Красовский Н.Н. Теория управления движением

---

100

".'

, .

~.

C

OBmEE

PEmEHHE

3A,Il;ATIH

OB

ynPABJIEHHH

[rn,

5

ECJUI

YCJlOBlIe

(15.9)

CHOBa

IICTOJlKOBaTh

KaK

YCJlOBlIe

MlIHlIMYMa

HOpMbI

p*

[u]

epYHKD;1I11

u('r),

paccMaTpllBaeMoH

KaK

:meMeHT

npocTpa,HCTBa

:Jt[t

.t{3]'

TO

oc

COOTHomeHllH

,(15.8), (15.9)

03Ha'IaIDT,

qTO

IICxo,n;HaH

3a;n;aqa

OnJITh

CBe,n;eHa K

np06JIeMe

MOMeHTOB.

IIp>>

3TOM

CJle,n;yeT

J1l1mh

npe;n;nOJIOlKlITb,

qTO

epYHKD;1I11

h(i)

(T)

HBJlJIIDTCJI

3J1eMeHTaMll

npocTpaHcTBa

2[t

.t{3]'

oc

§ 16:

IIpooJleMa

MOMeHTOB

, IJ;eJIb HaCTOHm;ero

naparpaljla

-

n03HaKOMHTb

TIHTaTeJIH C

OCHOB-

HhIMH

TeopeMaMH,

onpep;eJIHIOm;HMII

pemeHlIe

npOOJIeMhI MOMeHTOB,

K

KOTopoH,KaK

MhI TOJIbKO lITO

yoep;IIJIIICb,

CBOP;HTCH

3ap;alIlI

00

ynpaB-

JIeHlIII.

PemeHHe

ynoMHHyTOH

npOOJIeMhI CBH3aHO C

O)J;HOH

1I3

Bam-

HeHmlIX

TeopeM

ljlYHKIJ;IIOHaJIbHOrO

aHaJIlI3a

- C

TaK

Ha3hIBaeMoii

meope.MOU

Xalla - Ballaxa 0

pacnpOCTpaHeHlI1I

JIIIHeHHOro

ljlyHK-

IJ;IIOHaJIa.

MhI,

op;HaKO,

He

oyp;eM

yrJIYOJIHTbCH B

1I3YlIeHlIe

aToii

CBH-

31I,

HO

BhIBep;eM

pemeHlIe

npOOJIeMhI MOMeHToB,

onlIpaHcb

Ha

Te

HarJIHp;-

HhIe

npep;CTaBJIeHIIH,

KOTophIe

p;aHhI B

IIpIIJIOmeHIIII.

IITaK,

BepHeMcH

K

ypaBHeHIIHM

(15.4),

T.

e.

<p

[Mi]

('1')]

,

C,

(i

= 1,

...

, n), (16.1)

rp;e,

CJIep;OBaTeJIbHO,

HaM

Hap;JIemlIT

onpep;eJIIITb

HelI3BeCTHyro

onepa-

IJ;IUO

<p

C

HalIMeHbmeii

B03MO}KHOii

HOpMOii

p*

[<p].

Haiip;eM

CHalIaJIa

neo6xo8UJ1tble

yc.n.oeuJ/"

KOTOphIM p;OJImHa yP;OB-

JIeTBOpHTb JIIOOaH OnepaIJ;IIH

<p,

pa3pemaIOm;aH

ypaBHeHIIH

(16.1).

IIYCTb,

CJIep;OBaTeJIbHO,

<p

-

HeKOTopaH

TaKaH

OnepaIJ;IIH. PaCCMOT-

plIM

MHomeCTBO gil{h},

COCTOHm;ee

1I3

Bcex

BeKTop-ljlYHKIJ;lIii

h

(1:)

BIIp;a

n

h ('r) =

~

l,h[i]

(-r)

(t1X

<

-r

<

t(3),

(16.2)

i=l

rp;e nOCTOHHHhIe

l,

CTeCHeHhI YCJIOBlIeM

n

~

l,c, = 1.

(16.3)

i=l

BCJIep;CTBlIe

(16.1)

II

no

cBoiicTBaM

JIIIHeiiHOCTII

onepaIJ;1I1I

<p

Ha

ljlYHKIJ;IIHX h

(1:)

1I3

gil

IIMeeM:

<p

[h

(1:)]

= 1.

(16.4)

B caMOM p;eJIe,

n n n

<p

[h (:r)] =

<p

[~

l,h[i

l

] =

~

li<P

[Mil]

=

2]

l'll'

= 1.

i=l

i=l i=l

0003Ha'IIIM

Tenepb

CIIMBOJIOM @

{h}

MHomeCTBO ecex

ljlYHKIJ;lIii

h

(1:)

1I3

cOBoKynHocTII

."B[toc.te]

{h

(1:)},

KOTopble

yp;OBJIeTBOpHIOT

yc-

JIOBIIIO

(16.4).

STO

oyp;eT

MaOiKeCTBO oOJIee

mlIpoKoe,

'IeM

MIWmeCTJlO

.~~~--,-.,,-,-,.

',~''''.'

......

.;'.'

.....

'"."

":

....

;-::.'

:

..

-

::::;-',::

"::

...

:":'.':>,::>:';":

..

'.~

..

·,-,

"

...

-'

C

§ 16]

npOBJIEMA

MOMEHTOB

ljlYHKIJ;IIH

gil,

nopomp;eHHoe

KOHe'IHhIM 'tJIICJIOM

napaMeTpOB

l,.

COOTHomeHIIH(16.4)

II

(16.3)

n03BOJIHIOT

HaM

Tenepb

OIJ;eHIITb

HOpM)'

p*

[<pJ.

HanOMHIIM,

'ITO

HopMa

p*

[<p}

MomeT

OhITb

JIerKO

BhI'IIICJIeHa,

eCJIH

B npOCTpaHCTBe

ffi3

{h}

1I3BeCTHO «paCCTOHHlIe»

Pcp

OT

(<IIJIOCKOCTH»

@,

rp;e

<p

[h] = 1

p;o

«TO'IKII» h

(1:)

= 0

(pIIC.

16.1).

IIPII

aTOM (CM.

IIplJ-

JIOmeHlIe,

§§

52,

58)

p*

[<p}

=

-.

1

(16.5)

Pcp

(PaCCTOHHlIe

Pcp

onpep;eJIHeTCH

KaK

HlImHHH

rpaHb

inf

p

[h

(1:)

1.

Iwrp;a

h

(1:)

npooeraeT

Bce

MHomeCTBO @.)

.....

....

PIle. 16.1.

TaK

KaK

MHomeCTBO

gil

cop;epmlITCH

B

@,

3aKJIIO'IaeM,

'ITO

PCP

<

pO,

(16.6)

rp;e

pO

eCTb paCCTOHHlIe

OT

TO'IKII

h

(1:)

=0

p;o

MHomeCTBa

gil

(16.2),

(16.3),

onpep;eJIeHHOe, CJIep;OBaTeJIbHO,

YCJIOBlIeM (CM.

plIC.

16.1)

n

pO

=

min

p [h} =

min

p

[2]

l,Mil]

,

I

i=l

= p

[?l

n

L~h[i]J

= p

rhO}

UplI

2]

n

liC,

=

1.

(16.7)

t=l

i=l

BamHO

OTMeTIITb, 'ITO 3p;eCb, p;eiiCTBIITeJIbHO,

MIIHIIMYM

{}ocmuzaem-

n

CJ/,

Ha

HeKOTOpoii

ljlYHKIJ;1I1I

hO

(1:)

=

~

l~

h[il

('r),

cop;epnmm;eiiCH

:B

gil.

i=l

TaKyIQ

ljlYHKIJ;IIIO

hO

(1:)

HaaOBe:M

J·H,(,1l1~~Ul,.n.b1f,Oa.

·.··/:2;.+,

",.

t.-.,,'

.:

.....

::,.

",

,..:

..

,:.":

'~'

..

,

.'

.".':

..

: ...,

..

'

.,._:

:':

..

':

'.":.:'

':,'

:::

;~:,

C

'102

OBmEE

PEmEHHE

3A,D;AtIH OB

yrrPABJIEHHH

[rJI.

5

,D;oKaiKeM

Cym;ecTBoBaHHe

MHHHMaJII:>HOR

<PYH~HH

hO

('t). TIYCTI:>

CHa'IaJIa

<PYHK~HH

h[i]

('t)

.f/,UneUnO

neaaeUCUMb~,

T.

e.

nYCTI:>

JIHHeRHaH

KOM6HHa~HH

HX

(16.2)

MOiKeT

06pam;aTI:>CH

B TOlI{;n;eCTBeHHbIR HyJII:>

JIHml:>

rrpH

ZI

=

...

=

Zn

= 0.

Tor;n;ano

CBoRcTBaM

HOpMLI

p [h]

n

(CM.

§ 56)

HMeeM:

y[ZI'

...

,

Zn]

= P

[~

Zjh[i]]

>0

npH

zi

+

...

+

i=I

+

z;=/=

° =

00

npH

Z~+

..•

+

Z;

H,

60JIee

Toro,

limy

[ZI'

••.

,

Zn]

~

00.

KpOMe

Toro,

YUI"'"

Zn]

eCTI:>

HerrpepLIBHaH

<PYH~HH

OT

Zt·

Zn]

'

Ho

Tor;n;a

cym;ecTBoBaHHe

MHHHMyMa

Y

[Zl!

°

...

,

°

eCT]) CJIe;n;CTBHe

HSBeCTHLIX

CBORCTB

HenpepLIBHLIx

<PYH~HR.

TIpe;n;nOJIOiKHM

Tenepl:>,

'ITO

cpe;n;H

<PYH~HR

h[i]

('t)

TOJII:>KO

k < n

JIHHeHHO

He3aBHCHMLIX.

MLI

Bcer;n;a

MOiKeM

BLI6paTI:>

HyMepa~HIO

TaK,

'IT06LI

JIHHeHHO

He3aBH,CHMLIMH

OLIJIH

rrepBLIe

k

<PYHK~HH

h[j] ('t)

(j

= 1,

...

, k). OCTaJII:>HLIe

<PYHK~HH

h[i]

('t)

6y;n;yT,

CJIe.n;OBaTeJII:>HO,

JIHHeRHLIMH

KOMOHHa~HJlMH

OT

h[j]

('t):

k

Mi]

('t') =

2]

'At:h

Ul

('t'). (10.8)

;=1

Ho

Tor;n;a

n "

[i]]

[~.

[j]J

. ·

(16.9)

p

[

~

ijh

= P

~

Z;h

= Y[lI,

...

,

lk].

\=1

3=1

r;n;e

'IHCJIa

n

lj =

i;

+

2]

'Alil

l

i=k+I

BCJIe;n;CTBHe (16.3) H (16.8)

CTeCHeHLI

yCJIOBHeM

k n k

2]

ljc; +

2]

li

(Cl-

2]

Cj

'At

i)

= 1.

(16.10)

;=1

i=k+I

;=1

.

BeJIH'IHHa

Y

[Z~,

, Z;]

CHOBa

BCJIe;n;CTBHe

JIHHeHHOH

He3aBHCH-

MOCTH h[j] ('t) (j =

1,

,

k)

oy;n;eT

HerrpepLIBHoH

<pYH~HeH

OT

zj,

npH'IeM

lim

Y =

00

npH

z·i

+ ...+

z*:

~

00.

OTCIO;n;a OnHTI:>

[

·0

·0] u

BLIBO;n;HTCH

cym;ecTBoBaHHe

MHHHMyMa

Y

Zl"'"

Zk

;n;JIH

aTOH

BeJIH'IHHLI.

TeM

CaMLIM

CHOBa

yCTaHaBJIHBaeTCJI

cym;ecTBoBaHHe

MH-

HHMaJII:>HOR

<PYHK~HH

hO

('t).

Tenepl:>

H3

(16.5) H (16.6)

nOJIy'IaeM

HyiKHyIO

o~eHKY

p.

[<p]

>

-\-.

(16.11)

P

lI3

aTOR

o~eHKH

CJIe;n;yeT,

npeiK;n;e

Bcero,

'ITO

BeJIH'IHHa

pO

OOH-

aaTeJlI:>IlO

OTJlH'IIla

01'

HyJIH

1

eCJlII

TOJlbKO

ypallHeHIIJI

(16.1)

IIMelOT

-.::.~

"'"

,:-.'

C

§ 16]

lIPOBJIEMA

MOME:H'1'OB

103

pemeHlIe

B BII;n;e

OrpaHII'IeHHOR

JIIIHeHHOH

onepa~HII

<p,

a TOJII:>KO

Ta-

KHe

onepa~1I1I

Hac

H

HHTepeCYIOT

no

YCJIOBHJlM

3a;n;a'IH.

BBlI;n;y

BaiK-

HOC'fH

HepaBeHCTBa

pO

> 0,

KaK

HeOOxo;n;IIMoro

yCJIOBHH

;n;JIJI

pas-

pemHIIIOCTII

ypaBHeHIIR

(16.1), nO;n;TBep;n;HM

ero

em;e

Henocpe;n;CTBeHHo.

C

nOR

~eJII:>IO

npe;n;nOJIOiKlIlII,

'ITO

sa;n;a'Ia

IIMeeT

pemeHlIe,

HO

TeM

He

MeHee

pO

= 0. O;n;HaKO rrOCJIe;n;Hee

03Ha'IaeT,

'ITO

hO('t)

=0,

T.

e.

<p

rho

('t)] =

<p

{O]

= 0, a aTO

npOTIIBOpetIlIT

yCJIOBIIIO

(16.4),

IIOO

hO

('t) CO;n;epiKIITCH B

f}.

TIo;n;Bo;n;H

IITor

HamlIM

paccYlI{;n;eHIIJlM,

MLI

MomeM

C<pOpMyJIHpO-

BaTI:>

CJIe;n;ylom;ylo

JIeMMy,

KOTOpaJl

YKa3LIBaeT

HeoOXO;n;lIM:bIH

nplI3HaK

;n;JIH

paspemlIMocTII

npOOJIeMbI

MOMeHTOB

(15.4).

JI

e M M a 16.1.

OnmUMa.f/,bnaJL

onepaq,UJL

<po,

paapeUtalOUfaJL

npo6-

.f/,eJty

MOMenmoe,

cYUfecmeyem

mO.f/,blW

moz8a,

ri,oz8a

ebmO.f/,NJLemCJL

llepa-

eeNcmeo

pO

=

p[h

O

('t)]

>0.

(16.12)

IIpu

amOM

lleo6xo8uJtO

eblnO.f/,1iJLemCJL

YC.f/,oeue

p.

[<Dol

>

~.

(16.13)

p

(HanOMHIIM,

'ITO

s;n;ecl:>

pO

-

'I1Ir,JIO,

HaH;n;eHHoe

H3

yCJIOBHH

(16.7).)

JIeMMa

16.1 ;n;aeT

o~eHKY

cuuay

;n;JIH

HOpMbI

p*

[,<pol.

ECTecTBeHHo

nOCTaBHTI:>

Bonpoc:

HBJIHeTCH

JIH

aTa

o~eHKa

;n;OCTH1HHMOii?

lIHa'Ie

rOBOpJl,

BOSHHKaeT

Bonpoc

0

TOM,

Bcer;n;a

JIH

MOiKHO

rrOCTpOHTI:>

one-

pa~HIO

<pO,

pa3pemaIOm;yIO

rrp06JIeMy

MOMeHTOB

H

TaKyIo,

'ITO

. 1

p·[<p°j=-o'

(16.14)

p

OKa3LIBaeTCH,

'ITO

OTBeT

Ha

nOCTaBJIeHH:bIH

Bonpoc

HBJIHeTCH

nOJIO-

amTeJII:>HLIIII

H,

CJIe;n;OBaTeJII:>HO,

cnpaBe;n;JIHB

TaKOH

8ocmamo'mb~U

npH3-

Halt

paspemHMOCTH

np06JIeMLI

MOMeHTOB.

JI

eM

M a 16.2.

IIycmb

'J,UC.f/,O

pO,

onpe8eAeH,NOe

YCAOeueM

(16.7),

nMO;JICUmeAbNO.

Toz8a

np06.f/,eMa

MOMeNmoe

(15.4)

UMeem

OnmUMa.f/,b-

lloe

peUteH,ue

<po,

npU'J,eM

cnpaee8.f/,ueo

paeellcmeo (16.14).

,D;oKa1KeM

JIeMMy.

TIpem;n;e

Bcero,

3aMeTHM,

'ITO

B:bIrrOJIHeHHe

pa-

BeHCTBa

(16.4)

Ha

<pYHK~nHx

h ('t) BH;n;a (16.2), (16.3)

eCTI:>

He

TOJII:>KO

ueo6xo8uJtOe, HO

II

8ocmamo'J,Noe

yCJIOBHe

;n;JIJI

Toro,

'ITOOLI

onepa~HH

<p

;n;OCTaBJIHJIa

pemeHHe

ypaBHeHHR

(16.1). B

caMOM

;n;eJIe, nYCTI:>

yCJIO-

BHe

(16.4)

BLInOJIHeHO.

TIoKaiKeM,

'ITO

OTCIO;n;a CJIe;n;yeT

BLInOJIHeHHe

paBeHcTB

(16.1). TIYCTI:>

CHa'IaJIa

npH

BLIopaHHoM

i = S

Hille

eM

C

s

4=

0.

Tor;n;a,

nOJIaraH

B (16.2)

Zs

=

1lc

s

,

ZI

= °

npH

i

=/=

s,

T.

e.

nOJIaraH

h

('t)

= h[s] ('t)/c

s

'

nOJIY'IHM

cpa3Y

H3

(16.4)

<p

[h ('t')j =

<p

[_1

JP]]

=

_1

<p

[h[s]] = 1,

C C

s s

'ITO

H ;n;ORaS:bIBaeT

BbmOJIHeHHe

(16.1)

npH

TalWM

SHUqeHHH

~

=

s.

,,_-,-"-,---,,-"-,c.:.c~,-,c.;...cc....--.:..:....:c-,-"_':..':..c...'

.:....:..-"

.

...:._.~:-.:...

..

~.-'-

.

...:._.:~.::,:~~:~:::

•.

:

::

•.

',

,':'

",':,

.

:.,;.

',.:' .

~'

','

":"':'

''',',':''.

'k"'!

",,

C

.

',.,.

..

:..~f

104

OBllJ,EE

PEIDEHI1E

8A.D:A'{I1 OB

ynPABJIF.HJ1M

lrJl. 5

IIpeArrOJIOlliHMTerrepb,

'ITO

C

s

=

O.

ECJIH

6hI

rrpH

3TOM,

BOrrpeKH

Hame-

My

yTBepn'AeHHIO,

He

BhIrrOJIHHJIOCb

paBeHCTBO

(P

[heslJ

= 0, TO

He

MOrJIO

6hI

BhIrrOJIHHTbCH

H

yCJIOBHe

(16.4)

AJIH

/u060u

cPYHK[~HH

h

(t)

H3

fP.

B

caMOM

AeJIe,

AorrycTHM,

'ITO

cp

[hrslJ

= a

=1=

O.

IIo

CBOHCTBaM

JIHHeHHOcTH

orrepa~HH

cp

HMeeM:

n n

cp

[h

(t)J =

cp

[2]

li

Mil

] +

lscp

[MslJ

=

cp

[~

lih[il] +

Lsa.

(16.15)

t,cs

Ho

rrpH

C

s

= 0

TJHCJIO

L8

B (16.2), (16.3)

MOlliHO

BhI6HpaTb

/uo-

6bl.M

6e3

H3MeHeHHH

APyrHx

TJHCeJI Li' a 3'1'0

c03AaeT

rrpOTHBOpeTJHe

MelliAY

(16.4) H (16.15).

IITaK,

AeHCTBHTeJIbHO,

yCJIOBHe

(16.4)

06ec-

rreTJHBaeT

BhIrrOJIHeHHe

(16.1).

.

Terrepb

AJIH

AOKa3aTeJIbCTBa

JIeMMhI

OCTaeTCH

rrOCTpOHTb

orrepa-

~HIO

cpo

[hl,

IWTopaH

YAOBJIeTBOpHJIa

6hI

yCJIOBHHM

(16.4) H (16.14).

CAeJIaeM

3'1'0.

3aMeTHM,

rrpelliAe

Bcero,

'ITO

MHOllieCTBO

fP{h},

orrpeAeJIeHHOe

yCJIO-

BHHMH

(16.2)

JiI

(16.3),

eCTb

MHO}KeCTBO

MOClwe.

HarroMHHM

(CM.

§ 61),

'ITO

MHOllieCTBO

..Jtt

3JIeMeHTOB

h

(t)

H3

fE

{h (t)}

Ha3bIBaeTCH

rrJIOCKHM,

KorAa

(<UpHMaH»

hh

(r)

=

'A.h(l)

(t)

+

(1

-

'A.)

h(2)

(,r)

(-

00 <

'A.

<

00),

rrpOBeAeHHaH

TJepe3

JII06hIe

ABe

«TOTJKH»

h(l)

H

h(2)

H3

..Jtt,

JIelliHT

BceMH

CBOHMH

TOTJKaMH

hh

B

..Jtt.

IIOHHTHe

rrJIOCKOrO

MHOllieCTBa

rrepeHOCHT,

CJIe,zi;oBaTeJIbHO,

B

rrpOCTpaHCTBO

cPYHK~HH.

rrOHHTHe

06hITJHOH

rrJIOCKO-

CTH

TpeXMepHoro

rrpOCTpaHCTBa

(CM.

CHOBa

pHC.

16.1,

KOTOphIH,

KO-

HeTJHO,

HMeeT

BeCbMa

YCJIOBHhIH

CMhICJI).

TOT

cPaKT,

'ITO

MHOllieCTBO

fP

BHAa

(16.2), (16.3),

AeHCTBHTeJIbHO,

HBJIHeTCH

MOC1£UM,

rrpOBepeH

B § 61,

KYAa

MbI

H

OTChIJIaeM

TJHTaTeJIH.

IITaK,

HaM

HaAJIelliHT

rrOCTpOHTb

orrepa~HIO

cpO

[hJ,

KOTopaH

Ha

3JIeMeHTax

h

H3

rrJIOCKOrO

MHOllieCTBa

fP

rrpHHHMaJIa

6hI

3HaTJeHHe

1 H

HMeJIa

6hI

HOpMy

p*

[cpoJ

=

11

pO,

rAe

pO

- 3TO

KaK

pa3

paCCTOHHHe

0'1'

fP

AO

TOTJKH h

(t)

=

O.

,D;JIH 3'1'01'0

rrOCTpoeHHH

paCCMOTpHM

B

rrpOCTpaH-

CTBe

fE{h (t)}

ccPepy

&0

paAHyca

p =

pO

C

~eHTpOM

B TOTJKe h

(t)

-

O.

MHaTJe

rOBopH,

paCCMOTpHM

COBoKyrrHoCTb

Bcex

cPYHK~HH

h ('t)

H3

fE

{h},

KOTophIe

YAOBJIeTBOpHIOT

HepaBeHcTBy

p

[h

('t)J <

pO

(CM.

pHC.

16.1).

TaK

KaE

B

rrpocTpaHCTBe

fE

{h}

paCCTOHHHe

0'1'

(<UJIOCKOCTH»

fjiJ

AO

«TOTJKH»

h ('t) =0

paBHO

KaK

pa3paAHycy

pO

CcPePhI

&0,

TO

3Ta

ccPepa

H

rrJIOCKOCTb

fP·

«1£aCalOmCJl»

APyr

APyra,

H 3'1'0

KacaHHe

rrpOHC-

XO)J;HT B TOTJKe

hO

('t),

HBJIHIOIqeHCH

Jt£Ul-tUMa.l/,blWU

cPYHK~HeH.

HOA06-

HhIX

TOTJeK

KacaHHH

MOllieT

6hITb

He

OAHa,

HO 3'1'0

He

BalliHO.

BalliHO

TO,

'ITO

ccPepa

&0

He

HMeeT

TaKHX

CBOHX

BHyTpeHHHX

TOTJeK,

KOTophIe

JIelliaJIH

6hI

Ha

fP.

MHaTJe

rOBopH,

BalliHO

TO,

'ITO

B

COOTBeTCTBHH

C

YC-

JIOBHeM

(16.7) B

cOBoKyrrHoCTH

fP{h}

HeT

cPYHK~HH

h ('t),

AJIH

KOTO-

phIX

p

[hJ

<

pO.

r

~

.

§

161

ITPOBJTEMA

MOMEHTOB

105

Terrepb

MhI

MOllieM

BOCrrOJIb30BaTbCH

TeopeMOH

H3

cPYHK~HOHaJIb

Horo

aHaJIH3a

0

paaOe.l/,ellUU

8bLnY1£.I/,oeo

JttnO:JlCecm8a

&0

U

MOC1£oeo

M1l0-

:JICeCm8a

fP,

KOTophIe

He

HMeIOT

06IqHX

BHyTpeHHHX

TOTJeK.

COrJIaCHO

3TOH

TeopeMe,

rrpHBeAeaHOH

H

pa3'bHCHeHHOH

B

IIpHJIO}KeHHH

H

HB-

JIHIOIqeHCH

OAHOH

H3

reOMeTpHTJeCKHX

cPOPM

cPYHAaMeHTaJIbHOH

Teope-

MhI

Xaaa

-

BaHaxa

[6*J

(CM.

§

53

H § 61),

MOlliHO

rrOCTpOHTb

orrepa-

~HIO

cpo

[hl,

KOTopaH

paaOMJlem

ccPepy

&0

H

rrJIOCKOCTb

fP

rrOBepXHO-

CTbIO

@,

rAe

cpo

[hJ

= 1.

CTporHH

aHaJIHTHTJeCKHH

CMhICJI

rrOCJIeAHerO

reOMeTpHTJeCKOrO

060pOTa

peTJH

orrpeAeJIHeTCH

yCJIOBHHMH:

cpo

[h ('t)J = 1

rrpH

h ('t)

H3

fP,

(16.16)

cpo

[h ('t)J < 1

rrpH

h

('t)

H3

&0.

(16.17)

TaKaH

«pa3AeJIHIOIqaH»

orrepa~HH

cpo

[hJ

H

HBJIHeTCH

HCKOMOH

orre-

pa~HeH

cp,

pa3pemaIOIqeH

rrp06JIeMY

MOMeaTOB.

,D;JIH

AOKa3aTeJIbCTBa

3'1'01'0

yTBeplliAeHHH

BCJIeACTBHe

(16.16) AOCTaTOTJHO

rrpOBepHTb

JIHmb,

'ITO

p*

[cpOJ

= 1/

p

o.

A

AJIH

3TorO'B

CBOIO

OTJepeAb

AOCTaTOTJHO

rrpOBe-

pHTb,

'ITO

paCCTOHHHe

p~

0'1'

TOTJKH h

('t)

=0 AO rrJIOCKOCTH @

paBHo

pO.

IIpoBepHM

3'1'0.

TaK

KaK

fP

COAeplliHTCH

B

@,

TO

pc;>

<

pO.

Ho

p"

He

MOllieT

6hITb

CTporo

MeHbme,

TJeM

pO.

B

caMOM

AeJIe,

B

CJIyTJae

HepaBeH-

CTBa

pc;>

<

pO

MHOllieCTBO

@

COAeplliaJIO

6hI

TOTJKH h ('t), Y

KOTOphIX

P

[h

('t) J<

pO,

T.

e.

MHOllieCTBO

@

COAeplliaJIO

6hI

BHyTpeHHHe

TOTJKH

H3

CcPePhI

&0,

a 3'1'0

rrpOTHBOpeTJHT

HepaBeHcTBy

(16.17).

MTaK,

HaMH

rrOCTpoeHa

orrepa~HH

cpo

[hJ,

KOTopaH

YAOBJIeTBOpHeT

BceM

yCJIOBHHM

JIeMMhI

16.2.

Ha

3TOM

AOKa3aTeJIbCTBO

JIeMMhI

aaBep-

maeTCH.

CpaBHHTeJTbHaH

rrpOCTOTa

rrpHBeAeHHoro

AOKa3aTeJIbCTBa

He

AOJIlli-

Ha

BBOAHTb

TJHTaTeJIH

B

aa6JIylliAeHHe,

TaK

KaK

OCHOBHaH

TPYAHOCTb

3'1'01'0

AOKaaaTeJIbCTBa

OGTaeTCH

CKphITOH

B

HCrrOJIbaOBaHHOH

aaMHTeo-

peMe

XaHa

-

BaHaxa

0 CYUfecm80eanuu

rHrreprrJIOCKOCTH

@,

OTAeJIH-

IOIqeH

MHOllieCTBa

&0

H

fP.

3Ta

TeopeMa,

HMeIOIqaH

HCHhIH

HarJIHAHhIH

CMhICJI,

Tpe6yeT,

OAHaKO,

TpYAHoro

CTpororo

AOKaaaTeJIbCTBa,

KOTopoe

MOlliHO

HaHTH

B

Kypcax

cPYHK~HOHaJIbHOrO

aHaJIHa

a [6*,

14*

J.

Pe3IOMHpYH

Bce

rrpeAhIAyIqHe

paccylliAeHHH,

rrOJIyTJaeM

CJIeAYIO-

~YIO

OCHOBHyIO

TeopeMy

0

pei:rreHHH

rrp06JIeMhI

MOMeHTOB.

Teo

p e M a

16.1.IIp06.lleMa

JtWMenm08

UMeem

peutenue

(po

[h ('t)J

moeoa U

mQ.l/,b1£O

moeoa,

1£oeoa

'IUC.I/,O

pO

(16.7)

nQ.l/,O:JICUmMbno.

IIpu

amOM

1l0pM,a

p*

[cpoJ

OnmUMa.l/,blloL"i

paapeutalO/lfeU onepaq,uu

cpo

[hJ

OnpeOMJlemCJl

ua

pa8ellCm8a

pOI<

[cpoJ

='~

.

p

OCTaHOBHMCH

Terrepb

Ha

Borrpoce

0

TOM,

KaK

n,e

cPaKTHTJeCKH

CJIe-

AyeT

HCKaTb

orrepa~HIO

cpo

[h ('t)J

rrpH

pemeHHH

TOH

HJIH

HHOH

KOHKpeT-

HOH

rrp06JIeMhI

MOMeHTOB.

OTJeBHAHO,

TOT

crroc06,

KOTOphIH

MhI

'';

•.

,

.",';'

,,:,_~,-~~_.

,

..

'.'.-

.

'L

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,

__

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.;.','

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...

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••••

(2

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-:':"-1

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10ti

OEJ.IJ;EE

PEll1~HME

3AAA

1

1H

OE

YilPADJIEHlIII

[rJI.

5

npnM.OHHJIH

npH

nOCTpOeHHH

qP

[h] B ,o;OKaaaTOJLbCTBe

JIOMMM

16.2, ,o;JIH

aT

on

~OJIH

COBopmOHHO

HO

ro,o;HTCH,

TaK

KaK

OH

HocnT

xapaKTOp

'InCTO-

ro

MaTOMaTn'IOCKOrO

paccym,o;OHHH

0

cym;OCTBoBaHHH

cpo.

IIoaToMy

Con'Iac

M.bI

yKamoM

O,o;HO

O'IOHb

BamHOO

ceoucmeo

Ma/f,CUMY~W,

KOTO-

p.bIM

o6JIa,o;aoT

OnTHMaJIbHaH

paapomarom;aH

OnOpaIJ;HH

cpo

[h] H

KOTOpOO

HBJIHOTCH

CHJIbH.bIM

no,o;cnopbOM

npn

nOCTpOOHnH

onopaIJ;HH

cpo

B

acP<peKTHBHon

cPOpMO.

Ilpn

aTOM

OCHOBHyro

POJIb

6y,o;OT

nrpaTb

Ta

MHHHMaJIbHaH

<PYHKIJ;HH

n

h

O

(t)

=

~

I~Mi]

(t)

(16.18)

i=1

na

ceMOHCTBa

(16.2),

Ha

KOTopon

,o;OCTHraOTCH

MHHnMyM

(16.7). 3,o;OCb

'InCJIa

l~

CYTb

pomOHnH

aa,o;a'In

(16.7).

IIoJI03HO

nOMHHTb,

'ITO

no

onpo-

,o;OJIOHHIO cPYHKIJ;HH

hO

('1:)

oe

HopMa

p

[hO]

y,o;OBJIOTBOpHOT

paBoHcTBy

p[hO]

=

pO.

HyrKHOO

HaM

CBonCTBO

MaKcHMyMa

BMBo,o;nTCH

CJIO,o;yIO-

m;HM

o6paaoM.

PaapomaIOm;aH

onOpaIJ;nH

cpo

y,o;OBJIOTBOpHeT

paBoHcTBy

(16.4)

cpo

[hO

(1:)]

=

1,

(16.19)

npn'IOM

p*

[cpO]

= 1/

p

o.

ECJIH

Tonopb

paCCMOTpOTb

KaKyro-JIH60

,o;py-

ryIO

onepaIJ;Hro

cp,

OTJIH'IHYIO

OT

cpo,

HO

HMOIOm;yro

Ty

rKO

caMyro

HOp-

My

p*

[cp]

=

1/pO,

TO

no

HaBocTHoMy

CBoncTBy

HOpMM

(CM.

§ 57)

1

nOJIyqHM

cp

[hO]

-<

p*

[cp]

p

[!to]

=

popO

= 1. CJIo,o;OBa

TOJIbHO,

C

y'IOTOM

(16.19)

nMOOM

HepaBeHCTBO

cp

rhO]

-<

cpo

rhO]

= 1.

JtlTaK,

cnpaBe,o;JIHBO

CJIO,o;YIOm;OO

yTBopm,o;eHHo,

IWTOpOO

B

COOTBeT-

CTBnH

C

yCTaHoBnBmoncH

TOpMHHOJIOrHOn,

HaaOBOM

npHHIJ;nnOM

MaK-

cHMyMa.

II

p H H

IJ;

H n M a K C H M Y M

a.

OnmUMa/lbT-laSl

paapemalOUfaSl

onepalfUSl

cpo

[h

('1:)],

UMeloUfaSl

T-lOpMy

p*

[cpO]

= 1/

pO,

eblOMSlemcSl

cpeou ecex

opyeux

/lu/-leUT-lblX

onepalfuu

(p

[h ('r)] c mou

JlCC

T-lOP~tOU

p*

[cp]

= 1/pO

c/leoylOUfUM

ceoucmeOM

Ma/f,CUMYMa

lW

MUHUJlW/lbl-lO.U

fPYT-l/f,lfUU

hO

('1:):

cpo

rhO

(t)]

=

max

cp

[h

O

(t)]

npH

p*

[cp]

=

-0

1

.

(16.20)

'" . p

IIo,o;'IepKHOM,

'ITO

nOCJIO

onpo,o;OJIOHHH

MHHnMaJIbHOn

<PYHKIJ;HH

h°(ot-)

B

mnpOKOM

KJIaCCe

CJIy'IaOB

npHHIJ;Hn

MaKcHMyMa

nOaBOJIHOT

06M'IHO

60a

60JIbmHx

aaTpy,o;HoHHn

HaHTH

JIBH.bIH BH,o;

OnOpaIJ;HH

(pO

[h

(or)].

B

aTOM

'IHTaTOJIb

CMomOT

y60,o;HTbCJI

Ha

paCCMaTpnBaOM.bIX

HHme

npnMepOB

(CM.

§§

18-23).

O,o;HaKo,

KaK

M.bI

yBn,o;HM

na

aTHX

me

npHMepOB,

onpe,o;eJIeHHO

caMon

MnHHMaJIbHOn

<pYHKIJ;Hn

hO

(or)

MomeT

,o;OCTaBJIHTb HOMaJI.bIe

BbI'IHCJIHTeJIbH.bIO

Tpy,o;HOCTH.

IIoJIeaHo

om;e

OT-

MeTHTb,

'ITO

B pH,o;O

CJIy'IaOB

BMOCTO

'IHceJJ

it

onpe,o;OJIHIOm;HX

MHHH-

r.··,

('~.'_::.:

§

16]

IIPOBJIEMA

MOMEHTOB

107

MaJIbHyro

cPYHKIJ;Hro

hO(t)

,

y,o;o6Heo

nCKaTb

'InCJIa

k1,

,o;OCTaBJIJIIOm;HO

pomeHHe

aa,o;a'IH:

HanTH

n

VO

=

max

2j

kiCi

(16.21)

k

i=1

n

rrpH

yCJIOBHH,

'ITO

pO

[~

kiMi](t)J

-<

1.

t=1

qHCJIa

kt

pemaIOm;He

aa,o;aqy

(16.21),

6y,o;yT

CBJIaaH.bI

C

qHCJIaMH

l~,

onpe,o;OJIHIOm;HMH

MHHHMaJIbHYIO

cPYHKIJ;HIO

hO

(t)

(16.15),

paBeHcTBaMH

k~

= lio/po.

IIpH

aTOM

VO

= tlpo.

(B

CJIyqae

pO

= 0

nOJIy'IaeTCJI

n

sup

~

kio

i

=

00.)

,I(oKaaaTOJIbCTBO

nOCJIe,o;HHX

cooTHomeHHH

MbI

npe-

i=1

AOCTaBJIJIeM

qHTaTeJIro

B

KaqOCTBe

ynpamHOHHH.

<DYHIn~HIO

h"

(t)

=

n

=

2J

k~h[i]

(t)

=

hO

(t)/po

HaaOBeM

fJ/f,CmpeMMbHou

cPyHKIJ;HOH.

i:::::=l

IIpuMep

1"6.1.

Terrepb B

I<aQCCTBe

HJJJIIOCTpa~HH

paccMoTpHM peuIEiIIlIC

rrpHMepa 15.1, orrHpMlcb Ha TeopeMy 16.1 H

rrpHH~Hrr

MaHcHMyMa.

IIycTb

B

aTOM

rrpHMepe HpHBaJI S= S(t),

T]

=

T]

(t)

orrpe,n;eJIJIeTCJI

ypaBHeHHJIMl1

S(t) =

t,

T]

(t) = 1 - t (0 < t < 1),

Hoa¢¢H~HeHT

~

paBeH

e,n;HHH~e

H

c

1

= - c

2

= 1. Tor,n;a

HMeeM

CJIe,n;yIO~YIO

rrpo6JIeMY

MOMeHTOB

(15.3), (15.4):

1

~'t'uO('t')d't'=1,

~

(1

-

't')

u

O

('t')

d't'

= - 1,

(16.22)

o

°

max

Iu

O

('t') I=

min.

(16.23)

O~"~1

<J)YHJ{~lUl

h(l)

('r) =

't',

h(2)

('t')

= 1 -

't'

JIlmeHrrO HeaamICHMbI.

COrJIaCHO

aaKJIIOqeHHIO,

c,n;eJIaHHoMy

B

_'KoH~e

o6cym,n;eHHJI rrpnMepa 15.1, rrp06JIeMY

MO-

MeHTOB

Hap;o pacCMaTpHBaTb.

ap;eCb

B rrpOCTpanCTBe 2

{h

('t');

0'<

r

-<

1},

rile

HopMa p

orrpep;eJIJIeTCJI

paBenCTBOM

1

P [h ('t')] =

~

Ih ('t') I

d'l:.

o

BbIqHCJnIM BeJIH'lHHy

pO

(16.7).

11

MeeM:

1

pO"",

m/in

~Ill'l:

+

l2

-l2'1:

I

d'l:

npn

II

-

l2

= 1

o

lIJIH

1

pO

=

min

~

I

l'

+

l21

d'l:

npn

-

00

<'

2

<

00.

I,

0

"i'W

e

..;

'

:_,~~.

-"-'

.

'.'

.....

_..

. .

..

.

'.'

...

'.

~.~

...

....

: .

)'

..

,.~.~"

c

···

..

c!

':':,

~

»

108

OBru:EE

PEIIIEHHE

3A)J;Aqn

OB

YIIPABJIEHHH

[rJI.

5

IIocJIe~lU[H

Sa~aqa

llMeeT

pemeHlle

l~

= 0,5,

19

=

-0,5,

pO

= 1/4.

8'1'0

Bll~HO

liS

pllC.

16.2.

)J;eicTBllTeJIbHO,

samTp:rixoBaHHaJI

IIJIOrn;a~b,

paBHaJI

RaK

pas

MllHllMllSJilpyeMoMy

llHTerpaJIY,

6y.o;eT,

OqeBll~HO,

HallMeHbmeR

npllZ2 =

zg

=

-0,5'.

COrJIaCHO

TeopeMe

16.1

HopMa

p*

[<pO]

lICROMOR

onepalvuI

<po

paBHa

. 1/po =

4.

)J;pyrHMll

CJIOBaMlI,

JIMeeM

max

h.

Iu ('r)l <

4.

1:IT06hI

HaRTH

onepa~llIO

<pO,

T.

T

e.

q'r06bI

HaRTH

tPYHK~HIO

uO('r)

B (16.22), (16.23),

BOCnOJIbsyeMCJI

CBOMCTBOM

MaRcHMYMa(16.20).

8'1'0

yCJIOBHe

MaRcllMYMa

B

~aHHOM

ROHRpeT-

HOM

CJIyqae

sarrHChIBaeTCJI

TaR:

I

1/

max

~

(1'

- 0,5) u

(1')

d1'

=

u °

1

=

~

(1'

- 0,5) u

O

(1')

d1'

PIlC.

16.2.

°

npll

Iu

(1')1

<

4.

PemeHJIe

nOCJIe,ll;HeR

Sa~aqll,

OqeBll~HO,

~aeTCJI

tPYHK~HeR

UO

(1')

= 4

sign

(1'

- 0,5), ,

ROTOpaJI

RycOqHO-nOCTOHHHa

II

llMeeT

TOq}\y

paspbIBa npll

l'

= 0,5.

IfTaK,

B

3TOM

n npep;IiIp;ym;eM

naparpa<pax

MIiI

BLUlCHnJIH

CJIep;yIO-

m;ee.

3ap;aqa 06 yrrpaBJIeHnn nMeeT

CMIiICJI

np06JIeMIiI

MOMeHTOB.

TIpn

p;OnOJIHHTeJIbHIiIX~YCJIOBnJIX

co

CBOHCTBaMn

BIiIrryKJIOCTn (KaKoBIiIM

yCJIOBneM

JIBJIJIeTCJI

RaK

paa

Harne Tpe60BaHne MnHnMyMa HOpMIiI

p*

[u]

=

p*

[<Pul)

ynOMJIHyTaJI np06JIeMa paspernaeTCJI onepan;neii

<pO

(JInHeHHIiIM

¢YYi;quoHa.llO.A~),

OTp;eJIJIIOm;eH

BIiInyKJIOe

MHomeCTBO

.?J>{h}

(16.2),

(16.3)

n c<pepy

&0

{h}

(p

[h] <

pO).

B SaKJIIOqeHne 3Toro naparpa<pa

Cp;eJIaeM

em;e

Tpn

nOJIeSHIiIX

SaMeqaHnJI.

1)

lIs

TeopeMbI (16.1)

CJIep;yeT,

qTO

yCJIOBne

pO>O

eCTb

Heo6xop;nMoe

n

p;OCTaTOqHOe

yCJIOBne

P;JIJI

paspernnMOCTn np06JIeMIiI

MOMeHTOB.

Ho

BTO

yCJIOBne

06JISaTeJIbHO

BIiIrrOJIHeHO,

eCJIn

<pyHKn;nn

h[i]

('t) JIn-

n

HeHHO

HesaBncnMIiI,

n60

Torp;a p

[~lih[i]]

> 0

npn

li + ... +

l;

=1=

i

=1=

O.

CJIep;OBaTeJIbHO,

JInHeHHaJI HesaBncnMOCTb

<pyHRn;nH

h[i]

(c)

eCTb

(Jocmamo'/{,Hoe

yCJIOBne

paspernnMOCTn np06JIeMIiI

MOMeHTOB,

Ka-

KOBIiI

6IiI

Hn

6IiIJIn qnCJIa

Ci'

2)

IIycTb

<pyHKn;Hn

h[i]

(c)

He

JIBJIJIIOTCJI

JInHeHHO

HesaBncnMIiIMn,

HO

qnCJIa

Ci

B ypaBHeHnJIX

(16.1)

TaKOBIiI,qTO

pO

>

O.

Torp;a ncxop;-

HYro rrp06JIeMy

MOMeHTOB

MomHO

CBeCTn

K aHaJIOrnqHOH np06JIeMe,

rp;e,

op;HaKO,

6YP;yT

<pnrypnpoBaTb JInrnb

JInHeHHO

HesaBncnMIiIe

<pYHKn;nlJ

h[iJ

('t), BIiI6paHHIiIe

ns

cOBoKynHocTn Bcex ncxop;HIiIX

<PYHK-

n;nii

h[i]

(.).

B

caMOM

p;eJIe,

npep;nOJIOmnM

CHOBa,

KaK n BIiIrne,

Ha

CTp.

102,

qTO

JInHeHHO

HesaBncnMIiI JInrnb <pyHKn;nn h[j]

(.)

109

§

16]

rrpOBJIEMA

MOMEHTOB

(j

=

1,

...

,

k)

H BIiInOJIHeiIbI

COOTHOrneHnJI

(16.8).

ECJIn

n n

(16.24)

pO

=

min

p

[2J

lih[i]

(t)]

> 0

npH

~

liCi

==

1,

i=l

TO

06JISaTe.rrbHO

p;OJImHIiI

BIiInOJIHJITbCJI

COOTHOrneHHJI

k

C'i=

~

c/J..

ij

(i

>k).

(16.25)

;=1

,n;eHCTBnTeJIbHO,

B npOTnBHOM CJIyqae

COrJIaCHO

(16.9)

n

(16.10)

Monwo

6IiIJIO

6i.I BIiI6paTb qnCJIa

l;

= 0 n

li

=/=

0 TaR, qTo6IiI yp;OBJIeTBOpnTb

n

paBeHcTBy

(16.10)(T.

e.

paBeHcTBy

2J

liCi

=

1)

n nOJIyqnTb

pO

=

.

i=l

k

= p

[2J

l;

h[j] ] = O.TIpoTnBOpeqne C

(16.24)

n

p;OKaSIiIBaeT

(16.25).

Ho

;=1

B CJIyqae op;HOBpeMeHHoro BIiInOJIHeHHJI

(16.8)

n

(16.25),

OqeBHp;HO,

p;OCTaTOqHO

yp;OBJIeTBOpnTb Jlnrnb

nepBIiIMk

ypaBHeHnJIM

(16.1),

n60

OCTaJIbHIiIe

n - k paBeHcTB

BIiInOJIHJITCJI

Torp;a aBTOMaTnqeCKH.

TaRnM

06pasoM, rrp06JIeMa

MOMeHTOB

(16.1),

p;eHCTBnTeJIbHO,

CBOp;nTCJI

B

p;aHHOM

CJIyqae K np06JIeMe

<po

[Mil] =

Cj

(j

=

1,

...

, k), (16.26)

p*

[<pO]

=

min.

(16.27)

BamHo

nop;qepRHyTb

npn

3TOM,

qTO

qnCJIO

pO,

KOTopoe

MbI

HaHp;eM,

pernaJI B

COOTBeTCTBnn

C TeopeMoH

16.1

np06JIeMy

MOMeHTOB

(16.24),

(16.27),

COBnap;eT,

ROHeqHO,

C

qnCJIOM

pO,

ROTopoe nOJIyqaeTCJI

npn

perneHnn ncxop;HoH np06JIeMIiI

MOMeHTOB

(16.1),

(16.27).

8TO

HeTpyp;-

HO

Bnp;eTb n Henocpep;cTBeHHo,

nCXop;JI

ns

COOTHorneHnH

(16.9)

n

(16.10),

yqnTIiIBaJI,

qTO

BTopaH cyMMa B

JIeBOH

qaCTH

(16.10)

BCJIep;CT-

Bne (16.25)

eCTb

HyJIb. .

3)

IIpo6.1£eMY

.MO.MeHm06

(15.1),

(15.3)

MomHO

TpaKTOBaTb KaR

uaonepU.Mempu'/{,eCK,YlO

aa(Ja'/{,y

ns

Bapnan;noHHoro nCqHc,[[eHnH

(CM.

[4*],

CTp;

48),

rp;e Hap;JIemnT onpep;eJInTb MnHnMyM

<pyHRn;nOHaJIa

p*,

T.

e.

p*

[u ('t)] =

min

(16.28)

npn

p;OnOJIHnTeJIbHIiIX

yCJIOBHHX

til

~

h[i]'

('t) u

(t)

dt

=

Ci

(i=1,

...

,n).

(16.29)

tot

OnHcaHHoe B

3TOM

naparpa<pe perneHne np06JIeMIiI

MOMeHTOB

TpaR-

TyeTcH Torp;a KaK perneHHe yKasaHHoH HsonepnMeTpnqeCKOH sap;aqn,

'.

--:..:.~.

..

-L.-

;.

:.'.,:::

:'.,-:~

..

,<

: } .'. :','

"::

..

::

":':'.,

.:.::.:::

..

~:

~

..

:.':~

..

~:~~\

'"

';

"

P,:';:-

C'

110

OBII(EE

PEmEHHE

aAAA'!H

OB

YIIPABJIEHHH

lrJl. 5

IIOJIylIeHHOe,

op;HaRO,

He

RJIaCCH'IeCRHM

IIyTeM,

HO

MeTop;aMH

<PYHRD;HO-

HaJIbHOrO

aHaJIHsa.

TaRoR

IIOP;XOP; B

CJIy'Iae

sap;a'I

ynpaBJIeHHH

no-

JIeaeH

no

CJIep;YIOID;HM

IIpH'IHHaM.

BO-IIepBbIX,

ocym;eCTBJIHeTCH

BOS-

MomHOCTb

OXBaTHTb

TaRHe

CJIyqaH

p

[U

(r)],

Rorp;a

BCJIep;CTBHe

Hep;o-

CTaTO'IHOR

rJIap;ROCTH

<PYHRn;lIOHaJIa

p

nplIMeHeHlIe

RJIaCClI'IeCRlIX

TeopeM

BaplIaD;lIOHHoro

lIC'IHCJIeHlIH

aaTpYP;HlITeJIbHO.

BO-BTOPbIX

(u

fJmo

eaJICT-tee)

,

pemeHlIe,

onlIcaHHoe

BbIme,

p;aeT

HBHoe

yCJIOBHe

MlIHlIMYMa

(16.7),

06JIer'IaIOm;ee

pemeHlIe

1I

lICCJIep;OBamte

xpaeeou

lJaiJa"lu,

COCTaBJIHIOm;eR

OCHOBHyIO

TPYP;HOCTb

np06JIeM

ynpaBJIeHlIH,

oc06eHHo

B

Tex

CJIyqaHx,

Rorp;a

Tpe6yeTcH

npOCJIep;lITb

BJIlIHHHe

Rpae-'

BbIX

YCJIOBlIR

Ha

peayJIbTaT

ynpaBJIeHlIH.

Op;HaRo

HCHO,

'ITO

XapaI{Tep

pemeHlIH

H OCHOBHbIe

COOTHomeHlIH,

onpep;eJIHIOm;lIe

ero,

HBJIHIOTCH

aHaJIOrOM

COOTBeTCTBYIOIl~HX

RJIaCCH'IeCRHX

cooTHomeHHR.

TaR,

Ha-

npHMep,

BeJIH'IlIHbI

lil

<pHrypHpYIOID;He

B

COOTHomeHlIHX

(16.2),

(16.7),

MomHO

TpaRTOBa'fb

RaR

MH01RHTeJIlI

JIarpaHma,

HCnOJIbayeMbIe

B

BaplIaD;HOHHOM

HC'IHCJIeHHH

IIpH

pemeHHH

rJIap;ROR

Sap;a'IH

BHp;a

(16.28), (16.29)

Ha

YCJIOBHbIR

MHHHMyM.

ITo

npH1JlIHaM,

yRasaHHbIM,

B

'IaCTHOCTH,

BO

BBep;eHHH

(CM.

CTp.

15),

MbI

He

OCTaHaBJIlIBaeMCH

sp;eCb

Ha

aTOM

Bonpoce

IIop;p06Hee.

B

p;aJIbHeRmeM,

B § 18,

6yp;eT

paccMoT-

peH

ROHRpeTHbIR

npHMep

HOpMbI

p*

[ul,

Rorp;a

yRasaHHaH

CBHSb

npo-

HBJIHeTCH

oc06eHHo

npOapa'IHO.

l.IHTaTemo,

ROTOpbIR

He

yp;OBJIeTBO-

pHTCH

ynoMHHyTbIM

IIpHMepOM,

MbI

peRoMeHp;yeM

06paTHTbCH

R

Op;HOMy

Ha

RypCOB

BapHaD;HOHHoro

HC'IHCJIeHIIH

(CM.,

HanpHMep,

[4*]), a

TaR-

me

R

RypcaM,

cop;epmam;I1M

0606m;eHlIH

RJIaCCH'IeCRIIX

nOHHTHR

B

p;y-

i

I

r

xe

HOBbIX

MeTop;OB

JIHHeRHOro

(HarrpHMep,

[28*,29*]).

H

HeJIHHeRHOro

nporpaMMlIpOBaHlIH

§ 17.

PemeHue

aa~a1JlI

06

ynpaBJIemUl

BepHeMcH

BHOBb

R

HameR

aap;a'Ie

13.1

06

OIITl'IMaJlbHOM

yIIpaBJIe-

HHl'I.

B § 15

nORaaaHo,

'ITO

aTa

sap;a'Ia

CBOP;HTCH R

np06JIeMe

MOMeH-

TOB. B

npep;bIp;ym;eM

IIaparpa<pe

MbI

HSYlIIIJIII

IIpo6JIeMy

MOMeHTOB

H

C<pOpMyJIIIpOBaJIH

TeopeMy

16.1 H

npHHD;Hn

MaI<cHMyMa,

xapaRTepH-

aYIOm;I1e

ee

pemeHIIe.

Tenepb,

CyMMHpyH

peayJIbTaTbI

§§

15

II

16.,

no-

JIyqaeM

CJIep;yIOm;ee

yTBep)l{p;eHIIe.

Teo

p

eM

a 17.1. 3aiJa

t

ta

13.1

UMeem

peuteT-tue

moeiJa

u

mOJl,bXO

moeiJa,

1f,oeiJa

iJ/lJL

MUT-tU;I,W/lbT-tOU

rfiYH1f,1-fUU

hO

(r), nocmpoew,lOu

COMac-

1-tO

(16.7),

ebLnOJl,HJLemCJL

yc./ioeue

p

[hO

1=

pO

>

O.

If

T-tmeT-tCUeT-tOCmb

OnmUMa/lbT-tOeO

ynpa8lleflUJL %

[uOl

yiJo8llemeopJLem

paeel-lcmey %

[uOl

.-:

= 1/

pO,

U fimo ynpae/lerlUe o6/laiJaem

ceoucmeOM

Ma1f,CUMYMa

t~

~

h

O

'

('t')

UO

('t')

d't'

=

tat

max

U

~

h

o

'

('t')

U

('t')

d't'

tat

(17.1)

npu

p*

[u]

= 1/

p

o.

':.::'.:,

C

111

17]

PEIIIEHl'IE

aAp:A'-Ul

OB

YIWADJlEHlll'I

OnHpaHCb

Ha

ny

TeopeMy,

MbI

p;ap;HM

cen.lIac

paoolIee

npanIlJIo

P;JIJl

pemeHHH

Sap;a'IH

13.1.

ITpemp;e

Bcero,

BbITll1meM

HBHoe

BblpameHHe

P;JIH

BeRTop-<pYHRD;HH

n

h

('t')

=

2J

lih[i]

('t'),

i=l

cpep;lI

ROTOpbIX

lIm;eTCH

MHHHMaJIbHaH

<PYHRD;HH

hO

(or).

BBOP;H

060·

SHa'IeHHe

l P;JIH

BeRTopa-CTOJI6D;a

c

ROMnOHeHTaMH

It

(i

= 1,

...

,

n)

1I

BcnOMHHaH,

'ITO

h[i]'

('t)

CYTb

CTpOKH

h[i]'

[tfJ,

't'l

MaTpIlD;bI

H

[t/3,

't],

nOJIyqHM

P;JIH

BeRTopa-CTpORII

h'

('t) CJIep;YIOm;yIO

MaTpH'IHyIO

sanHCb:

h' ('t) =

l'

H

[tfJ,

'tJ.

,IJ;aJIee,

COrJIaCHO

(6.1)

HMeeM:

H

[t/3,

'tl = X

[t/3,

'tl B ('t).

TaR

RaR,

HaRoHeD;,

X

[t,

't1 =

X-l

h,

t l,

TO

OROH'IaTeJIbHO

nOJIy'IHM:

h' ('t) =

l'X-l

['t,

t{31

B ('t).

TpaHcIIoHHpyH

h'

('t)

no

CBORCTBaM

MaTpH'IHbIX

npOl1SBep,OHl1H,

Hail-

~eM:

h ('t) =

B'

('t)[X-l

['t,

t~Jl'l.

(17.2)

MaTpHD;a

[X-l

[t,

t{3Jl'

HBJIHeTCH

<pYHp;aMeHTaJIbHOH

MaTplID;eR

S [t,

t{3]

P;JIH

ypaBHeHHH

(CM. § 10, B

'IaCTHOCTII,

ypaBHeHIIe

(10.3))

S = -

A'

(t) s. (17.3)

CJIep;OBaTeJIbHO,

COMHOlliIlTeJIb

S ('t) =

[X-l

['t,

lpll'l

= S ['t,

l{3]Z

npH

aap;aHHOM

l

eCTb

He

111'0

HHoe,

RaR

HeROTopoe

lIaCTHoe

pemeIUle

ypaBHeHHH

(17.3),

COIIpH)I{eHHOrO

I{

ypaBHeHHIO

(5.2).

BOJIee

Toro,

npH

't =

t{3

HMeeM

S

(t{3)

= l.

Ho

Torp;a

ORaSbIBaeTCH,

'ITO

MHO}ReCTBO

,BeRTop-<pyHRD;HR

h ('t)

orrpep,eJIHeTCH

paBeHCTBOM

(h

('t) =

B'

('t) S ('t),

rp;e

BeRTop-<pyHRD;HH

S ('t)

CYTb

BceBOSMomHble

pemeHIIH

CHCTeMbI

(17.3).

TaRHM

06pasoM,

sap;a'Ia

BHp;a

(16.7)

06

onpep;eJIeHIIH

MIIHH-

MaJIbHOrO

aJIeMeHTa

hO

('t)

<pOpMyJIIIpyeTcH

TeIIepb

cJIep;yIOID;HM

06pa-

SOM:

HaRTH

(17.4)

pO

=

min

p

[B'

('t')

s

('t')]

= P

[B'

('t')

SO

('t')]

s

UpH

RpaenoM

YCJIOBHII

C'1i

(t{3)

=,

1,

•

,_¥~_'

••

~_;:

.........

~~.:.....1....__'_'

...

,.

."."

•••

.:.~.:,~

";";"

!

•••••

".,".;"

.';.,

c~

c

····;

'-'

112

DBllI)J;E

PEmEHHE

3A,lJ;AQH

DB

YIIPABJIEHHH

[rJI.

5

rAe

C -

BeKTOp-CTOJI6en;

{c

I

,

...

,

cn}

(15.2).

CMIilCJI

3aAaTJH (17.4)

3a-

KJIIOTJaeTCH,

CJIeAOBaTeJIbHO,

B

TOM,

TJT06Iil

H3

BCeX

ABHmeHHH

CorrpH-

meHHOH

CHCTeMIil (17.3),

CTeCHeHHIilX

KpaeBIilM

yCJIOBHeM

c's

(tfJ)

= 1,

BIil6paTb

MUl-lu.Ma/tblwe

ABHmeHHe

SO

('r), rrpHTJeM h

O

('r)

=

B'

('r)

SO

(T).

TaRHM

06pa30M,

TeopeMa

17.1

AOCTaBJIHeT

CJIeAYIOm;ee

rrpaBHJIO

AJIH

pemeHHH

3aAaTJH 13.1.

II

p a B H JI 0 M H H H

MaR

C a 17.1. /f/tJl

mozo

"lm06bL

pezuumb

3a8a"l.y 13.1 , c/te8yem

nocmpoumb

cucmeMy (17.3),

conpJl~el-ll-lYlO

l'i, 8al-l-

1/,OU,

u paccMompemb

Bel'i,mOp-(/jYl-ll'i,q,uu

h

(T)

=

B'

(T)

S

(T),

z8e

s

(T)-

88U~el-lUJl

CUCmeMbL

(17.3),

C1'(leCl-le1l1-lbLe

YC/tOBUeM

c's

(t~)

= 1. Cpe8u

amux

88U~el-lUU

l-la8/te~um

l-laumu

MUl-lUMa/tbl-lOe

88U~el-lUe

SO

(T),

8/tJl

l'i,omopozo

BUnOltl-lJlemCJl

YC/tOBue

(17.4).

3a8a"l.a 13.1

UMeem

pezuel-lue moz8a u

mOltbl'i,O

moz8a,'

l'i,oz8a

p

[B'

(T)

sO

(T)1

=

pO

>0.

IIpu

amOM

OnmUMMbl-lOe

ynpaB/tel-lUe

/1,0

(t)

UMeem

l-lOpMy

p*

[UO]

= 1 /

pO

u cpe8u

Bcex

8onycl'i,aeMbLx

y,npa8/tel-lUU

u (t) C

l-lOpMOU

p*

[u]

=

lI

p

o

Bu8e.ttJlemcJl

YC/t08UeM

Mal'i,Cu.MyMa

(17.1)

l-la

MUl-lUMMbl-lOU

(/jYl-ll'i,q,uu

hO

(T)

=

B'

(T)

SO

(T).

II

pHBeAeM

em;e

OAHY

4>OPMYJIHPOBKY

rrpaBHJIa

AJIH

pemeHlIH

3aAa-

TJH

13.1.

RaR

6IilJIO

OTMeTJeHO B § 16 (CM. c>rp. 107), BMeCTO 3aAaTJH

(16.7)

(HJIH

(17.4»

MomHO

pemaTb

3aMeHHIOm;yIO

ee

3aAaTJy

(16.21)

n

06

orrpeAeJIeHHH

3KCTpeMaJIbHOH

4>yHRn;HH h

a

(T)

=

~

k~

h[i]

(T).

i=l

::ha

BTopaH

3aAaTJa B

0603HaTJeHHHX

HaCTOHm;ero

rraparpa4>a

BIilrJIHAHT

TaK:

HaRTH

,,0

= max c's (tfl)

8

=

c'sa

(tfl)

(17.5)

rrpH

yCJIOBHH

P

[B'

(T)

S

(T)]

<1.

,1J;BHrKeHl-Ie

sa

(T),

pa3pemaIOm;ee

3aAaTJy

(17.5) H

rropomAaIOm;ee

al'i,CmpeMa/tbl-lYlO

(/jYl-ll'i,q,UlO

h

a

(T)

=

B'

(T)

sa

(T),

Ha30BeM

al'i,CmpeMa/tb-

I-tbLM

88u~ellueM.

HarrOMHHM,

'ITO

h

a

(T)

= h

O

(T)

/

pO

H

rrOTOMYSa

(T)

=

SO

(T)

/

pO.

TaKHM

rryTeM

rrpHAeM

R CJIeAYIOm;eMY

BIilBOAY.

II

p a B H JI 0 17.2.

/f/tJl

mozo

"lmo6u pezuumb

3a8a"l.y

13.1, c/te8y-

em nocmpoumb cucmeMy

(17.3),

conpJl~el-ll-lYlO

l'i, 8al-ll-lou, u

paCCMom-

pemb

Bel'i,mOp-(/jyl-ll'i,q,uu

h

(T)

=

B'(T)

S

(T),

nocmpOel-ll-lbLe

no

Bce803-

MO~l-lbLJn,

8BU~el-lUJlM

s

(T)

CUCmeMbL

(17.3) u

cmeCl-lel-ll-lbLe

l-lepaBel-lCmeOM

p (h(T)] < 1. Cpe8u

amux

(/jYl-ll'i,q,uu

l-la8/te~um

l-laumu

(/jYl-ll'i,q,alO

a

k

(T)

=

B'

(T)

sa

(T)

=

B'

(T)

S

[T,

tfJJ

k

a

,

nopo~8el-ll-lYJO

al'i,CmpeMMb-

llbLM

8BU:ncel-lUeM

Sa(T),

l-la

l'i,OmOpOM

80cmuzaemCJl

Mal'i,CuMyM

(17.5).

3a8u"ta 13.1

u.Meem

pezuel-lue

moz8a u

mOltbl'i,O

moz8a,

l'i,oz8a

c'sa

(tfJ)

=

,,0

<

00.

IIpu

amOM

onmUMa/tbl-lOe

ynpaBltel-lue U

O

(t)

UMeem

llOpMy

1-'*

(uoJ

=

,,0

u cpe8u

Bcex

8onYCl'i,aeMbLX

ynpaB/tel-lUU U

(tl

C

l-lOPJllUU

p*

[uJ =

,,0

8bL8e.ttJlemcJl

YC/tOBUeM

Mo.l'i,CuMyMa

1/,0.

al'i,cmpe-

I,

I~

ii

•

·

•

§ 17]

PEmEHHE

3A,lJ;AQH

DB

YIIPABJIEHHH

113

MMbl-lOU

(/jyl-ll'i,q,UU

h

a

(T):

~

he

(T)

a

O

(t)

dt

=

max

~

he

(T)

U

(t)

dt

=

,,0

(17.6)

ta.

U ta.

npa

p* [aJ =

,,0.

IIpHMe'IaHHe

17.1.

BeJIH'lHHa

p[B'('r)s('t'),]

JIBJIJIeTCJI q>yHRIJ;HeR

<D

[II"",

In] OT

rrepeMeHHWx

Ii

=

si

(t/3)'

CJIe~OBaTeJILHO,

3a~a'IH

(17.4), H (17.5)

CYTL

06Ll'IHWe

KOaeQHOMepawe

3a~a'IH

Ha

yCJIOBHWR

sKcTpeMyM.

IIyCTL

CRCTeMa (17.3)

TaKOBa,

'ITO

BLIpameHHe

p

[B'

('t')

s ('t')] =

= P

[B'

('t')

S ['t',

tfJ]

s (tfJ)]

o6pam;aeTCH

B

HyJIb

JImrrL

rrpH

s (til) = 0,

JIHaqe

rOBopH,

nyCTb

BeKTop-q>YHKIJ;HH

h(i)

[til' 't']

JIHHeRHO

He3aBHCHMW.

Tor~a

BeJIlIqRHa

p

[B'

('t')

s ('t')],

paCCMaTpRBaeMaJI

KaK

q>YHKIJ;HH

OT

BeKTopa

k = s

(tfJ)'

HMeeT

CMWCJI

HOP.M~

'Y

[k] B

n-MepHOM

UpOCTpaHCTBe

$n

{k}.

::ITO

06CTOHTeJIbCTBO'

eCTeCTBeHHO

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'TOrO

q>aKTa,

'ITO

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qHHa

p

HMeeT

CMWCJI

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B q>yHKIJ;HOHaJIbHOM

UpOCTpaHCTBe

${h

['t'H.

Ho

B

TaKOM

CJIyqae

BeJIH'IHHy

V =

c's

(til) =

c'

k

MOmHO

paCCMaTpHBaTb

KaK

3Ha-

qeHHJI

JIHHeRHOii

orrepaIJ;HH,

uopom~eHHOR

B

upOCTpaHCTBe

$n{k}

BeKTOpOM

C

(CM.

CTp.

398), a

BeJIH'IHHy

v

o

,

oupe~eJIeHHyIO

paBeHcTBoM

(17.5),

Ha~JIeiRHT

Tor-

~a

TpaKToBaTb

KaK

HOp.My

r*

[c]

BeKTopa

c B

rrpocTpaHcTBe

$-n{c},

COUpHiReHHOM

K

upocTpaHcTBy

$n

{k}.

liTaK,

rrpHxo~M

K

CJIe~YIOm;eMY

BbIBO~.

lfHmmcu6-

nocmb

p*

CUD]

onmw.£a.ltbnOZO ynpa6.ltenW/,

Uo

(t), paspewa1OUfezo saoatlY 13.1,

pa6na

nopMe

v*

[c]

6e/'i.mOpa

c 6

npocmpancm6e

$~,

conpS/,:HCenno;\£

/'i.

npocmpal-/,-

cm6Y

$n

{k},

zoe

nopMa

'Y

(k) onpeoe.ttena pa6encm60M

'Y

[k]

= P

[B'

('t') S

['t',t~]

k].

AHaJIOrH'IHbIe

3aMe'IaHHH

MOiRHO

C~eJIaTb

H K

CJIe~YIOm;eMY

HHiRe

UpaBHJIy

17.4.

IIp

H M e q a H H e

17.2.

IIycTb

B

3a~aqe

13.2

MHoroo6paaHB

@.

{:J3}

B03-

MOiRHbIX

KOHe'IHWX,

COCTOJIHHB:

:J3

COCTaBJIJIeT

rHUepUJIOCKOCTb

Xi

(til) =

xli

(i

=

1,

...

,m

<n). ::ITO

03HaqaeT,

'ITO

KpaeBbIM

yCJIOBHJIM

Xi

(tfJ)

=

X~

~OJImHbI

Y~OBJIeTBOpHTb

JImrrL

nepBwe

m

Koop~HHaT

1)

H

'ITO

OCTaJIbHbIe

Koop~HHa

TbI

MorYT,

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upHHHMaTLB

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tfJ

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Ho

Tor~a

3a~a'Ia

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K

oUpe~eJIeHHIO

TaKoroynpaBJIeHHH

uO

(t),

KOTopoe

y~OB'JIeTBO

pHeT

JImrrb

uepBWM

m

ypaBHeHHHM

(15.1)

(3Ha'IeHHH

Ci

rrpH

i>

m

~JIH

Hac

pOJIH

He

HrpaIOT)

H

HMeeT

UpH

aTOM

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%

[UO].

TaKHM

06-

pa30M,

B

paCCMaTpHBaeMOM

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rrOJIyqaeTCJI

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rrepBhIX

m q>yHKIJ;Hii:

h[i]

('t'),

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pemeHHH

KOTOpOR

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rrpaBHJIaMH.

qHTaTeJIb

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rrpaBHJIO

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aTOM

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.

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pemeHHH

3aAaTJH 13.1

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(15.1)

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rrpeo6pa-

30BaHIil

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ypaBHeHHH

(15.1)

t(3

~

H

[t(3,

1:J

U

(T)dT

= C (17.7)

ta.

1)

TOT

q>aKT.

'ITO

BbI6paHhI

In

nep8~X

KOOPAIl:HaT.

He

urpanU'IfIBaeT

('61J~HOCTI1.

u60

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BBeCTIl:

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nyMepUIJ;HIO.

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114

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[ttl,

texl.

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CMMCJIy

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H

[t{3,

't']

= X

[t{3,

t]

B

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= X

[t{3,

t

..

] X-I ['t', t

ex

] B

('t')

TorAa

nOJIy'IHM

t{3

~

X-I ['t',

t.,;}

B

('t')

u

('t')

d't'

=

X-I

[t{3,

t

..

] C =

e.

lex

I1HMMlI

CJIOBaMH,

ypaBHeHlIH

(15.1)

npe06pasyloTcR

TorAa

K CHCTeMe

t{3

~f[i]'('t')u('t')d't'=ei

(i=1,

...

,n),

(17.8)

t..

rAe

ff

i

]'

(.)

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BeKTopM-CTpOKlI

MaTplIIJ;i.I

X-I [" t

ex

] B

('r)

II

t{3

e = X-I

[t{3,

t

ex

]

x{3

- x

ex

-

~

t

ex

B

pRAe

CJIy'IaeB

MO)J\eT

(maSaTbCR,

'ITO

AJIR

ypaBHeHlIn

(17.8)

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yCJIOBlI1I

%

X-I ['t', t

ex

]w (.)d't'.

(17.9)

pemaTb

np06JIeMy

MOMeHTOB

[u]

=

min

YA06Hee,

'IeM

AJIR

CIICTeMM

(17.7). B

'IaCTHOCTH,

K

ypaBHeHlIRM

(17,8)

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CBOAIITb

SaAalJY

13.1,

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w (t) = 0 H

pe'lb

IIAeT

0

npHBeAeHHII

06'beKTa

B

co-

x

ex

•

CTORHlIe

x

(ttl)

=

0,

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KaK

TorAa

e = -

PememIe

SaAa'IH

(17.8)

npOBOAHTCR

no

HSBecTHoMy

HaM

nJIaHy,

npll'IeM

nplIAeTCR

paCCMaTpHBaTb

IlJYHKIJ;HH

n

h

('t')

=

~

lil[i]

('t'),

i=I

aaMeHRIOm;He

aAeCb

<PYHKIJ;lIH (17.2).

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KaK

II

BMme,

Ha

CTp.

111,

y6eAlIMcR,

'ITO

Tenepb

h('r)=B'

(.)

S

["

totH.

OTCIOAa

BMTe-

KaeT

npaBHJIO,

KOTOpOMy

HaAJIemHT

CJIeAOBaTb

nplI

pemeHHH

aaAa'IH

B

nOCJIeAHeM

CJIyqae.

II

p a B

II

JI 0 17.3.

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peULumb

aa()a"lY

13.1

npu

w (t) =0

U

:rf3

= 0,

c.f(,e()yem

paccJtwmpemb

ee/'i,mop-¢YH/'i,LfUU

h

(.)

=

B'

(.)

s

(.),

z()e

s

(.)

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cucmeMb~

(17.3),

cmeCHeHHbLe

YC.f(,OeUeM

s'

(t

ex

) x

..

= - 1. (17.10)

CpefJu

amux

()eU:JICeHUU

Ha()M:JICUm

Haumu

MUHUMa.f(,bHOe

()eu:JICel-lue

SO

(.),

().f(,.ft

/'i,omopozo

eblnO.f(,H.ftemC.ft

YC.f(,oeue

pO =

min

p

[B'

(.)

s

(.)]

= P

[B'

(.)

SO

(.)l.

(17.11)

s

3a()a

1

w

UJtteem

peULel-lUe

moz()a U

mO.fl,b/'i,O

moz()a,

/'i,oz()a

pO

>

O.

OnmuJtia.f(,blWe

ynpaMeuue

UJtteem

uopJrty

p*

[UO]

=

11

po

U cpe()u

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PEillEHHE

3A)l;A"I1I

OB

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115

ecex

()onyc/'i,ae.Mb~x

ynpaMeNUU U

(.)

C

HOpMOU

p*

[u]

=

lI

p

ll

eb~()M.ftem

C.f£

YC.f(,oeue.M

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(17.1)

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MUHUMa.f(,bHOU

rfiYU/'i,LfUU

hO

(.).

3aMeTIIM

em;e,

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II

3AeCb,

RaR

II

BMme,

saAa'Ia

(17.10),(17.11)

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6MTb

saMeHeHa

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06

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a/'i,CmpeMa.f(,bHOZO

ABII-

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s· ('r).

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p a B

II

JI 0 17.4.

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aa()a"lY

13.1

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w (t) =0 U

:rf3

= 0,

c.f(,e()yem

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ee/'i,mop-rfiYH/'i,LfUlO

h

(.)

=

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(.)

s

(.),

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s

(.)

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()eU:JICeHU.ft

cucme.Mbl

(17.3). Cpeou

amux

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(.),

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=

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(t

..

) x

..

=

sel'

(t

..

) X

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(17.12)

s

npu

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p

[B'

('r) s ('r)]

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1.

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e

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U

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ua

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npaeU/laMU

17.1-17.4

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ucxo().ft cpaay ua

aa()a

1

tU

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e ma6/lULfe 12.1

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(12.19» no MUlluMuaupye.Mou

UH-

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[u]

8b~pa:JICeHUe

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p [h

(.)]

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aameM

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ua

npaeU.l/,

17.1-17.4.

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I'

.

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%

[u]

= p*

[u]

-<

It

paspe-

filIMa

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II

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TorAa,

Kor,n;a

mini

p

[l'H

(tf), 't')] =

po

>-

~

npH

l'c

= 1. (17.13)

!J.

01'0

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B

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He

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m;en

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p

[h

('r)l.

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(12.19),

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(17.13)

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B

BHAe

t{3

mini

[max

u

~

l'

H

[t{3,

't']u

('t')

d't']

>-

_i_

t

!J.

ex

npH

l'

c = 1

II

%

[u]

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1.

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= IAI%

[ul,

nplIxo,n;IIM

K

aKBIIBaJIeHTHOMY

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tf3

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,

maxu[l'

~

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d't']>-1

(17.14)

I

t

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It,

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6M

HH

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l,

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f

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= 1.

3anHcb

(17.14)

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9 H 10).

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On

pep;

e JI e H H e 17.1.

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cucTfleM,a

(13.1) e

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t =

tOf.

liaxo8umcR- e

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xa.

U 80nycnaeMlile

ynpae.n,eliUR-

U = U (t)

(t

..

-<

t

-<

t)3)

(u.n,u

o606Ufelilible

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[u]

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f.L

(x

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06.n,acmblO

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f.L'

ta.,

t)3]

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X =

x,g,

e

na:JIC8ylO

us nomopblX

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6blmb

nepeee8elia n MOMelimy t -:-

t)3

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lIcnoJlbayH

¢OpMyJly

RomH

(5.6),

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~

[x~,

f.L,

t«.,

t)3]

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x = X

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+

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[u]

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f.L.

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p;aJlbHeimeM

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[u]

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=

p*

[UJ), TO

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[xa.,

f.L,

ta.,

t)3]

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npOCTpaHCTBe

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{x}

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f.L,

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t)3]H

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[U]

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(t)}

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(CM.

IIpHJlo-

llieHHe,

§ 61).

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Kallip;oe

ynpaBJleHHe

u'J..

(t)

= AU(lJ(t)+

+

(1

-

A)

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J

(t) (0

-<

A

-<

1)

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cop;epllilITCH

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e.

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f.L.

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(17.15)

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(t)

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(13.1) B

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=AX(1) +

(1

-

A)

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+

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-

A)

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~,

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e.

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HO

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f.L,

t

..

,

t/3]

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p*

{u}

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0

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1:']

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fl

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1:']

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(17.15)

tao

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p.

(.:

~~':'

117

§ 17]

PEmEHHE

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OB

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[6*1.

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qTO

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(B

HeCKOJlbKO

oc.n,a6.n,e1iliOU

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(CM.

[23*],

1'.1,

CTp.

372): us.n,lo60u

6ecnOli~liOU

c06onynliocmu om1£pumux <wnpecmllocmeu» , nonpu6alOUfeu

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@,

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e

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(t)

(k = 1, 2,

...

),

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e

amou

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(i

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t

fl

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i

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t« t«

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i

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00,

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(17.16). B

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~

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f.L,

t

..

,

t/31.

01'0

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e.

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[u]

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II

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(17.15)

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p;o-

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~

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f.L,

t

..

,

t)31.

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n),

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toc

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T.

e.

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1

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§ 61, a

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CTp.

41).

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Roro

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B § 38

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47

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·T01JRY g =

O.

Gp;HaRo

9TO

yCJIOBIIe

He

HrpaeT

cym;ecTBeHHOB:

pOJllJ,

eCJJH

IIOJIl>30IlaTbCH

3aIIHCbJO (17.17).

RpoMe

Toro,

YI{a3aHBOe

yCJIOBHe

Bcerp;a

MO,I{HO

BLIllOJIHHTL,

AOJIaH

lloAxOAHm;YJO

3aMeHy

v (t) = u (t) - U

o

(t).

Красовский Н.Н. Теория управления движением

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