trình bày về ứng dụng bất đẳng thức bất phân
Trang 1Chu'dng 3
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(J'ng d t.lng
Trang 2Nam 1979, M.J Smith [7] d§ ra mQt mo hlnh m6i cho bEd toan can bAng m1;tng giao thong dva tren nguyen ly can bAng Wardrop (J Wardrop [13]) va chung minh du<;Jcdi§u ki~n t6n t1;tican bAng cua m1;tnggiap thong tu0ng dli<Jngv6i di§u ki~n c6 nghi~m cua bai toan bilt dilng thuc bit~n phan.
San d6, nhi§u tac gia khac da ti~p t\lC phat tri~n mo hlnh va cac k~t qua v§ t6n t1;tinghi~m cua bai toan can bAng giao thong, chilng h1;tnG.Y Chen va N.D Yen [11, 12], P Daniel, A Maugeri va W Oettli [2], M De Luca [22], M Florian va N Hadjisawas [18], S Schaible [20], X.Q Yang va C.J Goh [8], B Ricceri [21], Q.H Anssari [28, 19], P.Q Khanh va L.M Luu [3, 4, 5], F Giannessi [26, 27], Tu0ng l1ng v6i moi d1;tngcua bai toan bilt dilng thuc bi~n phan, ta c6 cac mo hlnh can b~ng m1;tnggiao thong tu0ng ling, chilng h1;tn: can bAng m1;tnggiao thong da m\lC tieu [8], din bAng m1;tnggiao thong ph\! thuQc thbi gian [2], can bAng m1;tnggiao thong da tri [3, 4, 5],
Trong m\lC nay chung toi trlnh bay nguyen 15'can bAng Wardrop, cac mo hlnh
toan hQc cua bai toan can bang m1;tnggiao thong va cac dieu ki~n de mo hlnh m1;tngd1;ttcan bAng.
Nguyen ly can bAng Wardrop [13] (1952) du<;Jcphat bi~u nhu san : "MQt m1;tng giao thong du<;JcgQi la d1;ttcan bAng Wardrop n~u mQi ngubi d§u Iva chQn hanh trlnh c6 chi phi thilp h0n".
De ap d\lng nguyen ly nay mo hlnh toan hQc cua bai toan can bAng m1;tnggiao thong du<;Jcdua ra nhu san
3.1 Bai toan m~ng giao thong
?
Gia sv c6 mQt m1;tnggiao thong, trong d6 ngubi ta phai v~n chuyen hang h6a tv diem A, gQila di~m b~t d§.u (origin), d~n di~m B, gQila di~m cu6i (destination), theo mQt nhu c§.u hang hoa nao d6 C~p di~m A- B du<;JcgQi la c~p d§.u-cu6i Moi mQt dubng di tv A d~n B gQila mOt hanh trlnh n6i c~p A-B Ta ki hi~u W 1at~p h<;Jptilt ca cac c~p d§.u cu6i trong m<;Lngva P la t~p h<;Jp tilt ca cac hanh
trlnh trong m(;Lng.Gia sv W va P co hUll h(;Lnph&n tv Vdi moi c(Lpd&u cu6i
w E W, ta ki hi~u P w la t~p h<;Jptilt ca cac hanh trlnh n6i c~p d§.u cu6i w E W.
Ki hi~u dw la nhu Cali can v~n chuyen tv diem dati Aw den diem cuoi Bw Ta ki hi~u d la vect0 nhu ctiu trong d6 cac thanh ph§.n cua d la cac dw tu0ng ling va
Trang 3s6 thanh ph§,n cua d b~ng s6 ph§,n ta cua W Tren moi hanh trinh r E P, giii sa
co mOt luQng hanh hoa (flow) chuy@nqua, ki hi~u la Fr Ta gQi Fr la dong tren hanh trinh r VectClF ma cac thanh ph§,n la cac dong Fr tren hanh trinh, gQila vectCldong hay phuCingan luu thong 86 thanh ph§,n cua F b~ng s6 ph§,n ta cua
P Giii sa tren moi hanh trinh r E P co rang buOc v~ tiii nang
va v6i mai ci;ip d§,u cu6i w E W cac dong trong Pw luau thoa nhu c§,u, nghia 180
L Fr = dw.
rEPw
(3.2)
Xet ma tri;in s6 Kronecker <I>= «I>w,r)xac dtnh nhu sail
{
I, r E Pw;
Khi do h~ thllc (3.2) duQc vi@tl[;1inhusau
Neu mOt vecta dong F thoa man cac h~ thllc (3.1) va (3.4) thl F duQc gQi la
vectc5dong ch~p nh~n duQc (feasible flow vector) Di;it
K:= {FIAr < Fr < {tr,Vr E P,<I>F= d}. (3.5)
T~p K duQc gQila t~p ch~p nh~n duQc cua bai toan m[;1nggiao thong.
Nh~n xet 3.1.1 K la tt)p l6i, dong va bi ch(in va n€u <I>A< d < <I>{tthi K =I- cpo
Ch71ngmink Gia sa FI, F2 E K thl, vui mQi t E [0,1],
A < tFI + (1 - t)F2 < {to
va
tFI + (1 - t)F2 = tcpFI + (1 - t)cpF2= d.
V~y K la t~p 16i Tfnh dong, bt chi;inva khac rang cua K suy fa tu: tinh lien
tl,lCelm <I>va (3.1) 0
Trang 4B:= {s E Pw/Hs > As}.
VI H thoa man (3.7) nen Cq > Os, \/q E A, '\Is E B Do d6 t6n t<;1i"YwE R sao cho
inf Cq > "y > sup Os.
L§,ytuy ymOt vecta dong F E K Khi d6 vdi mQi r E Pw, n@uCr < "Yw thi r tj. A,
nghia la Hr = f-lr guy ra Fr - Hr < 0 va (Cr - "Yw)(Fr- Hr) > o Tllang t\i, n@u
Or > "Ywthi (Cr - "Yw)(Fr - Hr) > o guy ra
L Cr(Fr - Hr) > "Yw L (Fr - Hr) = "Yw(dw - dw) = o.
V~y
(C,F - H)= L L Cr(Fr - Hr) > o.
wEW rEP w
Cia S11ngll<;1cl<;1iH khang la vecta dong can b~ng Khi d6 t6n t<;1iW E W, q, s E
Pw sao cho Cq < Os,Hq < f-lq,Hs > As D$,t
<5:= min{f-lq- Hq, Hs - As},
va
Fq = Hq + <5,Fs = Hs - <5,Fr = Hr, \/r =1= q, s.
Khi d6 F E K va (C, FH)= <5( Cq - Os) < o V~y (3.7) khang thoa D
3.2 M~ng giao thong ph\1 thuQC thai gian
3.2.1 M6 hlnh kh6ng co rang buQc
Cia s11T c R va £,P(T),p > 11a khang gian cac ham f : T + R do dll<;1C tren
T va IlfilP khii tfch Lesbesgue tren T Xet ma hinh m<;1nggiao thong d tren vdi
gia thi@t t<;1imoi thai di~m t E T, dong tren hanh trinh r E P la Fr(t) ph1.lthuQc bi~n thai gian t, nhu cilu cua c~p dilu cu6i W E W la dw(t) va Ar(t), f-lr(t) la cac r~ng buQc tiii nang tren hanh trinh rEP Cia S11r~ng Fr(.), Ar(.), f-lr(.) E fl(T), vdi p > 1 va thoa man, hAu kh~p nai (ghi t~t la h.k.n) tren T,
Ar(t) < Fr(t)f-lr(t),\/r E P,
Trang 5cpF(t) = d(t).
vdi CP,F(.), d(.) dli<;1Cdtnh nghia CJm1.lC(3.1).
Khi d6 t~p chfip nh~n dli<;1C cua bai toan can b~ng m<,tnggiao thong dli<;1C vi~t l~i nhli sau
K:= {F(.) E cP(T)/[cpF(t) = d(t),
Ar(t) < Fr(t) < f-lr(t),VrE P], h.k.n T}. (3.9)
Nh~n xet 3.2.1 K Latt),pMi, d6ng, bi chiJ,nva compact y€u N€u gilLsit them
<I>A(t) < d(t) < CPf-l(t)h.k.n tren T th2 K -# 4>.
Kf hi~u c = cP(T) va c* = cq(T) (~ + ~ = 1) 1a khong gian d6i ngau tapa
cua c Vdi moi G E c* va F E c, ta dtnh nghia
Khi do chi phi tren m<,tnggiao thong 1a anh x<,tC(.) tll K vao c* va vectcl dong
can b~ng tren m<,tngdli<;1C dtnh nghia nhli sau
Dinh nghia 3.2.1 (Xem [2j) M(Jt vectd dong H(.) E K d'l1Qc gfJi La can b1ing n€u
H th6a man, h.k.n tren T,
[VWE W, Vp,S E Pw, Cq(H)(t) < Cs(H)(t)]
::::; [Hq(t)= f-lq(t)hay Ht(s) = As(t)]. (3.11)
M6i lien h~ giua nghi~m ciia bai toan can b~ng va bai toan bfit d~ng thac bi~n
?
phan the hif;jn (j dtnh 1y gall.
Dinh ly 3.2.1 Vectd dong H E K can b1ingn€u va chi n€u
A = {q E Pw/Hq(t) < f-lq(t)h.k.n T};
B = {s E Pw/ Hs(t) > f-ls(t) h.k.n T}.
Trang 6Tli (3.11) va dtnh nghla cua A, B suy fa, vdi moi q E A, s E B va h.k.n tren T,
Cq(H)(t) > Cs(H)(t).
Suy ra t6n t<;Li"Yw(t) E £(T) sao cho, h.k.n tren T,
inf Cq(H)(t) > "Yw(t) > sup Cs(H)(t).
Vdi F E K va r E Pw tliy y, n~u Cr(F)(t) < "Yw(t)h.k.n tren T thl Hr(t) =
J.lr(t)h.k.n tren T Suy ra Fr(t) - Hr(t) < 0 h.k.n tren T Do d6 (Cr(F)(t)
-1w(t),Fr(t) - Hr(t)» 0 h.k.n tren T N@uCr(F)(t) > "Yw(t) h.k.n tren T thl
I
I Hr(t) = Ar(t) h.k.n tren T Suy ra Fr(t) - Hr(t)/ > 0 h.~.n tren T Vi v~y
I (Cr(F)(t) -"Yw(t), Fr(t) - Hr(t) » 0 h.k.n tren T Cu6i cling,neu Cr(F)(t) = "Yw(t)
: h.k.n tren T thl (Cr(F)(t) - "Yw(t),Fr(t) - Hr(t))= 0 h.k.n tren T Suy fa, h.k.n tren T,
'E (Cr(F)(t), Fr(t) - Hr(t)» "Yw(t) 'E (Fr(t) - Hr(t)) = O.
Do d6, h.k.n tren T,
'E (Cr(F)(t), Fr(t) - Hr(t) » o.
rEPw
Nhli v~y
l (C(F)(t), F - H)dt > 0,
nghla la
((C(F), F - H) » 0, VF E K.
V~y H thoa man (3.12).
Gia S11ngli<;5Cl<;LiH thoa man (3.12) va H E K kh6ng thoa (3.11) Khi d6 t6n
Cq(H)(t) < Cs(H)(t); Hq(t) < J.lq(t);Hs(t) > As(t).
Vdi moi tEE, di;it
6(t) = min{J-lq(t)- Hq(t), Hs(t) - As(t)}.
Trang 7SHYfa, h.k.n tren T,
<5(t)> 0
va
Hs(t) - <5(t) > As(t); Hq(t) + <5(t)< f-lq(t).
DM
Fq(t) = Hq(t) + <5(t),VtE T;
Fs(t) = Hs(t)- <5(t),Vt E T;
Fr(t) = Hr(t), Vt E T, r =I-q, r =I-s
va
F(t) = H(t), Vt E T\E.
Khi d6 ta c6 F E K va
((C(H), F - H)) = IT( C(H)(t), F(t) - H(t) )dt
= r (C(H)(t), F(t) - H(t) )dt
JT\E
v-0
+ IE( C(H)(t), F(t) - H(t) )dt
= IE <5(t)[Cq(H)(t) - Cs(H)(t)]dt < O.
Di§u nay mall thuan vdi vi~c H thoa man (3.12). D
W,3'Yw E £ sao rho Vr E Pw, h.k.n tren T,
Cr(H)(t) <'Yw(t) * Hr(t) = f-lr(t), Cr(H)(t) > 'Yw(t) * Hr(t) = Ar(t). (3.13)
Gia sa trong mo hlnh m~ng giao thong d tren, cac vecteJdong F thoa man them rang buQc FED, VF E K va D c £ la t~p 16ithoa
Khi d6, dinh nghla v§ dong can b~ng cua bai toan m~ng giao thong du<)c phat bi~u nhu san.
Trang 8D!nh nghia 3.2.2 (Xem [2j) Vecta dong H E K nD a71rjcg9i to, can bang n€u
H th6a man
((C (H), F - H) ) > 0, \/F E K n D. (3.15) Quan h~ gil1a dong can b~ng cua bai toan m:;tng giao thong trong tntdng hQp c6 them rang buQc va bai toan b§,t d~ng thuc bi~n phan th~ hi~n d dinh ly sail
D!nh ly 3.2.2 (Xem [2j) Vectd dong H E K n D can bang n€u va chi n€u tan tf;li8(.) E £* saD cho
((8, F - H) )< 0,\/F E D, (( C(H) + 8, F - H) » 0, \/F E K.
(3.16: (3.17:
Chang minh Gia S11cac di~u ki~n (3.16) va (3.17) thoa man Khi d6
((C (H), F - H) ) > (( C(H) + 8, F - H) ) > 0, \/F E K nD.
guy ra H thoa man (3.15)
Gia S11ngllQcl:;tiH thoa man (3.15) Ta dl;tt
A = {(F,~) E D x R/~ < O},
B = {(F,~) E K x R/((C(H),F - H))< ~}.
Ta th§,y A i= cP va B i= cP. Ta chung minh An B = cP. Th~t v~y, n~u (F,~) E A
thl ~ < o Ma ( (C (H), F - H) ) > 0,\/F E K nD Suy ra (F, ~) ~ B Vi A =
DnR-va D la t~p 16i lien A la t~p 16i D6i vdi t~p B, gia sU:(Fl, ~d, (F2,~2) E B DnR-va
u E [0,1] tuy y Ta c6
((C(H), Fl - H) )< ~l, ( ( C (H), F2 - H) ) < ~2
va
((C(H),uF1 + (1- u)F2 - H)) = fr(C(H),uF1 + (1- u)F2- H)dt
= ufr(C(H),F1- H)dt
+(1 - u) fr( C(H), F2- H)dt
= u( (C(H), Fl - H)) +(1 - u)( (C(H), F2- H))
< U~l + (1 - U)~2'
Trang 9Suy ra B la t~p 16i Tit intD -I- cPsuy ra intA -I- cPoTa thiiy cae gia thi~t cua dinh IS'tach t~p 16i thoa man vai hai t~p A, B Do d6 t6n ti;ti (8, k) E £* x R sao cho (8, k) -I- 0 va t6n ti;ti 0; E R sao cho
((8, F) )+k1 < 0;,\f(F, 1) E A, ((8,F,))+k1 > 0;,\f(F,1) E B.
(3.18)
~ (3.19)
Tit (3.18) suy fa k > O N~u k = 0 thl ta chQn Fo E K n intD, khi d6 ((8, Fo))= 0; VI 8 -I- 0 nen t6n ti;ti FED sao cho ((8, F))> 0; (di@u nay mati
thuan vai vi~c H thoa man (3.18)) Do d6 khong miit tfnh t6ng quat ta c6 th~ gia s11k = 1 Khi d6 vai mQi 1 > 0 du nho va vai mQi FED, ta c6
N@uchQn 1 = ((C(H), F - H)) thay vao (3.19), thl vai mQi F E K ta c6
((8, F) )+( (C(H), F - H) » 0; (3.21 )
Neu chQn F = H thay vao (3.20), (3.21) thl vdi mQi FED, ta c6
((8, H))= 0;,((8, F) )< 0;,
va vai mQi F E K
((C(H)+8,F-H))> O 0
Dinh nghla 3.2.3 (Xem [2j) Gid s'llX la khang gian vecto tapa th'l,tC,K c X la Uj,pl6i va C : K -+X* finh xq, don trio
(i) Anh xq, C du(jc 99i la rinh xq, t'l,tadon di~u (pseudomonotone) ntu C thoa man vcJim9i x,y E K, (C(x),y - x» 0 -+(C(y),x - y)< 0;
(ii) Anh xq, C du(jc 99i la rinh xq, lien t'{LChemi ntu C thoa man vcJi m9i Y E K, finh xq, x ~ (C (x), y - x) la rinh xq, n'lla lien t'{Lctren K;
(iii) Anh xq, C du(jc 99i la rinh xq, lien t'{LC hemi d9C theo doq,n ntu C thoa man vcJi
m9i x, y E K, finh xq,~~ (C(~), Y -~) la n'lla lien t'{LC tren theo doq,n[x,y].
Trang 10D@ tlm di~u ki~n t6n t<;tican bAng cua bai toan m<;tnggiao thong, chung ta xet
hai dinh ly ve bat dang thlic bien phan Bali.
lai, khac q; va C : K -+ X* la anh xr,£thoa man
(i) Tan tr,£iU)pA c K thoa man A la tq.pcompact,khac q; va tan tr,£itq.p.B c K
thoa man B la tq.p Mi, compact sao cho \/x E K\A, 3y E B, (C(x), y - x)< 0; (ii) C la anh Xr,£lien t7j,Chemi.
Khi d6 tan tr,£ix E A sao cho vdi m9i Y E K
(C(x),y - x» o
Dinh ly 3.2.4 (Xem !2j) Cirl s71X la khang gian vectd tapa, K c X latq.p lai, khac q; va CK -+ X* la anh Xr,£thoa man
(i) Tan tr,£itq.p A c K thoa man A la tq.p compact, khac q; va B c K thoa man B
la tq.p lai, compact sao cho Vx E K\A,::Iy E B, (C(x), y - x)< 0;
(ii) C la anh Xr,£t'{ta ddn di~'Uva lien t7j,Chemi d9c theo dor,£n.
Khi d6 tan tr,£ix E A sao cho vdi m9i Y E K
(C(x),y - x» o.
Ap dl,mg Dinh ly (3.2.3) va Dinh ly (3.2.4) cho mo hlnh m<;tnggiao thong vdi
X = c va K la t~p ch§,p nh~n dU<;1c,xac dinh bdi cong thlic (3.9) Do nh~n xet (3.2.1) lien K la t~p 16i,dong va bi chi;tn.Suy ra K la t~p compact y@utrong c
Do do di~u ki~n (i) cua cac Dinh ly (3.2.3) va Dinh ly (3.2.4) du<;1cthoa man vdi
A = K va B = q; Bai toan can biing se co nghi~m n@uno thoa man di~u ki~n
Bali.
Dinh ly 3.2.5 Cirl s71K c c la tq.p chap nhQ,nd'l1(JC cua bai loan mr,£nggiao thong va C : K -+ c* la anh Xr,£chi phi Khi d6 bai loan can bang giao thong c6 nghi~m n€'Ucac di€'Uki~n sau thoa man
(i) C la anh Xr,£lien t7j,Chemi d6i vdi tapa mr,£nh tren K va tan tr,£itq.p A c K
latq.p compact, khac q; va tan tr,£itq.p B c K la tq.p lai, compact sao cho VH E K\A,::IF E B, ((C(H),F - H))< 0;
Trang 11(ii) C la anh X(l lien t'l,lChemi d6i vdi tapa ytu tren K;
(iii) C la anh X(l t'lja ddn di~u tren K va lien t'l,lChemin dQc theo do(ln.
3.3 M~ng giao thong da fiVC tieu phV thuQc thai gian
ThljC te, nguoi tham gia giao thong thuong Ilja chQn hanh trlnh v~n chuyen
hang h6a dlja tren nhieu tieu chuan, chang h(;Lnchi phI toi thieu va thai gian toi thi§u, Do d6 vi~c xet bai toan can b~ng m(;Lngv6i chi phI da ml).Ctieu la c:1n thi@t
Cia Slt cac ki hi~u va dOi tu</ng cua mo hlnh giOng nhu trlnh bay cua ml).C
(3.1) va (3.2) Cia Slt bay gia tren moi hanh trlnh r E P, chi phI la mot anh X(;L
Cr : K -+ £* phl) thuOc thai gian Ta d~nh nghia
C(H)(F - H) = (( (C1(H), F - H)), , ((Cr(H), F - H)), ).
D~nh nghia vecta dong can b~ng tren m(;Lnggiao thong du</c vi@t l(;Linhu sau.
h.k.n tren T, \/w E W, \/q, s E Pw,
[Cq(H)(t) - CS(H)(t) E -R~\{O}] =? [Hq(t) = f-lq(t) hay Hs(t) - As(t)].(3.22)
Quan h~ cua vecta dong can biing trong bai toan m<;Lnggiao thong va nghi~m cua bai toan bitt diing thu:c bi@nphan vecta du</cth§ hi~n trong d~nh ly sau
D!nh ly 3.3.1 Dieu ki~n din de vectd dong H E K din bang la vdi mQi F E K
Ch71ngmink Cia Slt H thoa man (3.23) va H khong thoa (3.22) Khi d6 t6n t(;Li
W E W, q, s E Pw va t~p E c £ c6 dO do duang sao cho, h.k.n tren E,
Cq(H)(t) - CS(H)(t) E -R~\{O},
Hq(t) < f-lq(t), Hs(t) > As(t).
Trang 12V6i moi tEE, d~t
o(t) = min{jl;q(t) - Hq(t), Hs(t) - As(t)}.
Khi do, h.k.n.tren E
o(t) > 0
va
o(t) + Hq(t) < jl;q(t),
Hs(t)- o(t) >s (t).
D~t
va
F(t) = H(t), Vt E T\E.
= JE(Ci(H)(t), F(t) - H(t) )dt
Vi Cq(H)(t) - CS(H)(t) E -R~\{O}, h.k.n tren E lien C(H)(F - H) E
3.4 M~ng giao thong da tr! ph\! thuQC thai gian
Xet mo hint m(;Lnggiao thong v6i cac kf hi$u va dinh nghia dll<;1C trlnh bay nhll
trong cac m\lC (3.1) va (3.2) Vecta dong F(.) E £, dll<;1CgQi Ia vecta dong ch§.p
nh~n dllQCn@uF thoa man, h.k.n.tren T,
F(t) E £',0 < Fr(t) < J-lr(t). (3.24) Cia 811t~p ch§.p nh~n dll<;1CK Ia mot anh XI;1xac dinh bai
K(H) = {F E R~\{O}/<I>FE B(d(H), E(H)),0< F < jl;, h.k.n T}. (3.25)
Trang 13vdi B(d(H), E(H)) 1a qua c§,utam d(H) ban kinh E(H).
Gia sit chi phi C 1a anh xl;tda tri tli K(H) vao £*, vdi
C(F)(t) = (C1(F)(t), , Cr(F)(t), , Cm(F)(t)) (3.26)
va vdi C(H) E £*, F E £
D~t
e(H)(.) E £, e(H)(t) E V(t) C mill [0, ILs(t)],"It E T.
s=l, ,m
Dinh nghia 3.4.1 (Xem [3, 4, 5]) Vecta dong H E K(H) d'l1{Jc gQi lil vecta dong
din bangyeu neu H thoa man, h.k.n tren T, VwE W, Vq, S E Pw,3c(H)(t) E
(C(H)(t) saD cho
[cq(H)(t) < cs(H)(t)] => (Hq(t) E [lLq(t)-e(H)(t), ILq(t)] hay Hs(t) E [0, e(H) (t)])
Dinh ly 3.4.1 (Xem [3, 4, 5]) Neu vecta dong H E H(K) lil vecta dong din bang yeu th'i H lil nghiifm cua bili toan gid bat dang thitc bien phan (Q VI) vdi
f(x, y) = 2M.e*(H), trong d6
M = h(lE(H)(t) +m.e(H)(t))dt. (3.28)
Chitng mink Gia sit H 1a dong can b~ng y@u.Khi do vdi moi w E W, d~t
A = {q E Pw/Hq(t)< ILq(t)- e(H)(t), h.k.n on T};
B = {s E Pw/Hs(t) > e(H)(t), h.k.n onT}.
Khi do ton tl;ti1'(t) thoa man, h.k.n tren T,
inf cq(H)(t) > 1'(t) > sup cs(H)(t),
vdi c(H)(t) E C(H)(t).
L§,y tuy yF E K(H), w E W, r E Pw, n@uer(F)(t) < 1'(t), h.k.n tren T thl
r ~ A guy ra Hr(t) E [lLr(t) - e(H)(tt), J-lr(t)],h.k.n tren T Do do, h.k.n tren T,
(er(H)(t) -1'(t), Fr(t) - Hr(t) - e(H)(t)» 0