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(19)

X

o

X

o

TaK

KaK

perneHHe

3THX

ypaBHeHHH,

Kor,na

OHO

cyIIJ,eCTByeT,

KaK

npaBHJIO

(HO

He

Bcer,na),

He

e,nHHCTBeHHO,

TO

3TO CJIyR<HT OCHOBaHHeM

,nJIH

nOCTaHOBKH

3a,natI

A H

5,

CepOpMyJIHpOBaHHbIX

Bblrne

B 9TOM

naparpaepe.

<PaKTHtIeCKH

np06JIeMa

MOMeHTOB

eCTb CHCTeMa

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OT-

HOCHTeJIbHO

HeH3BeCTHbIX

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(IS),

(19).

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Toro

tIT06bI

OT

ypaBHeHHH

(IS),

(19)

nepeHTH

K COOTBeT-

cTBylOIII,eH

(9KBHBaJIeHTHOH)

np06JIeMe

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He06xo,nHMO

c,ne-

JIaTb

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rnar,

COCTOHU~HH

B TOM, tITO

Bce

BXO,UHIII,He B

3THypaBHeHHH

H3BeCTHble

epyHKUHH

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B pH,UbI

no

Ka.

KOH-JIH60

nOJIHOH

Ha

OTpe3Ke

[xo,

Xl]

CHCTeMe

epyHKUHH

hd

x

),

h

2

(X),

...

, h

n

(X),

...

TIoCJIe

Toro,

KaK

TaKoe

pa3JIOR<eHHe

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CJIe,UyeT

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paBHHTb'

K09epepHI.J;HeHT.bI B JIeBOH H IIp-aBOH

tIaCTHX

paBeHcTB

(18)

H (19)

npH

O,UHHaKOBbIX

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h

k

(x). B

pe3YJIbTaTe

nOJIytIHM,

B0b6IIJ,e

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KOTopaH

H

npe,U-

CTaBJIHer

c060»

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.'

,/~-.

(2'

l<OJiEBATEJIbHA51 CHCTEMA

123

HHTerpaJIbHbIM:

ypaBHeHHflM

(lSY,

(19):

~gk(tl'

S,

-r)F(s,

or,

u(-r),

v(s),

w(s,

-r»dsd-r+

I,

~

[gkl(tl' -r)fl(-r, U

l

(t»

+gk2(tl,

-r)f2(-r, U

2

(-r))]dT=a

k

(tl)'

(20)

I.

gi.

(iI'

S,

T)

F

(s,

-r,

U (-r), v

(s),

W

(s,

or»~

ds

d-r+

I,

+~[g;n(tl'

-r)fl(-r, U

l

(-r»

+gkz(tl'

T)f2(-r, U

2

(-r)]

d-r=b

k

(tl)'

(21)

t'

•

k =

1,

2,

...

, n,

...

B

paBeHcTBax

(20)

H(21)

Bce

epyHKUHH

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3a

HCKJIlOtIeHHeM

U

(i),

v (x), W (x,

i),

u

1

(I), u

2

(I) .

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tITO

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perneHHH

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(1)-(5)

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C nOMOIII,blO MeTo,Ua pa3,UeJIeHHH

nepeMeHHblx

(a

B KOHetIHOM CtIeTe H

npH

npHMeHeHHH

HHTerpaJIbHbIX

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BaHRH),

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,B

pH,UbI

no

CHCTeMe

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h

k

(x).

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perneHHH

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,UaBaeMbIH'

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caM

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pH,Ua

no

HeKOTopOH

nOJIHOH CHCTeMe

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hk,(x).

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BOH 3a,UatIH

IIhypMa

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,UHepepepeHU·H-

aJIbHbIX

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saMeTHM

eIII,e, tITO ,H3JIOR<eHHaH 3,UeCb

npoue,Uypa

npH;Be-

,UeHHH K

np06JIeMe

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(20),

(21)

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K

JIH-

HeHHbIM

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60JIee

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nopH,UKa,

tIeM

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a

TaKR<e

60JIee

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pa3MepHocTH

npocTpaHcTBa,

tIeM

nepBaH.

I'

§

9.

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pacnpe)(eneHHOH

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cHcTeMoH

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CHCTeMbI,

T.

e.

ee

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paBHoBeCHH,

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UHeH

Q(x,

i),

0

~

x

~

n,

t

~

0,

KOTopaH

nO,UtIHHHeTCH

npH

0 <x <n

H

t > 0 BOJIHOBOMy

ypaBHeHHIO

~

iJ2Q

a

2

Q

at2

--:-

ax

2

(I)

co

cJIe,UylOIII,HMH

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H

rpaHHtIHbIMJi

yCJIOBHHMH:

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'

Q(x,O)-'-Qo(x),

al(x,

O)=Ql(X),

O~x~tt,

{2)

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t)

= U

(I),

Q (n,

t)

=

O,'i'~

O.

.(3)

H

§

9]

°

.

(::-:,

METO,n MOMEHTOB B TEOPHH ",i):;ABJIEHH5I

[r

JI.

2-

i24

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(t),

°

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t

~

T,

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T]

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p

.

T.

e. .

T

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II

U (t)

ilL:

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T].

~

Iu (t)

IP

dt

~

i,

(4)

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3a

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T

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My

(I),

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e.

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paBHoBecHH

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Q(X,

T)=O'ar(X,

T)=O,

O~x~n.

(5)

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'ITO

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n,

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(Henpe-

pbIBHO)

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1

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°

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= Q0 (n) = Q1

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0.

<l>yHKUHIO U (t)

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L

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T]

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I < p <

00.

p

PerneHHe

Q

(X,

t)

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(1) C YCJlOBHHMH (2), (3)

oy,neM

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MaTb

B OOOoIIJ,eHHOM CMblCJle

KaK

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npH

n --..

00

perneHtlH

Qn

(x,

t)

ypaBHeHHH

(1)-(3),

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ynpaB-

JleHHH

Un (t), )l.BaJK,nbI

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(0) =

U~

(0) =

0,

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L

p

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T]

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(2»)

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caTb

B BH)!.e

co

Q(x,

t)=

L.

(q~cosnt+

~

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sinnx+

n=1

co

t

+

L.

;

sin

nx

5

sin

n

(t--r)

U (1)

d-r,

(6)

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r,ne

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H

q~,

n =

I,

2,

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,-

K03<p<pHUHeHTbI pa3J10>KeHHH B pH)!.

no

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o

(i)

H Q

1

(x)

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n],

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q~,

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I,

2,

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, CYTb K03<P<PHUHeHTbI

pH)!.a

<l>ypbe

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Ha

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n],

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C

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[0,

n]

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0]

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e.

rpa<pHKH

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x).

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c: '

~>

125

, KOJlEBATEJIbHA5I CHCTEMA

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(I)

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(1),

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U

(t)

E

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T]

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~

x

~

ft,

°

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~

T.

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Toro

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yoe)!.HTbcH

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pa3a

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-

;J

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sin

n

(t--r)

U

(-r)

d-r

BOCnOJlb30BaTbCH

paBeHcTBoM

co

~

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nx=

1-~npH

xe

(0, n).

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~

n n

n=1

C

y'leToM

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ycnoKoeHHH

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Q(x,

T)=

= i:

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nx[q~COS

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nT+

ST

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n

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u

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n=1

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(7)

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(x,

T)

=

= i:

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nx

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nT

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cos

nT

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~

n cosn

(T

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u

(-r)

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n=1

0

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°< x < n.

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nx}

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[0,

n]

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sin

nx,

.

n =

1,2,

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T

S

~

sinn(T-~)u(-r)d-r+q~cosnT+

~

q~slnnT

=0,

(9)

o

T

S;

ncosn(T--r)u(-r)d-r-nq~sinnT+q~cosnT=O,

(10)

o

n=I,2,

..

,

YMHO>KaH

ypaBHeHHe

(9)

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<pHKcHpoBaHHoM

n

Ha

/ H CKJla)l.bIBaH

ero

C

ypaBHeHHe.M

(10)

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TOM

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n, a

3aTeM

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aH

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e

inT

H CHOBa pa3)!.eJIHH Bew.eCTBeHHbIe H

MHHMbIe

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:126

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[rJl.,2

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He06xo-

JJ;HMO

H /J.OCTaTOqHO

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yMoBm'i

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T

S

u(t)sinntdt=~q~,

n=I,2,

...

,

(11)

o

T

SU(t)COsntdt=-;n

q~,

n=I,2,...

(12)

o

TaKHM

06pa30M,

3a)l.aqa

06

onpe.n.eJIeHHH

ycnoKalfBalOm.ero

yrrpaB-

JIeHHH u (I)

OKa3aJIaCb

3KBlfBaJIeHTHOH

3a.n.aqe 0

pa3pelllHMotTH

6ec-

KOHeqHOHnp06JIeMbI

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(12).

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(t)

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l-np06JIeMOH

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u (t),

0:::;;;

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(II),

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II

u

(t)

II

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b [0, T],

T.

e.

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L [0, T]

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TO

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(II),

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{sin

nt,

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nt}

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T]

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p

ilpe)l.CTaBHM

T B BH)l.e

T=2nk+8,

r.n.e

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k=

=0,

1,2,

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0:::;;;8

< 2n.

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(II),

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C YQeTOM

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u

(t)

C rrepHO)l.OM

2n

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+e e

2n

~,U(t)sinntdt=(k+I)~'u(t)sinntdt+k

~u(t)sinntdt-

'0

0 e

2n

= S

qr

(t)

sin

nt

dt =

~

q~,

n=I,2,

...

,

(13)

o

2nk+ee

2n

~

u(t)cosn(dt=(k+I)

~u(t)cosntdt+k

~

u(t)cosntdt=

o '

'I

'U':

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I,

. I . n

,Iii

ill

S

qr(t)cosntdt=-2ilq~,

n=I,2,

...

,

(14)

o

r)l.e

(k+I)U(t)

nplf

t

E

[0,8),.

qr

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{

- ku

(t)

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t E

[e,

2n].

(15)

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li3

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(l4)<pyHKUHH

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(t)

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/

CJ

KO,TlEBATEJlbHASI

CJ,fCTEM,A

1'21

§

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(I3')H

(14) EH')l.RO,

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H

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n =

1,2,

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,

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(t)

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2n]

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(14)

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QeM

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C).

YhaK,

""

qr

(t) = ;

L.

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nt-,-

~'

q~cos

nt)

+C=

F

(t)

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(16)

n=

I '

B

CHJIy

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Q°(x) H Qi (x), ,pH)I.

B

(16)

CXO)),HTCH

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113

paBeHcTBa

(15)

TOr,lla

nOJIyQaeM,

QTO

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k

=1=

°HCKOMQe

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paBHO

(Ha

OTpe3Ke

BpeMeHH

[0, 2n])

k~

I

[F

(t)

+,Cl

rrpH

t

E[O,

8),

(17)

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(t)

=

I.

{

k

[F

(t)

+

C]

npH

t E

[8,

2n],

npHqeM

rrpH

t

60JIblllHe,

qeM

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T),

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U

(t)

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TIPH

k = °H

0:::;;;

8:::;;;

2n

paBeHcTBa

(13),

(14)

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3arrHCaTb

B BH)),e

2n

.

t n n

(18)

ft=

I,

2,

..

-,

v(t)smn

=7jqo'

S

o

2n 'n n

(19)

V

(t)

cos

nt

=

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qlo

n=I,2,

"',

S

o

r.n.e

u

(t)

npH

0:::;;;

t:::;;;

8,

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=

{

,°

npH

e <

t::::;;;

2n.

Ho

Tor)),a

v (t),

CHOBac

TOQHOCTblO )),0

aMHTHBHOH

KOHCTaHTbI C,

BOCCTaHaBJIHBaeTCH

0)),H03HaqHO:

v(t)=F(t)+C,

O~

t~2n.

(21)

Ho

npH

tE(e,

2n],no

onpe,lleJIeHIiIO

(20),

<PYHItUHfl

vet) )),OJI)I{Ha

6bITb

paBHa

HyJIIO.

TI03TOMY

B 3TOM

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(k = °HJIH

0<

T

~

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nOJIyqaeM,

qTO KOHCTaHTa C

paBHa-F

(t*),

!;,1l,e

t*-JIl06oe

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T.

e.

C=-FW),

t*'E

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2n],

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H HCKOMoe

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U

(t)

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T

c

lI

C T

po

r 0

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I

II

I

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I

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u

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128

C

METO.n. MOMEHTOB B TEOPHH' ynPABJIEHH51

[r

JI. 2

113

(21) npOCTO

nOJIytIaeM

(c

ytIeTOM

(22»

a(t)=F(t)-F(t*),

O~t~e,

t*E(e,

2n].

(23)

113

(21)

TaK)Ke

BH.n.HO, tITO

ynpaBJIeHHe

a(t)

no

4>opMyJIe

(23)

(B

cJIytIae

°

~

T < 2n)

cym.ecTByeT

Tor.n.a H TOJlbKO Tor.n.a, Kor.n.a

F

(t)

eCTb

KOHCTaHTa

Ha

oTpe3Ke

(e,

2n].

TaKHM

06pa30M,

Bce

MHO)KeCTBO HCKOMbIX

ynpaBJIeHHH

a (t)

np06JIeMbI

MOMeHTOB (11),

(12)

npH

npOH3BOJIbHOM

T =

2nk

+e onHCbIBaeTC5I CJIe.n.YIOUI.HM

o6pa30M.

ECJIH

k

=1=

0,

TO

a

(t)

.n.aeTC5I

epopMyJIOH

(17).

ECJIH

k = 0,

TO

a

(t)

.n.aeTC5I 4>OpMyJIOH (23)

npH

yCJIOBHH" tITO

co

F (t) =

~

L

(q~Sin

nt-

~

q~cosnt)

(24)

n=1

eCTb

nOCTOHHHaH

BeJIHtIHHa

Ha

HHTepBaJIe

(e,

2n],

paBHaH

F (t*),

r.n.e

t*-npOH3BOJIbHaH

TOtIKa

HHTepBaJIa

(e,

2n].

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)Ke

HatIaJIb-

HbIe

4>YHKUHH

Q0 (x) H Q1

(x)

TaKOBbI,

'lTO

F (t)

He

eCTb nOCTOHH-

HaH

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Ha

HHTepBaJIe

(e,

2n],

TO

ynpaBJIeHHH

a

(t),

ycnoKa-

HBaIOm.ero

cHcTeMy

(1)

3a

BpeMH

T < 2n,

He

cym.eCTByeT.

TaKHM

06pa30M,

.n.JIH

peweHHH

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OnTHMaJIbHOrO

ynpaBJIe-

HHH,

T.

e.

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a (t)

H3

MHO)KeCTBa

4>yHKUHH,

pewaIOm.HX

np06JIeMy

MOMeHTOB (11), (12)

('I'.

e.

ycnoKaHBaIOm.HX

cH,cTeMy

(1», B

,HaweM

paCnOpH)KeHHH

OCTaeTCH TOJIbKO

BbI60p

KOH-

CTaHTbI

C B

4>opMyJIe

(17)

npH

T

~

8n.

3Ty

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onpe.n.eJIHTb,

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TaK,

tIT06bI

npH

3a.n.aHHOM T

(T

~

2n)

MHliHMH3HpoBaTb

HOpMy

II

a

(t)

IIL

[0,

Tj,

r.n.e

p

a (t)

6epeTcH

B

BH.n.e

(17), HJIH

MHHHMH3HpoBaTb

JII060H

.n.pyroH

4>yHKuHOHaJI

or

a (t).

MO)KHO

TaK)Ke

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ycnoKoeHHH

T

min

(npH

yCJIOBHH, tITO

OHO

He

MeHbwe

2n),

no.n.6HpaH

C

B 4>OpMyJIe (17)

TaK,

tIT06bl

II

a

(t)

II

L

ro,

T

min]

~

I.

(25)

p

JIerKo

BH.n.eTb (HJIH

npOBepHTb),

tITO F (t) MO)KHO npe.n.CTaBHTb

K8K

I

1-

F(t)="2Qo(t)+T

I

S-

Ql(l:)dl:,

r

~

t~2n,

(26)

°

r.n.e

{

Qo(t)=

QoW

-Qo(2n-t)

npH

O~

t~

n,

11:

< t

~

2n,

(27)

- J Q1

(t)

npH

O~

t::;;;;;

n,

Qdt)=t

-,-Qd

2n

-

t

)

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n <t

~

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§

9]

,

I(OJIEBATEJIbHMI

CHCTEMA

129

c'

Ilpn

t > 2n.

4>yHKUHH

F(t)

npo.n.OJI:>KaeTCH C

OTpe3Ka

[0,

2n]

nepHO-

.n.HtIeCKH C nepHO.n.OM 2n. .

D03TOMY

npH

T =

2nk

+

e,

k =

1,2,

...

,

OnTHMaJIbHOe

ynpaBJIe-

HHe

a

(t)

npH

npOH3BOJIbHbIX

HatIaJIbHbIX

yCJIOBHHX

Q

o

(x), Q

1

(x)

MO)KHO

3anHcaTb

B BH.n.e •

. 1 1 -

1-

r

k+1

b·

Q

O(t)+2S

I

Ql(l:)dl:]

+C,

O~t~e,

0

a (t) =

.'

I '

{

'1[1-

.15-

]

I

k2"QO(t)+2"

Q

1

(l:)dl:

+C,

e<t~2n;

\ °

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t>

2n

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a

(t)

hpo.n.OJI)KaeTCH nepHo.n.HlJeCKH

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2n]

C

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C MO)KHO

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BOJIbHOrO

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)Ke

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06

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a

2

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2

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..

ax

2

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t>O,

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H B

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e.

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HeT

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napaMeTpaMH

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2b=RC+GL,

c=GR,

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a

3aTeM

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(x,

t)

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R

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t)

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a

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2

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,

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II

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01

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11

II

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Ii

IJ

I'

fj

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H

II

1

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i

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MET01,l

MOMEHT9B

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2

8,

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t)...::.

e:

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u

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R

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t)

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.

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(jt(x,O)=Qdx).

(35)

YCJIOBH5I

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T8K>Ke,

6e3

Orp8HJltIe-

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06m

H

oCTI1,

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a =

1,

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B

npoTHBHOM

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MeHHyIO

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(.13 (34)

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tITO eCJII1 B

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p8BJIeHI15I

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b • .

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. .

ort)

= ea U

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TO

38n8tI8

06

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B TOtIHOCTI1 CBeneTC5I K

38n8tIe

06

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b

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Qt (x).

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Me

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fIpli

T <

2n

.

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He

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He

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2n].

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u

(t)

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e

neJI5IeTC5I

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[ at J

u(t)=e

a

}QO(at)-2~

~

Qt('r)d-r

-F(t*),

(36)

rne

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Ii

t*E[T,

2cp].

fIpli

T

~

2n,

Korn8

T

MOMHO

npenCT8BHTb

B

BMne

T =

2nk

+

e,

rne

k-OnHO

1i3 tIHCeJI

1,2,

"',

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MOW

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IiMeer

Blin

u

(t)

=

(

b

t [

at

J

_J

k~le-a:

',~

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~

QtCr)dT +C,

tE

[0,

:],

J

(37)

-)

.'

I

-a-

b t [

I -

a'

at

-

·l

lie

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~

Qt(T)dT

+C,

tE

[:'

2:],

rne

Q

o

(t)

JI

Q

1

(t)

onp

e

n

e

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(27), (28).

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orpe3-

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2n, ¢YHKI1,1i5I u

(t)

C

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nOJIMaerC5I

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C

nepliOnOM

2n.

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6bITb

onpeneJIeH8

Jl3

YCJIO-

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npOH3BOJIbHOrO

38naHHoro

¢YHKUIiOH8JI8

OT

ynp8B-

JIeHJl5I (37).

'·:"

... /

C

§ 10]

KYCOtIHO-HEOnHOPO,LlHA51: CHCTEMA

lSI

I:

I

..

I

-:

I

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O)=Qo(x),

O::::;;;x::::;;;l,

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O)=Qd

x

}, O::::;;;x::::;;;l,

(3)

H

Q (0,

t)

= u

(t),

t~O,

(4)

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t)=

0,

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5*

METO.n;-MOM~HTOB

B TEOPHH

yClBJJEHH5I

132

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JI. 2

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1

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33,ll.aHHbletPYHKIJ;HH,

a

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UHH

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[0,

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tPYHKUHIO

u

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°

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t

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qT06bl

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t = T

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6bl

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min

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min

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pa3JIHqHblMH

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./

Pi

°< X < 1

1

,

P (x) =

P2

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(8)

{

npH

1

< X < 1

1

+1

2

==

1,

.

{k

1

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0<x<1

1

,

k

(X)

= k

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a

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1

< x < 1

1

+1

a

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1t

1

•

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1

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l

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1

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1

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a

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1

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t)

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e.

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