A Software Package for Optimal Control of Large-Scale Differential



      ! "#
$%&(' ) * "+, ")- /.01234)5 67&98:;;+,&(?&@ ACB D  E&( "+ ,FHGIEJ' G K HL MONPRQTSVUXWYMZ%[MZNC\^]_ZN`M-LbaYc;SedgfVhji_N k lJm_n!o@prqtsmvueq`wyx{z|m~” }vo@„•o@€ xvuw@€‚p}ƒuo@€‚„o(o@–IuV—V…o˜ue†Eq‰ueo‡9™{€ˆpyw@o@uepvšƒso@p›mvo(ue– q‰o@”„ŠœC†EuŒžX‹XŸg€ guŽ¡Xm¢ mrp€u‹%‘Ou€ˆ‡@mvpƒ’€ˆqt“’‘Ì£s¤\]Q W QDVW X9[_X:–uX9[ tqQ0ghi`’[_XaU@VW[_UoeOQ|W U)b9–)Q0YTe2? t 465 &(*@7A9;:B< `fZ Q|d)QDVQ0Z `f[_QDY–uX,`¨[ ]Qs`@h[ U)v(`’[_XaS YX:^FQ0Z3Q0V2[_Xa`f[_X:U)V›W Ufž[ † `@Z3? Q 54CD )AE  "@‘m£ ¤\]XaWe`@W X:S›v©hb:[ Xa¥b:QcW ]UuU@[_X:VdŸ[ t¥OQŸU@ž1W3[ Z_`f[ Q0d@tŽS0`@V † U@Z ØR–@Q0Z3t † Q0b:b¨žU)Z©W3v(`fbab9c [_Ufcv“UuYQDZ_`f[ Q›W XaµDQˆÙ&l&p WtuW3[_QDv“WD»¨`fVY‹]`@W(`fV˜`@YYX:[ XaU)Vi`fb\`@Y–f`@V2[—`fd)Qª[ ]i`f[#X:[¿X:W XaV]Q0Z QDV2[_b:tˆ¥i`@Z_`fbabaQDbÌ£FÃÄU † QD–)QDZ0»/žU)ZTb,`@Z3d)QcW S0`@baQ¿Ù&l&pü`@VYkl1n¸p‚W3tuW3[ Q0v#WÇ[ ]Q0Z3Q#X:W `›¥Z U)ebaQ0v eFQDS-`@hW Q[_]Q¿SDU)v“¥h[—`f[ XaU)V`@bMSDU)v#¥baQzŠX9[ tˆd)Z3U † WZ `@¥X:Yb:t † X:[ ]ˆ[_]Q¿YXc v“QDVW X:U)VÅU@ž&[ ]QœÙ&l&p‰W3tuW3[ Q0vª£
¤\]QˆYX:Í#S0hb9[ t˜b:XaQDW“X:V˜[_]Q‡S0U)v#¥h[_`f[_X:U)V˜U@ž&[ ]Q YQ0Z3X:–f`f[ X:–)QDW¨U@žJ[ ]Q¸SDU)V2[_X:VuhX:[ t#S0U)VW3[_Z `@XaV2[ WTj̏@`)r † X9[_]§Z3Q0W ¥OQ0S[
[_UÇ[_]Q1–@`fZ X,`febaQDW³ý§÷Ì£ ¤\]Q † U)Z3Ø#[ U6S0U@v“¥h[_Qo[_]QÄYQ0Z3X:–f`f[_X9–)Q1v(`’[_Z XG Š F{sjx3r HIFiý§÷JXaWsU@žJU@Z YQDKZ J(jm½ ¾  r»`@VY žU)Z¸[ ]Q¥Z U@ebaQDv“WqhVYQ0Z&SDU)VW3XaYQDZ_`f[ XaU)Vœ½T¾“S0`@VœeFQǖ)Q0Zt™b,`fZ d)QŸjžU)ZqQzŠ`fv“¥b:Q@»OžU)Z `@VkÙ&l&püW3tuW[_Q0v U@e[—`@X:VQ0Y™žZ3U)v [ ]QW QDv“Xc YX:W S0Z3QD[ Xaµ-`’[_XaU@VˆU@ž³`Ÿósl&püW3tuW3[ Q0vª»/½¾ XaWs[_]QÄ¥Z UuYhS[\U@žJ[_]Q¸Vhv©eOQ0Z
Ufž ósl&p|W\`fVY([_]QqVhv©eOQ0Z¨Ufž/W ¥i`f[ X,`@bOd)Z3XaY(¥OU)X:V[ W—rz£ º V“SDU)V2[_Z `@W3[0»[ ]QoS0U)v#¥h[_`f[_X:U)Vi`@b † U@Z Ø6žU)Z|[ ]QoW XaVd)baQ
W3]UuU@[_X:VdTv“Q[_]UuY¿XaW|U@ž~U)Z YQDZ J§j!½¾M ½ L’r
`@b:[ ]U)hd@]Ÿ[ ]QTv“Q[_]UuYŸXaW\VUf[1`@WoW3[—`febaQf»iZ U)ehW3[1U)Z\¥i`@Z `@bab:Q0b:Xaµ-`febaQf£ PRQ¸Z Q0YhS0Q¸[ ]QqSDU)v“¥h[—`f[ XaU)V`@biS0U)v#¥b:QŠX:[ t#U@ž[ ]Qqv©hb:[_X:¥baQoW3]UUf[_XaVd6v#QD[_]UY#žU)Z [_]X:Ws[ tu¥OQ¸U@ž¥Z U)ebaQ0v‰e2t#v#UYX9žtuXaVd&[_]QÄv#QD[ ]UY#[_UÇv(`@ØfQÄhW Q¸Ufž[_]QÄW3[ Z hS[_hZ3Q¸U@ž [_]Q¸S0U)V2[_X:VhX:[ t“S0U)VW3[_Z `@XaV2[ W¨[_U©Z3Q0YhS0Qq[_]Q&Vuhv©eFQDZ
U@ž/W3Q0VW3X:[_X9–uX:[ t#W U)b:h[_X:U)VW † ]XaS—] `@Z Q“VQ0QDYQ0YŒ[ U‡SDU)v“¥h[_Q#[_]Q“YQ0Z3X:–f`f[_X9–)QDW0£ ¤MUˆYU‡[_]X:W0» † Q§Z QDS-`@W[6[_]Q§S0U)V2[ XaVhX:[ t S0U)VW3[_Z `@XaV2[ WX:VR`§žU)Z3v † ]QDZ Q#U)Vb:t‡[_]Q¿v“`f[_Z3X9Šc–@Q0SD[ U)Z&¥Z3UYhSD[_W(Nj F{sjx3Or HIFiý§÷!Qr PSR `@Z Q\VQ0Q0YQ0Y»Z_`’[_]QDZ4[ ]i`@V[_]Q1QDV2[_XaZ3Qov(`f[ Z XT Š F{sjx3Or HIFiý§÷…£2¤\]Qov“`f[_Z3X9Šc–@Q0S[_U)ZÂ¥Z UuYc hSD[ W1`@Z QTYX:Z Q0S[_X:U)Vi`@bFYQ0Z3X:–f`f[_X9–)QDVW UQ0`@S—]‡S0`@VŸeFQTS0U@v“¥h[_Q0YŸ–uX,``W X:Vd)b:Q&W Q0VW X:[ X:–uX:[ t `@Vi`@b9tuW XaWD£i¤\]QVuhv©eFQDZÄUfžÑ–)QDSD[ U)Z M W PWRqW3hS—]‡[ ]i`f[o[_]QYX:Z Q0S[_X:U)Vi`@bW3Q0VW3X:[ X:–uX:[ XaQ0W1`@Z3Q VQ0QDYQ0YÆXaWÄW v“`@babm»¢U@ž4U)Z YQ0X Z J(jmY ½ Lr£~¤\]hWq[ ]QS0U@v“¥b:QŠX:[ tªU@ž|[ ]Qv#UYX9¡iQ0Y•v©hb9[_X9c ¥baQ¿W3]UUf[_XaVd•SDU)v“¥h[—`f[ XaU)VÆXaWTZ3Q0YhSDQ0YR[ Z U J(jm½¾-Y ½ L’rz» Z3U)hd)]b:tˆ[ ]Q“W `@v#Q(`@WT[_]`f[ U@žÂW3XaVd@baQTW ]UuU@[ XaVd@£ [1VžU)Z[_hVi`’[_Q0b9t)»i[_]QÇZ3Q0YhS[_X:U)V‡X:V•SDU)v“¥h[—`f[ XaU)V`@bS0U@v“¥b:QŠX:[ t S0U)v#Q0W“`f[“`œ¥Z X:S0IQ \|[ ]Q•W[—`@eXabaX9[ tŒU@žÄ[_]Qªv#UYX9¡iQ0Yšv©hb:[_X:¥baQ(W ]UuU@[_X:Vdk`@bad)U@Z X:[ ]v W h^FQ0Z3W1žZ3U)v [ ]QW_`@v#Q&baX:v“X9[—`f[ XaU)VWs`@WoW3XaVd)b:Q&W ]UuU@[ XaVd£ÃÄU † Q–)QDZ0»¢žU)Z\v(`fVt§ó4l&p W3tuW3[ Q0v#W0»MXaVSDbahYX:VdŸ[_]Q›`f¥¥baX:S-`f[ XaU)VW&YQ0W3S0Z X:eFQDYŽ]Q0Z3Q@» [_]X:WÇX:W6VU@[6`@VŒX:W W3hQ@»Ñ`@VY [_]Qov#UYX:¡iQDY#v#QD[ ]UuY#X:W4v“U)Z3Q\Z U)ehW3[4žU)Z|VU@VbaX:VQ-`@Z|¥Z U)eb:Q0v#W0£)¤\]XaW4XaW|YhQ1[ UT[ ]Q "

t

ti

k=1

k=2

k=3

.

.

.

k+1

k

t i+1

.

.

.

.

k=N c

ðj¬ºÍ¶Éø@ɎÜÝ¿X¾Œ»›³~¿XÀ­~½¹e¬®¾Œ»›±v³~Ò(°˜ÀÄ­ Ë ¬º»~«¬®¾f‘,rMX:WÂ`qW Ufž[ † `@Z3Q
¥i`@S—Ø’`@d@Q
žU)ZÑW3U)b:–uXaVdqlonqpœXaVX9[_X,`fb)–@`fbahQ³¥Z U)ebaQ0v#W0£fº[ hW QDW|–@`fZ X,`febaQzc U)Z3YQ0Z –@`fZ X,`febaQzc W3[ Q0¥ei`fS—Ø † `@Z3Y(YX9^~QDZ Q0V2[ X,`f[ XaU)V6žU@Z v©hba`@W0£’¤\]QobaX:VQ-`fZ W3tuW3[ Q0v#W[ ]i`f[Ç`@Z3XaW Q#`f[TQ0`@S—]k[ Xav#QW3[ Q0¥Œ`fZ Q¿W U)b9–)QDY † X9[_]ÆQ0X:[ ]Q0ZTYQDVW Q#U)ZTei`@VYQ0Y YXaZ3Q0S[TbaXaVQ-`@ZÄW3tuW3[ Q0v v#QD[ ]UYW0»U)Z † X:[_]œ¥Z3Q0SDU)VYX9[_XaU@VQ0 Y &Z3tub:U8–‡X:[ Q0Z_`’[_X:–@Q6v#QD[ ]c UYW0£@º V¿`fYYX:[ XaU)V[_UqW3U)b:–uX:VdÄ[ ]Q\lonqp‹jÌÔ@Û2rz» 465 &(*@7A9;:B< Xav#¥baQDv“QDV2[_WÑW3QD–@Q0Z_`fb]Xad@]b:t QDÍ#S0X:Q0V2[&`fbad)U)Z3X:[ ]v“W³žU)Zo¥FQDZ3žU)Z3v“X:Vd¿W Q0VW X:[ X:–uX:[ tª`fVi`@b:tuW3XaW0£¢lqhZ XaVd“X:V[ Q0d)Z `f[_X:U)VŸU)V `d)X:–@Q0VŸW ]UuU@[ XaVdX:V[ Q0Z–@`fbù 8» 465 &(*@7A9;:B< Z3Q0ghXaZ3Q0WoS0U@v“¥h[—`f[ XaU)V§UfžM[ ]QTZ Q0W3XaYh`@bU@ž [_]QoW[—`f[ Qo`@VY¿W QDVW X9[_X:–uX9[ t©Q0ghi`’[_XaU@VW1jÌÔ@Û)r cjmIÔ ]2rz»)`@W † Q0b:bi`@W4S0U)v#¥h[_`f[_X:U)V©U@žO[_]M Q 2)`’c S0U)eX,`@V † X9[_]›Z3Q0W ¥OQ0S[\[_U©[ ]QqW3[_`f[_QDWo`@VY§[ ]Q0X:Z
YQ0Z3X:–f`f[ X:–)QDW0£¤\]Q&W heZ U)h[ XaVQ

 S-`@b:baW
[ ]QThW QDZc YQD¡iVQDYˆZ U@h[_X:VQ0 W  •`@V Y   §[_U¿QD–f`@b:hi`f[ QT[ ]QTZ QDW XaYhi`@bJU@ž [_]Q6W3[—`’[_Q©QDguh`f[_X:U)VW&`@VYˆ[_]Q©Z3Xad)]2[¸W XaYQ6U@ž|[ ]Q`@YYX:[_X:U)Vi`@b/QDguh`f[_X:U)V”jmÛ2rz»FZ3Q0W3¥FQDSc [_X9–)Q0b9t)»O`f[¸`“d)X9–)QDV•[_X:v“Q&xz»¢žU)Z1d)X:–@Q0V•–f`@b:hQ0W1U@žÂ[ ]Q6¥`@Z_`@v#QD[ Q0Z3WÄs»FW[—`f[ Q0W1{|»¢W3[—`’[_Q

„"

daspk Direct Solver

% !!'&("

Iterative Solver

cNjacA Band

cNresA

cNpjacA

cNpsolA

cNresB

cNpjacB

cNpsolB

cUpjac

cUpsol

Dense

cNjacBbnd

cNjacBdns

j_cNsys

A

D

g_cNsys

I

F

O

R

cNsys

  !" # $

cUsysM

cUsysC

ðj¬ºÍ¶ÉV÷Ɏß!½¹³~¿X½¶»~¬®¾±·­Ý¸v°˜ÀÄÀº±·´¹ŒÁ*)
+,9Ç.-0/
1'2

YQ0Z3X:–f`f[ X:–)QDW\{ } »¢`@VYªSDU)V2[_Z3U)baW K UiXÌ£Q@£9»

w˜€‚wTjxzy {Ây { } y_sy K r×

jÌÔI-)r

¤\]Q#W heZ U)h[ XaVQ3   ªXaW¥Z3Q0¥Z3USDQ0W W3Q0YŒ[ ]Z U)hd)]$54CD )AE [_Uªd)QDVQ0Z `f[_Q¿[_]Q#žU)b9c baU † XaVd©[ † U“W heZ3U)h[ XaVQDW6jW QDQÇÎMXad£iÛ)r \/j!`2r 4  [ U¿QD–f`@bah`f[_Q F¢w 5 € FOw { 5 ~ F¢w { 5 } ~ F¢w f wÅ jÌÔ¿>2r K 5 ~ F{ F{ } Fi F K 5 žU)ZÑd@X:–)QDV6–f`@bahQ0WÑUfž[_]Q¨W Q0QDY©–)QDSD[_U@Z W {45 » { 5 } » s5 »@`@VY K 5 Uf`@VYŸje¢7r 6  1[_UqQ–@`fbahi`f[ Q 5 € F¢w ý 5 ~ F¢w ý 5 } ½ª jÌÔ¿¬2r F{ Fi{ } 5 » ý5 } £ žU)Z\d)X9–)Q0VŸ–f`@bahQ0W\U@ž [_]QTW3Q0Q0Y•v(`’[_Z X:S0QDW ý™ â—ÝÞ!ç ô @ð á q ä æ õ ô åÌð@åmÞÌä’ß ¤\]QRQDghi`f[_X:U)VWŸ[_]i`’[•v6hW3[•eFQÆW U@b:–)QDY‚U)V‚`”d@X:–)QDV‚W3]UUf[_XaVd”X:V2[_Q0Z–f`@bTù“S0`@VüeOQ ¥i`@Z[_X:[ XaU)VQ0Y(X:V2[_U©W3X9Š(¥i`@Z3[ W0»S0U)Z3Z Q0W3¥FU@VYXaVd©[_U6[ ]QqW X:v“X:b,`@Z4¥i`fZ3[_X9[_X:U)V“UfžJ[_]Qq–@Q0SD[ U)Z øs£~PRQQ–f`@bahi`’[_QQ-`fS—]k¥i`fZ3[TU@žs[_]QZ3Q0W3XaYhi`fbÂe2t™Z QD¥FQ0`f[_QDYb:t™S0`@bab:XaV9 d 4   † X9[_] S0U)V2–@Q0VX:Q0V2[ÂW3Q0Q0Y–)QDSD[_U@Z W0£8nÄWM`fVÇQzŠ`fv“¥b:Q@»-SDU)VW3XaYQDZ/[ ]Q|žU)hZ[_]¥i`@Z3[/Ufž[ ]Q³Z3Q0W3XaYhi`fb j!W3Q0Qª¤Ñ`@ebaQœ„8r£|¤\]Q•W3heZ U@h[_X:V3 Q 4   Æv©hW[#eFQŸS-`fbabaQDY”? ½ wjÌS Ô ~ S ½ zj!S ½ y  „8r3r [_X:v“QDW1[ U(QD–f`@b:hi`f[ Q[ ]Q6Z3Q0W X:Yhi`@b¨jÌ¿Ô ]2r\žU@ZÄQ0`@S—]ˆS0U@V[ Z U)b ¥i`@Z `@v#QD[_QDZÄX:V•[ ]Q6X:V[ Q0Z–@`fb ù £ þ U)v#¥i`@Z3XaVd•jÌ¿Ô ]2r\`@VYRjÌIÔ >2r
X9[1žU@babaU † W
[ ]i`f[ Œ € J£i¤\]QÇW QDQ0Y‡–@Q0SD[ U)Z {Œ 5 • 5 XaWoW3QD[1[_U [_]i`’[©¥i`@Z[U@ž\[ ]Q“–@Q0SD[ U)T Z øƒ[_]i`’[©S0U@V[_`@XaVW6[_]Q(W Q0VW X:[ X:–uX:[ XaQDW † X9[_]ŒZ3Q0W ¥OQ0S[[_U‡[ ]Q S0hZ3Z QDV[
S0U)V2[_Z3U)bO¥i`@Z_`fv“Q[_Q0ZD£¤\]QqW QDQ0Y§–@Q0SD[ U)Z { 5 } XaW³W Q[¨[_U[_]QÄS0U)Z3Z QDW ¥OU)VYX:Vd©¥i`@Z[ nhoÕÔÏop

8

„8

aX V/ø žU)bab:U

} £/ÎiU)Z&Q0`@S—]kSDU)V2[_Z3U)b`vˆ€ñ„)y…”…”…y_½?w»[_]Q#X:VX:[ X,`@b:Xaµ0`f[_X:U)VŸU@ž³[ ]Q#W Q0QDYk–@Q0SD[ U)Z K 5 † W¨žZ U)v

· uhR’jx3r4€ ¹ ‰ R   ÷ Œ ~žƒT…I‰ R   ÷  ~ ‘ ‹ ‘ Ž   – y“’ ‹ ’   · ~ ¹‰

º

jƒ  ƒF„r;~ӄ0‘#‰YR ‹  ÷   y ¼ y º ~ ‘ Ž, ‰ ‹ ø   y x ¹ y y ‹ R   ÷  ~ ‘ ‘ Ž  – ‰ R ‹  ÷   ¼ x ¹ › R ÷ “ y ’ y Q ’ ’  ‹  

† ] Q0Z3Q † QT]i`-–)QhW QDYŽjl]2r
`@VYRjN>2rz£ ¤\]Q0Z3QDžU)Z3Q@» [_U‡SDU)v#¥h[_Q¿[_]Q(Z QDW X:Yhi`@bsU@ž
[ ]Q“W3Q0VW3X:[ X:–uX:[ tÆQ0ghi`’[_XaU@VW † X9[_]kZ Q0W3¥FQDSD[ [_U¿[_]QS0U@V[ Z U)bJ¥`@Z_`@v#QD[ Q0ã Z ‰ R   ÷ Œ [_]QU@Vb:t›VU)Vµ0QDZ U“S0U)v#¥FU@VQ0V2[1U@ž K 5 X:W;uh: Ro€ „@£O¤ U S0U)v#¥h[ Q[ ]QTZ QDW XaYhi`@b/U@žÑ[ ]QW Q0VW X:[ X:–uX:[ t§Q0ghi`f[ XaU)VW † X:[ ]•Z3Q0W3¥FQDSD[¸[ U ‰ R   ÷  » † Q6W3QD[ uh: R\ € ƒãy ~kx ¹ £¤MUSDU)v#¥h[_Qq[ ]QqZ3Q0W X:Yhi`@b~U@žJ[_]Q&W3Q0VW3X:[ X:–uX:[ t“Q0ghi`f[ XaU)VW † X:[_](Z Q0W3¥FQDSD[ [_U}‰YR ‹  ÷   »6ƒO„6€ „)y-×9×:×9y°ƒF» – € Ôy-×9×:×9y—½Sy» † QW3QD[6   XaWsS-`fbabaQDY † X9[_]#YX:^FQ0Z3Q0V2[ `@Z d@hv“QDV2[_Wq[_U›U@e[—`@X:VˆQ0X9[_]QDZ&`›YQ0VW Q¿U)Z&`§ei`fVYQ0k Y 2)`@SDU)eXa`@V£~ºžs[_]3 Q 2)`@S0U@eX,`@V™X:W VU@[oei`fVYQ0Y~»OW3heZ3U)h[_X:V Q 6  ¿X:W\S-`@b:baQDY † X:[_]Ÿ[_]QTžU@babaU † X:Vd6W3Q0Q0Y•v(`’[_Z X:S0QDVW \ ý5 € B

Ž,«

ý 5 } € ?„ … B ø  y

ø y

† ] Xab:QˆX:ž&[ ]Q¶2)`@SDU)eX,`fVÏXaW§ei`fVYQ0Y~»@6 v(`’[_Z X:S0QDWV\

Ž,«

  ŽX:W§S-`fbabaQDY

·¸ BA º.»

† ]Q0Z3QB‚X:W³[_]Qx2)`@S0U@eX,`@V(ei`@VY † X:Y[_]Ÿ`fVY nhoÕÔÏoÕÔ C

† X:[ ]Å[_]QˆžU@babaU † X:VdRW3Q0Q0Y

·¸ B A .º » „ ý5 } € ? … ¹ ¼ y 

5 € ýñ y ¹  ¼

?

jmÛ@2r

jmÛ„8r

XaW³[ ]QqSDhZ Z3Q0V2[oXaV2[_QDd)Z_`’[_XaU@V(W3[ Q0¥›W3Xaµ0Qf£

ßiå âÌà@î3ð@åmÞÌä’ßEDFä’áaâî3ð@ß~ï—â—Ý

¤\]Q
`fS0S0hZ_`@St6U@ž[_]Q³W3[_`f[_QDWÂ`@VYÇW QDVW X9[_X:–uX9[_XaQDWMSDU)v#¥h[_QDY6e2tW465 &*@79;:=< X:WMW3¥FQDS0X9¡iQ0Y e2t™[_]Q¿Q0Z3Z U)Z&[ U)baQDZ_`@VSDQ0G W F8Ix H¿Ø
`fVK Y J)Ix HØ £ ¤\]Q0Z Q“`@Z3Q¿[ ]Z QDQ“v#QD[ ]UuYWTU@ž¨W3¥FQDS0X9žtX:Vd [_]QDW QT–f`@bahQDW\Xa} V i ))A*,+ £ „”]

¤ ]QoW X:v“¥b:Q0W[ÂhW3Q1X:WÑ[_U&[_`@Ø@Q\eOU@[_]LF8xIH¿ØF`@VY;J)xIH¿Ø¢[_U&eOQ\W S0`@b,`@Z3W0£)¤\]Q0W Q1hW3Q0Z|YQ¡iVQ0Y \ –f`@bahQDW † XababO[_]QDV•eOQThW QDYªžU@ZÄ`fbabF–f`@Z3X,`@ebaQ0WD»eOU@[_]ªW3[_`f[_Q`@VYªW3Q0VW3X:[ X:–uX:[ t@£ n W QDS0U)VY™U)¥[_X:U)VˆX:WÄ[_U(W U)b9–)Q6[ ]Q©W3[_`f[_Q©`@VYˆW3Q0VW3X:[_X9–uX:[ t•Q0ghi`f[ XaU)VW † X:[ ]ˆYX:^FQ0Z3Q0V2[ [_U)b:Q0Z `@VS0QDW0£¤\]Q1–@`fbahQDWMF8xIH¿Ø9¾&`@VY9J)xIH¿Ø:¾q`@Z Q1hW3Q0Y(žU)Z³`@b:biW3[—`’[_Qo–@`fZ X,`febaQDW³`fVY9F8xIH¿Øaû `@VN Y J)Ix H¿Ø,û¨`@Z QThW3Q0Y•žU@Zo`@babFW Q0VW X:[ X:–uX:[ t(–@`fZ X,`febaQDW0£ n¸WÄ`#[ ]XaZ3Y•U)¥[_XaU@V»[_]QhW3Q0Z¸S0`@V‡¥Z U–uXaYQÇ`“W3heZ U@h[_X:VQ † ]XaS—]ˆW3¥FQDS0X:¡Q0WOF8xIHØ|`@VY J)Ix H¿Ø³–f`@b:hQ0WqžU@ZqQ-`@S—]ÆU@žs[_]QW[—`f[ Q–@`fZ X,`febaQDW&`@VY™žU)Zq[ ]QS0U)W[TžhVSD[ XaU)Vˆ–f`@Z3X,`@ebaQ@£ º VŽ`@YYX:[_X:U)V»U@Z YQDZ6U@ž¨v(`@d@VX:[ hYQ¿XaVžU@Z v“`f[_X:U)V™žU)Z&[_]Q“W3[—`’[_Q0WD»M¥`@Z_`@v#QD[ Q0Z3W0»M`@VY S0U)V2[ Z U)b:W©v©hW[eFQ›¥Z U–uXaYQDY£Âp|Z Z3U)Z6[ U)baQDZ_`@VSDQ0W©žU)Z[_]Q›W3Q0VW3X:[ X:–uX:[ tƖf`@Z Xa`@ebaQDW©`@Z3Q S0U)v#¥h[ Q0Y•`@W\žU)bab:U † VW \ Q F’Ix HØTXaW§W3QD[§[ UŒ[_]R Q F8Ix H¿Ø P ÎiU)Z(`@bab1W QDVW X9[_X:–uX9[ t‹–f`@Z3X,`@eb:Q0WD»s[ ]QˆZ QDb,`f[ X:–@Q•[_U@baQ0Z `@VSDQ [ U)baQDZ_`@VSDQ&U@ž [_]QTS0U@Z Z QDW ¥OU)VYX:Vd¿W3[_`f[_Q&–f`@Z Xa`@eb:Q@£ Q J)Ix HØ X:W P ÎiU)Z³W Q0VW X:[ X:–uX:[ XaQDW † X:[ ]§Z QDW ¥OQ0S[1[ U©¥i`@Z `@v#QD[_QDZ W
k[_]Q&`feW U)b:h[_QÄ[_U)b:Q0Z_`fVS0 SDU)v“¥h[_QDY(e2t#YX:–uXaYXaVdT[_]O Q J)Ix H¿Ø[_U@baQ0Z `@VSDQÄU@ž~[_]Q1S0U)Z3Z QDW ¥OU)VYX:Vd©W3[_`f[_Qo–f`@Z Xa`@eb:Q e2t§[ ]QTQ0W3[ Xav“`f[_Q&¥Z U–uXaYQDYªžU)Z
[ ]i`f[o¥i`fZ_`@v#QD[ Q0Z0£ P ÎiU)Z§–f`@Z3X,`@eb:Q0W§[ ]i`f[•Z Q0¥Z Q0W3Q0V2[ˆb:XaVQ0`@Z›S0U@v©eXaV`f[_X:U)VW›U@žÇW QDVW X9[_X:–uX9[_XaQDW † X:[_]ÓZ Qzc W3¥FQDSD[©[ UˆXaVX9[_Xa`@b|S0U)VYX:[ XaU)VW(j!W3Q0Q›¤M`@eb:Q‡„8rz» [ ]Q›`@eW3U)bah[_Q(QDZ Z3U)ZÇ[ U)baQDZ_`@VSDS Q J)Ix HØ X:W§S0U)v#¥h[ Q0Yƒ`@WŸ` † Q0X:d)]2[_QDYüW3hv U@žSDU)Z Z3Q0W ¥OU)VYXaVT d J)Ix H¿ØT[_U)b:Q0Z `@VS0QDW(žU)Z([ ]Q W[—`f[ Q©–@`fZ X,`febaQDW¸YX9–X:YQ0Yœe2t™QDW3[_X:v(`’[_Q0W¸U@ž4[_]Q¿W3[_`f[_QDW0£J¤\]Q † Q0X:d)]2[_W`@Z3Q©[_]QSDUfc Q͓SDXaQDV[ W\U@žJ[_]QqbaX:VQ-`@ZsS0U@v©eXaV`f[_X:U)VW0 £ ‚JQD[U J  Vy J  y-×9×:×9Vy J ‘~eFQ¸[ ]Q&`@eW U)b:h[_Q1[_U)bc QDZ_`@VSDQ0W6žU@Z[ ]Q(W[—`f[ Q“–f`@Z Xa`@eb:Q0W&Š { {XW `@VYRS0U)VW3XaYŽ,QDZÇ « [ ]Q§`@Z3Z_`-t g € À¤Â Z » « Ž † ]QDZ Q Z { W Ž « £i¤\]Q0V•[_]Q`@eW3U)bah[_Q&[_U)b:Q0Z `@VS0QDWÄhW Q0Y•XaVŸS0U@v“¥h[_XaVd g XaWoW3À¤QDÂ[1Y [_U Z ÷ H: ÷ » ]QDZ _ U J  Vy J  y-×9×:×9Vy J ‘ [ Ä÷ ’ ] Q ^ : ÷ XaWo`@VªQ0W[_Xav“`f[ Q¸žU)Z ^ ÷ £ Ž,«  \ ^ \ † Ž «

nhoa`

¨ âßÝÞå!ÞÝÞåN©›äÜÛ&åãiâ § õOåmÞ æ ð@á ¨ ä’á ô å!ÞÌä’ß

º VRv“`@V2tœU)¥[ Xav“`@bÑSDU)V2[_Z U@bs¥Z U)eb:Q0v#W0»JU)e[—`@X:VXaVdª`@VkU)¥[ Xav“`@bÑW3U)bah[ XaU)VÆXaWTVUf[[ ]Q U)Vb9t•d@U2`@bÌ£¢¤\]QX:cV bihQDVS0QUfž4¥Z U)ebaQ0vÚ¥i`@Z `@v#QD[_QDZ WÄU)Vˆ[_]Q6U)¥[_X:v(`fb W3U)bah[_XaU@VŽj[ ]Q W U#S-`@b:baQDYªW3Q0VW3X:[ X:–uX:[ t›UfžÑ[_]QÇU)¥[_Xav“`@bFW U)b:h[_X:U)V¢r¨XaWo`fbaW U#U@žMXaV2[_QDZ QDW3[-£ QDVW X9[_X9–X9[ t›XaVuc žU)Z v“`f[ XaU)V“¥Z U–uXaYQDWo`6¡iZ W[3cU)Z YQDZ\`@¥¥Z U-ŠXav“`f[_X:U)V(U@ž/[ ]QqeOQ0]i`-–uXaU)Z¨U@ž/[_]Q&U)¥[_Xav“`@b W U)b:h[_X:U)V † ]QDVŸ¥i`@Z_`fv“Q[_Q0Z3W\`@Z3QqVU@[o`’[
[_]QDXaZ
U@¥[_X:v(`@bi–f`@b:hQ0W
U@Z † ]Q0VŸSDU)VW3[ Z_`@X:V2[_W `@Z QTW3baXad@][ b:t#–uXaU)ba`f[_QDY£º Z V i )A)*@+ » † Qv“`@Ø@QThW3QÇU@ž [_]W Q uQ0VW3X:[_X9–uX:[ t›¤\]QDU)Z QDv j!W3Q0Q  Ô8‘ržU)Z4VU)Vb:XaVQ0`@Z4¥Z U@d)Z_`@v#v#XaVd&¥Z3U)eb:Q0v#W † X9[_]“Q0ghi`@b:X:[ t¿`@V@Y mfU)ZsXaVQDghi`@baX9[ tS0U@Vc W3[ Z_`@X:V[ WV\ ç p úâ6å q|ÞÌï—â“ï—ä’ßiå!Þß ô ä ô Ýá ©Ÿç@Þ íqâî_âßiå!Þ!ð ú0áaâð@ßOç•ï—ä’ßiì dfe eHO(G ehg cjim A ifk âmå lon ? noð@ßOf ÝÞ!çuâî&åãi â ÛDð æ Þá ©›Üä Û¨õOî_Iä ú0áaâ æ Ý â lÂj ^ r æ ÞßiÞ æ BÞ t’@ jmÛ)Ô)r Ý ô úsr-â—ïåså…ä ? j ^ rsÌ € u y p~j ^ Ñ r †ut¢y õð@î3ð æ âå…âîzBÞ t’â / ç úV©ªåãiÁ â Ý@â—ïå ä’î—? Ý už{vW ú ð@ßO9 ç tì{vWOw o
Ö Ý—Ý ôuæ â“åãð@å Û-ä’î§#j u (y tr&€ „(-

mj y_2r§åãuÞaݝõOî_äIú0áaâ æ ãð’ݪðkáaäï ð@á æ ÞßiÞ æôuæx^ ¹ nGqÑãuÞÌï—ãŒÞaݟî_âÌà ô á9ð@î•ð@ßOçyqÑãuÞÌï—ãŽå…ä’ì àuâåãiâîq|Þåã‡ÞåÝ¿ð’Ý—ÝäïÞ!ð@å â ç k ðà@î3ð@ßuàuâ æô áéå!ÞöõOáéÞÌâîWÝ@â—ïå ä’î_ÝGz ¹ ð@ßOçL{ ¹ n³Ý0ð@å!ÞaÝ ë â—Ý6åãiâ Ýâ—ï—ä’ßOç™ä’î3çuâîÇÝ ô | ïÞÌâß~ï ©‡ï—ä’ßOç@ÞåmÞÌä’ßÝ o DãiâßRåãiâî_â(â~})ÞaÝåÝ#ð@ߋä—õiâß™Ý õiãiâî_†ï—âßiì å…âî_â çŸð@åo#j u (y trs€ jmy_2r&Ý ô ï—ã•åãð@8å Û-ä’î¿â Ý@âî°©›j#u/ytr {K¦åãiâî_âÇÞaݝð@ß ^ j#u y(trà{‚WOƒ ð@ßOç„zMj#u y(tr¥{…W ú y{|jlu/ytrõ{†W w n=qÑãuÞÌï—ã‚ð@î_âŒðÅáaäï ð@á æ ÞßiÞ æôæ ð@ßOçƒð’Ý—ÝäïÞì ð@å…â ç k ðà@î3ð@ßuàuâ æô áéå!ÞöõOáéÞÌâ î Ý@â—ïå ä’î—ÝÜä Û4õOî—Iä ú0áa⠈ æ ‡ ÔnŠ‰Io0‹ ô îzåãiâî æ ä’î_Vâ n ^ ²j … Vr nŒzMÜj …éVr nÂð@ßOç {4²j … r(ð@î_â›ï—ä’ßiå!Þß ô ä ô Ýá ©™ç@Þ íqâî_âßiå!Þ!ð ú0áaâ¿Þy ß  ð@ßO ç q¨â#ãIð Ý@â ^ jmy—)r€ ^ ¹ y zMj!y—2r&€ z ¹ y {4jmy—2r|Ž € { ¹ oŒC ߜð2ç2ç@Þå!ÞÌä’cß n,Û-ä’îÇð@áá/#j u (y tKr {Rn|åãiâî_â&ãiä’á9ç’Ý 3wV j#u/ytr€ L‘’

 j#u y(t rр qÑãiâî_ⓐ jlu/ytr&ÞaÝåãiâ6ä—õOåmÞ æ

 zMj#u y(trzy  {|jl/u ytry

ð@áÑï—äÝåõð@î3ð æ âå…âîzÞBt’â çbúV©(jlu/ytrn  #j u (y tr|X € lÂj ^ j#u/ytr r×

¤\]Q›YQDW XaZ3Q0YŽW3Q0VW3X:[_X9–uX:[_X:Q0W6`@Z Q([ ]hWYXaZ3Q0S[_b:tkU)e[_`@XaVQ0YRžZ3U)v [ ]f Q ‚/`@d)Z `@Vd)Q“v©hbc [_X:¥baX:Q0Z W\SDU)v#¥h[_QDYˆYhZ3XaVd¿[_]QÇW3U)bah[ XaU)VªU@žM[_]QÇU)¥[_X:v“X:µ-`f[ XaU)V(¥Z U)ebaQ0vª£in¸b9[_]U@hd)] [_]Qq`@eFU–@Q&v“Q[_]UuY§d)X9–)Q0W\U@Vb:t#W QDVW X9[_X9–X9[_X:Q0W¨UfžJ[_]Q&U)¥[_Xav“`@b¢W U@bah[ XaU)V † X:[_](Z Q0W3¥FQDSD[ [_UTS0U@VW3[ Z_`@X:V¥OQ0Z[_hZ3ei`f[_X:U)V»)Uf[_]QDZsW3Q0VW3X:[ X:–uX:[ XaQ0W|S0`@V“`@baW3UeOQoU)e[_`@XaVQDY † X:[_]¿v#XaVU@Z S—]i`@Vd@Q0WsXaV©[_]Q1¥Z U)eb:Q0v W QD[c h¥~£ þ U@VW X:YQ0ZÂ[ ]QoS-`@W3Q † ]QDZ QoW QDVW X9[_X9–X9[ t©U@žO[_]Q1U@¥[_Xc v(`fbOW3U)bah[_XaU@V † X9[_]§Z3Q0W ¥OQ0S[
[_U6W U)v#QÄ¥Z U)eb:Q0v ¥i`@Z `@v#QD[_QDZ³X:W¨YQDW X:Z Q0Y~£¤\]Q¸W U)b:h[_X:U)V XaWs[_U`@YY§[_]i`f[¨¥i`@Z `@v“Q[_QDZ
`@v#U)Vd[_]QqU@¥[_X:v“X:µ-`f[ XaU)V–f`@Z Xa`@eb:Q0W
`fVY›`@YYŸ`fV§Q0ghi`@bc X:[ tkS0U)VW3[_Z `@XaV2[[ U™Ø@Q0QD¥”[ ]Q›¥i`@Z `@v#QD[_QDZ6[_UœX9[_W©VU@v“X:Vi`@b4–@`fbahQf£Ñn
[[_]Q›U)¥[_X:v©hv W U)b:h[_X:U)V»Ñ[ ]} Q ‚ `fd)Z_`@Vd)Q•v6hb:[ Xa¥b:XaQ0Z6`@W W3USDX,`f[ Q0Y † X:[ ]‹[_]XaW#`@YYX9[_X:U)Vi`@b
SDU)VW[_Z_`fXaV2[ tuXaQ0b:YW
[_]QYQ0W X:Z QDY•W3Q0VW3X:[_X9–uX:[ t)£ ”

 ˜ e_š Œ{ › g 7 ˜ e •—–JNg ™

¤MU#Z hV‡`¥Z3U)ebaQDvÒX:V i )A)A*,+ [_]QhW Q0Z1v©hW[1W h¥¥b:t›W3heZ U@h[_X:VQ0W\[_]`f[ P P P P P

W3¥FQDS0X:žt9b¢`@d)W
žU)Z
[ ]QTU)¥[ Xav#Xaµ0QDZ\`@VYŸ[_]QX:V2[_Q0d@Z_`f[ U)ZVU W3¥FQDS0X:žt¸[_]Q4XaVX:[_Xa`@b8d)hQDW WM`@VYT[_]Q4eOU)hVYW/U)VT[_]Qs¥`@Z_`@v#QD[ Q0Z3WJŸ`@VYT[_]Q4S0U@V[ Z U)b:W K U Q–@`fbahi`f[ Qq[_]Q`@YYX:[ XaU)Vi`fbFSDU)VW3[ Z_`@X:V2[_WÇj̏feiro`fVY›[_]QDXaM Z 2)`@SDU)eX,`fVVW U Q–@`fbahi`f[ Qq[_]QTZ3Q0W X:Yhi`@b~U@žM[ ]QTv“UuYQDbQ0ghi`f[ XaU)VW6jmÔ@`2°r U Q–@`fbahi`f[ Qq[_]QTX:V[ Q0d)Z `@VYŸU@ž [_]Q`@YYX:[ XaU)Vi`fb~YX9^~QDZ Q0V2[ X,`@bFQ0ghi`f[ XaU)VRjmÛ2rz£

¤MU#Xab:bahW[_Z `f[_Q1[_]QThW3QÇU@žˆi ))A*,+ S0U)VW3XaYQDZ\[_]Q¿„Dc l¦]Q-`f[1Q0ghi`’[_XaU@V F ^ F ^ € Fix Fœ  „”>

jmÛ@Û2r

5

4

3

2

1

0

0

0.2

0.4

0.6

0.8

1 t(s)

1.2

1.4

1.6

1.8

2

ðj¬ºÍ¶É!û¶ÉVþ{¯»~¬ÄìºÛ·°@»~¬º¿X¾ ³~±·­~½À®»~­I¼ˆ¿˜³»~«¶±ê«¶±·°@»¯³ƒ¿˜¹eÀº±·Ãâɶß¿XÀ®¬®´&

=/

&

D(



BA C

.-

?

E5

9(

>

.-

F/

G

(

@

.-

H

B$



6( 3

65

I"

@


6A

JA

K

6

(

L"

M&

.+

N(

K

&

S

O



.-

P

.+




/



Q/

6R

T? U?

L&

.+

47

V& "

F7

V& 7

*Z

[

65

*8

O(

0$

W(

XY

\

F7

Y

]

Ô)Ô

O

.-

X&