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Some results on the optimal control with unilateral state constraints

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(1)Some results on the optimal control with unilateral state constraints Bernard Brogliato. To cite this version: Bernard Brogliato. Some results on the optimal control with unilateral state constraints. [Research Report] RR-5992, INRIA. 2006. �inria-00103775v2�. HAL Id: inria-00103775 https://hal.inria.fr/inria-00103775v2 Submitted on 12 Oct 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(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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(21) |.  # , . C'DA^hSL^hSvŠ DRQŠ G.Š ^h‹Ž‹ GF!x5G2mUwDAG.c0gTUw^hF^ !2j†UW^Žc0Yg[]Wcyx[‹hGF ¨VªB¬¨>sy¬J^ŽYUWcj—SwZ[^ŽUWjyx[‹ŽG!Œ”]jeFGŠ*cy]WšŠ DA^hnDtjy‹Ž‹hc†ŠkSkZ5S UWc]w^h~yc0]wc0ZASl‹hQ.UWjyšyG&^ŽYUWcj0nncyZ[YU'UwD[Gg<c0SWSw^Žx[‹hG&g[]WGXSwGYAnGvcyŒD[^h~yDAG]kmTGX~y]WGGmT^`SVUW]w^hx[ZTUW^Žc0YAS• C'D[GŸA]SVUJc0]Wm[G]kY[G2n%G2SwSWje]WQncyYAmT^ŽUw^hcyY5S Œ”cy]JUwD[G!cygTUW^ŽFjy‹n%c0YUw]Wcy‹)g[]WcyxA‹ŽGXFž^hY#¨VªB¬¨>se¬knXjeY x<GŒ”cy]WFZA‹hjeUwGXm j0S ©Ése{[“[| “AqyzT“5!q B«         0 x(t) A BR−1 B T x(t) ˙   λ(t) + =   −C T η(t) Q −AT η(t) ˙   . ¨ ¬. ¨”|R¬ ¨€ ¬ ¨‡ ¬ Š DAG]WG η(t) ∈ IRn “ 0 ≤ γ ∈ IRm “ γ T (Cx(T ) + D) = 0 “ β ∈ IRn “bjyYAm UwDAG—cyg[Uw^hFjy‹*ncyYUw]Wcy‹ h^ S~y^h‘yGXY(cyY 1 ^hYUwGX]w‘†je‹`S*Š D[G]WG η(·) ^hS j.Œ”Z[Y5n¦Uw^hcyYxRQ x(0) = x ¯0 , η(T1 ) = F x(T1 ) + C T γ + β = F x¯1 + C T γ + β = η1        0 ≤ w(t) = Cx(t) + D ⊥ λ(t) ≥ 0. »¨ q0¬ (JY^hF.gA‹Ž^`n%^ŽUbjySWSlZ[FgTUW^Žc0YŠ D[^`nD!je‹h‹hc†ŠkS)cyYAGbUwcLŠ ]w^ŽUwG*x<ceUwD¨VªX¬jeY5m—¨”|¬)^`SUwDAjeU UwDAG F!Z[‹–UW^ŽgA‹Ž^hG] ^hS¢jkFG2jySwZ[]wG0• ‰ ŒbYAceUvUwD[GXYx5cyUwD ¨Vª2¬JjeYAm#¨”|¬JDAjB‘0GUwcx5G!~yGYAG]je‹h^ !XGXmUwcx5G.FGXjyY[^ŽYA~eŒ”Z[‹>•vC'DA^hSL^hSJUwDAGŸA]SlULcyx[λdVGXn¦UW^Ž‘0GceŒ4UWD[^`S Š*cy]Wš:• ‰ YºUWD[^`SgAjeg<G]UwD[Gc0gTUw^hFje‹ Uw]j†dVG2n¦UWcy]WQtjeYAmºcyg[Uw^hFjy‹˜n%c0YUw]Wcy‹h‹ŽGX]vŠ ^h‹h‹bY[ceUx<G~y^h‘yGXY½jSwg5G2n%^`je‹bYAceUWjeUw^hcyY ^hY cy]mTGX]'Uwc‹h^Ž~0DUwGYUWD[Gg[]WGXSwGYUWjeUw^hcyY)•*C'D[G‘†je‹hZ[G2SkceŒ¢UwDAGFZ[‹ŽUw^hg[‹h^ŽGX]WS γ jeYAm β nXjeYx<G!njy‹hnZ[‹`j†UwG2mŒ”]wc0FžSwcyFG x<cyZAYAm[je]WQ—ncyYAmT^ŽUw^hcyY5S©Žª T“%/¦sT• †«?• ‰ Œ x(T ) ^`SkY[cyUJSlg<GXn^–Ÿ5GXm jeY5m F = 0 UwDAGY η(T ) = 0 • -ºGj0SwSwZ[FGvUwDAjeUkUwDAG OT‹hjeUwG]&n%c0YASlUw]je^hY0U&_ZAje‹h^–Ÿ<nj†UW^Žc0Y#¨ ∃ x SwZA1 nD UwD5j†U Cx + D > 0¬J^hS&SWj†UW^hSlŸAGXmtjeYAm1UwDAjeULUWD[Gg[]wc0x[‹hGF ¨lª2¬!¨»sy¬L^`S YAcy]WFjy‹»“ejLg[]Wcyg<G]wUVQvUwDAjeU ^`S ]WG‹`j†UwG2mUWcLUWD[G n%c0Y0UW]wc0‹Ž‹`jexA^Ž‹h^–UVQ&cyŒ[UwD[G ncyYASlUw]je^hY[GXm!SlQTSlUwGXF¨¥SwGG'Gy• ~A• © | “eab]Wcyg<c0Sw^–UWcyY |5•Žª«^hYtnj0SlG λ ^`SvjFGXjySwZ[]WG2¬%•-½G!n%c0Y†dVGXn%UwZ[]WGUWDAj†Uv]wG2jynDAjyx[^h‹Ž^ŽUVQjyYAm Y[c0]wFje‹h^ŽUVQD[c0‹hm Œ”cy] rwu ≥ 2 xRQUješR^hY[~ ^hY jeY#GXY[‹`je]W~yGXm#SlGU.cyŒLmT^`SVUW]w^hx[ZTUW^Žc0YAS!UwDAjeU.Š ^h‹Ž‹ x<G mTGXSWn%]W^hx5G2m(‹`j†UwGX]¨»SlGXGSwGXn%Uw^hcyY [•Éqe¬¦ • []WcyF jº~yGYAG]je‹ u(·) g<cy^hYUceŒ ‘R^ŽGXŠ DAc†Š GX‘yG]2“+UWD[G—n%c0YUw]Wcy‹h‹hjyx[^Ž‹h^ŽUVQg[]Wcyx[‹hGF Œ”cy]—¨>sy¬v]WGFje^hYASc0g5GXY(jeYAm½Š*GŠ ^Ž‹h‹˜Y[cyUUWj0nš‹hG^ŽUD[G]WG ¨»SlGXG (Jg[g<GYAmT^Žœ ¸ Œ”c0] Slc0F.Gvg[]WG‹h^ŽF^hYAje]WQD[^hY0USW¬%• ‰ YUWD[GSlG2_0ZAG‹:Š G&SwDAje‹h‹+mTGY[cyUwG T T T “ ˜ C) ˜ UwDAG UW]w^hg[‹hGjySWSlcTn%^`j†UWGXmUwcUwDAG&SwQTSVUWGFž^ŽY ¨”|R¬&¨¥j0¬&¨»nX¬%“[Š D[^`nDŠ*GSlDAjy‹Ž‹+cyŒ­UwGXY]WG%Œ”GX]'Uwc^hYx˜ UWD[=GSw(xGX_Z[,Gη‹)j0)S Uw(D[A,˜ G B,

(22)       !"#!$% &• ‰ Uk^`Skn%‹hGXjy]*UwDAjeUk^–Œ λ(t) ^hS j.FGXj0SlZA]wG0“RUwD[GXYŒ”]WcyF¨”|R¬&¨¥nX¬ ¨¥{¬ −C T λ(t) ∈ ∂ψΦ (x(t)) ¨¥ZASl^hY[~ ©É#s T“Aab]wc0g5cSl^ŽUw^hcyYºªy• [•hªyª«:c0Ym[^–£:G]WGYUw^`j†UW^Žc0YceŒ¢ncyYR‘yGœŒ”Z[YAn%Uw^hcyYAS¬'jeYAm SlZAg[g (λ) ⊂ {t| Cx(t) + D = 0} ¨ e¬ ‰ U^`SY[cyUwGŠ*cy]wUwDRQ UWDAj†UŠ D[GXY ^`SY[ceU—j½Œ”Z[Y5n¦Uw^hcyY ¨¥Gy• ~A• j¤L^Ž]jynFGXj0SlZ[]WG2¬%“˜UWD[GY UWD[G FGXjeYA^ŽY[~ cyŒLUWD[G ^hYAn‹ŽZASw^hcyY^hYt¨¥{¬bD5jyS˜Uwcx<GLSlUWjeUwG2m]Wλ^Ž~0cy]WcyZASw‹ŽQ0“ySwGG©hªe“[ORG2n¦Uw^hcyYsB«:cy]L© sR“Rg)• e{†«?•¢C'DAGYUWD[G&mTQRYAjeF^`nje‹<SlQTSlUwGXF ^hY ¨¥|¬v¨¥j¬&¨¥n2¬*FjBQ—x5G‘R^hGŠ*GXmjyS j.FGXjySwZ[]WGmT^–£:G]WGYUW^hjy‹+^ŽYAn‹ŽZ5Sl^hcyY ,¨ 0¬ ˜x(t) ∈ ∂ψ ˜ (˜ x˜˙ (t) − A˜ Φ x(t)) Š ^ŽUwD “[jyYAmŠ D[G]WGj.SlUWjeUwG]WG^hY[^ŽUw^`je‹h^hSWj†UW^Žc0Y—Fjeg[gA^ŽY[~^`S'F^hSWSw^ŽY[~^hY.¨ y¬¨¥‹ŽGUkZAS Y[cyUw^`n%G Œ”]WcyF ¨ e¬JΦ˜UWDA=j†U{˜UWxD[|CG˜x˜n%c0+Y0UWD]wc0‹ ≥^ŽY[0}gAZTU u(·) F!ZASVUx<Gjn%cyYUW^ŽYRZ[c0ZASvŒ”Z[YAn%Uw^hcyY ceŒ Uw^hF.G¨¥G‘0GY jeYAjy‹ŽQUw^`nX¬LGX‘yG]WQRŠ D[G]WG c0ZTUWSw^`mTG!SwZ[g[g ¬¦• *Jc†Š GX‘yG]vjySLŠ G.SlD5je‹h‹ SwGG!^ŽY½SlG2n¦UW^Žc0Y [“:^hYt~0GY[GX]Wjy‹c0Y[GDAj0SkUWc]WGXSwcy]wUkUWcj—F.c0]wG.n%c0F.gA‹ŽGœ Œ”c0]wFjy‹Ž^`SlF UWc(λ) n%c0]w]WGXn%Uw‹hQDAjeYAm[‹ŽG¨”|¬% • '4œ[n%GXgTU^–Œ j0m[mT^ŽUw^hcyYAjy‹¢ncyYASlUw]je^hYUWSvje]WGjymAmTGXmUwc ¨”|¬kUwcjySWSwZ[]wGUwDAjeU λ ^`S jFGXj0SlZ[]WGy“:UWD[GY½UWD[G.Œ”cy]WFjy‹Ž^`SwF ^ŽY ¨”|R¬LjyYAmºGXSwg5G2n%^`je‹h‹ŽQUWD[( G & )* &+  ,-.* / 10/

(23) vn%c0YAmT^ŽUw^hcyY ¨”|R¬!¨»nX¬vje]WG FG2jeY[^hY[~y‹hGXSWSX• - D[GXY m = 1 “RUWD[G&]wGX‹hjeUw^h‘yG&mTG~0]wGXG r ceŒUwD[GvUw]W^hg[‹ŽG (A,˜ B, ˜ C) ˜ ^hY¨”|0¬*^`S*UVŠ ^`n%GLUwD[G]WG‹`j†UW^Ž‘0GvmTGX~y]WGG e c. Œ W U [ D  G W U w ] h ^ [ g h ‹ G h ^ Y > ¨ y s  ¬ © 0 q 2 s & « ¥ ¨. Š A D † j . U Š  G  n e j h ‹ b ‹ W U [ D  G w ] X G h ‹ e j w U h ^ y ‘  G T m X G ~y]WGG^hYUwDA^hS!gAjeg<G]!^hS!ceŒ­UWGY(]WG%Œ”GX]w]WGXm rwu UWcj0SkUWD[Gn%cyY5SVUW]Wjy^ŽY(A, UWSJB, cy]mTC) G]v^hYUwDAG!c0gTUw^hFje‹4n%c0Y0UW]wc0‹‹h^ŽUwG]j†UWZ[]WG—©É#s B«”¬¦• -½G!Š ^h‹h‹ SWjBQUwDAjeULUWD[G!Uw]W^Žg[‹hG (A, B, C) D5jyS'jZAY[^–Œ”c0]wF›‘0GXn%Uwcy]']WG‹`j†UW^Ž‘0GLmTGX~y]WGG rwu ^ŽŒ UWD[G&Fj†Uw]W^Žœ CAr −1B ∈ IRm×m Š DAc0SwGLGXY0UW]w^hGXS'jy]wGJUwD[GSWnje‹`je]S “[^hS*Œ”ZA‹Ž‹)]jeYAš<“[jyYAm C Ak−1 B = 0 Œ”cy]kjy‹Ž‹ 1 ≤ i ≤ m “ 1 ≤ j ≤ m “[jeY5m k ≤ rwu − 2 • C Ar −1 B u(t) = argmaxu∈U. .  1 T T − u Ru + η(t) Bu = R−1 B T η(t) 2. wu. wu. i. j. i. j. ¾”¿À ¾­Á.

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ceŒUwGXY¤vjeF•bC'D[^`S je‹h‹hc†ŠkS Z5S*Uwc~yGXY[G]je‹h^ !GLSwcyFG&]WGXSwZ[‹–US'cyYUwD[GGœT^hSlUwGXYAn%G¨”c0]'Y[cyYT’?G%œT^`SVUWGYAnG2¬ ceŒ¢GYUw]WQ—SlUWjeUwGXS Œ”cy]*cTm[m]WG‹`j†UW^Ž‘0GLmTGX~y]WGGvSlQTSlUwGFSX“jyYAmUWcUjynšR‹hGkUwDAGLF!Z[‹–UW^Ž‘†jy]w^`jex[‹hGLnXjySwGy• ¢^ŽYAjy‹Ž‹hQjSlUwZ5mTQceŒ x<cyZAYAm[je]WQje]nS ^hS g[]Wcyg<c0SwGXm.ZASw^ŽYA~n%c0F.gA‹ŽGXF.GXYUWje]W^ŽUVQUWD[Gc0]wQ0• (J‹Ž‹5UwD[G2SlGkUwcRcy‹`S˜jy]wGk^hYUw]WcTmTZAn%G2m.Œ”cy]˜UwD[GJŸA]WSlUbUW^ŽFGJ^hYUWD[GLncyYUwGœU ceŒ cyg[Uw^hFjy‹ncyYUw]Wcy‹+Š ^ŽUwD^hY[G2_0Z5je‹h^–UVQSVUj†UwG!n%cyY5SVUW]Wjy^ŽYUWSJjeYAmje]WGg[]Wc†‘yG2m—UWc—Sw^Ž~0Y[^–Ÿ<njeYUW‹ŽQ^ŽFg[]Wc†‘yGvUwDAG_ZAje‹h^–Uj†UW^Ž‘0G šRY[c†Š ‹hGXmT~0G—ceŒJUwD[Gcyg[Uw^hFjy‹*Uw]j†dVGXn%Uwc0]w^hGXSX•C'D[Gjyg[g[]Wc0j0nD(je‹`SlcºgAjB‘yG2SUWD[GŠ*jBQ½Uwc†Š'je]m[SUwD[G mTG2Sl^h~yY(cyŒkUw^hFG%’ SlUwGgAg[^ŽYA~YRZ[FG]W^hnXje‹˜jy‹Ž~0cy]W^–UWD[FS&UwctSwcy‹h‘yGUWD[Gf*¼La ^ŽY ¨¥|¬%“UWD[G]WGxRQ½g[]Wc†‘^`mT^hY[~ Y[^`n%G—jy‹–UWG]WYAj†UW^Ž‘0GUwcF!Z[‹ŽUw^hg[‹ŽG 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^hYAmT^`njeUwcy] Œ”ZAYAn¦UW^Žc0Y.cyŒ:j&SwG%U ©És “0g)• sB«”¬¦“ ¨”UwD[G SwZ[x<m[^–£:G]WGYUw^`je‹AceŒ:UwD[G ^hYAmT^`njeUwc0]bceŒ+jn%c0Y‘0G%œ!SwG%U KψK©É#s [“yg)• &{ B«­¬%“0SwZ[g[g ¨ T ¬'¨­UWD[GJSwZ[g[g<cy]wUbKceŒ)jmT^`SVUW]w^hx[ZTUW^Ž∂ψ c0Y KT ©ÉsysR“ g)•hª|RsB«”¬¦“ x  0 ¨ x ^hS YAceU !GX]wc&jeYAmUwDAG'ŸA]WSlU4Y[cyY !G]WcvG‹hGFGYU4ceŒ<UwD[G'‘0GXn¦UWcy] x ^`S g5cSl^ŽUw^h‘yGB¬*“ < ¨­UWD[G'ŸA]SVU4Y[c0Y !GX]wc G‹hGFGXY0UbcyŒ:UwD[G ‘0GXn¦UWcy] x ^`Sbg<c0Sw^ŽUw^h‘yG2¬%“0Sw^ŽF^h‹hjy]w‹hQŒ”c0] ≺ jyYAm 4 “ σf (t) = f (t+) − f (t−) UwD[GbdVZ[FgceŒ<UwD[G Œ”ZAYAn¦UW^Žc0Y jeU UW^ŽFG “[f*¼ ¨¥x5c0Z[YAmTG2m—‘†jy]w^`j†UW^Žc0Y5¬¦“Ri ¸  f*¼ ¨¥]w^h~yDU n%c0Y0UW^ŽYRZ[c0ZAS ceŒ ‹hcTnje‹:x<cyZ[Y5mTGXm—‘Bjy]w^`j†UW^Žc0Y5¬¦“Ti ¸ O f ¼ f (·) ¨”]W^h~yDU'ncyYUw^hYRtZ[cyZ5S cyŒSwg5G2n%^`je‹+‹hcRnXje‹:x<cyZ[YAm[GXm‘Bjy]w^`j†UW^Žc0Y5¬¦“Tf*¼Ja ¨”x<cyZAYAm[je]WQ‘†je‹hZ[G&g[]Wcyx[‹hGF¬¦“ ‰ ¼La ¨¥^ŽYA^–UW^hjy‹+‘Bjy‹ŽZAG g[]Wcyx[‹hGF¬%“ δ UwD[G¤L^Ž]jynF.G2SlZA]wG.j†U t “ 0m = (0, . . . , 0)T ∈ IRm “ 0 = (0m)T “ I ^hSJUwD[G n−^hm[GYUw^ŽUVQFjeUw]W^–œ+• ¸ cyYUjyn¦U&UW^ŽFtGXSje]WGm[GY[cyUwGXm j0S τ ¨¥Š D[GY cyYA‹ŽQºcyY[Gn%c0YUWjyn%U^hSjyYAjem‹hQTSlG2mA¬Lcy] t Š n D[GYjSwGX_Z[GY5n%GceŒ'n%c0Y0Ujyn%U Uw^hFGXS'^`S'G%œ[jeF^hY[GXm)“ E (h) mTGYAceUwG2S UwD[GSwG%U ceŒ5dVZ[FgAS'ceŒ)UWD[GvŒ”Z[YAn%Uw^hcyY h(·) • e ^hkS UwDAG i−UWD—Z[Y[^ŽU ‘yG2n¦Uwc0]'ceŒ IRn U j!Œ”Z[YAn%Uw^hcyY^ŽY C ∞ (I) 0^`S'^ŽY[ŸAY[^ŽUwG‹hQmT^–£:G]WGYUW^hjyx[‹ŽGjyYAm—Š ^ŽUwD ncyFgAjyn%UkSlZ[gAg5c0]lU2i• 0. À À Ä#0[Í112. +.

(27) {.  # , . ! ¢º'%    *.  *¢%   +¢k C'DAGJY[GœU )GXF.Fj^`S ^ŽFg<cy]wUWjeYUbŒ”c0]bUWD[GJŠ*G‹h‹–’?g<c0SwGXmTY[G2SwS˜ceŒb¨”|R¬k¨»j0¬bSwGGXY—jyS jeY ‰ ¼La7¨ x(0−) = x “ η(0− ) = η ¬%“ jyYAm.Š ^Ž‹h‹Ajy‹hSwc&x5GkZ5SlGŒ”Z[‹[Œ”c0] UwD[GJ_ZAjy‹Ž^ŽUWjeUw^h‘yGkjyYAje‹hQTSl^`S¢ceŒ:UwD[Gkf*¼La(Slc0‹ŽZTUW^Žc0YASX“ySlGXGkSlG2n¦UW^Žc0YAS4|jeY5m0qT• -½Gkn%c0YASw^hmT0GX] jyYAm m = 1 ^ŽY )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. . „ ¶)¶ ˆ ) 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. !. Œ”cy]4Slc0FG. r≥1. •. C'DAG&ŸA]WSlUkjySWSlGX]lUW^Žc0YcyŒUWD[G )GXFFj^`S jSw^ŽFg[‹hGnjy‹hnZ[‹hjeUw^hcyYŠ ^–UWD ˜(1) • C'D[G (1) T jyYAm M (1) = −C A˜2 C T ‹hGXj0mT^hY[~tKjy]wš0c†‘gAje]jeFGUwG]Š GŠ'jeYUUWctn%c0Fg[ZTUwG—^hS M (2r+2) =A2(C =0)BB (2r+1) T • 2r+1 T A˜2 C ‰ Œ rwu = α + 1 “ UwD[GXY CAi B = 0 Œ”c0]je‹h‹ 0 ≤ i ≤ α − 1 •7uLZ[]c0xTdVGXA˜n%Uw^h‘yG(0^`SUw−c(C) SwD[c†Š=UWDA−C j†U M (2α+2) = “ •bf*Q—mT^h]wG2n¦Uknjy‹hnZ[‹`j†Uw^hcyYŠ*GvD5jB‘yGLŒ”cy]kjeYRQ^hYUwGX~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 Œ”c0‹Ž‹hc†ŠkS*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†Š YZASl^hY[~t¨lªyªX¬kjeYAmUWD[GY xRQ^ŽYAm[ZAn¦UW^Žc0Y)•"[]WcyF¨lªyªB¬'^–UJŒ”cy‹h‹Žc†ŠkS xRQ^ŽY5mTZAn¦UW^Žc0YUwDAjeU A˜(α) = 1 Š D[GX]wG L (Q) ^hS˜Slc0F.G'Fj†UW]w^Žœ.m[Gg<GYAmT^hY[~c0Y!UWD[GkFj†UW]w^Žœ Q • ‰ YAmTGXGXm‹hG%UbZAS˜jySWSlZ[FG'UWDAj†U A˜(i−1) Aα + Lα (Q) = α 1  • ' C [ D X G Y. Š – ^ 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+]wcRLceiŒ)(Q) ^`S'n%c0F.gA‹ŽGLUwGyi•(Q) A]wc0=F}L¨lª2i−1 se¬L(Q)A ¨”x<¬˜c0Y[+G (i−1) •4IJSw^ŽYA~¨VªBse¬ jeYAmSl^hYAn%G ˜ A˜2 A1 = A Q 1 2α m[GXmTZAnGXS:UWDAj†U M (2α+2) = X • (JSWSwZ[FG Œ”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 UWDAj†U CL (Q)BB T (AT )2α−i Œ”cy] je‹h‹ 1 ≤ i ≤ α ¨”UwD[G*UWG]WFSŠ ^ŽUwD α+1 ≤ i ≤ 2α Y[GG2mY[cyUx<G*ncyYASw^`mTG]WGXm CT = 0 i 2α j0S CAi B = 0 Œ”c0]vjy‹Ž‹ 0 ≤ i ≤ α − 1¬%•&C'DAGYtŠ*G.cyxTUje^hY M (2α+2) = X jeYAm (−1)i+1 CAi BB T (AT )2α−i C T UWD[GcyYA‹ŽQYAcyY !XG]WcUWG]WF ^hY—UWD[^hSJSlZ[Fž^hS*Œ”c0] i = α •bC'D[GX]wGŒ”cy]WG M (2α+2) i=1 α+1 α T T α T• CcGYAmUWD[Gg[]WcRceŒV“[^ŽULnXjeYx5G!SlD[c†Š Yje~je^hYxRQ^ŽYAm[ZAn¦UW^Žc0Y ZASl^hY[~¨lª2sy=¬ UW(−1) DAj†U L (Q)CA^`SkBB n%c0F.g<(A c0SwGXm) cyCŒUWG]WFSkceŒ UWD[G'~yGXY[G]je‹Œ”cy]WF Aj BF cy] FB T (AT )j “eŠ ^ŽUwD 0 ≤ j ≤ i−2 jeY5m F ^`S4SwcyFG FjeiUw]W^–œ+• ‰ YAmTGG2mc0Y[G'DAjyS L (Q) = i i−3 X T “ y j ' S l S A D † c. Š  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. ”Œj=0 c0]kje‹h‹. 1≤i≤α. jyS']WGX_Z[^h]WGXm+•. •!uLYAG!n%c0YAn%‹hZAmTG2SLŒ”]wc0F ¨lª2s0¬¦•C'DRZAS. CLi (Q)BB T (AT )2α−i C T = 0. ¾”¿À ¾­Á.

(30) 

(31)  . . )GXFFjºªn%cyYUW^ŽYRZ[G2SJUWcD[cy‹`mŒ”c0]SlQTSlUwGFSvŠ ^ŽUwDj—ZAY[^–Œ”c0]wF ]wGX‹hjeUw^h‘yG!m[G~y]WGGjeY5m m ≥ 1 ¨¥]wG2nje‹h‹¢UwDAjeU C ∈ ¬vSl^hYAnG CAi B = 0 Œ”cy]je‹h‹ 0 ≤ i ≤ rwu − 2 ^hYUWD[^`S&FZ[‹ŽUw^h‘†je]W^hjyx[‹ŽGnj0SlG.je‹`Slc5• ‰ YºSwZAnDºjnj0SlG0“+DAjB‘R^ŽYA~ IRm×n jy]wGL^ŽYAm[Gg<GYAmTGXYUX“[Š D[GX]wG C ^hS UwDAG i−UwD]wc†Š cyŒ C “ m×m Œ”Z[‹h‹–’?]WjyY[š^hF.gA‹Ž^hGXS*UWDAj†U*UwD[G&‘0GXn¦UWcy]S (2r ) M CiT i jeY5m B DAj0∈S'I]WRjyY[š 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 ^hYUw]WcRm[ZAn%G'jJnXjeY[c0Y[^hnXje‹RSVUj†UWG*SwgAjynG˜]WGg[]WGXSwGYUj†Uw^hcyY&Œ”c0] UWD[G'SlQTSlUwGF ˜ C, ‰ U˜ Š D) ]WGg[]WGXSwGYUWjeUw^hcyY Š D[^hnD Fješ0GXS.UWD[GSlcy’?nXje‹h‹ŽG2m !XG]Wce’°mTQRYAjeF^`nS.G%œTg[‹h^`n%^ŽUw‹hQ jegAg5(G2A, je˜]2• B, ^Ž‹h‹kx<G ZASlGŒ”Z[‹kŒ”c0]Slc0FG SwZ[xASwGX_Z[GYU!m[G‘yGX‹Žc0g[FGYUWS¨¥SwGGje‹`Slc ]WGFje]Wš½q^Ž‘[¬w¬¦•.Acy]!jUw]W^ŽgA‹ŽG (A,˜ B, ˜ C) ˜ “4Š ^–UWD(^hY[g[ZTU λ ∈ IRm “ cyZ[Uwg[ZTU A “ y j A Y — m A Z [ Y – ^ ” Œ 0 c w ] ž 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“¢jeYAmºUwDAGŒ”Z[‹h‹–’?]WjyY[šºSlUWjeUwGUW]WjyYASVŒ”c0]wFjeUw^hcyY½FjeUw]W^–œ½^hS § ¨¬¦• -ºG -½Gnje‹h‹˜^ŽUUwDAG Tm GXY[ceUWG z¯˜T = (z , . .z˜.−, z ) • ‰ YUwDAGSlG2_0ZAG‹>“TUwD[GncyFg<cyY[GXY0US'ceŒ z˜ jeYAmceŒ z Š ^h‹Ž‹)x<ceUWD x5GW˜m[GY[zcy˜Uw=GXmWj0˜S x˜z “AGœ[n%Gg[U 1 ^ŽŒ'YAceU!n‹ŽG2je]Œ”]WcyF UwD[Gn%rcyYUWG%œRUX• OR^hYAn%GUWD[G !XG]WcmTQRYAjeF^`nSgA‹hjBQTSjeY^hFg5c0]lUjeYU]Wcy‹hG^ŽYUWD[G—SwQTSVUWGFiSŠ*G—je]WG mTG2je‹h^ŽY[~Š ^ŽUwD)“[UwD[G&Œ”c0‹Ž‹hc†Š ^ŽYA~!^`S ceŒ¢^ŽYUWG]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  . „ ¶)¶ ˆ )GU jyYAm‹hG%U •4C'D[GvUW]WjyYASVŒ”GX]˜Œ”ZAYAn¦UW^Žc0YUWDAj†Ukncy]W]wG2Slg<cyYAmAS n×n x5GSwZAnD—UWDAj†U LT L = Q UwcUwD[Gkc0g5GX]WjeUwc0] Qλ 7→≥ 0w ^hY¨”|R¬ ^hS˜LGX∈_ZAIjyR‹TUwc H (s) = C(sI −1 ˜ 2n − A) ˜ B ˜ = G(s)G(−s)(1 − H ∗ (s)H(s))−1 ©ÉqysB«>“ Š D[GX]wG G(s) ^`S¢UwD[G'UW]WjyYASlŒ”G]4ceŒ5UwD[G'UW]w^hg[‹hG (A, wλ Ž ^   Y > ¨ e s % ¬ • h ^ ¢ S W U [ D ' G Uw]jeYASlŒ”G]4FjeUw]W^–œcyŒ5UWD[G'Uw]W^hg[‹ŽG (A, B, L) “ B, C) H(s) h ^ S w U A D L G W U W ] y j A Y l S ” Œ  G * ]  F e j w U W ] – ^  œ y c. Œ W U [ D J G W U w ] h ^ [ g h ‹ G ©Éqe|[“Tg)•És#0z†«?• ∗ T T T “T^»• Gy• UWD[GjymBdVc0^ŽYU SlQTSlUwGXF™Uwc H (s) ) ¸ cyY5Sl^`mTG]vY[c†Š UWD[GUVŠ*c¹+¤›njeYAcyY[^`njy‹)Œ”cy]WF(−A SLj0SwSwcT,n%L^`j†UW,GXBmUWc— UWD[G.SwQTSVUWGFSL^ŽY ¨»sy¬vjyYAm(¨”|R¬¦(A, “+]WGXB, Swg5G2L)n¦UW^Ž‘0G‹hQy• []WcyF )GXFFj ªcyY[G D5jyS jeYAm ξ˜ ∈ IR2(n−r ) • A]wc0F UWD[GŒ¥j0n¦UUWDAj†U H (s) = G(s)G(−s)(1 − ξ ∈ IRn−r e j A Y m w S Ž ^ A Y  n — G W U [ D G ° ’ T m QRYAjyF.^`nS.n%c0]w]WGXSwg<cyYAm[SUwcºUwDAGYRZ[FG]j†UWcy] B(s)wλceŒJUwD[GUw]jeY5SVŒ”GX]Œ”Z[Y5n¦Uw^hcyY H ∗ (s)H(s))−1 ξ1 “RŠ*GLmTG2mTZAnG UwDAjeU˜UWD[GJ]WcRceUWS˜ceŒ+UWD[GJg<cy‹hQRY[cyF^`je‹ je]WG FcTmTGXS ceŒ+UWD[GJFj†UW]w^Žœ A˜ •bOR^hYAn%GJUwD[G G(s) ξ cy]mTGX]*cyŒ B(α)B(−α) ^`S g[]WGXn^hSwG‹hQ 2(n − rwu ) “[UwD[G]WB(α)B(−α) GXSwZ[‹ŽU*Œ”c0‹Ž‹hc†ŠkSX=• 0.

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G%U&ZASvn%cyY5Sl^`mTG]LUwDAG.SwQTSVUWGF}^ŽY ¨”|R¬¨»j0¬!¨¥nX¬LjeY5m ‹ŽGUvZ5SJŒ”c0]w~0G%ULŒ”cy]LUwD[G!Uw^hFG!x<G^hY[~UwDAjeUv^ŽUvFjBQ]WGg[]WGXSwGYUkUWD[G YAGXn%G2SwSWje]WQn%c0YAmT^ŽUw^hcyYAS'cyŒUWD[GFj†œT^ŽF!Z[Fg[]W^hYAn%^hg[‹hGy• (kY^hF.g<cy]wUWjyYUkg5c0^ŽYU ^hS*Ÿ5]WSlU UwcZ[YAm[G]SVUjeYAm—UwD[Gm[QY5jeF^hnXS cyŒvSwZAnD jºmTQRYAjyF^hnXje‹kSwQRSlUwGXF ^ŽYR‘0cy‹h‘^hY[~ n%c0F.gA‹ŽGXF.GXYUWje]WQ’?Sw‹`jynšRY[GXSWSncyYAm[^–UW^Žc0YAS• [c0]^ŽYASlUWjyYAn%G0“˜Š DAjeU^hS.UWD[G FG2jeY[^hY[~tcyŒ λ ≥ 0 ^ŽŒ λ ^hS.Y[cyUjºFGXj0SlZ[]WGº¨»mT^`SVUW]w^hx[ZTUW^Žc0YAje‹'FZA‹–UW^Žg[‹h^hG]S!FjBQG2jySw^Ž‹hQ x5GY[GG2mTGXm#Uwc½^hY0UWG~0]WjeUwG ¨¥|¬¨»j0¬¨»nX¬w¬l®™CcUwD[^`SGXYAm ‹ŽGUZAS]wG2nje‹h‹˜SwcyFGŒ¥jyn%UWS!jex<cyZTUUWD[GD[^h~yD[GX]c0]WmTGX]SlŠ*GGXg[^hY[~ g[]WcRnGXSWS,¨ *&u&ORa ¬j0S ^ŽU^`S^ŽYUw]WcTmTZAnGXm ^ŽY ©hªe“ s2«?“ Š DA^hnD g[]Wc†‘^`mTG2S.j½]W^Ž~0cy]WcyZASŒ”]jeFGŠ*cy]Wš UwcSlUwZAmTQ(UwD[GmTQRYAjyF^hnXS^ŽY ¨”|R¬¨¥j0¬¨»nX¬%“ jyYAm GœUWGYAmASkUWD[GŠ*G‹h‹–’?šRY[c†Š YŸA]SVU&jeYAmtSlG2n%cyY5mcy]mTG]vSlŠ*GGXg[^ŽYA~g[]WcRnGXSWS¨»SlGXG]WG%Œ”G]WGY5n%GXSJ^ŽY ©Žª%« jyYAm©hªyªe“ /q[• e« Œ”c0]j—YAcyYT’?Fj†UwDAGFj†UW^hnXje‹4^hY0UW]wcTmTZ5n¦Uw^hcyY<¬¦•i cyZA~yD[‹hQSlg<GXjyšR^ŽY[~5“+UwDA) G *&u&ORa ^hSjSwg5G2n%^ŽŸ5nmT^Ž£<GX]wGXYUw^`je‹4^hYAn%‹hZASw^Žc0Y)“ Š DAc0SwGSwcy‹hZTUw^hcyYASjy]wGmT^hSlUw]W^hx[ZTUw^hcyY5S“*jeYAm#Š D[^hnD g5GX]wF^ŽUWS.UWcº~0^Ž‘0GjºF.G2jeY[^hY[~ºUwcºUwD[G SwQTSVUWGF ¨”|R¬¨”GXSwg<GXn%^`je‹h‹hQ • -½GmTGY[cyUwGjyS T (I) UWD[GSwG%UceŒ'mT^`SVUW]w^hx[ZTUW^Žc0YASceŒ mTGX~y]WGG UWD[Gg<c0Sw^–UW^Ž‘0GY[G2SwSvceŒ λ Š D[GXY ^–U^`SY[cyUjFG2jySwZ[]wGB¬¦) ¸ É © y s 2 s & «. Š [ D ` ^  n D e j W ]   G y ~  G A Y  G  ] † j w U 2 G m R x Q i O f ¼NŒ”Z[YAn¦UW^Žc0YAScyY “ Š D[nc0SwGtSwZAnnGXSWSl^h‘yGmTGX]w^h‘†j†UW^Ž‘0GXSg<c0SWSlG2SwSjeY n+1 I jyxASwcy‹hZTUwGX‹ŽQ(ncyYUw^hYRZ[cyZAS!gAjy]lU¨¥m[GY[cyUwGXm jyS [·]¬UwDAjeU^hSjy‹hSwc½i ¸ O f*¼žc0Y I • C'D[G]W^Ž~0DUmTGX]w^h‘†j†UW^Ž‘0GcyŒ [h] ^`S m[GY[cyUwGXmj0S hˆ (1) = d [h] (t) = lim [h](t+σ)−[h](t) •˜C'D[GSwG%UkcyŒ¢SwZAnDŒ”Z[YAn%Uw^hcyYAS ^`SkmTGXY[ceUWGXmjyS F∞ (I; IR) = σ→0 dt σ ‰ A “. Š Ž ^ w U D • Yg5je]wUw^`n%Z[‹`je] d ˆ (k−1) (k) ˆ [m# h xRQ jº] Œ”∈Z[Y5RCSLBV (I; IR)} ∩k∈IN Fk (I; IR) Fk (I; IR)• =‰ Œ {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 “*^ŽU—DAj0Sj (I) F ∈ F∞ (I; IR) .FŒ”Z[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^Žx[Z[Uw^hcyYAjy‹mTGX]w^h‘†j†Uw^h‘yG0“AjeY5m m[GY[cyUwGXS h T , ϕi = −∞ ϕ d[Fˆ (n−1) ], ∀ ϕ ∈ C0∞ (I) D UWD[GLOUW^ŽGX‹–U»dVGXS cy]˜m[^–£:G]WGYUw^`je‹5FGXjySwZ[]WG ~yGXY[G]j†UWGXm.xRQ.j&Œ”ZAYAn¦UW^Žc0Y z ceŒ+‹hcTnje‹Ax5c0Z[YAmTG2m‘†je]W^hjeUw^hcyY© s2«?•¢C'dzDRZAS T (I) m[GY[cyUwGXSJUwDAG!SwG%ULcyŒ˜je‹h‹ O[nDŠ'je]wU !0¡5mT^`SlUw]W^Žx[Z[Uw^hcyYASvSlZ5nD UwDAjeUJUWD[G]WGGœR^`SlUWSvjŒ”Z[YAn%Uw^hcyY F ∈ F (I; IR) SlZAnDnUwDAjeU ∞ • )GU n x<G&UwD[GSwFje‹h‹ŽG2SVU ^hY0UWG~0G] SlZ5nDUWDAj†U T ∈ T (I) “AŠ*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^Ž]jynLFGXj0SlZ[]WG D5jyS'mTG~0]wGXGvs[“^ŽUWS m[G]W^Ž‘†j†UW^Ž‘0G DAj0S'mTG~0]wGXG [“RGUWne• ¸ c0YUw^hYZAcyZAS˜Œ”Z[YAn%Uw^hcyYAS*DAjB‘0GvmTGX~y]WGG Az “4jeYAm(m[^hSWn%c0Y0UW^ŽYRZ[c0ZAδS&0 Œ”Z[Y5n¦Uw^hcyY5SDAjB‘0G—mTGX~y]WGG ªy•ºC'δD[˙0GŒ¥j0n¦UUWDAj†U!UwD[GmT^`SVUW]w^hx[ZTUW^Žc0YASŠ*GŠ*cy]WšºŠ ^ŽUwD#cy]W^Ž~0^ŽYAjeUwG Œ”]WcyF f*¼ Œ”ZAYAn¦UW^Žc0YAS“+^`Sn%]WZAn^hjy‹Œ”c0]vUwD[GnDAjy]Wj0n¦UwGX]w^ !XjeUw^hcyY cyŒbUwD[GX^Ž]SwZ[g[g<cy]wUX•.OR^ŽY5n%G.UwD[GY[cyUw^hcyYºceŒ*jSwcy‹hZTUw^hcyY Œ”c0]vUwD[GŒ”c0]wFje‹h^`SlF ^hYºŠ DA^hnD½Š GSwDAje‹h‹ GF!x5G2m ¨¥|¬v^`Svn]wZ5n%^`je‹>“ jeYºG%œ[jeFg[‹hG^hS&g[]Wc†‘R^hm[GXm^hY mTG%Uje^h‹4^ŽY jeg[g<GY5mT^–œ fv• -½GL^hYASw^hSlU*DAG]WGJcyY—UwD[GLŒ¥jyn%U UWDAj†U*UwD5j†U*UwD[G&Slc0‹ŽZ[Uw^hcyYAS˜UwDAjeU Š ^Ž‹h‹:x5G&n%c0YASl^`mTGX]wG2mY[G%œRU2“Rjy]wGLceŒSwg<GXn%^`je‹<x<cyZ[YAm[GXm ‘†jy]w^`j†UW^Žc0Y)• ‰ Y jeY[cyUwD[GX]kŠ*cy]m[S“RUWD[G^h]JmTGX]w^h‘†j†UW^Ž‘0GXS mTcY[cyULn%c0YUWje^hYjeYRQSw^ŽYA~yZ[‹`je]  Gx<GXSw~yZ[G^hYUwG~0]Wjyx[‹hGvg5je]wUX• ORGXG ¤LG%Ÿ5Y[^–UW^Žc0Y ª&jyYAmGX_ZAjeUw^hcyYº.¨ 0s0¬¦• C'D[GLn%cy]WG'ceŒ<UwD[G *&u&ORa‹h^ŽG2S4^ŽYUWD[GkmTGŸAY[^ŽUw^hcyYceŒ<UwDAG Œ”cy‹h‹Žc†Š ^hY[~SwG%UbceŒ<UWjyY[~yGXYU˜n%cyYAGXSX•  G%U x5GJj&YAcyY[GXF.g[UVQ n‹ŽcSlG2mncyYR‘yGœ—SlZAxASlGU ceŒ IR • -ºGm[GY[cyUwG&xRQ T (x) UwD[G&UjeY[~0GYUkn%c0Y[G&ceŒ Φ jeU x ∈ IR mTGŸAY[GXmΦ xRQ Φ ¨Vª2{0¬ TΦ (x) = cone(Φ − {x}) Š DAG]WG cone(Φ − {x}) mTGY[cyUwG2SUWD[G n%c0Y[G~0GY[GX]WjeUwG2mxRQ Φ − {x} “˜mTGŸAY[G2m#jyS.^hY © &X« UWctUješ0G—^hY0UWc½j0nncyZ[YU ncyYASlUw]je^hYU*‘R^hcy‹`j†UW^Žc0YAS“TjyYAm cone(Φ − {x}) ^`S ^–USkn%‹hc0SwZ[]WGy'• 1v^Ž‘0GY jn%‹hc0SwGXmY[cyYAGFgTUVQ—ncyYR‘yGœ—SlGU Φ “TŠ*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 ).. HkcyUwGvUwD5j†U 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*^ŽYUw]WcTmTZAnGLY[c†Š UVŠ*c.FjeUwD[GXFj†Uw^`njy‹<Œ”cy]WFjy‹Ž^`SwFS ceŒUwD[G *vu&OTa¢• Acy]*UwDA^hS'Š*GL]WG‹hQZ[g<cyYUwD[GSwg5G2n%^`je‹+SVUj†UWG SwgAjynGv]WGgA]wG2SlGXY0Uj†UW^Žc0Y—^hY½¨lª|¬%“TUwDAG]wG2jySwcyYŒ”c0]'UwD[^`S'gAjy]lUW^hnZ[‹hjy]knD[cy^`n%Gvx<G^hY[~G%œTg[‹`je^hY[GXm^hY½©hª%«?•  X¥‚Xƒ „ X”€  ƒ X )¶ Y  ¶ „ ( .X¥‚ 

(44) ¢^ŽYAm + jyYAm ξi ∈ Tr−1 (IR+ ) (1 ≤ i ≤ n − r) SWj†Uw^`SlŒ”Q^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¦UW‹ŽQ½¨Vª e¬'^hSJ]wGXŠ ]w^ŽUlUWGY jyS  •'C'D[G]WG‹`j†UW^Žc0YAS −1 ˜ = 0 c0] D˜ ˜x − Bλ ~0^Ž‘0GY^hY¨VªXr¬ DAjB‘0G'Uwcx<Gk^hY0UWG]Wg[]WG%UwG2m^ŽYD˜UWzD[Gk−Œ”c0W˜‹Ž‹hc†A˜Š W˜^hY[~Slz˜GXYA−SwGyW˜§ ¢Bλ ^hYAmY[c0Y[Y[GX~0j†xUW^Ž−‘0G A˜ ]wG2je‹Ž’>‘†jy˜‹ŽZ[G2=m!iJ0 jymTc0YF.G2jySwZ[]WGXS ]wGX‹hjeUw^h‘yG—Uwc Š D[^`nD(UwDAG  Gx<GXSw~yZ[GFGXj0SlZ[]WG dt jyYAm(UWD[GOUw^hG‹ŽU¥dVG2SFGXjySwZ[]WG d{z } g5cSwSwGXSWS dµi (1 ≤ i ≤ r) i m[GYASw^–UW^ŽG2S dt jeYAm d{z } ]wG2Slg<GXn%Uw^h‘yG‹hQSwZAnDUwDAjeUX§ 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 Swcy‹hZTUw^hcyY5S!cyŒJUWD[GmT^`SVUW]w^hx[ZTUW^Žc0YAje‹*Œ”cy]WFje‹h^hSwF Š ^h‹Ž‹kx<G SlDAc†Š Y(Uwc x<G mT^hSlUw]W^hx[ZTUw^hcyY5SceŒvm[G~y]WGGg<c0SWSl^hx[‹hQ `‹ je]W~yGX]*UwDAjyY 1 •  G%UJZASkY[c†Š7^hYUw]WcTmTZAn%GjSwGXncyYAmŒ”c0]wFje‹h^`SlFŠ DAc0SwGSlc0‹ŽZTUW^Žc0YAS je]WG&Œ”Z[YAn%Uw^hcyYASX“[^hYAmTGXg5GXYAmTGYUW‹ŽQceŒ UWD[GJ]WG‹`j†UW^Ž‘0GJmTGX~y]WGG rwu •4C'D[^hS Š ^h‹Ž‹<x5Gv_Z[^–UWGJZASwG%Œ”ZA‹<Š DAGYUWD[GJc0gTUw^hFje‹:n%cyYUW]wc0‹Ag[]Wcyx[‹hGF ^`S˜GXFx<GXm[mTG2m^hY0UWcUWD[G *&u&ORa “ASwGGSlG2n¦UW^Žc0YAS [•ÉqR“ A• {T•  ² [‚ „ ²  X ² „ ²ZY ƒ 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}^ŽY ¨»sysy¬JjyYAm(¨»s!y¬JDAjySJUwcx5G.^ŽYUWG]Wg[]wGUwG2m ^ŽYUwD[G!Œ”cy‹h‹Žc†Š ^hY[~—SwGY5SlG0§^hYAmtY[cyY[YAG~0jeUw^h‘yG]WGXjy‹–’?‘†je‹hZ[GXm iJjymTc0Y#FGXj0SlZ[]WG dµi ]WG‹`j†UW^Ž‘0G—UWc Š D[^hnD UwDAG  Gx<GXSw~yZ[GFGXj0SlZA]wG dt jeYAm#UwD[GtORUw^hG‹ŽU¥dVGXSF.G2jySwZ[]WG dzi g5cSwSwGXSWS m[GYASw^–UW^ŽG2S dt jeYAm dz ]wG2Slg<GXn%Uw^h‘yG‹hQ—SwZAnDUwDAjeU ˙ − 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+äŽÕ¦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. ÚÉÊΘähÐBÄ2ÞVÛ»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]2“TF.c0]wG&^hYUwZ[^ŽUw^h‘yG&Š'jBQUwcSwGGvUwD[GFG2jySwZ[]wGmT^Ž£:G]WGYUw^`je‹:Œ”c0]wFje‹h^`SlF“T^`S*Uwcn%c0YASw^hmTGX]*UwD[G&Œ”c0‹Ž‹hc†Š ^ŽYA~A§. ¢^ŽY5m. 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. SwZAnDUwD5j†U. •••. ¨>sy¬. •••. Š ^ŽUwDUwD[G^hYAn‹ŽZASw^hcyYAS jeY5m. λ. ^`S'^hY ¨lª"¬¦§. dνi ∈ −∂ψT i−1 ({z1 }(t− ),...,{zi−1 }(t− )) ({zi }(t+ )). Œ”c0]kje‹h‹. Φ. ¨>s#¬. 1≤i≤r. >¨ ser¬ ‰ YcyUwD[GX]LŠ*cy]m[S ^`S&j—mT^`SVUW]w^hx[ZTUW^Žc0YŠ D[cSlG!F.G2jySwZ[]WGgAjy]lUv^hS jyYAm ^ŽUWSvg5cSl^ŽUw^h‘R^–UVQ^`SJZAYAmTG]SlUwcRcTmjySJUwD[G g<c0Sw^–UW^Ž‘R^ŽUVQ ceŒ dν •!C'λD[GF.G2jySwZ[]WGXS dν jy]wG!FZ[‹ŽUw^hg[‹h^ŽGX]WSvŠ D[cSlG.FGXdνjeYAr^ŽY[~^hYUwDAGncyYUwGœRUvcyŒ˜cygTUW^ŽFjy‹Ž^ŽUVQ Š ^Ž‹h‹ x5G FjymTGn‹ŽG2je] ‹`j†UwGXr]¨¥SwGG ¸ cy]Wcy‹h‹`je]W^ŽG2S jeiY5mr[“AjeYAm]WGFje]Wš 0¬%• []WcyF ¨>s†¬'jyYAm ¨»s#0¬*^–U Œ”c0‹Ž‹hc†ŠkS*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‹Ž‹ UwDAjeUL~0^Ž‘0GYj—n‹ŽcSlG2m n%c0Y‘0G%œY[cyYAGFgTUVQSlGU jyYAm “<g[]WcBœ ^`SkUWD[G!n‹ŽcSlG2SVU T g<cy^hY0U&UWc z˜ ^hY K “)^hYtUWD[GF.GUw]W^hnmTGŸAY[G2mtxRQ M •C cx<G%UwUwGK]ZAYAmTG]MSlUWje=Y5m M¨.yzy¬v>jy0YAm ¨.[ªB¬JM^Ž[K; U^hS&z˜Z5]SlGŒ”Z[‹4Uwc]WGXnXje‹h‹ D[GX]wGJUwD[GLGX_Z[^h‘Bjy‹ŽGXYAn%G2S4Œ”c0]˜UVŠ*c‘yG2n¦UWcy]SbceŒ jyg[g[]Wcyg[]W^`j†UwGkm[^ŽFGY5Sl^hcyY)“ M = M T > 0 jeYAm K jn‹ŽcSlG2mn%c0Y‘0G%œSlGUX§ g[]WcBœ [K, y] ⇐⇒ x = g[]Wced (K; y) ⇐⇒ x = argmin 1 (z − y)T M (z − x − y ∈ −M −1 ∂ψK (x) ⇐⇒ x = M •(JSJŠ G.SlDAjy‹Ž‹¢SlGXGy“5UwM ” Œ 0 c v ] y j Ž ‹ ‹ D[G!FGXjySwZ[]WGmT^Ž£<GX]wGXY0UW^hjy‹ Œ”z∈K c0]wFjy2‹Ž^`SlF ^`SkZASwG%Œ”Z[‹¢^hY v ∈ K y) ⇐⇒ hx − y, v − xi ≥ 0 UwDAGn%c0Y0UWG%œRU ceŒ4cyg[Uw^hFjy‹ n%c0YUw]Wcy‹>“[SlGXGSlG2n¦Uw^hcyY [•ÉqT• ‰ YAmTGG2m^–UL^hSkj.Œ”c0]wFjy‹Ž^`SlF Š D[^hnD]WG%Uje^hYAS cyYA‹ŽQUWD[G . FGXjySwZ[]WG gAjy]lU/ceŒbUWD[G.m[^hSlUw]W^ŽxAZTUw^hcyYAjy‹¢Œ”cy]WFjy‹Ž^`SwF“:^>• G0•&UwD[GSlUWjeUwGXSvmT^`SwncyYUw^hYRZ[^ŽUw^hGXSX•K c0SlUv^hFg5c0]lUjeYUw‹hQ^–US&Slc0‹ŽZTUW^Žc0YAS&je]WG Œ”Z[YAn%Uw^hcyYASkceŒ UW^ŽFG^hYAmTGXg5GXYAmTGYUW‹ŽQceŒUwD[GmTGX~y]WGG&ceŒ λ •bf*cyUwDŒ”cy]WFje‹h^hSwFS*jy]wG&]WG‹`j†UWGXm—UwD[]WcyZ[~0D—UWD[GvŒ”cy‹h‹Žc†Š ^hY[~5• „ ¶  ¶ ‚X”ƒX ¶)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  (JYG%œ[jeFg[‹hGJ^`S4UW]wG2j†UWGXm^hY—mTG%Uje^h‹5^hY—SwGXn%Uw^hcyY|A• |A•ÉsvŠ D[^`nDD[GX‹ŽgAS Z[YAm[G]SVUjeYAmT^hY[~D[c†Š ¨lª e¬°’¨lªXry¬b^hS˜^ŽYUWG~y]j†UWGXm ^hY Uw^hFGjeYAm D[c†ŠUWD[GSlUWj†UWG.dVZ[Fg Fjyg[g[^hY[~0S .¨ [ªX¬.¨ yzy¬.Š*cy]Wš:• HJceUW^hnGUWDAj†Ua˜]wc0g5cSl^ŽUw^hcyY ª jeg[g[‹h^hGXS.Uwc½UWD[G Swcy‹hZTUW^Žc0YAS˜cyŒ+UwD[G&f*¼Ja jyS Š GX‹Ž‹>• ‰ U*^`S˜YAceUwGXŠ c0]lUWDRQUWDAj†U'^–Œ UwD[G&mT^`SVUW]w^hx[ZTUW^Žc0Y z˜ ^`S*jSwcy‹hZTUw^hcyYcyŒ)UwDAGvmT^`SlUw]W^Žx[Z[Uw^hcyYAjy‹ gA]wc0x[‹ŽGXF ^hY¨Vª e¬'¨Vª"0¬ ¨Vª2ry¬4jeY5m^ŽŒ:UwD[G'Œ”Z[Y5n¦Uw^hcyY ζ(·) s y¬'¨»se|¬%“eUwDAGY ˜ ^hS˜jSlc0‹ŽZTUW^Žc0Y!cyŒ5UWD[GkFGXj0SlZA]wG*g[]Wcyx[‹hGF ¨»s0se¬ ¨»! ‰ e j h ‹ . F  c V S b U X G y ‘ X G w ] R Q. Š [ 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 – ^ Œ ` ^ 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 ” Œ 0 c ]wFje‹h^`SlFS4je]WG ˜ {˜ G2_zZ[}(·) ^Ž‘†jy‹Ž=GXY0ζ(·) U*cyY[GJUwc!GXjynD—ceUwDAG]2• A]wc0F›[0, j ¸ Tcy1Y] UW]wc0‹AGYA~y^hY[GGX] g<cy^hYU ceŒ‘λ^hGŠ“UWD[G . ]wG2je‹ /JSlc0‹ŽZTUW^Žc0Y^`S˜UWD[GLSwcy‹hZTUW^Žc0YceŒ UWD[G&mT^hSlUw]W^hx[ZTUw^hcyY5je‹<Œ”cy]WFje‹h^hSwF• *Jc†Š GX‘yGX]bUwDAGvFG2jySwZ[]wGvmT^Ž£<GX]wGXY0UW^hjy‹<Œ”cy]WFjy‹Ž^`SwF Š ^h‹h‹+g[]wc†‘0G Uwcx<Gv_Z[^ŽUwG&ZASwG%Œ”ZA‹:Uwc Œ”c0]wF!Z[‹`j†UwGjyYGœUWGYAm[GXmcygTUW^ŽFjy‹)n%cyYUW]wc0‹<gA]wc0x[‹ŽGXF• ²  Y X”ƒ 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]kSwcy‹hZTUW^Žc0YAS'FjBQg5cSwSwGXSWS jynXn%Z[F!Z[‹`j†Uw^hcyY5S'ceŒ<dVZ[FgAS“Ax[ZTUJc0Y[‹ŽQcyYUwD[G‹hG%Œ­ULceŒ4jeYRQ “AjyS']W^Ž~0DU 0j nnZ[FZ[‹`j†UW^Žc0YAS'njyY[Y[ceU GœR^`SlU'xRQ—m[G%ŸAY[^ŽUw^hcyY •bC'D[GQGXYAn%c0FgAjySWS*]w^h~yDUl’°jeY5je‹hQ0UW^hnLSlc0‹ŽZ[Uw^hcyYAS&© |0{B«?t•b∈C'D[[a,GvŒ”b)cy‹h‹hc†Š ^ŽY[~^`S gA]wc†‘0GXm^hY ©Žª%«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[]Wc†‘yGXmUwDAjeU4Z[Y[^`_0ZAGY[G2SwS D[c0‹hmAS ^ŽY!UwD[Gkn‹hj0SwS¢cyŒ5]WG~yZA‹hjy]¢Swcy‹hZTUw^hcyY5S cyY[‹hQy• ( n%]WZAn%^`je‹TSlUwGg^`S¢Uwc&Z[YAm[G]SVUjeYAm UWD[GJ]WG‹`j†UW^Žc0YASwD[^Žg—x5GUVŠ GXGY U jeY5mUwD[G&n%‹`jySWSbcyŒ Slc0‹ŽZ[Uw^hcyYAS ceŒ+UWD[G *&u&ORa • ‰ Y5mTGG2m^ŽU*^`S˜g<c0SWSw^Žx[‹hGkUWDAj†U*]wG2jynDAjyx[^h‹Ž^ŽUVQ cyŒ x¯ Œ”]WcyF x¯ ^ŽY ¨»s0¬¦“JDAcy‹`m[SŒ”c0]j u Š DA^hnD7x5GX‹Žc0Y[~0SUwc j#SwG%U U cyŒmT^`SVUW]w^hx[ZTUW^Žc0YAS• *Jc†Š GX‘yG]—^hS—UwD[GX]wG j ncy]W]wG21Slg<cyY5m[jeYAn0Gkx<G%UVŠ*GGYUwDA^hS U jeY5m Tr−1([0, T1])® (kYG‹hGFGYU'ceŒ jeYASwŠ GX] Š ^h‹h‹+x5G&~y^h‘yGXY^hYSwGXn%Uw^hcyY+A• qT• (kY ^hFg5c0]lUjeYULgA]wc0g5GX]lUVQŠ D[^`nDD[GX‹ŽgASvZ[Y5mTG]SVUjeYAm[^ŽY[~D[c†Š UwDAG *&u&ORa7Š*cy]WšTSk^`SkUWD[G.Œ”cy‹h‹Žc†Š ^hY[~5•vHJceUw^`n%G!ŸA]SVULUwDAjeU Š*G&FjBQŠ ]w^ŽUwG dν ^hY¨»s e¬&¨»#s 0¬'jyS i ,¨ sy¬ dνi = gi (t)dt + dJi , Š D[GX]wG jeYAm ^`S.jeY#jeUwcyF^`nFGXj0SlZA]wGŠ ^ŽUwD n%c0Z[YUWjexA‹ŽGjeYAmc0]Wm[G]jex[‹hGSlGU.cyŒJj†UWcyFS + ~0GY[GX]WjeUwG2mgxRiQ∈UwDAFG∞]w^h~y(IR DUkn%;c0IR) Y0UW^ŽYRZ[c0ZAS¢dJdVZ[iFgŒ”Z[YAn%Uw^hcyY J •bC'D[G&Œ”cy‹h‹Žc†Š ^hY[~.D[cy‹`m[SX• i „ ¶  ¶ ‚ X¥ƒ 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'DRZASX“XUWD[G ncyFg[‹hGFGYUje]W^–UVQv^ŽY¨”|¬ ¨¥n2¬+^`S ~0^Ž‘0GYjJF.G2jeY[^hY[~kŠ ^ŽUwD¨,0{y¬ jyYAm¨.&†¬%•¢C'D[G*n%c0Y[G ¸ at^hY—¨.&†¬SlDAc†ŠkS Š DAjeU ¸ a UwD[G&FGXj0SlZA]wG2S dν SwjeUw^`SVŒ”Q¨”UwD[G&]W^Ž~0cy]WcyZAS˜FGXjeYA^ŽY[~.ceŒ UwD[G&^hYAn%‹hZASw^Žc0Y—^hY—UWD[G&]w^h~yDUl’?DAjyYAmR’°Sl^`mTGkcyŒ*¨,†¬ ^hS G%œTg[‹`je^hY[G2m^ŽYt©Žªe“Rab]Wcyg<c0Sw^ŽUw^hcyiY ª%«:cy]L© sR“g)• e{†«”¬¦• ‰ U ^`S ^ŽFg<cy]wUWjeYUbUWcY[cyUw^`n%GJUwDAjeU ~y^h‘yGXY {z }(t−), ..., {z }(t−) “ i−1 UwDAGºFGXj0SlZA]wGº‘Bjy‹ŽZAG½mTGXg5GXYAm[ScyY {z }(t+) • -ºGSwDAje‹h‹vncyFG½xAjynš cyY ¨,&y¬^hY7SwGXn%Uw^hcy1Y7|A•ÉsR• []WcyF ¨.!y¬—^–U i Œ”cy‹h‹hc†ŠkS¢UwDAjeUbUwD[GJY[cyYT’°j†UWcyF^`n g5je]wUWS4cyŒ5UWD[G  jy~y]jeY[~0G F!Z[‹ŽUw^hg[‹Ž^hG]Sbjy]w

(59) G !XG]Wc&jy‹ŽFcSVUbGX‘yG]WQRŠ D[G]WG*G%œ[n%GXgTUbŒ”cy] g (·) • ¸ cyY5SlG2_0ZAGYUw‹hQvUwD[

(60) G  jy~y]jeY[~0GbFZA‹–UW^Žg[‹h^hG]S¢j0SwSwcTn%^`j†UWGXm&UWcvj0m[mT^ŽUw^hcyYAjy‹RncyYASlUw]je^hYUWS Uwcvx5G'^hFg5cSlG2mcyYUwD[G'g[]WcyxA‹ŽrGXF“ je]WGg[Z[]WG‹hQtj†UWcyF^`ne•C'D[^`SFjBQG%œTg[‹`je^hYºŠ DRQ UWD[GQºDAjB‘yG.x<GGY ^h~yY[c0]wG2mt^hY½UwD[G‹h^ŽUwG]j†UWZ[]WGSw^hYAn%G.UwDAGn%cyY5mT^–UW^Žc0Y UwD5j†U λ x<Gj.F.G2jySwZ[]WGvD5jyS je‹hŠ'jBQRS x<GGY^hFg5cSlG2m+• „    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]*^ŽFg<cy]wUWjeYU*UwcRcy‹:^`S. ^`S ^`mTGYUW^hnXje‹h‹ŽQ X! G]WcA§ z (t). . UwD[GvŒ”cy‹h‹hc†Š ^ŽY[~$ ¸ a Š D[^hnDD[cy‹`m[S*cyY—Uw^hFGL^hYUwG]W‘†je‹`S. 1. “ [τ − , τ ). 1. >0. “RcyY—Š 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 Š ]w^ŽUwGºGX_Z[^h‘Bjy‹ŽGXYUw‹hQ g (t) ^hYASlUwG2jym cyŒ λ(t) •›C'D[^`S ¸ aFcyY[^ŽUwc0]WSUwD[G½G‘0cy‹hZTUw^hcyY cyŒUwD[G FZA‹–UW^Žg[‹h^hG] λ(·) Œ”cy] t ∈ [τ − , τ ) • C'D[GFj†rUW]w^ŽœcyŒ UwD[^`S ¸ a ^hS'UWD[G‹hGXj0mT^ŽYA~Kje]Wšyc†‘g5je]jeFG%UWG] M (2r ) • -ºG UwDAG]WG%Œ”cy]WGDAjB‘yGjeUcyZ[]m[^hSwg5cSwjy‹˜j n%c0Fg[‹ŽGUwGmTQRYAjyF^hnXje‹˜SwQTSVUWGF Š DA^hnDje‹h‹hc†ŠkS&ZASUWc ~y^h‘yGjFGXjeYA^ŽY[~UWcUwD[G SwQRSlUwGXFž^hY#¨¥|¬¦“<Œ”c0]LjeYRQ^ŽYA^–UW^hjy‹¢m[jeUWjjyYAmŒ”cy]vje‹h‹ t ≥ 0 •JC'D[^`SL^hSLjŸA]SVUvn%]WZAn%^`je‹¢SVUWGg Œ”c0]kUwDAGZ[YAm[G]SVUjeYAmT^hY[~cyŒ UwDAGY[GXnGXSWSwjy]wQncyYAmT^ŽUw^hcyY5S'ceŒc0gTUw^hFje‹h^–UVQ0•  ¶ „ ¶  » T„ ³   / - T 9!  ,  1R(JPR@ R@   D' d H({x}(t), {η}(t)) = 0    @;  %  " dt „ ¶)¶ ˆ Œ”cy‹h‹Žc†ŠkSŒ”]wc0FG2jySwQ½nXje‹`n%Z[‹`j†UW^Žc0YASjeY5m½UwD[G—Œ¥jyn¦U!UwDAjeUUwDAGSwZ[g[g<cy]wUceŒ UWD[G—FGXj0SlZA]wG2S dν ^`S!j !XG]Wc i )GXx5G2Sl~0Z[G&FGXjySwZ[]WGLc0Y • [0, T1 ] *kc†Š*G‘0G]*je‹hcyY[~UwD[G&cyg[Uw^hFjy‹<Uw]j†dVGXn%Uwc0]wQ.UwDAG *LjeF^Ž‹ŽUwc0Y[^`jeY—ZAYAmTG]W~ycRG2S dVZAF.g5S j†U j†Uwc0FS cyŒ ¨¥SwGG ¸ cy]Wcy‹h‹`je]WQ λ 0¬%• wu. À À Ä#0[Í112.

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