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�
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
,
(-(.$/ 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+c_a >zjbdf rEE^_a[%nTgrB]vp
$v)+[EE m6iqY[bhpnTgEn*]vzv[1] s ]v E]vflkBng2\/g+ijY[])\/griqbhw)grfiqkckEf&ijYTg+iw)grm68]eTpq] s iqk!puijgriq]mT]vw]Np|
pjgrzjtw)kEm s bliqbdkEm*pOkr&gEpqtc\Ln[iqkrijbdwpuijgE[bdflbli|tkEFbhpqkEfhg+iq] s p|ig+iqbdkEm*grzjtpukBfle[iqbdkEmTp:kE&gw)fdgBpqp:kE&e[m[bdfdgriq])zgrf
s t>m*gr\2bdwvgrfputopuiq])\/pvkEzj]n[zj]vw)bdpq])fdtE8m[kEmTflbdm[]vgEz]v EkEfdeoijblkBm +grzjbdgriqbdkEmTgEfblmT]v>eTgrfdbliqbd]vpgrzj]wkBmTpqbs
]vzq]
s
mTp|igr[bdfdbi|t2wzjbij])zjbdg%gEzq]3gEfdpqkEbd E]vmn[n[fdbdwvg+ijblkBmTpwvgrm/8] kEe[m
s
blm!\2]NwYTgrm[bhw)pKkEzK]vfl]Nw¡iqzjbhw)grf
w)blzweTbip)
¢A£>¤F¥j¦§ +¨Eª© grzjbdgriqbdkEmTgEfbdm[]v>eTgEflbliqbd]vpv s bl«F])zj])m>iqbhgrfObdmTwfdeTpqblkBmTpv&iqkBn*kBflkBEbhw)grf s ]vEzj])]Bp|igr[bdflbli|t
ijY[])kBzqt! kEze[m[bdfhg+iq]vzjgEf s tcmTgr\2bhw)grfpqtop|ij])\/pv
¬K ®¯*°%²±³r´ µU¶·¸|¹<º<¶»²¼¾½¿[ÀcÁ-® Â|ÃUµN¶Ä½¡µ>³EÅNÁ¶Äµ»<ƲǸ|µU¶Äºq³UÈÉjÊË¡ËN³rÌ+¹CÁµUÍ|¸¡³EÇ:ÁµU¶Ä¸|ÎhÏн)¸|Îĸ|·¸|µrÑÃNµU¶·vÆh¹<¸|ÃUµU¶Ä½¡µBÏ¿d¹
Ò
Ó®²±Ô® ÕUÖ¡µN¸|Æh±ÎÄ×r¸|ºq³Ør ®Å)ÙÚ°%½¡µ)»<Ûr½¡µUµU½»u³Ü¡Ý¡ÝÞÁ|·¸|µvÃU¸ßU¸Îhà¯*ÃU¹á½¡×r¸³%â¡ã¡â¡âÊ.ÅNÁ¶Äµ»<Æh²ºCä¶Ä¸|¹u³Ì+¹<ÁµUÍ|¸¡³
å ¸u¹áµ+Á¹áß>Ïå ¹á½¡æ¡ÎĶçÁj»C½)ѶĵU¹á¶çÁÎÄ×r¸|ºqÏ¿l¹
,
2 0
,
/3 0
,
(
,
])i$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 bijblkBmTpLmvwvpjpqgEblzj]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&¾mw)kEmTpqbs Z)zj] s ]Np bdmNBe*g+iqbdkEm*p +grzjbhg+iqbdkEmTm[])fdfl]Np s t>m*gr\2bd>e[]Np)
¾]vpgEn[n[fdbdwvg+ijblkBmTppu]ijzqkBe[ E]vmBi s gEmTp-fd]vpwbdzwe[blijpvfl]Nw¡ijzqbh>e[]vp1kEe6fd]vppqtop|ijZ)\2]vp-\vwvgrm[bh>e[]vpv
§
N
¥
LÔ
m v>eTg+ijblkBmTpK +gEzqbhg+ijblkBm[m[])fdfd]vpvrblm*wfdeTpubdkEm*p
s
bl«!vzq]vm>iqbd])fdfl]Np)
s
])Bz-ijkEn8kEfdkEBbd>e[]B+iqYvkr
zjbd]
s
]fdg2puijgrTblfdbi"%n*kBe[z3fd]vppqtcpuiqZv\2]vp
s
tcmTgE\Lbh>e[]vp-eTm[blfhg+i")zgreox&
1 2
:/
,
X-YT] puijgrTblfdbi|t kEp|ig+iqbdkEm*grzjt pqkEfdeoiqbdkEm*p!kr
s
t>m*gr\2bdwvgrf%putopuiq]v\2pwkBmTp|ijbijeoiq]Np6g E]vzqt bl\2n8kEzqijgEmBi
ijkEn[bhw%bdm6n[n[fdbl] s g+ijY[])\/g+ijbdwvp3grm s43 mTEbdm[])]vzqbdm[T i¾bdp-1])fdf65cm[k+miqYTgri3blmiqYT]w)gEpq]%krOgLfdgEzqB]
w)fdgBpqpkrKm[kBm[fdblm[]Ngrz s b«F])zj])m>ijbdgEf]NBe*g+iqbdkEm*p)*iqY[]Lpun8]vwiqzje[\ kr87flbdm[]Ngrzjb9v] s 7kBn*]vzjgriqkEzp s ])iq]vzq\2bdm[]vp
ijY[]4: tBgrnTe[m[k+ puijgrTblfdbi|t kEzbdmTpuijgE[blfdbli|t krgrm ]N>e[blfdbd[zqbde[\6X-YTbdpbhp5cm[k+mWgEpiqY[]4: tBgEn[e[m[k+ p
fdbdm[]vgEzqb;9vgriqbdkEm \2])iqY[k s=<>>*>o 0 ^?CA@k+]v E])zN&\/grmctbd\2n*kBzuigrm>in[zjkE[fd])\/p%bdmÞ]vm[Ebdm[]v])zjblm[CBápq])]
<D -v[1Br1 0 -)^T+>KEo 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$n8kBpjpu]Npqp-bdm>iqzjblmTpqbhwm[kEm*pu\2kckrijY[m[]vpjpY[bhwYAw)gEm[m[kri¾*]zj])\2k+ B] s >t6wYTgEm[E]kE:w)k>kBzu
s blmTgriq]Np)kBz%>t6 ]v] s *gEw5FX-Y[]2\2k s ])fhpwkBzqzj]vpqn8kEm s bdm[$iqkApue*wY n[zjkE[fd])\/p%grzj]2w)grfdfd] s e[m[bdfdgriq])zgrf
s t>m*gr\2bdwvgrf&pqtcpuiq]v\/pkBzmTkEmoRzq]vEe[fhgrz s tcmTgr\2bhw)gEfFpqtop|ij])\/pv: fhgrzjE]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\ kr¾]) EkBfle[iqbdkEm. +grzjbhg+iqbdkEm*grfbdm[]NBe*grfdbijbl]Np)pq])]
]B* <0 - 0 v^?RWX-YT]>e[]Np|ijblkBm.kr¾puijgE[bdflbli|tWkr¾puijgriqbdkEmTgEzqtWpukBfle[iqbdkEmTpkr]v EkEfdeoijblkBm. +grzjbdgriqbdkEmTgEf
bdm[]NBe*grfdbijbl]NpbhpL\eTwY\2kEzj]wkB\2n[flbhw)griq] s iqkÞ8]bdmc E]vpuiqbdBgriq] s KgEpLbliLbhpLiqY[]6w)gBpu]blm E]vm[])zgrf kEz
Yctc[zjbss tcmTgE\Lbhw)gEfputopuiq]v\2pv[pq])]%]ET <ro*[#?R
X-YT]Kzq]Npu]NgrzwY%]«FkEzqijpFijk s ]v E])fdkEn%B])m[]vzjgEfr\/g+iqYT])\/g+ijbdwvgrfBgEn[n[zjkBgBwY[]vp8ijk¾puiqe s t%puijgrTblfdbi|tkrTpuijgriqbdkEmo
gEzqtpukBfleoijblkBmTpkE:]v EkBfleoijblkBm +gEzqbhg+ijblkBmTgrf&bdm[]v>eTgEflbliqbd]vp¾grzj]%zq]vfdgriqbd E])fdt!m[]v ipu]v])\/p-iqY*g+i3ijY[]IHTzp|i
w)kEm>iqzjbd[eoiqbdkEmgBw¡ijblm[$blm6ijY[bdppq])mTpq]-gEp3\/g s ]ctKJ¾eTbiqiqm[]vz < 0
&+^T8EL? kBz¾g!w)fdgBpqpkrOnTgrzgr8kEfdbhw
+gEzqbhg+ijblkBmTgrfrbdm[]NBe*grfdbijbl]NpbdmM@3bdfl8])zqipqnTgBw]vpv AkBzq]zq]Nw])m>ijfltBN: tBgrnTe[m[k+ ¾gEn[n[zjkBgBwY%gEm sIO
zg5Ek+ op5cblbl
:gB`ogEflfd]6blmc +grzjbhgrmTw)]$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
blmTe[m[bdfdgriq])zgrf
s
tcmTgr\2bhw)gEf*pqtcpuiq]v\/p8B<pu]v]
<
rFNc>^[[c8NcoEN?gEm
s
iqY[]zj] ])zj])m*w]vpwbliq]
s
ijY[])zj])bdmF m <NN?pueP!w)bl]vmBi3w)kEm s bliqbdkEm*p1 kEz3puijgrTblfdbi|t$krfdbdm[]vgEz]) EkBfle[iqbdkEm UgEzqbhg+ijblkBmTgrf8bdm[]v>eTgEflbliqbd]vp
Y*gU E]*]v])m s ])zjbd E] s [grm s ]%m[k+Hn[zjkEn8kBpq]¾ijk/p|ije s t$mT]vw]Npqpjgrzjt!wkBm s bliqbdkEmTpv
¾eTzLgEbl\ bdm.iqYTbdpLnTgEn*]vzLbhpiqkWp|ig+iq]m[]vw)]vpjpqgEzqtWwkBm s bliqbdkEmTpkrgEpqt>\2noijkrijbdwp|igr[bdflbli|tÞ kEz/g w)fdgBpqp
kEe[mTblfhg+ij])zgrf s tcmTgr\2bhw)gEfKputopuiq])\/pvLkEzj]n[zj]vw)bdpq])fdtE ]/wkBmTpqbs ]vziqY[]2n[zjkE[fd])\,QI:])iRTSVUWYX[Z
UWK\^]_I`ba8]Lg$n[zqkBn*]vzv&w)kEmc E])x6gEm s fdk+]vz¾pq])\2blRw)kEm>iqbdmce[kEeTp3 e[mTwiqbdkEm: ]iIcdSeUW X ZfUW X
*]
g2mTkEm[fdblmT]vgrz¾wkEm>ijblmce[kBeTpkEn8])zg+ijkEzN^: ]i(ghiKUW X
*]Ebd E]vm^jÞ]w)kEmTpqbs ])zijY[]k § £ mlng!ho#Q
pbdm s gLe[m[bhBeT]wkEm>ijblmce[kBeTp1\2gEn[n[bdm[qrZts nquo1pueTwYijYTg+i
v s
v q i,w^x
yz#{
n|!} _I`~uUW X o ~ B|NF
s nquoiK n Ro }& q^ | ~ B<*F
v s
v q n quo_c
ns n
quouo }M
s n quou_R
n
o R ns n quouo
|e}
i,UW X } gT]Bq^
|e}
B0 F
gEm s s n$| og h B ^F
X-YT] UgEzqbhg+ijblkBmTgrf&bdm[]NBe*grfdbi|tblmB
0
Fw)gEmAgrfhpuk28] kEzj\e[fhg+ij] s gEp1ijY[]pq]iqC +grfde[] ss b«F])zj])m>iqbhgrf ]v>eTg+
ijblkBmQ
v s
v q nquo_c ns nquooi Y R ns nquoo} gT]Bq^ |e} B<*F
YT])zj]
R s
]vm[krij]vpiqYT]%pqe[
s
bl«8]vzq]vmBijbdgEf&krR1
ibhpgEpjpue[\2]
s
iqY*g+iYc n|
oi
Y
R
n|
oKbC]B:iqY[]%kBzqbdEbdm
|
kE iqYT]putopuiq])\bdpgijzqbd cbdgEfp|ig+iqbdkEm*grzjt!pukBfleoijblkBmkE BC*F
R
|
iqYT])m[BC*FKzq]
s
eTw]NpKijkgp|igrm
s
gEz
s
putopuiq])\ kE
s
b«F])zj])m>ijbdgEf*]NBe*g+iqbdkEm*pgEm
s
bli1bhp1])fdf5cm[k+m
B<pu]v]3]ET
<
EL?FObdm2iqY[bhpwvgEpq]-iqYTgribl
|
bdpgrm/bhpukBfdgriq]
s
9v])zjkkrc grm
s
bhpgBputc\2noijkriqbhw)gEflfdtpuijgE[fd]ErijY[])m
ijY[])zj]%]xobhp|ip h=| pueTwYijYTg+i
n
c
} }|
o
}&
i n|e}
h
}
YT])zj] Vnc }
}| o s ]vm[krij]vp¾iqYT]2}1zjkEec1])z s ])Ezj])]Lkr cÚbliqYÞzq]Npun8]vwiijk$ijY[]/kEn8])m Tgrfdf S
]NgKi UWYX S#gagEm s,|
¾eTzBkBgrfblmiqYTbdpLnTgEn*]vzbhpiqkÞE])mT])zgrfdb9v]/iqY[bhpágr\2kEe*pzj]vpqe[fliijk ijY[]\2k s ]vf1bdm Bá^FBC*F¡.kEzj]
nTzq]Nwbhpu]vfltBofl])i-e*p s ] H*m[]%iqY[]\/gEn[n[bdm[ gEp
ngVo Sbg lng c ngVoo
YT])zj] l S n.v !#" _ Ro%$'&gEm s(.v)!#"6s ])m[kEiq]NpiqYT]bs ])m>iqbli|t\/gEn[n[bdm[!kEmKUWYXFjÞ]blfdfn[zqk+ B]
ijYTg+i3bl | bdp3gEmbhpqkEfhg+iq] s 9v])zjk2kr* grm s bhp3gEpqtc\Ln[iqkrijbdwvgrfdfltp|igr[fd]EcijY[])m6ijY[])zj]%]xobdpuijp+ h | pue*wY
ijYTg+i
n }}| o }& i n|!}h,
X-YTbdp-zj]vpqe[fliwkEm*p|ijbijeoiq]Np1iqY[]\/grbdm6zq]NpueTfikriqY[]n*grn8])z3grm s bhp-Ebd E]vmbdmAX-Y[])kBzq]v\Ôo
m6`c]vwiqbdkEmAoc1]¾zj]vwvgrfdf*ijY[] e[m s gE\2])m>ijgEf8nTzqkBn*]vzuijbl]NpkEiqYT]%}1zjkEec1])z s ])Ezj])]1]¾bdflf&e*pu]blmiqYTbdp
n*grn8])zN m `c]Nw¡iqbdkEm 0 ] s bhpjweTpjpiqY[]/\/gEblmÞn[zjkEn8])zqiqbd]vp%kriqY[]/kBn*]vzjgriqkBz l m.`c]Nw¡ijblkBm ^*]
nTzq]Npu]vmBigAw)kEmTw)bdpq]2zq]v >bd]) krpqkE\2]2zq]Nw])m>izj]vpqe[flijpbdm p|igr[bdflbli|tijY[])kBzqt kr-e[m[bdfdgriq])zgrf s tcmTgr\2bhw)grf
pqtop|ij])\/pv m `c]Nw¡ijblkBm o]!bdm>iqzjk s eTw]2ijY[]!y:kEbdmTwvgrz"/kEn8])zg+iqkBzgEpjpukow)bdgriq] s ijkAn[zqkB[fd])\ B<*F m
`o]vw¡ijblkBm D [1]nTzqk+ B]¾kBe[z-\2gEblm6zj]vpqe[fliv
mLijY[bdpKnTgrn8])zNr1] s ]v E])fdkEn grfhpuk pu]v E])zgrfLgrn[nTzqk>gEwY[]NpijkwkE\2n[e[iq]-iqYT]m>eT\8])z Vn }- }| o¡
wvgrm*]]NgEpqblfdt]) +grfdeTgriq] s iqYT])m/gYH*zjpuiKgrn[n[zjkBgBwYwkBmTpqbdpui:iqk%eTpq]-pqkE\2]1n[zgEw¡ijbdwvgrfczj]vpqe[fip:kr s ])Bzq]v]
ijY[])kBzqtiqk w)kE\2n[eoij] n } }| o B<pu]v]$`c]vwiqbdkEm D F¡X-YT]$grn[nTzqk>gEwY bhpkE1n*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]3bhpKkEnTgrzqiqbhwe[fhgrzblm>iq]vzq]Np|i kEzKijY[]¾puiqe s tkE wkB\LnTfl]v\L]vm>ijgrzjbli|t s tcmTgE\Lbhw)gEf*pqtop|ij])\/p < D ?R
kEzj]E])mT])zgrfdfltBob R=>/? YT])zj]@ s ])m[kEiq]Npg/wfdkBpq] s w)kEmc E])xpq]ivTiqY[]vmA] s ]) E]vflkBnAg!pu]NwkBm s
gEn[n[zjkBgBwY[Bápq])]`o]vw¡ijblkBm FiqY*g+ieTpq]vpg2zq]Npue[flikr^J¾e[bliuijm[])z <EL? Y[bhwYzj] s eTw]Np-iqY[]w)kE\2n[eoig+ijblkBm
kE Vn }-A}| oiqkLiqYT]kEm[]%kr VnCB }}| oYT])zj]D5Bbhp s ]"HTm[] s ct
Bngo g l ?FE ng HG In| o.gVo }
YT])zj]F@ h bdpiqY[]wfdkBpq] s wkBm> B]x¾w)kEm[] s ] HTm[] s ct@ h \*JK hML @ gEm sNG In| o s ])mTkriq]NpFiqYT]CO>gEwkB[bhgrm
\/griqzjbx$kEc g+i |
R bhp2gÞw)kEmc E])x.grm s wkBmBijblmce[kBeTp e[mTwiqbdkEmiqY[]vm 1] s ]) B])fdkEn gijY[blz s grn[n[zjkBgBwY Bápq])]`c]Nw¡ijblkBm
>Fct pqY[k+blmTijYTg+i Pn } } | o%w)gEmW*]$w)kE\2n[eoij] s ct eTpqbdm[pukB\2]!grn[n[zjkEnTzqbhg+ij],7":tBgEn[e[m[k+
eTmTw¡ijblkBm7+
X-YT]LwvgEpq]krflbdm[]vgEzwkE\2n[fd])\2]vmBigrzjbi|tn[zjkE[fd])\/pbhp
s
bdpjweTpjpq]
s
bdmW`o]vw¡ijblkBm[ETpbdmTgEflfdtE&bdmÞ`c]Nw¡ijblkBm
N[Ke*pubdm[ÞkEe[z2\/grbdm zj]vpqe[fliijkEE])iqY[]vz2bijYzj]vw)])m>iLbdmc +grzjbdgEmBiLzq]NpueTfip B<pu]v]
<
#?F¡11]n[zjk+ E]pukB\2]
bdmTpuijgE[blfdbli|t/iqY[]vkEzj])\/pv
`o]) E]vzjgEfF]x[gr\2n[fd]vpgEm
s
gEn[n[fdbdwvg+ijblkBmTpgrzj]¾Bbl B])m6puk/gBp1iqk2blfdfdeTp|ijzjgriq]iqYT]%iqY[]vkEzj]iqbhw)gEf&zq]NpueTfip)
A§ v[§ EKm$iqY[]%pq]v>e[])f*iqYT]pjw)gEfdgEzn[zjk s eTw¡i1kEm UW X bhp s ])m[kEiq] s ct
}
(BábijY!iqYT]gBpqpqkowbhg+iq] s
mTkEzj\
"F(p[kBz | *]Lpu])i S ]g i UW X S g< aB¾X-YT])m ]gCi UW X S #ga
gEm s ]NgKi,UW X S#ga
X-YT]1bs ]vmBijbi|t\2gEn[n[bdm[%kEmUW X bdp s ])m[kEiq] s ct .v)! "F: ]iF@ *]3g¾m[kEmT])\2noi|tw)flk>pu] s wkBmc E]xpqe[Tpq]i
kE UW X OX-Y[] s eTgrf8pu])ikr @ bhp s ]"HTm[] s ct @8S ]Ti,UW X S }g |!}. g i(@a>X-YT]zj]vw]NpqpqbdkEm
w)kEm[]kr @ bhp s ] HTm[] s ct @ x S \)K h & n@ g h o3Y[])zj]Mg h bhp¾gEm>t])fd])\2]vmBikr @ Yj ] s ])mTkriq]
ct*? iqY[]bdm s bhw)griqkEz1 e[m*w¡iqbdkEmAkE<@ ob<]E*? ngo | blg i @ grm s *? ngo _I`0b gi @.
pTkEz3g e[mTwiqbdkEm i& nUW X ~UW(o1] s ])m[kEiq]%ct !ngVoiqYT]Ezg s bd])m>ikr g+ig i UW X
jÞ] s ])m[kEiq]1ct vn#"*}%$ o iqY[] s bhp|igrmTw)] zjkE\g¾n*kBblm>i " i UW(X%iqk%g¾pq]i $ & UWX&UiqYTgriObdp vVn'"}($ oS
8:*),+-/. "10 r
2 3
()) 0)Ô54W" ()61(87 96::4<;
: ]iA & UW X 8]grm kEn8])m grm s 8kEe[m s ] s pq]iv: ]i = i>& nK~uUW X o@?8
h n K~uUW X oL*]ABbl B])m
`o]iBAC n4o S ]Ngi SD= ngVo | a$grm s C n4o S ]Ngbib S FEG C ngVo | a!YT])zj] G C ngVo
s
])m[kEiq]vpLijY[]2O>gEw)kE[bhgrm g+ijzqblx kr= griAg
s
]"HTm[]
s
ct
nG C o4HG SJI CLK
INMO
n
go¡
}#P
i ] }";;}LQ
a>
AC n4oR? C n4o S2grm s | i= n 4oiqYT])mAC n4obdp3gHTmTbij]pu])igrm s iqYT]}zjkEec1])z1iqkBn*kBflkBEbhw)gEf
s ])Ezj])]krT=AbliqY6zj]vpqn*]Nw¡i-iqkA gEm s | bhp1])fdf s ] H*m[] s ct/iqY[]% kBzq\e[fhg
Pn = } } | o U
M
-/VXWZY\[T]^
: n' FE G C ngVoo }
YT])zj]2pqEm nquo kBzIq | grm s pqEm nquo kBzIq | /AkBzq]2B])m[]vzjgEflfdtE&bl9==S_ Z UW X bdp
w)kEm>iqbdmce[kEe*p1gEm s,| i= n 4oiqY[]vmiqY[]}1zjkEec1])ziqkEn8kEfdkEBbdwvgrf s ]vEzj])]%krT=AbliqY6zj]vpqn*]Nw¡iiqk grm s
| bhp-]vflfls ] HTmT] s Bápq])]%]ET <>?GF-gEm ss ])m[kEiq] s ct n= } } | o¡
: ]ieTpmTk+ zj]vw)gEflf&pqkE\2]%n[zjkEn8])zqiqbd]vp-kriqYT]%iqkEn8kEfdkEBbdwvgrf s ])Bzq]v]%]%bdflfe*pu]%fhg+iq]vzblmijY[bdp3nTgrn8])zN
`a
| i= n 4o1grm s Pn= } }| ob | iqY[]vmijY[])zj]]xobhp|ip gKiKpqeTwYiqYTgri= ngo |
cRa
: ]iIRSRd|e} Def ZfUW X ~ n'g} goYrZ R n'g}go¡8]Lw)kEm>iqbdmce[kEe*ppqeTwYAiqY*g+iv8 kBz¾]NgEwY g ihd|e} C
kBm[]%YTgEp | i,R n'g} 4oX-Y[])mijY[]\/grn g rZ Vn R n'g}"o } }| o-bhp3wkBmTp|igrm>i-kEm8d|!} C
iXa
j ]YTgU B]
Vn.v !#" } } | o
jka
| i= n 4o1grm s L==| iqYT])m
Pn L = } }| o Pn= } }| o
gEm
s
Vnu+L
= } }|
o
nu
No X Vn
= } }|
o
Ra :]i(
! &
8]grm6kEn8])m6pu])ipue*wY$ijYTg+iA C n
4o
& !
KX-YT])m
Vn= } } | o Pn= } !} | o
Xa :]iATi UW XX *]gLm[kBmTpqblm[Be[fhgrz-\/g+iqzjblxFOX-Y[]vm n
A
}
}|
o
^ : n FE A o¡
a :]i= S d }
_MZtUW 8]g2wkEm>ijblmce[kBeTp e[mTw¡ijblkBm`ceTn[n*k>pu]ijYTg+i=
nu
ob
|
grm s = n
_ob |
X-YT])m
Pn
= } } _*d
} |
o
|
8\)8=
nu
o(=
n
_o |e}
Vn= } } _*d}| o _NH8) = n& o | : = n_o |e}
Vn= } } _*d}| o H8) = n& o =| : = n_o |e
-:( 1
3
:K)Ú"Þ-%Ô
2
-:
: ]i RSUW X Z UWK\^]`ba¾8]¾gnTzqkBn*]vzv>wkEmc B]xLgEm s fdk+]vzpq])\2bwkBm>iqbdm>eTkEeTpO\/gEn[n[bdm[Tj ] s ])mTkriq]
ct vZ0 ]NR^a%iqY[] s kB\/grbdmkE R1[bC]B
vZ0 ]NR^aST]NgKi UWYX4SR ngVo+ _I`ba
jÞ] s ]vm[krij]ct R.iqY[]w)kEmc E])x$pue[ s bl«8]vzq]vm>iqbhgrfkER1K~]vwvgrfdf&iqY*g+i
R ngVo T] i UWYX S*R n o R ngo }u g }&V i,UWXVa
jÞ] s ]vm[krij]ct4 n Ro1iqY[] s kE\/grbdmkE R gEm s >t n RoiqY[]zgrm[B]¾kE R1[bC]B
n Ro ST]NgKi UWYX4S R ngVobS a } n RoS \ M - ! " R ngVo
~3]vw)gEflfFijYTg+i
n Ro &=vZ0 ]NR^a & n$ R o BD F
: ]iCiUWYX *]2Bbl B])mj ]/wkBmTpqbs ]vziqY[]2 +grzjbdgriqbdkEmTgEfbdm[]NBe*grfdbi|tAn[zjkETfl]v\ Q pblm s g=iUWX pue*wY
ijYTg+i
g }u g_R n o R ngo |!}. i UW X BCF
yzqkB[fl]v\ BRFKYTgEpge[m[bh>e[]¾pqkEfdeoiqbdkEm B<pu]v]]B* < 0 0
?GFOiqYTgri1] s ])mTkriq]>t lneo¡OX-Y[]kBn*]vzjgriqkBz
l
SUWXZtUW(X~ rZ l neo
bhp1iqYceTp3]vflfl
s
] HTm[]
s
i3bdpwfd]vgrz1ijYTg+i
l<nUW X o & n Ro B>F
gEm s l neo n.v !#" _ Ro $ & 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)griqkBz 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 Cl ? gY 698;:
-)?
#g N
: ]i SUWX Z UWYX.8]6g w)kEm>iqbdmce[kEeTp\/grn[nTblm[Wgrm s wkBmTpqbs ]vzijY[]blm[]N>eTgrfdbi|t nTzqkB[fl]v\,Q4pblm s
gKi,UWXpqeTwYiqYTgri
n
gVo }uM
g _R n o R n
g o |!}. i UW X BEF
ibhp-wfd]vgrziqY*g+in[zjkE[fd])\ B$EFbdp-]NBeTbl +grfd])m>i1iqkiqY[]%m[kBm[fdblm[]Ngrz1]v>eTgriqbdkEmQpblm s
gKi,UW X pue*wY!iqY*g+i
g
Cl n
g n
goo
|!
BuvF
§k§ á § `
'!e"6
l
IV !! G
l<!Cl<
M
}&VV}
Ai UW X
§§
: ]i
V}
i UW X 8]Ebd E]vm j ]pq]iYg S ln
ogrm s g IS
ln
oj ]Y*gU E]
g
[V}
gV_R
n o R n
gVo
|e}&
i,UW X BuENF
gEm
s g } g _R n8o R ng o |e}& i UW X BuN*F
`o]iuijblmT Stg Ablm BuELF2grm s fStg blm BuN*F1]AkE[ijgrbdm iqYT]Azq]vfdgriqbdkEmTp
g bV}g g "
R ng No_R ngVo | grm s
g }g g N R ngVo_R ng No | zjkE\bhwY.] s ] s e*w]!ijYTg+i
g g g g ) M + i kBflfdk+3p1iqYTgri
l n o Cl neo> M
: ]ieTpmTk+
s
]vm[krij]ct
SUWYX Z
! "
iqY[]pq]iqC +gEfle[]
s
kBn*]vzjgriqkBz
s
] HTm[]
s
>t
ngVo T]Z=Ki,UW X S g = }u g_R n o R ngo |!}. 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= o ln = o }. q |e BuN*F
krij]¾ijYTg+i
ngo ]
_R nuo#a ngo }& g i UW X }
pqk%iqY*g+i
bhpK\/g+xobd\/grfT\2kBm[krijkEm[]B
izj]vpqe[flijpKiqYTgri kEzgrmctq |
rijY[]kBn*]vzjgriqkBz n.v
!#"_Kq
o%$'&
bhpgL]vflf
s
] HTmT]
s
pubdm[Bfl])C +grfde[]
s
kEn8])zg+ijkEzN
(%-:( 7 2 01( 2/&3 0
: ]ieTp H*zjpuizq]Nw)grfdfgLE]vm[])zgrfF]xobhp|ij])mTw)]grm s e[m[bh>e[])mT]vpjp-krOpukBfleoijblkBmTp1zq]NpueTfiMB<pu]v]]B* <UN?GF
£c§ £ `
" R SUW X Z UW,\^]_I` a )K4V ! V+ ""u$G-!-
%- " c SUWYXAZ 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+( $) d|!} _I` o#~ BuUF
s nquoiK n Ro }& q^ | ~ Bu>F
v s
v q nquo_c ns nquouo }u s nquo_R n o R ns nquoo |!}V i UWYX } G q | ~ BuEF
s n$| og!h BCrF
£ .o
`
-Ne M'e
c SUWXAZtUWX
)#G$. L
c ngVo^A(g_ !ngo_c
& ngo } gKi UW X }
'!
ATi UW XeX
#
i& nUW X ~UW(o
4+
c & " 'G$G-!-
G
c
& ngVo c & no g ' }. g } i UW(X }
% "!#6!
|
'e"
c M!G!-!- IV+
cb_
I.VV N+*+ G'!
^ M
& AYg
}
gV_
X-YT] +gEzqbhg+ijblkBmTgrfFbdm[]v>eTgEflbli|t!bdmCBuEFwvgrmAgrfhpqk8]zqbliuij])mAgEp1ijY[] s bl«8]vzq]vm>iqbhgrf&bdmTwfdeTpqblkBm
v s
v q nquo_c ns nquouoi Y R ns nquouo } g[]Eqid|!} } BCoNF
`oe[n[n8kBpq]ijYTg+i-iqYT]%gBpqpqe[\2noijblkBmTp1krX-Y[]vkEzj])\ %grzj]¾pjg+ijbdpuHT] s grm ss ]vm[krij]¾cts n& ~ug h oiqYT]eTm[bd>e[]
pqkEfdeoijblkBm kryKzjkE[fd])\ lng h o Bápq])]BuNFBá^FuF : ]i$eTpY[]vzq] gEfdpqk s ]vm[krij]ct nc } R o!iqYT] pq]ikE
puijgriqbdkEmTgEzqt!pqkEfdeoijblkBmTp1krB|EF¡obC]B
nc }RoT]
gKi UW X S c n
go }
gV_R n8o R n
g o |e}. i,UW X a
krij]¾ijYTg+i
nc } Ro^ ]
gKi UW X S
g, ln
g c n g oa
pTkEzghIi, n Ro¡[1] s ])m[kEiq]ct nghLoijY[] ")"G
ng h o S ]s n ~g h o#~ | a
: ]i$e*p!m[k+ gEpjpqe[\2]AbijY[kEeoiflk>pqp!krE]vm[])zgrfdbli|t iqYTgri$ijY[]Aiqzjbd >bhgrfpukBfle[iqbdkEm | bhp$g.puijgriqbdkEmTgEzqt
pqkEfdeoijblkBmkEB|NEF[b<]E
c n$| oi Y R n$| o BCE*F
X-YT])m
s n& ~ | oF |e
jÞ]2\/gUtmTk+ s ] H*m[]/gEpblm <NN?:iqYT]2puijgE[bdflbli|tkriqY[]Liqzjbd >bhgrfOpqkEfdeoiqbdkEm X-Y[]/p|ig+ijblkBmTgrzjtApukBfleoijblkBm
| bhpwvgrfdfl] s puijgE[fd]bl1pq\/grfdfn8])zqiqeTzqTgriqbdkEmTpkEKiqY[]2bdm[bliqbhgrfKwkBm s bliqbdkEm[s nq.hLo | fd]vg s iqkApqkEfdeoiqbdkEmTp
YTbdwYzq]v\/grbdmbdmiqY[]m[]vblBYc*kBzqY[kck s kE | kBz3grfdf q^ | on[zj]vw)bdpq])fdtQ
£D N§
`
'!A -G)"#G- !G g[ | "+,.,) .*)" G'e""e"Au% * V- %
% # | 'e"#$ n o | #-!' 'e% g h iC n$ Ro G' #g h B 'e
"-$ s n
~ug h o u%V *)" lng h o "# s nq#~ug h o } q |e
bdm$g s[s
bliqbdkEm/ijY[]ijzjgr{|]vwiqkEzjbd]vpOkE&iqY[]¾n*]vzuije[zq8]
s
pukBfleoijblkBmTpgEzq]¾g+iuijzjgBw¡ij]
s
ct
|
iqYT])m!1]¾pjgUtijYTg+i
ijY[]puijg+ijblkBmTgrzjt!pqkEfdeoiqbdkEmbdpgBputc\2noijkriqbhw)gEflfdt$puijgE[fl]Bon[zj]vwbhpq])fdtVQ
£D N§
c
'! -G)"#G- !G g | KL#Y6$G .*)",% GI #6) KV+
'e"#$ | #-! '['e% A ghAiC n Ro GG' #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#.*)"% GIV #.*)"
X-YT] kBflfdk+bdm[zj]vpqe[fliE])mT])zgrfdb9v]vpiqkAe[m[bdfhg+iq]vzjgEfOputopuiq])\/pijY[]:tBgEn[e[m[k+ p s blzj]vwi\L])iqY[k s /X-Y[]
gEn[n[zjkBgBwY!\/g5E]Npgrm]vpjpu]vm>iqbhgrf&eTpq]%krOpukB\L]gEeoxoblfdbdgEzqt2 eTmTw¡ijblkBm i &*nUW(X~uUWYo
£c§ £ c
-N*!L"8G'! G'!IL#-Y$e8u%
'e M6- 'e"G' V+G
'!+
-Ne M'e'e"M $ | V+ i& nUW X ~uUW8o #-! 'KG'!
n| o | ~
ngo |e}. g,i n$ R ok? "!}g8b | ~
#
c ngo } !ngou_R ngo R ng ! ngouo |e} gKi, n$ Rok? !
'! [G'!#"-$ &%$%'& (*)+ #.*)"
£ .o
c -, #-(V! nNo V+ n*o G
'!"."e#- 'e ."V,&%K[ *VG
/ S d|e} ZtUW "%G /
n$| o | V+
/ nquo |!}. q i n|e} V+ #-!'G'! ngVo
/ n g<o }. gKi
n RoF? !
'! 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 |e}& i R ngo }& gKi
n RoR? 4!
GG45% @ + .#8VV"Y$" + "" V+ R*? 'e"KL#-Y$ n1 o G
'e"
G(V 'e
c ngo } B!ngVo8 |e}. g i2@ ? !+ g !ngoYi2@ }. g i2@ ? ! ' L#
V # "e#- (G'! !$ngVo i ? ngVo }. g i @>?> ! ' ' '! ."# G'!I L"V" $ u%
GG' $! ^6AG'!)-V+*#4&% @
6", "V" #C&%
'! "7 [G89
