ࡱ> z b}ȈPڣdvLB(>#.#.`<>?????~?~???~?????|~???>?|~? ><>`>?~????????nWJUzfip4ePNG  IHDRMl`dgAMA|Q pHYs.#.#x?vdIDATh;6`*TM@"\$GSL:a"Å 2\L+Í Đ?IβYK"?VX6=,)A1/Eh9 8'u+G]Q|-H5" (Zb(1T}T8&ۨQ˓\ٸ'"Q2Zjp1+[کAC]\b?f%UF5Uѭ%O@L ΜFf DjkLYjU/.DTujohY$iJ?ҷe+PTLE)ƃ"V>5iJU jaYjRTߴjJ^Tށ҃H5JL\+ 'W7.ceExTk%Z%K$GR+cJd e[T]+5Tcb(..b]@B,R75G-jH֎Ijj,mTAaà.WCiRWDkbwQQR7Hv1TG;K\I! 6FC0 aFM2ZH5UESx:Ԥ7#X#m&"ތ%O ưkm@ .W j7Q R{Qaj}jz-vsQa/@g.M݁CU2W0I:b5֪#ĔX[My{G PQ. @1jYҽcMHsm,EnPPhneWr,GYrϴp³Ԡ)Ņxd鍲{QO,ը=$wkIfq`!Sv}y߈CTjR, ?lnS],u@%'(>l.SJW Ý0T\‰=n9 xHr?O5oR=1 Clu>n%7,x^yjX8+ n7Luf.-sx\/7@VK%sK:<貐9w!岾edѐc/Bvhj7VV)eA/]^5kd9ZWLDkC6'/ ;~NwSMxSc;tvI7=IENDB`nNѾ\V"*zPNG  IHDRB,gAMA|Q pHYs.#.#x?vIDATHǵ1n PG29J^,[^#02b I"}|y[ ymt4^A7tX:XX빆5 1kǮ sWᚖ*͛a[7ozԸ:y3FϽiKNbN^،hD K*zI.Zb])UAP.5[<x MSQ?5q-XkZqt 6*eZ5я^Ң] hםOJjBi~Ӛid[&olT?(Ո7ukϽo87!:81ں3XtS*c(-'9鷛w n栋>JXUPauA`A~96ԡe}}UCQgVIENDB``! pK)o) D@RxMP qfk 9 AQ<pQ&G'9eřf~+S|o~?BH]0PK9AL#0 Q59~ 4 7SD P<tLHx!q0 sśs *4WxY<`kذd+9{HA[:m.!5ҨŮvbn[_YHfpA&>2/'0zMґCN+]˟2< (-Ox#.#.|~~~~??????~~~~????n 8 ARo,PNG  IHDR^O=gAMA|Q pHYs.#.#x?vIDATHǵ1 P3nϰ߱4X2KV̼"e!3<\@)>=*p@*Avod %D6lPeXX Lxغ'3WeP3Y X8 l ]Z .tL5[[mcXX7a ;yQ )6&i.XmlQ,:xa (ޅ xOŹ7yXCcexMm (h@a(IENDB`n_SQJuPNG  IHDRl0VrgAMA|Q pHYs.#.#x?v(IDATh=n0`.Q;UWءusoЫn]%da$$*Q&b $~ޣi'j’6oPMv6RFF뮆,+ݍro/a(Uv68]G06.9qyÐ#u߈} 65цNkNbdIPlM:lԏ2FFR7r)ΐCT`:=TYLũQF n(ɢ5.5eTcZgyaC?7*͋,Ʈ|;0 6ca]v )+0j0L⫮_FyQ h `5 3Fsɰ%`jf 0d p ]ͬѢ{*HFIz0cAc&72Zʸ0( a`p׸R $CWSgxqxF%dƙ2r2fJF9a5䓌2HC[3tq6CPbŴl Nov NȀ1C3n^lX5过 %SF|Xkόsx?Y1Pn͸`{h'<1-[ }ħPsF{f(o7,s[nmw/q3^ѐfnK"[10 krpIENDB`n@azymM>޻PNG  IHDRIlgAMA|Q pHYs.#.#x?vIDAThڽn6`-6JLWH"^)@Z\y{Wl- d43JEolggeD #VVQͭ>::L9ԙoPqT JwFtQTF)c)s %A"Ap5#6<ߑ QJ5uzEuHѺ(Jc G;Od+NUTcMBd|IC)G Oa;3ܝsEJ^SD|=4d=E J])60{Ks\DTGgDM MPQZOĨ9S8^\PqFj-48j):Qa4b;BB8{Pvu`z_5Q_jy Ǘ&K Dk+O`=$o &{(*9F6`&;QQCi¨^VmP v"JFlSԍ(aEҸMD8?FINrLhUBحpy-`KXO!{e+%DTEm)QrnSO95n~.1"pwZWSs2|䨎Ί˩nވCwjitrK`ToT.+JeVT-ŔhV0*,(b:*NnϓMvL3<ʴ@IGuӜP0Q⑦L)_*hy/됁 )BUYu*?W%8u.xép";w5嗀r*d:|e1^?EH}9N5Ec݌u5E(q :ݘQ['4ES[x- &",A.LIT<2ןMIfkR^%NM'lc=~'ۣD5ʽ"]%^pUe)c(ѽ*@^Ixr_8<]-1ߗ䃩xPj|!RGQ%}xJJTݫG Z/IENDB`n4@;[blgߺPNG  IHDRSkigAMA|Q pHYs.#.#x?vIDATXõM6X DJYf1c*fe(`v ѡ`{ninQU?p'lxjO;q~sDx? nnu`5fXAB*3Pl-_kΉg~Z%wy!xcAppD_]yr#—O<>O#Crz"NE_6yk2Mo[ m\*Qr 4L1u %ME,҃.ҬͅU}hupw.ͨsW :%@'c,ʮS=倖U࣏&Su>IV](Ϗ' xpqU>[<\wKc`q 5^DQ~:(<:(vkQ@I5\(LyCDAC9=\DP=EZQ6 /~z=U1{Nc{(@KUQp1#\bWAp7\lk<|?wIibU288$SLcU*jisfU )Q3$x~) lhYJ}\hEchW{&8Q<QY4M+8!D.f҅tzpҍ<5WX}@AXgk"˦_/ṫu/ ~ |V[j;/u?8l;3 75#_9#8w #kݜ)| 7Y .pSN%N\ ֭6E -*{@هn]pL8*U"y{?UQDoU+pUT bphpKb9=N."sK٫spź2/\.79EƜD.Qk_K΂kPss; G?nWTѕ}nq%JF?3Nl%!AQQJx[,p*z8)GPvXx ۢW$7S%RcΩ>*/*~N xý,F|J8Yw7ҟS{M 4#|UjPM3x;{:+$>mcDYk cr8d;E}(O8[(wZGImXt&?|* /A'+Oc .Atp <{{ Oz#{(pO /\'{kn"eMpP/$h~ 7Zgp2_YsƨO{,_y)?-Ӽ,IENDB`nf݌S ^IN"PNG  IHDRd gAMA|Q pHYs.#.#x?vIDATh=6`-U0@I7E> ^˿2w4v?^&r;k`P m= `?V׬-cmİun^^Y~S-+yCZ~e^a~^EXS(a89mZӇp unZՉtJtW'4o n^l8ط:Y]f&ί|9i/~ϫ:a;OgY2~CIENDB`no4Z'{̔ PNG  IHDRhdgAMA|Q pHYs.#.#x?vIDATX= `,Kq DRXɬR#UJk lG ~HP4Bv#dlh3MF ZvhLL1M5N!w冦bMzm3+l~k|J}ШZP!1[cE=Ο{Fmljޙ/8!򱬭g9qpmYڐlσGM "-Ɩ,4pNӝxh)f]+%{HA[6 N ,Q[]#jt ^0| n׸56ń*k -8K emIcfks6BWvwMGZ."ا]&UȵT紮є&px0UmMs j=;vKQ._Ƅ)^ڙmb z VNE4dS G4LHc?d iPjN|p\m Eօ=h^a5h/ \Q܍^%ӨQQ-&fRjإ9 uոDf\vؖeЧMըF7V4L n Ԛ,v}YۣaXV5wYP骪y$&X"]pQc<5? 0#dm%NHצ-ߠkK,ik]^ 3A/hp]yR4jMXMU>287+e*jV4\JQkAt%mگ@xZUC0ۓQZ[Ps^g>ЂӖԞ(1rpnQVg*Oj]u_<wY6-m𻚘Uv G'kfnjiQu5>imxO..hөW.rk/Э& *ujPv yh> Bʭ;IENDB`zhP90͎(S|#.#.0(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(x(0(ztSMGN(xK#.#.????<???????????~???>8?n?A՚"#ZLPNG  IHDRSGgAMA|Q pHYs.#.#x?vIDATXõ;8 RN4#LAJNеkWQ^snƀ%-HV*' @sgfoO ydXǹ=ȑ ЅKV]sK9]d4*qXpAnV*^>:AmTP&_3ATSqq] occu)RTB9W9x]6*s+aӆ2@MrUwKA"q6\sqZ] 54D jM$aruG'tzaDjS*Ww&+؋>MLF?v".!391w뿈4#bh0Ǥ5nws7A\[O{ft1<]{uUK)W[}9%uRM+*ȝܢn`r*p3w&LrΕWgLzvs=PK\/4+o|>tW7jٯpqR%Orlw30_^ϿwQPcUB _F6F}{|O?wUEԏ)mzX9 ߝ 9RY}Y'10P^IENDB`z*oŔjؾPT^&](KK#.#.z1܂gC(B:#.#.????????n!x^i_S6PNG  IHDR_&gAMA|Q pHYs.#.#x?v~IDAThOFp  ]]+R|l >w\V& R̛Y6ŋ 8F3ͦnslՖ]Y:V_Oix> ƧcqfO?!sN: O*up'%nE7Wd .*nE`MRlmlUA!-;qw+J, a#z>;ofADau_?J)ʅ|*ehVA:Z1#/e3`>YI5!,j%7 h%f0ͳjՄrR߭C{ɍ,ղKRR\2˳0sP+Wdl&VJܞq}nkӒY̜\kADfb@մąP b|],h= KP5bpsMx( }IV{\.΢Xm`I*ndYP)wn,ʞ8b1U\/\.*}+޹ F⬋aR ,=g+YOE2 rl jg7jVY8m`QUu*Q֯q<_(M-.a~?YkE|[f񽺹4hߜ +fXT5gM8KEiʲMWm`AđXcE7,> )YrW,YWVw2PHc KűYkܪo âdYeu}n XͤX(VsECia 2?օX}F; )ʢXueXohUz+eլM`|酕IS"sC5׻b9=kYt},5(>κ e`z΢ w*kE|_eJY*+t'J1* _-z.蛦#KVgrk¨cg,Qi(爌iJ+Rc dɽ:zU*dg ̪4gvVCUVtnf}V,Aѭz8ˊcVOYǙMe4,ʕͬ2ʘ+UwVXIU"r`fMYr<3kVS,}kwfb-D4,q2 F6QNqe;+9],I.XIM9]6d :}OaS3&*Y;Xú?VDkY教 6Na*UgX'{VGEXX0?c f`@|%}BGr1ViXn0 #cl00¿k0?BNǂx xa)ʒXY+kVgUof:fb w#Y6‹'1#5Yj,Y ,,Uzм,ARoj UsvhŇJN֬Su6fѧ9iQoeIGt6Na*CdXO|cb.XM:wr|K0#5QYY^$zQ /)_Ք >mdU5&Xkt!aQMVJ|G4$Xgu溜g |ƗP kcd,&rN^Dŗ(4Sf~p|Ub54 5bI4z/v`J'qVƂǬ!,IƬz;Aywm5c[w_mXZ6]/t_;b{{a1c9a5e=-K Ӝzqe&5%kbr:JMӪTjx4SS9{|C,Wcbd? +J|Dz!, T3tV*fmV˦bV?QǭYu_d15¿SpUvYoQ6JdAutUǣ51"<×&d9}"ZësWrK/Y$V_yGCursaYC2yӷG>rH/÷6Y2*P-fE{2eXz`ns]Tۓg4QdnIENDB`nc,OpIh5QPNG  IHDRm_zgAMA|Q pHYs.#.#x?vIDAThO6pPl* ]vU@4, r`9W`ْFrqy6$$ju>abHSR!b-9/8iCĐT*~CD_✯rDN 6D\4g|֍sCDF㈫sqW9AS9*'h)TD)\NqS_t 8g9E81~Ct99`Jgx{>cDRz0I'N Ҳmᤄp(@A!m99doƗ&jZxq>^ 3r19-8! 4B889q8pCi 3ω#!A33  G8.d}2v1F]TO?/8F8.d甎w2 S%S,9 i83^&p`wr*SW1& ᚎPNp[8z.'e8p˹<7qt9^pU NoLŌEU9Nf" ]cۼǑ\Ϊycǁq ϡT#Tk:ga'))lĥm;3T[NbSi.CUghfa6BGZ[tSJNrFa93VfG}S󹁵7#:ĹH1sRE$Rؖ| ot>d--fſb2͙f4YʝWmU5-ה`2LNvg5A@Àvx*NI/9|f'9P귉")]-]"ºq8o8о3zbƹ @U~2o|!)~+a9xZWzYA @a%a# DFW JWg|W<9l='phɲ\4x\ࢶDgǼ#r$-yPkX/l Aɴ_N#8.2dEЭw2mR&,&Yx{1F\<Y;lr1No`3ϣ1v3pFK0l~K58f[j8i잆甶D4tVkSq^b  G\'$uqlpr=>YB~@88pY[Ž' ''KKNxS.8Lhʤ*9,lXrIsᘐZ٧Xظa"\3'5w+NsV'>YF\K#r.9]Nu^E\Tѵ9;|Kd,Fvk|WYXX݂x*(c>9*Ze3p=;F /4(,Sx'qp95X4n;˂psֆKWk ev=S.3tޗ ?rDO۟<`jGU՟>\aaXNgަOm[ljFL3ٴ}0?s0G8#=ʡ3?-/|['>j(h}8x/p.ڗrh8}5%~ ')nF6[}⹡"&߁X%r4PᯗP JݬK6'LlQJ%"R26vX5;vyCuDLضO5Z1 <|G )[V jdB(xEE ]}_&ux FZ;ai`?qBRdcs39{q͆[I[aUAEڵ ZzeM>e-j #=K1\c#2l9{MvYZk.ØꭕBmkѹXZFk=Ψ4-XC/ֈCFTφ1udSŮC΋*v㟲.~l9.sk*Ror؅;-x~Ë@;z;*;S%h5l'cߓB-뛢oӥ%qs^Ĝz˱Zo15,4a^l|Tևyח|Cر\([ ÎuB˦nNkzKn>ŎfA)B##1s|h.eum_ <Kdw+ IENDB`n $i7L)PNG  IHDR_;gAMA|Q pHYs.#.#x?vIDATh=6` .\H"VkV2| Z82 ĈG&3Fl⵩ODq3_mUIץJž9a_vva'}Š=+WXiv b!^=iIʥ8?k`M`cőˬ$*,Q4c_P&ri07 xj+ #Iv@Hp+zgirŪ v),w2Xц >gYe0*8@Af]s릛2SN/aNܒ>|Y0 J'2T/*VE)`a*ANae4X73.YX03VWzVaUi K3كCCrƄ]B2нdNue!`P0oXS.qX`c& ; k VW*Xbc`X&ama港=q_,ƤMh TEڙ+)$+twc3XrȰڛf΍þ |_>aGQv0O?R +ÕA$ ˓4po6 ݠ25 >a+WW[naps!>wC 3DakQU0?>@0SGwB f># M, 1C(0w` 6͝0T7) +C`,0j2 RV`S Ma8F4U F"P04۫g &{6 vAXƵajc!]>m )#N`0A? i04|,ᗎa&aCCΰb,ܘ5ʬ7%HƁO).u0VU[1aWSsq?EPa+nsycB~O9~Rr% b5z ]fr kT0'SEaK1jOw>pRolV (_Ly :\ \k"Sv^/_C6BL`cGJ<]a F!CPf%f`4Dwi] +aO >:X 0f8`kq|&/aH16 Xxrk&0 SQiϑR3X2EDE?oEd\qZr7^{(2, .\nQt(Գ 0].xz|C>Y;xlAX)al~`_ X)<F[e7 Eڲ)Cka1qIiʛ03)4B5g?C#.`! >` WpQEZx6TCJ'W n`ZX%`&XŞs3˰=e20.#U$;S'7t}Ux;ÞdC+Xj /% 6m&س??<?<xx<px8<~>z-WWx7z7q׭ ּ(S)#.#.>x?||> ? ?   ?|8?p~?nE0oWY9ݲQtCPNG  IHDR[gAMA|Q pHYs.#.#x?vIDATh=6pSS )Yr9"`IGSh;_A p ??%+fO%b>JQ&)  +\#- oI2ˆob ەlUNQ(֟Ђ "Bo:.datœgmN8S} y5Guor _uLf@Lo/4 ~f}1?W)aKF 1z-"t̿fM7sDh|2aB KD8pCdJ(jL<[e!'9|pXEDOrBhPJ>#cBav]8P%T2{-L\Ƅu=ST赩$W98Z/*K a0d'-t> s`.#NpAwqAѪ#pBYNPh3[+lpЍȝu AQ(jD8 B^eXd.Ac)YO mL$â!j㓻ЀE(j:T<8bgA#r23n}o#&VdX !gXi}PC7 ͂EF(9mBO9'=ZQB-GEAB瓩@2?¸L$ &!- -7  B(}h@L.ǿ24SnVp^('0IL;BpdP 5VW'iƏzKmz= ~*ePĘ9?'2+ P?*#1gx}AQ쑞T턑$O\9ˠ(&>J5j[ l٪0A%FIEߨ Yx3Wn)&0+ $@BiBO³I `V&G4[KlUgXNU>c#e55:uЋ~CBfZ^ͰZkD`Ń~=g%@RʞpjaNat hT9>{Pg@%R/'-ׁxDZRhuNof9cJX'pm;zJY|vppKUgA`2v*`iA-)NE9j-Z 41@ TW?v?"vrQ :, Vrz 4ˌXD])T@#n?鲼_ Ӂzk0q͓/\Edj,Fs!B+ywE\^泼+/! \׶ ciϞC6 }??C5fL#mvw(["S}k,fGU?bhIENDB`z6K a Wڿ(_O#.#.???????????????????????????zI V=-FH[(-)H#.#.>8~8?~~~???? p<z)g`;i3(B>#.#.??x????????????n չ,l)t.DPNG  IHDRtdV <gAMA|Q pHYs.#.#x?v IDATxͮ:,ABG`ݼݲI3A4op 14g&&UˎqzG槻+˿IT |exR( ggGBգ(uVNwQOkmW?^.% Beg{xU/}x/=6ޏzz$_CyUM3NJy #o3{݆?@AO{y|^?RGs;yf1hrCޡżS61S 6 &qN|l^v/gFoE~4lbp6{x;әMox2n^BlkEpycY|nq1DyU?Dܞ\teY ]hEa8'{g[%;RN_lށ/4#:žj{w4{ÂO,yA*8+>;6F8^(.w[!J B NiV FM?D|\N%?|>\YP:y*/X3˻y #E9)vNAݪ%ok$-PIL+~Ւ@W f7PӼK=̟lx^,ě AߘC?8r0G*Z ]BԝiKOpuzq` ,5Z-yI$p,ޓ[yoSK"o9/YO4TM/@45y|M゗DA^|ȓʗ(/-\T8>oKsɎͤՀAwv/;x(omx]zuocyM:Żu^K 2w9_hc$xkwQ'`^ xmLt^7-y K=x]Ѱ̩0 a# *  酦ɨcYEu\qг^0X]SXmM*ba 5`_5bN+xO E2q |jj{F(Hj: ^^˾j̫yi?h:{ fZ2½TS=f0j1d^;vW5IFq 6(SbrJc%x*%yOıw,f2Jގ'Ϡ%aK7fL\q;_( 8o=JZ-yqTUbskoZ'4-4a>o`'O͖K TSC^z+xn'7;ւϳx\؈p5/4iM8";$O̎STwZ0O1\mѡ2I"IoD!Ci^3@$JQ7 oW^=2?%/W6역~ /]u|)p7= ^xϬ}n_ۼja7(er[x$?͝qtrNB2u'YEpWw#ibޠ؝'xyzO}5/x^ {Hx|=`Ac醷DDb+*텼8%wBN#RPub9cE%ˁ:({[KޙcD Ԏx'Nj-4W$|X7ymOA)^1J/=V0*^T:ygHWJz+8p~~V?RL.NW\I x+[¿KK2*h8Rs{[‘o[=jsZO kxm0gx b/@ Z]~B`_K^>x̃uX o">Y 1b0/]=l5c/ [t2y;<^xKMg"}^ 12:wgu|^Vy5;"??xߙ}kxO)51^896Ex>hsqoIB/Lnv!yEy?d*7xۊZ} ;"ob+ |\ik/фz^rxgÉ$ x/a;{Px5V*566SCʕ2;gy,/Y`bp)x+]/Pba&ٌUX ;㼺TGynQG1XV .B,ڂ./YtOI8_yOo-%N?I/\pF@-Fmd5Ym޽u},wcl900o;i+`7ywyW^-/$WcӣxѭQ\8?=1-. k Eȋ-Us?a~MXo/Ni^#J~'j戍+>Œ;|_1zw♽-&[rwRW?aߐGJ/)Z8(a ކ6 Ӹc/ NoUEc.g1g?81ԉߪ[40iQЀh%_'x37ŋ;\8|-7or^b}j7n/PV"'=^PY6eHJk!A_tVJ#m'|р@&6?ꬔ`³+zմ7U ~B;n{YJ㺞7n߈@8m1J8Z1l);o&2p^' x^ě7}S2b\TLLbxוRޯ&[gAcIENDB`z%> @/I/S([O#.#.>>>??~???p8|||~>>>??????n 6/%88-)u,qPNG  IHDR_gAMA|Q pHYs.#.#x?vIDATh=8`!l' m9` ZmZt*HTWVRĭlCl(f5=T5~2?Ed]MHV#Fkv( -ȊK"*+^8H_'OHB~lqhl(D/VHlbC sIn )08r !(g7!Tpz"uwH"hH~ rO[Mס"D!Fd= dwmI#"b"KvGz&8H,A" )ZHc RBS=P!wzfFӑ!nOʪ1c#_Zt8!;q}=5HEw|$yg`RH+s89\%؄j6 S%Vt Έ4AL>zgMFޘ(5B#UuÛ%HwXkCNH2棇J@ɝc>zV.HD}p6 2B _X=~2]yjhQV#c>Hc/#tDdrzbP0_nX!`ԉmL6H{\ =0$fEBxF}BLmK&}K)` 3ZFĈĞ`>7zC1C?c>!$AI!B`]ŘCio Tu?acWCF0RB'q)ϺGȰ),mZȂIG PȮu$W_Go.aqu\nc˜v#)|"vB0 vP)D|,CX6O |UeKܧʳAdXޓ |H"q@BsNCv}W_pu("6Nz[.#m];3 Alj֑d,~B0 kT =NZk!G2^x'=\D?/}k߈ҳ_F䛿ނ\SU~Fd8;H3UFdzKqdSnAZoAx!K|șy2{?lE֖o- _##Z4IENDB`nqO!}u52wMPNG  IHDR ZUgAMA|Q pHYs.#.#x?vIDATx흽8eM0:h}lEkk̾lA%zޠ^E # T*]@+-%R"k[en_DsHNY;7_DkxUƏDD]; ^?NivWOµ*{4rvz?ÿADl;?{p|~ߕUOrc]OO/I[-q_??ZI3_Nj^\iT\z7wu*͡WҦx-I2~ V^9~QQ$cf~l/Z^ ?gHf;^l%+?mgC;dv|$ !dW;f MkN 7 AQ2~"1;u)dw~fD|zs)cCfu 1 {kUOH'aզCH~^ےF8V;d(RmMwV-[9C+Ψڠƒ|ɵ5):z9JHB|t[g Zzj~u9:O!~jZZC}Yԅ,sIA-Ђ:_?n{jZRNS]C%52h^/{>q(J? UR`Ƴ/.F(֠ _)‡U"7D$/BsC<8NZOi4⻔xC_DIUGSJN#~Gotiy*qй+Nj߫\Oy`-"5o㗡/W9ui>Ե,+ Z.+tEiC5a*f|n/T|TE Ƭɞ4^gϜz_T#a6DkŇaodx;U]j|x W=‡E%Eyv.E_Q^ԆqI|\ JS\u['B90}-j!>UMkPO{#2?;XVu|AgL"T(SOQ8QFF =>&^qŠW*Q5Z-'FUIDLJaU'ѻ1{/y,'&Xt!?aT@CuN; e|JdGVi(:GQѭG|:N.OC]C\?Bꪨ2Wc'~3~ Re>N//"/UYT|"w+a8DE.V-wЈV}y֐"NO}4=|B?e /O*vٟRRR\#K|)kٿ'L$RiǵYoKas;߯X'#H Zۛ ̼Zz_AM *jmڕmy))k_sX,ĵ\pi4zq*'z'xC/BOq2'? ~m7{;C ۯ7+~/%a=fA.񏜒DF /ݳbJ[E*'-!o@]Y_:*$qE*B*O z< |Q>.?w;㗿}J?>a|g[(}A?ez<',d/ d{j r?J҄Lg<f}%qmJ4weiĒP;i)o!,M4PDM*n]|ࡔ_IC%Kژٔh.q@یLc%j"sЂ,Txf^$~UugθMX80~3+7ҋk' MWЮXPvem3nS Nӹ_}伇LY&̮E՜ނ 5>z:BqmJ4u+LJU^?/a`$Q% nV'hB۔hh;eP77 iJE[3/VmJ4gagݳ a |WSI/5M))ԗ #f&tmJ4Gi+m״UAud5DMdp6.a=hғz{&q%N|7{?`/Z?ke5DMؿIuM]"KzįG\fcO?WG:s_,~MO53d~Sh ~)]2Tic@fY?󰖨..e>T|! A4$<۩}Y?bS7{ _;af(A-QRc KYz9&5zҪG( ý^]AV?!>-{?YCkLBu@ ejES¯?i%6>׫58*~·Z5_2T/9_ \[Ï/ f Ki6<Bo?l6ʾ믇a;SGu7mEa[cv+jߪN@V":n"۫$Bǟ}^{]_QNo^ jz?DjJ5Q6KrM9JgSb?YvP$Q˹ wE׾S0Mds.w;Lr?"W}{o1s l$QalJT#sQaɕu1~3>v{Iԉ؀(_9=;?ᳫMۑm_Z]1{ujٔM]taZ{xnwőǸx%Q|ǥ޲|:, E:himJdk[q>t%aI(U޴iώnI(ɺ37[2(ys{0gruBϒѲ {N0 zs[ٰɺd9YǏjsX7fĈq8.v`i,ɭYBjX]5Fb¿BsBFֆF2HΌo _ƿ|>`'/Ouu3﯁yؔ^^\۳>[Wf[}]*yd߸iſqSʣ=w7qZ{2{oHy{2nM'Ma7DٓɃa?>;'nzWwKK/d>*>Ƿ\xOޖrOM^^ySM[\G+%gO8bȰjG_*$=l΋_MpӞq{!j Z^ܓ)-#ַrO( =x= ~./tP_췧[|FOn׍0>fg O7t H"z𙇹Nk`jv^oȈYz.b wOf?f%8tB̂X<sVOQ |fm|P Mo?sPZ{R/~"|<' Kn/o{M2NIJAhK؏Z]"BrzI{KB~/Hb-oo6j}1ց<7g_xԅWiX_7߿z>~~LnʾG E~P?W3plsӨ:Tpyǎ2>:;:c6Z+|:p]5miK%;0>Ryc%w?Qf[/6>BPxϑhhS_u |-+≷sQ|5|¥?Q4B|<>~puu|'sw.߹Un]J~ Mx=pKWfc1ǝX~x`W>\J2_ǯ?‡g P>3}}||.ৄ |(¯{ H _ ¬{x3̃^~;|Y!^~}c ~Tz R?Xo>%pNKno|!eس^B}*CP{K_tsO"]n|1 aЈ&4"MK+O) uLj'_ݡxpu+ǭ>,M>E>]%1hڌ1⻸ N lkP_?ߔ`/JO-UiKkok.)1~ LtK6(<3~ K2WJWg@Jیn6 l%Vn\|9T"ʤKpm|rί^[4IHܠ/%ŤHتufPJt Y>(>[<2<}myM ჹExqRǗ<>x_a-|: Xa=jw݌OZ>rKo-];}zɪ .8n~ OO.}a???'_PwwBD{;z*~vWW|&۠"z'۰o"{vwOŹ*]:]??GU+gHIENDB`nڸ _U6PNG  IHDRSAgAMA|Q pHYs.#.#x?vWIDATXí1n:P.\@{]io@.\ WQ^AFrř4 8'sHFJ mhKmTUI|H5ЍE:_Mz?lE׫WA.gXMQhE1B7w^lEyVYq\`B-T4 *8XD j>woՑiaQ8m_68*oETTIE$ki4cEeq\גh~fŊĚ6"J8DCiUSޯ:/b]GV3mSgnOےxˇ$)לhyQ}I45=ǢxNDxCRPYjxXz$"^!Pcw\m"^!`{QTx&/%(ODG^<8./쐖UC3mTKC":rFiݝ⍽P#.#.??p8?~<8???~~?~~~?????????`?`ppp08? ~?n&MgK>'PNG  IHDRS\^gAMA|Q pHYs.#.#x?vIDATXױ0PPRߐ:~+Ed1.\/qVkݕ܅g8hWCxum 0]A&Jb(z"=ec8EQ/nEcU,T{R)VM&d)L؅t! 9Q 4XaLLx` Wybȅc6AsBUq0LC> fb!8R^'gz $uw8)sQ?7hmZhGq ZqWd [[X.5{z4j_16A3&|p Apj 7,7?GlhG,IW,, ] @p@Q|][MB?xPzx,pV*=|a6IDJ+m oC-3M lQe .q]\crP ))ӽX6*,k7 \" `~ 1Ux.(CEAm= {P&*5 O:*~/E'<&($߅[|/A\9u!V- l[** Ӳeq[0XW!ީEX\q`p/pXN ,$Ip;pS.v$fC3ő=p4G8(ުqasK_&3IBE y L[-Fs}4,>I1b%_뢠Rg}.ŔvN袠kR5XR/RhqqwBlÊ !SDIUa5bڀL:*U3*URURx%xU _^)ˑpn]tgIENDB`nsE\%9?PNG  IHDRbSv0kgAMA|Q pHYs.#.#x?v_IDATHǕ=n0` rIzlHA x)#* \ c #CW]8TG[mAdH>>H̷AWN/6=t\>] +춓qM 0z{SKsg=)ʬc łQPEp:p4ixp˧E=q==^Cq3vLQg7j oSZLU\E;!L{iG!$O-D3j Z 5hsZ~4>#F16mԦ29 )gu]첎i!뺊ԉ C'=dgH6/65oI)P9$-+I{'=qpgUƤ!_ԃ&m+\;t_CUOz.rzY7zÇNu=LItê>f}ϵ%aAwgZ:g%K)Z5- z)!Cҹ߬Z)lj[̀:PnCqkQo۠yaZSk>c=aڠ+[Կh<2K:I.뉆؃!IwvEl N}ִж@ZEӊ%8=E@{Ҵ#Qwt=헴EGm=1i),f|Ant4SivP[>=Zwek<.ܷKMm`z/TQo{ʝ7km?!UIENDB`z)mv.?D5P(S:#.#.8|~~~?????x|p}p8<? >>0> ~~~~~~????????zmDܿ 9K#(O,#.#.((((((((((?((((((?((((((?((((((((?((((?((~?(|(8(((((?((((?((((?(?(?((((((((((((((((?(?(((((((((zmseӇiJJ(O,#.#.((((((((((?((((((?((((((?((((((((??((((~(?((((((?(?(?>(>(~(~(?~(~(~(~(?~(~(?~(~(~(~(~(~(~(>(>(>(?(?(((((?(??(~((((((((nVw`[;WGr#PNG  IHDRy}|H3gAMA|Q pHYs.#.#x?vIDATh=8p.E5bʆ>\ņr 2H|d6K$Ė~')%zqwEën_ ~?pjq7_+AYs쫪~6>P' ^|l( ݗ hެ"YqJmY <qH"l]1/ pi_-Ɨw5&No(/oMথO^D Kߖ^ !S  e^ ^TB^ :d@~ -/!, `Kh)/!,n%4 %A76m2 >9q`bk6F@1ρdDPuV tOw68x%9t6&㺻O&u@~Οs@:o.8R2!H6c> wH'НNNdH/*Byħv,Ž.!&•Yۘ={rex] bh5?A2uYlCuDЂМ hRESǛzstE@=$0XB"芨ӧǁ~Qy 7"Cı;"Bq h8.Ej*q)R.ESdT^w#V pZT NG<;Q[-MI`c,ē9Bfn`b#)rE 37~nmnL9qeAN~5-S]{?"VpTx`?Z%bX(ށcop&8PBPy.N 0d-,|hj ~Pv C}3 Kn'\;ddIxOm1hյ> @xЇdx Sd;WeVv5L(Y_َ0W[ :~!U1cވ`./D?ER Qza̕+*r S2 f0E~; Oabp@>~Z"ᝄ;mS]|˂Yn@76(s`2ES[u_jM_tשtْ;7ԟ ;pD:9}K!s E)F Ecx6%K{r {|$R`yOg>8-_! <|+ XH2aK)-&^`\g.sQz!R 32IQ$([)EX,7R_ o`!E6="l[2PJ{$jpUEJUalI0Ɲ];|1= S$Oѳ`6EOMְgb)z|^ P*_uIENDB`zV5 ;m%(:@#.#.?p8~~?~~~???????`?`ppp08? ~?~??n[ -!)T㸳m!mPNG  IHDR^4SJgAMA|Q pHYs.#.#x?vIDATxݿ6p:fl];f>J@er (qGnqCyAlKsCpB'I#A"b2&&b``30v 8v.7FјHƲ`YkV+,R(&.݂1Af' .݃pY+?0VY);6a뉀jiU?Ջ0`13k ]6F?ph1 L۟bbv^ {@b5}*B0Cr+fl;g{-`)/ fҘįQ@_H;XEDf@J~vdYƎ2dYe &f]0VhlәɄU0Cb>If=[S@\ҍ-x#St hѵ&5T-0+ 1]y'm9/K0Vd,C`aM M3uZ@0rΎ:o;5Dm0l$z!/U[PZ\cSm׶!`*hcnހ1?SUd]tm>1Q]g0~6h]J6 [a5+2ޜE=%0fJj-"bwxba\k?U'Ѱ/AĊ+^(FgHWl#5bVk¬AX cyޫ$5X(vw5 ,fpe \qj]T.Gl1q.ΦUALaXPt1Ze0=a$,s( kDcq09G#a([¸–0c$&Fca70&>G0rG &/~8X n]y/ҐYgF!AbPp\/}z0 bȇݐa o\;ʸI2AXBwCΛ)qO1=ayJ;pO' C{0 I_&vvxtLu|THZڗ6͢q|8 n֑t9bPn؛}2cai :@x93yV UӔ>aNoLf(L{m?2ya"cla5,=k2CH`ycaV&"јO;#pL4Vwq1h~' f> ֓sv~] I]H OŦX;cҁ&\=u?vzŻ<\C9~ pF^Ѱ, EvзG1;1Oa?90B/졾fulzMED&.)=iꑀZ|IVN[ ;颰lDYIUtzjD'm$&u &'a' (SߒNm&iOnީAʹ3?,O%﫛ceIQL(7a[ՆN *88EVf*Y` 9ʝ¤t KjWVBYg bnM`u1dk!0~W3?+G{]'V#3?=9J ljOp{BSl`X0 ļ'>-=3ugT˖2PMW#q=wF`1=4+ 'U0>oKZ0[gqm S>wmr {Nq q@Xԩ Avhv Й0A7?IENDB`O( -   Equation Equation.30,Microsoft Equation 3.0/ 00DTimes New Romantt 0DArialNew Romantt 0" DSymbolew Romantt 00DSystemew Romantt 0 A .  @n?" dd@  @@``   p  1?7Y D &0@r$b}ȈPڣdvLBb$WJUzfip4eb$;pk}5L捚gث5 b$NѾ\V"*z2$pK)o) r$MґCN+]˟2< b$8 ARo,wb$_SQJub$azymM>޻H)b$@;[blgߺ<q$b$݌S ^IN"n*b$4Z'{̔ w2r$hP90͎6r$tSMGNW=b$A՚"#ZLGPBr$*oŔjؾPT^&]Gr$1܂gCdKb$!x^i_S6eNb$,OpIh5QkZWb$+Xͦ_b$$i7L)# |cb$dU_iSlr$ph'!uQ<4aor$WWx7z7q׭ ּ5rb$0oWY9ݲQtCM2ur$6K a Wڿ{r$I V=-FH[|r$g`;i31 b$չ,l)t.D>r$%> @/I/SBb$6/%88-)u,q?b$qO!}u52wMRb$ڸ _U6lr$uj8!2Yarq :b$&MgK>'Cb$sE\%9?<r$)mv.?D5Pr$Dܿ 9K#ur$seӇiJJu$$$b$w`[;WGr#^r$V5 ;m%[$$$b$-!)T㸳m!mc  0AA`fff33f3@1FlO ʚ;Sk8ʚ;g4HdHdp 0!ppp@ <4!d!d@w 0t<4dddd@w 0t <4BdBd@x 0t TEXPOINTINIT.USEAMSFONTSTrue.EMBEDFONTS False.USEBOLDAMS False((DEFAULTDISPLAYSOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \newcommand{\gr}{\partial} \begin{document} $ $ \end{document}  TEX2PS~latex $(base).tex; dvips -D $(res) -E -o $(base).ps $(base).dviTEX2PSBATCHlatex --interaction=nonstopmode $(base).tex; dvips -D $(res) -E -o $(base).ps $(base).dvi.DEFAULTWIDTH3240DEFAULTHEIGHT100>(DEFAULTMAGNIFICATION1.32DEFAULTFONTSIZE10<___PPT10 2? % "$& ) + /!#%'*,   0` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $ (    6 P  T Click to edit Master title style! !  0   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0t ``  X*  0X `   Z*  0 `   Z*H  0޽h ? ̙33 Default Design 0 (    Npy˼y˼ .   n*  a00aa  Nqy˼y˼ 3 .  p*  a00aad  c $ ?  4  Npy˼y˼ 9 f  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S   Ty˼y˼ r   n*  a00aa   T@y˼y˼ r3   p*  a00aaH  0ηo~ ? ̙3380___PPT10.wx h(  h h N9y˼y˼ .   x* a00aa h NJy˼y˼ 3 .  z* a00aa h TCy˼y˼ r   x* a00aa h TMy˼y˼ r3   z* a00aaH h 0ηo~ ? ̙3380___PPT10.w} 0 -%0 (     6h P n6GRADIENT ALGORITHMS for COMMON LYAPUNOV FUNCTIONS77  0`  "CDC  03  <  =Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A.*> x33.(  <*   6Roberto Tempo IEIIT-CNR, Politecnico di Torino, Italy67 n 33(,  H  0޽h ? ̙33p4 0 ))@0 )(    6C h ;PROBLEM   B(F,$D 0 g7Motivation: stability of uncertain and switched systems88l $  -$ ,$D 0  BK` GAnalytical results:ffCT `U;  $# /$   B<`U;  uA hard to come by (beyond ) require special structure BnB  dAtxp_figaLD SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $2\times 2 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH496PICTUREFILESIZE1798&l  5,  . 5, ,$D 0  BLV +  @ LMI methods: ff  BZ/ 5,  n: can handle large finite families provide limited insight;n;nl  9 0 9,$D 0@   /   B_ U  A Our approach:  Bc/  r> gradient descent iterations handle inequalities sequentially?n?   Bg~9 q3Goal: algorithmic approach with theoretical insight"4/8 wk ,wk+  BDmwk ]Given Hurwitz matrices and matrix , find matrix :^^Ph   dAtxp_fig:=b SOURCEF\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \begin{document} $Q>0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 55.8756PICTUREFILESIZE2590`   dAtxp_figmb SOURCEF\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \begin{document} $P>0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH556PICTUREFILESIZE2174  dAtxp_fig` : TL SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $A_i$, $i\in\I $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH846PICTUREFILESIZE3538l  <1p + ^Atxp_fig\z SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $PA_i+A_i^TP\le -Q\ \, \forall i $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 202.758PICTUREFILESIZE 11726H  0޽h ? ̙33 w ___PPT10W +5D ' : = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(+8+0+0 +1 0 "x"P> 0 !(     <T h GMOTIVATING EXAMPLE0 2 dA txp_fig# %0 $D 0h SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $P_2A_2+A_2^TP_2= -P_1 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH201.8758PICTUREFILESIZE 11726Il 9> `  = > 9` ,$D 0 0  dA txp_fig9 ` xp  SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $P_mA_m+A_m^TP_m= -P_{m-1} $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH249.8758PICTUREFILESIZE 13262 4  B > G  7. . .hl  a i  > a i ,$D 0r2   B1 g qr2 5 B B1 a i l l p ; p,$D 0 (  ^Atxp_figM0x SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $A_iA_j=A_jA_i \quad \forall i,j $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 184.756PICTUREFILESIZE9162     # X,$D  0   <   z& quadratic common Lyapunov function' '  h   c $A ??    &  Bp Z*In the special case when matrices commute:++ 6 B$,$D 0 i7Nonlinear extensions: Shim et al. (1998), Vu & L (2003)88l   <  ,$D 0 1  dAtxp_fig ld SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $P_1A_1+A_1^TP_1= -I $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH189.8758PICTUREFILESIZE 10862 :  <6  (Narendra & Balakrishnan, 1994) , H  0޽h ? ̙33___PPT10+|fD{' : = @B D6' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*; %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*< %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2 %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*= %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*> %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6 %(+8+0+6 0 +  0 B:`(@(  ( ( <  #ITERATIVE ALGORITHMS: PRIOR WORKP$  ( BM  KAlgebraic inequalities:ffS ( B|  AAgmon, Motzkin, Schoenberg (1954) Polyak (1964) Yakubovich (1966)"0BZ  _l M  ( M,$D 0 ( BM  HMatrix inequalities:ff ( BB  /Polyak & Tempo (2001) Calafiore & Polyak (2001)006 H ( 0޽h ? ̙33~___PPT10^+SDB' : = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+D9 0 *3"3p600 2(  0 0 6  R $GRADIENT ALGORITHMS: PRELIMINARIESP%  8 @ 40 0 ^Atxp_fig/@8 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $f $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE694P 0 < @  convex differentiable real-valued functional on the space of symmetric matrices, VV  0 ^A txp_fig[ Oz SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $f(R)\le 0\Leftrightarrow R\le 0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH175.8756PICTUREFILESIZE7698S l b  50b ,$D 0  0 <!  H Examples: "  0 Bh& B b  w%(need this to be a simple eigenvalue)&& T i  00#   0 ^Atxp_fig drj SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $f(R):=\lambda_{\max}(R) $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH165.8756PICTUREFILESIZE7366 .0 <1 i 21.wl B  60B ,$D 0'  0 Bx5   E( is Frobenius norm, is projection onto matrices)*F ; 0  0 dA txp_fig\NF SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $\|\cdot \| $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH316PICTUREFILESIZE1722 0 dAtxp_fig :2 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $+ $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH184PICTUREFILESIZE962 0 dAtxp_fig1"\NF SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $\ge\! 0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 28.756PICTUREFILESIZE1262 T  9  10# B # b   0 ^A txp_fig  bZ SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $f(R):=\|R^+\|^2 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH147.8756PICTUREFILESIZE8062 /0 <X  9  22.H 0 0޽h ? ̙33___PPT10+qD~' : = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*50%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*60%(+\ 0 FT>TX80 S(  8 8 ^Atxp_fig$D 0Z SOURCE>\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \newcommand{\gr}{\partial} \begin{document} $\gr_P v=2(AS+SA^T),\ \, S:=(PA+A^TP+Q)^+$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH457.8758PICTUREFILESIZE 24062 98 64g  R $GRADIENT ALGORITHMS: PRELIMINARIESP%  F @ :8  ;8 ^Atxp_fig/@8 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $f $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE694P <8 <r @  convex differentiable real-valued functional on the space of symmetric matrices, VV =8 ^A txp_fig[ Oz SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $f(R)\le 0\Leftrightarrow R\le 0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH175.8756PICTUREFILESIZE7698l e j  X8e j ,$D 07T  @  M8# e   8 <  0@  = Gradient: f3 8 ^Atxp_fig; ( | SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $\partial_P v(P,A)=Axx^T+xx^TA^T $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH271.8758PICTUREFILESIZE 13742  8 B܊  j  B( is unit eigenvector of with eigenvalue )CC, 8 dAtxp_fig"  d\ SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \begin{document} $\lambda_{\max}(R) $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 80.8756PICTUREFILESIZE3714 8 dAtxp_fighS  tl SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \newcommand{\gr}{\partial} \begin{document} $x $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE390 8 dAtxp_fig< , vn SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\K}{\mathcal K} \newcommand{\KL}{\mathcal K\mathcal L} \newcommand{\Ki}{\K_\infty} \newcommand {\R}{\mathbb R} \renewcommand{\P}{\mathcal P} \newcommand{\J}{\mathcal J} \newcommand{\I}{\mathcal I} \newcommand{\gr}{\partial} \begin{document} $R $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 15.8754PICTUREFILESIZE806L i  D8#   E8 ^Atxp_fig