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Tpe6yeTcH

peWHTh

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\

103

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7]

.,.

C

HallaJIhHhIM

H

Hy)l{HO

MeHT

pOBaHHOH

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IJ.HH, T.

O,ltHOMepHOrO

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TerrJIOn~BO)l;HOCTH

Q=

Q",

0 <x <

1,

t >

0,

(5)

yCJIOBHeM

Q(x, 0) = Q

o

(x),

O~x

< 1,

(6)

rpaHHllHbIMH

yCJIOBHHMH

Q(0, t)

=U

(t),

t~O',

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t)=O,

t~O.

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U

(t),

0

~

t

~

T,

llT06hl

B MO-

BpeM~HH

t = T BbInOJIHHJIOCh yCJIOBHe

Q(x,

T)=O.

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1.

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ITpe.n.nOJIO)l{HM

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HaM

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B «llaCTHOM»

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pac-

TeMnepaTyphl

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e.

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CJIOBaMH,

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o

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t

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0

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Horo

pacnpe)l.eJIeHHH

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t

~

T,

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.

1

u(t)=5

u

(t,

s)Qomd~.

(11

)

o

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u(t,

s),

O~t~T,

O~s<

1, ocyrn.ecTBJIHIOrn.ee

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pacnpe)l.eJIeHHH.

B3HToro

B BH)l.e

c5-<PYHKIJ.HH

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O~x

< 1,

O~

s < 1, eCTeCTBeHHO

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¢Yft-

OaMeftmaAbftbtM ynpa8JleftUeM. -

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BhITeKaeT

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lIto

B JIHHeH:HhIX

CHCTeMax

HeT

He06xo.n.HMOCTH

cpa3Y

HCKaTh

ynpaBJIeHHe.

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pacnpe)l.eJIeHHH

Q 0

(x),

O~x

< 1,

a

Hy)l{HO

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HaH:TH

ynpaBJIeHHe

U (t,

s).

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~

t

~

T,

0

~

s <

1,

oCYll\eCTBJIHIOrn.ee

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pTHoro»

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pacnpe)l.eJIe-

HHH,

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B BH)l.e c5-epyHKIJ.HH: Q

o

(x)

=c5

(x-s).

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3Toro

OCTaeTCH TOJIhKO BOCnOJIh30BaThCH npOCTOH:

¢opMyJIoii

(11).

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nOHH-

THe

B

)l.aJIhHeiirneM

6y.n.eT

o606rn.eHo.

.uoKa3aTe.TIbCTBO cnpaBe.n.JIHBOCTH

¢OPMYJIhI

(II)

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npocTo.

B caMOM .n.eJIe,

nycTh

u(t,

s),

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l,eCTI~.

-)

~,

/

'-'\

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B

TEOPHH

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104

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.-

Q

(x,

t)=

~

H6

Kax

pe3KoB

Q(x,

T.

e.

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<POpMyJIOH

BJIHeT

JIeHHH

3paqHO.

n03liUHH,

nYCTb

.n.aHO

ero

Ha

C

nOMOUJ,blO

KpHBaH

Q0 (x)

npHHUHny

nYJIbC.

K

HyJIIO,

ypaBHeHHHMH,

qy

0 rarneHHH

(16)

(15)

on-

_

(12)

(13)

~

s< 1,

Horo

pacnpe.n.eJIeHHH

Q

o

(X),

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B

BH.n.e

.•

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_(10).

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perneHHH

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ypaB-

HeHHH

B

qaCTHbIX

npOH3Bo.n.HbIX

.

I t

Q

(x,

f)

=

~

K

1

(X,

t,

1')

Q

o

(1')

d1')+

~

K (X,

f,

-c)

u

(-C)

d-c.

o 0

Tor.n.a

B

qaCTHOM

cJIyqae,

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Q

o

(x)

=6

(x-s)

H

u(t)=u(t,

s),

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<

1,

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<

1,

rrpH

f=T

nOJIyqHM

1 T

Q

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T,

s)=

~

/(1

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T,

1')6

(1')-s)d1')+

~

K(x,

T,

-c)

u

(-c;

S)dT. (14)

o 0

·OTclO.n.a

no

yCJIOBHIO,

TaK

KaK

Q

(x,

T,

s)

= 0, °

~

x < 1, °

HMeeM

T

K

i

(x,

T,

s)+

~

K(x,

T,

-c)

u

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s)d-c=O,

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O~s<l.

o

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Tenepb

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HBJIHIOw:eHcH

<paKTHqeCKH

pe.n.eJIeHHeM

6-<PyHKUHH,

no

KOTOpOH

1

Q0 (x) =

~

Q0

(11)

6

(x-1')

d1'),

o~x<

I,

o

a

TaKJKe

<POpMyJIOH

(12).

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Q0 (t) H U (t), BbIpa)KeHHble

paBeHCTBaMl1

(16) H (12), B

o6UJ,ylO

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(13).

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qTO

u (t),

BblpaJKeHHOe

<POpMyJIOH

(12),

eCTb

HCKOMoe

yrrpaBJIeHHe.

.lI.eHcTBHTeJIbHO,

nOJIyqHM

I 1

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t)=

~Kl(X,

t,

1')

~

Qo(~)6(~-1')ds+

o 0

I J

+

~

K

(x,

t,

-c)

~

u

(-c,

~)

Qo

(s) dsd-c. (17)

o 0

MeHHH

nopH.n.oK

HHTerpHpoBaHHH B

060HX

CJIaraeMblX

nOCJIe.n.HeH

<POpMyJIbI,

6y.n.eM

HMeTb

I I

Q

(x,

t)

=

~

Qo(6)

SKI

(x,

t,

1')

6

(s-1')

d1')ds+

o 0

I - t

+5

Qom

S

K(x,

t,

-c)

u

(-c,

s)d-cd~,

(18)

II

P

-~')

-~-

4>YH.D:AMEHTAJIbHOE

ynPABJIEHHE

lOS

CJIe.n.yeT,

qTO

I T

Qo

m

[K

1

(x,

T,

s)

+

~

K(x,

T,

-c)u(-c, s)d-c]ds. (19)

o 0

no

<popMyJIe

(15)

BbIpaJKeHHe,

CTOHlI,J,ee

B

KBa.n.paTHblx

cKo6-

nOCJIe.n.Hero

paBeHcTBa,

paBHO

HyJIIO

npH

JII06bIX

x H S

H3

OT-

°

~

x < 1,

~

s<

1.

3TO

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qTO

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°

T)=O,

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ynpaB'

u (t), on

pe.n.eJIHeMOl!

(12), ocyw:ecT-

rarneHHe npoIi3BOJIb-

Horo

HaqaJIbHOrO

pacnpe.n.e-

v

~

I.JL-'

\ I I

/11

;C

Qo(x),

°

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x <

1,

qTO

Ii

Tpe6oBaJIOCb

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.

<l>H3HqeCKH

3TO

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PHe.

2.7.1.

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npo-

3.n.eCb

Mbl

<paKTHqecKH

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CHCTeM.

BcaMOM

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pacnpe.n.eJIeHHe

Q

(x,

0) = Q0

(x).

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Y3KHe

npHMoyrOJIbHble

«HMnYJIbCbI»

(pHC.

2.7.1).

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HanpHMep

HMrryJIbC

Ha.n.

TOqKOH

x = Si'

raCHTCH

cooTBeTcTBylOw:ero

ynpaBJIeHHH

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BCH

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cyMMa

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TO,

no

cynepno3HUHH, H ynpaBJIeHUe,

raCHIQee

BCIO

KpHBylO,-

6y.n.eT

paBHocyMMe ynpaBJIeHHH,

raCHIQHX

KaJK.n.bIH

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HM-

B

npe.n.eJIe,

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umpHHa

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npHxo.n.HM

K

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(11).

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cHTyaUlIH

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MeCTO,

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MbI

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C

co.n.epJKaIQHMH

qaCTHble

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no

BpeMeHH

60JIee

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qeM

nepBblH.

PaCCMOTpHM,

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3a.n.a-

HaqaJIbHbIX

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.n.JIH

BOJIHOBOrO

ypaBHeHHH

Q=Q",

O<x<l,

t>O,

(20)-

C

rpaHHqHbIMH

yCJIOBHHMH

Q (0,

t)

= u (t),

t~O,

(21)

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t~O,

(21a)

H

HaqaJIbHblMH

yCJIOBHHMH

,

Q

(x,

0) =

Qo

(x),

O~x~

1,

(22)

Q

(x,

0) =

Ql

(x),

O~x~

1.

(23),

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HaRTH

TaKoe

ynpaBJIeHHe

u(t),

O~t~T,

npH

KOTOpOM

KOJIe6aTeJIbHaH

CHCTeMa,

onHCbIBaeMaH

ypaBHeHHHMH

(20)-(23),

'.'

I~

l/)6

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ABJIB HIHl

[fJI.

.

ug(t,~o)

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JIa

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':'BUH

(24)

(25~

(28)

(29)

(27)

oy-

2:

npHiUJIa

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B

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Q

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T)=O,

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1,

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T)

= 0, 0

~

x

~

1.

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(t,

~o)'

o

~

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T,

O~

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~

1,

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Qo

(x) =

(')

(x-~u),

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x

~

1,

O~

~o<1,(26)

Ql(X)=O,

O~x~1.

(llOqeMy

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3Toro

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CKa3aHO

n03LKe B

3TOM

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lIaM

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~),

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1,

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1,

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1,

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(t),

ycnoKaHBaIO~ee

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. HbIX Hat:IaJIbHblX pacnpe)l.eJIeHHHX

(22),

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u (t) =

~

ug

(t,

~o)

QU

(~3)

d~o

+

~

u~

(t;~l)

Q1

(~l)

d~1"

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o'

0

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o

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~),

u

1

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U~(t'~l»

(30a)

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(30)

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u

(t,

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=

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~o)'

u

1

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~l»

H BeKTopa Hat:IaJIbHbIX'

yCJIOBHH

(Q

0

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e. HHTerpHpoBaHHe

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H

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I ,

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u(t)=(uo(t,

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1

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Q!(~l»'

(300)

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B )l.aJ1bHeihueM.)

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(30),

T.

e.

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TH

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llo

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t)

BOJIHOBOrO

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*) 0

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107

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00

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....

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t -

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1

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e.

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Бутковский А.Г. Методы управления системами с распределенными параметрами

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