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He

)l.JIH BH)l.a (25), a )l.JIH BH)l.a (20).

TaK,aH

nOCTaHOBKa

HeKoTophle

peaJIhHhle

cHTyaUHH,

BCTpet:IaIOIII.HecSI

HanpHMep,

B

peaJIhHWX

nJIa3MOTpoHax

Ot:IeHh

Tpy)l.HO

nyry

HenpepwBHo.

cI>aKTHt:IeCKH

ee

nepeMemaIOT

cKati-

K

3a)l.atie

ynpaBJIeHHH.

DyCTh

K

MOMeHTy

Bpe-

•

MeHH

t = t

TpeoyeTcH

C03)l.aTb

3anaHHoe

pacnpeneJIeHHe

Q

(x,

t

)=

m

=

Q*

(x).

TaK

KaK

1

«I

_

Q*(x)=n

L

Q*(k)coskx,

(26)

k=-«I

r.ne

1t

•

Q*(k) =

~

Q*

(x)

cos

kx dx,

o

TO

H3

cpaBHeHHH

(26) H (23)

nOJIyt:IaeM

m-I

- 2

~

2 2

a (k) = k

2

Q*

(k)'e

k

tm=

~

Up

[e

k

tp+

1

-

ek tp]

COS

kx

o

•

(27)

p=l

DpOOJIeMa

(13L HBJlHeTCH

BeChMa

oo~eH

nOToMy,

t:ITO

B

HeH

HBJIHIOTCH

n POH3BOJlhHWMH

Bce

TpH

Haoopa

{up}, {x

p},

{t

p~'

Y'13

Hee

JKe CJIe-

)l.yeT

H

KOHTHHyaJIbH35I

npOOJIeMa,

eCJIH tix - 6 H

~t

p

---+-

O.

ECJIH

p

nOCJIe)l.OBaTeJJbHOCTH

{11t

p}

,H

{~xp}

3a)l.aTb,

TO

nOJIyt:IHM

nHCKpeTHWH'

aHaJIOr

JIHHeHHOH

npOOJIeMW

MOMeHTOB.

cI>HKcHpyeM,

HanpHMep,

HaoopbI

{Up}

H

{t

p

}'

Tor)l.a

(27) MOJKeT

CJIyJKHTh

)l.JIH

onpeneJIeHHH

HeH3BeCTHhlX

{x

}.

DycTb

uk=l,

p

I1t

k

=~t

=

const,

t

k

=

(k-I)

~t,

'k

=

I,

...

,

n.

Torna

paBeHcTBo

(27)

npHHHMaeT

BH)l.

p-l

k

2

Q*(k)e""(P-l)df

~

,

a* (k) = .

';'k't>t

=

~

ek-mAI

cos

kx

m

. (28)

I-e

m=l

OTCIOna

BHneH

OnHH

H3

He06xonHMhIX

npH3HaKOB

CYIII.eCTBOBaHHH

perneHHH

3a)l.at:IH,ObICTponeHCTBHH.

LI.eHcTBHTeJIbHO,

eCJIH

OfITHMaJIbHOe

1

149

F

i48

METO.l(' MOMEHTO'B B TEOPHH

\llPABJIEHH~

[r

JI.

'2'

nb

6bICTpO,neHCTBHIO

ynpaBJIeHIifecyrn.eCTByeT,

TO

-'

p-l

I

Q*

(k) I

~...!..

~

e-k'

[(p-I)-m]

<1t

1

_:e'-

kl

<11

-...::::

k

2

.l-. .

.

m=]'

Ho

B

npaBOH

qaCTH

nOCJIe,nHero

HepaBeHcTBa

CTOlfT

cyMMa

qJIeHOII

reOMeTpHqeCKOH

npOrpeCcHH.

n03TOMY

nOJIyqaeM

I

Q*

(k) I

~

~2

[I-e-

k

'

(p-

I)

<11].

(29)

3ToeCTb

OrpaHHqeHHe

H

Ha

:>KeJIaeMOe COCTOHHHe,

If

Ha

BpeMH

tp =

(p-I)

tt.t

nepexo,nHoro

npo,:!-ecca.

OqeBH,nHO,

qTO

MHHHMaJIbHoe

BpeMH

tp min

He

MO:>KeT

OKa3aTbCH

MeHbIIIe,

qeM·

BeJIHqHHa

max

~21n[I-k2IQ*(k)I]-I.

k

BepHeMCH

K

cHcTeMe

(28).

3,necb

TaK:>Ke HMeIOTCH

pa3JIHqHbie

B03MO:>KHOCTH. K

MOMeHTy

B

peMeHH

tp

cOBepIIIaeTcH

(p-I

)-H

IIIar

TOqeql!Oro

HCTOqHHKa.

CJIe,noBaTeJIbHO,

3a

(p-l)

IIIarOB

MO:>KHO

«yHH-

qTO:>KHTb»

p-l

JII06b1x

rapMoHHK,

HanpHMep

nepBbIx.

B 3TOM

CJIy-

qae

3a,naqa

ynpaBJIeHHH

CBO,nHTCH

K

peIIIeHHIO

CHCTeMbI

(28)

npH

k =

0,

I,

...

,

p-

2 C nocJIe,nyIOrn.eH

MHHHMH3au.HeH

no

M.

Hanplf-

Mep,

eCJlH

cOBepIIIeH

O,nHH

IIIar,

TO

p=2.

r,ne

,nOJI:>KeH

OKa3aTbCH

HCTOqHHK H KaKOBO

ero

MHHHMaJIbHO

B03MO>Jmoe

BpeMH

npe6bIBaHHH

TaM,

QTo6bl

YHHqTO:>KaJIaCb

k-H

rapMOHHKa?

B CHCTeMe

(28)

OCTaeTCH

O,nHO

ypaBHeHHe,

pelliaH

KOTopoe

Haxo,nHM

Xl

=

~,

t

2mln

= +

~2

In[l~k21

Q*

(k)

1l-

1

•

npH

p>

2 CHCTeMa

(28)

MO:>KeT

6bITb

pellieHa

qHCJIeHHblMH MeTO-

,naMH C

npHMeHeHHeM

BbIqHCJIHTeJIbHblX

MaIIIHH

H,

B qaCTHOCTH,

aHaJIOrOBOro

Mo,neJIHpOBaHHH.

O,nHa

H3 B03MO:>KHbIX H,neH Mo,neJIHpOB-aHHH OCHOBblBaeTCH

Ha

3aMeHe

nepeMeHHoH

no

<popMyJIe

x

k

=

arccos

1;k'

npH

3TOM

CHCTeMa

(28)

ne~exo,nHT

B

cHcTeMy

CJIe,nyIOrn.ero

BH,na:.

p-l

a,*(k) =

~e-k'mMTk(1;m)'

k=O,I,

...

,p-l,

(30)

.

m=1

r)i.e T

k

(1;)-nOJIHHOM

LJe6blI1IeBa

k-ro

nopH,nKa.

Ho

,nJIH nOJIHHOMOB

LJe6bIllieBa

HMeeTCH

peKyppeHTHoe

COOTHOIIIeHlfe

[349]

T

k

+

l

(1;) =

2GT

k

(G}-T

k

-

i

(1;),

(31)

KOTopoe

MO:>KeT

6blTb

JIerKOBOcnpoH3Be,neHo

Ha

3JIeKTpHqeCKOH

ceTKe

C

o6paTHoH

CBH3bIO

(pHC.

2.11.1).

LJepe3

S

0603HaqeH

<PYHKU.HO-

HaJIbHblH

npeo6pa30BaTeJIb

HanpH:>KeHIHl,

npeo6pa3yIOrn.HH

HanpH:>Ke-

HHe

V B S (V). ECJIH

Ha

KpaHHHH

JIeBblH

Y3eJI

no,naTb

Hanpfl:>KeHHe

PJ

"

HEJIHHEAHAH

TIPOBJIEMA MOMEHTOB

I,

a

Ha

caMOM

npaBoM

Y3JIe

.

peryJIHpOBaTb

HanpmKeHHe

TaK,

lJ

110-1

HHKa.

V

o

=

B

qTo6bl

BTOqKe

1

HanpH:>KeHHe

CTaJIO

paBHblM

VI =

1;,H,

KpoMe

Toro,

nOJIO:>KHTb S

(V)

=

2V,

B..

= 2- 21;,

TO

HanpH:>KeHHH B

Y3JIaX

ceTKH

r

TOqHOCTH

6y,nyT

y,nOBJIeTBOpHTb

COOTHOrneHHIO

V

k

+

l

(1;) =

21;V

k

(1;)-

V

n

-

l

(1;).

OmpaBJlHHCb

OT

CXeMbI

pHC.

2.11.1,.

y:>Ke HeTpY.ll.HO

cocTaBHTb

nOJI-

HyIO

cxeMy

Mo,neJlHpOBaHH5I

ypaBHeHHH

(30).

PaccMoTpHM

Tenepb

CJlyqaH,

Kor,na

CKOpOCTH

nepeMern.eHHHHCTOq'..

3aBe,nOMO 6C!lJlbIIIHe

If

MO:>KHO

npeHe6pe':lb

MOJleKYJI$lpHbIM

SIY

n

)

(n-1)

Vn-f'

n

PRe.

2.11.1.

nepeHOCOM

TenJia

B

cpaBHeHHH

C

KOHBeKTHBHbIM.

Tor,na

BMeCTO

ypa~HeHHH

Q= Q"

+15

[x-

iV

(.)

d.]

Hy:>KHO

paccMaTpHBaTb

ypaBHeHHH

Q=cS[x-iV(.)d.]..

Q(x,O)=O.

·PerneHHe

3THX

ypaBHeHHH

HMeeT

BH,n

I

Q (x,

t)

=

~

15

[x-S

(.)]

d.,

S(t)=

~

v(.)d

•.

o

I

B

,

B

'.

"

r

METO~

MOMEHTOB B TEOPI1I1

~~~BJlEHI1Sl

'

[f

Jl:

2'

·1'50

I1yCTb

Tpe6yeTcH

B MOMeHT

BpeMeHH

t = T

nOJlytIHTb

Q

(X,

T)

~

=

Q*

(X).

Tor.n.a

T

Q*

(X)

=

~

{j

[X-S

(-r)]

d-r.

o

flo

HHTerpaJl

B

npaBOH

'laCTH

nOCJle.n.HerO

paBeHCTBa

nOCJle

aaMeHbl

S,

(-r)

= z nepeXO.n.HT B CJle.n.YIOllJ,HH

HHTerpaJl:

S(T)

( 1

X E [§ (0), S

(1)],

S

S'[ar~s(z)]{j(z-x)dz=1

S'[arcS(x))

,

S(O)

~

0,

XE[S(O), S (T)].

Q*(x)

=

S'

Tare

S (x)) •

§ 12.

npHMeHeHHe

KOHelJHblX

HHTerpaJlbHblX

npeo6paaoBaHHH

B 3TOM

naparpa<pe

paccMaTpHBaeTcH

MeTO.n.

pemeHHH·

aa.n.atI

OnTHMaJlbHOrO

ynpaBJleHHH

CHCTeMaMH C

pacnpe.n.eJleHHbIMH

napa-

MeTpaMH,

oCHoBaHHbIH

Ha

npHMeHeHHH

KOHetIHblX

HHTerpaJlbHblX

npeQ6paaoBaHHH

[43].

IlycTb,

HanpHMep,

ynpaBJlHeMbIH

o6'beKT

C

pacnpe.n.eJIeHHbIMH

napaMeTpaMH

onHCbIBaeTCH

ypaBHeHHHMH

a

2

Q

aQ

a

2

Q'

aQ.

aaJ2+b

ar

= A (x)

ax

2

+B

(x)

ax

+C(x)

Q

+F

(x, t)

(1)

C

KpaeBbIMH

yCJIOBHHMH

aQ

' )

(oQ)

(

a

l

ox

+~lQ

x=x,

=w

l

(t); a

2

oX

+~2Q

x=x,

=w

2

(t)·

(2)

KaK

6bIJIO

nOKaaaHo

BbIme,

3<p<peKTHBHbiM MeTo.n.OM

pemeHHH

aa-

.n.3tI

OnTHMaJIbHOrO

ynpaBJleHHH

HBJIHeTCH

anna

paT

l-np06JIeMbi

MO-

MeHTOB. CBe.n.eHHe aa.ll.atIH

ynpaIiJleHHH

K

np06JleMe'MOMeHTOB

MO>KeT

6bITb

oCyllJ,eCTBJIeHO

Ha

OCHOBe paaJIO>KeHHH HMnYJIbCHQH

nepexo.n.-

HOH <pyHKI.I.HH CHCTeMbi

(.n.JlH

ynpaBJleHHH

BO·BpeMeHH)

HJIH <pyHKI.I.HH

fpHHa

(.n.JlH

ynp~iBJIeHHH

B -

rrpOCTpaHCTBe)

no

KaKOH-JIH60

CHCTeMe'

~

JIHHellHO

HeaaBHCHMbiX

<pyHKUHH,

HanpHMep'

no

CHCTeMe

co6CTBeH-

HbiX

<pyHKUHH

KpaeBOH

aa.n.atIH.

IlpH

3TOM

.n.JIH

nOJIytIeHHH

B aa.n.aH-

HOM

npOCTpaHCTBe

ynpaBJleHHH

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B01HH-

KaeT

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a.n.a

tIa

OTbiCKaHHH

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~o

C

r'

C'~:';

i

I2l

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151

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Ha

KOTOPOM .n.OCTHraeTCH YCJIOBHbiM

MHHH"

MyM

HOPMbi

HeKOToporo

:JJleMeHTa B COnpH>KeHHOM

npOCTpaHCTBe.

. B HaCTOHllJ,eM

naparpa<pe

CBe.n.eHHe

K'

l-np06JleMe

MOMeHTOB

OCYllJ,eCTBJlHeTCH C nOMOllJ,bIO

KOHetlHbiX

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npe06paaOBa-

HHH.

TaKof!

no.n.xo.n. He

Tpe6yeT

HBHoro

aHaHHH

HMrryJlbCHOH

nepe-

XO.n.HOH

<PYHKI.I.HHHJlH <pyHKI.I.HH

fpHHa

H o.n.HOTHneH

KaK

.n.Jl51

CJIy-

qaeB

ynpaBJleHH5I

BO

BpeMeHH,

TaK

H B

'npOCTpaHCTBe.

IlocJIe

nOJlytleHHH

l-np06JIeMbi

MOMeHTOB

paCCMaTpHBaeTCH

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aHa-

JIHTHtleCKHX

<P0PM

ynpaBJIeHHH,

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B KJIaCCe

L~(x)

[Xl'

x

2

],

.

r.n.e r

(x)-HeKOTopbIH

Bec.

Pa-:CMOTpHM

.n.Be

aa.n.atlH

nepexo.n.a

CHCTeMbI (1), (2) Ha

HeKoToporo

HatIaJlbHOrO

COCT05lHH5I

Q (x, 0) =

Qo

(x),

aQ(x

O)=Ql(X)

(3)

at

' ,

aa.n.aHHoe

KOHetlHOe

Q(x,

T)

=

Q;

(x),

aQ

(x

T)

==

Q;

(x).

(4)

at'

,

3 a.n. a

tI

a A.

YKaaaHHblH

nepexo.n.

.n.OJI>KeH

npoHaoHTH

aa

cTporo.

<pHKcHpoBaHHoe

BpeMH

T

TaK,

tIT06bI

HopMa

ynpaBJleHH5I

B

L~(Xl

I

npHH5IJla

MHHHMaJlbHOe

aHatIeHHe.

3 a.n. a

tI

a

6.

YKaa8HHbIH

nepexo.n.

.n.OJl>KeH

npoHaoHTH

aa

MHHH-

MaJlbHO

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BpeMH

T

npH

yCJIOBHH, tITO

HopMa

ynpaBJIeHH5I

L~

(X)

OrpaHHtleHa

HeKoTopbIM

tlHCJIOM.

OrpaHHtIHMCH

CJIytIaeM,

Kor.n.a <pyHKI.I.HH F (x,

t)

B (1)

npe.n.cTa-

BHMa

B

BH.n.e

F

(x~

t)

= u

l

(t)

VI

(x)+

U

2

(t)

V

2

(x),

(5)

H ,nOO'lepe.n.HO

paCCMOTpHM

aa.n.atIH

ynpaBJleHHH

B

npOCTpaHCTBe

(ynpaBJlHIOllJ,He

Boa.n.eHCTBH5I VI (x) H V

2

(x);

<pyHKUHH

U

l

(t),

U

2

(t);

WI

(t)

H W

2

(t) aa.n.aHbI) H

BO

BpeMeHH

(ynpaBJleHHH

U

l

(t),

U

(t),

2

WI

(t)

H W

2

(t);

<pyHKI.I.HH VI

(X)H

V

2

(x) aa.n.aHbI).

,UJIH CBe.n.eHHH

nOCTaBJleHHblX

aa.n.atI K

l-np06JIeMe

MOMeHTOB

nocTpoHM

TaKoeHHTerpaJIbHOe

npe06paaOBaHH~,

tIT06bl

nOCJIe

npH-

MeHeHH5I

ero

K CHCTeMe

(1)-(3)'

cooTBeTcTByIOllJ,aH

CHCTeMa B

npo-

cTpaHcTBe

Ha06pa>KeHHH

He

co.n.ep>KaJIa .n.H<PiPepeHUHaJlbHbIX

onepa--

I.I.Hf!

no

x.

'uJIH

3Toro

)l.H<p<pepeHI.I.HaJIbHbIH

onepaTOp

,

L

o

= A (x)

:x

22

+B (x)

:x

+C (x) (6)

t

nOMoli\bIO

no.n.CTaHOBKH

p (x) = r (x) A (x),

q (x) = r (x) C (x),

(7)

r.n.e

r (x) =

ex

p

[S

~

(B

- ::)

dx

J.

(8)

METOA

MOMEHTOB

'13

TEOPI1I1

Y~~JIEHI151

[PJl,

~

\52

.

nepeBe~eM

B

~OPMY

\{a[

aJ

}.\

LO.=r(x)

ax

p(x)ax

+q(x)

=,L.

(9)

HOBWH

onepaTOp

L

nO~qHHHeTCH

~opMYJIe

fpHHa

au

aU)Jx,

(u,

Lv)-(Lu,

v)

=

[

p

(

u

ax-

v

ax

x,

(10)

H

npH.

yCJIOBHHX,

aHaJIOrHqHbIX

(2),

HO

O~HOPO~HbIX,

caMOCOnpH}f{eH,

T.

e.

(u,

Lv)

= (Lu,

v).

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HCKOMoe

HHTerpaJIbHOe

npeoopa30BaHHe

npe~cTaBHM

B

BH~e

x.

LxY

= Y

(A.)

=

~

Y (x) W

(A.,

x) dx = (y, W).

(12)

x,

Tor~a

~aJIbH,eHrneHga~a~IeH

H6JIHeTCH

onpe~eJIemfe

H~pa

W

(A.,

x).

npHMeHHH

onepaTop

Lx

K (9) H

H:cnOJIb3yH

~OPMYJIY

fpl;:IHa

(10),

nOJIyqHM

Lx

[+-

Ly

J= (

+-

Ly,

W)

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+-

W)

~

W)

[.

( W)

ay

. a

(W)]

x.

=

y,L-

+ p -

--py-

- . (13\

(

y r

ax ax

r

x,

. 'J

.

,UJIH

Toro

qTOObI

npeoopa30BaHHaH

CHCTeMa

He

co~ep}f{aJIaon6

paIJ,H:H

~H:~epepeHIJ,HpOBaHHH

no

x,

~OCTaTOqHO

nOJI0}f{HTb

(y,

L~)=-A.,(y,Wl

(14)

H

npHCOe~H:HHTb

COOTBeTcTByIOmHe

(2),

HO

O~HOpo~Hble

KpaeBbI~

yCJIOBH:H. .

0003HaqaH

W =

rW*

H

yqHTbIBaH,

qTO

~YHKUH:H

Y B

paBeHcTB~

(14)

np0H:3BOJIba.aH,

nOJIyqaeM

~JIH

onpe~eJIeHHH

H~panpeoopa30Bil

HHH (12)

CJIe~YIOmyIO

caMOCOnpH}f{eHHyIO

3a.n.aqy

111TYPMa-JlHYBH:JIJI~.::

d

.[

dW*]

....

LW*=dx

p(x)tlX

+q(X)W*=-A.'r(x)W*,

(15)

dW*)

(dW*)

(

a

l

-d-

+

~l

W*

=

0,

au

-d

+

~,W*

. = 0. (16)

X

x==x,

x

x==x.

.

np"

3TOM

BWpa}f{eHHe

B

KBa~paTHbIX

CKOOKax

B

.paBeHcTBe

(13)

paBHo

' , ,

!'

(pW*)x==x.

-

~l

(pW*)X=XI'

(17)

-,

. 1

<:~

.

....,

,

c>

~''!.l!]

I(OHEqHbIE

·I1H'TErPAJIbHbIE nPEOBPA30BAHH51

\53

H:

a,

=F

0,

H:

paBHo

eCJI

If

a

l

=F

°

1',

(dW*.)

+1'1

(

dW*)

(18)

o'!'

--

p-

-

p-

.

~,

dx

x=x.

~l

.

dx

x=x.'

~~JIH

a.i

=a,

=

0:

TaKHM

oopaSOM,

3a~aqa

onpe~eJIeHHH

H~pa

CBeJIaCb K

OTWCKa·

HHIO COOCTBeHHbIX

3HaqeHHH

H

coocTBeHHbIX

~YHKIJ,HH

3a~aqH

(15), (16).

, .

H3

oomeH

TeopHi

caMOCOnp~}f{eHHbIX

~H~~peHUHaJIbHbIX

one-

paTopOB

[I88]

CJIe~yeT,

qTO

eCJIH

npM

p (x) >°B CHCTeMe (15), (16)

q (x) <0,

a~

+

~~

> 0,

a~

+

~~

> 0,

al~l

<0,

a2~'

~

0, .

TO

Bce

COO-

CTBeHHbIe

3HaqeHHH

nOJIO}f{HTeJIbHbI.

ECJIH

}f{e

q (x)

==

O'~l

=~,

= 0,.

Ttl

nepBoe

COOCTBeHHoe

3HaqeHHe

paBHO

llyJIIO.

MHO}f{eCTBO

COOCT6eH-

HhIX

3HaqeHHH

{AM

CqeTHO,

rtpHqeM

'.

~~., ~~

=

n'

[5

(

~

y/'

dX)-'.

(19)

x,

,

KaH<~DMY

coocTBeHHoMy

3HaqeHHIO

COOTBeTcTByeT

O~Ha

coocTBeH-

liaH

~YHKUHH

W~

=

W~

(A.

n

,

x),

HCKJIIOqaH YCJIOBHH

nepHO~HqHOCTH,

I

r~e

HX

MO}f{eT ObITb

~Be,

onpe~eJIHeMaH

C TOqHOCTbIO

~o

nOCTOHH-

Horn

MHO}f{HTeJIH H3 (15) H (16).

llJIH

~aJIbHeHrnero

Ba}f{HbIMH

f1BJIHIOTCH

CJIe~yIOIIJ,He

~Ba

CBon-

CTBa

(188].

CHcTeMa

{W~}

opTOrOHaJIbHa

C BeCOM r (x) .

Ha

OTpeSKe

[Xl'

x,],

..

T. e. '

(rW~,

W~)

=

II

W~

IIi;

(X)

{)nm'

(20)

r~e

HopMa

(X. )

1/'

II

W~

IlL'

=

t·~

[W*

(A.

n

, x)], r

(x)

dx t ,

(21)

r

(X)

x,

)

a

()nm-CHMBOJI

KpoHeKepa.

.

CHcTeMa

{Wri}

nOJIlia

B

KJIacceL~(x)

[Xl'

x;],

T.

e.

He

cymeCTByeT

HH

O~HOH

~YHKUHH

H3

L~(x)

[Xl'

x,],

3a

HCKJIIO'leHHeM

HyJIeBOH

TO'lKH

:noro

npocTpaHcTBa,

opTOrOHaJIbHOH

BceM

~YHKUHHM

CHCTeMbI

{W~}.

llaJIee,

HeTpY~Ho

nOKa3aTb,

qTO

eCJIH M

(A.,

x) H N

(A.,

x)

-JIH-

HeHHo

He3aBH:CHMble

perneHH:H

ypaBHeHHH

(15),

TO

H~PO

npe06pa30-'

BaHHH

(12)

W*

(A.

,

x)=

[a

l

dN

(~~.

Xl)

+

~IN

(A.

n

,

Xl)]

M

(A.

n

,

X)-

n

-

[a

l

dM

(;;'

Xl)

+

~IM

(A.

n

,

Xl)

JN

(A.

n

,

X).

(22)

npH'

3TOM COOCTBeHHble

3HaqeHHH

MOfYT

ObITb

HaH~eHbI

H3

perneHH.H

155

",~.

''·'

:

....,...!

l54

C

..

:./

METO.n. MOMEHTOB B TEOPHH YflF.'f\·BJlEHHSI

[r;ll.i2,

CJle.n.YIOII\ero

xa

paKTepHCTHtIeCKOro

ypaBHeHHH:

dW*

("n.

x

2

)

+A

W*

(A

)-0

Ch

. dx

t'z

n' X

z

. -

•.

(23)

TaKHM

o6pasoM,

HCKOMoe

HHTerpaJIhHOe

npeo6pasoaaHHe

HMeeTBH,n

(12)

x.

LxY

= Y

(An)

= 5y (x) r (x)

W*

(An'

x) dx,

(24)

X,

r,ne

H,npO

W*

(An,

x)

onpe,neJIHetcH

BblpaJKeHHeM

(22), a BeCOBaH

<PyHKUHH

r

(x)

-

BblpaJKeHHeM

(8). n

pH

3TOM,

KaK

CJIe,nyeT

H:S

(13)

H (14),

,nH<p<pepeHUHaJlhHoMY

BblpaJKeHHIO

L Q

cooTBeTcTByeT

B

npo-

cTpaHCTBe

Hs06paJKeHHH

o

-A~Q

(An)

+-

R

(An),

(25)

r,ne

R

(An)

onpe,neJIHeTCH

Bblpa)KeHHeM

(17) HJIH (18) B saBHCHMOCTH

01' BH,na

KpaeBhlX

YCJlOBHH R

(An)

==

O.

06paTHoe

npe06pa30BaHHe

MOJKeT

6hITh

HaH,neHOKaK

opToro-

HaJIhHoe'

paSJIOJKeHI1e

OpHrHHaJIa

y (x) B pH,n

no

CHCTeMe {W*

(An.

x)},

HMeHHO:

-1

- _

_.

W*

(An.

x) -

Lx

y

(An)

= y (x) - L

/I

W

~

liz

Y

(An)·

(26)

'n

npHMeHHM

nOCTpoeHHoe

npeo6paSOBaHHe

(24) K (1) - (4).

nOJIytIHM

dZQ

dQ

z--

a

dt2

+bdT=-AnQ+f(A

n

,

n,

(27)

- . -

dQ

(An.

0) -

Q

(An'

0) = Q

o

(An),

dt

_ = Q

(An),

l

(2-8)

.

Q.(A

T)=Q*(A)

dQ(An.T)=Q*(A)

0

n'

J..J. 1 -,

(29)

nt

n

c,ne

{(An'

t)

= R

(An'

t)

+U

l

(I)

VI

(An)

+U

z

(I) V

2

(An). (30)

PeuraH

sa,natIy

KOIIlH

(27), (28),

Haxo,nHM

t

Q

(An'

t) = 5{(An'

i)

On

(I

-:-i)

dT

+

o ,

+

Q.

(A.)

[I-A:

fa.

«)

d']

+Q.

(A.)

aG;

(I),

(31)

r,ne

o

n

(I)

~

/n

lt

- e

z

n2

t

A

In

(;32)

.

····

::",,'.>:

c

KOflEtIHhlE

HHTErPAJlhHhIE

nPEOBPA30BAHH5I

If

Ziz2-KOpHH

ypaBHeHHH

aZ2+bz+A~=0,

~

=

Vb2-4aX~.

t~l

B

napa60JIHtIeCKOM

cJIytIae

npn

a = 0

I 2

I

-b"n

t

On

(I)

=7)e

.

(33)

B

rHnep6oJIHtIeCKOM

(a

*

0)

n

pH

b = 0

On

(I) =

:r~

sin

~~'!.-

I. (34)

'),.n

yay

a

n

pH

b *0 HaH,neTCH

TaKOH

HOMep

N,

tITO An

~

,~_,

KaK

TOJIhKO

2 y a

.n

~

N.

B 3TOM

CJIyqae

.

b'

t

I

.!-e

-2a

sh

!J.

n

I,

n <

N,

l

On

(I) = {

!J.

n

2a (35)

b t

A'

2

-2a

.

tin

!J.~

e

S1ll

2a

t,

n~N,

r,ne

~~=

-

j~m

j = V

-I.

JI.R<p<PepeHUHpyH (31),

nOJIytIHM

dQ

('),.n.

t)

dt'

=

t

d S

-.

- - 2

-.

dO

n (I)

=

dt

f

(An'

i)

On

(I-i)

di-Qo

(An)

AnOn

(I) +

Ql

(An)

a([t-.

(36)

o

B MOMeHT 1= T (31) R (36) AOJIJKHbI

COBna,naTh

C

BeJIHtIRHaMH

(29).

l1s

3THX

yCJIOBRH

Haxo,nRM

CJIe,nYIOII\YIO

cHcTeMy:

~l*OnhVI

~An)

+

(U

2

:O

n

hV

2

(An~

.

Q~

(An)+S

(Am

T)=Pl

(An'

7')'1

dt

(u

1

*Onh

VI

(An)

+

d!

(U

2

*Onh

V

z

(A.

n)

-

'f'

(37)

=

-Q*

(A

) +dS

('),.n.

T)

= P

(A

T)

. 1 n

dt

2

m.

'

B KOTOpOH _

- t [ t ]

S

(Am

1)=

-

~

(R

(An'

T)

On

(I-T)

dT-

Q

o

(An)

1

~A~

~

On

(T)

di

.~

+Ql

(An)

aO

n

(I). (38)

a - iiH,/l,eKC T

03HatIaeT,

tITO

CBepTKH

H

npOH3Bo,nHhle

01'

HHX

BhltIHC-

JIeHbJ B MOMeHT 1=

T.

,PaCCMOTpHM

Teneph

CJIyqaH,

KOf,na

ynpaBJIeHHe

OCYII\eCTBJIifeTCH

.g'

npOCl'paHCTBe

•.

B CHCTeMe (37)

HeR3BeCTHblMH

HB;IlHIOTCH

VI

(A,,):-H

V

,

156

ME'tOJJ:MOMEHTOB B TEOPMM

yCABJlEHMSl

[f

JI. 2

2

(An)'

.amI

TOrOtlrOObI

9Ta

CHCTeMa

OblJla

pa3

pernHMa,

B

HeKoTo,~

pbIH

MOMeHT t = T

npH

npOH3BOJIbHOH

npaBoH

tlaCTH,

Heooxo.n.!iMO

H JJ:OCTaTOtlHO, tlTOObI

BpoHcKHaH

Ll

(An'

t)

=

(u

1

*G

n

)

I

!£

(

G)

dt

u

1

* n

(u

2

*G

n

)

d

dt

(u

2

*G

n

)

(39)

BbIQMCJIeHHbIH B 9TOT

MOMeHT,

OTJIHtlaJICH

OT

HyJIH

npH

Bcex

n =

= 1, 2, ...

ECJIH

B MOMeHT

t=

T

BpOHCKHaH

(39)

paBeH

HyJIIO

XOTH ObI

,lI,JIH

OJJ:HOrO

n =

N,

TO CHCTeMa (37), BOOOlII,e

rOBopH,

Hepa3pernl1Ma.

HCKJlIOtleHHeM

HBJlHeTCH

CJIytiaH,

KorJJ:a P1 H Pa

OKa3aJIHCb

TaKH-

MH,

tlTO

P

1

('J'-N.

T)

(U

1

*G

n

)T

(ua*Gnh

(40)

d -= d •

Pa('J'-N,

T)

dt (ul*Gnh

dt

(U

2

*G

n

)T

OTCIOJJ:a

BbITeKaeT,

tlTO «MOMeHTbI

,HeynpaBJIHeMOCTH»

MO:>KHO

yCTaHoBHTb

KaK-TaKHe

KOpHH

ypaBHeHHH

Ll

(An'

t)

= 0,

n=-1,2,

... ,

(41)

JJ:JlH

KOTOpbIX

He

BbmOJIHHeTCH

YCJIOElHe (40).

,UaJIee,

TO:>KJJ:eCTBeHHoe

paBeHcTBo

BpoHcKHaHa

(39)

HyJIIO

Heo6-

XOJJ:HMO

H JJ:OCTaTOtlHO

JJ:JlH

JIHHeHHOH 3aBHCHMOCTH

cBepToK

(u1*G,:,)

H(U

2

*G

n

).

DpHMeHHH

K nOCJIeJJ:HHM

npeoopa30BaHHe

JIanJIaca,

y6e:lK-

,lI,aeMCH, tlTO B 9TOM

cJIytlae

JIHHeHHO

3aBHCHMbI

H

caMH

!J>YHKII.HH

U

1

(t)

-H

U

a

(t).

Ho

3TO

03HatlaeT,tlTO

!J>aKTHtleCKH HMeioTCH ,1Ie'JJ:B-a

pacnpeJJ:eJIeHHbIX

B

npocTpaHcTBe

ynpaBJIeHHH,

a

OJJ:HO.

CHcTeMa

B

9TOM

cJIytlae

HeynpaBJIHeMa.

HCKJIIOtleHHH

'COCTaBJIHIOTCJIytlaH,

BbIpa:>KaeMbIe

paBeHcTBoM

(40), H

napaOOJIHtleCKHH

(npH

a=O),

eCJIH

BMeCTO JJ:BYX yCJIOBHH

(4)

nOTpeooBaTb

BbmOJlHeHHH

TOJIbKO

oJiHoro-

JIIOOoro.

DpeJJ:nOJIO:>KHM,

tlTO

npH

t = T CHCTeMa (37) COBMeCTHa H

ee

perneHHHMH

HBJIHIOTCH

V

1

(An)

= f-tnl·(T), - V

2

(An)

=

f-tna

(T)

(42)

~

.;

HJIH,

B

pa3BepHyTOM

BHJJ:e,

XI

f-tn1

(T)

=

~

V

1

(x) W

*

-(An'

x) ; (x) dx,

x,

x.

f-tn2

(T)=

~

Va

(x)

W*

(An'

x) r (x) dx,

;'

_.J4~)

~.

_

'1.

•

",'

x,

T·f:";·n"

PaBeHCTBa

(43) npeJJ:craBJIHIOT

cooaH

OeCKOHetlHoMepHyKl.npoCl-

,ileMy

MQMeHTOB

JJ:JIH

onpeJJ:eJIeHijH.

ynpa~JIeHHii

.J/1Lx}

.

~~/,~d~.

CJHEtlHbI~

HHTEfPAJlbHbIE

nPEOBPA30B'AHMH

1'57

B

IlHH

MHHHMH3HPYIOlII,HM

9JleMeHTOM

n,pOOJIeMbI MOMeHTOB

x.

f-tn=~

V(X)W~(x)r(x)dx,

n=l,

...

,N,

(44)

x,

CJIytlae

OpTOrOHaJIbHOCTH

CHCTeMbI

{W

n

(x)}

HBJIHeTCH COOTBeTCT-

ByIOlII,aH

flaCTHtlHaH

cyMMa

OpTOrOHaJIbHOrO

pa3JIO:>KeHHH !J>YHK-

V (x) *): '

N

fln

Vmin

=L.IJ

'w

n

1I~2

Wn

(x).

(45)

n=I

, ' (x)

B

COOTBeTCTBHIIC

§ 3

perneHHe

l-npOOJleMb~

(44)" OOJIaJJ:aIOlII,ee

MHHHMaJIbHOH

HOpMOH B Lf

(x)

[Xl'

X

a

],

HMe@T

BHJJ:

'N

N

Iq-I

1 1

1"

Vmin

(X)

= 1"

L.;~

W n

(X)

c;ign

L.;~

W n (X), p +- =

1,

(46)

n=1

q

r

I

rJJ:e

;o-BeKTOp,

Ha

KOTOpOM JJ:OCTHraeT

MHHHMyMa

HHTerpaJI

XII

N

Iq

(47)

I

n~\

;n

W

n (x) r '(x) dx

npH

yCJIOBHlf

N

~'

;n f-tn=

1.

(48)

n=1

BeJlHtlHHa

9Toro

MHHHMyMa

paBHa

A,

a

HopMa

ynpaBJIeHHH

l-p

L =

II

Vmin (x)

IIL~

('Il

x

"

XI]

= A-p-.

(49)

*)

Pel1leHHlIMH

B

L;

(X) [Xl> X2]

CHCTeMbl

(44)lI<1lJlllIOTCli

He

TOJlbKO

CYMMbI

BH,l\a

(45)

C

66J1bl1lHM.

qeM

N. qHCJlOM

qJleHOB,

HO

H

Bblpa)/{eHHlI,

HanpHMep.

TaKoro

BH,l\a:

•

N'

1\/2

N

V (x) =

~

t:\;~

Wn

(x),

sign

{;\

sgw

n (x).

(A)

I

r.n.e

cnoc06b1

onpe,lleJleHHlI

qHCJla

A.

H

BeKTopa;~

H3J10)/{eHbI

B § 3.

lleikTBH-

TeJlbHO,

TaK

KaK

Wn

(x)

EL;(x), TO Wn

(x)

E

L~/(a..)'

CJle,llOB8TeJlbHO,

pel1leHUe

v

(x)

B

L~

(Xl

HMeer

BH,l\' (A).

Ho

TaK

KaK

v (x) E

L;

(X)'

TO V

(x)

E

L;

(X)'

O,llHaKO

BO

Bcex

cJlyqallX

Ilv(x)1I

2

~lJvmln(x)IIL2

.

...

L,

(Xl " ,

(Xl

d

Бутковский А.Г. Методы управления системами с распределенными параметрами

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