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¦G9f¦G9¡J9rPX§ ¬³D«9XJf³fP9g`TfrrpâTf·G¡ X9 9`#f 99T9f¦Gf¦ # I PXGr1w f.X9«Jfrr°·T89f.9T«Jr TXr Í TD"¡GIrDTfgp"TG9f&¢¦f¡TX9 9 ,fLGp9#Ùc§O*XLTG.paTf¡X±L 9fGf wf¢ ¨ þ²X9«Jfrr°1TX)¡GIrDTXrgp wAfPfrfTAGk"GATX Í Þ8¡8£m§µJ×TØPØP× Ñ ß`Xf¢ T[§rµ×ØØPáPÙ8ff`L¦X,9G 99 f X)¡GIrDTfgpÛ9f19rL
Gk X9 9 TD f}TXrG¡*X 9T9Lw*9f"T9§
Í ×TØPØØIÙ Ñ ÞL 8m§ Í ×TØPØP×PÙ&TXß5XTf¢L 8m§ Í ×ØØáIÙcµ r `LXLX³L9LPfXw¦XXc9rPX"* Xfg9K?¦f`T*«JT9¡TfwP {x(t − iτ ), u(t −µ iτf),9. . . , } u(l) (t − iτ ), i = (i , . . . , i ), i , l ∈ Z+ } ¨ ¨ Xk«8GXT i1 τ1 +1 · · · + i`` τ` jI iτ §I¶- GXT.9f± T`IXL¡ K(δ] §&*f¡#¡#·δf1P, X. . .P, δ` g 9 â ? #rP9Lw9 ¨ K ¦G9T«9f¢Xµ ¨ fPfGr(¡*f Xf¦X¦XT[µ ¨ fr"¦fg9rXr¡T.¢«I a(δ]b(δ] =
k=0
Æ&D¡G$fff L °G¡+GÛTÈ9f w9. = = = =
f (x(t), x(t − τ ), u, u(t − τ )) h(x(t), x(t − τ )) , ϕ(t), t ∈ [−maxi τi , 0] u0 (t), t ∈ [−T, 0]
Í¿Ù
¨ f x ∈ Rn GfT9f9T}«TT9frµ u ∈ Rm ¡9f(ff¦GX ¡9f(P¦Gf¦G§¯*f(¦fG p þ?f ¨ âX9P#yL∈ °R G¡GT9±GXTÚI)9f « P µ §5*X T ), ®Gf99τrP = (τ1 , . . . , τ` ) τTiX∈[0, 9f1 ,TL∈«TTR9f x(t − τ ) µ-X³TXrP¢¦Xr.w § {x(t − τ ), . *f P91.T. , x(tTX− τ` )}T9 PL9ff¡r,9fu h T9¢¦X Pa.TX f n ϕ : [−max i (τ ¦ffþ?f ¨ PPI¦X¦Xw¦fX TiX),f0]r¡T→ G RDGg9rPX§ *f# T*rXg9¯w¦fDcDwP"f#«J¡Tf ¡ GfT9,? §O/Lx9T n C := C([−max (τ ), 0], R ) i¡ir"Tf ¡9f L9ff¡£ff¦fw¦fXc9 u(t) ff¦Gr*XGr¤59P9¥I¦XTà`k«#PG ra ¦ff¡¥I¦f&Pr¦G9rP-§*f& MTfrP¦Xa"ff¦Gw¦fDcD ¡GfT9(? § CU ÃfT`XP?Ópc«¯`f¡ ¨ 9þ¯"I«¢PT X±f9`p1T"rfT8GP9gDfrrpÈTfG¡ X9 9 §À°P9¦fg9r«Pµ T9ÈfP9g`TfÚr T?# ¨ #9}τi T.DG¡p9rf¢P¦Xτif?#9fGp9 ff¦G [P¦Gf¦G`Xk«?rP¦f§X wPGXfr±8P w ¨
,
Í ×PÙ
i+j=k
∂φ δi f + ∂x(t − iτ ) i X Í ∂φ + u(r+1) (t − jτ ) (r) ∂u (t − jτ ) j,r X X ω˙ = κ˙ ix dx(t − iτ ) + ν˙ j du(r) (t − jτ ) + φ˙ =
X
i
+
Ä8 Df
j,r
X i
κix dδ i f +
X j,r
νj du(r+1) (t − jτ )
X = spanK(δ] {dx}
Yk = spanK(δ] {dy, dy, ˙ . . . , dy (k−1) }
*f
U = spanK(δ] {du, du, ˙ ...}
.
Ù
Í Ù
Íÿ Ù Í áIÙ ÍÐ Ù
w X (Yk + U) ∩ X = (Yn +Í U) ÞL∩X}m§µX×kØ≥ ØI×nÙc§
rankK(δ] (Yn + U) ∩ X ≤ n
T9"( W _XZVS[Z
§ Ö A¯¶ÀpãÀpÁ8¯*ÀÃÁ
τ0 ∈ (0, Tµ)¦XaXT ` Wµ 3 τ0 W ⊂ [0, ∀τT1X∈ µ`9fT)9 ®G¡p W : τ1 6=¦Xτa09∀ϕ , ϕ ∈ C t ≥ 0 0 1 XJ µ u f∈C y(t, ϕ , u, τ1X)Ta6=TL y(t, ϕ τ¦f0) U 1¯ 0©,u, G f X 8 9 ¨ f¦G¯wPAy(t, 9f*ϕ,fu, r¡τTG)w¦fXc9rP µJXf ¡9fff¦G ϕ XG¡k? § u τ
T `L
ai (t)bj (t−iτ )δ k
¨ f9 ¨ 8D«Gf δ k1 . . . δ k` ? δ k §?*f89f¢ #Á8? 9f9ÚTXÚ1³ V.`ÃGPL-§*X K(δ] T Xr*DJf"aTfþ#£w9" fG¦fJ« ¨ g °G Df Í Æ&Pf-µ ¿ÏIÐG¿ Ù §X¶- GfT9 K(δ] fLGG¦f §*f} M P¦f9T5 ξ ∈ K} ¦ffLGf¦fr span r K(δ] {dξ ¡9f: ¦XfLGG¦f N M µ N = {w ∈§ M ¡:∃a(δ] = a(δ , . . . , δ ] ∈ K(δ] f¢P1¦fX fG`¦f a(δ]w PaTf∈f¢ N } N M Í Þ" &N TX Xk«?rX¢"aTfþ¥P¦DTG9 m§µ rank N K(δ] ×TØØI×Ù § Ä8r¤`P¡T àw¦fX X φ(x(t X f− mwPiτ), u(t − (l) jτ ), . . . , u (t − jτ )) K P i P ω = (r) κ dx(t − iτ ) + ν du (t − jτ ) j x G Xi f9fXJ9¦faT ¨ j,r Í Þ8¡8£m§µJ×TØPØP× Ñ ß`M Xf¢ 8T[§rµD×TØPØáPÙ
×f§ÁÃA¯*ÀÃÁ 8 ÁÄ Ò Â¶ÀpãÀpÁ8Â Ä Ì pÀ Á}Àp*ÀÃÁ Ö
x(t) ˙ y(t) x(t) u(t)
ra +rb i≤rX a ,j≤rb X
µ
À°Êf¡± ¨ ² D¡G.9f²fPf TLG 9Tff¢·T1ff¦f
¦G9f¦G9f9PaJ9rPàwP f paJ
XwP T9rL
Gkà PP9Gp9.§ *f·XPfr D³` 9T$wà`?fL¡T G Ì aL1 #T[§ Í ¿ÏÏ Ù §A¬ ¨ }f ¨ XT(wP)Gp9 TfwP Í ¿ Ù#X·T ¨ G ®G9µTJ¯Pp¯GTµJ ¯Dff¦G [P¦GX¦GGkI Gg¤59I¡T-¥I¦XTX*TM9fw9.
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ã±ýf©± .m§ 210
F (δ, y, . . . , y (k) , u, . . . , u(J) ) := = F (y(t − i0 τ ), . . . , y (k) (t − ik τ ), u(t − j0 τ ), . . . , u(J) (t − jl τ )) = 0 ,
*f1Pf [w9L f ¨ w9 X §&*f,d˜xGi ,XXgi9rP=âT1, . . . ,Í n¥I¦XT Í ¿¿ ÙpÙ X si rLfr*XT ∂h(si i ) ¡*
ÍÔ Ù
∂x
¦X9âXT(I)X ¨ f¡9ÝPr«P Í ¿ Ùcµ T¡)T¡pD Í Ô Ù µVP (y(t), ¦Xu(t)) 9ÝXT.r"G9r«TJ9r« I««. II¦fP¦Xt§D*f"w¦fXc9rP ¡L9T F L9ff¡r.ga}T9¢¦fLI9§
spanK(δ]
(si −1)
, . . . , hi
)
∂x
¾
Í¿ Ù
.
K(δ] i ½= 1, . . . , p bi (δ] ∂x ¾ (s −1) (s −1) ∂(h1 ,...,h1 1 ,...,hi i ) §X*f9 wPPµ spanK(δ] ∂x Pm PJ (si ) (j) bi (δ)dhi + r=1 j=0 cj,r (δ)du µXwr L µ ∈ fspan {d˜ x# xsG } JT≥ 0¢ 1 ,f.r.¢P.f,d˜ K(δ] 1 +···+s i «LT 9 ¡ 9 f J « J 9 T f ` 9 r X ¨ u § Î Xµ rfwJ¦fDcDr TX S cj,r (δ] ∈ K(δ]
hnno *f,fGT¡ fG9TÚf,X? T*fP×G§ë×G§ ¿ §MrÚÆ&PP( T[§ Í ¿ÏPÏÏ ÙwP9f TDT¢¦D&¦Xg*wPÃÄ
Gp9§ ¶- `ÚT
frLXX «cP ¨ r/P9 f §¶ r Gf9f J9r® ¨ g9 ∂f fIj 9∈r K r×n ∂x ¶
(s −1)
∂(h1 , . . . , h1 1
*I¦X)f ®GÈfX©9`IXL¡ bi (δ] ∈ (s ) µ ¦D969XJ ¡È ∂hi i
q UfWanJhWY x «#"Gp9 `9fwP Í ¿ ÙcµX ®G¡paP9¢ TX.T.`.GD¦fX J¦D≥9È 0 XT#)9ffr¢PI`Pf?f T µ J+1 V C `× U T T"I1 PrIC# µX9 ®G¡paÚrfX¦G
¦G9f¦G 9f9PaJ9rPMVfGp96TMXw9 Í Ô Ùc§
µ
½
(si )
bi (δ]dhi
+
γ m X X
cj,r (δ]dur(j) −
r=1 j=0 s1 +···+s X i
−
ÍÏ Ù
Í ¿kÿ Ù
aj (δ]d˜ xj = 0
,
j=1
VPP § Ö X )Tw¦fXc9rPXT9 aj(δ] ∈ K(δ] 9 f ¦ L Û L P P X f 1 T X ¨ )X«P) II¦fP¦X j,i k 9f(rXff¢PT«#rI¢P ¦X9 Äf(I G ` D G X & w M 9 f f ¦ f G ¦ ¯ 9fff¦GTXrfr¡T s1 DJ V¦XXc9rP-µfÚT`J«²¥I¦XTgp f¡f T` GX} J+1 §P*f VXXLG}T5¥I¦XT (s −1) (s ) ∂(h1 , ..., h1 1 ) ∂(h1 , ..., h1 1 ) Í ¿kÿ ÙcµT`C f× ¢"C¥IU¦X?9©9XµJ¡ PPf [w9 rankK(δ] = rankK(δ] . §}ff?f¢.f Ò X LrLLXµ ¨ LPG9r ∂x ∂xÍ ¿ ØIÙ M ¦X9,XT V¦XXc9rPX (s ) ξi (t) ∈ K dξi = bi (δ]dhi i + À
∂h1 f ¨ ,GXf §À°XG¦Dc«Pµ P P P TX (j) m J s1 +···+si w ∂x ≡ 0 GfT9 I sf1L= r0Xff¢PJ9« cj,r (δ]dur − j=1 aj (δ]d˜ xj r=1 j=0 si µ²VPa (s ) P1¢<i¦X≤ a(p9XJ ξi (δ, hi §-i * ,x ˜f,Lu,w¦f.X. c. 9, u(J) ) G=?0fG`XP i =µ 1, . . . , p x ˜ (s −1) (s −1) µIX ξi ¨ P¦fLf PP9rj ∂(h1 , ..., h1 1 , . . . , hi i ) j > s + · · · + s dξ = 0 1 i i rankK(δ] = Í ¿¿ Ù µ 9 f¡a·¡" `Ifr ∂x d˜ xjG¦Xj²> s1f+ ·«J··+ sri ¨ (s1 −1) (si ) Í I ± Ù 9 ) ¡ f `f¢ÝrfT9r k ¿ ÿ ∂(h1 , ..., h1 , . . . , hi ) . = rankK(δ] rXG`XfPàJ« ?$G Dd˜ fxg9rP-§"*?¦Xµ ¨ ∂x X«PG9rf K(δ] T ¶- (sp −1) µ ¨ f9 (s1 −1) S = (h , ..., h , . . . , h ) h p 1 i G?*fT8Tf`T*r 1 §X*f (s ) si = 0 ξi (δ, hi i , x ˜1 , . . . , x ˜s1 +···+si , u, . . . , u(J) ) = 0 , Í ¿ áPÙ Í ¿ ×Ù ∂S = s1 + · · · + s p = K ≤ n . rankK(δ] ¨ fa 9¢f ¨ g9 Í ¿ Ù f9GG¦D T rff¦f À
µ∂x 9f9â®G²L9LPfX)w¦fX X ¦G9f¦G}¥I¦XT K < n ¦Xa(9XJ g1 (δ, x), . . . , gn−K (δ, x) § (s ) (s −1) (s −1) n−K ) ξi (δ, yi i , y1 , . . . , y1 1 , . . . , yi i , Í ¿JÐ Ù rankK(δ] ∂(S,g1 ,...,g =n ∂x Ì }LfgpµfP9fG¦X 9f"fTaJ9rP u, . . . , u(J) ) = 0. * f 9 f ¦ w P 9 ¦fr9rf¢r i, 1M≤ i w≤ p Í Ùc§ p r f f f ¦
G ¦ 9 f G ¦ 8 I ¥ X ¦ J 9 r P X * T X 9 Ô ˜1 = h1 x ∂f ∂x
=
X
... x ˜ s1 x ˜s1 +1 ... x ˜s1 +s2 ... x ˜s1 +···+sp x ˜s1 +···+sp +1 . .. x ˜n
∂fj δ k ∈ K(δ] . ∂xi (t − kτ )
¥
(s −1)
= h1 1 = h2
(s −1)
= h2 2
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.
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r¢X9T 9g99τi ,w9 1, . . . , ` T-µX""f&rfTw9ÊTGXff¦G [P¦GX¦G ¥I¦XJ9rPX Í ¿ÿ ÙwP9f"f9«?rP¦X* -§
p −1) = h(s p = g1
= gn−K 211
*f"¥I¦XTX Í ¿kÿ Ù&¢« (s ) ai (δ]dyi i
+
fXTX
γ m X X
µ Xτ˜±9:= fLf{T f¦Gi1 [− P¦GTfi¦f0 ,". . ¥I. ,¦XTJiq9rP− X TÍ i×T0ØI}Ù iT= fX1,T. .`. , p¦X±¡GPrw±f §5¬³9Jf TD9 w9#8f¢¦Xkτ«TiLX.¬fI`9¢ ` ≥ 2 Í ×TØPØ Ð ÙOw£9fT5f¢*f¡P§JÀ
µf&X?G¡ATXrP¢¦XO8ff9?Tfrrank(M }X¢¦fr)J«J= 0TX(¬³f?`9¢ Í ×ØØ Ù&TX(¡*f9 wPrV}P¦G§ Ð ¬ ¨ r£f ¨ DJ fTrfT(¡Gf rDTf²¢Pf¡rÝτI1 , . ..X, τ¨ ` f¢Ú9XJ Í ¿ ÙTÛ` XPGr X¦G99-G ¨ r9rL ¢P §¯*f¡LG ¡f9Tf·w9 X(ff¦f ¦G9f¦Gτ˜·¥I¦XJ9X Í ×TØIÙ¦Xf¢ÝfÈfTaJ9rPÛwP Ö c9rP
cj,r (δ]du(j) r =
r=1 j=0
=
i sX l −1 X
(j)
ai,l,j (δ]dyl
,
Í ¿Ô Ù
l=1 j=0
¬âr f)rI¢faTr°Pµ ¨ P¦fL±DJ(9f `?fL¡T¡ P$fÈ V [DTX ¡GÈT#9f T`J«/¥I¦XTaiX(δ) ÛT9/9G¦XrXrP§¶ ai (δ] = P P kµ k X a δ c (δ] = c δ a (δ] = i j,r j,r i,l,j k k Pk k k §£Äfr&9ffg¤59I LXL a δ k i,l,j k TX±9f Í G k f`f¢T`J«I δ ¢aÙ8rfT" PfrXT∆ Di1T, . . . , ∆iq DJf 9f9PI §OÀ
τL1 ,. .., τf` 99Lr T , . . . , T i1 iq µ ¡·`?fL¡T} aGi(δ] ai,l,j ¢9.c©j,r(δ] Dµ£9XJ µ¯(δ] 9f±rXf¦G
¦G9f¦G(¥I¦XTδX PPaT¦fXG¡k"«TT9frµ 9f ¨ ¥I¦X µ ¨ f GXTf¡GIr°`9∆ Ti0µTD δ0 99`D δ0GrX¢ X ¡"©D§OÀ
¡9f PfrDJ D Ti0 DJ"f Lf9f . .,Tτ`§ G5¡TGi0I, .r.D.T,XTriqgp(TτM1,X. " ¶- `XLXL k r ¨ rLr ai (δ] XG ∆ ® i0 Í aGÊTV δ Ù (g,¡g9f ¥I¦X9k PL#TLfk¢1 ,.X. . , k` §*f ∆ i0 ff¦G [P¦Gf¦G*¥P¦DJD9`∆Xi1 fr,f.¢". ., ∆Í iq¿Ô Ù Gr` ¨ 9g (si )
yi
x ˜˙ 1 ... x ˜˙ s1 −1 x ˜˙ s1 ˙ x ˜s1 +1 . .. ˙ s1 +s2 −1 x ˜ ˙ x ˜s1 +s2 ... x ˜˙ s1 +···+sp y˜1 y˜2 ... y ˜ p˜ x ˜(t)
(s1 −1) −1 = f˜i (∆−1 , i0 ∆i1 , . . . , ∆i0 ∆iq , y1 , . . . , y1 (si −1) (si ) (γ) . . . , yi , . . . , yi , yi , u, . . . , u )
Í ¿Ï Ù
(s −1) yi (t) = f˜i (y1 (t), . . . , yi i (t), u(t), . . . , u(γ) (t) (s −1) (s ) y1 1 (t − Ti1 + Ti0 ), . . . , yi i (t − Ti1 + Ti0 ), (γ) u(t − Ti1 + Ti0 ), . . . , u (t − Ti1 + Ti0 ), ..., (s −1) (s ) y1 1 (t − Tiq + Ti0 ), . . . , yi i (t − Tiq + Ti0 ), (γ) u(t − Tiq + Ti0 ), . . . , u (t − Tiq + Ti0 )). (si )
Í ×TØIÙ
¶- (T11 − T10 , . . . , T1q − T10 , . . . , Tp1 − Tp0 , . . . , Tpq − Tp0 )tr = M (τ1 , . . . , τ` )tr ,
Í× ¿ Ù
¨ f M ¡ (1q + · · · + pq) × ` P9¢J9r® TD tr Gff aTD`IT §O¬,f ¨ w9v¦f¡J9fGP9gDfrrp,rv¡#wP τ1 , . . . , τ ` (9fwr ¨ f¢ f9`IrP-
hna¸fnkbcZVS[Z
=x ˜2 =x ˜ s1 = f˜1 (x ˜˙ s1 (t − τ˜), x ˜, x ˜(t − τ˜), u, . . . , (γ) u , u(t − τ˜), . . . , u(γ) (t − τ˜)) =x ˜s1 +2 =x ˜s1 +s2 = f˜2 (x ˜˙ s1 +s2 (t − τ˜), x ˜, x ˜(t − τ˜), u, . (γ) . . . , u , u(t − τ˜), . . . , u(γ) (t − τ˜)) = f˜p (x ˜˙ s1 +···+sp (t − τ˜), x ˜, x ˜(t − τ˜), u, . . . , u(γ) , u(t − τ˜), . . . , u(γ) (t − τ˜)) =x ˜1 =x ˜s1 +1 =x ˜1+s1 +···+sp−1 = ϕ(t), ˜ t ∈ [t0 − maxj τ˜j , t0 ]
Í ×P×Ù
}ILXT ¨ Xa#«£f8P¢XfGp Í ¿ Ù(y(t), Tu(t)) ¡pD¯9f`k«}f¦faT`G w ¦X9 XT±T"G«JJ9«³G t0I¦f≥P¦Xmax §D*?i τ˜¦XjµfXrXf¦G
¦ff¦G`X«?¦f}T£f G f?#fT(ff¢¦XÚf,rfXfrÈT? f¡9"¢Pr«£9f9TL¯f frXTD TX τi ¨ fT}rGTr#¡GIrDTf"¢f9τ˜TrP§ τ1 , . . . , τ ` §f¬X9X9« Ö ¦ff`Pf ¨ XT XT ¨ 9f³¦Dprank(M Xk« ) = `X ¨ ?¦XD« JPp Gr¤`P q§≥ Æ&`D¡G#fI±f ` ∆ ¥I¦XJ9rPX Í ×TØPÙcµOw ¨ fiq¡9 § «TT¦XJ9³J f®G9rL`PrI µ Í ×ØPiqÙM¢P≥r«1£ ¥I¦XTw t0 ≥ T Ti1 , . . . , Tiq (s )
(s −1)
yi i (t0 ) = ξ(y1 (t0 ), . . . , yi i (t0 ), u(t0 ), . . . , u(γ) (t0 ), (s −1) (s ) y1 1 (t0 − Ti1 + Ti0 ), . . . , yi i (t0 − Ti1 + Ti0 ), (γ) u(t0 − Ti1 + Ti0 ), . . . , u (t0 − Ti1 + Ti0 ), ..., (s −1) (s ) y1 1 (t0 − Tiq + Ti0 ), . . . , yi i (t0 − Tiq + Ti0 )), u(t0 − Tiq + Ti0 ), . . . , u(γ) (t0 − Tiq + Ti0 )).
f Xf,P`k« Í × ¿ Ù8TD ¡XTfP9Tr¥I¦XT¯©wI (s +j) yi i (t) µ µ9f 0T9( ≤Gj<r·iq¡G− . r.à , τr` 1 ,.X I1rDiTX=r.{1, ¢f..9., Tp}rPµAgTXτ1 § M
Í× Ù À
£fL [`PrI ¡8afP.¡T9¢"f¦X¢#G ¦f9X ®GX t0T¯r-L °G9«JJ9r«*I«Pr«
rank(M ) = `
µDJµ-9f9T9 hnno A¶- Xap <¢Pr`«³Xà9TL·rfT GXfrr LI rank(M ¨ Xa )Ê τi 212
9fG XXg9rPÞ¡ T[§ Í ×TØØI×Ù µfg X T9¯þ?f ¨ -§ Î ¨ «µ TX fTMfτP19g`Tfτ2§ τ2Üf·wP Í ¥D§ *fâff¦G [P¦GX¦G1¥Iτ¦X1 J9rP/ Í ¿Ô ÙÙ
Í ¨ Xa²·`#P9Xr«³I³af?If¢w ®fTLf Ùcµ9fGr¤5P9Tf¢ Í × Ù ¨ g9 t90≥ `(max cO9i s i −1)T ¢«Of ¨ ¥I¦XJ9rPX5w Ti1 , . . . , Tiq ¨ f¡9ÇârXf`XfPµ X dy(j) , j ≥ 0 rfT9r#XG`XGP8k« f¦f i j
K
iq ≥ 1
d¨ y − du˙ + 2δ1 δ2 (x1 )du = = 2(δ1 (x2 ))2 δ1 δ2 dy + 2δ1 δ2 (x1 )dy, ˙
d (s −1) ξ(y1 (t), . . . , yi i (t), dtj u(t), . . . , u(γ+j) (t), (s −1) y1 (t − Ti1 + Ti0 ), . . . , yi i (t − Ti1 + Ti0 , (γ+j) u(t − Ti1 + Ti0 ), . . . , u (t − Ti1 + Ti0 ), . . . , (si −1) y1 (t − Tiq + Ti0 ), . . . , yi (t − Tiq + Ti0 ), u(t − Tiq + Ti0 ), . . . , u(γ+j) (t − Tiq + Ti0 ))|t0 . (si +j)
yi
TX ¨ )²XT,9f9)T9à ¨ LPfL¡T¡µ TX g9·9`XGf¢ PfrDJ∆D0 ∆1 =X δ1 δ2 ¨ T9f ¨ (9rL
GkG§ T0 = 0 ·T1 =¸τ1 + τ2 *I¦Xµ 0 0 ¨ raTXþ ¿ µ?XfL¢P M= T9fT}fP9g`1 T1f§ Ì ¨ f¢ f"f"X9}XTM9f"f9?TATÆ&9r ¿ µ ¨ (T²¦X#9faXTf¢P#}«J¡Tf x ˜1f= µ · ¨ r9( yG=É x1 x ˜2 = y˙ = (δ1 (x2 ))2 + u
(t0 ) =
Í× Ù
Å}Xr
(s +j)
¡GP¡r±©9(wPL
i yi µ(t)Xap 9f T*f#¥I¦XTX"¡GP09rw≤ j < q−1 q GTr Í 9f9£L T , .¦D. . , Ê Tiq−T Pi1−T .`i0² ÝTiD0 TI©·¢PrPXT¡GP9rDTfgpfÙc§
µ µ *¢f9Tr Ö X =I`τ9if i r= . .¡,G`IrDTfrfT GXfrank(M ±¦ff¡¥I¦f). G1, T.( Í T ?«Ir PfXJ9rPX T − T , . . . , T − T i1 i iq i 0 0 ¦D®f f³f#P * LLD¦faJ9L T9#¢f9Tr GkG9Ù §A*I¦XµAr µ Gr¡GPrDfrPµ τi¨ fi¡9= 1,P . .X. r,`*f"f9?p§ ¥
˜˙ 1 (t) = x ˜2 (t) x . x ˜˙ 2 (t) = 2(˜ x2 (t) − u(t))˜ x1 (t − T1 ) + u(t) ˙ y(t) = x ˜1 (t)
Í× Ï Ù
t?RJY8¸`]rWy x˙ 1 (t) x˙ 2 (t) y1 (t) y2 (t) x
ÿ §Þã Ò ¶ Ö À°â9f¡c9rP ¨ ,r¦XaJ99f,X9ÈIâ f³ ®f XrTX f ¨ 9XJ ¨ þÚPX«TTfrrp Í XDªJLXTaTL 9LGP9gDfrr°1wLf¢¦f¡T LGGX9L 9aÙ)G?ÚXT f 9T9rLfr ¡GIrDTfgp/.f G¡k XTaTL 9aµ1«? «a9f§5*I¦Xµ`9fTG.9ff,L 9fGfwP p9rX¢PX9«JTfgpX GP9gDfrr°wP¯fPfrG G¡? Í ÞT[§µ5×ØØP× Ñ ß`XTf¢T[§rµ ×TØPØáIÙ8ff"`L¦XGLf 9f¡GP9rDJ frrpTM9fGk#X9L 9§
¬"Xk«
¬³"X«
TD
= = = =
x22 (t − τ1 ) + u(t) x1 (t − τ2 )x2 (t) x1 (t) ϕ(t), t ∈ [−τ, 0]
y˙ = (δ1 (x2 ))2 + u y¨ = 2δ1 (x2 )δ1 δ2 (x1 )δ1 (x2 ) + u˙ = = 2(δ1 (x2 ))2 δ1 δ2 (x1 ) + u˙
= = = = =
−x2 (t − τ1 ) x1 (t − τ2 ) x1 (t) x2 (t − τ2 ) ϕ(t), t ∈ [−T, 0]
Í ØPÙ
Í ¿ Ù
y˙1 = −δ1 x2 y¨1 = −δ1 δ2 x1
1 0 0 δ1 §±*XÜrff¦f TX ∂(S,h(s1 1 ) ,h(s2 2 ) ) = ∂x δ1 δ2 0 ¦G9f¦G}¥I¦XTX*fT*wP0ÉδT29
t?RJY8¸`]W x x˙ (t) 1 x˙ 2 (t) y(t) x(t)
Í× Ô Ù
Í d¨ y1 = −δ1 δ2 dy1 . δ1 dy2 = −δ2 dy˙ 1 ⇔ dy2 = −δ1−1 δ2 dy˙ 1
Í× ÿ Ù
×Ù
¬±?¦XXk« µ µ TX TX(9∆ I¦X10µ ∆11 = δ1 δ2 µ ∆21 = δ1 µ
∆22 = δ2
T10 = T10 = 0 T11 = τ1 + τ2 1 1 µ µ X T20 = T21 = τ1 T22 = τ2 M = 0 0 −1 1 ¨ faL¡¯T59fþ×G§*?¦X τ1 TX τ2 T9*fP9g`Tf§
Í ×TáIÙ
Ì LX#Lf. ®fTLfµ ¨ ±TÚ ¦Xr1T¡ ¦f TÈfÚ«TT¦f³9fÈ ¨ Ê â¢Pw9 f ®Gfg²ff¦G [P¦Gf¦f²¥I¦XJ9rPX µ y¨ −δ y91 1 ¦f= 19δf2¦G § M r X ¦ a J 9 µ } ¡ J 9 G ¦ ¨ −1 yVP2=X−δ "1Gpδ92y˙ 1 ¨ r #w¦fX Í Ö X XrfTX*fP DP-µI×TØPØ ¿ Ù µJ?L9rf¢ √ µ µ TX §P¬³}τX1 = 1frτ 2 = 2 ϕ1 (t) = et ϕ2 (t) = t + 1 TX µ1 (T11 − T10 )|t0 := y¨1 (t0 ) + y1 (t0 − T11 + T10 ) µw2 (T22 − T21 )| := + t0 ®G 22T9 §&8 `cy92(tO0µ)+ fy˙w1¦f(tX0 − XT T ) t = 4 21 0 p ©9.wP TX
¸ dy (s1 ) · dy˙ − du = ∂(S, h1 ) dx1 = dx2 ∂x d¨ y − du˙ · ¸ 1 0 dx1 0 2δ1 (x2 )δ1 = dx2 2(δ1 (x2 ))2 δ1 δ2 4δ1 δ2 (x1 )δ1 (x2 )δ1
.
Í× Ð Ù
Æ&T9µJ9f}J9r® ∂(S,h(s1 1 ) ) X9fþ ×J« TD(9fG6 ¨ ∂xþI#X9«JTXrP P9K(δ] frf¢
T11 − T10 = 1 +
213
(2)
T22 − T21 =
1
11
10
0
2
2
0.8
1
0.7
0
22
21
0
0.6
−1
0.5
−2 2
µ
µ
1
0.4 −3
0.3 −4
0.1
−6
0
−7 −8 −2
0
2
4
−0.1 −1
−0.5
T −T 11
10
0 T −T 22
0.5
1
21
Ì r¢D§ ¿ § q fU WMoaG_5\ S[Z
X rGTrâfàX·Xr 9?Taµ rPXTµ?f"T9T¡ f*9?TawP DJ(TÚ`±Ú Ì r¢D§ ¿ §*X µ (T`22G −9T219)f f¡&? fTf¢ 2 µ˙ 2 (TÍ 22 TX− d T )| := (y (t ) + y ˙ (t − T + T ))| 21 t 2 0 1 0 22 21 t T9f&0¦fDdt¥P¦XP9rL °G9«JJ9r«PaÙArX 0 T22 − T21 ¦X}`" 9?w*9fµGGX§ p
(2) Ì ¢X−§ 1¿ §
áX§}ÆÃÁÆ¶Å Ö ÀÃÁ Ö
¬³LXk«LTXr?©9fL¡GIrDTXrr°,T&fLL ¡T¢X9L 9ArLfffTG ¨ rL¦fg9fr PX9TI*LG¡G§ Ö aJ³rLXJ9Ý.X ¨ )?¡ÝTÝ ®?9DT ff¦G [P¦Gf¦G19f9I9T T(9f ?.µ9f w9 T ¨ f¡9$G ¡Gf²¡GIrDTXrgpÊT 9f Gk XTaTL 9a§ Ì ±Lf fG¡ ¨ g9 w ¨ «J¡TXr&XDTaTL 9aµP9fff¦G [¦ff¦G}¥I¦XJ DT`}¦XG9c9r9fP9gw X}«Jr¦X¯ X [¡T¢I&w96 ¦f9(fT9f§ ¬³ÈX«P²wP¦XT$f °T¢f9T¡²r¡â afaþ·9f.¡GIrDTfgp1T89fG¡kàX9 9 ¨ f¡9àrLrXTX#f³w· ®Gf¡ rT¡ ¦f¡J .TM9f"rfX¦G
¦G9f¦G8TX§ Ð §8Æ ¾ ÁÃ}¬ ¶ Ä
0.2
−5
rþP¦fafµ¶§µ Ã9rJ«5µ §§ Í ×TØPØP×PÙc§ÀpGP9rDTfr r°.XTG¡T¯rfTG¡kP °Gr¤`P9?§ »cQ½ MQ,RJSwU *enT_XS[hnJ]`» rdxTz µ Ð Ôf¿ § Æ&f-µ Ò §ã²§ Í ¿Ï?Ð?¿ Ù § Ì 9*f¢P¯TXX¯TX§ ¶XGP-f*GL Ò § Æ&P9µ §µ Ò aG-µA§Aãà§ Í ¿ÏÔ Ùc§*ff¦f XX ³G¦fXrf¢1f9XrwPGp9J« 9rX¢X§ ^?»9½Q enJ_XS[hnJ] M¸`S[ZVY rd µ ÐTÿ ØT Ð á § Æ&P9µ §µ²ã?P¢Xµ²Æ§ Î §rµ Ò aGµà§²ãà§ Í ¿ÏÏPÏ Ù § nJ_X]gZV_`WaRJh \9nJ_DSmhnT]MbclkbcS
WYb A½_àRT] sPW~hRTZ<\ bWSmS[ZV_IsTd MW9\SmGhW._5nJS°WcbZV_Ý\9nT_XS[hnJ]8RJ_`·ZV_ocnThcYRJu S[Z
A plot of the function µ (T −T ) for t =4 A plot of the function µ (T −T ) for t =4
ã Á Ö
*f¡ ¨ 9þ ¨ P¦fX`ÛIÊ9fÈÁ8TX  Taa Ö af?r fL¡TX rPrGwPT¡µ X Ö ¨ G¡ÚÂ}TaaÚÆ&P¦fX mµTX1X Ö ¨ G¡ Ì ¦fXfTwP Ö 99T¢P¡£ÂTa9«?¡f T9fG f¦f9¢ã±J9fJ9T-ãfGrX¢#Æ&P9§
Â Ì ÂÁÆ Ö }X¢¦frJ«JXµ`ã ܬ³fI`¢Dµ § Ö aJ rLXJ9rP TXÝ¡GIrDTfgp T9f·f¡âX9L wP fffL °G¡Gp9§ hnJjZ¡baZ
214