®)lou{l ~Lywx qF.|}loq~v~ ~ |loq z||lz| A uw>zXxoHq¸ò~ywx|uw~©z :uwloq8|lq

o}~v~ ~ |}loq²|loq8~}qp\´

M

q|Ax q8o~|}q:R

EC j`0oD|}loqq|)~ |loq ulou|·.~N|uwvxo~x xo|uw~ q:oq ~X`\´

¼vÀ £ ¼ =1 $'& R EC

j`0oH?/s/ /!33$y y/M2an 9 6J 91

k~.o~!¹Lq|loq8~}qp r.®hqAqq z²}q8xoy|D|lz!|Zywuy8vywzLu{zy¹vu{~Lu|~Lyx|}u~L?®)u|}l

|lqA~Lyx|}u~L~ z>~L|uwp?zyo~N|}~yvo}~ µ ywqp\Âz z¶qFxoy|J|lz|Duw¹qF|}loq)xoouNxoqq8}J~

y{z}u{zyZ¹vu{~Lu|~ywx|}u~L@´kmloqFqm}q8xoy|}z µ q ~x ~qozp>yqmuw'¨r«¯Í|loq8~}qp

o´L³Z|}lz!|A®q²qFzywy µ qyw~!® ~m|}loq}z8q¶~ ~Lpoyq|q8oq8}c

¼vÀ

£ ¼

1 #_y9/a?/

/|?6G2?47 nH5 ys/7(s7/5 9H4/S0n

(5k

la| !(jnljnmposqL#jmpoAopq lj¤ o(|_I

/a?/ 4(9

1(jIqL#jLoso6| 5 6

¢ 7

jnlj~9osqL#j~osoY~3

N77/;?Y ?//65 |($c /B4(9y)!_(

7

s/MD67?3lG

ny"?3s/7c/F?/H K6n9 5/yJp(3/4M(=yyy ajlq$#o(

?Y

p/9/4y96F5 gyyy1 @?/ ?/6y/99a?/ F7H5|

s7(s7/5 U?/y/p/9/4B96F5 yyyy2pn; 9U?/

H/5 /un

m

jIq_qo6|

¤

42/4$?/p/r/;/c

jIq¤ o"|

jIo

"?

jIq{q\o"| de8f

$

~9(!(jIqL#o

7

jIqL#os_

£ À@À()À2 ¼vÀ £ ¼ = ¦ uw|q8q ~ywyw~!®A u}q8|yw ~Lp |loq8~}qp/Âvuwq²®qalzd¹Lq

qlouµ u|q xl\z~Lyx|}u~L@´

§²~ qS|}~)o}~!¹qJxoouNxoqoqF8ÂF®hquwN|}~ xqDz¶~Lq8}z|~L }~pR|loqhq|H$'&/R EC j`0o

|~.u|}qy

Âvz

®)uwyylo~!® |lz|Du|huwD~N|z|uw¹qmz

~q Nxoq8L|}y>z

pu|}Dz²xouNxoqO:vq

H~uwL|F´

°aq:oq²|}loq.~Hqz!|}~ u'|}loq ~ywy~!®)uwo>®zdXc

F%$'&=R

ECj`0o H$'&/R ECj` o

j C4q**q Ho_ jA C4q**qA@H4opq

®)loq8q A

uw|}loq|z z ¹vu{~Lu|~ywx|uw~'~ |lq ~ywy~!®)uwo>q Nxz!|}u~L

¤ | I

$

k

~3

7 jIq$#o

m A

jIqsmpo( A jIqsmpo!

jIq$#o A

jIqpmpoAq ¯8N³

(14)

®)loq8q

7 jIq$#o6|

7

jIqL#o\?

4¡

¢

jIqsmpo ¯+)N³

®)u|l'|q8puwzy@~ u|uw~ A

jIq$a=o(| ¤ ~Azywy=I´

(A~|u{q¬|lz|Zu9~L |}ÂL|loq xo|}u~L

zq ulou|·a~N|}uvxo~LxÂL|lq?|}loq¬uw|}zo¸

|}zoq~Lx)~N|

xo|uw~

7

}z!|}uw

$<¬}xop|uw~'r´

(A~|u{q¶zy{~.|}lz!|FÂz~

uwo²|~|loq8~}qp ªÂL|loq¶¹vuw}~Nu|~ywx|uw~*~

|luwhq Nxz|uw~z

µ

quN|}q}oq|q

zm|}loq~ywx|uw~\~

zo~

µ

ywqp$~

~N|}~Ly@®)u|}l :ou|qlo~}u·8~ aÂ®)loq}q

|lq vzpuw8u{Lu¹Lq µ |loq xo|uw~ !

ÂJ|lq*uw|zN|}zoq~Lx~L|u{Lu¹Lq µ |lq

xo|uw~

7

z |}loq u{~xoN|)}z|q²u{muw¹q µ $´

q|)xo~!® o}~!¹q)|}lz!| uw®qywy q:oq Â\1/21D|lz| j

C

q**q

H

oqvu{|}8ÂuwxoouNxoqÂ

z Xµ qyw~oNA|}~ F%$'&/R EC

j`0o ~j

C

q*q H

ot H$'&/R EC

j`0o´kmlqqvu{|qq ~Ly¸

yw~!®A uwqF|}y }~p |loqo}q¹vu~Lx}qp?zs8S´ j

C

q*q

H

o²z µ q'uwL|}q}o}q|q z>z

¹qF|}~¶~ ywzLuw8zy¹vuw}~Nu|~Lyx|}u~L ~ q8~xoyq ~o|uwpzy~N|}~Lyo}~ µ yq8p8´B<)¸

~L uo|}~kmloq8~}qp&ª®hqlzd¹qquw|q8qz xoouNxoqq8}´Akmloq z|A|lz|aqFzlX~Lp>¸

H~oq8L|a~ j

C

q*q

H o µ qyw~oNm|~R

EC Â ~ywy~!®A u}q8|yw }~p/|}loq zL|)|lz!|¬qFzl

~Lp~LoqN|>8z µ q'uN|}q}oq|q z|loq\~ywx|uw~©~ z :ou|}ql~}u·8~©~L|uwp?zym~N|}~Ly

o}~ µ ywqp$®)u|}lo~x|:zy@~L|)z uw}lou|}·.~N|}uvxo~Lxmu|zN|}zoq~Lxm~L| xo|uw~@´

(A~!®.Â

}~p |lq

q:ou|uw~ ~

z*¹q8|~L¬¹vuw}~Nu|~Lyxo|uw~`z

y{z}u{zy¹vuw}~Nu|\~¸

ywx|uw~@ÂSu|¶u{¶|}}zuLlL|

~}®z

|~'q8q|}lz!|.za:oq

~LuN|¶~

|}loq~Lq8}z|~ /u{¶z¹q8|~L

¹vuw}~Nu|?~Lyxo|uw~*~ q Lxz!|uw~ ¯F!¸8L³ÂNz ~v¹Lqq8yLÂLzv?¹vu{~Lu|?~ywx|uw~~ ¯8¸

8N³huw)z :oq H~uwN|A~ |}loq.~Hqz!|}~´

<ANzuw\®hq.xq²|lq~}qp&ª|~uwN|q8o}q|)qFzl~Lp>H~qN|A~ j

C

q*q H ohzLm|lq

¹!zywxoq x|}u~L`~ zX~o|uwpzy~N|}~yo}~ µ ywqp\´(A~!® |~o}~!¹q²|lz!| $u{a~N|z|uw¹q

ywq|Ax)~pox|}qzxoq8 µ ~xo ~L;7

ãÂ®)loq}q¶|loq.o~Lp$uw q:oq zL

7 7| C q*q H 9|

%$'&

: E<$

jIqsmpo

jIqpmpo jIqsmpo

| IK

:< 5 6

kk

: <7

jIj~9osqL#j~oso3Y~O

4¡

¢

5 6

kk

: <

jIj~9osq~9o3Y~

JILK

:< 5 6

kk: < 7

jIj~9opq$#(j~9oAo3Y~O

4¡

¢ 5 6

kk: <

jIj~9osq4~9o3 ~

Q_dLeuf

:<

4¡

¢

*,

5 6

kk: <

jIj~9osq4~o

jIj~9osq4~9o2Y~

Q 5 6

4¡

¢

*,-

kk

: < ~ 7

7

| 5 6

U kk: <

Y~

Y

|jA+

kk:

6 < o 7

7

|jA+

kk6 o

Y9|Y

(15)

@îu|l ¤£ £ zL £¤ z

¤£ a £ ´ §

'|}loqzpq.®zd?®q~Lxoy

lzd¹Lq²o~!¹Lq

|lz!|

jIqsmpo

jIqpmpoQ Y7

kmloq.o}~Hq|~ ~LN|z|}u~L ~Lyyw~!®A)uq¶|loqq8¹Nuw~xhuwoq Lxzywu|}uqFAz}q¶|}xoq ~AzN

jIqsmpoZuwX`\´

M

#&%'%")(*8+-,² K 2 ¶,2 DJ

UNAP bUd Xs[Z yd\`d `cXAW d c]dW

@ qo~!®f|x\|}~|lqzo~duwp?z!|uw~'~ |}loq¹vu{~Lu|'~Lyxo|uw~~ |}loq~xooywq |}qp

~ 5HC3E1q Nxz!|}u~L?¯FoÂw8L³´@ qxq|}upq u{q|uw·8z|uw~@´ q|x qo~|q | a

|lq|}upq|}q ~ |}loq>|uwp>quwN|q}¹!zy¤ q$aò´ § ~L qA|}~~ µ |}zu z zoo}~dup?z|uw~ A

~ |}loq x|}u~L#A

zL~u{z!|q |~|}loqa|uwp>qauN|}q}¹dzy uw}}q|}u·Fz!|uw~ÂÂN®hq¶zoo}~dup?z|q

|lq vzpu{ Ijmpo6|"!

jIq$#o µ

IjnmBo(|<Ijnmpo ! jIq$#osq

z |}loq²|uwpq q}u¹!z|uw¹q²~ |loq.¹!zyxoq xo|uw~

m A

jIjnmpopqsmpo"|

m A

jIjmposqpmpo\ A

jIjmposqpmpo!

jIq$#osq

µ A

jI ! jIqL#o

qsm

o A

jIqpmpo

ywxoLu|loqFq\zo~duwp?z!|uw~²u |loq'|qp ~ q Nxz!|}u~L*¯8¸}FL³Â®q*~ µ |}zuw ~L

zv?|}upqBm t~ ¤ qBqp=q*j o=c

A

jIqpmpo | JI

$

k 7 jI !

jIqL#o qL#o ¯+d³

4¡

¢ , A

jI !

jIq$#oBqpm=o jAO 7BoBA

jI !

jIq$#okqsm\=o

z |}q}p>uwzy~L u|uw~

A

jIq$a=o(| ¤ ¯+9+L³

kmloqzoo}~dup?z!|}u5HC3E |}qp&~ q Nxz|uw~¯n+o¸ +9+³m8z µ quwN|q8o}q|}q zL)|lq

vzp>u{¶o}~zppuwoq Nxz!|}u~L ~Az|~lz|u{ u{q|q¶|uwpq.o~ µ ywqp$|}z|}uo?z|

|uwpqBm

| ¤ Âoz Lu¹Lq µ

|loq vzp>u{A~ |}loq~LN|uwNx~xm|}z!|}qMI

IjAjB9oBo(| IjBo ! 9 :

<

jIj;opq$#(j=osoBq

Ij¤ o6|_I

t]

¯+N³

|loq u{q|q¶|uwpqzs8L~!¹axp>\o}~qFOr Â

r

jsjB o=o6|r

jBo6|| , H|3q

r

jsjB o=o6|Sr

jBo(| |f } Bq ¯+!rv³

(16)

|loq²q¹!zyxz|uw~

xo|uw~

1

jIq ¤ o6| 3 C ¢ 7 9 :

<

jIj osqL#j oAo ¯+ªL³

<Azyw~~Lxy |~ z |}}z|q8 u |lq\~N|uwvxo~x~ µ yq8p\ÂZz |}}z|qL ~ |}lou{ uw}}q|}q

o}~ µ ywqp u{ q:oq zLhzpzoouwo }~p ~ ¤ q =qBq**qj oBoK[uvFw"[xz |~|lq

q|"($'&=&

´

<Q®mzd.|~~yw¹qmvxopq}uw8zywy|lou{o}~

µ

yq8p<o~!® uwD|loq

~Lyyw~!®)u´tvuqm|lq¬|}~lz¸

|u{o}~vq8}

~NqFo~|

qHq

~L|lq~N|uwvxo~xJ|}z|q~~|}loq)~LL|}~Lyu|uwH~L}u

µ

ywqZ|}~

~L|}x|Dzq8¹qN|J|}qqAzL~u{z!|}q |}~²|lou{o}~qF8´k~²q8zLl>z}~ |luwJ|}q8q¯n+rN³uw¹qF

zo~ µ z µ uywu|L´ Nxz!|}u~L¯n+ª!¸D+L³¬zX|lvx µ quwN|q}o}q|}q zLaz'|~lz|u{~Lzp\Â

®)loq8qX¯+ª³¶u{²|}loq xo|uw~ |}~ µ qpuwouwp>uw·q z ¯+L³¶u{.|}loqq|~ ~|zuwN|}8´ §

¨wFªF«òÂy5¬zxo}uq.z \~LqFuwo~>xq²|lou{m|qFloouNxoq²®)u|}l\z>ywuq8z)~ µ yq8p\´

UNnM T `d=] V(`cX]{T ]_^`cbd HXA[Z cd\`d `cXsW d (V+,Ud ^;,HTZ `cX ] T

§*~L qh|~>q|)~Lv¹q}q8q)}q8xoy|}®qaoq8q |~ q:q¬|}loq²¹dzyxq xo|uw~ A

~)zN

muw |loquwN|q}¹!zyB¤ q$aUØ´?@ q q:oq A

¯n+oF³¬z ¯+9+L³m|~Lq|loq8¬®)u|lX|}loq ~ywy~!®)uwo?|}q}p>uwzy~ u|uw~ ~Lazv mmuwX|lqy{z|

qpq8L|Oj o q$ak

A

jIqsmpo6| I

$

k 7 jI !

jIq$#osja mposqL#opjaS mpo ¯+N³

< p~L|)z!|}xo}zy9®zd|~?q|Az\uN|}q}~Lywz|uw~'~ A

~zN mZ®~xoy lzd¹q µ qq'|}~

xqywuoqFzAuwN|q8H~y{z!|}u~L|}q8louNxoqF´@(Aq8¹q|lqywq8}m|loqo~v~ )~ |lqqFxy|)uw¹q8'~L

|lq.ywzL|Aq8|}u~L'zq²pxl'uwp>yq8)xuwo|}loq.uN|}q}~Lywz|uw~*uwN|}~ xq z µ ~!¹Lq´

¼vÀ

£ ¼

1X@?/)9 c jA

jIqsmposo 6p/p/9/4>> cy Z J(D"7y

?3lGkyyyxjIqpmpo "? ?/B65a/ D"7yF?HGcy(96: ?1

£ À@À( ¦ kmloqAo}~v~ uwupuwywzD|~|loqao~v~ ~ |loq8~}qp ´Jkmloq zL|h|lz|Z|lq xo|uw~

z}q µ ~Lxo q ~Lp>qF ~Lp µ ~xo q q8})~ |loq.uw|}zL|zoq8~x)~L|8Âo®)luwl\u{)u qHq¸

q8L|A~ ´

M

UN) hg]{T {dD_dTHZd yd[L,D`

@ q.o~!® |}xo'|}~|}loqzo~duwp?z!|uw~*|loq8~}qpc

¼vÀ £ ¼ 1M6yB>tzUJ/A

jIqpmpo=24/n/BD A

jIqpmpo$J>7*4 96F5 X`

4 D/(4=D

yNJ A?/s/A=?/:*4a 9 J 91

£ À@À( ¦ E kmloq~Lq8p |loq zpuyw jA

jIqsmposo4u{q Lxuw~N|uwvxo~x8´kmloq}q ~LqLÂ µ

<¬}~LyuHkmloq~Lq8p\Â|lq}q)quw|}hzx µ q Nxoqqx~ A ~ |}loq¬q Nxoqq=~'A

Q|}lz!|~o¸

¹q8Lq8yw~v8zywy?xou~Lpyw|}~~pq xo|uw~ ?´ q|)x)~!¹Lq¬|}lz!|)|}loq²¹q8|~L xo|}u~L

| j C

q

v

q**q

H

o}~!¹q¶|lq.|loq8~}qp\Âuwq ARlz µ qq8o}~!¹q |~ µ q.|}loqxouNxoq ulou|·~N|uwvxo~x

¹qF|}~m¹vuw}~Nu|~Lyxo|uw~'~ ¯8¸}FL³´

q| µ qzR

C

j`0o xo|uw~@Âz jIq

mpohz>yw~zy9p?zvuwpxp$~ A

" ~A~pqxZuw

z)´k@îu|}lo~x|yw~L}a~ Lqoq8}zyu|\®hqz xooH~Lq|}lz!|²|}louw²p?zvuwpxpuw.|}uw|8Â5¬qq

|lq}qqu{|¬z?H~Lu|}u¹LquwL|}qLq { ¤ xl\|}lz!| ~¶zvjIqsmpomuw\|loq µ zywy4qN|}q}q

uw jIq

msoZz ~ }z uwx ®q.lzd¹q

jA

"{opjI(q

mpo { jA

"{opjIqsmpos

(17)

°aq:oq¬o~!® |}loqAH~uwN|U~ I q

m Âvxl>|lz|jI q

m ouwhz²p?z!upxop ~ A ~|lq

yw~Lq

q| ´Dtvuwq A

~v¹q8Lq8Dxou

~Lpyw?|~

~L *Âu|

~ywyw~!®AZ|}lz!|xjI q m oD~o¸

¹q8Lq8|}~jI(q

mpo}´tuq)!

jIqL#ou{ µ ~x q Â ~L ?y{z}qmq~xoLl@ÂyjI X!

jI qL#o q m

oÂ µ q8y~LoL|}~ Âoz ~q Nxoq8N|yw ~Lp/|lq q:u|}u~L~ jI q

m oD®hq.lzd¹q

A

jI q m o" jI q m o %

A

jI !

jI qL#o q

m o">jI !

jI q$#o q

m os

¯+L³

¬uwo>|loq.qvqFu~L ¯+F³Âz

8Lqquouw\p>uw

|lq.q Nxz!|}u~L ¯+³Â®hq²~

µ

|}zuw

¤ | I

$

k

~3 A jI q

m o\

7

jI !

jI q$#o qL#o

4¡

¢ A

jI !

jI qL#o q

m o

kjs{} 7 oBA

jI !

jI qL#o q

m oA

Q I

$

k

~#>jI !

jI q$#o q

m o >jI q

m o 7

jI !

jI qL#o q$#o

4¡

¢ A

jI !

jI q$#o q

m o

A

jI !

jI q$#o q

m o ¯+N³

tvuwq uw)zR

C

j`0o x|}u~L®hq²lzd¹Lq ~LA~Lp>q uw ¤ q*òÂ

>jI !

jI q$#o q

m o >jI q m o(|

>jI !

jI q$#o q

m o!

jI qL#o\

m

>jI !

jI q$#o q

m os

tvx µ |}u|}x|uwo|luw)y{z|Aqo}q8}u~L*uw ¯+L³mz ' u¹vu uwo µ ®q²~ µ |zuw

¤ Q JI

$

k~ >jI !

jI q$#o q

m o!

jI qL#o

m jI !

jI q$#o q

m o 7

jI !

jI qL#o q$#o

4¡

¢

,

A

jI !

jI q$#o q

m o A

jI !

jI q$#o q

m o

(A~!® ®hq²ywq| '|}q |~uw:ou||}~q|

¤ Q JI

$

k~ >jIq

mAoi!

jI q$#o

m

>jIq

mpo 7

jIq$#o

4¡

¢

,

A jIq

mpo A jI(q

mponq

®)lou{l*qF|z µ ywuwloqFZ|}lz!| uwmz¹vuw}~Nu|?x µ ~Lyx|}u~L*~ ¯FoÂw8L³´kmlq²}zpq¶z}xopqN|

z µ q.xq |}~?o~!¹Lq¶|lz| uwAzyw~?z¹vuw}~Nu|*xoq8¸Ý~ywx|uw~@´Dkmluw)q8 m|}loqo}~v~

~ |lqa|}loq~Lq8p\´

M

:<*m*

§ .|}louwzq8®qZo}~!¹q Âxuwo¬¹vu{~Lu|a|qFloouNxoq88ÂF|loq~v¹Lq}qq~ |}loqZvxopq8u{zy

zo~duwp?z!|uw~`o~L~Nq uw ¨wFªd«Ø´a'o~L²|lou{²oxo}H~LqÂ@u|.®zLaq8qF}z}'|~\qv|}q |lq

(18)

y{z}uw8zym¯U}zy{z³¬¹Nu{}~Lu|X~Lyx|}u~LX|}~\z'¹Lq8|}~¶¹vuw}~Nu|X~Lyxo|uw~ zL¶u|®mz²zyw}q8z

~Loq²u ¨w8d«z

¨èd«u\z?yuwlN|yw

u7-Sq}qN|A~N|}qv|8´

§

|)®h~Lxoy

*µ

q.uN|}q}q8|uwo|}~?q8q¶u

|lou{)®~p8z

pq|l~

~yw~~xoy

µ

qzooywuq

uw

|lq¶~N|qN|m~ NzpqA|}loq~LL´Z±hywzLuw8zy¹Nu{}~Lu||}q8louNxoqFhlzzywqFz µ q8qxq |}~

o}~!¹q~v¹q8Lqq~ uw}}q|}q>}loq8p>q ~¶oxoxou|²q¹!zLuw~ LzpqF µ Emz uØÂ'zy{~Loq

z tv~L}zd¹vu{z¯UqqB/21,31)¨«È³z µ ~x|zywyuwqJz kmuµ zywy@¯Uqq²¨w8d«Í³´6'o~L|loqFqmLzpq88Â

uq)|lqA¹!zywxoq xo|uw~u{Zz.}zywz xo|uw~@ÂNu|Z®zLH~L}uµ ywq)|~xq¬ywzLu{zy¹vu{~Lu|

~ywx|uw~@´J~L}uµ ywqmqv|q8uw~J~ |lou{®~p8~Lxoyµ q ~q ~LDLzpq8®)u|}l?¹qF|~LJ¹dzyxq

xo|uw~.uw |}lou{zLq|loq?xq~ ¹qF|}~²¹vuw}~Nu|X~Lyxo|uw~`®~xy µ q?q8qF}z}´ @ q

|luc8 p>~Lq'o}q8uwqyw |~ |loq8zq~ |®~ oy{zdq8}·q8~¸òxop ouwq8q®)u{q q|}q}puou{|u{

Lzpq8mz |}loqzLq¶~ o~o¸Ø·8q}~>xop$Nzpq88´

¥ .Àv¿ ¼ Á ¼ . ¢

kmloq\zx|}lo~>z}qu q µ |}q |~ <´"5¬zxo}uq ~>louw~ppqN|}z xqF|}u~L~ uwp¸

o}~!¹q8p>q8N|8´

2)!2)"2Ah2)

¨w« !JÂ/r}$9 /?YyÂ(< o u{~L¸@ qFywqLÂ

)q

®~N~

±hu|LÂ±zywu

~LuwzÂ98vv´

¨,+d«" $# !%&%&')(*",+.-/Â 7 nH5"ys(9E72siy+LD/k+ 9

s/U72s /M599n 4/5Â § kz}z|uw~~L <¬x|~Lpz|u{±h~N|}~yØÂF

¯8LL³ÂN@´@F10S2+´

¨!«234576289%;:< 6)=<?>5@DÂ2\p/D/=?/5/96=?/:6y/

9 9 7 A/N$5/Âvu0< ¹!zq8hu*°avzpu{ ²zpq8z <¬ooywu¸

8z!|uw~Âk²´yEmz}z)z <´y5¬zxo}uqLÂq 8´ÂEhuwp8NlzxqFÂFrÂvo@´L10S*)Lªo´

¨r«.AB34%C D!Âync9/ 24FEM3/4Euy)9/ :5kD 7G89ÈÂotvo}uoLq¸

Hq8y{zÂEq8ywuÂ98r´

¨èªd«*IJ3K5L !%C D:#5M(*JJ+.-/Â| 7H5|n}$9G2/p Du/yD6 7c/:9

2s/7/ 5y93 g4D/5kÂ # q8}z|uw~)q8q8z}lÂJL ¯8LL³Âo@´Dª+0

ªr´

¨!«*N1PO/Q!-R&RS<UT.<%C: WVVSDÂO p/D/ 47r7(s"!3H5 9?/? 65 3p

/u0$9)(65 72sprL655kÂ<Aoyuwq Xz!|lqp?z!|}uw8Dz # |}upuw·8z!|}u~L@Â2)

¯8LL³ÂN@´vX0N´

¨ãF«*N1KO/QD-RSR&<YT.<%C: WVVSZY[/\O/]P>SÂ3 7H5>F"?396 6(

^/p/n;/ uyÂ.t § <a C~Lxo}zy±h~N|}~Ly¶z # |uwpu·Fz!|uw~Âx+3+R¯8rN³Â

o´@rN10HFoL´

¨!«2$AB.O4%C%96._B`Na5b576c(**[/*[,5d<Â /!e099/ D *4

tv~uq|Ây+L¯8+³ÂN@´@a0N´

¨!«2TJf>5d!Â6>/p/"/ $3/D/F5k56hg672s2//"$//L{4$9

^ 2F? 4ÂC~xo}zy!~ A~!Lzytv|}z!|}uw|u{zyLtv~uwq|LÂ8rN.¯Frv³Â8o@´Nª10Lo´

¨w*)!« Â '6hg6 i9/,/gF7H5kGinÂ¹~Ly^´rL\~ \~Lo~L}zol²~L tv|}z!|}uw|u{

z <¬ooywuq ~ µ z µ uywu|LÂ±hlzpzBi 5¬zyyØÂ98L´

¨w«2TJf>5djkl$ !%C%C !#<<QSÂy/F5 /y?/4? y L / $4/Bnu

7 /p/"/ $3/D/F5 yX'hg'5a-3/ÂA\u||}quwyxoq8 q8tlv®hq8u·mHq}quwou¸

Lxoo qnHq8}u{loq}xooNp?z!|}loqp?z|u8qFÂ+\¯8LLª³Âvo@´@!X0c+4+´

(19)

¨w2+d«28D5LÂA 7H5|\ysJ(7//"/$9/D/F5 yy7(s*//(2

yÂol°|}loq8u{Âo°aqz|}pqN|)~

yqF|}u{zy@z

ywq8|}~Lou{

ouwoqq8uwoÂ

§

p¸

Hq}uwzy±h~ywyq8q¶~

tuwqqÂokq8lo~yw~Lz

\q

uwuoqLÂ@8LN´

¨w8!« 8 5L%&"6nl /A5@V

<6 Z m _-5L SÂF#"ysO97c/ -p/4

z ±h~LN|}~yØÂv°aq8qp µ q88Lo´

¨wr«.=K3$JA$ D 5LJÂ)'99n 4D/5;>9 //ÂDk)&}qN|}uwq 5azywy^Â

ywq®~N~ ±hyu7-9Â@Fr´

¨wFªd«* B _*-5L k8K, < DX5L<DÂ@'99 s:2?Y4 72p

rL655 47r7(si9?L´h°az |8Â8Lo´

¨w8!«.=[, D&VÂ{ 4Bnc)42/4"?F"DF?as7(s7/5;9

7 /p/"/ $3/D/F5 y 72p*/p/Â 5¬~x|~L~xo}zy~ p?z!|}loqp?z!|}uw8Â8¯8v³Â

o´or3)Nª0Nr+´

¨wdF«.==kTJ?>5dS6kT$ !/Q9 V ! Tm9l. !/ DÂK 7 nH5\44789n/ !78c

/9/F/n!Âc< ¹!zq <¬ooywuq ~ µ z µ uywu|LÂ98'¯FN³Âo´@Fª10H!!´

¨w8!«m (4<-&VS%C%C5L ! k2"IJ5L9%C%oÂ B7r7(s"!3H59;?/Ny/U9 n9

B$9:^H/s/y5/M"?0n/ØÂ|qFl@´}qH~|8Â §(¬ §<ÂCxoyw`8L´

¨w8!«J5d !%oÂ(!B/65 72sprL655k'(5 yH5 5 72 7/E96 4/5k:"?

5)7 56hg6 -L /ÂHt § <a C~xo}zyS~LX±h~LN|}~yØÂHF\¯8vªL³Âvo@´L?0vN´

¨,+)!«TB,TB =< !Â=//49 g0s_9|0pa9k /4;4/5k:"?3 5)7X78

L5/D/Â § kz}z|uw~~ <Axo|~p?z|u{±h~N|}~yØÂSr¯FN³Âvo@´10H8r´

¨,+« ,k <Â)/965 ^/s/y_/unckp6?/6!Â9uw`}~ µ z¸

µ uwywuw|u{\q|lo~ ¬u<¬ooywuq z!|}loqp?z|u{8Â <´k²´Emzxlz!¸Ý)qu Âoq ´Â<)z qpu{

}q8}8Â98N)oÂNo@´@Fo 0+!v´

(20)

:<2A

0.

¢8£

À Á ¥¢i,

À(. w

;

¼ * ,¼ ¥ ¼N¿ ,Í¡ ¼ Á ¼ ¢ ¼ £ +,. ,Í¡8¢-,Í¥¥ À(. ¢8£ À m¡ ½ ¡8¢ ¼ w

+´w °aNzpuw8)~ |loqZ°²±mt ´¶´.´²´.´²´.´²´.´.´²´.´¶´.´²´.´.´²´.´²´.´²´.´.´¶´

+´,+ < pu{uµ yq~N|}~y{mz ouwq8q8®)uwq²~Hq\yw~N~L'|z!|}qLuqF ´.´²´.´²´.´.´¶´ ª

+´ HDzywxoq xo|uw~z ~L|uwp?zywu|q Nxz|uw~ ´²´.´¶´.´²´.´.´²´.´²´.´²´.´.´¶´ ª

w.¼ ¥¢ À £6,Í¤ @,Í¡F¥ À ¡-,Í¢ ½ ¡ À @¢i,À(.

o´w °aq:ou|uw~Xz o~Lq8|}uqF´¶´.´²´.´²´.´²´.´.´²´.´¶´.´²´.´.´²´.´²´.´²´.´.´¶´

o´,+ kmloq.¹vu{~Lu|~ywx|uw~'~ |lqD°²±mtR´²´.´.´²´.´¶´.´²´.´.´²´.´²´.´²´.´.´¶´

= * *£ À , ¤¢-,À/.îÀ( ¢ ¼ o¤ ¼> . ¥¢-,À/. 08;

r´w kmloq u{}}q|q¶|}upq.o}~ µ ywqp-´.´²´.´²´.´²´.´.´²´.´¶´.´²´.´.´²´.´²´.´²´.´.´¶´ 4+

r´,+ § N|q8H~y{z!|}u~L~ |lq uw}}q|}q¶|uwp>q¹dzyxq x|}u~L ´²´.´.´²´.´²´.´²´.´.´¶´ F

r´ ±h~v¹Lq}qq¶qFxoy|´²´.´.´¶´.´²´.´²´.´²´.´.´²´.´¶´.´²´.´.´²´.´²´.´²´.´.´¶´ F

À(.

¥ ¡i,

À/.

¡

0L=

(21)

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - B.P. 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 - 38330 Montbonnot St Martin (France)

Using Viscosity Solution for Approximations in Piecewise Deterministic Control Systems

HAL Id: inria-00072982

https://hal.inria.fr/inria-00072982

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Using Viscosity Solution for Approximations in Piecewise Deterministic Control Systems

Francesco Moresino, Odile Pourtallier, Mabel M. Tidball

To cite this version:

Francesco Moresino, Odile Pourtallier, Mabel M. Tidball. Using Viscosity Solution for Approximations

in Piecewise Deterministic Control Systems. RR-3687, INRIA. 1999. �inria-00072982�

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Using Viscosity Solution for Approximations in Piecewise Deterministic Control Systems

Francesco Moresino — Odile Pourtallier — Mabel Tidball

N° 3687

Mai 1999

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - B.P. 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 - 38330 Montbonnot St Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France)

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399