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PEillEHIIE IIFOBJIEM:bI HABJIIO,IJ;EHlln

§ 31.

IIOCTaHOBlm

BaAa1JH

Tenepb

MbI

nepeXop;HM

R

H3yqemuo

3ap;a1JH 0

HaOJIIOp;emUl

ROOp-

p;HHaT

p;BHmym;eroCH

OO'I>eRTa,

onHCbIBaeMoro

JIHHeHHbIMH

p;mp«pepeH..,

~HaJIbHbIMH

ypaBHeHHHMH.

IIpH

HCCJIep;OBaHHH

MHorHXnpOOJIeM

Ha

TeopHH

ynpaBJIHeMbIX

p;BlDI<eHHH

TpeoyeTcH

3HaHHe

TeRym;HX

COCTOH-

ImH

x (t)

CHCTeMbI.

81'0

BaiImo,

HanpHMep,

B

CJIyqae,

Rorp;a

ynpaBJIe-

HHe.

U

«popMHpyeTcH

no

npHH~Hny

oopaTHoH

CBH3H

B BHp;eu

[t]

=

= u [Xl(t),

...

, X

n

(t)]

(CM.

§ 28).

Op;HaRo

1JaCTO

no

yCJIOBHHM

3ap;a1JH

HeJIb3H

npep;nOJIaraTb,

'ITO

Bce

RoopP;HHaTbI

Xi

(t)

(i

= 1, ... , n) n-

,.

, l

MepHoro

«pa30Boro

BeRTopa

X (t)

p;ocTynHbI

HenOCpep;CTBeHHOMy

H3-

MepeHHIO.

TpYP;HOCTb

npHMoro

H3MepeHHH

TOH

HJIH

HHOH

Roopp;HHaTbI

Xi

(t)

MomeT

B03HHRHYTb

BCJIep;CTBHe

OTCyTCTBHH

H3MepHTeJIbHOrO

Ra-

HaJIa,

CBH3bIBaIOm;ero

HaOJIIOp;aTeJIH

C

COOTBeTcTByIOrD;HMH

:meMeHTa-

'11

MH

p;BHmym;erocH

OO'I>eRTa. B

p;pyrHx

CJIy1JaHX

3Ta

TPYP;HOCTb

MomeT

ObITb

OOyCJIOBJIeHa

OOJIbIIIHMH

nOMexaMH

B

H3MepHIOm;HX

yCTpOH-

CTBax

Ii

B

RaHaJIaX

CBH3H.

HanpHMep,

Hepep;RO

TaRIie

nOMeXH

HaOJIIO-

-,

'll,

p;aIOTCH

npH

onpep;eJIeHHH

«pa30BbIX

Roopp;HHaT

Xj

(t), P;JIH BbI1JHCJIe-

HHH

ROTOpbIX

npHxop;HTCH

p;H«p«pepeH~HpOBaTb

pe3YJIbTaTbI

H3Mepe-

HHH,

.B

1JaCTHOCTH BbI1JHCJIHTb

npOH3Bop;HbIe

dl'Xi

(t)jdt

k

no

BpeMeHH

01'

p;pyrHx

Roopp;lIHaT

Xi

(t).

IIpH

3TOM

OiliHORa

H3MepeHHH

nOJIyqaeTcH,

RaR

npaBHJIO,

TeM

OOJIbIIIe,

1JeM

B:bIIIIe

nopHp;OR

npOH3Bop;HOH.

B

CO-

OTBeTCTBHH

C <lTHM

oyp;eM

p;aJIbilie

paCCMaTpHBaTb

CJIy1JaH,

Rorp;a

B03-

MomHO

1I3MepHTb

Henocpep;CTBeHHO

II

C

yP;OBJIeTBOpHTeJIbHOH

T01JHO-

CTbIO

JIHilib

HeROTopbIe

Roopp;HHaUI

Xi'

(j

= 1, ... , m < n)

(HJIH,

00-

J

JIee

oom;o,

HeROTOp:bIe

«PYHR~HH

gj

[Xl

(t),

...

, X

n

(t)] 01'

Xi

(t»

H

Ha

OCHOBaHIiH.

3THX

H3MepeHHH

TpeoyeTcH

BbI1JHCJIHTb

Hep;OCTaIOm;He'

3Ha1JeHHH

p;pyrlIx

«pa30BbIX

Roopp;HHaT

Xi

(t).

IITaR,

paCCMOTpHM

HeRoTopyIO

cHcTeMy,

onHCbIBaeMYIO

ypaBHe-

HIieM

1;

(t)

= f

[t,

X

(t)]

(x

=

{Xl'"''

x

n

}),

(31.1)

npH1JeM

p;onycTHM

CHa1JaJIa,

RaR

31'0

CJIep;yeT

H3

ypaBHeHHH

(31.1),

'lTO

ynpaBJIHIOm;He

B03p;eHCTBIiH

u (t)

OTCyTCTByIOT.

3p;eCb

f -

1I3BeCT-

HaH

BeRTop-«pyHRD;HH,

onpep;eJIeHHaH'

p;HHaMH1JeCRlIMH

CBOHCTBaMH

OO'I>eRTa.

IIYCTb

HMeeTCH

B03MOiImOCTb

H3MepHTb

Roopp;lIHaTbI

Yj (t)

(j

=

1,

...; m)

HeROToporo

BeRTOpa

Y (t),

CBH3aHHoro

C

«pa30BbIM

F'>

~~~~'-~

§

31]

nOCTAHOBHA

3AAA'IH

265

BeRTopOM

X (t)

COOTHOIIIeHlIeM

Y (t) = g ft, X (t)]

(31.2)

lIJlH,

B

nop;pOOHOH

3anHClI,

Yj

(t) =

gj

ft,

Xl

(t),

...

, X

n

(t)]

(j

=

1,

... , m <

n),

(31.3)

rp;e

gj

-

H3BeCTH:bIe

«PYHR~lIH,

onpep;eJIeHHbIe

ROHcTpyRIweH

1I3Me-

pHIOm;lIX

yCTpOHCTB.

TaR

RaR

M:bI

npep;nOJIOmHJIlI,

'ITO

m <n,

TO

ypaBHemIH

(31.3)

He

yp;aeTcH

pa3peIIIHTb

OTHOCHTeJIbHO

Bcex

BeJIH1JHH

Xi

B

BHp;e

«PYHR-

~HH

Xi

=

qi

[t,

YI"'"

Ym1

(i

=

f,

...

, n).

CJIeP;OBaTeJIbHO,

onpep;e-

JIeHHe

«p

a

30Boro

BeRTopa

X (t)

Ha

OCHOBaHl'lH

JIHilib

MrHOBeHHoro

3Ha-

1JeHHH

H3Mep

H

eMoro

BeRTopa

Y (t) B

p;aHHOM

CJIy1Jae

HeB03MomHO.

Op;-

HaRO,

OOXOP;H

B03HHRIIIYIO

TPYP;HOCTb,

MomHO

nonbITaTbCH

onpe,z:r;eJIHTb

3Ha1JeHHH

ROOpp;HHaT

Xi

(t),

IiCXOP;H

He

TOJIbRO

H3

MrHOBeHHbIX

3Ha1Je-

HHH

Yj (t),

HO

yqHTbIBaH

H

HeRoTopyro

npep;:bICTOpHIO

Yj

(t)

(t _

{}

<

't < t)

HaMeHeHHH

aTHX

BeJIH1JHH.

Torp;a,

CJIep;OBaTeJlbHO,

B03HH-

RaeT

3ap;a1Ja

00

onpep;eJIeHHH

TaI\OH

onepa4uu

CPi

ft, Y

(t)],

ROTopaH,

Oyp;yqH

BbInOJIHeHHOH

Hap;

BeRTop-«pYHRD;HeH

Y

(t)

= {Yj (t)},

(t - {} < 't < t),

nOpOfRp;eHHOH

B

COOTBeTCTBHH

C

ypaBHeHHeM

(31.2)

p;BHfReHHeM

CHCTeM:bI (31.1),

p;aCT

B

pe3YJIbTaTe

BeJIH1JHny

Xi

(t).

TaRaH

onepa~HH

p;OJIfRHa

ObITb

HaHp;eHa

P;JIH

Ramp;OH

HHTepeCYIOm;eH

Hac

Roopp;HHaTbI

Xi

(t).

113bICRaHHe

onepa~HH

(Pi

ft, Y ('t)] H

COCTaBJIHeT

npep;MeT

3ap;aQH

o

HaOJIIOp;eHHH

CHCTeMbI

(31.1)

no

BeJIH1JHHaM

Yj (31.3).

IIpemp;e1JeM

.

«POPMYJIHpOBaTb

3ap;a1Jy,

p;oroBopHMCH

0

CJIep;yIOm;HX

TepMHHax

H

000-

3Ha1JeHHHX.

qHCJIO

{}

>

0,

onpep;eJIHIOm;ee

OTpe30R

BpeMeHH

t -

{}

< 't < t,

Ha

ROTOpOM

Y1JHTbIBaeTCH

npep;bIcToplIH

Y ('t),

oyp;eM

C1JHTaTb

no-

CTOHHHbIM.

81'0

1JHCJIO JIHOO

3ap;aeTCH

no

yCJIOBHHM

3ap;a1Jn"

HCXOP;H,

HanpHMep,

H3

peaJIbHO

B03MomHOH

p;JIHTeJIbHOCTH

3anOMHHaHHH

3Ha-

1JeHHH

Yj

('t) B

BbI1JHCJIHTeJIbHI>lX

yCTp

oHcTBax

,

JIHOO

BeJIH1JHHa

{}

onpep;eJIHeTCH

H3

p;OnOJIHHTeJIbH:bIX

TpeooBaHHH,

conpoBomp;aIOm;HX

aa,z:r;a1JY

0

HaOJIIOp;eHHH.

BeRTop-«pYHJ(~HIO

Y

(t)

(31.2) (t -

{}

< 't < t)

oy,z:r;eM

Ha3:bIBaTb

cuma.ILOM.

)J;oroBopHMCH

0

CJIep;yIOm;eH

TepMHHO-

JIOrHH.

o n

pep;

e

JI

e H H e 31.1.

Ec.lLu

lte1f,OmOpbLu

CU2lta.IL

Y ('t)

MO-

~em

pea.ILUaOBambC.fl

B

cucme,"Ate

(31.1), (31.2)

npu

alta'leltUU

Xi

(t) =

=

lj,

mo

6yfJeM

ltaabLBamb

e20

CU2lta.ILOM,

ltecyUfUM

Xi

(t) =

lj,

U 060-

alta'lamb

CUMBO.ILOM

{y

(t)

I

Xi

(t) =

lj}.

Ec.lLU

npu

fJaltltOM

alta'leltUU

rjJaaOB020

Be1f,mOpa

X (t)

pea.ILuayemc.fl

CU2lta.l/,

Y ('t), mo

C1f,a~eM,

'lmo

amom

CU2lta.IL

OmBe'laem

fJaltltOMY

rjJaaOBOMY

Be1f,mopy

X (t).

ECJIH

ROHRpeTHoe

3Ha1JeHHe

Xi

(t) =

lj

BeJIH1JHH:bI

Xi

(t)

He

oyp;eT

HrpaTb

cym;ecTBeHHOH

pOJIH,

TO

CHMBOJI

lj

B

{y

('t)

/''Ci'

(t) =

lj}

oy-

,z:r;eM

onycRaTb

H

nHcaTb

npOCTO

{y

(.)

I

Xi

(t)}.

IIpH

3TOM

CJIep;yeT

.....

'.

;

..

, .'

..

::

'.:::

~:'

:-:;

:.::::'

...

.,:

...

>.';

-.:::':

:

..

:.:.::;

.:

~

:~.:

;

..

:>:~

~~;":Y:

c

Q

\~~

.

.'

2(i6

rrOCTAHOBKA

H

PEIIIEHHE

IIPOBJIEMbI

HABJIIO,n:EHlUI

[rJI.

8

Bcer,n;a

rrOMHHTb,

'ITO

3,n;eCb

HMeeTClI

B BH,n;y

BeKTOp-epyHltQHlI

Y

('"C),

orrpe,n;eJIeHHall

Ha

OTpe3Ke

t -

'fr

-<

't

-<

t,

'fr

=

const.

OlJeBH,n;HO,

3HalJeHHe

X

(t)

orrpe,n;eJIlIeT

e,n;HHCTBeHHbIM

06pa30M

OT-

BelJaIOrrJ;H:U:

eMy

CHrHaJI

y ('t).

BbI1lCHeHHe

06paTHO:U:

3aBHCHMOCTH

H

COCTaBJIlIeT

rrpe,n;MeT

3

a,n;a'lH0

Ha6JIIO,n;eHHH,

KOTOpall

epOpMynllpyeTclI

CJIe,n;yIOIqHM

06pa30M.

3

a,n;

a lJ a 31.1.

IIycmb

fJ8UJlCYUfUUCit

06U1£m

onUCbL8aemCit

ypa81-leuueM

(31.1), npU'leM

fJocmynnbL

Ua:MepenUlO

ana'lenUit

8e1£mop-

fPyn1£lfuu y (t) (31.2). Tpe6yemcit

naumu

onepalfulO

Cj)i

[t, Y ('t)J,

8bL-

'lUCMUOUfYlO

1£oopfJunamy

Xi

(t)

no

cuena.ILY

y ('t),

ma1£

'lmo

8bLnO.!Lnitem-

Cit

pa8encm8o

Cj)i

[t,

{y

('t) I

Xi

(t)}]

= Xi (t),

(31.4)

,;a1£o8o

6bL

nu

6bL.lLO

pea.ILUa08a8uteeCit

8

cucme:Me

(31.1) ana'lenue

Xi

(t)

U

1£a1£08

6bL

nu

6bL.IL

Cuena.IL,

necyUfuu Xi (t).

OrrepaIJ;HIO

Cj)i,

y,n;OBJIeTBOpllIOIqyIO

yCJIOBHIO

(31.4),

Ha30BeM

paapeutalOUfeu

OnepaIJ;He:u:

(,n;JIlI

3a,n;a'lIf

31.1).

ECJIH

ypaBHeHHlI

(31.1) H (31.2) JIHHe:U:HbI,

T.

e.

eCJIIf

1\

x = A (t)

X,

(31.5)

Y = G (t)

X,

(31.6)

:

111

r,n;e A (t) H G (t) -

MaTpHIJ;bI

{aii (t)}

(i,

j =

1,

... , n), {gik (t)}

(i-

1, ... ,

m;

k = 1, ... ,

n),

TO

pa3pemaIOrn;He

OrrepaIJ;IfIf

Cj)i

TaKme

~;.

CJIe,n;yeT

HCKan

B

epopMe

.lLUneUnbLX

orrepaIJ;H:U:.

8TO

06'1>lICHlIeTClI

TeM,

)."

'ITO

B ,n;aHHOM

CJIylJae

CIfrHaJI

y ('t)

CB1I3aH

C

BeKTopOM

X (t) JIIfHe:U:HO:U:

3

aBHCIfMOCTbIO

Y ('t) = G ('t) X ['t,

t]x(t),

(31. 7)

r,n;e X -

epyH,n;aMeHTaJIbHall

MaTpHIJ;a

CHCTeMbI

(31.5).

PaBeHcTBo

(31.7)

HaTaJIKHBaeT

cpa3y

Ha

CJIe,n;yIOrn;ee

pelIIeHHe

3a-

,n;alJH 0

Ha6JIIO,n;eHIfH.

IIycn

,n;JIlI

orrpe,n;eJIeHHOCTH

y ('t) -

CKaJIlIpHall

rrepeMeIIHall.

BlJ6epeM

Ha

OTpe3Ke

t -

'fr

-<

't

-<

t n

pa3JIH'lHbIX

3Ha-

lJeHIf:U: 't =

'ti'

IIoycJIOBHlIM

3a,n;a'lH

MbI

MomeM

H3MepIfTb

BeJIH'lHHbI

y ('t

i)'

Tor,n;a

H3

paBeHCTBa

(31.7) CJIe,n;yIOT n

ypaBHeHH:U:

,n;JIlI

HeIl3BecT-

HbIX

epaaoBbIx

Koop,n;IfHaT

n n

~

gs

(1:'j)

xsd't'j,

t]

Xi

(t) = Y

('t'j)

(j

= 1,

...

,n),

(31.8)

S=1

i=1

npIf'leM

no

YCJIOBIfllM

3a,n;a'lIf

3HalJeHHlI

gs

('ti),

paBHO

KaK

H

3Ha'leHHlI

Xsi

hi>

tl

3JIe~eHTOB

Xsi

['t,

t]

epyH,n;aMeHTaJIbHo:u:

MaTpHIJ;bI

X

[T,

t].

npe,n;nOJIaraIOTClI

If3BeCTHbIMH.

§

31]

ITOCTAHOBKA

3A,D;A'IH

267

ECJIH

orrpe,n;eJIIfTeJIh

D

CIfCTeMbI

(31.8)

OTJIHlJeH

OT

HyJIlI,

TO,

pa3-

pe

m

all

3TY

CHcTeMy,

rrOJIy'lHM:

n

Xi(t) =

~~Llijy('tj)

(i=1,

...

,n),

(31.9)

j=1

r,n;e

Ll

ii

-

COOTBeTcTByIOrn;He

MHHOpbI

orrpe,n;eJIHTeJIlI

D.

TaK

KaK

rrpaBall

'IaCTb

Kam,n;Oro

H3

paBeHCTB

(31.9)

MomeT

paCCMaTp

IfB

aTbClI

KaK

pe3YJIbTaT

JIIfHeHHO:U:

OrrepaIJ;IfH

<Pi'

BbIITOJIHeHHO:U:Ha,n;

HenpepbIB-

HbIM

CHrHaJIOM

y ('t) (t -

'fr

-<

't

-<

t),

TO,

,n;e:U:CTBHTeJIhHO,

MbI

yKa-

·3aHHMM

nyTeM

nOJIylJaeM

pelIIeHHe

3a,n;a'lIf

31.1 0

Ha6JIIO,n;eHHIf.

Cy-

rn;eCTBOBaHHe

TaKoro

npOCToro

pemeHHlI

3

a,n;a'IIf

,

Ka3aJIOCb

6bI,

,n;eJIlleT

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HCCJIep;OBaHHe

ee

MaJIO

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TeM

He

MeHee

B

rrOJIh3Y

3Toro

HCCJIe,n;OBaHHlI,

,n;aHHoro

B CJle,n;YIOrn;IfX

rraparpaepax,

MomHO

rrpIfBeCTH

TaKIfe

coo6pameHHlI.

Ha:u:p;eHHoe

TOJIhKO

'ITO

pemeHIfe

(31.9)

lIBJllIeTClIJIIfmh

O,n;HIfM

H3

60JlhmOrO

'IHCJIa

B03MomHbIX

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Ha

3a,n;alJy,

H OHO

MomeT

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3aTbClI

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HaIfJIYlJmero

pemeHIfll,

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Te

HJIH

HHbIe

,n;orrOJIHHTeJIbHbIe·

06CTOllTeJIbCTBa,

conyTcTByIOIqIfe

Ha6JIIO,n;e-

HHIO

(CM.

§ 32).

KpoMe

Toro,

06rn;ee

HCCJIe,n;OBaHIfe

rrp06JIeMbI

Ha6JIIO-

,n;eHHlI

nOJIe3HO

BCJIe,n;CTBIfe TeCHO:U:

CB1I3H

Mem,n;y

3TO:U:

3a,n;alJe:u: H

3a-

,n;alJe:u:

06

ynpaBJIeHIfIf,

H3YlJeHHo:u:

BO

II

'IaCTIf

KHIfrH

(CM.

§§

15-17).

HaKOHeIJ;,

3a,n;a'la

0

Ha6mo,n;eHHIf

n03BOJIlIeT

BeCbMa

eCTeCTBeHHbIM

06-

pa30M

IfCTOJIKOBaTh

MeTo,n;bI,

paccMaTpIfBaeMbIe

B ,n;aHHo:u:

KHIfre

C

no-

3HIJ;If:U:

MaTeMaTHlJeCKO:U:

TeopHH

Hrp

(CM.

§ 37).

II

pHBe,n;eM

Terrepb

rrpocTo:u:

rrpIfMep,

IfJIJIIOCTPIfPYIOIlWH

Ha

Mexa-

HH'leCIW:U:

Mo,n;eJIIf

rrocTaHOBKy

3a,n;a'lH

31.1.

IIpH.lIep

31.1.

PaccMoTpRM

,n:BHllieHlle

nIpOCROlHPIeCROro

MaHTHlma,

omI-

caHHoro

B § 3 (CM.

II.

4°),

HO

TeIIep,b

B ORpeCTrrOCTH

HHlliHero

YCToiiqHBOrO

IIOJIOllieHHH.

(iho

03Ha'iaeT,

'ITO

Ha

pHC.

3.4 OCb OZ,

CBH3aHHaH

co

cTeplliHeM

OA,

HaIIpaBJIeHa

BHll3.)

ECJIR

B

Ra'IeCTBe

o606IIJ;eHHbIx

Koop,n:HrraT

BLl6paTb

CHOBa

yrJILl

ql

H

q2'

OIIHCLIBaIOIIJ;He

nOBopOTbI

BHemHeii

H

BrryTpeHrrcii

paMOR

COOTBeTCTBerrrro, TO

ypaBrrerrHH

MaJILlX

ROJIe6aHIIll

MexarrH3Ma

B

mmCHrrOM

IIpu6JIII-

llieHRH

6YP;yT

HMeTb

BR,n:

1'2

= -

-2-

xl

-

-2-

x4,

C1

z

(12

Xl

=

X2,

g""-

wcri

J

(31.10)

wcri

1(""-

.1:

s

-:X4,

3:

4

=

-2-X2--2

Xs,

C1 C1

Z

2

rp;e

Xl

=

ql'

X

s

=

q2'

3TH

ypaBrrerrHH

IIOJIy'IaIOTCH

TparrC<popMaU;Heii

ypaBrrerrHii

(3.15),

ROTopaH

06yCJIOBJIerra

H3MerreHHeM

BLlpallierrHH

(3.13)

,n:JIH

nOTerrU;lIaJIb-

HOll

arreprRR,

BLl3BaHrrLlM

cMei:IJ;eHHeM

u;eHTpa

THllieCTH

CUCTCMLl

BIIlI3.

BCJIep;-

CTBlle

O'ieBUp;HOCTII

OTMe'IeHrrLlX

H3MeHerrHii:

rra

BLlBo,n:e

ypaBHeHull

(31.10)

oc-

TarraBJIliBaTbCH

He

6y,n:eM.

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

'.','.

'':_._:~''_'-':''-~'''':'~'~_'----,-~-.l~

",,','

.

'-,----,---"

':",

.

-:::

..

'~',

'

".'.

":':':E'~<:~~:~~:

268

nOCTAHOBHA H PBmEHHE IIPOEJIEMhI

HAEJIIO,UEHMJI

[rJI.

8

IIpe~IIOJlOiKIIM,

qTO

II.MeeTcH

B03140iRHOCTb

1l3MepHTb

yroJl

Xl

(t)

OTKJlOHe-

HIlH

BHemHeH

paMKII

OT

ropll30HTaJIbHOH

IIJlOCKOCTIl.

MHaqerOBOpH,

B

AaHBOM

cJlyqae

CllrHaJl

y

('r)

eCTb

<pym{~lIH

Xl

('r).

CJleAOBaTeJlbHO,

MaTpll~a

G B

ypaB-

aeHllll'

(31.6)

06pam;aeTcH

3)J,eCb B

BeKTop-cTPORY

{1, 0, 0,

O},

a

caMO

3TO

ypaB-

HeHlle

IIpllHllMaeT

BIl~

y

('r)

= [1, 0, 0, °1X

('r)

=

Xl

('r).

II

yCTb

Ha

OCHOBaHllIi

H3MepeHllH

ClirHaJla

y

('r)Tpe6yeTcH

BbIqllCJllITb

yroJl

IIoBopoTa

BHYTpeHBeH

paMKIi

x

a

(t),

a

TaRiRe

yrJloBble

CROpOCTR X2 (t) =

~l

(t),

x,

(t) =

~a

(t).

B

Ta-

ROM

cJlyqae

3aAaqa

31.1 0 Ha6J1IO)J,eHllH

CllCTeMbI

(31.10)

MOiReT

6bITb

cqlOPMYJlIi-

pOBaHa

CJIe~YIOm;lIM

06pa30M.

Tpe6yeTcH

HaHTIl

JlIiHeHHbIe

orrepa~lIll

<Pi

[t,

Y

('r)

1

(i

=

2,

3, 4),

B~~C

JlHIOm;lle

ROOp~IlHaTbI

X

2

(t), Xa (t), X

4

(t)

rro

CHraaJlY

y

(-r)

=

Xl

(-r),

TaR

qTO

BLI-

rrOJlHHIOTCH

paBeHCTBa

<Pi

[t,

Y (-r)] =

<Pi

[t,

Xl

(-r)] =

Xi

(t)

(i

=

2,

3, 4),

RaKOBbI

6bI

HIl

6bIJIIl

peaJIH30BaBmHeCH

B CHCTeMe (31.10)

3HaqeHHH

Xi

(t).

lIepeH,u;eM

renepb

K CJIylIaIO, Kor,u;a

HaOJIIO,u;aeMbIH

OO'l>eKT

no,u;-

BepmeH

ynpaBmHom;eMy

B03,u;eHCTBHIO,

T. e.

paCCMOTpHM

cHcTeMy,

~mHCbIBaeMYIO

ypaBHeHHeM

j;

(t) = f

It.

x (t), U (t»),

(31.11)

.,'1

r,u;e,

KaK

OObIlIHO.

x - cPa30BbIH

n-MepHbIH

BeKTop, U -

r-MepHbIH

\

BeKTop ynpaBJIHIOm;HX

CHJI.

B

aTOM

CJIyqae

,u;JIH

BbIlIHCJIeHHH

KOOp,u;H-

'"

HaT Xi (t)

(i

= 1,... ,

n)

OO'l>eKTa

yme

He,u;OCTaTOlIHO

3HaTb JIHmb CHr-

HaJI

y ('r) (31.2)

npH

t

~

it < T < t,

HO

HyrIma

em;eHHcP0pMaIJ;HH 0

f~

npe,u;bICTopHH ynpaBJIHIOm;HX

CHJI

Uj

(T)

(j

= 1•...,

r)

(HanpHMep,

npH

t - it <

't~t).

B OTJIHlIHe

01'

CHrHaJIa y (T).

apryMeHT

KOToporo

't

Me-

;::,:

HHeTCH

B npe,u;eJIax ompesna t -

it

< T < t. 3HalIeHHe

B03,u;eHcTBHH

i,

I

U = U (t) B YlIHTbIBaeMyI61npe,u;bIcTopHIO

CHJI

U (T) lie

en.rtlO1(,aemcH.

8'1.'0

OO'l>HCHHeTcH

CJIe,u;yIOm;eH

npHlIIifHoH. B

OOJIbmOM

lIHCJIe CJIylIaeB

aa,u;alIa 0 Haomop;eHHH.

T.

e. aap;alIa 0 BbIlIHCJIeHHH TeKym;HX

KOOp,u;H-

HaT

Xi

(t) ,u;BHmym;erocH

OO'l>eItTa.

B03HHKaeT B

CBH3H

C aap;alIeH

00

ynpaBJIeHHH

ero

p;BHmeHHeM

no

npHHIJ;Hrry 06paTHOH

CBH3H.

Kor,u;a

U

It]

= U

Ix

(t)l.

lIpH

aTOM

3HaHHe Koop,u;HHaT

Xi

(t) B

MOMeHT

t

Heooxo,u;HMO

KaK

pa3

P;JIH

Toro,

lITOObI BbIpaOOTaTb ynpaBJIeHHe U

It]

'B

8mom

:J/Ce

MOMeHT

BpeMeHH.

Ho

Torp;a, BKJIIOlIaH 3HalIeHHe U

It]

B

COCTaB

BeJIHllHH U (T)

(T;;>

t - it),

rro

KOTOpbIM

BbIlIHCJIHIOTCH

KO-

op,u;HHaTbI

Xi

(t),

npH,u;eM

It

JIOrH1I0CKH IIOpOlIHOMy

Kpyry.

IfTaK,

eCJIH Ha

OO'l>eKT

,u;eHcTByIOT

ynpaBJIHIOm;He

CHJIbI

Uj

(t).

TO

B KalIeCTBe

CU3lia.rta.

rro

KOTOpOMy

onpe,u;eJIHIOTCH

BeJIHlIHHbI

Xi

(t).

oy,u;eM

BbIOHpaTb

rrapy

BOKTOP-cPYHRIJ;HH

{y (T). U

(T)},

npHlleM

Bcer,u;a

oy,u;eM

rrpe,u;nOJIaraTb, lITO

apryMeHT

y

cPYHKIJ;HH

y (T)

MeHHeTCH

B

npe-

,u;eJIax ompesna t -

it

< T < t, a y

cPYHKIJ;HH

U (T)

apryMeHT

3a-

KJIIOlIeH B rrpe,u;eJIax nO.rtyuumepea.rta t - it < T < t.

lIo

aHaJIOrHH

C onpe,u;eJIeHHeM

31.1 CKarIteM, lITO CH3Ua/t {y (T), U (T) I

Xi

(t)

=

11}

HeceT

Xi

(t) =

11.

eCJIH

aTOT

CHrHaJI peaJIH3yeTcH B

CHCTeMe

(31.2),

1'1

(31.11)

rrpH

Xi (t) =

11.

cT

~:.

§ 31]

nOdTAHOIlHA

3A,rtAqlf

I

269

I

OrpaHHllHBaHCb TOJIblW JIHHeHHbIM CJIylIaeM. CcPOpMyJIHpyeM Te-

rrepb

CJIe,u;YIOm;yIO

aap;alIy 0

HaOJIIO,u;eHHH.

3 a,u; a q a 31.2.

llycmb

ynpaOltHe,?,taH

cucmeJlta onucbleaemcH

I

ypaeuenueJlt

.I

I

x = A

(t)x

+B (t)u (31.12)

I

U

f)ocmynHib/,

U3MepenulO

Slial(,eUUJ{,

eelbmopa y (t) (31.6) U

3Ual(,eUUH

Obl-

pa6ambleaeMblx ynpae.rtHlOUfux

CU.rt

U (t). Tpe6yemcH

uaumu

J1UUeuliYlO

onepatfUlO

fiJi

It.

{y

(T).

U(T)}),

Bbl1(,UC.rtHlOUfYlO

noopf)uuamy

Xi

(t) no

cueua.rty

{y

(T).

U

(T)}.

man

1(,mo

BbllW.rtuHemcH

paeeucm80

fiJi

It,

{y

('r), U

('1')

I Xi (t)}] = Xi (t). (31.13)

lbanoeo

6b~

flU

6bl/tO

pea.rtUS08a8lueeCH

e

cucme.itte

(31.12) 3na'lenue

Xi

(t)

U nanoo

6bl

UU

6bl/t

cU3ua.rt.

uecyUfuu

Xi

(t).

11

3,u;eCb

orrepaIJ;HH

fiJi>

y,u;OBJIeTBOpHIOm;He

yCJIOBHIO

(31.13).

Ha-

30BeM

paspeUtalOUfU,Mu

onepatfUHMu

(,u;mI

3a,u;U'IH

31.2).

3aMeTHM

CJIe,u;yIOm;ee.

RJIacc

cPYHRIJ;HH

y

(t)

H U (t) orrpep;eJIHeTCH

KJIaCCOM

eOS,MO:J/CUblX

yrrpaBJIeHHH. B aap;aqe 31.2 rrpep;nOJIaraeTcH.

qTO

orrepaIJ;HH

fiJi

,u;OJIlliHbI

ObITb

BbIllOJIHHMbI

Ha,u;

JIIOObIM

CHrHaJIOM {y (r),

U

(T)},

KOTOpbIH MomeT peaJIH30BaTbCH rrpH B03MomHbIX

ynpaBJIe-

HHHX U (t).

3ap;alIY 31.2

MomHoTpaHcwopMHpoBaTb

H

,u;JIH

CJIyqaH o606Ufeu-

liblX

B03MomHbIX ynpaBJIeHHH

dU

(t). Kor,u;a p;BHmeHHe orrHCbIBaeTCR

ypaBHeHHeM

dx

= A (t)

xdt

+B

(t)

dU

H Kor,u;a paccMaTpHBaeTcR

CU3ua.rt

{y

('r).

dU

(T)}.

MbI.

o,u;HaKo, He

OYAeM

cPOPMYJIHpOBaTb

3,u;eCb

nOAPOOHO

TaKyIO

OOOOIIJ;eHHyIO

aa,u;aqy

o

HaOJIIO,u;eHHH.

rrpep;OCTaBJIHH

aTO

qHTaTeJIIO.

IIpll,uep

31.2.

PaccMoTpMM

CHOBa

ABHiReHlIe

rHpOCROmlqeCROro

MaHTHMRa,

HO

Terrepb

YiRe

B ORpeCTHOCTII

BepXHero

HeycTOHqllBOrO

rrOJIOiReHllH

paBHOBe-

CRH,

T.

e.

HMeHHO

B TOM

COCTOHHIlII,

Ral{

3TO

H306pameHO

~a

pHC.

3.4

B § 3.

flYCTb,

Ral{

H

TaM,

MexalIH3M

yrrpaBJIHeTCH

MOMeHTOM

U,

rrpUJIOiReHHbIM

R

BHem-

HeH

paMRe.

YpaBHeHIIH

ABHil{mUIII

CHCTeMbI B

rrepBOM

IIpH6mHHeHlIH

IIpHBel~eHLI

yme

B § 3

If

lIMeIOT

BH)J,

(3.15),

T.

e.

Xl =

X2,

X2 =

-2-

g'A,

Xl

-

WCli

1 J

-2-

Xi

+

--2

U.

Cl

2

Cl

mCl

2

t31.14)

WCl

I

g'A,

Xs

=

x4,

~4

=

-2-

X2+

-2-

Xa.

Cl

2

Cl

2

II

pe,ll;IIOJI

OiRllM

,

qTO

AocTyIIHa

H3MepeHHIO

CROpOCTb X (t) =

~l

(t)

Bpam;eHHH

2

IlHemHeH

paMI{Ii,

a

TaRiRe

rrpHMeM,

qTO

MomHO

H3MepHTb

3HaqeHHH

U (t)

BbIpa6a-

TbIBaeMoro

yrrpaBJlHIOm;ero

MOMeHTa.

CJle)J,OBaTeJlbHO, B

AaHHOM

CJIyqae

y

(-r)

=

=X

2

('r)

Ii

MaTpH~a

G B

ypaBHeHlIlI

(31.6)

06pam;aeTcll

BBeRTop-cTpoRy

{O,

1,0,0},

a

caMO

3TO

ypaBHeHHe

rrplUIHMaeT

BR,ll; y

(-r)

= [0,

1,0,

0, J X (t) = :r2

(t).

L ............:..•.•

:_-...:.

__

..:

.•

"::,'

:::,'.':

" "

,'.,

:.'

':

.:

...

:

'.':.~:.

:.'

..

:::....~.

~~.~.:~.:.

r

~~'~,>

:~70

rtOC'1'AHOl3RA

11

PEmEHUE

rtP013JIEMbl

HABJlIO)J;EHl1ff

ern.

B

IIycTb

Tpe6yeTcH

onpe,l\eJIllTb

yrJIhI

Xl

(t)

II

X

3

(t)

nOBopOTa

06ellx

paMoK

II

yrJIOByIO

CKOpOCTb

BpaIIJ;enllH

BnyTpenneR

paMKH

x

4

(t) =

.1

3

(t).

3a,l\a'la

31.2

Q na6JIIO,n;eHllIl CIlCTeMbI (31.14)

MomeT

6hITb

C,!>OPMYJIHpOBaHa Tor,n;a

CJIe,n;YIOIIJ;IIM

o6pa30M.

Tpe6yeTcH

HaRTH

JIlmeRHLle

onepaU;llH

fIJi

[t,

{X

2

(T), U

(Tn],

BLI'IIICJIHIO-

lI~IIe

Koop,n;IIHaTbI

Xl

(t),

X3

(t)

II

X

4

(t)

no

ClilrnaJIY

{X

2

(T),

U(T)},

TaK

'ITO BhIllOJI-

HHIOTCH

paBeHCTBa

(Pi

[t,

{X2

(T),

It

(T)I

xi

(tn]

=

xi

(t)

(i

=

1,3,4),

I<aKOBO

6bI

HH6hIJIO

peaJIll30BanmeeCH

D CIlCTeMe (31.14)

3HR'IeHlle

Xi

(I).

§ 32.

3aAa'la

06

OrrTDMaJIbHOM

Ha6moAeHDH

B

npeAbIAym;eM

naparpa<pe

ObIJIa

C<POPMYJIlIpOBaHa

3aAa1Ja

0

BbI-

1JHCJIeHHH

<pa30Boro

BeRTopa

X (t)

ABHi:l\ym;erocH

OO'LeRTa,

HCXOAH

H3

3Ha1JeHHH

CHrHaJIa

y

('t)(t

- tt

-<

't

-<

t)

(HJIH

{y

('1:),

U

('tm.

ITpH

8TOM

Ha

HCROMble

pa3pemaIOm;He

onepan;HH

<Pi

He

HaJIaraJIHCb

HHRaRHe

AOrrOJIHHTeJIbHbIe

TpeOOBaHHH,

nOMHMO

OCHOBHoro

yCJIOBHH

(31.4).

KaR

MbI

yBHAHM

HIDRe,

B

TaROM

CJIy1Jae

3aAa1Ja

0

HaOJIIOAeHHH

HMeeT,

BOOOm;e

rOBopH,

OeC1JHCJIeHHOe

MHOJReCTBO

pemeHHH

{<pd.

Tenepb

MM

C<pOpMyJIHpyeM

OAHO

yCJIOBHe

OnTHMaJIbHOCTH,

ROTopoe

!

"

.~

1_1'1

BMAeJIHeT

cpeAH

8THX

pemeHHH

onpeAeJIeHHMe

onepan;HH

<p?

(i

=

1\

= 1,... , n),

oOeCne1JHBaIOm;He

B

H3BeCTHOM

CMMCJIe

HaHJIy'Imee

Ra-

1JeCTBO

HaOJIIOAeHHH.

8TO

yCJIOBHe

OnTHMaJIbHOCTH

MOJRHO

BbIBeCTH

H3

C

JIeAyIOm;HX

coOOpaJReHHH.

(MbI

OrpaHHtIHMCH

HHi-Re B

8TOM

rraparpa-

<pe

JIHmb

CJIy1JaeM

CBoooAHoro

ABHJReHHH

OO'LeRTa,

RorAa

ynpaBJIeHHe

u

oTcyTcTByeT.)

i

<~

BbIme,

B

§31,

npeAnOJIaraJIOCb,

'ITO

onepan;HH

<Pi

[t, Y

('1:)]

npo-

IN;

!

~'~:

,

~!11

H3BOAHTCH

HaA

CHrHaJIOM

y ('t),

H3MepeHHMM

H

nOAaHHMM

Ha

8TY

onepan;mo

oe3

onmORH.

B

AeHCTBHTeJIbHOCTH,

OAHaRO,

CHrHaJI

y ('t)

H3BeCTeH,

RaR

npaBHJIO,

JIHmb

C

HeROTopOH

norpemHOCTbIO.

CJIeAOBa-

TeJIbHO,

HaAo

npeAnOJIaraTb,

'ITO

onepan;HH

<Pi

npOH3BOAHTCH

He

HaA

CHrHaJIOM

y ('t),

CBH3aHHMM

C

BeRTopOM

X

(t)

COOTHomeHHeM

(31.2),

HO

BbIllOJIHHeTCH

OHa

<paRTH1JeCRH

HaA

HeROTopMM

CHrHaJIOM

z ('t) = Y

(,;)

+

11

('t) (t - tt

-<

't

-<

t), (32.1)

BRJIIOQaIOm;HM

noepeuulOcmb

11

(,;).

B

COOTBeTCTBMH

C

8TMM

TaM,

rAe)

D03MOJRHa

nyTaaHn;a

MeJRAY

CUrHaJIaMU

y

(,;)

H Z

(,;),

<pYHRn;UIO y ('t

fYAeM

Ha3blBaTb

uoeaJ/,bTiblJl1,

CHrHaJIOM,

<pYHRn;UIO Z (,;)

Ha30BeM

pe-

aJ/,bTiblJl1,

CUrHaJIOM.

ITorpemHocTb

11

(,;),

eCTeCTBellHO,

AOJIJRH3. C1JHTaTbCH

HeU3BeCT-

HOH,

oAHaRo

MOJRHO

npHHHTI>,

1JTO

H3

<pu3HtIeCRUX

YCJIODUH

npon;ecca

HaOJIIOAeHMH

BbITeRaeT

HeRoTopaH

On;eHRa

8TOH

rrorpemHocTu.

TorAa

MOJRHO

nOCTaBUTb

3aAa1JY

00

onpeAeJIeHHU

OnTHMaJIbHOH

pa3pemaIO-

m;eH

orrepan;HH

<p~[t,

Z

(,;)],

BM1JUCJIHIOm;eH

ROOPAHHaTy

Xi

(t) C

Hau-

MeHbmeH

D03MOJRHOH

omUOROH

(j)

rrpu

3aAaHHOH

On;eHRe

norpemHO-

CTM

11

(T).

8TUM

CJIOBaM

HeooxoAHMO

npUAaTb

T01JHMll

CMbICJI

•

.nJIH

c

§ a2]

3A,UA'IA

OB

orrTIIMAJlbHOM

HABJIIOJJ:lnIHlI

271

Toro

'lTOObI

CAeJIaTb

8TO

M

JIy1Jme

nOHCHHTb

Ham

1l0P;XOA

R

npOOJIeMe

OnTHMaJIbHOrO

HaOJIIOAeHHH,

MbI

HCTOJIRyeM

npon;ecc

HaomOAeHUH

RaR

HeiwTOpyIO

Hrpy.

B

COOTBeTCTBMH

C

8TMM

MbI

BpeMeHHO

paCCTaHeMCH

C

3aAa'IeH

0

HaOJIIOAeHUH

H

COBepmHM

8RCRypC

B

TeopHIO

lIrp.

8Ta

MaTeMaTH1JeCKaH

TeopUH,

HMeIOm;aH

CBOHM

npeAMeTOM

MCCJIe-

p;OBaHHe

ROH<pJIlIRTHMX

cHTyan;UH,OMJIa

nOCTpoeHa

B OCHOBHOM B

Te-

1JeHMe

nOCJIeAHMX

copOKa

JIeT.

C

ee

::meMeHTaMH

MOJRHO

n03HaROMMTb-

CH,

HanpUMep,

no

RHHraM

[2*,16*].

M:bI

npHBeAeM

COBceM

RpaTRO

JIHlJIb

HeROTOp:bIe

OCHOBHbIe

nOHHTMH

TeopUH

lIrp,

KOTopbIe

rroHaAooHTcH

AJIH

BMHCHeHMH

CBH3U

8TOH

TeopUH

C

HaililIMM

3aAaqaMH.

I

IhaR,

paccMoTpMM

OAHy

U3

TUnU'IHMX

cHTyan;HH,

C

ROTOpLIMH

HMeeT

AeJIO

TeopHH

Hrp.

8Ta

cHTyan;HH

onpeAeJIHeTCH

OObI'IHO

KaR

Hrpa

AByX

JIHn;.

B

TaROH

Hrpe

Y'IaCTByIOT

ABa

HrpORa,

npeCJIeAyIOm;HX

npOTHBO-

I

noJIoJRHMe

n;eJIu.

KaJRAbIH

HrpOK

MOJReT

COBepUlaTb

AeHCTBUH,

JIeJRa-

m;ue

B

npeAeJIaX,

p;onYCKaeM:bIX

npaBUJIaMH

HrpbI.

ITpHMeM,

'ITO

HCXOA

urpbI

xapaKTepu3yeTcH

HeROTOp:bIM

1JUCJIOM

~,

ROTopoe

yCJIOBHO

lIa30-

BeM

ebwepblUleM

nepBoro

HrpORa

(HJIH

npouepblUleJl1,

AJIH

BToporo).

IIoRa3aTeJIb

~

TaROB,

1JTO

B

HHTepecax

nepBoro

lIrpORa

CAeJIaT.b

eTO

B03MOJRHO

o

OJIbUlHM

, a B

HHTepecax

BToporo,

HarrpOTHB,-

CAeJIaTb

3Ha1JeHUe

~

B03MOJRHO

MeHbmHM.

B

8TOM

H COCTOHT

KOH<pJIHRT.

IIpH-

Mep:

napTUH

Hrp:bI

B

UlaxMaTbI,

rAe

MO}l~HO

npHHHTb

~

= 1 B

CJIY'Iae

BMHrp:bIma

nepBoro

HrpORa,

~

= 0 B

CJIyqae

HU1JbeH

H

~

=

-1,

eCJIH

napTuIO

B:bIHrpaeT

BTOPOH

HrpOR.

. .

KaJRAbIH

U3

HrpOKOB

CTpOHT

CBOH

AeHCTBHH,

pyROBOACTByHCbI\a-

RUMH-T.O

COOOpaJReHHHMH.

8TH

COOOpaJReHHH

COCTaBJIHIOT

TO,

'ITO

npHHHTO

Ha3bIBaTb

cmpameeuR,MU

HrpOROB.

M:bI

orpaHHtIHMCH

JIHUlb

t

TaKUMH

CJIy'IaHMU,

KorAa

MOJRHO

CTporo

O'IepTHTb

Te

KJIaCCbI

{a}

H

{~}

pa3JIU1JHbIX

CTpaTerHH

a H

~,

KOTOpbIe

MoryT

BbIOHpaTb

COOTBeT-

CTBeHHO

nepBLIH

H

BTOpOH

HrpoKH.

3AeCb

CIIMBOJIbI

a

II

BB

pa3JIlI1JH:bIX

Hrpax

MoryT

0003Ha1JaTb

caMbIe

pa3Hooopa3HbIe

OOT'CHTbI:

8TO

MoryT

ObITb

1JIICJIa,

BeRTopbI,

<pyHRn;IIII,

onepan;IIII,

HaoopLI

CJIOBeCHbIX

HH-

CTpyKn;HH,

nporpaMMLI

cJIyqaiiHbIX

onepan;HH

JiI

T.

l~.

I1pIIHHMaeTCH,

1JTO

BbmrpbIUl

~

HBJIHeTCH

:U:3BeCTHOH

<pyImn;:u:eH

~

[a,

~]

OT

a

H3

{a}

II

OT

~

H3

{B}.

BaJRHO

nOAqepRHyTb

CJleAYIOm;ee

YCJIOBIIe:

Iwm,JJ:LIH

II3

HrpOKOB

npII

B:bIOOpe

cBoeH

CTpaTerIIH

lie

UTirjJopMupOeaTi

0

TOM,

Ka-

KyIO

CTpaTerIIIO

H36epeT

ero

napTHep.

IhaK,

paCCMaTpHBaeMaH

HaMH

Hrpa

ABYX

JIHn;

xapaRTepH3yeTcH

CJIeAYIOm;IIMII

1JepTaMII

.

.naHbI

3a-

naCbI

CTpaTerIIH

{a}

II

{~},

OTI~YAa

ux

MoryT

1JeprraTb

nepBbIH

II

BTO-

POH

HrpoKu,

II

BbIopaHa

<pYHKn;IIH

~

[a, Bl,

orrpeAeJIeHHaH

AJIH

Bcex

a

U3

{a}

U

~

U3

{~}.

3aAa1Ja

nepBoro

IIrpoKa

-

BLIOOPOM

a

MaRCIIMH3U-

pOBaTb

~,

3aAa'Ia

BToporo

IIrpOKa

-

BbIOOPOM

~

MUHHMII3IIpOBaTb

~.

TaRuM

00pa30M,

paccMoTpeHHe

IIrpoBoH

cHTyan;UII

npIIBOAHT

1\

3a-

Aa1Je 0 MUliUMance

~

(HJlII

0

ManCUJI1,lllie

~).

YT01JHIIM

8TY

3aAaqy.

BYAeM

paccyi-HAaTb

CHaqaJIa

C

T01JRII

3peHIIH

BToporo

urpoRa.

(8TO

ORaJReTCH

YAOOH:bIM

AJIH

nepexop;a

nOTOM

R

3aAa

q

e 0

HaomoAeHHII.)

'.'

••.•

""",>

.....

:,>.,

"";,:~;'.':'

...... ' ·,·c··;·.::···'.·.··

.~.

':l.:,·"

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

..

~:~:~~;'

"

:

'.:'

~::';!

C

272

rIOCTAHOBHA

11

P~iJlIEHlm

IU'OBJIEMbI

HABJIIO,o:EHH

fl

frn.

8

IlTal\,

BTopOHHrpOI\

BbIOOPOM

~

AOJUl\eH

CTpeMHTbCH

npeAeJIbHO

yMeHb-

IIIHTb

CBOH

npOHrpbIIII

~.

ITYCTb

OH

BbIOpaJI

HeKoTopyIO

CTpaTerHIO

~

=

~a.

TorAa,

O'leBHAIlO,

caMbIH

HeOJIarOnpHHTHbIH

AJIH

HerO

CJIy-

'laH

ocym;ecTBHTCH,

ecmI

nepBbIH

HrpOK

BbIoepeT

TaKyIO

CTpaTerHIO

a = at..,

AJIH

KOTOpOH

~

[at..,

~a]

=

max

~

[a,

~a

I = X

[~a]

no

BceM

IX

a HS

{a}.

B

caMOM

AeJIe,

O'leBHp;HO,

BeJIH'lHHa

X

[~a]

OSHa'laeT

caMbIH

OOJIbllOH

npOHrpbIll,

KaKOH

TOJIbKO

MOllieT

nocTHrHyTb

BToporo

Hrpo.-

Ka,

eCJIH OH

HsoepeT

CTpaTerHIO

~a.

ITPH

aTOM

BalliHO

nOA'lepKHyTb,

'ITO

OT

yKasaHHoro

npOHTp'bIlla

BTOPOH

HrpOK

HHKaK

He

rapaHTHpo-

BaH,

TaK

KaK

no

yCJIOBHHM

HrpbI

OH

He

MOllieT

npeAyraAaTb,

KaKyIO

CTpaTerHIO

BbIOepWl'

nepBbIH

HrpOK.

JReJIaH

npep;eJIbHO

OCJIaOHTb

rpo-

Sflm;yIO

eMy

HenpHHTHOCTb,

BTOPOH

HrpOK

P;OJIJI\8H

HCKaTb

TaKyIO

CTpa-

TemIO

~o,

P;JIH

KOTOpOH

X

[~Ol

=

min

X

[~]

no

BceM

~

HS

{~}.

11TaI<,

P;JIH

BToporo

HrpOKa

npep;CTaBJIHeTCH

paSYMHbIM

BbIOHpaTb

CTpaTe-

ruIO

~O

HS

yCJIOBUH

.

mjnmax~[a,p]=x[PoI.

(32.2)

f3

IX

,

'YKaaaHHaH

CTpaTerHJI

~o

oOeCne'lHBaeT

eMy

npOHrpbIll

He

OOJIblle,

~

:1\,

'leM

BeJIH'lHHa

(32.2),

KaK

ObI

HH

p;eHCTBOBaJI

nepBbIH

HrpOK.

ITPH

3TOM

i

I

lli

B

npeAeJIaX

npaBHJI

HrpbI

HeJIbSH

yKaSaTb

p;pyryro

cTpaTeruIO

~

HS

{~},

KOTOpaJI

rapaHTHpOBaJIa

ObI

BTOpOMy

HrpoKy

npoHrpbIll

MeHb-

l:

llHH,

'leM

BeJIH'lHHa

(32.2).

I1TaK,

CTpaTerHH

~o,

HaHp;eHHaH

HS

yCJIO-

BHH

(32.2),

HBJIHeTCH

B

HSBeCTHOM

CMbICJIe

OnTHMaJIbHOH

P;JIH

BToporo

JirpoKa.

3aMeTHM

em;e

CJIep;YIOm;ee.

Bbille

MbI

MOJI'laJIHBO

npeAnOJIa-

i;:

~~';

raJIH,

'ITO

npH

Kallip;OM

BbIoope

~

=

~a

cym;ecTByeT

MaKcHMyM

ll)i

X

[~a]

=

~

[a

h

,

~a],

p;OCTHraeMbIH

Ha

HeKOTopOH

CTpaTerHH

a

h

H3

,r

l

{a}.

Op;HaKO

MOiHeT

CJIy'lHTbCH,

'ITO

TaKoro

aJIeMeHTa

a

h

B

Haoope

CTpaTerHH

{a}

HeT.

Torp;a

paccymp;eHHH

CJIep;yeT

npOBop;HTb

no

TOMy

iHe

ImaHy,

lITO

H

BbIlle,

HO

rrop;

BeJIHqHHOH

X

[~a]

Hap;o

nOHHMaTb

X

[~(j

I =

sup

~

[a,

~a

I.

TaI\HM

oopaSOM,

B OOm;eM

cJiylIae

onmUMMb-

a.

!toU

CTpaTerHeH

llToporo

nrpOKa

oyp;eM

HaSbIDaTb

TaKyIo

CTpaTerHIO

~O,

KOTOpaJI

rapaHTHpyeTeMy

rrpOHrpbIll

He

OOJIbllHH,

'leM

X

[PO]

=

min

sup

1;

[a,~],

(32.3)

f3

a.

H

oOJIaAaeT

rrpH

aTOM

TeM

CBOHCTBOM,

lITO

HHKaKaH

p;pyraH

CTpaTerHH

He

MOllieT

rapaHTHpOBaTb

MeHbllHH

rrpOHrpbIIII

1).

PaCCYllip;aH

aHaJIO-

rWIHbIM

00pa30M

sa

nepBoro

HrpOKa,

yoep;HMcH,

lITO

ero

orf'fHMaJIbHaH

1)

06paTHM

BHHMauHe

'lHTaTeJIJI,

'ITO

3,L\eCb IIO

~

MhI

IIO-IIpeJRHeMy

lIrn;eM

MI1HIfMYM,

T.

e.

npe,L\IIOJIaraeM,

'ITO

cyrn;eCTByeT

CTpaTerHJI

~o

H3

{~},

Ha

ROTOpOii:

.

numHHll

rpaHb

,L\OCTHraeTCll. Iif

rra

'Ie

, MbI

IIpOCTO

IIpllHHMaeM,

'ITO

paCCMaTpHBae-

Mall

aall:a'la

06

OflTUMam,IIOR

CTpaTeI'IUI

~o

pCmOHHJl.

ne

llMeCT,

H

TaRne

CJIyqall

\'13

paCCMOTpeHIUl

Bo06rrw

HCHJT!O'IaCM.

AnUJIOrIf'ruh/c

co06pa1f(OIIIUI

OTIIOCllTCli II

II

CJIyqaIO

(32.4).

(}:-

32]

3A

ALIA

013

OrrTHMAJIbHOM

HABJIIO.D:EHHU

273

CTpaTerHH

aO

p;OJIlliHa

BbIOHpaTbCH

HS

yCJIOBHH

~

'Iji

[an]

=

max

'Iji

[a],

rp;e

'Iji

[aa] =

inf

[aa,

~].

at

f3

CTpaTerHH

aO

oYAeT

rapaHTHpoBaTb

nepBoMy

HrpoKy

BbIHrpbIll

He

MeHbllHH,

lIeM

'Iji

[an]

=

max

inf

~

[a,

~I,

(32.4)

a.

f3

~

H

HHKaKaH

p;pyraH

CTpaTerHH

He

MomeT

rapaHTHpOBaTb

eMy

OOJIbIDHH

BbIHrpbIll.

3aMeTHM,

'ITO,

BOOOm;e

rOBopH,

'Iji

[an]

=1=

X

[~O],

HO

OOH-

'SaTeJIbHo'lji

[aOI

-<

X

[~O],

KaK

31'0

p;OKaSbIBaeTCH

B

TeopHH

Hrp

(CM.

[2*

J).

It

oocymp;eHHIO

3Toro

OOCTOHTeJIbCTBa

MbI

em;e

BepHeMCH

B § 37,

a

ceHlIac

nOKa

OrpaHH'lHMCH

Cp;eJIaHHbIM

saMelIaHHeM.

Ha

aTOM

nOKa

Hall

KpaTKHH

3KCKypC

B

TeopHIO

Hrp

SaKOHTJ:HM H

BepHeMCH

.K

sap;alIe

o

HaOJIIOp;eHHH.

r

)

I1TaK,

paCCMOTpHM

p;HHaMH'lecKyIO

cHcTeMy

1;

= A

(t)

x, (32.5)

npep;nOJIaraH,

'ITO

pesyJIbTaTOM

HSMepeHHH

ee

p;BHllieUHH

HBJIHeTCH

pea.ttbllbLU cuma.tt

z ('t) = G ('t) x ('t) +

L1

('t)

(t -

1'}

-<

'1:

-<

t),

(32.6)

cop;eplliam;HH

norpellHOCTb

L1

('1:).

B

p;aJIbHeHlleM

y~OOHO

CTJ:HTaTb,

'ITO

BeKTop-<pYHKIJ;HH

Y

('1:)

H

L1

('1:)

HBJIHIOTCH

3JIeMeHTaMH

h

('1:)

He-

KOToporo

<PYHKIJ;HOHaJIbHOrO

rrpOCTpaHCTBa

fliJ

{h

('1:);

t -

1'}

-<

'1:

-<

t},

B

KOTOpOM

onpep;eJIeHa

l£opMa p [h

('1:)

J,

H

rrpHHHTb,

lITO

OIJ;eHKa BOS-

MOlliHOH

nOMeXH

L1

('1:)

BbIpalliaeTCH

KaK

pa3

B

BHp;e

HepaBeHCTBa,

CBH-

SbIBaIOm;ero

HOpMy

p

[,1

('1:)],

T.

e.

p

[L1

('1:)]

-<

(');

(32.7)

j'

l'

3p;eCb

°-

nOJIOlliHTeJJbHaH

rrOCTOHHHaH.

81'0

npep;nOJIOllieHHe

HB-

JIHeTCH

eCTeCTBeHHbIM,

TaK

I,aK,

IIop;oHpaH

pa3JIHlIHbIe

HSBeCTHbIe

HOpMbI

p,

MomHO

OXBaTHTb

OOJIbllOe

'IHCJIO

HHTepeCHbIX

CJIylIaeB.

Ha-

npHMep,

eCJIH

no

yCJIOBHHM

SaAa'lH

HSBeCTHO,

lITO

rrOMexa

L1

('1:)

B

JIIO-

OOH

MOMeHT

'1:

He

rrpeBocxop;HT

no

MOp;yJIIO

BeJIH'lHHy

0,

T.

e.

eCJIH

II

L1

('1:)

II

-<

0,

(32.8)

TO

B

Ka'leCTBe

npOCTpaHCTBa

fliJ

CJIep;yeT

BbIopaTb

MHOllieCTBO

r6

{h

('1:);

t -

1'}

-<

'1:

-<

t}

HerrpepbIBHbIX

<PYHKIJ;HH h

('1:)

C

HOpMOH

(CM. § 60) P

[h

('1:)]

---:

max

II

h

('1:)

II

(t -

1'}

-<

'1:

-<

t).

Torp;a

He-

1:

paBeHCTBO

(32.7) H

oyp;eT

OSHa'laTb

orpaHHlIeHHe

(32.8).

3arrHcb

OIJ;eH-

KH

P;JIH

L1

('1:)

B

<popMe

(32.7)

oOJIerTJaeT

MaTeMaTHlIeCKoe

HCCJIep;OBaHHe

3ap;a

I

IH.

.',"

','.

<':"'~'.-:

." ' :::

:...

~::-;

-:?:.

,';'::' :

..

~

:.:~?:r:\

••

,

1

:i

'

r

\,~~:,

~

r

274

IIOCTAHOBRA

11

PEIllEHI1E

lIPOBJIEMbI

HAEJIIO,D.EHI18:

[rJl.

8

MTaR,

nyCTb

B

CIICTeMe

(32.5), (32.6)

MoryT

peaJIlI30BaTbCJI

JII061i1e

3Ha'IeHIIJI

cPa30Boro

BeKTopa

x (t)

11

JII06aJI

nOMexa

~

(or),

CTeCHeHHaJI

YCJIOBI1JIMII

(32.7).

ClIrHaJI

Z

(or)

(32.6),

peaJIII3YIOIIJ;IIHCJI

B

CIICTeMe

(32.5) (32.6)

nplI

HeKOTopOM

3Ha'IeHIIII

Xi

(t)

II

nplI

yCJIOBlI1I

(32.7),

6y-

;a;eM

Ha31i1BaTb

peaJIbHhIM

ClIrHaJIOM,

l-leCYUfUM

Xi (t),

11

0603Ha'II1M

ero

CIIMBOJIOM

{z

(or)

I Xi (t)}.

IIYCTb

nOKa

3Ha'IeHlIe

t

cPIIKClIpOBaHO.

By;a;eM

IICKaTb

TaKyIO

one-

paU;I1IO

cp?

[t, Z

(or)],

KOTOpaJI

nplI

onpe;a;eJIEiHIII1

Koop)J;JIHaThI

Xi (t)

;a;-a-

BaJIa

61i1

HalIMeHbrnyIO

B03MOIKHYIO

ornlI6Ky

B

Ca.MOM

l-le6.fl,aeonpUR-ml-lOM

C.lLY'IJ,ae

peaJIbHOrO

ClIrHaJIa

{z

(or)

IXi (t)},

KaKOH

TOJIbKO

MOiI\eT

BCTpe-

TIITbCJI.

)J;JIJI

HCTOJIKOBaHIIJI

:'noro

BlilpaIKeHI1JI

HaM

II

nOTpe6yIOTCJI

paCCMOTpeHHhIe

TOJIbKO

'ITO

nOHJITI1JI

TeoplIH

I1rp.

IIOCTPOIIM

IIrpy,

KOTOpyIO

MOIKHO

CBJI3aTb

C ;a;aHHhIM

npou;eCCOM

Ha6JIIo;a;eHIIJI.

IIYCTb

Ha6JIIO;a;aTeJIb

I1CnOJIHJIeT

POJIb

BToporo

IIrpOKa.

B

Ka'IeCTBe

nepBoro

I1rpOKa

Mlil

6y;a;eM

nepCOHI1cPI1U;I1pOBaTb

Ha6JIIO-

;a;aeMYIO

CI1CTeMy.

BeJII1'IIIHOH

S,

KOTOpaJI

xapaKTepI13yeT

3;a;eCb <<llpO-

IIrpIilIll»

Ha6JIIO;a;aTeJIJI,

6y;a;eT

CJIYIKHTb

BeJIII'II,ma

I

w;j,

r;a;e

Wi

-

onIlI6Ka

Ha6JIIO;a;eHIIJI,

paBHaJI

\

,\'

Wi

=

cp

[{z

(or)

I

Xi

(t)}] - -Xi (t). (32.9)

CTpaTerI1JIMI1

a

nepBoro

IIrpOKa

eCTeCTBeHHO

C'II1TaTb

Tenepb

Bce-

l'iil

'

~

B03MOIKHlile

ClIrHaJIIiI

z

(or)

(32.6)

1I3

fB,

r;a;e

nOMexa

~

(or)

,CTeCHeHa

YCJIOBI1eM

p

[~

(or)]

< 6, a

CTpaTemJIMII

B

BToporo

I1rpOKa

Ha;a;JIeIKIIT

C'II1TaTb

BceB03MOIKHhIe

JII1HeHHhIe

onepaU;III1

cp

[z

(or)],

onpe;a;eJIeH-

HhIe

Ha

npOCTpaHCTBe

fB

ClIrHaJIOB

z

(or).

Tenepb

;a;OCTaTO'IHO

BOCnOJIb-

",

..

30BaTbCJI

paccYIK;a;eHlleM,

npI1Be;a;eHHIiIM

Ha

CTp.

272,

'IT061>I

nplIHTI1

'~

K CJIe;a;YIOIIJ;eMY

BIiIBo;a;y.

B

Ka'IeCTBe

onmUMMbl-lOU

onepaU;1I1I

cpo;,

~I

IWTOpaJI

rapaHTlIpyeT

Ha6JIIO;a;aTeJIIO

HalIMeHl>rnlIH

B03MOIKHI>IH

npo-

uepbWJ,

~

= I

Wi

I,

Ha;a;JIeIKI1T

BIiI6paTI>

OnTIIMaJII>HYIO

cTpaTerIiIO

BO

Ha6JIIO;a;aTeJIJI

B

paccMaTplIBaeMoH

3;a;eCI>

IIrpe.

IIPII

aTOM

MaKCII-

MaJII>HIiIH

npOI1rplilrn,

KOTOplilH

rp03lIT

Ha6JIIO;a;aTeJIIO

nplI

BI>I60pe

(P~

[t,

z(or)]

=

BO,

6y;a;eT

onpe;a;eJIJITI>CJI

paBeHCTBOM

'X

[~O]

=

min

sup'

w;/,

(32.10)

cp

z

II

HIIKaKaJI

onepaU;IIJI

cp

[z

(or)]

He

MOIKeT

rapaHTI1pOBaTI>

HaM

OIJ1lI6Ky

MeHl>rnyIO,

'IeM

BeJIII'II1Ha

(32.10).

STy

BeJIII'IIIHy

'X

[BO]

(32.10)

6y-

.n.eM

0603Ha'IaTb

CIIMBOJIOM

woo

MTaK,

YCJIOBI1MCJI

C'IIITaTb

HalIJIy'IrneH

onepaU;lIeH

cp

Ty

onepaU;IIIO

cp?

1I3

fB*, ;a;JIJI

KOTOpOH

BeJIII'II1Ha

sup

I

Wi

I

;a;OCTlIraeT

MIIHIIMYMa,

T.

e.

;a;JIJI

KOTOpOH

1)

sup

I

Wdcp~,

{z

(T)

I

xdt)}]

1 =

min

sup

I

wdCP,

{z

(T)

I

xdt)}]

1 (32.11)

z

cp

z

1) CIfMBOJI Wi

[ep,

{z (or) I

Xi

(t)} 1

IIo,n;qepKlIBaeT

3.I\eCr"

qTO

lleJIH'IHHa

(,li

eCTh

¢YHI{f~HJr

(TO'lHee

rPYHR~IfOHaJI)

OT

ep

If

Z ('t').

I

r~

~.'

I

§ 32]

3A)l;A

tIA

OB

On'I'HMAJIhHOM

HABJIIO)l;EHHH

275

!

no

BceM

B03MOIKHIiIM

JIIIHeHHIiIM

onepaU;HJIM

cP,

KOToplile,

KaK

H3BeCTHO,

COCTaBJIJIIOT

rrpOCTpaHCTBO

fIi3*

(CM.

§ 59).

ECJIH

B

yCJIOBlI1I

(32.7)

6>

0, TO

BeJIH'IHHa

(32.11)

HalI60JIl,rneH

ornlI6KlI

IWi

/,

KOTOpaJI

MO-

IKeT

peaJIlI30BaTbCJI

rrplI

IICnOJIb30BaHlIlI

onepaU;1I1I

cp?

;a;JIJI

BIiI'IIIC-

JIeHIIJI

Xi (t),

06JI3aTeJIbHO

rrOJIOIKHTeJIbHa,

T.

e.

npI1

caMOM

JIy'IrneM

BIiI60pe

onepaU;1I1I

HelI36eIKHIiI

ornlI6KlI

Wi

=1=

O.

ECJIII

me

nOMexa

~

(or)

OTCyTcTByeT

(1'.

e.

z

(or)

= Y

(or»

II

onepaU;IIJI

cp?

pernaeT

3a;a;a-

'IY

31.1 0

Ha6JIIO;a;eHIIII,

TO

eCTeCTBeHHO,

'ITO

W

O

=

O.

O'IeBH;a;HO

TaKme,

'ITO

y;a;OBJIeTBOpIITeJIbHIiIM

~

CJIe;a;yeT

C'II1TaTb

JIlIrnb

TaKoe

perneHlIe,

;a;JIJI

KO-

Toporo

Wo<

00.

TaKHM

06pa30M,

no-

JIy'IaeM

CJIe;a;YIOIIJ;YIO

3a-

~(t)

;a;a'IY

06

OnTIIMaJIbHOM

$('t)

Ha6JIIO;a;eHIIII

.

3 a;a;

a'I

a.

32.1

IIycmb

o

't

t

------

't

8euJlCYUfuuCR-

o6'be/'i,m

onu-

Cb18aemCR-

ypael-lel-lUeM

(32.5),

npU<teM

peaY.fl,bma-

mOM

UaMepel-lUR-

amoeo

8euJlCel-luR-

R-MR-emCR-

pea.fl,bl-lblU

CUeT-la.fl,

Z

(or)

(32.6),

e8e

noMexa

cmeCl-lel-la

oepal-lU'IJ,e7-lue},t

(32.7). Tpe6yemcR- l-laumu

onepalfUlo

cp?

ft,

z

(or)],

/'i,omopaR-

(npu

KaJIC80M

t

ua

paCCMampueae-

MOU

o6.fl,acmu

ua},Wl-lel-lUR-

epeMe7-lu)

y808.fl,emeopR-em

YC.fl,OeUlO

sup

I

CP?

[t,

{z

(t') I

Xi

(t)}]

-

Xi

(t) I=

min

sup

I

cP

[{z

(T)

I

xi

(t)}]-

Xi(t) I

k t

cp

z

(32.12)

no

eceM

eOaMOJICl-lblM

pea.fl,uaa/fUR-,M,Z

(or)

U no

eceM

e03MOJICl-lblM

onepa-

lfUR.M

cpo

Onepa/fulO

cp~

[t,

z

(or)]

6y8eM

l-la3b18amb

onmuMMbltOU paapeutalo-

Ufeu

onepalfueu (;a;JIJI

3a;a;a'IlI

32.1).

06IIJ;lIe

co06pameHIIJI,

BlilCKa3aHHlile

Blilrne

B

aTOM

naparpacP

e

,

MhI

npOIIJIJIIOCTplIpyeM

ceii'Iac

Ha

O'IeHb

npocToM

npI1Mepe.

C

aTOH

u;e-

JIbIO

paccMoTpHM

MaTepHaJIbHyIO

TO'IKy

m,

cB060;a;HO

W:llI}Kyn~YIOCJI

no

npJIMOH

£

(pIIC.

32.1)

II

OnIICI>IBaeMYIO,

CJIe;a;OBaTeJIbHO,

ypaBHeHIIJlMI1

':£;1

= X

2

'

1;2

= 0 (Xl =

£,

X

2

=

~).

(32.13)

'."

IIpe;a;noJIOIKIIM,

'ITO

I13MepJIeTCJI

KOOp;a;IIHaTa

Xl

(or)

= £

(or)

(t

-

{}

<

or

< t),

II

Ha

OCHOBaHlI1I

3'roro

1I3MepeHlIJI

Tpe6yeTCJI

BhI-

'IlICJIIITb

CKOpOCTb,

~

(t)

= X

2

(t)

TO'IKII

m B

MOMeHT

t.

HCHO,

'11'0

no-

CTOJIHHyIO

CKOpOCTb

X

2

(t)

paBHOMepHO

;a;BIIIKYIIJ;eHCJI

TO'II<II

m B ;a;aH-

HOM

CJIy'Iae

MomHO

BIiI'IIICJIJITb

no

061i1'IHOH

cPopMyJIe

X2

(t) =

xI(t)

-

::I(t

-

tt)

(32.14)

PlIC.

32.1.

.....

..

,.

......

,

..

:,'

..

'

..

::"'::>:-'::~:i::~

C

..

,

' '

276

noCT

AH01JHA

11

PEi1IEHIIE npOBJI1!;Mhl HAi3JIIO.D:EHlliI

[rJI.

8

TaR

RaR

npaBaH

qaCTb paBeHCTBa

(32.14)

eCTb

JlHHeiiHaH

onepa-

n;HH

Ha,n;

CIIrHaJlOM

y (r) =

Xl

(r)

(t - 1t

-<

't

-<

t),

(32.15)

TO

MomHO

CRa3aTb, 'ITO JlHHeiiHLIe

onepa~HH

'PI

[t,

Y (,;)] =

<PI

[t,

Xl

('t)] =

Xl

(t),

<P2

[t,

Y (,;)] =

cpd

t

,

xd,;)]

pemaIOT

3,n;eCb

3a,n;aqy

=

'f

1

[y

(t) - Y (t

-1t)]

=

1

. =

'fr

[xdt)-xd

t

-")]

(32.16)

31.1

0

HaOJlIO,n;eHHH

<pa30Boro BeKTopa

X (t) =

{Xl

(t), X

2

(t)}

no

CHrHaJlY Y

(,;)

(32.15).

TaKHM

oopa30M,

MJiI

BH,n;HM,

'ITO 3a,n;aqa 0

HaOJlIO,n;eHHH

no

H,n;eaJlbHoMy CHrHaJlY

Y ('t) =

Xl

('t) pemaeTCH

3,n;eCb

TpHBHaJlbHO. ,Il;eJlO,

o,n;HaKO,

cpa3Y

OC-

JlOmHHeTCH,

KaK

TOJlbKO

MJiI

nepeii,n;eM K

3a,n;aqe

0

HaOJlIO,n;eHHH

X (t)

no

peaJlbHoMy

CHr~aJlY

z ('t) = Y ('t) + L\('t) =

Xl

('t) + L\(.-) (t

-1t

-<

't

~

t),

(32.17)

~.

co,n;epmam;eMY

nOMexy

L\

('t).

')~

B

caMOM

,n;eJle,

npe,n;nOJlOmHM,

HanpHMep,

'ITO nOMexa

L\

('t) CTec-

'li.'

HeHa JlHmb

CJle,n;YIOm;HM

HHTerpaJlbHJiIM OrpaHHqeHHeM:

t

t:

~

I

L\

(,;) I

d,;

-<

6,

(32.18)

t-&

a

BO

BceM npOqeM MomeT

OJiITb

COBepmeHHO npOH3BOJlbHOii. B qaCT-

HOCTH,

3HaqeHHH

L\

(t)n

L\

(t - 1t)

npH

OrpaHHqeHHH

(32.18)

MoryT

~

OJiITb

CKOJlb

yro,n;HO OOJlbmHMH

no

aOCOJlIOTHOii

BeJlHqHHe.3aMeTHM,

KCTaTH, 'ITO BeJlHqHHa B JleBoii qaCTH

(32.18)

HMeeT

CMJiICJI

HOpMJiI

P [L\('t)]

<PYHR~HH

L\('t),

eCJlH

:ny

<PYHK~HIO

paCCMaTpHBaTb

KaK

aJleMeHT npOCTpaHCTBa

2[t-&,t]

(CM.

§

56).

.

ECJlll

H

Tenepb

,n;JlH

BJiIqHCJleHHH

CKOpOCTH

X

2

(t)

MJiI

nOnJiITaeMCH

CHOBa

BOCnOJlb30BaTbCH <popMyJloii, aHaJlOrHqHOii

(32.14),

T.

e.

eCJlH

MJiI

nOJlOmHM

"

X2 (t) = z (t) -

~

(t -

'fr)

_

Xl

(t) -

Xl

(t

-

'fr)

+

t:,.

(t)-

t:,.

(t

-

'fr)

_ (t) + '

-

'fr 'fr

- X2

U>2,

TO

omHOKa

U>2'

paBHaH

[L\

(t)

-

L\

(t - 1t)

]/1t,

MomeT OKa3aTbCH

CKOJlb

yro,n;HO oOJlbmoii.

CJle,n;OBaTeJlbHO,

nOCTpoeHHLIe BbIme

onepa-

n;HH

CPl

H

CP2

(32.16)

B

cJlyqae

Haomo,n;eHHH x

(t)

no

peaJlbHoMy CHrHllJJY

z ('t)

(32.17)

npH

yCJlOBHH

(32.18)

He

,n;aIOT

CKOJlbKO-HHOy,n;b

y,n;OBJleT-

BOpHTeJlbHOrO pemeHHH. Me}K,n;y

npOqllM,

aTO

CBH3aHO

C

TeM

OOCTOH-

TeJlbCTBOM,

'ITO BeJlHqHHa B

JleBOH

qaCTH HepaBeHCTBa

(32.18)

HMeeT

Q.

§

32]

3A.D:A'IA

OB

OnTHMAJIbHOM HABJIIO.D:EHllli

277

CMJiICJI

HOpMbI B npOCTpaHCTBe

2[t-&,t]

HHTerpHpyeMbIx

<PYHK~Hii,

a

onepan;HH

CPl

H

CP2

(32.16)

onpe,n;eJleHbI H

orpaHll'IeHbI

B oOJlee

Y3KOM

npocTpaHcTBe

~[t--l},t]

HenpepbIBHbIX

<PYHK~HH.

HanpoTHB,

eCJlH

npe,n;~

nOJlaraTb, 'ITO nOMexa

L\

('t)

CTeCHeHa

OrpaHHqeHHeM

.I

L\

(.-)

I

-<

6,

(32.19)

KOTopoe OTBeqaeT RaK

pa3

OrpaHHqeHHIO

HOpMJiI

p

[L\

(1:)]

B npOCTpaH-

CTBe

~[t-&,t]

(CM.

BbIme, CTp.

273),

TO

omHOKH

npH

HCnOJlb30BaHHH

one-

pa~Hiicpl

H

QJ2

(32.16)

,n;JlH

BbJ1IHCJleHHH

BeJlH'IHH

Xl

(t) H x

2

(t)

no

CHrHaJlY

(32.17),

KaK

HeTpy,n;HO

npOBepHTb, He npeB30H,n;YT

COOTBeTCTBeHHO

()

H

26/1t.

(8TH

omHORH nOJlyqalOTCH

npH

L\

(t) = 6,

L\

(t

- 1t) = - 6.)

HH-

me

oy,n;eT

nOKa3aHO

(CM.

§

33),

'ITO

onepa~HH

(32.16)

B

cJlyqae

orpa-

HHqeHHH

(32.19)

KaK

pa3

H

HBJlHIOTCH

3,n;eCb

OllTHMaJ.IbHbIMH

,n;JlH

3a-

,n;aqH32.1.

IIMeHHo,

MomHO

npOBepHTb, 'ITO KaKyIO

ObI

,n;pyryIO JlH-

HeiiHyIO

onepa~HIO

cP

(H

,n;ame

HeJlHHeiiHyIO

onepa~HIO),

OTJlHqHYIO

OT

onepa~HH

CP2'

MbI

HH

BblOpaJlll,

MomHO

no,n;oopaTb TaKyIO

nOMexy

L\

('t)

(32.19)

H TaKoe ,!VlHmeHHe TO'IKH m,

npH

KOTOpbIX omHOKa

U>2

npH

BbIqHCJleHHH

CKOpOCTH

no

<popMyJle X

2

(t) z

cP

[z

(t)]

oy,n;eT

OOJlbme, qeM 26/1t. AHaJlOrHqHaH

npOBepKa

B03MomHa H

,n;JlH

onepa-

n;HH

CPl

(CM.

HHme,

§

33).

TaKHM

oopa30M,

OKa3bIBaeTCH, 'ITO onepan;HH

<PI

H

CP2

(32.16)

HBJlHIOTCH

OllTHMaJlbHbIMH pa3pemaIOID;HMH onepan;HH'-

MH

B

3a,n;aqe

32.1

,n;JlH

CHCTeMbI

(32.13)

B

cJlyqae

peaJlbHOrO CHrHaJla

(32.17)

npH

OrpaHH'IeHHH

(32.19).

BepHeMcH

Tenepb

onHTb K cJlyqaIO, Kor,n;a nOMexa

L\

('t)

CTeCHeHa

YCJlOBHeM

(32.18),

H nOCTapaeMCH no,n;oopaTb

onepa~HIO

cpa,

ROTopaiI

B OTJlH'IHe

OT

onepa~ml

CP2

(32.16)

,n;aBaJla

ObI

H

npH

OrpaHHqeHIIH

(32.18)

ROHeqHYIO

omHoKy,

He

npeBOCXO,n;Hm;yIO

onpe,n;eJleHHLIX npe,n;e-

JlOB

(,n;JlH

onepa~HH

QJ2

(32.16),

KaR MbI BH,n;eJlll BbIme, BepXHHH

rpaHb

omHoKH

npH

YCJlOBHH

(32.18)

paBHa

00).

Roop,n;HHaTa

Xl

('t) CBH3aiIa

CO

CKOpOCTbIO

x

2

(t)

COOTHomeHHeM

Xl

(1:)

=

Xl

(t) + X

2

(t) ('t - t).

(32.20)

By,n;eM HCKaTb

onepa~IIIO

cpa

[t,

Z ('t)] B

<pOpMe

t

cpa

[t,

z

(,;)]

=

~

z

(,;)

V

(t,';)

d,;

(32.21

)

t-i/o

H nOTpeoyeM CHaqaJla,

qTOObI

npll

L\

('t) = 0

onepa~HH

(32.21)

,n;a-

BaJla TOqHbIH pe3YJlbTaT 1),

T.

e.

qTOObI

BbInOJlHHJlOCb

YCJlOBHe

t

cpa

[t,

Xl

(,;)]

= \

[Xl

(t) + X2 (t) (,; -

t)]

V (t, 't) d,; = X2

(t).

t~i/o

1)

MLI

ymfAlIM

Imme,

B § 33,

'ITO

aTO

Tpe60BaIme

JIBJIJIeTCJI Heo6xo8u.MbZM

YCJIOBHeM, KOTOPOMY

AOJImHa

YAOBJIeTBOpJITb

Belt/wa

onmU,i\WJ1,bHaa

paapema-

lOu/aa

OIIepa~lIJI

AJIJI

3aAa'IH

32.1.

_~~~......,

.J'.'

...•

:.'

...

··c.·

..

·.~c_.~··_··~·_·_·~·_·~·.:-'

•

.:L:c:.c:>

".,,'

..

;

••

,-:.'.:.'

'.':

c

····

","

278

IIoc'rAHOBRA

H

PEIIIEHHE

IIPOBJIEMbI

HABJIIOAEHHH

[rJI.

8

,IJ;JIH

BMIIOJIHeHHH

:3TOrO

yCJIOBHH

IIpH

m06MX

Xl

(t) H X

2

(t)

He-

06XO,IJ;HMO H ,IJ;OCTaTOlJHO,

tiT06M

BMIIOJIHHJIHCb

paBeHCTBa

t t

~

V

(t,

1:)dt =

0,

~

(1:

- t) V

(t,

1:)

d1:

=

1.

(32.22)

t-8-

8TH

paBeHCTBa

,IJ;0CTaBJIHIOT

ypaBHeHHH

,IJ;JIH

HeH3BeCTHoH

4JYHK~HH

V (t,

1:).

C

TaKHMH

ypaBHeHHHMH

MbI

Yllie

BCTpelJaJIHCb

BIJme

npu

U3Y.-

lJeHUU

aa,IJ;alJU

06

ynpaBJIeHUU

(CM.

(8.5) U § 9).

PemaH

ux

YKa3aHHbIM

TaM

cnoc060M,

yBH,IJ;UM,

'ITO

yCJIOBUHM

(32.22)

y,IJ;OBJIeTBOpHeT,

HanpH-

Mep,

4JYHK~UH

12

(

{}

)

v(t,1:)-W

1:-t+

T'

(t-'I'}:::::;;1::::::;;t).

(32.23)

TaK

CIIpaBe,IJ;JIUBO

HepaBeHcTBo

t

,

~

.1.(1:)

V

(t,

1:)

d1:

1-<

1-8-

t

-<

max

IV

(t,

1:)

I

~

1.1.

(1:)

I

dt

-<

max

IV

(t,

t)

16

(32.24)

-r

t-8-

-r

.Jl

6 .

U

TaK

KaK

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4JYHK~UU

V (t,

1:)

(32.23)

max

Iv(t,

T)

1=

ofr2'

TO

OITIu6-

Ka

Iffi,1

npu

BbIlJUCJIeHUU

X

2

(t)

onepa~ueH

(32.21) C

4JYHK~ueH

v (t,

T)

(32.23) B

CJIy'Iae

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(32.18) 6Y,IJ;eT

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rpaHb,

paBHylO

KaK

pa3

66!1'}2.

lIT06bI

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aTOM,

~OCTaTOlJHO

paCCMOTpeTb

nOMexu

.1.

(t) BU,IJ;a

15

.1.

(1:)

= e

npu

11:-

t I

-<

e,

.1.

(1:)

= 0

npH

11:-

t

I>

e

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e

~

O.

(32.25)

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(32.22)

y,IJ;OBJIeTBOpHeT

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WYHK~UH

v (t,

1:)

= -

:2

sgn

(1:-

t +

~)

, (32.26)

npUlJeM

H3

HepaBeHCTBa

(32.24) CJIe,IJ;yeT

Tenepb,

lJTO

MaKcHMyM

OITIu6-

KH

Iffi2 I

npH

BbIlJHCJIeHHH

X

2

(t)

CHOBa

onepa~HeH

(32.21),

HO

Yllie

C

4JYHK~HeH

v (t,

T)

(32.26), 6Y,IJ;eT

paBeH

46/fP.

(8TOT

MaKcHMyM

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CTHraeTCH,

HanpHMep,

Ha

nOMexe

.1.

(T)

BH,IJ;a (32.25)

npH

,IJ;oCTaTOlJHO

MaJIOM

e > 0.)

IhaK,

B

CJIylJae

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(32.18)

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(32.21)

npH

BM60pe

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v (t,

T)

(32.26) ,IJ;aeT

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pe3yJIbTaT,

lJeM

npH

BbI60pe

4JYHK~HH

v (t,

T)

(32.23).

Bo06m;e

H3

MaTepHaJIa

§ 16

MM

Yllie

3HaeM,

lJTO

HeJIb3H

nOCTpOHTb

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v (t,

T),

KOTopaH

§ 33]

PEIIIEHllE

3AAA

L

IH

OB

OTITHMAJIbHOM

HAIiJIIOAEHHH

279

y,IJ;OBJIeTBOpHJIa

6bI

yCJIOBHHM

(32.22) H

npHTOM

y,IJ;OBJIeTBOpHJIa

6M

HepaBeHcTBy

1 4

sup

Iv

(t,

1:)

I<

f.lO

=

ofr2

•

(32.27)

-r

OnHpaHCb

Tenepb

Ha

HepaBeHcTBo

(32.24) H

paCCMaTpHBaH

nOMeXH

.1.

(T)

BH,IJ;a (32.25) H

aHaJIOrHlJHMe

HM,

HeTpy,IJ;HO

BMBeCTH,

lJTO

HeJIb3H

HaHTH

onepa~HIO

(32.21),

KOTopaH

B

CJIylJae

OrpaHHlJeHHH

(32.18)

npH

BMlJHCJIeHHH

X

z

(t)

no

CHrHaJIy

Z

(1:)

,IJ;aBaJIa

6M

omH6Ry

MeHbITIylO,

lJeM

46/fj-2.

A

OTCIO,IJ;a

B CBOIO OlJepe,IJ;b BMBO,IJ;HTCH 3

aKJIIO'IeHHe

, lJTO

npH

BbI60pe

v (t,

T)

(32.26)

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(32.21)

l1BJIHeTCH

onmU.MaJlb-

noil

B

CJIylJae

nOMeXH

.1.

(T),

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(32.18).

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rOBopH,

onepa~HH

(32.21), (32.26)

HBJIH8TCH

3,IJ;ecp

onmu-

MaJlbnOU paapeUWlOuteU

onepa~HeH

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3a,IJ;alJH 32.1.

lIHTaTeJIb

3aMeTHJI,

BepOHTHO,

lJTO

onpe,IJ;eJIeHHe

onTHMaJIbHOH

pa3-

peITIaIOm;eH

onepa~HI1:

CPOi

BH,IJ;a (32.21)

6bIJIO

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C

paccylli,IJ;eHHH-

MH,

aHaJIOrHlJHbIMH

TeM,

C

KOTOpbIMH

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He

HBJIHeTCH

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OHO

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np06JIeMaMH

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JIH-

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CHCTeMax.

8Ta

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nO,IJ;p06HO

B § 36

nOCJIe

pemeHUH

3a,IJ;alJH 32.1

06

onTHMaJIbHOM

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,IJ;aH-

HorOB

§ 33.

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3,IJ;eCb

MbI

OrpaHHlJHBaeMCH

JIHITIb

,IJ;aHHMM

6erJIMM

3

aMe'IaHHellf.

§ 33.

PemeHHe

aaAalJH

06

OnTHMaJIbHOM

Ha6JIIoAeUHH

MbI

nepeXO,IJ;HM

Tenepb

K

pemeHHIO

3a,IJ;alJH 32.1. BY,IJ;eM, CJIe,IJ;OBa-

TeJIbHO,

paccMaTpHBaTb

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(t)

CHCTeMM

(32.5),

rrpe,IJ;nOJIaraH,

lJTO

H3MepeHHe

ero

,IJ;aeT

peaJIbHbIH

CMrHaJl

Z

(T)

= G

(T)

X

(T)

+

.1.

(r) (32.6),

Hecym;HH

nOMexy

.1.

(r),

OrpaHHlJeHHylO

HepaBeHCT-

BOM

(32.7).

Tpe6yeTcH

HaHTH

onopa~mI

cpoi

[t, Z

(T)]

(i

= 1,... ,

n).

KOTopMe

B

Kalli,IJ;MH

paccMaTpHBaeMbIH

MOMeHT

BpeMeHu

t

BOCCTaHil-

BJIHB8IOT

RPJIWIHHM

Xi

(t) C

HaHMeHbITIeH

B03MOlliHOH

OITIH6KOH

ffii =

CPy

[t,

{z

(1:)

I

Xi

(t»]-

Xi

(t)

(33.1 )

B

caMOM

He6JIarOnpHHTHOM

CJIylJae

CHrHaJIa

Z

(T).

IIpelli,IJ;e

Bcero,

He06xo,IJ;HMO

3aMeTHTb,

lJTO

onepa~HH

CPi

[t,

Z

(T)l,

KOTopaH

npH

Kalli,IJ;OM t

06eCnelJHBaeT

KOHelJHYIO

BepXHIOIO

rpaHb

,IJ;JIH

MO,IJ;yJIH Iffii I

omH6KH

ffii (33.1),

06HaaTeJIbHO

,IJ;OJIiKHa

y,IJ;OBJIeTBOpHTb

yCJIOBHJO

CPi

[t,

{y

(1:)

I

Xi

(i)}]

=

Xi

(t).

(33.2)

.,

.•.:

..•.•.

_.~

Красовский Н.Н. Теория управления движением

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