Higher order Moreau's sweeping process: Mathematical formulation and numerical simulation

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Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation

Vincent Acary, Bernard Brogliato, Daniel Goeleven

To cite this version:

Vincent Acary, Bernard Brogliato, Daniel Goeleven. Higher order Moreau’s sweeping process: Math- ematical formulation and numerical simulation. [Research Report] RR-5236, INRIA. 2004, pp.60.

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a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Higher order Moreau’s sweeping process:

Mathematical formulation and numerical simulation

Vincent Acary — Bernard Brogliato — Daniel Goeleven

N° 5236 – version 2

version initiale June 2004 – version révisée Mai 2005

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˙

x(t) =Ax(t) +Bλ(t)§]y@luafnosozaNnuy{sotaNl/]×{a¦noy&a4{y@~@a,t~pZ&my@`7{t~pyªnu]aª®y{so`

Cx(t)≥0^¯

\]a!_kApV`7tNl7tla4`½IaNmaN8t~pAnoy³¾_tlnusot~½qmnuty{p{.mt~¼IaNsuaNp@not{t~pVqlty{p

,

pV`7aN8nu]aZ]_t~@]_a4s

y@so_a4sl§/a4aN}_tp_}suymaWlul

/

§]_tw]¾{~y¢§.l¦ql¦nuyZtpAnua4@sonua¨nu]_alukmlnoa4` §]t~a7suaWl}IaNnutp_nu]_a&q_pt¸·

w¢noa4sM4y{plnustpAn,y@p¾nu]_aElnonua@¯ ¥ pnu]a&}_suy@}VyAlaW¿ª®y{so`7{~twlu` nu]_a@so{p_{a`q¸not~}_ta4s

λ ^tl

_tlnusot~½qmnuty{p§]_t]}VyAluluaNloluaNl«lu}VaWt=8Vf_aNy@`7}VyAlt~nuty{ptpAnuy`7aN{luq_soaNl'§]_tw]lo¢nutwlª®ktp4~qluty{pl

tpAnuy&ªô`7t~kZy{ª«y@pA@a©Zy@p_aNlN¯¦Ànut`7a·ºlnoa4}_}t~p_pGq_`7a4sotN${y{sot~nu]_`tl,4y{plnusoqÁnoaNZ{pt~nol

}suy@}VaNsnot~aWl.soap{~kG»NaNC¯\]_y@q_{]!no]_a¨4y{pG{aNsu@a4p4adnoy¢§/{so_ldly@~qmnot~y@ply{ªno]_a¨4y{pAnutpGq_y{qVl·=nut`7a

lukmlnoa4`tldp_yn¦k{a4nfy@`}~a4nua{´Vnu]a¨pVkGlutwl.l]y¢§.lnu]V¢ndno]_apGq_`7aNsutw4{lu]a4`7a}Iy@lolaWluluaNl.lnusoy{p

}suy@}VaNsnot~aWl4¯

\]_a}{}VaNs2tly@suAp_twlaW{l(ª®y@~y¢§.l4¯$iGy{`7a`&¢no]_a4`&¢notNAnuyGy@l2{sua'}_suaWlaNpAnuaN¨tpluaNnuty{p_¯À

lu}IaNtwI4{p_y{p_tw4{lno¢noadlu}{4asuaN}_suaWlaNpAno¢not~y@p7§]_t]&twlMqVla4ª®q_ª®y@s2nu]afluq_½luaNvAq_a4pAn'_a4{aN~y@}_`7a4pAnol

twltpAnusoyG_qaWtpluaNÁnot~y@pZm¯/\]ay{sosoaNlu}Vy@pmtp__t¸¼Ra4soa4pAnutwt~p4~qVlty{pª®y{so`&tlu`þtwltpAnusoyG_qaW

{p`7y{nut×nuaWtpluaNnuty{pE´m§]_a4soadt`7}Vy@snpAn/}suy@}VaNsnot~aWl/{p§/a4~·=}Iy@luaNmpaNlol«soa,l]_y¢§p¯$iGaWÁ·

not~y@p³!tlma4@ynuaW¿nuy!nu]aEmaNlut{pËp¾no]_aE{pkmltwl,yªnut`7a·ºlnoa4}_}_tp_pGq_`7aNsutw4{«lu]a4`7a{¯ ¥p

(8)

luaNnuty{p!pË}_}~tw4nuty{p¾yª/nu]_a&a4©Anoa4p_aNl§/a4aN}_tp_}suymaWlul,twl}_soaNlua4pAnoaNC¯ É y{pVqlty{pVly@lua

no]_a}{}VaNst~pZluaNÁnot~y@pZG¯

ZÇ N¡_Ç ¤ \]a,ª®y{~y¢§tp_&p_y{no¢not~y@p!tlqVlaW

IRtwlnu]asuaWCt~pa{´

IR⁺ twlnu]_ap_y{p_pa4@nut{a,soaN

tp_a{´

x_i ^twl.no]_a ino]±y@`}Iy{pa4pAnfy{ª«&{aWÁnuy@s

x∈IRⁿ¯,\]_asoa4w¢not~@amaN{soa4a½Ian§/a4aNpZn§'ylt{p{l

w ^pV λ ma8p_tp_¾nu]_a!y{q_nu}_qmn{pËno]_a!t~p_}qmn&yª¦¾lukGlnuaN`!´Mtl&_a4p_y{nuaN8{l

r¯8\]_a!tpmtw4¢noy{s ª®qpÁnot~y@p¶yª¦±luan

K twl¨maNp_ynoaN8{l

ψ_K(·)^´«p ∂ twl¨nu]_a!y@pG{a©ËpVkGlutwlluq_½I_t¸¼Ra4soa4pAnutw¢not~y@p¯

\]a4~yAlaW³y@pA@a©Ë]Gq_/y{ª¦¿la4n

K tl7maNp_ynoaN¶½Ak

co(K)^¯ ^¹]_aNp K tl7¿p_y{pa4`7}mnk³y@luaN

4y{pG{a4©8luan&nu]a4pµnu]ap_y{so`&.4y{p_a!noy

K tlEma4p_y{nuaW @l

N_K(·) = ∂ψ_K(·)^¯ a©mtwy@{s}_]_tw4{

tp_aWv@qVt¸not~aWlsoamaNp_ynoaN¾{l

^¯ P_y{s{aWÁnoy{s

x^´ x 0`aWplfno]¢n{~2aNp@nosutaNl

xi = 0^y@s,nu]_a

8Vsoln«p_y@p_»4aNsuyaNp@nosuktwl

>0^¯ x≥0`7aN{pl2no]¢naNp@nosutaNl

xi ≥0^¯ In

ma4pynuaWl2no]_a

n×nt_a4pAnut~nk

`&nusot¸©C´

0ⁿ _a4p_y{nuaNlnu]_a@aNnuy{s

(0,0, . . . ,0)^T ∈ IRⁿ ^{p 0n = (0ⁿ)^T^¯a4n M ∈IR^n×n ^½Va

lukG`7`a4nusot¦pVE}Iy@lut¸not~@a¦_a8p_t~nua`&nusot¸©C´Anu]_a}suy×©mt`7nuty{py@}VaNsonuy@s

proxM[K;.]tlma8p_aWE½Gk

proxM[K;x] = argmin_z∈K(z−x)^TM(z−x)^¯ ^¥^ª M=In

´I§'aluan

prox[K;.] :=proxIn[K;.]^¯

\]a¦pynonuty{pl

hx, yi:=x^Ty ^pV kxk:=√

x^Tx §t~{luy7½Va,qluaNC¯

7£{ ÇRÄ ³ \]aª®y{y¢§t~p_ {4suy@pAkG`&l{sua¿qlaWµnu]suy@q_{]_y@qmnEno]_a¾}}Ia4s É y{`7}_a4`7a4pAnsot¸nk

x«suy@½_~aN`

,É x / ´ tp_aN{s É y{`7}_a4`7aNp@nsot¸nk±xMsuy@½_a4`

, É x /

´ÍM{sutw¢not~y@p ¥paNvAqt~nk

,Í ¥/ ´ Xfs·

_t~p{sukZÐft¸¼Ra4soa4pAnot{(Ï«vAqnuty{p

,

X¦ÐfÏ

/

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,

Ð.ÀdÏ

/

´eZaW{luq_soaÐdt~ªÌ·

ª®aNsuaNpAnutw ¥ p4~qluty{p

,

eÐ ¥/ ´ É y{`7}_a4`7a4pAno{sut~nkiGkmlnuaN`

,É i / ´ t~p_aWs É y{`7}_a4`7aNp@nsot¸nkiGkmlnoa4`

, É i / ¯

,+(.-#,+(%/0&04

¥

pËno]_twl¨luaNnuty{pË§/a}_soaNlua4pAnª®q_p`7a4pAno{'{pkmltwl,nuyGy@l§]_t]Ë§t'½IaE]_aN~}_ª®q_'tp¶luanunut~p_

no]_a]_t{]_aNsy{sma4slu§'aNa4}_tp_7}_soymaNlol'ª®y{so`&tlu`!¯

¥

p}{snot4q_ws.7w{lolyª$mtlnusot½_qmnuty{pVlnu]n.tl

}Iynoa4pAnutw4@luly{ªluy{qmnuty{pVlyªnu]_amkGp{`7tNltl}_soaNlua4pAnoaNt~pZma4not=¯

¡_È Ç Ã ¡_( Ã a4n/qVl/_a4p_y{nua¦½Ak

B(IR)nu]a¦|/y{soa4

σ {aN½_spE~a4n

dR^½Va, IRⁿ−¢q_aW

d{my@p±`7aN@lq_soa{´tT¯a{¯7!|'y@suaN$suaN{q_ws¦`7aN{luq_soaluq]¿nu]n

dRi(K) <+∞ª®y@sa4@a4soky{`7}@Án luan K⊂IR ^,^lua4aa{¯^¯ ²^@ ^3$/ ^¯ ^a4n A∈ B(IR)½Ia{t{aNp´m§/a,lo×k&no]¢n

{A_i}^mi=1

twl.W8p_t~nua}{snot¸not~y@p

y{ª A^t~ª A_i∈ B(IR)^´ A_i∩A_j =∅, i6=j ^p ∪^mi=1A_i=A^¯ an.qlpy¢§*maNp_ynoa,½Ak

P(A)^no]_aluan

y{ª 8pt¸noa}sunut~nuty{ply{ª

A¯M\]_a`7ymmq_ql`7aN{luq_soa,yª

dR twlma8Vp_aN½Gk

,

laNa¦a@¯V¯

2

A

37/

|dR|(A) = sup

{Ai}^m_i=1∈P(A) m

X

i=1

kdR(Ai)k,∀A∈ B(IR).

an

dµ ½VaZ¿suaW~·=¢q_aW³.{_y{p³`7aN@lqsua@¯ an&ql7maNp_ynoa½Gk

L¹_loc(IR, dµ; IRⁿ) nu]alu}{4ay{ª

dµ·=ym4{~ktp@noa4@so{½_~a

IRⁿ·=¢{~q_aWª®qpÁnot~y@pl4¯QXfp_a&lu×kml¦nu]V¢n

dR ]{l!maNplut¸nksoa4w¢not~@anuy

dµ

(9)

}suy¢GtwmaN¨nu]V¢n«no]_a4soada4©mtlnol'

,

4@lul«yª

/

ª®q_pnuty{p

R⁰_µ∈L¹_loc(IR, dµ; IRⁿ)luq]7nu]V¢n

dR=R⁰_µdµ^´

t=¯a{¯

dR(A) = Z

A

R⁰_µdµ,∀A∈ B(IR).

¥ª

dµtwlCp_y@p_p_a4A¢not~@a${p

dRtwl½luy{qmnoa4kfy@pAnutpAqy{qlR§t~nu]soaNlu}VaWÁnCnuy

dµ^´×tT¯^a{¯ A∈ B(IR), dµ(A) = 0⇒ dR(A) = 0´Rnu]a4pEw{lolutN$mt~soaNn,4y{pluaNvAq_aNpa¨yªMno]_a aN½VaWl@q_a·º.{_y{pm·ºb.t²{ymmkG`nu]_a4·

y@suaN`

,

lua4aZ\]_aNy{soa4` _¯

-

_¯Ý{Ztp

2

A

3$/

aNpluq_suaWlnu]_a!a4©Gtwlnua4pVa!yª,¿qp_tvAq_a

,

w{lolyª

/

ª®q_pVÁnuty{p

R⁰_µ∈L¹_loc(IR, dµ; IRⁿ) lqV]Eno]¢n

dR=R⁰_µdµ^¯«Ê.aNsua R⁰_µ(t) =dR

dµ(t),

§]a4soa

dR dµ

maNp_ynoaNl$no]_a.ma4sot×nut{a

,

ma4plut~nk

/

y{ª

dR§t~nu]&soaNlu}VaWÁn$nuy

dµ^¯ ^¥p}Vsunutwq_wsW´{lut~p4a

|dR|

twl2p_y@p_p_aN@¢not~@ap

kdR(A)k ≤ |dR|(A),∀A∈ B(IR)´@nu]_aNsuaa4©mtlnolM,qp_tvAq_a

,

w{lol$yª

/

ª®q_pnuty{p

θ_R∈L¹_loc(IR,|dR|; IRⁿ) luq]Eno]¢n

dR=θ_R|dR|^¯

Ã Ã T¡ Ã ¡_( Ã Ç £ Ã N an

I ½IasuaW$p_y{p_·UmaN{aNp_a4s¢noa¨tpAnua4so¢

,

pyna4`7}mnk¿p_y{s

soaN_qaWnoy7lut~p_@~a4nuy@p

/

´_{pEan

{K(t);t∈I}½Va¨ªô`7t~ky{ªpy{p_aN`}_nky@luaNy{pG@a©Ey@p_aNlN¯

imq_}_}Iy@luanu]V¢n

dµ tl¨p_y@p_p_a4A¢not~@ap

dR tl½luy{qmnoa4~ky@pAnutpAqy{ql§t¸no]³suaWl}IaNnnoy

dµ^¯¾|/k

4y{pG{aNpAnuty{p´G§/a,lu]{~C§sot¸noa

dR=R⁰_µdµ∈K(t) on I ^, ^/

noy7`aWpnu]V¢n

R⁰_µ(t)∈K(t), dµ−a.e. t∈I. ^, ^/

Ç «Ç

ô

Ç >$ <=>?I# $# <= >$# 5@ #&#" ">O#"A

A ∈ B(IR) A⊂I

dR(A)∈co(∪^τ∈AK(τ)). ^, ^/

ÇÇ ¤ iGq}_}VyAlanu]n

, /

twl¦lunutwl.8aWC¯ ¥ª

A∈ B(IR)^´ A⊂I ^p dµ(A) = 0^nu]a4p dR(A) = 0

{pt~n¨twl¨4~aWsno]¢n

( ) ]_y{w_l¨ltpa

0 ∈ co(∪^τ∈AK(τ))^¯ ^an A ∈ B(IR)^´ A ⊂ I lq]³nu]n

0< dµ(A)<+∞^¯2\]_a4p R_µ⁰(A)⊂co(∪^τ∈AK(τ)){pEnu]Gql

,

laNa,\]a4y{soa4`m¯üG¯@t~p

2

@

37/

1 dµ(A)

Z

A

R⁰_µdµ∈co(∪^τ∈AK(τ))

lutpa

co(∪^τ∈AK(τ))twl2y@luaN¨py@pA@a©C¯ ¥nMsoaNluq_~nol(nu]n

dR(A) =R

AR⁰_µdµ∈co(∪^τ∈AK(τ))

lutpa

dµ(A)>0p]_a4soa

(10)

£Ç Ç Ç È Ã È _¡m ô¡mÇ a4n

I ma4p_y{nua.¦p_y@pm·U_a4{aNp_a4s¢noa«soaN{mt~pAnuaNsu¢{

,

p_y{nMa4`7}mnk

py{s.soaNmqVaNnoyltp_@~a4nuy{p

/

¯'|/k

u∈BV(I; IRⁿ)t~ndtwl.`7aN{pAnnu]n

C > 0 ^lq] no]¢nª®y@sE98pt¸noa±laWvAq_a4p4aNl

t0< t1< ... < tN ^,N so½_t~nussok

/

yª}Vy@t~pAnlyª

I´_§/a,]×{a

N

X

i=1

ku(ti)−u(ti−1)k ≤C.

an

J ½Iaf¨lq½_t~pAnoa4so×{y{ª

I¯2\]_afsuaWIpAq`½Ia4s

var(u, J) := supPN

i=1ku(ti)−u(ti−1)k^´G§]_aNsua

no]_a¿luq_}_soa4`¨q_` tlno²@a4p¹§t~nu] soaNlu}IaNÁnnuy¶{~dnu]_a8p_t~nualaWvAq_a4p4aNl

t0 < t1 < ... < tN ^,N

{su½t¸noso{suk

/

yª(}Iy{tpAnoly{ª

J´_twl4{~aNEnu]_a¢{sutw¢not~y@py{ª

u^tp J^¯

ÀdpGk |'Íª®q_pnuty{p*]{l! y@q_pAno½~ala4ny{ª¨mtwlu4y{pAnutpGq_t~nk¹}Vy@t~pAnol!{p twl!`yAln!a4@a4sokG§]_a4soa

_t¸¼Ra4soa4pAnutw½~a@¯À |/Í ª®q_pÁnot~y@pËma8paN¿y{p

[a, b]⊂I }Iy@lolaWluluaNlf~a4ªÌn·U~t`7t¸nltp

(a, b] p¿sot~@]@nu·

t`t~noltp

[a, b)¯ËeZy@suaNy¢{aNsN´(nu]_aª®q_pnuty{pl

t 7→ u(t⁺) := lims→t,s>tu(s) ^pV t 7→ u(t⁻) :=

lims→t,s<tu(s) soa¦½Iyno]Z|'Í°ª®qpÁnot~y@pl4¯

¾ama4p_y{nua¿½Ak

LBV(I; IRⁿ) no]_al}@a¿yªª®q_pVÁnuty{pVly{ª~ymN~kµ½Iy{q_p_aN ¢{sutw¢not~y@p´.t=¯a@¯ y{ª

½Iy{qpmaN¢sotw¢nuty{py{pa4@a4sokEy@`}V{Án.luq_½_tpAnua4so¢Cy{ª

I^¯

¾amaNp_ynoa½Gk

RCLBV(I; IRⁿ) no]_alu}{4ay{ª'sot~@]An·ºy{pAnot~pGq_y@qlfª®q_pÁnot~y@pl,yªyGN~k±½Vy@q_pmaW

¢{sutw¢not~y@p¯ ¥ ndtwl.²Gp_y¢§pno]¢ndt~ª

u∈RCLBV(I; IRⁿ) ^p [a, b] maNp_ynoaNl.&4y{`7}@Ánfluq_½_tpAnua4so¢

y{ª

I^´mnu]a4p u(t) 4p½IasuaN}_suaWlaNpAnuaNEt~pno]_a,ª®y{so`

,

lua4aa@¯V¯

2

@

37/

u(t) =J^u(t) + [u](t) +ζu(t),∀t∈[a, b],

§]a4soa

J^u twlzq_`7}³ª®q_pVÁnuty{p´

[u] ^tlp ½luy{qmnoa4k¾4y{pAnutpGq_y{qlª®q_pnuty{p p

ζ_u tl¿lutp_{qm·

wsEª®q_pnuty{p¯Ê.aNsua

J^u tlZzq`} ª®q_pnuty{p tp¹no]_a¾lua4pVlaZno]¢n

J^u twlEsot~@]An·ºy{pAnot~pGq_y@qlpV

@t~@a4p!pGk

t1, ..., t_N ^yª Ju

lqV]no]¢n

PN

i=1kJu(ti)− Ju(t⁻_i )k+ε > var(Ju,[a, b])^´ [u]tl/p½luy{qmnoa4k4y{pAnutpGq_y{ql«ª®q_pnuty{pt~pEnu]_a lua4pVladnu]n'ª®y@s'aN{a4sok

δ >0luq]&no]¢n

PN

i=1k[u](βi)−[u](αi)k< ε´@ª®y@spGk 4y{~aWÁnot~y@pZyª«mtwl®zy@t~pAnfluq_½_tpAnuaNsu¢wl

(αi, βi]⊂[a, b](1 ≤i≤N) lqV]Znu]n

PN

i=1(βi−αi)< δ^´

{p

ζu

twl¨lutp_{q_wsª®q_pnuty{p³tp³no]_alua4plua&no]¢n

ζu

tly@pAnutpAqy{ql¨pË½Iy{qpmaN³×{sutw¢not~y@p

ª®qpÁnot~y@py@p

[a, b] luq]nu]n

ζ˙u= 0 `yAlnaN{a4sokG§]_a4soa¦y{p

[a, b]^¯

|/k

u∈RCSLBV(I; IRⁿ)t~ntl`7aN{pAn«nu]V¢n

utl¨sut{]An·ºy@pAnutpAqy{qlMª®q_pÁnot~y@py{ª(lu}VaWtwR~ymN~k

½Iy{qpmaN!¢{sutw¢not~y@p´_t=¯a@¯

utldyª2½Vy@q_pmaW!¢sotnuty{pZpZNp!½Ia§sut~nnoa4p@lnu]alq`y{ª2¦zq`}

ª®qpÁnot~y@p³pË{p{½luy{qmnuaN~k¾y{pAnot~pGq_y@ql,ª®q_pnuty{pËy@p¾aN{aNsuk¾y{`7}@Ánluq_½_tpAnua4so¢My{ª

I^¯ZiGy´t~ª u∈RCSLBV(I; IRⁿ)^nu]_aNp

u= [u] +Ju ^,

/

§]a4soa

[u] tl¦ym4k!{½luy{qmnuaN~k!y@p@not~pGq_y@qlª®qpÁnot~y@p¿4{~aNZnu]_a½luy{qmnuaN~kZ4y{pAnutpGq_y{qlfy@`¨·

}Iy{pa4pAnyª

u^p Ju

twlq_p_twvAq_a4k_a8p_aWq_}nuy&7y{pVlnpAn½Ak

Ju(t) = X

t≥tn

u(t⁺_n)−u(t⁻_n) = X

t≥tn

u(tn)−u(t⁻_n) ^, ^/

(11)

§]a4soa

t1< t2 < ... < t_n < ... maNp_ynoanu]a¨y@q_pAno{½_~k`&pGk}Vy@t~pAnolfyª«_tloy@p@not~pGq_t~nkyª

u ^t~p I^¯

@t~@a4pt~p

, / ¯

Ã ® Ã Ã ¡( Ã a4n

u∈LBV(I; IRⁿ)½Ia¦@t~@a4p¯ ¾a,_a4p_y{nua,½Gk

du nu]aiGnuta4~nôzaNl`7aW{luq_sua

@a4p_aNsonuaW½Gk

u ^,^laNa¦a@¯^V¯ ²^A ³ ^pV ²^A ^3$/ ¯MaNNRnu]nª®y{s

a≤b^´ a, b∈I du([a, b]) =u(b⁺)−u(a⁻),

du([a, b)) =u(b⁻)−u(a⁻), du((a, b]) =u(b⁺)−u(a⁺), du((a, b)) =u(b⁻)−u(a⁺).

¥ p!}{snot4q_wsW´A§/a,]×{a

du({a}) =u(a⁺)−u(a⁻).

Py{s

u∈LBV(I; IRⁿ)^´ u⁺ ^p u⁻ maNp_ynoa¦nu]_a,ª®q_pVÁnuty{pVl.ma8Vp_aN½Gk

u⁺(t) =u(t⁺) = lim

s→t,s>tu(s)),∀t∈I,

{p

u⁻(t) =u(t⁻) = lim

s→t,s<tu(s)),∀t∈I.

¥ª

u, v∈LBV(I; IRⁿ)^nu]_aNp u^Tv∈LBV(I; IR) ^pV

d(u^Tv) = (v⁻)^Tdu+ (u⁺)^Tdv= (v⁺)^Tdu+ (u⁻)^Tdv. ^, ^/

anql.{luysuaW4Rnu]V¢n

2(u⁻)^Tdu≤d(u^Tu) = (u⁺+u⁻)^Tdu≤2(u⁺)^Tdu. ^, ^/

Ã ¡( Ã È Ã Ã ô¡ £Ç \]_a±`&¢noa4sot{d@t~@a4p tp¹nu]tlZiGaWÁnuty{p 4p¹½Ia¾qlaWµnuy

F : IR×IRⁿ → IRⁿ ^½Va C¹^·

`&{}_}_tp_³pµan

_t¸¼Ra4soa4pAnutwCtp4~qluty{p Ptp

u∈RCLBV(IR; IRⁿ)luq]Eno]¢n

du+F(t, u(t))dt∈ −NC(u(t)). ^,?0A/

\]a¦lua4pluady{ª

,I0 /

dµlqV]nu]V¢n/nu]_a

`7aW{luq_suaWl

du ^pV dt soa¦{½ly@~q_nua4kE4y{pAnutpGq_y{ql/§t~nu]!soaNlu}VaWÁnnuy

dµ^{p

du

dµ(t) +F(t, u(t))dt

dµ(t)∈ −N_C(u(t)), dµ−a.e. t∈IR. ^,K- ^/

(12)

0

bdynoanu]ndno]_a¨4y{p4a4}mndy{ª«ly@~q_nuty{pmyGaNldp_yn¦ma4}Ia4pyª2nu]a¨]_y@t4ayª2nu]ap_y{pp_a4A¢nut{a¨.{my@p

`7aW{luq_sua

dµ lut~pVanu]_aEsut{]An·U]{p±lut_a&yª ,.-

/

twlZy{pa{´({p±nu]_aE_a4plut¸not~aWl4{p¿½Ia&y{½_notp_aW

y@p_a.ª®soy{`y{nu]_aNs'½Gk7`q¸not~}_tw4¢not~y@pE§t~nu]!p_y@p_p_aN@¢not~@a.ª®q_pVÁnuty{p¯ ¥nsoaNluq_~nol«ª®suy@` x«suy@}VyAlt~nuty{p

-

no]¢n

du(A) + Z

A

F(τ, u(τ))dτ ∈co([

τ∈A

−NC(u(τ))), ^,.-A-/

ª®y@sa4@a4sok

A∈ B(IR) luq]nu]n

dµ(A)<+∞^¯

anqlpy¢§ soaN4{~Cluy{`7a,t~`7}Iy{suno{p@nd4y{pluaNvAq_aNpaWl/yªluq]Z`&¢nu]a4`&¢notNC`7ymma4=¯ ¿a]V×{a

u(t⁺)−u(t⁻)∈ −NC(u(t)),∀t∈IR ^,.- ^/

{p

u⁰_t+F(t, u(t))∈ −NC(u(t)), a.e. t∈IR.

ÀNy@somtp_&noyno]_ap_y{nonuty{plft~pAnosuymmq4aN{½Vy¢@a{´

u⁰_t maNp_ynoaNldnu]_a7ma4pVlt~nk!yª

du §t~nu]±soaNlu}IaNÁnfnuy no]_a aN½VaWl@q_a,`7aN{luq_soa

dt^¯

ôN4 ô(Ç Ã Ã ×¡_ Ã ÈË Ä £{Ç an

I ½Ia.no]_ady@}VaNpsuaWVt~pAnuaNsu¢{{t{aNp

½Gk

I =]α, β[

§]a4soa

α∈IR^{p β∈IR∪ {+∞}^¯ ¾aluan.wly

I˜= [α, β[.

Êda4soa

C₀^∞(I) _a4p_y{nuaNl¦no]_alu}{4a¨y{ª'soaN~·U×{~qaN

C^∞(I)·U`7{}_}_tp_@l,§t~nu]y@`7}{nluq_}_}Iy{sunp

D⁰(I)tlnu]a/lu}{4a«y{ªVim]G§/{sno»«mtwlnosut½_qmnot~y@ply@p

I¯(aW4{~@nu]n(ª®y@s

T ∈ D⁰(I)^´¢no]_a ^,{aNp_a4st~»NaN

/

_a4sot~¢¢not~@afy{ª

T twlma8paN½Gk

hDT, ϕi=−hT,ϕ˙i,∀ϕ∈C₀^∞(I).

\]a

,

{aNp_a4st~»NaN

/

maNsut¢¢not~@a¦y{ª(y@somaNs

ntwl/nu]_aNp!{t{a4p½Gk

DⁿT =D(Dⁿ⁻¹T) (n≥2),

no]¢n.twl

hDⁿT, ϕi= (−1)ⁿhT, ϕ⁽ⁿ⁾i,∀ϕ∈C₀^∞(I).

Py{s

a∈I´_§'amaNp_ynoa,½Ak

δa

nu]_aÐftso@¦_tlnusot~½qmnuty{pZ¢n

a´ma8p_aN½Gk

hδ_a, ϕi=ϕ(a),∀ϕ∈C₀^∞(I).

bdynoafno]¢n

δ_a=DH(.−a)^§]_aNsua Htlnu]_aÊdaN×Gtlutwmadª®qpÁnot~y@p

H(t) =

1 if t≥0 0 if t <0 .