Some results on the optimal control with unilateral state constraints
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Some results on the optimal control with unilateral state constraints Bernard Brogliato. N° 5992 Octobre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5992--FR+ENG. Thème NUM.
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(kYG%[jeFg[hGJcyjZ[Y5n¦Uw^hcyYUwDAjeU*x<GhcyYA~0SUwcUWD[G&n%cyY5Sl^`mTG]WGXmSlgAj0n%GLce SwcyhZTUw^hcyYAS'^`S'~y^hyGXY^hY (kg[g<GY5mT^fv4ab]wGX^hF^Y5je]WQ]wG2SlZAUS'cyYn%cyYUW]wc0`jex[^hh^UVQcy*¨»s0¬'je]WG&^Y (kgAg5GXYAmT^ ¸ W ¶ X T X ) ¶ Y Y · Y¶ Z² Y ¥ T ² GYUW]wQUw^hF.GvjeYAmSlUWjeUwG¨»SVUjUwGJcyYUwDAGkx5c0Z[YAm[jy]wQcyhc*GXm.xRQj&x5c0Z[YAm[jy]wQ je]nX¬%eG%T^U Uw^hF.GkjyYAm.SVUjUwG¨»SVUjUwG c0YUWD[G'x5c0Z[YAm[jy]wQcyhc*GXmxQ!jeY.^YUwGX]w^hcy]4jy]Wn2¬¦yncyYUWj0n¦U¢Uw^hF.GkjyYAm.SVUjUwG¨»SVUjUWG cyYUwD[G!x5c0Z[YAm[jy]wQ n%c0F^Y[~]wc0F}jyY^hY0UWG]W^c0]Jje]nX¬%AUWcyZAnDUw^hF.GjeY5mSVUjUWG¨¥SlUWjeUwG.n%cyF^hY[~]WcyF jeYt^YUwGX]w^hcy]vje]n jeY5mcyhhc G2mxRQjeY^hYUwGX]w^hcy]'je]nX¬%XdVZAYAn¦UW^c0YUW^FGjeYAmSVUjUWG.¨¥GYUw]WQyTn%c0YUWjyn%UXRG%T^UXRUWcyZAnDUw^hF.G&jeY5mSlUWjUWG2¬% ¸ a ¨¥ncyFg[hGFGYUje]W^UVQgA]wc0x[GXF¬¦ ¸ a ¨¥^hY[G2je] ¸ a ¬% ¨UwD[G ^hYAmT^`njeUwcy] ZAYAn¦UW^c0Y.cy:j&SwG%U ©És 0g) sB«¬¦ ¨UwD[G SwZ[x<m[^£:G]WGYUw^`jeAce:UwD[G ^hYAmT^`njeUwc0]bce+jn%c0Y0G%!SwG%U KψK©É#s [yg) &{ B«¬%0SwZ[g[g ¨ T ¬'¨UWD[GJSwZ[g[g<cy]wUbKce)jmT^`SVUW]w^hx[ZTUW^∂ψ c0Y KT ©ÉsysR g)hª|RsB«¬¦ x 0 ¨ x ^hS YAceU !GX]wc&jeYAmUwDAG'A]WSlU4Y[cyY !G]WcvGhGFGYU4ce<UwD[G'0GXn¦UWcy] x ^`S g5cSl^Uw^hyGB¬* < ¨UWD[G'A]SVU4Y[c0Y !GX]wc GhGFGXY0Ubcy:UwD[G 0GXn¦UWcy] x ^`Sbg<c0Sw^Uw^hyG2¬%0Sw^F^hhjy]whQc0] ≺ jyYAm 4 σf (t) = f (t+) − f (t−) UwD[GbdVZ[Fgce<UwD[G ZAYAn¦UW^c0Y jeU UW^FG [f*¼ ¨¥x5c0Z[YAmTG2mjy]w^`jUW^c0Y5¬¦Ri ¸ f*¼ ¨¥]w^h~yDU n%c0Y0UW^YRZ[c0ZAS ce hcTnje:x<cyZ[Y5mTGXmBjy]w^`jUW^c0Y5¬¦Ti ¸ O f ¼ f (·) ¨]W^h~yDU'ncyYUw^hYRtZ[cyZ5S cySwg5G2n%^`je+hcRnXje:x<cyZ[YAm[GXmBjy]w^`jUW^c0Y5¬¦Tf*¼Ja ¨x<cyZAYAm[je]WQjehZ[G&g[]Wcyx[hGF¬¦ ¼La ¨¥^YA^UW^hjy+BjyZAG g[]Wcyx[hGF¬% δ UwD[G¤L^]jynF.G2SlZA]wG.jU t 0m = (0, . . . , 0)T ∈ IRm 0 = (0m)T I ^hSJUwD[G n−^hm[GYUw^UVQFjeUw]W^+ ¸ cyYUjyn¦U&UW^FtGXSje]WGm[GY[cyUwGXm j0S τ ¨¥ D[GY cyYAQºcyY[Gn%c0YUWjyn%U^hSjyYAjemhQTSlG2mA¬Lcy] t n D[GYjSwGX_Z[GY5n%Gce'n%c0Y0Ujyn%U Uw^hFGXS'^`S'G%[jeF^hY[GXm) E (h) mTGYAceUwG2S UwD[GSwG%U ce5dVZ[FgAS'ce)UWD[GvZ[YAn%Uw^hcyY h(·) e ^hkS UwDAG i−UWDZ[Y[^U yG2n¦Uwc0]'ce IRn U j!Z[YAn%Uw^hcyY^Y C ∞ (I) 0^`S'^Y[AY[^UwGhQmT^£:G]WGYUW^hjyx[GjyYAm ^UwD ncyFgAjyn%UkSlZ[gAg5c0]lU2i 0. 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(27) {. # , . ! ¢º'% *. *¢% +¢k C'DAGJY[GU )GXF.Fj^`S ^Fg<cy]wUWjeYUbc0]bUWD[GJ*Gh?g<c0SwGXmTY[G2SwSceb¨|R¬k¨»j0¬bSwGGXYjyS jeY ¼La7¨ x(0−) = x η(0− ) = η ¬% jyYAm. ^hAjyhSwc&x5GkZ5SlGZ[[c0] UwD[GJ_ZAjy^UWjeUw^hyGkjyYAjehQTSl^`S¢ce:UwD[Gkf*¼La(Slc0ZTUW^c0YASXySlGXGkSlG2n¦UW^c0YAS4|jeY5m0qT -½Gkn%c0YASw^hmT0GX] jyYAm m = 1 ^Y )GXFFj ªe R = In wu * @)' + * ! ; #) * - (A,˜ B, ² (7 ˜ C) ˜ M (2) = −CBB T C T < : r # = 1 * - &()! , ' $ * @ ' #*; + % wu 0 B, C) r ≥ 2 '
(28) +*/
(29) ; ' )- ˜ ˜ ˜ (A, (A, B, C) M (2r ) = (−1)r CAr −1 B(CAr −1 B)T (= ˜ C˜ A˜r−1 B). wu. . ¶)¶ ) GU¢ZAS4m[GY[cyUwG. A˜ =. . A BB T Q −AT. . =. . A˜1 A˜3. A˜2 A˜4. . wu. jyYAm. A˜r =. (r) A˜1 (r) A˜3. wu. (r) A˜2 (r) A˜4. wu. !. cy]4Slc0FG. r≥1. . C'DAG&A]WSlUkjySWSlGX]lUW^c0YcyUWD[G )GXFFj^`S jSw^Fg[hGnjyhnZ[hjeUw^hcyY ^UWD ˜(1) C'D[G (1) T jyYAm M (1) = −C A˜2 C T hGXj0mT^hY[~tKjy]w0cgAje]jeFGUwG] G'jeYUUWctn%c0Fg[ZTUwG^hS M (2r+2) =A2(C =0)BB (2r+1) T 2r+1 T A˜2 C rwu = α + 1 UwD[GXY CAi B = 0 c0]jeh 0 ≤ i ≤ α − 1 7uLZ[]c0xTdVGXA˜n%Uw^hyG(0^`SUw−c(C) SwD[c=UWDA−C jU M (2α+2) = bf*QmT^h]wG2n¦UknjyhnZ[`jUw^hcyY*GvD5jByGLcy]kjeYRQ^hYUwGX~yG] i ≥ 1 (−1)α+1 CAα BB T (AT )α C T α ≥ 1 ¨Vª0ª2¬. (i) (i−1) (i−1) A˜1 = A˜1 A + A˜2 Q (i+1) (i) (i) A˜2 = A˜1 BB T − A˜2 AT. U c0hckS*UwDAjeU (2α) A˜2 =. 2α−1 X i=1. (2α+1) A˜2 =. (i) (1) (−1)i+1 A˜1 BB T (AT )2α−i−1 − A˜2 (AT )2α−1. 2α X. (i). (1). (−1)i A˜1 BB T (AT )2α−i + A˜2 (AT )2α. (a). ¨VªBsy¬. (b). i=1. C'D[^`SJnXjeY x<GSwD[c YZASl^hY[~t¨lªyªX¬kjeYAmUWD[GY xRQ^YAm[ZAn¦UW^c0Y)"[]WcyF¨lªyªB¬'^UJcyhckS xRQ^Y5mTZAn¦UW^c0YUwDAjeU A˜(α) = 1 D[GX]wG L (Q) ^hSSlc0F.G'FjUW]w^.m[Gg<GYAmT^hY[~c0Y!UWD[GkFjUW]w^ Q YAmTGXGXmhG%UbZASjySWSlZ[FG'UWDAjU A˜(i−1) Aα + Lα (Q) = α 1 ' C [ D X G Y. ^ W U D (i) (i−1) i−1 i i−1 ˜ ˜ A1 = (A + Li−1 (Q))A + A2 A + Li−1 (Q) jyYAm A˜(2) = A2 + BB TQQ=UWD[AGLgA+]wcRLcei)(Q) ^`S'n%c0F.gAGLUwGyi(Q) A]wc0=F}L¨lª2i−1 se¬L(Q)A ¨x<¬c0Y[+G (i−1) 4IJSw^YA~¨VªBse¬ jeYAmSl^hYAn%G ˜ A˜2 A1 = A Q 1 2α m[GXmTZAnGXS:UWDAjU M (2α+2) = X (JSWSwZ[FG cy]+UWD[G4Uw^hFGbx<G^hY[~ (i) (1) (−1)i+1 C A˜1 BB T (AT )2α−i C T +C A˜2 (AT )2α C T i=1 UWDAjU CL (Q)BB T (AT )2α−i cy] jeh 1 ≤ i ≤ α ¨UwD[G*UWG]WFS ^UwD α+1 ≤ i ≤ 2α Y[GG2mY[cyUx<G*ncyYASw^`mTG]WGXm CT = 0 i 2α j0S CAi B = 0 c0]vjy 0 ≤ i ≤ α − 1¬%&C'DAGYt*G.cyxTUje^hY M (2α+2) = X jeYAm (−1)i+1 CAi BB T (AT )2α−i C T UWD[GcyYAQYAcyY !XG]WcUWG]WF ^hYUWD[^hSJSlZ[F^hS*c0] i = α bC'D[GX]wGcy]WG M (2α+2) i=1 α+1 α T T α T CcGYAmUWD[Gg[]WcRceV[^ULnXjeYx5G!SlD[c Yje~je^hYxRQ^YAm[ZAn¦UW^c0Y ZASl^hY[~¨lª2sy=¬ UW(−1) DAjU L (Q)CA^`SkBB n%c0F.g<(A c0SwGXm) cyCUWG]WFSkce UWD[G'~yGXY[G]jecy]WF Aj BF cy] FB T (AT )j e ^UwD 0 ≤ j ≤ i−2 jeY5m F ^`S4SwcyFG FjeiUw]W^+ YAmTGG2mc0Y[G'DAjyS L (Q) = i i−3 X T y j ' S l S A D c. Y e j < x c y 0 G L ( w S w S [ Z F v G w U A D e j U b ' C [ D G Y (i−1) L (Q)A + A˜ Q L (Q) = Aj BF + FB T (AT )j L (Q) = i−1. i−1. 2. i. j=0. i−3 X. (i−1) Aj BFA + FB T (AT )j+1 + A˜2 Q. j=0 c0]kjeh. 1≤i≤α. jyS']WGX_Z[^h]WGXm+. !uLYAG!n%c0YAn%hZAmTG2SL]wc0F ¨lª2s0¬¦C'DRZAS. CLi (Q)BB T (AT )2α−i C T = 0. ¾¿À ¾Á.
(30)
(31) . . )GXFFjºªn%cyYUW^YRZ[G2SJUWcD[cy`mc0]SlQTSlUwGFSv ^UwDjZAY[^c0]wF ]wGXhjeUw^hyG!m[G~y]WGGjeY5m m ≥ 1 ¨¥]wG2njeh¢UwDAjeU C ∈ ¬vSl^hYAnG CAi B = 0 cy]jeh 0 ≤ i ≤ rwu − 2 ^hYUWD[^`S&FZ[Uw^hje]W^hjyx[Gnj0SlG.je`Slc5 YºSwZAnDºjnj0SlG0+DAjBR^YA~ IRm×n jy]wGL^YAm[Gg<GYAmTGXYUX[ D[GX]wG C ^hS UwDAG i−UwD]wc cy C m×m Z[h?]WjyY[^hF.gA^hGXS*UWDAjU*UwD[G&0GXn¦UWcy]S (2r ) M CiT i jeY5m B DAj0∈S'I]WRjyY[ m ² '.+! ' . wu. H ' :. . rwu = 5 CA4 B = 1. x˙ 1 (t) = x1 (t) + x2 (t) x˙ 2 (t) = x3 (t) x˙ 3 (t) = x4 (t) + x3 (t) x˙ 4 (t) = x4 (t) + x6 (t) x˙ 5 (t) = x5 (t) + x1 (t) x˙ 6 (t) = u(t) w(t) = x1 (t) ≥ 0. +. M (10) = −1. ¨lª"¬. . C'D[^`S¢^hS¢jLnXjeY[c0Y[^`nje )GU¢Z5S ^hYUw]WcRm[ZAn%G'jJnXjeY[c0Y[^hnXjeRSVUjUWG*SwgAjynG]WGg[]WGXSwGYUjUw^hcyY&c0] UWD[G'SlQTSlUwGF ˜ C, U˜ D) ]WGg[]WGXSwGYUWjeUw^hcyY D[^hnD Fje0GXS.UWD[GSlcy?nXjehG2m !XG]Wce°mTQRYAjeF^`nS.G%Tg[h^`n%^UwhQ jegAg5(G2A, je˜]2 B, ^hkx<G ZASlGZ[kc0]Slc0FG SwZ[xASwGX_Z[GYU!m[GyGXc0g[FGYUWS¨¥SwGGje`Slc ]WGFje]W½q^[¬w¬¦.Acy]!jUw]W^gAG (A,˜ B, ˜ C) ˜ 4 ^UWD(^hY[g[ZTU λ ∈ IRm cyZ[Uwg[ZTU A y j A Y m A Z [ Y ^ 0 c w ] F w ] X G h e j w U h ^ y & G T m G 0 ~ w ] X G G T ^ k U W ] X G 0 j [ m & S © | B« ? § w ∈ IRm r≥1 z˙1 (t) = z2 (t) z˙2 (t) = z3 (t) z˙3 (t) = z4 (t) z˙r−1 (t) = zr (t) ˜ −1 z˜(t) + C˜ A˜r−1 Bλ(t) ˜ z˙r (t) = C˜ A˜r W ˙ = A˜ξ ξ(t) + B ˜ ξ(t) z (t) ξ 1 w(t) = z1 (t) + D, z1 (t) = Cx(t) z(0− ) = z0 .. . ¨lª|R¬. mTQRYAjyF^hnXS¢jeYAmºUwDAGZ[h?]WjyY[ºSlUWjeUwGUW]WjyYASVc0]wFjeUw^hcyY½FjeUw]W^½^hS § ¨¬¦ -ºG -½Gnjeh^UUwDAG Tm GXY[ceUWG z¯˜T = (z , . .z˜.−, z ) YUwDAGSlG2_0ZAG>TUwD[GncyFg<cyY[GXY0US'ce z˜ jeYAmce z ^h)x<ceUWD x5GW˜m[GY[zcy˜Uw=GXmWj0˜S x˜z AG[n%Gg[U 1 ^'YAceU!nG2je]]WcyF UwD[Gn%rcyYUWG%RUX OR^hYAn%GUWD[G !XG]WcmTQRYAjeF^`nSgAhjBQTSjeY^hFg5c0]lUjeYU]WcyhG^YUWD[GSwQTSVUWGFiS*Gje]WG mTG2jeh^Y[~ ^UwD)[UwD[G&c0hc ^YA~!^`S ce¢^YUWG]WGXSlUX ² # F 0 m = 1 .+ ' z− )+ z˜− ! !.! '#6! . + ' (A, B, C) */ )+ (A, ˜ B, ˜ C) ˜ * *> +
(32) & + @$ * - , ' )#.! ' +# .! A )+. 9! -*; ! H ' : σ A˜ξ A M& $ .! , A˜ξ mσ. & @E; # & . Aξ. '& $ .! . . mσ −σ. +. σ. A #* & @E; # & ξ . ¶)¶ )GU jyYAmhG%U 4C'D[GvUW]WjyYASVGX]ZAYAn¦UW^c0YUWDAjUkncy]W]wG2Slg<cyYAmAS n×n x5GSwZAnDUWDAjU LT L = Q UwcUwD[Gkc0g5GX]WjeUwc0] Qλ 7→≥ 0w ^hY¨|R¬ ^hSLGX∈_ZAIjyRTUwc H (s) = C(sI −1 ˜ 2n − A) ˜ B ˜ = G(s)G(−s)(1 − H ∗ (s)H(s))−1 ©ÉqysB«> D[GX]wG G(s) ^`S¢UwD[G'UW]WjyYASlG]4ce5UwD[G'UW]w^hg[hG (A, wλ ^ Y > ¨ e s % ¬ h ^ ¢ S W U [ D ' G Uw]jeYASlG]4FjeUw]W^cy5UWD[G'Uw]W^hg[G (A, B, L) B, C) H(s) h ^ S w U A D L G W U W ] y j A Y l S G * ] F e j w U W ] ^ y c. W U [ D J G W U w ] h ^ [ g h G ©Éqe|[Tg)És#0z«? ∗ T T T T^» Gy UWD[GjymBdVc0^YU SlQTSlUwGXFUwc H (s) ) ¸ cyY5Sl^`mTG]vY[c UWD[GUV*c¹+¤njeYAcyY[^`njy)cy]WF(−A SLj0SwSwcT,n%L^`jUW,GXBmUWc UWD[G.SwQTSVUWGFSL^Y ¨»sy¬vjyYAm(¨|R¬¦(A, +]WGXB, Swg5G2L)n¦UW^0GhQy []WcyF )GXFFj ªcyY[G D5jyS jeYAm ξ˜ ∈ IR2(n−r ) A]wc0F UWD[G¥j0n¦UUWDAjU H (s) = G(s)G(−s)(1 − ξ ∈ IRn−r e j A Y m w S ^ A Y n G W U [ D G ° T m QRYAjyF.^`nS.n%c0]w]WGXSwg<cyYAm[SUwcºUwDAGYRZ[FG]jUWcy] B(s)wλceJUwD[GUw]jeY5SVGX]Z[Y5n¦Uw^hcyY H ∗ (s)H(s))−1 ξ1 R*GLmTG2mTZAnG UwDAjeUUWD[GJ]WcRceUWSce+UWD[GJg<cyhQRY[cyF^`je je]WG FcTmTGXS ce+UWD[GJFjUW]w^ A˜ bOR^hYAn%GJUwD[G G(s) ξ cy]mTGX]*cy B(α)B(−α) ^`S g[]WGXn^hSwGhQ 2(n − rwu ) [UwD[G]WB(α)B(−α) GXSwZ[U*c0hckSX= 0.
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(40) #'&(*)+&,!-. G%U&ZASvn%cyY5Sl^`mTG]LUwDAG.SwQTSVUWGF}^Y ¨|R¬¨»j0¬!¨¥nX¬LjeY5m GUvZ5SJc0]w~0G%ULcy]LUwD[G!Uw^hFG!x<G^hY[~UwDAjeUv^UvFjBQ]WGg[]WGXSwGYUkUWD[G YAGXn%G2SwSWje]WQn%c0YAmT^Uw^hcyYAS'cyUWD[GFjT^F!Z[Fg[]W^hYAn%^hg[hGy (kY^hF.g<cy]wUWjyYUkg5c0^YU ^hS*5]WSlU UwcZ[YAm[G]SVUjeYAmUwD[Gm[QY5jeF^hnXS cyvSwZAnD jºmTQRYAjyF^hnXjekSwQRSlUwGXF ^YR0cyh^hY[~ n%c0F.gAGXF.GXYUWje]WQ?Sw`jynRY[GXSWSncyYAm[^UW^c0YAS [c0]^YASlUWjyYAn%G0 DAjeU^hS.UWD[G FG2jeY[^hY[~tcy λ ≥ 0 ^ λ ^hS.Y[cyUjºFGXj0SlZ[]WGº¨»mT^`SVUW]w^hx[ZTUW^c0YAje'FZAUW^g[h^hG]S!FjBQG2jySw^hQ x5GY[GG2mTGXm#Uwc½^hY0UWG~0]WjeUwG ¨¥|¬¨»j0¬¨»nX¬w¬l®CcUwD[^`SGXYAm GUZAS]wG2njehSwcyFG¥jyn%UWS!jex<cyZTUUWD[GD[^h~yD[GX]c0]WmTGX]Sl*GGXg[^hY[~ g[]WcRnGXSWS,¨ *&u&ORa ¬j0S ^U^`S^YUw]WcTmTZAnGXm ^Y ©hªe s2«? DA^hnD g[]Wc^`mTG2S.j½]W^~0cy]WcyZAS]jeFG*cy]W UwcSlUwZAmTQ(UwD[GmTQRYAjyF^hnXS^Y ¨|R¬¨¥j0¬¨»nX¬% jyYAm GUWGYAmASkUWD[G*Gh?RY[c YA]SVU&jeYAmtSlG2n%cyY5mcy]mTG]vSl*GGXg[^YA~g[]WcRnGXSWS¨»SlGXG]WG%G]WGY5n%GXSJ^Y ©ª%« jyYAm©hªyªe /q[ e« c0]jYAcyYT?FjUwDAGFjUW^hnXje4^hY0UW]wcTmTZ5n¦Uw^hcyY<¬¦i cyZA~yD[hQSlg<GXjyR^Y[~5+UwDA) G *&u&ORa ^hSjSwg5G2n%^5nmT^£<GX]wGXYUw^`je4^hYAn%hZASw^c0Y) DAc0SwGSwcyhZTUw^hcyYASjy]wGmT^hSlUw]W^hx[ZTUw^hcyY5S*jeYAm# D[^hnD g5GX]wF^UWS.UWcº~0^0GjºF.G2jeY[^hY[~ºUwcºUwD[G SwQTSVUWGF ¨|R¬¨GXSwg<GXn%^`jehhQ -½GmTGY[cyUwGjyS T (I) UWD[GSwG%Uce'mT^`SVUW]w^hx[ZTUW^c0YASce mTGX~y]WGG UWD[Gg<c0Sw^UW^0GY[G2SwSvce λ D[GXY ^U^`SY[cyUjFG2jySwZ[]wGB¬¦) ¸ É © y s 2 s & «. [ D ` ^ n D e j W ] G y ~ G A Y G ] j w U 2 G m R x Q i O f ¼NZ[YAn¦UW^c0YAScyY D[nc0SwGtSwZAnnGXSWSl^hyGmTGX]w^hjUW^0GXSg<c0SWSlG2SwSjeY n+1 I jyxASwcyhZTUwGXQ(ncyYUw^hYRZ[cyZAS!gAjy]lU¨¥m[GY[cyUwGXm jyS [·]¬UwDAjeU^hSjyhSwc½i ¸ O f*¼c0Y I C'D[G]W^~0DUmTGX]w^hjUW^0Gcy [h] ^`S m[GY[cyUwGXmj0S hˆ (1) = d [h] (t) = lim [h](t+σ)−[h](t) C'D[GSwG%Ukcy¢SwZAnDZ[YAn%Uw^hcyYAS ^`SkmTGXY[ceUWGXmjyS F∞ (I; IR) = σ→0 dt σ A . ^ w U D Yg5je]wUw^`n%Z[`je] d ˆ (k−1) (k) ˆ [m# h xRQ jº] ∈Z[Y5RCSLBV (I; IR)} ∩k∈IN Fk (I; IR) Fk (I; IR) = {h ∈ Fk−1 (I; jyIR) : ^hh := dt A Y m S 0 ~ G [ Y X G W ] e j w U 2 G ¦ n w U h ^ y c Y *^UDAj0Sj (I) F ∈ F∞ (I; IR) .FZ[0(I;YAn%UwIR) . ^hcy-Y /t=g5jeRCSLBV ]wUmTGXY[ceUWGXm(I;j0IR) S {T }(·)T =∈ [TFˆn(n) & e j A Y m j . F 2 G y j w S [ Z W ] G 0 / A g e j w ] U [ m G [ Y y c w U X G m y j S SlZAnD UwDAjeU ](·) T R +∞ mTGXY[ceUWGXS'UWD[G!mT^`SlUw]W^x[Z[Uw^hcyYAjymTGX]w^hjUw^hyG0AjeY5m m[GY[cyUwGXS h T , ϕi = −∞ ϕ d[Fˆ (n−1) ], ∀ ϕ ∈ C0∞ (I) D UWD[GLOUW^GXU»dVGXS cy]m[^£:G]WGYUw^`je5FGXjySwZ[]WG ~yGXY[G]jUWGXm.xRQ.j&ZAYAn¦UW^c0Y z ce+hcTnjeAx5c0Z[YAmTG2mje]W^hjeUw^hcyY© s2«?¢C'dzDRZAS T (I) m[GY[cyUwGXSJUwDAG!SwG%ULcyjeh O[nD'je]wU !0¡5mT^`SlUw]W^x[Z[Uw^hcyYASvSlZ5nD UwDAjeUJUWD[G]WGGR^`SlUWSvjZ[YAn%Uw^hcyY F ∈ F (I; IR) SlZAnDnUwDAjeU ∞ )GU n x<G&UwD[GSwFjehG2SVU ^hY0UWG~0G] SlZ5nDUWDAjU T ∈ T (I) A*GSlGU T = Dn F +. +. +. n. n + 1 if n ≥ 1 deg(T ) = 1 if n = 0 and E0 ({T }) 6= ∅ 0 if n = 0 and E0 ({T }) = ∅. ¨VªBqy¬. C'DAGv¤L^]jynLFGXj0SlZ[]WG D5jyS'mTG~0]wGXGvs[^UWS m[G]W^jUW^0G DAj0S'mTG~0]wGXG [RGUWne ¸ c0YUw^hYZAcyZASZ[YAn%Uw^hcyYAS*DAjB0GvmTGX~y]WGG Az 4jeYAm(m[^hSWn%c0Y0UW^YRZ[c0ZAδS&0 Z[Y5n¦Uw^hcyY5SDAjB0GmTGX~y]WGG ªyºC'δD[˙0G¥j0n¦UUWDAjU!UwD[GmT^`SVUW]w^hx[ZTUW^c0YAS*G*cy]Wº ^UwD#cy]W^~0^YAjeUwG ]WcyF f*¼ ZAYAn¦UW^c0YAS+^`Sn%]WZAn^hjyc0]vUwD[GnDAjy]Wj0n¦UwGX]w^ !XjeUw^hcyY cybUwD[GX^]SwZ[g[g<cy]wUX.OR^Y5n%G.UwD[GY[cyUw^hcyYºce*jSwcyhZTUw^hcyY c0]vUwD[Gc0]wFjeh^`SlF ^hYº DA^hnD½ GSwDAjeh GF!x5G2m ¨¥|¬v^`Svn]wZ5n%^`je> jeYºG%[jeFg[hG^hS&g[]WcR^hm[GXm^hY mTG%Uje^h4^Y jeg[g<GY5mT^ fv -½GL^hYASw^hSlU*DAG]WGJcyYUwD[GL¥jyn%U UWDAjU*UwD5jU*UwD[G&Slc0Z[Uw^hcyYASUwDAjeU ^h:x5G&n%c0YASl^`mTGX]wG2mY[G%RU2Rjy]wGLceSwg<GXn%^`je<x<cyZ[YAm[GXm jy]w^`jUW^c0Y) Y jeY[cyUwD[GX]k*cy]m[SRUWD[G^h]JmTGX]w^hjUW^0GXS mTcY[cyULn%c0YUWje^hYjeYRQSw^YA~yZ[`je] Gx<GXSw~yZ[G^hYUwG~0]Wjyx[hGvg5je]wUX ORGXG ¤LG%5Y[^UW^c0Y ª&jyYAmGX_ZAjeUw^hcyYº.¨ 0s0¬¦ C'D[GLn%cy]WG'ce<UwD[G *&u&ORah^G2S4^YUWD[GkmTGAY[^Uw^hcyYce<UwDAG cyhc ^hY[~SwG%Ubce<UWjyY[~yGXYUn%cyYAGXSX G%U x5GJj&YAcyY[GXF.g[UVQ ncSlG2mncyYRyGSlZAxASlGU ce IR -ºGm[GY[cyUwG&xRQ T (x) UwD[G&UjeY[~0GYUkn%c0Y[G&ce Φ jeU x ∈ IR mTGAY[GXmΦ xRQ Φ ¨Vª2{0¬ TΦ (x) = cone(Φ − {x}) DAG]WG cone(Φ − {x}) mTGY[cyUwG2SUWD[G n%c0Y[G~0GY[GX]WjeUwG2mxRQ Φ − {x} mTGAY[G2m#jyS.^hY © &X« UWctUje0G^hY0UWc½j0nncyZ[YU ncyYASlUw]je^hYU*R^hcy`jUW^c0YASTjyYAm cone(Φ − {x}) ^`S ^USkn%hc0SwZ[]WGy' 1v^0GY jn%hc0SwGXmY[cyYAGFgTUVQncyYRyGSlGU Φ T*GSlGU TΦ0 (z1 ) = Φ, TΦ1 (z1 ) = TΦ (z1 ), TΦ2 (z1 , z2 ) = TTΦ1 (z1 ) (z2 ), TΦi (z1 , ..., zi ) = TT i−1 (z1 ,...,zi−1) (zi ), i ≥ 0. Φ. ¾¿À ¾Á.
(41) r.
(42) [ ² ( . . . m≥2. )+. Φ = (IR+ )m. ' :. i l l TΦi (z1 , ..., zi ) = ×m l=1 TΦ (z1 , ..., zi ).. HkcyUwGvUwD5jU TIR+ (x) =. jeY5m. . IR IR+. if x > 0 if x ≤ 0. TIR (x) = IR. [ ² ( . R!*RK. H ' 2&BA9+ C @. ,# ' *)+ .! # . &(!. -, * ! $. TΦi−1 (z1 , ..., zi−1 ). *; : A
(43) . ∂ψT i−1 (z1 ,...,zi−1 ) (zi ) = {w ∈ IRn : hw, v − zi i ≥ 0, ∀v ∈ TΦi−1 (z1 , ..., zi−1 )}. + ' & + ! # ! SA K * Φ. TΦi−1 (z1 , ..., zi−1 ). #. zi. . TΦi−1 (z1 , ..., zi−1 ) = IR ⇒ ∂ψT i−1 (z1 ,...,zi−1 ) (zi ) = {0}, Φ. TΦi−1 (z1 , ..., zi−1 ) = IR+ and zi > 0 ⇒ ∂ψT i−1 (z1 ,...,zi−1) (zi ) = {0}, Φ. TΦi−1 (z1 , ..., zi−1 ). = IR. +. and zi ≤ 0 ⇒ ∂ψT i−1 (z1 ,...,zi−1 ) (zi ) = IR− . Φ. )GU ZAS*^YUw]WcTmTZAnGLY[c UV*c.FjeUwD[GXFjUw^`njy<cy]WFjy^`SwFS ceUwD[G *vu&OTa¢ Acy]*UwDA^hS'*GL]WGhQZ[g<cyYUwD[GSwg5G2n%^`je+SVUjUWG SwgAjynGv]WGgA]wG2SlGXY0UjUW^c0Y^hY½¨lª|¬%TUwDAG]wG2jySwcyYc0]'UwD[^`S'gAjy]lUW^hnZ[hjy]knD[cy^`n%Gvx<G^hY[~G%Tg[`je^hY[GXm^hY½©hª%«? X¥X X X )¶ Y ¶ ( .X¥
(44) ¢^YAm + jyYAm ξi ∈ Tr−1 (IR+ ) (1 ≤ i ≤ n − r) SWjUw^`SlQ^hY[~.UwDAGmT^hSlUw]W^hx[ZTUw^hcyY5je+GX_ZAjeUw^hcyYAS z1, ..., zr ∈ Tr−1(IR ) . Dz1 − z2 = 0 Dz2 − z3 = 0 Dz3 − z4 = 0 . . ¨lª y¬. Dzr−1 − zr = 0 ˜ −1 z˜ = C˜ A˜r−1 Bλ ˜ Dzr − C˜ A˜r W ˜ ξ z1 Dξ = A˜ξ ξ + B. ˜ −1 [D(r−1) Dz1 − {z2 } + . . . + λ = (C˜ A˜r−1 B) ˜ −1 {z} +D Dzr−1 − {zr } ]+ Dzr − C˜ A˜r W. À À Ä#0[Í112. ¨Vª 0¬.
(45) ª2z. # , . jyYAm . d{z1 } − {z2 }(t)dt ∈ −∂ψTΦ0 ({z1 }(t+ )) + d{zi } − {zi+1 }(t)dt ∈ −∂ψT i−1 ({z1 }(t− ),...,{zi−1}(t− )) ({zi }(t )). . Φ. ¨Vª2r0¬. d{zr−1 } − {zr }(t)dt ∈ −∂ψT r−2 ({z1 }(t− ),...,{zr−2 }(t− )) ({zr−1 }(t+ )) Φ (C˜ A˜r−1 B) + ˜ −1 [d{zr } − C˜ A˜r W ˜ −1 {z}(t)dt] ∈ −∂ψ r−1 T ({z1 }(t− ),...,{zr−1 }(t− )) ({zr }(t )). K cy]WGn%cyFgAj0n¦UWQ½¨Vª e¬'^hSJ]wGX ]w^UlUWGY jyS 'C'D[G]WG`jUW^c0YAS −1 ˜ = 0 c0] D˜ ˜x − Bλ ~0^0GY^hY¨VªXr¬ DAjB0G'Uwcx<Gk^hY0UWG]Wg[]WG%UwG2m^YD˜UWzD[Gk−c0W˜hcA˜ W˜^hY[~Slz˜GXYA−SwGyW˜§ ¢Bλ ^hYAmY[c0Y[Y[GX~0jxUW^−0G A˜ ]wG2je>jy˜Z[G2=m!iJ0 jymTc0YF.G2jySwZ[]WGXS ]wGXhjeUw^hyGUwc D[^`nD(UwDAG Gx<GXSw~yZ[GFGXj0SlZ[]WG dt jyYAm(UWD[GOUw^hGU¥dVG2SFGXjySwZ[]WG d{z } g5cSwSwGXSWS dµi (1 ≤ i ≤ r) i m[GYASw^UW^G2S dt jeYAm d{z } ]wG2Slg<GXn%Uw^hyGhQSwZAnDUwDAjeUX§ Φ. i. dµi. dµi. d{zi } dt (t) − {zi+1 }(t) (t) ∈ −∂ψT i−1 ({z1 }(t− ),...,{zi−1 }(t− )) ({zi }(t+ )), Φ dµi dµi dµi − a.e. t ∈ IR. jyYAm. (1 ≤ i ≤ r − 1). ˜ −1 [ d{zr } (t) − C˜ A˜r W ˜ −1 {z}(t) dt (t)] (C˜ A˜r−1 B) dµr dµr ∈ −∂ψT r−1 ({z1 }(t− ),...,{zr−1 }(t− )) ({zr }(t+ )), Φ. dµr − a.e. t ∈ IR.. ¨»syz0¬ ¨>sTªB¬. C'D[G SwcyhZTUw^hcyY5S!cyJUWD[GmT^`SVUW]w^hx[ZTUW^c0YAje*cy]WFjeh^hSwF ^hkx<G SlDAc Y(Uwc x<G mT^hSlUw]W^hx[ZTUw^hcyY5Scevm[G~y]WGGg<c0SWSl^hx[hQ ` je]W~yGX]*UwDAjyY 1 G%UJZASkY[c7^hYUw]WcTmTZAn%GjSwGXncyYAmc0]wFjeh^`SlF DAc0SwGSlc0ZTUW^c0YAS je]WG&Z[YAn%Uw^hcyYASX[^hYAmTGXg5GXYAmTGYUWQce UWD[GJ]WG`jUW^0GJmTGX~y]WGG rwu 4C'D[^hS ^h<x5Gv_Z[^UWGJZASwG%ZA< DAGYUWD[GJc0gTUw^hFje:n%cyYUW]wc0Ag[]Wcyx[hGF ^`SGXFx<GXm[mTG2m^hY0UWcUWD[G *&u&ORa ASwGGSlG2n¦UW^c0YAS [ÉqR A {T ² [ ² X ² ²ZY X» ¶ ( .X¥
(46) ¢^hYAm jeYAm ξi ∈ F∞(IR+ ; IR)(1 ≤ zi ∈ F∞ (IR+ ; IR) (1 ≤ i ≤ r) w S A Z n D w U A D e j U ¨ X ¬ i ≤ 2n − r) ¨»s0sy¬ dzi − zi+1 (t)dt ∈ −∂ψ (zi (t+ )) (1 ≤ i ≤ r − 1) . . i−1 TΦ (z1 (t− ),...,zi−1 (t− )). + ˜ −1 [dzr − C˜ A˜r W ˜ −1 z(t)dt] ∈ −∂ψ r−1 (C˜ A˜r−1 B) T (z1 (t− ),...,zr−1 (t− )) (zr (t )). jyYAm. ¨»s!0¬. Φ. »¨ se|¬ C'DAG!SwQRSlUwGXF}^Y ¨»sysy¬JjyYAm(¨»s!y¬JDAjySJUwcx5G.^YUWG]Wg[]wGUwG2m ^YUwD[G!cyhc ^hY[~SwGY5SlG0§^hYAmtY[cyY[YAG~0jeUw^hyG]WGXjy?jehZ[GXm iJjymTc0Y#FGXj0SlZ[]WG dµi ]WG`jUW^0GUWc D[^hnD UwDAG Gx<GXSw~yZ[GFGXj0SlZA]wG dt jeYAm#UwD[GtORUw^hGU¥dVGXSF.G2jySwZ[]WG dzi g5cSwSwGXSWS m[GYASw^UW^G2S dt jeYAm dz ]wG2Slg<GXn%Uw^hyGhQSwZAnDUwDAjeU ˙ − A˜ξ ξ(t) − B ˜ξ z1 (t) = 0, dt − a.e. t ∈ IR ξ(t). i. dµi. dµi. jyYAm. dzi dt (t) − zi+1 (t) (t) ∈ −∂ψT i−1 (z1 (t− ),...,zi−1(t− )) (zi (t+ )), dµi − a.e. t ∈ IR (1 ≤ i ≤ r − 1) Φ dµi dµi. . ˜ −1 [ (C˜ A˜r−1 B). ¾ÄJÛ>ÂBÚÉÊ+äÕ¦Ô»ÜkÎÇÉÚÉÊ»Ü&Ë {z } = z eÅVÞlÎÐBÊ>Å i. i. zi. ¨»s0qy¬. dzr ˜ −1 z(t) dt (t)] (t) − C˜ A˜r W dµr dµr. ÚÉÊÎähÐBÄ2ÞVÛ»ÚÉÕ¦Ä&ÎÄ2Ñ. dzi. ÚÉÊ Î ÙÛ»ÚÉÅVÇ Û ¥ÅVÊÜ ÅlÎÊ»ÐBԻŦà . ¾¿À ¾Á.
(47) ª0ª.
(48) . ¨»sy{0¬. ∈ −∂ψT r−1 (z1 (t− ),...,zr−1 (t− )) (zr (t+ )), dµr − a.e. t ∈ IR. Φ. (kYAceUwDAG]2TF.c0]wG&^hYUwZ[^Uw^hyG&'jBQUwcSwGGvUwD[GFG2jySwZ[]wGmT^£:G]WGYUw^`je:c0]wFjeh^`SlFT^`S*Uwcn%c0YASw^hmTGX]*UwD[G&c0hc ^YA~A§. ¢^Y5m. jeYAmFGXj0SlZA]wG2S. z1 , ..., zr , ξ1 , ..., ξn−r ∈ F∞ (IR+ ; IR) dν1 , . . . , dνr dz1 = z2 (t)dt + dν1 dz2 = z3 (t)dt + dν2 dz3 = z4 (t)dt + dν3 dzi = zi+1 (t)dt + dνi dzr−1 = zr (t)dt + dνr−1 ˜ −1 z(t)dt + C˜ A˜r−1 Bdν ˜ r dzr = C˜ A˜r W ˙ ˜ξ z1 (t) ξ(t) = A˜ξ ξ(t) + B. SwZAnDUwD5jU. . ¨>sy¬. . ^UwDUwD[G^hYAnZASw^hcyYAS jeY5m. λ. ^`S'^hY ¨lª"¬¦§. dνi ∈ −∂ψT i−1 ({z1 }(t− ),...,{zi−1 }(t− )) ({zi }(t+ )). c0]kjeh. Φ. ¨>s#¬. 1≤i≤r. >¨ ser¬ YcyUwD[GX]L*cy]m[S ^`S&jmT^`SVUW]w^hx[ZTUW^c0Y D[cSlG!F.G2jySwZ[]WGgAjy]lUv^hS jyYAm ^UWSvg5cSl^Uw^hR^UVQ^`SJZAYAmTG]SlUwcRcTmjySJUwD[G g<c0Sw^UW^R^UVQ ce dν !C'λD[GF.G2jySwZ[]WGXS dν jy]wG!FZ[Uw^hg[h^GX]WSv D[cSlG.FGXdνjeYAr^Y[~^hYUwDAGncyYUwGRUvcycygTUW^Fjy^UVQ ^h x5G FjymTGnG2je] `jUwGXr]¨¥SwGG ¸ cy]Wcyh`je]W^G2S jeiY5mr[AjeYAm]WGFje]W 0¬% []WcyF ¨>s¬'jyYAm ¨»s#0¬*^U c0hckS*UwDAjeU ˜ −1 [D(r−1) ν1 + . . . + Dνr−1 ] + νr λ = (C˜ A˜r−1 B). ¨,yz¬. dνi ({t}) = {zi }(t+ ) − {zi }(t− ) ∈ −∂ψT i−1 ({z1 }(t− ),...,{zi−1 }(t− )) ({zi }(t+ )), (1 ≤ i ≤ r − 1). jeY5m. Φ. .¨ [ªB¬ -½G]WGXnjy UwDAjeUL~0^0GYjncSlG2m n%c0Y0G%Y[cyYAGFgTUVQSlGU jyYAm <g[]WcB ^`SkUWD[G!ncSlG2SVU T g<cy^hY0U&UWc z˜ ^hY K )^hYtUWD[GF.GUw]W^hnmTGAY[G2mtxRQ M C cx<G%UwUwGK]ZAYAmTG]MSlUWje=Y5m M¨.yzy¬v>jy0YAm ¨.[ªB¬JM^[K; U^hS&z˜Z5]SlGZ[4Uwc]WGXnXjeh D[GX]wGJUwD[GLGX_Z[^hBjyGXYAn%G2S4c0]UV*cyG2n¦UWcy]Sbce jyg[g[]Wcyg[]W^`jUwGkm[^FGY5Sl^hcyY) M = M T > 0 jeYAm K jncSlG2mn%c0Y0G%SlGUX§ g[]WcB [K, y] ⇐⇒ x = g[]Wced (K; y) ⇐⇒ x = argmin 1 (z − y)T M (z − x − y ∈ −M −1 ∂ψK (x) ⇐⇒ x = M (JSJ G.SlDAjy¢SlGXGy5UwM 0 c v ] y j D[G!FGXjySwZ[]WGmT^£<GX]wGXY0UW^hjy z∈K c0]wFjy2^`SlF ^`SkZASwG%Z[¢^hY v ∈ K y) ⇐⇒ hx − y, v − xi ≥ 0 UwDAGn%c0Y0UWG%RU ce4cyg[Uw^hFjy n%c0YUw]Wcy>[SlGXGSlG2n¦Uw^hcyY [ÉqT YAmTGG2m^UL^hSkj.c0]wFjy^`SlF D[^hnD]WG%Uje^hYAS cyYAQUWD[G . FGXjySwZ[]WG gAjy]lU/cebUWD[G.m[^hSlUw]W^xAZTUw^hcyYAjy¢cy]WFjy^`SwF:^> G0&UwD[GSlUWjeUwGXSvmT^`SwncyYUw^hYRZ[^Uw^hGXSXK c0SlUv^hFg5c0]lUjeYUwhQ^US&Slc0ZTUW^c0YAS&je]WG Z[YAn%Uw^hcyYASkce UW^FG^hYAmTGXg5GXYAmTGYUWQceUwD[GmTGX~y]WGG&ce λ bf*cyUwDcy]WFjeh^hSwFS*jy]wG&]WG`jUWGXmUwD[]WcyZ[~0DUWD[Gvcyhc ^hY[~5 ¶ ¶ XX ¶)Y
(49) R 0 (z , ..., z , ξ) ∈ (T (IR+ ))n A # & -, A R JG=R@GR@ M * r r−1 H ' :1 {z}(0−) = z0 ∈ IRn z1 = {z1 } ∈ F∞ (IR+ ; IR) zi ∈ Ti−1 (IR+ ) (2 ≤ i ≤ r), ξ = {ξ} ∈ ) + . & , . . . A . 1 @1 > + (F (IR ; IR))n−r ({z }, ..., {z }, ξ) {z}(0−) = z ∈ IRn + ˜ ∂ψ r−1 dνr ({t}) = {zr }(t+ ) − {zr }(t− ) ∈ −C˜ A˜r−1 B T ({z1 }(t− ),...,{zr−1 }(t− )) ({zr }(t )). Φ. ∞. 0 H ' :. 1. r. A. (w1 , ..., wr , ξ) ∈ (F∞ (IR+ ; IR))n $ % (z1 , ..., zr , ξ) ∈ (Tr−1 (IR+ ))n zi = w i +. i−1 X j=1. À À Ä#0[Í112. & , A . Q : > *. )+. X. z1 := w1. tk ∈E0 (wj ). (i−j−1). − (wj (t+ k ) − wi (tk ))δtk. . 0. {z}(0−) = z0 ∈ IRn. (2 ≤ i ≤ r),. .
(50) ªBs. # , . {z}(0−) = z0 ∈ IRn (JYG%[jeFg[hGJ^`S4UW]wG2jUWGXm^hYmTG%Uje^h5^hYSwGXn%Uw^hcyY|A |AÉsv D[^`nDD[GXgAS Z[YAm[G]SVUjeYAmT^hY[~D[c ¨lª e¬°¨lªXry¬b^hS^YUWG~y]jUWGXm ^hY Uw^hFGjeYAm D[cUWD[GSlUWjUWG.dVZ[Fg Fjyg[g[^hY[~0S .¨ [ªX¬.¨ yzy¬.*cy]W: HJceUW^hnGUWDAjUa]wc0g5cSl^Uw^hcyY ª jeg[g[h^hGXS.Uwc½UWD[G SwcyhZTUW^c0YAScy+UwD[G&f*¼Ja jyS GX> U*^`SYAceUwGX c0]lUWDRQUWDAjU'^ UwD[G&mT^`SVUW]w^hx[ZTUW^c0Y z˜ ^`S*jSwcyhZTUw^hcyYcy)UwDAGvmT^`SlUw]W^x[Z[Uw^hcyYAjy gA]wc0x[GXF ^hY¨Vª e¬'¨Vª"0¬ ¨Vª2ry¬4jeY5m^:UwD[G'Z[Y5n¦Uw^hcyY ζ(·) s y¬'¨»se|¬%eUwDAGY ˜ ^hSjSlc0ZTUW^c0Y!cy5UWD[GkFGXj0SlZA]wG*g[]Wcyx[hGF ¨»s0se¬ ¨»! e j h . F c V S b U X G y X G w ] R Q. [ D X G w ] * G y c Y U ` ^ b S e j ` w S c % n h X G e j. ] W U A D j U ^ ` ^ b S & j F X G 0 j l S A Z w ] 0 G w U [ D X G Y 5 x y c w U D 0 c ]wFjeh^`SlFS4je]WG ˜ {˜ G2_zZ[}(·) ^jy=GXY0ζ(·) U*cyY[GJUwc!GXjynDceUwDAG]2 A]wc0F[0, j ¸ Tcy1Y] UW]wc0AGYA~y^hY[GGX] g<cy^hYU ceλ^hGUWD[G . ]wG2je /JSlc0ZTUW^c0Y^`SUWD[GLSwcyhZTUW^c0Yce UWD[G&mT^hSlUw]W^hx[ZTUw^hcyY5je<cy]WFjeh^hSwF *Jc GXyGX]bUwDAGvFG2jySwZ[]wGvmT^£<GX]wGXY0UW^hjy<cy]WFjy^`SwF ^hh+g[]wc0G Uwcx<Gv_Z[^UwG&ZASwG%ZA:Uwc c0]wF!Z[`jUwGjyYGUWGYAm[GXmcygTUW^Fjy)n%cyYUW]wc0<gA]wc0x[GXF ² Y X X )¶ Y 0 0 ≤ a < b ∈ IR ∪ {+∞} A ( *; @$) # ' & -, z ∈ (Tr−1(IR+))n R(J R R > * %&
(51) ! t ∈ [a, b) ' . > ( )( )( ) {z}(0−) = z0 ∈ IRn [a, b) # & -, A . $ > A ' +. R(J R2 R :$. 2&B!9 '# ' .! , . [t, σ) (σ > 0). {z}. . [t, σ). # -.!". i GX~yZ[`je]kSwcyhZTUW^c0YAS'FjBQg5cSwSwGXSWS jynXn%Z[F!Z[`jUw^hcyY5S'ce<dVZ[FgASAx[ZTUJc0Y[QcyYUwD[GhG%ULce4jeYRQ AjyS']W^~0DU 0j nnZ[FZ[`jUW^c0YAS'njyY[Y[ceU GR^`SlU'xRQm[G%AY[^Uw^hcyY bC'D[GQGXYAn%c0FgAjySWS*]w^h~yDUl°jeY5jehQ0UW^hnLSlc0Z[Uw^hcyYAS&© |0{B«?tb∈C'D[[a,Gvb)cyhhc ^Y[~^`S gA]wc0GXm^hY ©ª%«jeYAmn%c0YAn%GX]wY5S*UwD[G ¼La ^hY¨Vª¬&¨Vª y¬&¨Vª2ry¬¦ ²R¶ ² & ! '# C˜A˜r−1B˜ > 0 + m = 1 !9 z ∈ IRn * ) * R(JKSR@S=R @ >. .
(52) . {z}(0−) = z0 ∈ IRn. ; . X. X. %&
(53) & -, $. '. . z1 (·) ≡ {z1 }(·) ≥ 0 IR+ XX {z¯ ˜}(0+) = z¯ ˜00 √ X X.X k{˜ z}(t)k ≤ ect k˜ z0 k, ∀ t ∈ IR+ . . )+. c>0 %&
(54) ) & - , R(JKI=R@IR: z˜2. %' 0 &(!.8:& Q+ .56 +QA + n ∞. z˜1 2 h˜ z , ϕi, ∀ϕ ∈ C0 (IR ; IR ). . 0 z0,i =. )+. 0. >,. *. z˜0.
(55) . 0 z0,r =. {˜ z }(0− ) = z˜0 ∈ IRn. * @. h˜ z 1 , ϕi =. i−1 TΦ (z0,1 , ..., z0,i−1 ); z0,i , ∀ 1 ≤ i ≤ r − 1,. , . ˜ −1 ˜A ˜r−1 B) (C. r−1 TΦ (z0,1 , ..., z0,r−1 ); z0,r. 0 z0,l = z0,l , (r + 1 ≤ l ≤ 2n).. Ub^`S g[]WcyGXmUwDAjeU4Z[Y[^`_0ZAGY[G2SwS D[c0hmAS ^Y!UwD[Gknhj0SwS¢cy5]WG~yZAhjy]¢SwcyhZTUw^hcyY5S cyY[hQy ( n%]WZAn%^`jeTSlUwGg^`S¢Uwc&Z[YAm[G]SVUjeYAm UWD[GJ]WG`jUW^c0YASwD[^gx5GUV GXGY U jeY5mUwD[G&n%`jySWSbcy Slc0Z[Uw^hcyYAS ce+UWD[G *&u&ORa Y5mTGG2m^U*^`Sg<c0SWSw^x[hGkUWDAjU*]wG2jynDAjyx[^h^UVQ cy x¯ ]WcyF x¯ ^Y ¨»s0¬¦JDAcy`m[Sc0]j u DA^hnD7x5GXc0Y[~0SUwc j#SwG%U U cymT^`SVUW]w^hx[ZTUW^c0YAS *Jc GXyG]^hSUwD[GX]wG j ncy]W]wG21Slg<cyY5m[jeYAn0Gkx<G%UV*GGYUwDA^hS U jeY5m Tr−1([0, T1])® (kYGhGFGYU'ce jeYASw GX] ^hh+x5G&~y^hyGXY^hYSwGXn%Uw^hcyY+A qT (kY ^hFg5c0]lUjeYULgA]wc0g5GX]lUVQ D[^`nDD[GXgASvZ[Y5mTG]SVUjeYAm[^Y[~D[c UwDAG *&u&ORa7*cy]WTSk^`SkUWD[G.cyhc ^hY[~5vHJceUw^`n%G!A]SVULUwDAjeU *G&FjBQ ]w^UwG dν ^hY¨»s e¬&¨»#s 0¬'jyS i ,¨ sy¬ dνi = gi (t)dt + dJi , D[GX]wG jeYAm ^`S.jeY#jeUwcyF^`nFGXj0SlZA]wG ^UwD n%c0Z[YUWjexAGjeYAmc0]Wm[G]jex[hGSlGU.cyJjUWcyFS + ~0GY[GX]WjeUwG2mgxRiQ∈UwDAFG∞]w^h~y(IR DUkn%;c0IR) Y0UW^YRZ[c0ZAS¢dJdVZ[iFgZ[YAn%Uw^hcyY J bC'D[G&cyhc ^hY[~.D[cy`m[SX i ¶ ¶ X¥ X )¶ Y F
(56) R,0 z(·) A ) & , ? Z 22 R@! ) ; .¨ !0¬ gi (t) = 0, dt − a.e. t ∈ IR+ , (1 ≤ i ≤ r − 1),. ¾¿À ¾Á.
(57) ª.
(58) . ¨.e|R¬. gr (t) ∈ −∂ψT r−1 ({z1 }(t− ),...,{zr−1 }(t− )) ({zr }(t+ )), dt − a.e. t ∈ IR+ ,. +. Φ. 0 ≤ z1 (t+ ) ⊥ dνr (t+ ) ≥ 0,. ! . ¨.0q0¬. t ∈ IR+. ¨.y{¬. 0 ≤ z1 (t+ ) ⊥ gr (t+ ) ≥ 0, dt − a.e. t ∈ IR+ . TΦi−1 ({z1 }(t− ), ..., {zi−1 }(t− )) 3 {zi }(t+ ) ⊥ −dJi ({t}) ∈ ∂ψT i−1 ({z1 }(t− ),...,{zi−1 }(t− )) ({zi }(t+ )). . 1≤i≤r. ++# . t ∈ IR. +. ¨.&y¬. Φ. . C'DRZASXXUWD[G ncyFg[hGFGYUje]W^UVQv^Y¨|¬ ¨¥n2¬+^`S ~0^0GYjJF.G2jeY[^hY[~k ^UwD¨,0{y¬ jyYAm¨.&¬%¢C'D[G*n%c0Y[G ¸ at^hY¨.&¬SlDAckS DAjeU ¸ a UwD[G&FGXj0SlZA]wG2S dν SwjeUw^`SVQ¨UwD[G&]W^~0cy]WcyZASFGXjeYA^Y[~.ce UwD[G&^hYAn%hZASw^c0Y^hYUWD[G&]w^h~yDUl?DAjyYAmR°Sl^`mTGkcy*¨,¬ ^hS G%Tg[`je^hY[G2m^Yt©ªeRab]Wcyg<c0Sw^Uw^hcyiY ª%«:cy]L© sRg) e{«¬¦ U ^`S ^Fg<cy]wUWjeYUbUWcY[cyUw^`n%GJUwDAjeU ~y^hyGXY {z }(t−), ..., {z }(t−) i−1 UwDAGºFGXj0SlZA]wGºBjyZAG½mTGXg5GXYAm[ScyY {z }(t+) -ºGSwDAjehvncyFG½xAjyn cyY ¨,&y¬^hY7SwGXn%Uw^hcy1Y7|AÉsR []WcyF ¨.!y¬^U i cyhhckS¢UwDAjeUbUwD[GJY[cyYT°jUWcyF^`n g5je]wUWS4cy5UWD[G jy~y]jeY[~0G F!Z[Uw^hg[^hG]Sbjy]w
(59) G !XG]Wc&jyFcSVUbGXyG]WQR D[G]WG*G%[n%GXgTUbcy] g (·) ¸ cyY5SlG2_0ZAGYUwhQvUwD[
(60) G jy~y]jeY[~0GbFZAUW^g[h^hG]S¢j0SwSwcTn%^`jUWGXm&UWcvj0m[mT^Uw^hcyYAjyRncyYASlUw]je^hYUWS Uwcvx5G'^hFg5cSlG2mcyYUwD[G'g[]WcyxArGXF je]WGg[Z[]WGhQtjUWcyF^`neC'D[^`SFjBQG%Tg[`je^hYº DRQ UWD[GQºDAjByG.x<GGY ^h~yY[c0]wG2mt^hY½UwD[Gh^UwG]jUWZ[]WGSw^hYAn%G.UwDAGn%cyY5mT^UW^c0Y UwD5jU λ x<Gj.F.G2jySwZ[]WGvD5jyS jeh'jBQRS x<GGY^hFg5cSlG2m+ H R * -2& m ≥ 2 + C˜A˜r−1B˜ $ */&
(61) ) .! )# ² ( [
(62) R9. . . H ' "
(63) @ A
(64) # *' $ +$ + x(·) H ' ,! !.! z˜− + $ .! */R: * - +&B! 9++ ' ! '
(65) A B : C-, D * ' + C @ ,# *! &, H ' . & (λ) )+ 2& ! (dν ) M! &( A + +! A , i * ; * @ # / # & -, ' * " $ +! % Cx ¯ +D ≥0 (w(·)) = 0 [0, T ] 0. (kYAceUwDAG]*^Fg<cy]wUWjeYU*UwcRcy:^`S. ^`S ^`mTGYUW^hnXjehQ X! G]WcA§ z (t). . UwD[Gvcyhhc ^Y[~$ ¸ a D[^hnDD[cy`m[S*cyYUw^hFGL^hYUwG]Wje`S. 1. [τ − , τ ). 1. >0. RcyY DA^hnD. ¨,0¬. ˜ −1 z˜(t+ ) + C˜ A˜r−1 Bλ(t) ˜ 0 ≤ λ(t) ⊥ C˜ A˜r W ≥0. Yo¨.!0¬cyY[G½n%c0Z[hm ]w^UwGºGX_Z[^hBjyGXYUwhQ g (t) ^hYASlUwG2jym cy λ(t) C'D[^`S ¸ aFcyY[^Uwc0]WSUwD[G½G0cyhZTUw^hcyY cyUwD[G FZAUW^g[h^hG] λ(·) cy] t ∈ [τ − , τ ) C'D[GFjrUW]w^cy UwD[^`S ¸ a ^hS'UWD[GhGXj0mT^YA~Kje]Wycg5je]jeFG%UWG] M (2r ) -ºG UwDAG]WG%cy]WGDAjByGjeUcyZ[]m[^hSwg5cSwjyj n%c0Fg[GUwGmTQRYAjyF^hnXjeSwQTSVUWGF DA^hnDjehhckS&ZASUWc ~y^hyGjFGXjeYA^Y[~UWcUwD[G SwQRSlUwGXF^hY#¨¥|¬¦<c0]LjeYRQ^YA^UW^hjy¢m[jeUWjjyYAmcy]vjeh t ≥ 0 JC'D[^`SL^hSLjA]SVUvn%]WZAn%^`je¢SVUWGg c0]kUwDAGZ[YAm[G]SVUjeYAmT^hY[~cy UwDAGY[GXnGXSWSwjy]wQncyYAmT^Uw^hcyY5S'cec0gTUw^hFjeh^UVQ0 ¶ ¶ » T ³ / - T 9! , 1R(JPR@ R@ D' d H({x}(t), {η}(t)) = 0 @; % " dt ¶)¶ cyhckS]wc0FG2jySwQ½nXje`n%Z[`jUW^c0YASjeY5m½UwD[G¥jyn¦U!UwDAjeUUwDAGSwZ[g[g<cy]wUce UWD[GFGXj0SlZA]wG2S dν ^`S!j !XG]Wc i )GXx5G2Sl~0Z[G&FGXjySwZ[]WGLc0Y [0, T1 ] *kc*G0G]*jehcyY[~UwD[G&cyg[Uw^hFjy<Uw]jdVGXn%Uwc0]wQ.UwDAG *LjeF^Uwc0Y[^`jeYZAYAmTG]W~ycRG2S dVZAF.g5S jU jUwc0FS cy ¨¥SwGG ¸ cy]Wcyh`je]WQ λ 0¬% wu. À À Ä#0[Í112.
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