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Математическая модель дискретной оптимизации динамики пучка заряженных частиц.

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Y,l1,K 517.97:621.384
BeCTHllK C1l6ry. Cep. 10, 2006, Bblil. 2
E. jI,. KOmU1ta
MATEMATlrQECKA5I MO,l1;EJIb ,l1;HCKPETHO:ti OIlTHMH3AIJ;HH
,l1;HHAMHKH Ily"tJKA 3AP5I)I(EHHblX "tJACTHIJ; *)
1.
BBep;eHHe.
B
pa60Te CTPOIfTCfl MaTeMaTlf'IeCKM MO,n;eJIb OnTIfMIf3aIJ,IfIf ,n;IfCKpeTHbIX
CIfcTeM, KOTOpafl MO)KeT CJIY)KIfTb ,n;JIfl pellleHIffl 3a,n;a'-I OnTIfMIf3all,IfIf ,n;IfHaMIfKIf 3apfl)KeH­
HbIX '-IaCTIfIJ, B yCKOpIfTeJIflx.
MeTO,n;bI ,ll;lfCKpeTHOM. OnTIfMIf3all,IfIf ,n;IfHaMIf'-IeCKIfX CIfCTeM
lllIfPOKO npe,n;cTaBJIeHbI B MaTeMaTIf'-IeCKOM. JIIfTepaType, B '-IaCTHOCTIf B pa60Tax
[1- 3],
H
IfX npIfMeHeHIfe ,n;JI.H: onTIfMIf3aIJ,IfIf ,ll;lfHaMIfKIf ny'-IKOB 3ap.H:)KeHHbIX '-IaCTIfIJ, B YCKopIfTeJUIX
npe,n;CTaBJI.H:eTC.H: IfHTepeCHbIM.
O,n;HaKo ,n;JI.H: pellleHIf.H: pa3Ho06pa3HbIX 3a,n;a'-I onTIfMIf3aIJ,HH
,n;HHaMHKH ny'-IKOB He06xo,n;HMO C03,n;aHHe CneIJ,IfaJIbHbIX Mo,n;eJIeM. onTHMH3aIJ,HH, KOTopble
6bI Y'-IHTbIBaJIH CneIJ,H<pIfKY H3Y'-IaeMbIX np06JIeM. Pa3pa60TKe TaKHX Mo,n;eJIeM. nOCB.H:lIJ,eHbI
pa60TbI
[4- 7]. B
HaCTO.H:lIJ,eM. CTaTbe npo,n;OJI)KaeTC.H: IfCCJIe,n;OBaHHe B 3TOM HanpaBJIeHHH H
pa3BIfBaeTC.H: H,n;e.H: COBMeCTHOM. onTHMH3aIJ,HH nporpaMMHoro H B03MYlIJ,eHHbIX ,n;BH)KeHHM. .
OnHcbIBaeTC.H: <PYHKll,IfOHaJI ,n;OCTaTO'fHO 06lIJ,ero BH,n;a, n03BOJI.H:IOlIJ,HM. OIJ,eHIfBaTb Pa3JIH'-I­
Hble ,n;HHaMIf'-IeCKHe xapaKTepHCTIfKIf ny'-IKa.
2.
IlocTaHoBKa 3ap;aQH. MaTeMaTHQeCKaj{ MOp;eJIb onTHMH3aU;HH. PaccMoTpH M
cIfcTeMY ,n;HcKpeTHblx ypaBHeHHM. CJIe,n;yIOlIJ,erO BH,n;a:
x(k
y(k
+ 1)
+ 1)
= f(k,x(k),x(k -1),x(k - 2),u(k)),
(1)
= F(k , x(k),x(k -1) , x(k - 2),y(k),u(k)),
(2)
k = 0, ... ,N -1,
r,n;e x(k) - n-MepHbIM. <pa30BbIM. BeKTOp, xapaKTepH3YIOlIJ,HM. nporpaMMHoe ,n;BIf)KeHIfe·
y(k) - m-MepHbIM. <pa30BbIM. BeKTOp B03MYlIJ,eHHOrO ,n;BIf)KeHIf.H:; u(k) - r-MepHbIM. BeKTop·
f(k)
f(k, x(k), x(k - 1), x(k - 2), u(k))
n-MepHM
BeKTopHM
<PYHKIJ,lf.H:.
F(k) = F(k,x(k),x(k - 1),x(k - 2),y(k),u(k)) - m-MepHM BeKTopHM <PYHKIJ,H.H:. OT­
HOCIfTeJIbHO f(k) npe,n;nOJIaraeM, '-ITO npIf K~OM k E {O, 1, ... , N} OHa onpe,n;eJIeHa H
HenpepbIBHa Ha MHO)KeCTBe Ox x Ox x Ox x U(k)
no BceM CBOIfM aprYMeHTaM
(x(k), x(k - 1), x(k - 2), u(k)) BMeCTe C '-IaCTHbIMIf npoIf3Bo,n;HbIMIf. By,n;eM TaK)Ke C'-IIf­
TaTb, '-ITO npIf Ka)K,n;OM k E {O, 1, ... ,N} F(k) onpe,n;eJIeHa If HenpepbIBHa Ha MHO)KeCTBe
Ox x Ox x Ox x Oy x U(k) no cOBoKynHocTIf aprYMeHToB (x(k), x(k -1), x(k - 2), y(k), u(k))
BMeCTe C '-IaCTHbIMIf npoH3Bo,n;HbIMIf ,n;o BToporo nOp.H:,n;Ka BKJIIO'-IIfTeJIbHO. 3,n;ecb Ox - 06­
JIaCTb B R n , Oy - 06JIaCTb B Rm, U (k) - KOMnaKTHoe MHO)KeCTBO B Rr, k = 0, 1, ... ,N - 1.
IIpH 3TOM npIfMeM, '-ITO .H:K06IfaH
Jk = J(k,x(k),x(k -1),x(k - 2),y(k),u(k))
OTJIIf'feH OT HyJI.H: npIf BCex If3MeHeHH.H:X
YpaBHeHIfe
(1)
= I~:g]
I
k, x(k), x(k - 1), x(k - 2), y(k), u(k).
onIfCbIBaeT ,n;11HaMIfKY nporpaMMHoro ,n;BH)KeHIf.H:, ypaBHeHIfe
(2) -
B03­
MYlIJ,eHHOe ,n;BIf)KeHIfe.
IIpe,n;noJIaraeM ,n;aJIee, '-ITO Ha'-IaJlbHoe yCJIOBIfe
CIfCTeMbI
(2)
onIfCbIBaeTC.H: MHO)KeCTBOM
Mo -
x(O) = Xo
3a,n;aHO, If Ha'-IaJIbHOe COCTO.H:HIfe
KOMnaKTHbIM MHO)KeCTBOM B
Rm
HeHYJIeBOa
*) Pa60Ta BblilOJIHeHa IIpll qmHaHcoBoH IIOp,p,ep.>KKe POCCllHCKoro ¢oHp,a ¢YHp,aMeHTaJIbHbIX llCCJIep,OBa­
HllH (ilpoeKT Ng 03-01-00726).
© E. ,l1,. KOTllHa, 2006
30
y epbI.
IIocJIe,l1,oBaTeJIbHocTb BeKTOpOB
.::€HJ1 eM H 0603HaqaTb ,l1,JI5I KpaTKOCTH
{U(O), u(I), ... , u(N - I)} 6Y,l1,eM Ha3bIBaTb ynpaB­
u, a cooTBeTcTBYIOUJ,yIO eMY nOCJIe,l1,OBaTeJIbHOCTb
{x(O),x(I), ... ,x(N)} - TpaeKTopHeH nporpaMMHoro ,l1,BH)l(eHH5I H 0603HaqaTb
x(xo,u). 0603HaqHM x(k) = x(k,xo,u(k)) <pa3OBoe COCT05lHHe nporpaMMHOH qac­
~I Ha k-M mare. AHaJIOr11qHO, nOCJIe,l1,OBaTeJIbHOCTb BeKTopOB {y(O) , y(l), .. . , y(N)} 6y­
e~I Ha3bIBaTb TpaeKTopHeH B03MYIlI,eHHOrO .n;B11)l{eHH5I 11 0603HaqaTb y = y(x, Yo, u), 3.n;eCb
= y(O). <I>a30BOe COCT05lHHe qaCT11~bI Ha k-M ware 0603HaqHM y(k) = y(k, x, Yo, u).
MHO)l{eCTBO TpaeKTop11H y = y(x, Yo, u), COOTBeTCTBYIOIlI,HX HaqaJIbHOMY COCT05lHHIO xo,
_-npaBJIeH11IO u 11 Pa3JI11QHbIM HaqaJIbHbIM COCT05lH115lM Yo E M o, 6Y,l1,eM Ha3bIBaTb aHCaM6JIeM
BeKTOPOB
I
=
:-paeKTopHH, HJIH nyqKOM TpaeKTopHH, HJI11 npOCTO nyqKOM. <I>a30BOe COCT05lHHe nyqKa Ha
- M ware 6Y,l1,eM Ha3bIBaTb TaK)I{e CeqeH11eM nyqKa TpaeKTop11H11 0603HaqaTb
Mk,u
= {y(k)
: y(k)
= y(k, Yo, x(k), u(k)), Yo
Y npaBJIeHH5I, y.n;OBJIeTBOp5lIOIlI,l1e yCJIOB115lM
u(k) E U(k), k =
E
Mk,u,
T. e.
Mo}.
0,1, ... ,
N -
1, 6y.n;eM Ha3bI­
BaTb ,l1,OnycTHMbIM11.
YpaBHeH115lM11 T11na
MO)l{eT 6bITb on11caHa .n;l1HaMHKa 3ap5l)l{eHHbIX qaCT11~ B YCKO­
(1), (2)
p 5lIOIlI,eH HJI11 <POKycHpYIOIlI,eH cTpyKType
[2, 8, 9]. 11
np11MeHHTeJIbHO K 3a.n;aqaM <pOPMHPO­
BaEH5I ,l1,llHaM11K11 3ap5l)l(eHHbIX qaCT11~ nyqOK TpaeKTopHH MO)l{HO TpaKToBaTb KaK nyqOK
3ap5l)l{eHHbIX
qaCT11~.
Ha TpaeKTOp115lX C11CTeMbI BBe,l1,eM <PYHK~HOHaJI KaqeCTBa, n03BOJI5IIOIlI,HH O,l1,HOBpeMeHHo
o ueH11Tb ,l1,l1HaM11KY nporpaMMHoro 11 B03MYIll,eHHOrO .n;B11)l(eHHHH npoBo.n;HTb 11X cOBMecTHYIO
onTHM113a~l1IO :
N-l
J(u)
=
L
J
+
i.pk(x(k),Yk,U(k)) dYk
k=l Mk,'U
3 .n;eCb
i.pk = i.pk(x(k), y(k), u(k)) -
<PYHK~115l, KOTOpa51 np11 Ka)l{.n;OM
k
nx
x
ny
BMeCTe C qaCTHbIM11 np0113Bo.n;HbIM11,
peH~HpyeMa5I <PYHK~l151 Ha MHO)l{eCTBe
~\th , u,
(3)
g(YN) dYN.
MN,'U
o npe.n;eJIeHa 11 HenpepbIBHa Ha MHO)l{eCTBe
(x(k),y(k),u(k))
J
= 1, ... , N.
k E {O, 1, ... , N}
U (k) no BceM CB011M aprYMeHTaM
9 = g(y(N)) - HenpepbIBHo ,l1,ll<p<pe­
x
ny, Yk - nepeMeHHa5l11HTerpHpOBaHl151 no MHo)l{ecTBY
HameH3a.n;aqeH 5IBJI5IeTC5I MHH11MH3a~1151 <PYHK~HOHaJIa
(3)
no BceM .n;onYCT11MbIM ynpaB­
.1eHH5IM.
3. BapUaD;U.H <PYHKD;uOHaJIa.
PaccMoTpHM ,l1,OnycT11Mble ynpaBJIeHH5I
BeTCTBYIOIlI,lle 11M TpaeKTop1111 0603Ha'-IllM
x(xo, u)
11
x(xo, u),
u
H
U.
COOT­
a COOTBeTCTBYIOIlI,l1e TpaeKTO­
pHH B03MYIll,eHHbIX .n;B11)l(eH11H ­
y(x, Yo, u)
H
fj(x, Yo , u).
(4)
flu(k) = u(k) - u(k) 6y.n;eM Ha3bIBaTb BapHa~HeH ynpaBJIeHl151 u Ha k-M mare,
flx(k) = flx(k,x(k)) = x(k,xo,u) -x(k,xo,u) - npHpaIll,eHHeM TpaeKTopHH npo­
r paMMHoro .n;BH)l(eHl151 x(xo, u) Ha k-M mare, Pa3HOCTb fly(k) = fly(k, y(k)) = fj(x, Yo, u) ­
y(x, Yo , u) - np11paIll,eH11eM TpaeKTop11H B03MYIll,eHHOrO .n;B11)l{eHl151 Ha k-M mare. COOTBeT­
CTBeHHO flu, flx 11 fly 6Y,l1,eM Ha3bIBaTb Bap11a~HeH ynpaBJIeHH5I u H npHpaIll,eHH5IM11 Tpa­
€ KTOP11H x(xo, u) 11 y(x, Yo, u). B CHJIY CBOHCTB HenpepbIBHOCTH, JJflxJJ -+ 0 np11 JJfluJJ -+ 011
IflyJJ. -+ 0 np11 JJfluJJ -+ 0 paBHOMepHO no Yo E Mo, r.n;e JJfluJJ
= k=O,l,max
JJflu(k)JJ, 3.n;eCb
.
... ,N-l Pa3HOCTb
Pa3HOCTb
iflu(k)JJ = j(flu(k) , flu(k)).
HopMa
flx
H HopMa
fly
onpe.n;eJI5IIOTC5I aHaJIOrHqHO. 31
<5x(k), <5y(k)
u.
0603Ha'-lHM
BapHau;mi TpaeKTOpUH CHCTeMhI
(1), (2)
rrpH ,n;orrycTHMoll BapH­
aU;HH ~ u H ,n;aHHOM
BBe,n;eM B paccMoTpeHHe CJIe.n;yIOIIJ;He ypaBHeHH.H:
<5x(k
Bf(k)
Bx(k) <5x(k)
+ 1) =
<5x(k
<5x(k
Bf(k)
+ 1) =
Bf(k)
+ 1) = Bx(k) <5x(k) + Bu(k) ~u(k),
Bf(k)
+ Bx(k _
Bf(k)
Bx(k) <5x(k)
<5y(k + 1)
<5y(k
+ 1) =
=
Bf(k)
1) <5x(k - 1)
BF(k)
Bx(k) <5x(k)
BF(k)
Bx(k) <5x(k)
<5y(k
+ 1) =
+
Bf(k)
BF(k)
+
1)
+
BF(k)
By(k) <5y(k)
BF(k)
1) <5x(k - 1)
BF(k)
+ Bx(k _ 2) <5x(k - 2) +
CHCTeMY ypaBHeHHll
(5), (6)
u.
= 0,
BF(k)
Bu(k) ~u(k),
BF(k)
By(k) <5y(k)
+
(1), (2)
k
= 1,
BF(k)
Bu(k) ~u(k),
= 2, ... , N
Ha30BeM CHCTeMOll B BapHaU;H.HX ,1l.JI.H CHCTeMhI
BapHaU;H.HMH TpaeKTopHll CHCTeMhI
yrrpaBJIeHHH
+
k
(5)
+
k
<5y(k) -
k = 2, ... , N - 1;
BF(k)
Bu(k) ~u(k),
+ Bx(k _
k = 1,
+
+ Bu(k) ~u(k),
2)
BF(k)
By(k) <5y(k)
+ Bx(k _ 1) <5x(k -
BF(k)
Bx(k) <5x(k)
Bf(k)
+ Bx(k _
Bf(k)
= 0,
+ BU(k) ~u(k),
1) <5x(k - 1)
+ Bx(k _ 2) <5x(k -
k
(6)
- 1.
(1), (2),
a
<5x(k),
rrpH ,n;orrycTHMoll BapHau;HH ~u H ,n;aHHOM
ITPH 3TOM C'-IHTaeM, '-ITO
<5x(O)
HeTpy,n;HO rrOKa3aTh, '-ITO
=0
H
<5y(O)
= O.
II~x-<5xll H II~y-<5YII eCTh BeJIH'-IHHhI 60JIee BhICOKoro rrOp.H,n;Ka
Yo E Mo. 9TO CJIe.n;yeT H3 rrpe,n;cTaBJIeHHll ~x(k, x(k))
MaJIOCTH, '-IeM II~ull, paBHOMepHO rro
H ~y(k,y(k)) rro <p0pMYJIe TellJIOpa, ypaBHeHHll
PaccMoTpHM oTo6p~eHHe MHO)KeCTBa
TOpH.HMH
(4),
Mk,u
(5), (6)
H KOMrraKTHOCTH MHO)KeCTBa
B MHO)KeCTBO
Mk ,u,
HCXO,n;.HIIJ;HMH H3 0Il.HHX H Tex )Ke TO'-leK MHO)KeCTBa
Mo.
orrpe,n;eJI.HeMOe TpaeK­
Mo.
0603Ha'-lHM ero
y(k) = y(y(k)).
(7)
ITo rrOCTpoeHHIO H B CHJIY rrpe,n;rrOJIO)KeHHll 0 HerrpephIBHoll ,1l.H<p<pepeHU;HpyeMocTH rrpa­
BOll qaCTH YPaBHeHH.H
,n;H<p<pepeHU;HpyeMoe
6HaH rrpe06pa30BaHH.H
det
. 32
(2),
<popMYJIa
oTo6p~eHHe
(7)
(7)
Mk,u
B
orrpe,n;eJI.HeT B3aHMHO o,n;H03Ha'-lHOe HerrpephIBHO
Mk,u.
TaK KaK
y(k)
[2]
= y(k) + ~y(k, y(k)),
MO)KeT 6hITb rrpe,n;CTaBJIeH B BH,n;e
(~~~~~)
= 1+
divy~y(k, y(k)) + 0 (11~y(k, y(k))11) , .
.HKO­
Lle
=~
OLlYi(k,y(k))
~
0 ·(k)
.
· A (k (k))
dIVyu.y
,y O'IeBI1,lI,HO, 'ITO
(8)
y~
i=l
IdivyLly(k, y(k)) - divyby(k , y(k))1 eCTb BeJIl1'-lI1Ha
IILlull, paBHOMepHO no k = 1,2, ... ,N 11 Yo E Mo.
60JIee BbICOKOrO no­
pR,ll;Ka MaJIOCTI1, 'IeM
[2],
I1pI1Be,n;eM 3,n;eCb JIeMMY, ,n;OKa3aHHYIO B
JIeMMa 1. I1YCTb
A( a)
= {aij} -
KOTOpa5I nOHa,LJ;o6I1TC5I B ,n;aJIbHeilIneM.
Heoco6M KBa,LJ;paTHM MaTpl1l1,a nOp5I,n;Ka
n,
3JIeMeHTbI
KOTOPOH 3aBI1C5IT OT napaMeTpa a. Tor,n;a I1MeeT MeCTO COOTHOllIeHl1e
S ( A-l( )oA(a))
p
a oa
=~
IAI
oIA(a)1
oa .
3 a Me'! a H 11 e. TaK KaK CJIe,n; OT np0I13Be,n;eHI15I KBa,n;paTHbIX MaTpl1l1, He 3aBI1CI1T OT
nop5I,n;Ka I1X YMHO)KeHI15I 11 CJIe,n; OT CYMMbI MaTpl1l1, paBeH cyMMe CJIe,n;OB 3TI1X MaTpl1l1"
Ha OCHOBaHl111 JIeMMbI
1
TO
MO)KHO 3anl1CaTb
sp (~A-l
OAi) = S (~OAi A:-l) = ~ _1
~ lOP ~ 0
~
~ lA-I
a
i=1
a
i=1
olAil
0 .
a
~
i=1
I1PI1 npe06pa30BaHl1l1 Bapl1all,l1l1 I1CCJIe,n;yeMoro <pyHKlI,110HaJIa 6y,n;eT I1CnOJIb30BaHa
JIeMMa
2.
CnpaBe,n;JII1BO CJIe,n;yIOllI,ee paBeHCTBO:
divyby(k + 1)
+J
-1
(k)
(OJ(k)
oy(k) by(k)
= divyby(k) +
oJ(k)
oJ(k)
1) bx(k - 1) +
+ ox(k) bx(k) + ox(k _
oJ(k)
OJ(k))
+ ox(k _ 2) bx(k - 2) + ou(k) Llu(k)
3,n;ecb
by(k)
= by(k, y(k)),
divyby(O)
= Sp
,
k = 0, ... ,N-1.
(9)
= O.
,II; 0 K a 3 aTe JI b C T B O. I1cxo,n;5I 113 paaeHcTBa
.
'
dlVyby(k + 1)
(Oby(k + 1))
oy(k + 1)
= Sp
,II;aJIee, I1CnOJIb3Y5:l ypaBHeHI15I B Bapl1all,l15IX
(6)
Oby(k + 1) oy(k) o (OF(k)
of(k)
= oy(k) ox(k) bx(k) + ox(k _ 1) bx(k - 1)
(8)
11 ypaBHeHI15:l
(2),
(ObY(k + 1) (OF(k))
oy(k)
oy(k)
,n;JI5I CI1CTeMbI
of(k)
+ ox(k _
(2),
I1MeeM
-1) .
(10)
Haxo,n;I1M
2) bx(k - 2)
+
OF(k))
oy(k) by(k) .
I1PI1 3TOM
o (OF(k)
)
oy(k) oy(k) by(k)
m
0
= ~ 8Yj(k)
(OF(k))
oy(k) bYj(k)
+
of(k) oby(k)
oy(k) oy(k)'
OCTaJIbHble np0l13Bo,n;Hble onpe,n;eJI5IIOTC5I aHaJIOrl1'IHO.
33
TIo,n;CTaBJUI.H nOJIy~eHHble
~aeM
BblIIIe BblpruKeHH.H B
(10)
H
HCnOJIb3y.H JIeMMY
1,
nOJIy­
(9).
(3)
HaJi,n;eM npHpaIIJ,eHHe <PYHKIIMOHaJIa
III = I(it) - I(u)
J
+ J
=~ (
npH ,n;onYCTHMbIX ynpaBJIeHH.HX U H
Mk,u
g(YN )d,YN -
(7),
TIpHMeH.H.H npe06pa30BaHHe
J
'P, ('i(k), 'ilk, it(k))ay, -
J
it:
'P, (x(k), y" U(k))dY,) +
M k ,1J.
g(YN )dYN.
onpe,n;eJI.HeM
divyb"y(k, y(k)) H y~HTbIBa.H,
~TO BeJIH~HHbI II~x(k,x(k)) - b"x(k,x(k))II, II~y(k,y(k)) - b"y(k,y(k))II, Idivy~y(k,y(k))­
divyb"y(k,y(k))1 HMeIOT nOp.H,n;OK MaJIOCTH 60JIee BbICOKHH, ~eM II~ull, paBHOMepHO no Yo E
,1l,aJIee, Bbl,ll;eJI.H.H JIHHeHHble
Mo
k
H
~JIeHbI no ~x(k), ~y(k), ~u(k),
= 1,2, ... ,N, HMeeM
~I =
H + o(ll~ull).
(11)
B (11)
(12)
3,n;eCb
divy(<P~ (x(k), y(k), u(k))b"y(k, y(k)))
.
dlVy(g(y(N))b"y(N, yeN)))
C IJ;eJIbIO
=
= 8~~Z) b"Y,(k, y(k)) + <Pkdivy(b"y(k, y(k))),
8g(y(N))
8y(N) b"y(N, yeN))
.
+ g(y(N))dlVy(b"y(N, yeN))).
npe06pa30BaHH.H BapHaIJ;HH <PYHKIJ;HOHaJIa BBe,n;eM BcnOMOraTeJIbHble nepeMeH­
Hble, y,n;OBJIeTBOp.HIOIIJ,He CJIe,n;yIOIIJ,eH Pa3HOCTHOH CHCTeMe:
q(k)
T
8F(k)
T
P (k) = JkP (k
T
I (k)
l
=~
~=k
'34
= Jkq(k + 1) + <pk,
(i
II Jjp (2 +
T
J=k
8Jk
8<Pk + 1) 8y(k) + q(k + 1) 8y(k) + 8y(k) , .
8F(i)
1) 8x(k)
+
i
.
T(.
+ j=k J j , '/,
II
k = N -1, N - 2, ... ,1' l
=
k+2,
[N _ 1
3,ll,eCb
p(k) -
k=1, ... ,N-3
k = N _ 2 N _ l'
= (O~~/:/)),
m-MepHbIM BeKTop,
,(k) -
q(N)
= g(y(N)),
n-MepHbIM BeKTop,
JIaeM 3aMeHY nepeMeHHbIX B CHJIY CHCTeMbI
YN-I,
k
eCJm
> i-I,
()
13
tII
'-1 Jj = 1,
Tor,ll,a
j=k
IIpe06pa3yeM nOCJIe,ll,HHM HHTerpaJI B BapHau,HH
HOM
II (. )
"
npH KOHe9:HbIX YCJIOBHHX
pT (N)
i-I
) of('/,)
OJi)
O<.pk
+ 1 ox(k) + j=k q '/, + 1 ox(k) + ox(k) ,
(2),
(12),
,T(N)
q(k) -
= O.
CKaJIHp.
HCnOJIb3YH nepeMeHHble
(13).
C,ll,e­
H, nepeM,ll,H K HHTerpHpoBaHHIO no nepeMeH­
nOJIY9:aeM
J
(pT(N)6y(N, yeN))
+ q(N)div y 6y(N, yeN))) IN-IdYN-I.
MN-l,u
,lJ;JIH KpaTKOCTH BMeCTO
H HCnOJIb3Y-S JIeMMy
2
yeN, YN)
J {(
yeN). 3aMeHHH 6y(N)
div y 6y(N), HMeeM
6Y,ll,eM nHcaTb
,ll,JIH npe06pa30BaHHH
no q:,opMYJIe
(6)
T
of(N - 1)
OJN-I )
IN-IP (N) oxeN _ 1) + q(N) oxeN _ 1) 6x(N - 1)+
MN-l,u
T
of(N - 1)
OJN-l )
+ ( IN-IP (N) oy(N _ 1) + q(N) oy(N _ 1) 6y(N - 1)
.
+ IN_lq(N)dlv y 6y(N - 1) +
T
of(N - 1)
OJN-l )
+ ( IN-IP (N) oxeN _ 2) + q(N) oxeN _ 2) 6x(N - 2)+
of(N - 1)
OJN-l )
}
T
+ ( IN-IP .(N) oxeN _ 3) + q(N) oxeN _ 3) 6x(N - 3) dYN-l +
J(
T
of(N - 1)
IN-1P (N) ou(N _ 1)
+
OJN-l
+ q(N) ou(N -1)
)
~u(N - l)dYN-l.
(14)
MN-l,u .
(14) C HHTerpaJIOM no Ce9:eHHIO MN-1,u B BapHau,HH (12).
6x(N - 1), 6y(N - 1), div y 6y(N - 1) H y'fHTbIBM COOTHOIlleHHH (13),
CJIQ)KHM HHTerpaJI
'fJIeHbI npH
C06HpaH
3anHilleM
nOJIY9:eHHYIO CyMMy B BH,ll,e
J{
,T(N' - 1)6x(N - 1) + pT (N - 1)6y(N - 1) + q(N - 1)div y 6y(N - 1) +
MN-l,u
35
T
8F(N - 1)
+ ( IN-IP (N) 8x(N _ 2)
+
8F(N - 1)
T
IN-IP (N) 8x(N _ 3)
(
(
OCTaBMM
T
8F(N - 1)
IN-IP (N) 8u(N _ 1)
I N- 1
MN -
(5), (6)
M
2 ,'U B BapMau,MM
J
)
8JN- 1
+ q(N) 8x(N _
3)
8JN-l
+ q(N) 8u(N _
)
1)
}
6x(N - 3)dYN-l
8<PN-l)
+ 8u(N _
1)
+ IN-I,
(15)
~u(N - 1) dYN-l.
(15) OT MHTerpMpOBaHM5I no YN-l
6x(N - 1),6y(N - 1), div y6y(N - 1) no
6e3 M3MeHeHMu M nepeu,n;eM B <popMYJIe
YN-2.
K MHTerpMpOBaHMIO no
<p0pMYJIaM
8JN-l
+ q(N) 8x(N -2) 6x(N - 2) +
,n:aJIee, 3aMeH5I5I
(9) M CKJIa,n;hIBruI
(12), MMeeM
nOJIY'IeHHOe BhlpIDKeHMe C MHTerpaJIOM no Ce'IeHMIO
{pT(N - 2)6y(N - 2) + "?(N - 2)6x(N - 2) + q(N - 2)div y(N - 2) +
T
8f(N - 2)
T
8F(N - 2)
8JN-2
+ ( IN-2/ (N - 1) 8x(N _ 3) + IN-2P (N - 1) 8x(N _ 3) + q(N - 1) 8x(N _ 3) +
T
8F(N - 1)
+IN- 1 JN-2P (N) 8x(N _ 3)
.
8JN-l)
+ IN-2q(N) 8x(N _
T
8f(N - 1)
+ ( IN-2/ (N - 1) 8x(N _ 4)
8F(N - 2)
T
+ I N- 2P (N -
8JN-2
+ q(N - 1)8x(N
_ 4) ) 6x(N - 4) } dYN-2
3) 6x(N - 3) +
1) 8x(N _ 4)
+
+ IN-l + IN-2,
r,n;e
(
T
8F(N - 2)
IN-2P (N -1) 8u(N _ 2)
8JN-2
+ q(N - 1) 8u(N _ 2)
T
+ IN-2/ (N -
8f(N - 2)
1) 8u(N _ 2)
+
8<PN-2)
+ 8u(N _ 2) ~u(N - 2)dYN-2.
IIOBTop5I5I 3TY npou,e,n;ypy, CBe,n;eM Bce MHTerpMpOBaHMe K MHTerpMpOBaHMIO no MHO­
)KeCTBY Ha'IaJIhHhIX COCT05IHMU
36
Mo.
"Y'IMThIBruI,
'ITO Ha'IaJIhHhle 3Ha'IeHM5I BapMau,MU
6x(0)
= 0,6y(0) = 0 11
<p.YHKU;110HaJIa
div y6y(0)
(3):
J
N-l
61 = ~
= 0,
(T
JkP (k
rrOJIy'-IaeM CJIe,n;yIDIIJ;ee Bblpa)KeHl1e ,n;JIR Bapl1aU;1111
8F(k)
+ 1) 8u(k) + Jkr
T
(k
8f(k)
+ 1) 8u(k) +
k-O M k ,,,
8Jk
8<Pk )
+ q(k + 1) 8u(k) + 8u(k)
4.
(16)
dYkfl.U(k).
YCJIOBHe OnTHMaJIhHOCTH. BBe,n;eM CrreU;l1aJIbHyID Bapl1aU;I1ID yrrpaBJI€HI1R
= 0,
fl.u(k)
k
= 1,2, ... , N
- 1,
k
1= j,
fl.u(j)
Ey,n;eM Ha3bIBaTb ee ,n;orrycTI1MoH ,n;JI5I ,n;orrycTI1Moro yrrpaBJIeHI15I
T aKoe, ~TO rrpl1
0
u,
[2]
1= O.
eCJII1 cYIIJ;eCTByeT
"£
>0
~ c~ "£
u(j)
+ cfl.u(j)
E U (j).
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OHaJIa
(16),
rrOJIy~aeM CJIe,n;yIDIIJ;YID TeopeMy.
TeopeMa. ,LI;JI5I Toro
~T06bI
yrrpaBJIeHl1e
UO
= (uO(0),uO(1), ... ,uO(N -1)) 6bIJIO OrrTI1­
M aJIbHbIM, He06xo,n;I1MO, ~T06bI rrpl1 Bcex ,n;orrycTI1MbIX Bapl1aU;115IX yrrpaBJIeHI15I BbII10JIH5IJIOCb
HepaBeHcTBo
J (
J
kP
T(k 1)8F(k, x(k), x(k-1), x(k-2), Yk, uO(k)) +
8u(k)
+
Mk,,,o J
+
T(k
kr
+ q(k +
+
1) 8f(k, x(k), x(k-1), x(k-2), uO(k))
8u(k)
+
1) 8Jk
8u(k)
+
8<Pk(X(k),Ykl UO (k)))d fl. (k) 0
8u(k)
Yk u
~,
k = 0,1, ... ,N - 1.
J k = J(k,x(k),x(k - 1),x(k - 2),y(k),uO(k)) 11 p(k + 1),r(k + 1),q(k + 1) y,n;OB­
(13) rrpl1 OrrTI1MaJIbHOM yrrpaBJIeHl111 11 COOTBeTCTBYIDIIJ;I1X eMy
OrrTI1MaJIbHbIX TpaeKTOpl15IX x(xo, uO) 11 y(x, Yo, uO).
5,. 3aKJIIOQeHHe. TaKI1M 06Pa30M, B HaCT05IIIJ;eH pa60Te rrocTpoeHa ,n;OCTaTO~HO 06IIJ;a5I
3,n;ecb
ileTBOp5IIDT COOTHOllleHI15IM
MaTeMaTI1~eCKa5I Mo,n;eJIb OrrTI1MI13aU;1111 rrporpaMMHoro 11 B03MYIIJ;eHHbIX ,n;BI1)KeHI1H B ,n;I1C­
KpeTHbIX CI1CTeMaX, KOTOPa5I MO)KeT 6bITb I1CrrOJIb30BaHa ,n;JI5I pellleHI15I 3a,n;a~ OrrTI1MI13aU;1111
,n;I1HaMI1KI1 3ap5I)KeHHbIX ~aCTI1U; B pa3JII1~HbIX yCKOp5IIDIIJ;I1X 11 <poKYCI1PYIDIIJ;I1X CTpyKTypax,
B ~aCTHOCTI1 ,n;JI5I OrrTI1MI13aU;1111 ,n;I1HaMI1KI1 3ap5I)KeHHbIX ~aCTI1U; B JII1HeHHOM YCKopl1TeJIe C
Tpy6KaMI1 ,n;pe:i1<pa.
(16)
I10JIy~eHHoe aHaJII1TI1~eCKOe rrpe,n;CTaBJleHl1e Bapl1aU;1111 <PYHKU;110HaJIa
11 YCJIOBl1e OrrTI1MaJIbHOCTI1 rr03BOJI5IIDT CTPOI1Tb pa3JII1~Hble HarrpaBJIeHHble MeTo,n;bI MI1­
HI1MI13aU;1111 I1CCJIe,n;yeMoro <PYHKU;110HaJIa.
MO)KHO paCCMaTpl1BaTb KaK
3a,n;a~y
OTMeTI1M TaK)Ke, ~TO I1CCJIe,n;yeMyID rrp06JIeMY
OrrTI1MI13aU;1111 B YCJIOBI15IX Heorrpe,n;eJIeHHOCTI1 rro Ha­
'-IaJIbHbIM ,n;aHHbIM.
Summary
Kotina E. D. Discrete optimization mathematical model of charged particles beam dynamics.
The mathematical model was suggested that allows conducting simultaneous optimization of
programmed motion and ensemble of perturbed motions in discrete systems. Analytical expressions
37
for functional variations are suggested that help constructing various directed methods of optimiza­
tion. Given mathematical model can be used in the optimization of the dynamics of charged
particles in linear accelerators.
JhlTepaTypa
1. nponou A. H. 9JIeMeHTbI TeopRR OnTRMaJIbHbIX ~cKpeTHbIX npo~eccoB . M.: HaYKa, 1973.
255 c.
2. 06C.R'H,'/-/,U1C06 /1,. A. MaTeMaTwlecKRe MeTO~I ynpaBJIeHRH nyqKaMR. J1.: Ib,n;-Bo J1eHRHrp.
YH-Ta, 1980. 228 c.
3. 06C.R'H(ttU1C06 /1,. A. Mo,n;eJIRpOBaHRe R onTRMR3~H ~HaMRKR nYQKOB 3apH:>I<:eHHbIX QaCTR~.
J1.: Ib,n;-Bo J1eHRHrp. YH-Ta, 1990. 310 c.
4. Kotina E. D., Ovsyannikov A. D. On simultaneous optimization of programmed and per­
turbed motions in discrete systems/ / Proc. of 11th Intern. IFAC Workshop. Vol. 1. Oxford, UK,
2001. P. 187- 189.
5. KomU'l-ta E. /1,. ,n:RcKpeTHaH 3a,n;aQa onTRMR3~ c cyMMapHbIM nOKaJaTeJIeM KaQeCTBa / /
Tpy~I Me)l{,LJ;yHap. ceMRHapa «,n:RHaMRKa 3apH)l{eHHbIX QaCTR~ R onTRMR3a~RH». BDO-2001.
,
CapaToB, 2001. C. 51-53.
6. Kotina E. D., Garbuzova S. A. Optimization of longitudinal motion of charged particles
in drift-tube linear accelerator/ / Proc. of Intern Workshop "Beam Dynamics & Optimization" .
BDO-2002. St.Petersburg, 2002. P. 135-141.
7. Kotina E. D. Discrete optimization problem / / Problems of Atomic Science and Technology.
2004. N 1. P. 147- 149.
8. MypU'lt B . n., BO'ttJape6 B. H., Kywu'tt B. B., lPeJomo6 A. n. J1RHeHHbIe YCKopRTeJIR ROHOB:
B 2 T. T. 1: IIpo6JIeMbI R TeopRH. M.: AToMR3,n;aT, 1978. 264 c.
9. CO.ft06'be6 JI. 10. Mo,n;eJIRpOBaHH.e }KeCTKo<poKycRpyIO~ero KaHaJIa JIRHeHHOro YCKopRTeJIH
npOTOHOB Ha 9IJ;BM / / TpY~I 1-ro BcecoI03. COBe~aHRH no yCKopRTeJIHM 3apH}KeHHbIX QaCTR~.
M., 1968. T. 2. C. 431-435.
CTaTbfl IIOCTYlUiJIa B pe,a;aKD;Iuo 11 ,a;eKa6pfl 2005 r.
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