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a 8 a 12

JIHHEiIHbIE OllEPAlI;IHI B BERTOPHbIX

DPOCTPAHCTBAX

§ 49.

HOpMllpoBaHHoe

JIBUeHUOe

npOCTpaUCTBO

PaccMoTplIM

cOBoKynHocTI.

.Ai'

n

BceB03MomHliIX

n-MepHIiIX

BeKTO-

pOB,

KOMnOHeHTIiI

KOTOpI.IX

HBJIHIOTCH

p;eHCTBIITeJII.HIiIMII

TfIICJIaMII.

8JIeMeHTliI

3TOH

cOBoKynHocTII

.Ai'

n B

p;aJII.HeHmeM

6yp;eM

0603HaTfaTI.

CTPOTfH!,IMII

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6yKBaMlI

a,

h,

c,

...

,

x,

y,

Z,

.

••

ECJIII

He

orOBopeHO

npOTIIBHoe,

6yp;eM

TpaKTOBaTh

paCCMaTplIBaeMhIe

BeK-

TOpliI

KaK

BeKTopbI-CTOJI6IJ;bI,

XOTH

II

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IIX

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P;JIH

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B

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{aI'

a

2

, • •

.,

an}.

MHp;eKC

I

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/

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ai,

hi,

c ,

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0603HaTfaTh

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([3*],

CTp.

49;

[24*],

CTp.

11)

CJIep;YIOIIJ;lIe

npaBIIJIa

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II

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IIX

Ha

p;eHcT-

I

-,

BIITeJII.HbIe

TfIICJIa.

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P;BYX

BeKTopOB

X = {x;}

II

Y = {y;}

eCTh

BeKTOp

Z = {Zi},

I

~

:

:~!

:

''II:

KOMnOHeHTI.I

KOToporo

Zi

CYTI.

CYMMbI

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IWMnOHeHT

t'

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II

Yi

BeKTopOB

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II

y,

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e.

Z = X + Y = {Xi + y;}.

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a

Ha

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X

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a)

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eCTI.

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a

Ha

;~;

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I

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1I3

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+ Y = Y + X

(I<OMMYTaTIIBHocTI.);

2)

v-

X +

(y

+

z)

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+ Z

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3)!B

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n

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1I3

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X = {Xi}

1I3

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n

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= {

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x,

rp;e

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a X -

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~,

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1·

X = X P;JIH

I<amp;oro

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X

1I3

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n;

1) B

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peQb.

I

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§

49]

BOPMlIPOBAHHOE

JIlIHE:fiHOE

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395

3)

a (x + y) =

ax

+

ay

II

(a

+

p)

X =

ax

+

By

(,n;lIcTpn6y-

'fliIBHOCTh).

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n

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)

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[

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-

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paBeHCTBa

(49.1)

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[x]

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I<aK

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=

Yi

-

Zi'

TO

(49.2)

p

[x]

=

[~

x;]""

t=1

HopMy

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x,

onpep;eJIHeMylO

rPOpMyJIOH

(49.2),

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3TOH

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0603-

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x

I-

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onlIcaHHoe

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(49.2),

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HB-

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lIHaTfe.

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npoIJ;ecce

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x

1I3

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C

TaI<IIMII

paccToHHlIHMII

P ({y;}, {Zi}),

I<OTOpliIe

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CJIep;YlOIIJ;HMII

CBOHCTBaMII:

1) P ({Yi},

{Zi})

= P ({Zi}, {Yi});

2)

P ({Yi}, {Zi}) = 0,

eCJIII

Bce

Zi

= %

3)

P({Yi},

{Zi})

> 0,

eCJIII

He

Bce

Zi

paBHIiI

Yi;

4)p

[{Xi}'

{Yi}']

<P [{Xi}, {z;}] +

P[{Yi};

{z;}].

(49.3)

'

...

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,

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B

BEK'l'OPHbIX

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lrJl. 12.

396

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YCJIOBHJI

(49.3)

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x

H3

JV'

n,

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meHHH BeKTopa

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Ha

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a

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pa3.

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JV'

n orrpep;eJIeHa

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[xl,

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corrOCTaBJIeHO

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y.rr;oBJIeTBOpJIIOm;ee

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1) p

[x]

= 0,

ecmr

x = 0;

2)

p[x]>O,

ecmI

x

=1=

0:

(49.4)

3)

p[x+y],<p[xl+p[y]

(rrpaBHJIO

TpeyrOJIbHHKa);

4)

p[ax]

=

\a!p[xl;

EBKJIH.rr;OBa

HopMa

(49.2)

y.rr;oBJIeTBOpJleT,

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yCJIOBHJIM

(49.4).

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p

[x].

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p

[x]

=

max

(I

xli,

...

, IX

n

I)

(49.5)

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,~

n

i,

.~.~

(49.6)

p

[X

1=

~

I

Xi

I·

, '

~

i=l

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1JTO

BeJIH'IHHhI

(49.5)

H

(49.6)

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y.rr;OB-

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(49.4),

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pa

x.

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.rr;OCTaBJIJIeT

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[xl

rrpH

rrOMOm;H

paBeHCTBa

n

(49.7)

p

[xl

=

[~ll

Xi

IPfP

,

r.rr;e

p - KaKoe-HH6y.rr;b

1JHCJIO

H3

rrOJIyHHTepBaJIa 1

'<

p <

00.

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p = 1

H3

(49.7)

rrOJIy'IaeM 'IaCTHhIH

CJIy1JaH

(49.6),

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'IaeM

eBKJIH.rr;OBY

HOpMy

(49.2).

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yCJIOBHH

1),

2), 4).rr;JIJI

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(49.7)

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rrpo-

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n

l!p

n .

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n

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[~

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Xi

+

Yi

IV]

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[p

I

Xi

IP]

+

[p

I

Yi

IP]

_1

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,=1

Ha3hIBaeMoro 06hI'lHO HepaBeHCTBOM MHHKOBCKOrO,

MOmHO

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B

KHHre

[14*]

(CTp.

499).

3aMeTHM em;e, 'ITO

yCJIOBHJI

····,

'.'.

'.

:

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c

§ 50J

JlHHEi1HbIE

OnBPA!J;trti

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(49.4)

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HerrpephIBHyIO

3aBHCHMOCTb

BeJIH1JHHhI

p

[xl

01'

BeKTopa x.

MHomeCTBO TO'IeK x =

{Xi}'

HMeIOm;HX

HOpMy p

[xl

'<

e,Ha3:M-

BaeTCJI C<pepOHC

pa.rr;HYCOM

e.

IIpH

e = 1 c<pepa Ha3hIBaeTCJI

e.rr;HHH'I-

HoH.

HarrpHMep,

B CJIy'IaJlX

(49.2),

(49.5)

H

(49.6)

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B .rr;ByMepHoM npOCTpaHCTBe

JV'

2

or-

paHH'IHBaIOTCJI

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l.Z'z

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1,

2, 3 (pHC.

49.1).

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pH

H3MeHeHHH p

01'

1

.rr;o

00

JIH-

I

HHJI, OrpaHH'IHBaIOm;aJl

e.rr;HIl:H1JHYIO

I

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p

[xl<

1,

B

CJIy1Jae

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I

(49.7)

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01'

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0 BeK1'opHoM rrpoc1'paH-

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6e3 BhI6paHHoH B HeM onpe.rr;eJIeHHOH HOpMhI,

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me

pe1Jb

noH.rr;eT

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n-Mep-

HOM

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6y.rr;eM

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ero

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5i3

n

.

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aa.rr;aHO

n-MepHoe

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npOCTpaHCTBO

JV'

no )J;onYCTfIM,

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=

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(50.2)

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HH

6hIJIH BeRTOp h H

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~;

2)

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[h + g1 =

~

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l,

(50.3)

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61>1

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f(

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n'

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r·::

,-'r

(7'\

""--'"

\:..~:

.

~

398

.JIHHEkIHbIE

OnEPAD.IHr

B

BEHTOPHbIX

DPOCTPAHCTBAX

[rJl,

12

§

511

HOPMA

JIHHEfiHOfI

OilEPAIJ;flff

399

CrrpaBep;JIHBO

CJIep;yIOm;ee

naiI<HOe yTBepmp;eHHe (CM., HarrpHMep,

[14*],

CTp.

180).

Teo

p

eM

a

50.1.

JI106aR

./tUl-leiiT-taR

onepalfUR

ep

l-lao

eelf,mOpaMU

'f

h

U3

,AI'

n

UMeem

euo

ep

[h]

=

h'u

= u

1

h

1

+

...

+

unh

n

, (50.4)

zoe

U -

Cl-loea

l-lelf,omopbUl

eelf,mop

U3

,AI'

n'

l1pHBep;eM paccymp;eHlfH, rrop;TBepmp;aIOm;He

crrpaBep;JIHBOCTb

Teo-

peMhI

50.1.

BCHRHH BeR'fOp h

U3

,AI'n MomeT 6LITb

rrpep;CTaBJIeH

B

BHp;e

!t =

!t1e(1)

+ h

2

e(2)

+ ... + kne(n),

rp;e

hI'

h

2

,

...

, h

n

-

ROMrrOHeHTLI

h,

a

eel),

e(2),

...

,

ern)

- 6a3uc-

l-llile

BeRTophI

BHp;a

e(i)

=

{O,

...

,0,

1,0,

...

,

O}

(1

CTOHT

Ha

i-M MecTe). B

qaCTHOCTH,

eCJIlI n =

3,

TO

MomeT 6LITb e(l) =

i,

e(2)

= j H

e(3)

= k, rp;e

i,

j,

k Opl'OHOpMHpOBaHHLIe BeRTophI TpeXMep-

Horo

eBKJIHp;OBa

rrpOCTpaHCTBa,

ah

1

, h

2

, h

3

-

rrpOeKD.HH

BeKTopa h

Ha Koopp;HHaTHLIe

OCH,

HarrpaBJIeHHH

KOTOpLIX

orrpep;eJIHIOT

i,

j,

k.

Terrepb

JIHHeHHaH orrepan;HH

ep

01'

h C yqeTOM

(50.2)

H

(50.3)

MomeT

. 6LITb pacrrHcaHa TaK:

ep

[k]

=

ep

[h

1

e(l)

+ h

2

e(2)

+ ... + kne(n)] =

=

h

1

ep

[e(!)] + h

2

ep

[e(2)]

+ ... +

hnep

[e(n)]. (50.5)

"

0603HaqHB

ep

[e(i)] = Ui

(i

=

1,

...

, n) H TpaRTyH Ui KaK KOMrro-

'"

HeHThI

HeKoToporo

BeRTOpa

U,

rrOJIyqHM

H3

(50.5)

paBeHCTBO

(50.4),

'ITO H rrOp;TBepmp;aeT crrpaBep;JIHBOCTb TeopeMLI

50.1.

113

(50.4)

CJIep;yeT, 'ITO JIHHeHHyIO orrepan;HIO

ep

[h]

BrrOJIHe

orrpe-

p;eJIHeT

HeROTophIH n-MepHLIH BeKTOp U Jl3,A1'

n'

OqeBHp;HO,

crrpaBep;JIHBO

~

JI

06paTHoe,

T.

e.

BCHKOMy

BeRTopy U

H3,A1'

n

OTBeqaeT

BrrOJIHe

orrpe-

p;eJIeHHaH

JIHHeHHaH orrepan;HH

ep

[h]

BHp;a

(50.4).'

TeopeMy

50.1

Ha3LI-

BaIOT

TeopeMoH

06

06m;eM

BHp;e

JIHHeHHOH

orrepan;HH

ep

[h] B rrpOCTpaH-

CTBe

,AI'

n'

OTMeTHM

OP;HO

BamHoe

06CTOHTeJIbCTBO.

113

paBeHcTBa

(50.4)

CJIep;yeT, 'ITO

MHomeCTBO

TO'IeK

{h;}

H3

,AI'

n,

yP;OBJIeTBOpHIOm;HX

yCJIOBHIO

ep[h]

=

a,

rp;e a - RaRaH-JIH6o

rrOCTOHHHaH,

HBJIHeTCH

rH-

n

rreprrJIOCIWCTbIO

2J

Ui

h

i

=

a.

B

~raCTHOCTJI,

B CJIyqae p;ByMepHoro

rrpo-

i=l

CTpaHCTBa ,AI'2

MHO}ReCTBO

TOqeK

ep

[h] = a - rrpHMaH,

'B

CJIyqae

~J1!'

3

MHomeCTBO

TO~.reK

cP

[k]

= a -

rrJIOCKOCTb.

§ 51.

HopMa

JIHHeHHOH

orrepaIJ;HH

B

3TOM

rraparpatPe

paccMaTpHBaeTcH BamHoe

rrOHHTHe

HOpMhI JIH-

HeHHOH

orrepan;HH

cP

[h]. H opMa orrepaIJ;HH

ep

[h]

eCTb

qHCJIO,

KOTopoe

rroRa3IiLBaeT,

HaCKOJIbKO

BeJIHKH

MoryT

6LITb 8Ha'IeHHH

tPYHKIJ;HH

ep

rro cpaBHeHHIO C

HOpMOH

P

apryMeHTa

h.

XapaKTepHcTHKa 3Ta orrH-

paeTCH Ha cpaBHeHHe HOpMLI BeKTopoB

h C

a6COJIIOTHOH

BeJIH'IHHOH

3Ha'IeHHH onepan;iHI

ep

[h] Ha HHX. 1103TOMy HopMa orrepan;JIH

ep

orrpe-

p;eJIHeTCH

nocJte

BLI60pa B rrpOCTpaHCTBe

,AI'

n onpeoeJteruwu MeTpHKH.

B

COOTBeTCTBHH

C

3THM

6yp;eM paCCMaTpHBaTb p;aJIbme JIHHeHHble

HOpMHpOBaHHLIerrpOCTpaHCTBa

fla

n

.

RaKOBa 6LI

HH

6LIJIa JIHHeHHaH orrepan;HH

ep

[h], Bcerp;a MomHO

yRa-

3aTb

'IHCJIO

v,

06eCrre'IHBaIOm;ee HepaBeHCTBO

I

ep

[h]1

<vp [h]

(51.1)

rrpH

Bcex

h

H3

,Al'n' HaUMel-tbutee

n;i

'IHCeJI v,

yp;OBJIeTBOpHIOm;HX

Hepa-

BeHcTBy (51.1),H BLI6HpaeTCH B Ka'IeCTBe

nopMb~

p*

[ep]

onepalfuu

ep.

OKa3LIBaeTCH, 'ITO HopMa

p*

[ep]

MomeT 6LITb HaHp;eHa TaR.

PaCCMOT-

pHM B

flaT!,

MHomeCTBO

BeKTopOB h

H3

ep;HHH'IHOH

CtPepLI

p [h] <

1.

(51.2)

<!>yHKn;HH

TJ

= I

cp

[h] I HerrpepLIBHa rro h H, CJIep;OBaTeJIb.. ,

P;OCTH-

raeT

Ha OrpaHH'IeHHOM 3aMKHyToM MHomeCTBe

(51.2)

CBoero

IIaH60JIb-

mero

3Ha'IeHHH

TJo

=

max

I

ep

[h]l.

31'0 3HaqeHHe

TJo

H orrpep;eJIHeT HOp-

1

My

orrepan;HH

ep

[h],

T.

e.

)

p*[ep] =

max

(I

ep[h]

I rrpH

p[h]

<

1).

(51.3)

110p;'IepKHeM

em;e

pa3,

'ITO orrpep;eJIeHHe HOpMLI

p*

[ep]

orrepan;HH

ep

cym;eCTBeHHO

3aBHCHT

01'

BLI60pa

HCXOP;HOH

HOpMLI P [h] BeKTopOB

h.

COrJIaCHO TeopeMe

50.106

06m;eM

BHp;e

JIHHeHHOH

orrepan;HH

Kam-

p;aH

TaKaH orrepan;HH

ep

B

.Af'

n orrpep;eJIHeTCH B

eOOTBeTCTBHH

C paBeH-

CTBOM

(50.4)

HeKOTopbIM n-MepHbIM

BeKTOpOM

u. CrrpaBep;JIHBO, KaK

OTMe'IaJIOCb

BLIme, H 06paTHoe: JII060H n-MepHhIH BeKTOp U orrpep;eJIH-

eT

HeKoTopyIO JIHHeHHyIO orrepan;HIO

ep.

IIpHrrHmeM

BeKTopy

U,

orrpe-

p;eJIHIOm;eMY

orrepan;mo

cp,

l-lOpMy

p*

[U],

pailHyIO HopMe

(51.3)

COOT-

BeTcTByIOm;eH orrepan;HH, T. e.

rrOJIOmHM

p*

[u]

=

p*

[ep].

OKa3LI-

BaeTCH,

'ITO BeJIH'IHHa

p*

[U] yp;OBJIeTBopHeT BceM Tpe60BaHHHM

(49.4),

rrpep;1>HBJIHeMLIM

K HopMe B

n-MepHOM

BeKTopHOM rrpOCTpaHCTBe.

CrrpaBep;JIHBOCTb

nora

BamHoro

yTBepmp;eHHH MhI rrpHMeM

3p;eCb

6ea

p;OKa3aTeJIbCTBa, OrpaHH'IHBillHCb 06cymp;eHHeM HeKOTopLIX rrpOCTLIX

rrpHMepOB.

)J;oI<a3aTeJIbCTBO

MomHO HaHTH B

Kypce

[9*].

TaKHM 06pa30M, lIMeeM

CJIep;YIOm;yIO

cHTyan;HIO.

IIycTb

p;aHLI

p;Ba

3K3eMrrJIHpa

,AI'~)

H

,AI'~)

n-MepHoro

BeKTopHoro rrpOCTpaHCTBa.

BBep;eM B rrpOCTpaHCTBe

,AI'~l)

HeKoTopyIO HOpMy p [h]

OP;HHM

H3

crro-

C060B, YKa3aHHLIX B §

49.

IIoJIy'IHM

HOpMl1pOBaHHoe rrpOCTpaHCT-

BO

:JiJ

n

.

Ramp;LIH BeKTOp U

113

rrpOCTpaHCTBa

,AI'~)

COrJIaCHO

CRa3aH-

HOMy BhIme orrpep;eJIHeT

JIHH6HHYIO

orrepan;HIO

CPu

[h]

(50.4).II

pHrrHIIleM

1)

TOT

.paRT,

'11'0

paBeHcTBo

(51.3),

p;eitcTBI1TeJIbHO,

onpep;eJmeT

'IHCJIO

,,0

=

p*

[ep

1,

KOTopoe

JIBJIJIeTCJI

rraHMeHbIDHM

cpep,H

'IHCeJI,

yp;OBJIeTBOpJIJOlIVIX

HepaBeucTBy

(51.1),

'IHTaTeJIb

MOlKeT

npoBeplu'b

caMOCTOIlTeJIbHO.

Красовский Н.Н. Теория управления движением

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