• Aucun résultat trouvé

Necessary Conditions of asymptotic stability for unilateral dynamical systems

N/A
N/A
Protected

Academic year: 2021

Partager "Necessary Conditions of asymptotic stability for unilateral dynamical systems"

Copied!
55
0
0

Texte intégral

(1)

HAL Id: inria-00071417

https://hal.inria.fr/inria-00071417

Submitted on 23 May 2006

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

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Necessary Conditions of asymptotic stability for unilateral dynamical systems

Daniel Goeleven, Bernard Brogliato

To cite this version:

Daniel Goeleven, Bernard Brogliato. Necessary Conditions of asymptotic stability for unilateral dy- namical systems. [Research Report] RR-5171, INRIA. 2004. �inria-00071417�

(2)

ISRN INRIA/RR--5171--FR+ENG

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

Necessary Conditions of asymptotic stability for unilateral dynamical systems

Daniel Goeleven — Bernard Brogliato

N° 5171

Avril 2004

(3)
(4)

,

(-(.$/ 01(. 23 0

46587:9<;>=3?A@;B=C;EDF;>7HG>IKJL;BMN7O5*MNPHJLMN@FQ8=C9R5TSU@WV

X-Y[Z)\2]%^L_a`cbd\e[fhg+ijblkBm6]ikEnoijbl\2bhpqgriqbdkEm

s ]pqtcpuiqZv\2]vpwkB\2n[fl])xo]vp

yKzqkE{|]i3}1bln8kEn

~3grnTn*kBzui s ]zj]vwY[]vzjwYT]m8€:o‚+ƒc‚_a„†…>zjbdf ‡rˆEˆE^_a[‚%nTgr‰B]vp

Š$‹Œv)Ž+[EE‘“’ m6iqY[bhpnTgEn*]vzv”[•1] s ]v…E]vflkBng2\/g+ijY[])\/griqbhw)grfiqkckEf&ijYTg+i–w)grm6—8]eTpq] s iqk!puijgriq]mT]vw]Np|˜

pjgrzjtw)kEm s bliqbdkEm*pOkr™&gEpqtc\Ln[iqkrijbdwpuijgE—[bdflbli|tkE™FbhpqkEfhg+iq] s p|ig+iqbdkEm*grzjtpukBfle[iqbdkEmTp:kE™&gw)fdgBpqp:kE™&e[m[bdfdgriq])zgrf

s t>m*gr\2bdwvgrfputopuiq])\/pvš–›œkEzj]n[zj]vw)bdpq])fdtE”8m[kEmTflbdm[]vgEz†]v…EkEfdeoijblkBmœ…+grzjbdgriqbdkEmTgEfblmT]v>eTgrfdbliqbd]vp–grzj]“wkBmTpqbs ˜

]vzq]

s š ’

mTp|igr—[bdfdbži|t2wzjbžij])zjbdg%gEzq]3gEfdpqk‰Ebd…E]vmš„†n[n[fdbdwvg+ijblkBmTpŸwvgrm/—8]™ kEe[m

s

blm!\2]NwYTgrm[bhw)pKkEzK]vfl]Nw¡iqzjbhw)grf

w)blzweTbžip)š

¢A£>¤F¥j¦§ Ž+¨ŒE‘ª© grzjbdgriqbdkEmTgEfbdm[]v>eTgEflbliqbd]vpv” s bl«F])zj])m>iqbhgrfObdmTwfdeTpqblkBmTpv”&iqkBn*kBflkB‰Ebhw)grf s ]v‰Ezj])]B”p|igr—[bdflbli|t

ijY[])kBzqt!™ kEze[m[bdfhg+iq]vzjgEf s tcmTgr\2bhw)grfpqtop|ij])\/pvš

¬K­ ®¯*°%­²±³r´ µU¶·¸|¹<º<¶»²¼¾½¿[ÀcÁ-® Â|ÃUµN¶Ä½¡µ>³EÅNÁ¶Äµ»<ƲǸ|µU¶Äºq³UÈÉjÊË¡ËN³rÌ+¹CÁµUÍ|¸¡³EÇ:ÁµU¶Ä¸|ÎhÏн)¸|Îĸ|·¸|µrÑÃNµU¶·vÆh¹<¸|ÃUµU¶Ä½¡µBÏ¿d¹

Ò

­ Ó®­²±Ô® ÕUÖ¡µN¸|Æh±ÎÄ×r¸|ºq³Ør­ ®Å)ÙÚ°%½¡µ)»<Ûr½¡µUµU½»u³Ü¡Ý¡ÝÞÁ|·¸|µvÃU¸œßU¸Îhà¯*ÃU¹á½¡×r¸³%â¡ã¡â¡âÊ.ÅNÁ¶Äµ»<Æh­²ºCä¶Ä¸|¹u³†Ì+¹<ÁµUÍ|¸¡³

å ¸u¹áµ+Á¹áß>Ïå ¹á½¡æ¡ÎĶçÁj»C½)ѶĵU¹á¶çÁÎÄ×r¸|ºqÏ¿l¹

(5)

,

2 0

,

/3 0

,

(

,

Œ ‘ ])i$grzqiqbhwfd]nTzqkBn*k>pu]e[m[]A\ijY[k s ]A\/g+ijY)\/g+ijbd>e[]6n8])zj\2]iuigrm>i s ]6™ kBzq\“e[fl]vz s ]vp

w)kEm s bžijblkBmTpLmvwvpjpqgEblzj]vp s ]6p|igr—[bdflbliAgBputc\2noiqkEiqbh>e[] s en8kEbdm>i s v>e[bdflbd—[zj]bhpukBf se[m[]AwfhgEpjpq] s ]

pqtop|ijZ)\2]vp s tcmTgE\Lbh>e[]vpe[m[bdfhg+ivzjgEeox&š¾mw)kEmTpqbs Z)zj] s ]Np bdmNBe*g+iqbdkEm*p …+grzjbhg+iqbdkEmTm[])fdfl]Np s t>m*gr\2bd>e[]Np)š

¾]vpgEn[n[fdbdwvg+ijblkBmTppu]–ijzqkBe[…E]vmBi s gEmTp-fd]vpwbdzwe[blijpvfl]Nw¡ijzqbh>e[]vp1kEe6fd]vppqtop|ijZ)\2]vp-\vwvgrm[bh>e[]vpvš

§

NŒ

¥

 ŒL‘Ô’

m v>eTg+ijblkBmTpK…+gEzqbhg+ijblkBm[m[])fdfd]vpv”rblm*wfdeTpubdkEm*p

s

bl«!vzq]vm>iqbd])fdfl]Np)”

s

])‰Bz-ijkEn8kEfdkE‰Bbd>e[]B”+iqYvkr˜

zjbd]

s

]fdg2puijgr—Tblfdbži"%n*kBe[z3fd]vppqtcpuiqZv\2]vp

s

tcmTgE\Lbh>e[]vp-eTm[blfhg+i")zgreox&š

(6)

1 2

“ :/

,

Ÿ

X-YT] puijgr—Tblfdbži|t kE™p|ig+iqbdkEm*grzjt pqkEfdeoiqbdkEm*p!kr™

s

t>m*gr\2bdwvgrf%putopuiq]v\2pwkBmTp|ijbžijeoiq]Np6g …E]vzqt bl\2n8kEzqijgEmBi

ijkEn[bhw%bdm6„†n[n[fdbl] s ›œg+ijY[])\/g+ijbdwvp3grm s43 mT‰Ebdm[])]vzqbdm[‰Tš ’i¾bdp-•1])fdfž˜65cm[k+•miqYTgri3blmiqYT]w)gEpq]%kr™OgLfdgEzq‰B]

w)fdgBpqp†kr™Km[kBm[fdblm[]Ngrz s bž«F])zj])m>ijbdgEf]NBe*g+iqbdkEm*p)”*iqY[]Lpun8]vwiqzje[\ kr™87flbdm[]Ngrzjb9v] s 7–kBn*]vzjgriqkEzp s ])iq]vzq\2bdm[]vp

ijY[]4: tBgrnTe[m[k+…œpuijgr—Tblfdbži|t kEzbdmTpuijgE—[blfdbli|t kr™grm ]N>e[blfdbd—[zqbde[\6šX-YTbdpbhp5cm[k+•mWgEpiqY[]4: tBgEn[e[m[k+… p

fdbdm[]vgEzqb;9vgriqbdkEm \2])iqY[k s=<‡>ƒ>”‡*>o” 0 ^?CšA@†k+•]v…E])zN”&\/grmctœbd\2n*kBzuigrm>in[zjkE—[fd])\/p%bdmÞ]vm[‰Ebdm[]v])zjblm[‰CBápq])]

<D ”-‚vˆ[”1‚B‚r”1‚ 0 ”-‚)^T”‚+ƒ>”K‡Eo” 0*0 ?GFblmc…BkEfd…E]/bdm[]v>eTgEflbliqbd]vpLbdm iqY[]vblz2\/g+ijY[])\/griqbhw)grf™ kEzj\eTfdgriqbdkEm.grm s

w)kEmTpq]v>e[]vm>iqfdt$n8kBpjpu]Npqp-bdm>iqzjblmTpqbhwm[kEm*pu\2kckrijY[m[]vpjp•Y[bhwYAw)gEm[m[kri¾—*]zj])\2k+…B] s —>t6wYTgEm[‰E]kE™:w)k>kBzu˜

s blmTgriq]Np)”kBz%—>t6™ ]v] s —*gEw5Fš“X-Y[]2\2k s ])fhpwkBzqzj]vpqn8kEm s bdm[‰$iqkApue*wY n[zjkE—[fd])\/p%grzj]2w)grfdfd] s e[m[bdfdgriq])zgrf

s t>m*gr\2bdwvgrf&pqtcpuiq]v\/pkBzmTkEmo˜Rzq]v‰Ee[fhgrz s tcmTgr\2bhw)gEfFpqtop|ij])\/pvš:„ fhgrzj‰E]–wfhgEpjpkE™ e[m[bdfhg+iq]vzjgEf s tcmTgr\2bhw)grf

pqtop|ij])\/pLw)grm—*]zj])n[zj]vpq])m>iq] s e[m s ])zLiqY[]™ kBzq\/grfdbhpu\ kr™¾])…EkBfle[iqbdkEm.…+grzjbhg+iqbdkEm*grfŸbdm[]NBe*grfdbžijbl]Np)”Ÿpq])]

]Bš‰*š <0 ”-‚ 0 ”‚v^?RšWX-YT]>e[]Np|ijblkBm.kr™¾puijgE—[bdflbli|tWkr™¾puijgriqbdkEmTgEzqtWpukBfle[iqbdkEmTpkr™†]v…EkEfdeoijblkBm.…+grzjbdgriqbdkEmTgEf

bdm[]NBe*grfdbžijbl]Np“bhpL\eTwY\2kEzj]wkB\2n[flbhw)griq] s iqkޗ8]bdmc…E]vpuiqbd‰Bgriq] s ”KgEpLbliLbhpLiqY[]6w)gBpu]blm ‰E]vm[])zgrfŸ™ kEz

Yctc—[zjbss tcmTgE\Lbhw)gEfputopuiq]v\2pv”[pq])]%]Eš‰Tš <‡rˆo”*‡[‚#?Rš

X-YT]Kzq]Npu]NgrzwY%]«FkEzqijpFijk s ]v…E])fdkEn%‰B])m[]vzjgEfr\/g+iqYT])\/g+ijbdwvgrfBgEn[n[zjkBgBwY[]vp8ijk¾puiqe s t%puijgr—Tblfdbži|t–kr™TpuijgriqbdkEmo˜

gEzqtpukBfleoijblkBmTpkE™:]v…EkBfleoijblkBm…+gEzqbhg+ijblkBmTgrf&bdm[]v>eTgEflbliqbd]vp¾grzj]%zq]vfdgriqbd…E])fdt!m[]v•š ’i–pu]v])\/p-iqY*g+i3ijY[]IHTzp|i

w)kEm>iqzjbd—[eoiqbdkEmœgBw¡ijblm[‰$blm6ijY[bdp–pq])mTpq]•-gEp3\/g s ]—ctKJ¾eTbžiqiqm[]vz <‡ 0

”&‡+^T”8‡EL?™ kBz¾g!w)fdgBpqp†kr™OnTgrzgr—8kEfdbhw

…+gEzqbhg+ijblkBmTgrfrbdm[]NBe*grfdbžijbl]NpbdmM@3bdfl—8])zqipqnTgBw]vpvš ›AkBzq]Ÿzq]Nw])m>ijfltB”N: tBgrnTe[m[k+…¾gEn[n[zjkBgBwY%gEm sIO

zg5Ek+…op5cblbl˜

:gB`ogEflfd]6blmc…+grzjbhgrmTw)]$ijY[])kBzqt.Y*gU…E]6—*]v])m ‰E]vm[])zgrfdb;9)]

s

iqk ]v…EkEfdeoijblkBm…+grzjbhg+iqbdkEm*grf-blmT]v>eTgrfdbliqbd]vp/grm

s

w)kEzjzq]Npun8kEm

s

blmT‰e[m[bdfdgriq])zgrf

s

tcmTgr\2bhw)gEf*pqtcpuiq]v\/p8B<pu]v]

<

‚r”F‚N‡c”>^[”[ƒc”8‚Nc”o‡E‡N?gEm

s

iqY[]–zj]™ ])zj])m*w]vpwbliq]

s

ijY[])zj])bdmFš ’m <‚N‡N?pueP!w)bl]vmBi3w)kEm s bliqbdkEm*p1™ kEz3puijgr—Tblfdbži|t$kr™fdbdm[]vgEz])…EkBfle[iqbdkEm…UgEzqbhg+ijblkBmTgrf8bdm[]v>eTgEflbliqbd]vp

Y*gU…E]–—*]v])m s ])zjbd…E] s ”[grm s •]%m[k+•Hn[zjkEn8kBpq]¾ijk/p|ije s t$mT]vw]Npqpjgrzjt!wkBm s bliqbdkEmTpvš

¾eTzLgEbl\ bdm.iqYTbdpLnTgEn*]vzLbhpiqkWp|ig+iq]m[]vw)]vpjpqgEzqtWwkBm s bliqbdkEmTp“kr™–gEpqt>\2noijkrijbdwp|igr—[bdflbli|tޙ kEz/g w)fdgBpqp

kE™e[mTblfhg+ij])zgrf s tcmTgr\2bhw)gEfKputopuiq])\/pvšL›œkEzj]“n[zj]vw)bdpq])fdtE” •]/wkBmTpqbs ]vz–iqY[]2n[zjkE—[fd])\,QI:])iRTSVUWYX[Z

UWK\^]_I`ba“—8]Lg$n[zqkBn*]vzv”&w)kEmc…E])x6gEm s fdk+•]vz¾pq])\2bl˜Rw)kEm>iqbdmce[kEeTp3™ e[mTwiqbdkEmš: ]iIcdSeUW X ZfUW X

—*]

g2mTkEm[fdblmT]vgrz¾wkEm>ijblmce[kBeTpkEn8])zg+ijkEzNš^: ]i(ghiKUW X

—*]“‰Ebd…E]vmš^jÞ]w)kEmTpqbs ])zijY[]k Ž § ‹£ mlng!ho#Q

pbdm s gLe[m[bhBeT]wkEm>ijblmce[kBeTp1\2gEn[n[bdm[‰qrZts nquo1pueTwYijYTg+i

v s

v q i,w^x

yz#{

n|!} _I`~uUW X o ~ B|‚NF

s nquo€iK Rƒo }&„ q^… | ~ B<‡*F

† v s

v q n quo‡_ˆc

ns n

quouo }‰MŠ

s n quou‹Œ_R

n‰

o Š R ns n quouoŽ…

|e}„‰

i,UW X } gTš]Bšq^…

|e}

B0 F

gEm s s n$| og h’‘ B ^“F

X-YT]…UgEzqbhg+ijblkBmTgrf&bdm[]NBe*grfdbži|tblm”B

0

Fw)gEmAgrfhpuk2—8]™ kEzj\e[fhg+ij] s gEp1ijY[]pq]iq˜C…+grfde[] ss bž«F])zj])m>iqbhgrf ]v>eTg+˜

ijblkBm•Q

v s

v q nquoŒ_c ns nquoo€i ŠY‚ R ns nquoo} gTš]Bšq^… |e} B<*F

(7)

•YT])zj]

‚ R s

]vm[krij]vpiqYT]%pqe[—

s

bl«8]vzq]vmBijbdgEf&kr™‡R1š

’

ibhpgEpjpue[\2]

s

iqY*g+iYc n|

o€i

ŠY‚

R

n|

oKbCš]Bš:iqY[]%kBzqbd‰Ebdm

|

kE™ iqYT]putopuiq])\bdpg“ijzqbd…cbdgEfp|ig+iqbdkEm*grzjt!pukBfleoijblkBmkE™ BC*Fš

’

™R

|

iqYT])m[BC*FKzq]

s

eTw]NpKijk“g“p|igrm

s

gEz

s

putopuiq])\ kE™

s

bž«F])zj])m>ijbdgEf*]NBe*g+iqbdkEm*pgEm

s

bli1bhpŸ•1])fdf5cm[k+•m

B<pu]v]3]Eš‰Tš

<

‚˜EL?FObdm2iqY[bhpŸwvgEpq]-iqYTgribl™

|

bdpgrm/bhpukBfdgriq]

s

9v])zjkkr™•c grm

s

bhpŸgBputc\2noijkriqbhw)gEflfdt“puijgE—[fd]E”rijY[])m

ijY[])zj]%]xobhp|ip h=| pueTwYijYTg+i

n

c

} }|

oƒ

}&„

i n|e}

’h

}

•YT])zj] Vnc }

}| o s ]vm[krij]vp¾iqYT]2}1zjkEec•1])z s ])‰Ezj])]Lkr™ cڕbliqYÞzq]Npun8]vwi–ijk$ijY[]/kEn8])m —Tgrfdf S

]NgKi UWYX S#gagEm s,| š

¾eTz“‰BkBgrfblmiqYTbdpLnTgEn*]vz“bhp“iqkމE])mT])zgrfdb9v]/iqY[bhp“™ágr\2kEe*pzj]vpqe[fli“ijk ijY[]\2k s ]vf1bdm Bá^“F˜BC*F¡š.›œkEzj]

nTzq]Nwbhpu]vfltB”ofl])i-e*p s ] H*m[]%iqY[]\/gEn[n[bdm[‰ gEp

ngVo Sbg Š lŒng Š c ngVoo

•YT])zj] l S n.v !#" _ ‚ Rƒo%$'&gEm s(.v)!#"6s ])m[kEiq]NpiqYT]bs ])m>iqbli|t\/gEn[n[bdm[‰!kEmKUWYXFšŽjÞ]•blfdfn[zqk+…B]

ijYTg+i3bl™ | bdp3gEmbhpqkEfhg+iq] s 9v])zjk2kr™* grm s bhp3gEpqtc\Ln[iqkrijbdwvgrfdfltp|igr—[fd]E”cijY[])m6ijY[])zj]%]xobdpuijp+ h | pue*wY

ijYTg+i

n }“}| o  }&„ i n|!}“h, ‘

X-YTbdp-zj]vpqe[fli†wkEm*p|ijbžijeoiq]Np1iqY[]\/grbdm6zq]NpueTfžikr™iqY[]n*grn8])z3grm s bhp-‰Ebd…E]vmbdmAX-Y[])kBzq]v\ԁoš

’ m6`c]vwiqbdkEmA‡o”c•1]¾zj]vwvgrfdf*ijY[]–™ e[m s gE\2])m>ijgEf8nTzqkBn*]vzuijbl]NpkE™iqYT]%}1zjkEec•1])z s ])‰Ezj])]–•1]¾•bdflf&e*pu]–blmiqYTbdp

n*grn8])zNš ’m `c]Nw¡iqbdkEm 0 ”•] s bhpjweTpjp–iqY[]/\/gEblmÞn[zjkEn8])zqiqbd]vp%kr™iqY[]/kBn*]vzjgriqkBz l š ’m.`c]Nw¡ijblkBm ^*”•]

nTzq]Npu]vmBigAw)kEmTw)bdpq]2zq]v…>bd])• kr™pqkE\2]2zq]Nw])m>izj]vpqe[flijpbdm p|igr—[bdflbli|tœijY[])kBzqt kr™-e[m[bdfdgriq])zgrf s tcmTgr\2bhw)grf

pqtop|ij])\/pvš ’m `c]Nw¡ijblkBm o”•]!bdm>iqzjk s eTw]2ijY[]!y:kEbdmTwvgrz"/kEn8])zg+iqkBzgEpjpukow)bdgriq] s ijkAn[zqkB—[fd])\ B<*Fš ’m

`o]vw¡ijblkBm D ”[•1]–nTzqk+…B]¾kBe[z-\2gEblm6zj]vpqe[flivš

’ mLijY[bdpKnTgrn8])zN”r•1] s ]v…E])fdkEn grfhpuk pu]v…E])zgrfLgrn[nTzqk>gEwY[]NpijkwkE\2n[e[iq]-iqYT]m>eT\—8])z Vn }- }| o¡š ’™

wvgrm“—*]]NgEpqblfdt])…+grfdeTgriq] s iqYT])m/gYH*zjpuiKgrn[n[zjkBgBwYwkBmTpqbdpui:iqk%eTpq]-pqkE\2]1n[zgEw¡ijbdwvgrfczj]vpqe[fžip:kr™ s ])‰Bzq]v]

ijY[])kBzqtœiqk w)kE\2n[eoij] n } }| o B<pu]v]$`c]vwiqbdkEm D F¡šX-YT]$grn[nTzqk>gEwY bhpkE™1n*grzqiqbhwe[fhgrzbdm>iq])zj]vpuibl™

R.0/ ! " 1 ”[•bliqY2/ ! " 1 s ])mTkriqbdm[‰2ijY[]blm s bdwvg+ijkEz1™ e[mTwiqbdkEm6kr™ UWYX3-”[pqbdmTw]bdmiqY[bhpw)gBpu]’Q

54ngVo 7698;:•]˜g4 } c<4ngVo#a ‘

X-YTbdpw)gEpq]3bhpKkE™nTgrzqiqbhwe[fhgrzŸblm>iq]vzq]Np|iŸ™ kEzKijY[]¾puiqe s t“kE™ wkB\LnTfl]v\L]vm>ijgrzjbli|t s tcmTgE\Lbhw)gEf*pqtop|ij])\/p <‚ D ?Rš

›œkEzj]‰E])mT])zgrfdfltB”obž™ R=>/? •YT])zj]@ s ])m[kEiq]Np†g/wfdkBpq] s w)kEmc…E])xpq]iv”TiqY[]vmA•] s ])…E]vflkBnAg!pu]NwkBm s

gEn[n[zjkBgBwY[Bápq])]`o]vw¡ijblkBm ƒFiqY*g+i†eTpq]vp†g2zq]Npue[flikr™^J¾e[bliuijm[])z <‡EL? •Y[bhwYzj] s eTw]Np-iqY[]w)kE\2n[eoig+ijblkBm

kE™ Vn }-A’}| oiqkLiqYT]kEm[]%kr™ VnCB }“}| oŸ•YT])zj]D5Bœbhp s ]"HTm[] s —ct

Bƒngo  g Š l ?FE ng ŠHG IŽn| o.gVo }

•YT])zj]F@ h bdpiqY[]wfdkBpq] s wkBm>…B]x¾w)kEm[] s ] HTm[] s —ct@ h  \*JK hML @ gEm sNG IŽn| o s ])mTkriq]NpFiqYT]CO>gEwkB—[bhgrm

\/griqzjbžx$kE™c g+i | š

’™ R bhp2gÞw)kEmc…E])x.grm s wkBmBijblmce[kBeTp“™ e[mTwiqbdkEmiqY[]vm •1] s ])…B])fdkEn gœijY[blz s grn[n[zjkBgBwY Bápq])]œ`c]Nw¡ijblkBm

>“F–—ct pqY[k+•blmT‰ijYTg+i Pn } } | o%w)gEmW—*]$w)kE\2n[eoij] s —ct eTpqbdm[‰œpukB\2]!grn[n[zjkEnTzqbhg+ij],7":tBgEn[e[m[k+…

™ eTmTw¡ijblkBm‡7+š

(8)

X-YT]LwvgEpq]kr™flbdm[]vgEzwkE\2n[fd])\2]vmBigrzjbži|tn[zjkE—[fd])\/p–bhp

s

bdpjweTpjpq]

s

bdmW`o]vw¡ijblkBm[ETšpbdmTgEflfdtE”&bdmÞ`c]Nw¡ijblkBm

‚Nˆ[”Ke*pubdm[‰ÞkEe[z2\/grbdm zj]vpqe[fli“ijkE‰E])iqY[]vz2•bžijYzj]vw)])m>iLbdmc…+grzjbdgEmBiLzq]NpueTfžip B<pu]v]

<

‚#?F¡”1•1]n[zjk+…E]pukB\2]

bdmTpuijgE—[blfdbli|t/iqY[]vkEzj])\/pvš

`o])…E]vzjgEfF]x[gr\2n[fd]vpgEm

s

gEn[n[fdbdwvg+ijblkBmTpgrzj]¾‰Bbl…B])m6puk/gBp1iqk2blfdfdeTp|ijzjgriq]–iqYT]%iqY[]vkEzj]iqbhw)gEf&zq]NpueTfžip)š

v[§ ŒE‘K’m$iqY[]%pq]v>e[])f*iqYT]–pjw)gEfdgEzŸn[zjk s eTw¡i1kEm UW X bhp s ])m[kEiq] s —ct

†

}

‹(Bá•bžijY!iqYT]–gBpqpqkowbhg+iq] s

mTkEzj\

"Fš(p[kBz | ”*•]Lpu])i S ]˜g i UW X S g< “aBš¾X-YT])m  ]˜gCi UW X S #g’a

gEm s ‚  ]NgKi,UW X S#gސ“a ‘

X-YT]1bs ]vmBijbži|t\2gEn[n[bdm[‰%kEmUW X bdp s ])m[kEiq] s —ct .v)! "Fš: ]iF@ —*]3g¾m[kEmT])\2noi|tw)flk>pu] s wkBmc…E]xpqe[—Tpq]i

kE™ UW X šOX-Y[] s eTgrf8pu])iŸkr™ @ bhp s ]"HTm[] s —ct @8S ]Ti,UW X S † } … |!}.„ g i(@”a>šX-YT]†zj]vw]NpqpqbdkEm

w)kEm[]kr™ @ bhp s ] HTm[] s —ct @ x S \)K h & n@ Š g h o3•Y[])zj]Mg h bhp¾gEm>t])fd])\2]vmBi–kr™ @ šYj ] s ])mTkriq]

—ct*? iqY[]bdm s bhw)griqkEz1™ e[m*w¡iqbdkEmAkE™<@ ”ob<š]Eš*? ngo  | bl™g i @ grm s *? ngo  _I`0bž™ gi @.š

pTkEz3g“™ e[mTwiqbdkEm i& nUW X ~UW(o•1] s ])m[kEiq]%—ct !ngVoiqYT]‰Ezg s bd])m>ikr™ g+iŽg i UW X š

jÞ] s ])m[kEiq]1—ct vn#"*}%$ o iqY[] s bhp|igrmTw)]Ÿ™ zjkE\g¾n*kBblm>i " i UW(X%iqk%g¾pq]i $ & UWŽX&”UiqYTgriObdp vVn'"’}($ o€S

8:*),+-/. "ŽŠ10

2 3

()) 0)Ô54W" ()61(87 96::4<;

: ]iA & UW X —8]œgrm kEn8])m grm s —8kEe[m s ] s pq]ivš: ]i = i>& nK~uUW X o@?8

h n K~uUW X oL—*]A‰Bbl…B])mš

`o]iBAC n4o S ]Ngi  SD= ngVo | a$grm s C n4o S ]Ngbib S FEG C ngVo | a!•YT])zj] G C ngVo

s

])m[kEiq]vpLijY[]2O>gEw)kE—[bhgrm ›œg+ijzqblx kr™= griAg

s

]"HTm[]

s

—ct

nG C o4HG SJI CLK

INMO

n

go¡”

}#P

i ] }"‘;‘‘;}LQ

a>š

’™

AC n4oR? C n4o S2grm s | i= 4oiqYT])mAC n4obdp3gHTmTbžij]pu])i†grm s iqYT]}zjkEec•1])z1iqkBn*kBflkB‰Ebhw)gEf

s ])‰Ezj])]–kr™T=A•bliqY6zj]vpqn*]Nw¡i-iqkA gEm s | bhp•1])fdfž˜ s ] H*m[] s —ct/iqY[]%™ kBzq\“e[fhg

Pn = } } | U

M

-/VXWZY\[T]^

: n' FE G C ngVoo }

•YT])zj]2pq‰Em nquo  “™ kBzIq | grm s pq‰Em nquo  Š ™ kBzIq | š/›AkBzq]2‰B])m[]vzjgEflfdtE”&bl™9==S_ Z UW X bdp

w)kEm>iqbdmce[kEe*p1gEm s,| i= 4oŸiqY[]vmiqY[]}1zjkEec•1])ziqkEn8kEfdkE‰Bbdwvgrf s ]v‰Ezj])]%kr™T=A•bliqY6zj]vpqn*]Nw¡iiqk grm s

| bhp-•]vflfl˜s ] HTmT] s Bápq])]%]Eš‰Tš <‚˜>?GF-gEm ss ])m[kEiq] s —ct n= }  } | o¡š

: ]ieTpmTk+• zj]vw)gEflf&pqkE\2]%n[zjkEn8])zqiqbd]vp-kr™iqYT]%iqkEn8kEfdkE‰Bbdwvgrf s ])‰Bzq]v]%•]%•bdflfe*pu]%fhg+iq]vzblmijY[bdp3nTgrn8])zNš

`a

’™ | i= 4o1grm s Pn= }  }| ob | iqY[]vmijY[])zj]]xobhp|ip gKiKpqeTwYiqYTgri= ngo | š

cRa

: ]iIRSRd|e} Def ZfUW X ~ n'gŒ} goYrZ R n'gŒ}go¡”—8]Lw)kEm>iqbdmce[kEe*p†pqeTwYAiqY*g+iv”8™ kBz¾]NgEwY g ihd|e}

kBm[]%YTgEp | i,R n'gŒ}‚ 4ošŸX-Y[])mijY[]\/grn g rZ Vn R n'gŒ}"‘o }  }| o-bhp3wkBmTp|igrm>i-kEm8d|!}

iXa

j ]YTgU…B]

Vn.v !#" }  } | ‘

jka

’™ | i= 4o1grm s L==| iqYT])m

Pn L = }  }| o  Pn= }  }| o

(9)

gEm

s

VnuŠ+L

= }  }|

nuŠ

No X Vn

= }  }|

o

š

Ra :]i(

! &

—8]grm6kEn8])m6pu])i†pue*wY$ijYTg+iA C n

4o

&  !

šKX-YT])m

Vn= }  } | Pn= }  !} | o ‘

Xa :]iATi UW XX —*]gLm[kBmTpqblm[‰Be[fhgrz-\/g+iqzjblxFšOX-Y[]vm n

A

‘}



}|

^ : n FE A o¡š

a :]i= S dŠ }

_MZtUW —8]g2wkEm>ijblmce[kBeTp™ e[mTw¡ijblkBmš`ceTn[n*k>pu]–ijYTg+i=

nuŠ

ob

|

grm s = n

_ob | š

X-YT])m

Pn

= } Š } _*d

} |

|

8\)8=

nuŠ

o(=

n

_o |e}

Vn= } Š } _*d}| _NH8) = n&Š o | : = n_o |e}

Vn= } Š } _*d}| Š H8) = n&Š o =| : = n_o |e‘

-:( 1

3

:K)Ú"Þ-%Ô

2

-:

: ]i RˆSUW X Z UWK\^]`ba¾—8]¾gnTzqkBn*]vzv”>wkEmc…B]xLgEm s fdk+•]vzŸpq])\2bž˜wkBm>iqbdm>eTkEeTpO\/gEn[n[bdm[‰Tšj ] s ])mTkriq]

—ct vZ0 ]NR^a%iqY[] s kB\/grbdmkE™ R1”[bCš]Bš

vZ0 ]NR^aST]NgKi UWYX4S’R ngVo+ _I`ba ‘

jÞ] s ]vm[krij]–—ct ‚ R.iqY[]w)kEmc…E])x$pue[— s bl«8]vzq]vm>iqbhgrfkE™R1šK~]vwvgrfdf&iqY*g+i

‚ R ngVo T] i UWYX S*R n‰ o Š R ngo … † }u‰Š }&„V‰ i,UWŽXVa ‘

jÞ] s ]vm[krij]–—ct4 Rƒo1iqY[] s kE\/grbdmkE™ ‚ R gEm s —>t RƒoiqY[]zgrm[‰B]¾kE™ ‚ R1”[bCš]Bš

 Rƒo ST]NgKi UWYX4S ‚ R ngVobS a } Rƒo€S \ M - ! " ‚ R ngVo ‘

~3]vw)gEflfFijYTg+i

 Rƒo &=vZ0 ]NR^a &  n$‚ R o ‘ BD F

: ]iCi”UWYX —*]2‰Bbl…B])mšj ]/wkBmTpqbs ]vz–iqY[]2…+grzjbdgriqbdkEmTgEfbdm[]NBe*grfdbži|tAn[zjkE—Tfl]v\ Q pblm s g=iˆUWŽX pue*wY

ijYTg+i

†g Š }u‰Š g‹Œ_ˆR o Š R ngoŽ… |!}.„‰ i UW X ‘ BCƒF

yŸzqkB—[fl]v\ BRƒFKYTgEpge[m[bh>e[]¾pqkEfdeoiqbdkEm B<pu]v]†]Bš‰*š <‚ 0 ” 0

ˆ?GFOiqYTgri•1] s ])mTkriq]†—>t lŒneo¡šOX-Y[]†kBn*]vzjgriqkBz

l

S’UWŽXZtUW(X~ rZ l neo

(10)

bhp1iqYceTp3•]vflfl˜

s

] HTm[]

s š ’

i3bdp†wfd]vgrz1ijYTg+i

l<‡nUW X o &  Rƒo B>’F

gEm s l neo n.v !#" _ ‚ Rƒo $ & no ‘

’™ R ? •Y[])zj]@ bhpgm[kBm[])\2noi|t!w)flk>pu] s w)kEmc…E])x/pq]i3gEm s ? s ]vm[krij]vpiqYT]–bdm s bhw)griqkBzŸ™ e[m*w¡iqbdkEm

kE™<@ ”ciqY[]vm

l l l ?

•YT])zj] l ? s ])m[kEiq]vp1ijY[]n[zjkr{|]Nw¡iqbdkEmkEn8])zg+ijkEz-kEm>iqk@•Y[bhwY6bdp s ]"HTm[] s —ct/ijY[]%™ kEzj\eTfdg!Q

#g ŠCl ? gY 698;:

-)?

#g Š N ‘

: ]i SUWŽX Z UWYX.—8]6g w)kEm>iqbdmce[kEeTp“\/grn[nTblm[‰Wgrm s wkBmTpqbs ]vzijY[]blm[]N>eTgrfdbži|t nTzqkB—[fl]v\,Q4pblm s

gKi,UWŽXpqeTwYiqYTgri

† n

gVo }u‰MŠ

g ‹Œ_ˆR o Š R n

g o … |!}.„‰ i UW X ‘ BE’F

’ibhp-wfd]vgrziqY*g+in[zjkE—[fd])\ B$E’FŸbdp-]NBeTbl…+grfd])m>i1iqk“iqY[]%m[kBm[fdblm[]Ngrz1]v>eTgriqbdkEm•Qƒpblm s

gKi,UW X pue*wY!iqY*g+i

g

ŠCl n

g Š n

goo 

|!‘

Bu‚vˆ’F

Ž §kŸ§ Œá § `

'!e"6

l

IV !! G

l<!‰ŠCl<

‰MŠ

}&„V‰V}

Ai UW X ‘

Ž

§§

‘

: ]i

‰V}

i UW X —8]‰Ebd…E]vmš j ]pq]iYg S l‡n‰

ogrm s g IS

lŒn

ošƒj ]Y*gU…E]

†g

Š[‰V}

Š

gV‹‡_ˆR

n o Š R n

gVo …

|e}&„

i,UW X Bu‚E‚NF

gEm

s †g Š } Š g ‹Œ_ˆR n8o Š R ng o€… |e}&„ i UW X ‘ Bu‚N‡*F

`o]iuijblmT‰ Stg Ablm Bu‚E‚LF2grm s fStg blm Bu‚N‡*F”•1]AkE—[ijgrbdm iqYT]Azq]vfdgriqbdkEmTp

†g Šb‰V}g Š g Š

R ng No_R ngVo | grm sŠ

†g Š }g Š g Š R ngVo€_R ng No | ”™ zjkE\•bhwY.•] s ] s e*w]!ijYTg+i

g Š g g Š g ) ‰MŠ ’i™ kBflfdk+•3p1iqYTgri

l n‰ o ŠCl neo> ‰MŠ ‘

(11)

: ]ieTpmTk+•

s

]vm[krij]–—ct

SUWYX Z

! "

iqY[]pq]iq˜C…+gEfle[]

s

kBn*]vzjgriqkBz

s

] HTm[]

s

—>t

ŒngVo T]Z=Ki,UW X S †g Š = }u‰Š g‹Œ_ˆR o Š R ngoŽ… |!}.„‰ i UW X a ‘ Bu‚ 0 F

jÞ]pq])]–iqYTgri

$ &

n= o  l n= o ‘ Bu‚)^“F

’i3bhpgrfhpqkL]NgEpqt2iqk!pq])]–iqY*g+i

nq o $ & nqL= lŒn = o }.„ q |e‘ Bu‚N*F

†krij]¾ijYTg+i

ngo  ‚ ] ‘

_ˆR nu‘o#a ngo }&„ g i UW X }

pqk%iqY*g+i

bhpK\/g+xobd\/grfT\2kBm[krijkEm[]Bš

’

iŸzj]vpqe[flijpKiqYTgriŸ™ kEzgrmctq |

”rijY[]†kBn*]vzjgriqkBz n.v

!#"€_Kq

o%$'&

bhpgL•]vflf

s

] HTmT]

s

pubdm[‰Bfl])˜C…+grfde[]

s

kEn8])zg+ijkEzNš

(%-:( 7 2 01( 2/&3 0

: ]ieTp H*zjpuizq]Nw)grfdfgL‰E]vm[])zgrfF]xobhp|ij])mTw)]grm s e[m[bh>e[])mT]vpjp-kr™OpukBfleoijblkBmTp1zq]NpueTfžiMB<pu]v]–]Bš‰*š <‚UN?GFš

£c§ Ž £ `

Œ" R SƒUW X Z UW,\^]_I` a )K4V ! ˆV+ ""u$G-!-

%- Œ" c S’UWYXAZ UWYX ) 4$G-!- M !".I#-!','e% I"

iKUW c_

g h i UW X )A 'e" ’#M- -! s i

h n d|!} _I` o#~uUW X o #-!','e

v s

v q i,w

xyz#{

n|!} _I`~uUW X

o ~ Bu‚ D F

s I#N'u+( $’) d|!} _I` o#~ Bu‚UƒF

s nquo€iK Rƒo }&„ q^… | ~ Bu‚˜>’F

† v s

v q nquoŒ_c ns nquouo }u‰Š s nquo‹Œ_ˆR o Š R ns nquoo … |!}‡„V‰ i UWYX } G q€… | ~ Bu‚˜E’F

s n$| og!h ‘ BC‡rˆ’F

(12)

£ .oŽ

`

-N’e M'e

c SUWŽXAZtUWŽX

)ƒ#G$. L

c ngVo^A(g_ !ngo‡_c

& ngo } „ gKi UW X }

‡'!

ATi UW XeX

#

i& nUW X ~UW(o

4+

c & ƒ" 'G$G-!-

G

c

& ngVo Š c & no g Š ' }.„ g } i UW(X }

% "!#6!

|

'e"

c M!G!-!- IV+

cb_

I.VŽV N+*+ G'!

…

^ M

& † Š AYg

}

gV‹_

‘

X-YT]–…+gEzqbhg+ijblkBmTgrfFbdm[]v>eTgEflbli|t!bdmCBu‚˜E“FwvgrmAgrfhpqk“—8]•zqbliuij])mAgEp1ijY[] s bl«8]vzq]vm>iqbhgrf&bdmTwfdeTpqblkBm

v s

v q nquoŒ_ˆc ns nquouo€i ŠY‚ R ns nquouo } g[š]Ešq€id|!} } BC‡o‚NF

`oe[n[n8kBpq]†ijYTg+i-iqYT]%gBpqpqe[\2noijblkBmTp1kr™X-Y[]vkEzj])\ ‚%grzj]¾pjg+ijbdpuHT] s grm ss ]vm[krij]¾—cts n&‘ ~ug h oŸiqYT]–eTm[bd>e[]

pqkEfdeoijblkBm kr™yKzjkE—[fd])\ lng h o Bápq])]Bu‚NF˜Bá^“FuFš : ]i$eTpY[]vzq] gEfdpqk s ]vm[krij]œ—ct nc } R o!iqYT] pq]ikE™

puijgriqbdkEmTgEzqt!pqkEfdeoijblkBmTp1kr™ŽB|‚˜E“F¡”obCš]Bš

nc }Rƒo€T]

gKi UW X S †c n

go } Š

gV‹Œ_R n8o Š R n

g o€… |e}.„ i,UW X a ‘

†krij]¾ijYTg+i

nc } Rƒo^ ]

gKi UW X S

g, lŒn

g Š c n g oa ‘

pTkEzŽghIi, Rƒo¡”[•1] s ])m[kEiq]—ct nghLoijY[] ")"G

ng h o S ]˜s n ~g h o#~ … | a ‘

: ]i$e*p!m[k+• gEpjpqe[\2]A•bžijY[kEeoiflk>pqp!kr™‰E]vm[])zgrfdbli|t iqYTgri$ijY[]Aiqzjbd…>bhgrf–pukBfle[iqbdkEm | bhp$g.puijgriqbdkEmTgEzqt

pqkEfdeoijblkBmkE™€B|‚NE’F”[b<š]Eš

c n$| o€i ŠY‚ R n$| o ‘ BC‡E‡*F

X-YT])m

s n&‘ ~ | oF |e‘

jÞ]2\/gUtmTk+• s ] H*m[]/gEp–blm <‚NN?:iqYT]2puijgE—[bdflbli|tœkr™ŸiqY[]Liqzjbd…>bhgrfOpqkEfdeoiqbdkEm šX-Y[]/p|ig+ijblkBmTgrzjtApukBfleoijblkBm

| bhp–wvgrfdfl] s puijgE—[fd]“bl™1pq\/grfdfn8])zqiqeTzq—TgriqbdkEmTp–kE™KiqY[]2bdm[bliqbhgrfKwkBm s bliqbdkEm[s nq.hLo  | fd]vg s iqkApqkEfdeoiqbdkEmTp

•YTbdwYzq]v\/grbdmbdmiqY[]m[]vbl‰BYc—*kBzqY[kck s kE™ | ™ kBz3grfdf q^… | ”on[zj]vw)bdpq])fdtQ

£D  N§

`

'!A -G)"#G- !G g[ | "+,.,) .*)" G”'e""e"Au% ƒ* V- %

% # | 'e"’#$  n o | #-!' 'e•% g h iC n$‚ Rƒo ƒG' #g h B ˆ'e

"-$ s n

~ug h o u%€V *)" lng h o "€# s nq#~ug h o }„ q … |e‘

(13)

’

™bdm$g s[s

bliqbdkEm/ijY[]†ijzjgr{|]vwiqkEzjbd]vpOkE™&iqY[]¾n*]vzuije[zq—8]

s

pukBfleoijblkBmTpŸgEzq]¾g+iuijzjgBw¡ij]

s

—ct

|

iqYT])m!•1]¾pjgUtijYTg+i

ijY[]puijg+ijblkBmTgrzjt!pqkEfdeoiqbdkEmbdpgBputc\2noijkriqbhw)gEflfdt$puijgE—[fl]B”on[zj]vwbhpq])fdtVQ

£D  N§

c

'! -G)"#G- !G g | KL#Y6$G .*)",% GI #6’) KV+

'e"’#$ | #-! '['e•% A ghAiC Rƒo ƒGG' #g!h [G'!"- s n~ughNo u%

*)"" lng!hLo %- G

8;6

3 x

#s nq#~g!hLoY |e‘

£D  N§

i

'!"-G) #G- !G g  | I-e#.*)"% IV #.*)"

X-YT]“™ kBflfdk+•bdm[‰zj]vpqe[fli‰E])mT])zgrfdb9v]vp†iqkAe[m[bdfhg+iq]vzjgEfOputopuiq])\/p–ijY[]:tBgEn[e[m[k+… p s blzj]vwi\L])iqY[k s š/X-Y[]

gEn[n[zjkBgBwY!\/g5E]Npgrm]vpjpu]vm>iqbhgrf&eTpq]%kr™OpukB\L]gEeoxoblfdbdgEzqt2™ eTmTw¡ijblkBm i &*nUW(X~uUWYoš

£c§ Ž £ c

-N*!L"8G'! G'!IL#-Y$e8u%

'e M6-“ 'e"ƒG' V+G

'!+

-N’e M'e€'e"M ’$ | V+ i& nUW X ~uUW8o #-! 'KG'!

n| | ~

ngo |e}.„ g,i  n$‚ R ok? "!}g8b | ~

#

†c ngo } !ngou‹Œ_R ngo Š R ng Š ! ngouo … |e}‡„ gKi, n$‚ Rƒok? !

'! [G'!#˜•"-$ &%$%'& (*)+ #.*)"

£ .oŽ

c -, #-(V! nNo V+ n*o G

'!".ˆ"e#- 'e ’."V,&%K[ *VG

/ S d|e} ZtUW "%G /

n$| | V+

/ nquo |!}.„ q i n|e} V+ #-!'G'! ngVoŽ…

/ n g<˜o }.„ gKi

 RƒoF? !

'!Ž ’VG /

n&‘o +* +I)"

/ no 78;:*)˜]Z ngVo S/B>#g a n… | o ^#-G.*)"

G0, #-Y$ n21 o G 'e 3 G(V ^'e

†c ngo } !ngou‹_

† } !ngou‹Ž… |e}&„ i ‚ R ngo }&„ gKi

 RƒoR? 4!

GG45% @ +’ .#8VV"Y$" + "" V+ R*? 'e"KL#-Y$ n1 o G

'e"

G(V 'e

†c ngo } B!ngVo‹8… |e}.„ g i2@ ? !ˆ+ g Š !ngoYi2@ }.„ g i2@ ? ! ' L#

V # "e#- (G'! Š !$ngVo i ? ngVo }.„ g i ‚ @>?> ! ‡' ' '!’ ."#˜ G'!I L"V" $ u%

ƒGG' $! ^6AG'!)-V+*#4&% @

6", ’"V" #C&%

'! "7 [G89

Références

Documents relatifs

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt -