Красовский Н.Н. Теория управления движением

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459): (1)

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116)

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O

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(t) B

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(15.1).

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17.3,

17.5),

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1,

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pHC.

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=

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~

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p

O).

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til

[SO

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(x

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- X

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t"J X

CX

) =

~

s'o

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d1'

t

cx

npH

u =

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no

u

(1')

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p*

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11

pO,

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ex

] x ,

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e.

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X npaBlfJI,

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386),

r.rr;e

p;aH

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Horo

TaM

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47.1.

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§

17

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MhI

He

oy.rr;eM

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3TH

npaBHJIa

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p;onycKaeMhIX ynpaBJIeHHH dU (t), TaK KaK

nOCJIe

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§

1831'0,

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He

.rr;OJImHO

saTpy.rr;HHTh

qH-

TaTeJIH. BnpO'IeM, OTp;eJIhHhIeaaMe'IaHHH,

OTHOCHm;HecH

K TaKoH

ne-

peq,opMyJIHpOBKe p;aHHhIx npaBHJI,

p;eJIaIOTCH

HHme

no

xop;y pemeaHH

cooTBeTcTByIOm;Hx aap;a'I

(CM.

§

23).

-;;.

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CD

r.fl,aBa

6

§ 18. YnpaBJIeHUe C MHHHMaJlbHOii 3HeprHeii

8THM

naparpaq,oM

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aa.rr;a'IH

4.2

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MaJIhHOM

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npH

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81'01'

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paa Torp;a, Korp;a B

aa.rr;a'Ie

4.2

nOTpeoyeTcH

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'f,

%[ul

=

[~llu(t)112dt]

.

(18.1)

t

cx

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HaaBaHHH

BeJIH~mHhI

(18.1)

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B §

12).

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(18.1)

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§

17,

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e.

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TOM,

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He

p;eJIaTb, TaK KaK yKaaaHHbrii:

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yme

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12.

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me

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p

[h

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c

122

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6

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(18.1),

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e.

t~

lh

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[k

Cr)]

=

[~

Ilk

(t)

11

2

dt]

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(18.2)

t

...

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(18.1).

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13.1,

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x

(u]

=

p*

[u]

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(18.1).

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17.1

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pemeHHH

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8

h',

tj3]

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(17.3) H

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c (15.2),

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lO

=

{l~,

...

,

l~},

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BeKTOp-

epYHKU;HH k

(or)

=

B'

(.)8

[,:, tj3].'l (l'c =

1)

HMeeT

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MY,

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e.

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pO

=

min

p [B! ('I) 8 ['I,

tj3].l]

= P

[B'

(1:)

8

[1:,

tj3]·lO] 1

I •

npH

yCJIOBHH

l'c

=

1.

TaK

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p (18.2)

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OAHoBpeMeHHO

co

CBOHM

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p2,

TO

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k

O

(1:)

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tj3].lO

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t~

:

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npHl'c=1.

(18.3)

t I

ex

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B (18.3),

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<1>

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,

In)

OT

nepeMeHHbIX

li H

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K

aaAaqe:

n n

q>

(ll'

.•.

,

In)

=

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min

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2j

liCi

= 1. (18.4)

i,

j=1

I

i=1

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(1:),

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1

n

IJ;HH k

(1:)

= h

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<1>

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i=1

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nO.l/,OJICUmMbltOU

H

nOBepXHOCTH

<1>(l1l

•••

,

In)

=

p2

=

const

> 0

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3JIJIHnCOHAbl

B

npocTpaHcTBe

{lll

...

,

In}'

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n

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pa3

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123

§ 18]

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...

,

lnJ.

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me

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<1>

=

p2

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paMH.

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aaAaqa

(18.4) COCTOHT, CJIep;OBaTeJIhHO, B

OTblCKaHHH

n

IJ;HJIHHApa

<I>

=

(pO)2,

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~

1.

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i=1

¢(l"lZ,l./)=;oZ

-.

t

z

PIIC.

18.1.

'\

PlIe. 18.2.

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a

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124

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6

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([23*],

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1,

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°

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npuulfun

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§

16).

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p;aHHOM

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o

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(18.5)

t«

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lex.

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1(3

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("t')

11

2

d"t'

=

(1}pO)2

=

(VO.)

2

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(18.6)

lex.

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ax;cmpe-

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(18.4)

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no

Baplla~1I0HHOMY

HCqHCJIeHlIIO

(CM., HanpHMep,

[10*],

§

25).

IIycTh

Ha

qllCJIa

BeKTop-iPYHK~lIH

y

(-r)

=

{Yl

(-r),

...

, Yn

(-r)},

yp;OBJIeTBO-

pHIOm;lIX yCJIOBlIHM:

b

~Gj("t',

y("t'»d"t'=Cj

(j=1,

...

,m),

(18.7)

a

Tpe6yeTcH HaHTlI

BeKTop-iPYHK~Hm

yO

(-r)

=

{y~

(-r),

..•

,

Y~

(-r)},

P;JIH

KOTOpOH

iPYHK~lIOHaJI

b

I [y("t')] =

~F("t',

y("t'»d-r

a

npllIDIMaeT

9KCTpeMaJIhHOe

3HaqeHlle. TaKaH aap;aqa

Baplla~HOHHoro

HCqHCJIeHHH

Ha3hIBaeTCH

H30nepHMeTpHqeCKoH. OKa3bIBaeTCH,

qTO

cnpaBep;JIlIBO yTBepJI,p;eHlle. ECJIlI

yO

(-r)

- pemeHHe 1I30nepllMeTpll-

qeCKOu

3ap;aqH,

TO

Cym;SCTByIOT

nOCmOR,UUb/,e

qHCJIa

A.l'··

.,

A.

m

-

(:":

§

18]

•

YIIPABJIEHlIE C MHHlIMAJIhHOfl 9HEPrHEfl

125

HeOnpep;eJIeHHhIe MHOmHTeJIlI

JIarpaHma

- TaKlIe,

qTO

KpllBaH

yO

(-r)

HBJIHeTCH

9X;CmpeMa.ltblO

P;JIH

iPYHK~lIOHaJIa

b m .

IdY("t')]

=

HF("t',

y("t'»)

+~

A.pj("t', y("t'»)Jd't'.

a

;=1

He06xop;HMMM

YCJIOBlIeM

9KcTpeMyMa nOCJIep;Hero

iPYHK~HOHaJIa

HB-

JIHeTCH

06pam;eiIHe B ayJIh

ero

Baplla~lIH

b

11.

m .

,

[""

(BF

""

BG;) ]

\ LJ

""1)":

+LJ

A.j

r

6Yi

d"t'

= 0,

J . 1

Y.

. 1 Y.

a

.=

1=

rp;e 6Yi

(-r)

-

npOU380.ltbUb~e

HenpephIBHo-p;lIiPiPepe~HpyeMhIe

iPYHK-

n;lIlI,

p;OCTaTOqHO

MaJIbIe

no

aOCOJIIOTHOH

BeJIHqHHe.

1I3

npOHaBOJIhHOCTlI

BapHa~HH

6Yi

(-r)

11

OCHOBHOH

JIeMMbI

BapHa~HoH

Horo

lIcqllCJIeIDIH

([10*],

§ 9) BhITeKaeT,

qTO

iPYHKn;HH

y~

(-r)

HBJIHIOTCH

pemeIDIeM

ClICTeMhI

ypaBHeHlIH

m

BF

.""

~

BG;

i}":' +

LJ

"';

T

Y.

;~1

Y.

_

-

0 . -

(l

-

1 )

,

...

, n .

18 8)

( .

A06aBJIHH K ypaBHeHHHM

(18.8)

YCJIOBlIH

(18.7),

MbI

nOJIyqllM n +m

co-

OTHomeIDIH

P;JIH

onpep;eJIeHHH

TaKoro

me

qHCJIa

HeH3BeCTHhIX

y~

(-r),

...

,

y~

(-r),

A.

1

,

.•.

,

A.m.

B HameM CJIyqae

(18.5),

(18.6)ypaBHeHllH

(18.8),

OqeBHp;HO,

npllIDIMaIOT

BHP;

h?

("t')

+

2'A.u?

("t')

= 0

(i

=

1,

...

,11,),

rp;e

h~

H

u~

- KOMnoHeHTbIBeKTopoB h

O

H

UO

COOTBeTCTBeHHO.

OTCIO-

p;a

nOJIyqaeM

ep;lIHcTBeHHMM

06pa30M aHaqeHlIe

P;JIH

UO

(-r):

.

1

UO

("t')

= -

21.

hO

("t').

(18.9)

Teneph

OCTaeTCH

BMqHCJIlITh nocToHHHyIO

'A..

3Ta

nOCTOHHHaH

on-

pep;eJIHeTCH

lIa

YCJIOBHH

(18.5),

(18.6)

C yqeTOM

(18.9),

oTKyp;a

CJIep;y-

eT,

qTO

-

'A./2

=

(VO)2

=

1J(pO)2.

TaKHM 06pa30M,

UO

(-r)

=

(vO)2B!

(-r)

S

[-r,

t{j]

1°.

(18.10)

ECJIlI Teneph

em;e

nop;CTaBlITIt B

(18.10)3HaqeIDIH

BeJIlIqHH

I?,

paapemaIOIIJ;lIX aap;aqy

(18.4),

TO

nOCJIe

Bcex Heooxop;HMbIX BMqllCJIe-

IDIH

OKOgqaTeJIhHO

nOJIyqllM:

11.

u~

(t)

= }

~

Djkckhji

[t(3,

t]

(i

= 1,

...

,

r),

(18.11)

j,

h=;;;;l

(>

~,>

126

CBO~WTBA

OIITHMAJIhHOrO

YIIPABJIEHHff

[rJl,

6

rAe

" t(:l

D =

det

{aii};

aii

=

2l

~

h

ik

[t~l,

't']

h

ik

[t{3,

't']

d't',

k=l

tot

hit

-

8JIeMeHThI

MaTpHIJ;hI

S'

['t',

tj3]

B

{-t)

H D

ik

-

aJIreOpaHlIeClme

AOrrOJIHeHHJI

B

D.

8TO

yrrpaBJIeHHe,

UO

(t) =

{U?

(t)},

eCTecTBeHHo,

cOBrraAaeT

C

TeM

yrrpaBJIeHHeM

(9.10),

KOTopoe

OhIJIO

HaHAeHo

B § 9.

Bai-KHO

OTMeTHTh

CJIeAyIOm;ee:

ooe

aaAalIH

(18.4) H (18.5), (18.6),

C

KOTOPhIMH

MhI

BCTpeTHJIHCb,

AorrYCKaIOT

eAHHCTBeHHoe

pemeHHe

rrpH

YCJIOBHH

JIHHeHHOH

HeaaBHCHMOCTH

BeKTop-4!YHKIJ;HH

h[il'

{'t),

KOTophIe

JlBJIJlIOTCJI

CTpOKaMH

MaTpHIJ;bI

S'

(,:,

tj3]

B ('t').

OAHaKo,

KaK

MbI

yme

aHaeM

(CM.

§§

16, 17),

AeJIO

CBOAHTCJI

OOJlaaTeJIbHO

K

CJIYlIaIO

JIHHeHHOH

HeaaBHCHMOCTH

4!YHKIJ;HH h

ijl

{'t)

(j

= 1,

...

, m),

rAe

m

-<

n, eCJIH

TOJIbKO

aa,IJ;alIa 13.1

HMeeT

pemeHHe,

a

,IJ;JIJI

8Toro

Heoo-

XO,IJ;HMO

H ,IJ;OCTaTOlIHO,

lITOOhI

pO

>

O.

OTcIOAa

CJIeAyeT,

lITO

HCX0,IJ;HaJl

aa,IJ;alIa

13.1

rrpH

yCJIOBHH

MHHHMyMa

HHTeHCHBHOCTH

x

[u]

(18.1)

HMe-

eT

e,IJ;HHCTBeHHoe

pemeHHe,

eCJIH

pO

>

O.

IhaK,

crrpaBe,IJ;JIHBO CJIe,IJ;yIO-

m;ee

aaKJIIOlIeHHe.

OnmlM£aJ1,blWe

ynpae.aenue

OJ1,R,

aaoa'J,u

13.1

npu

YCJ1,08UU

M,unlM£yltta

unmenCU8nocmu x

[u]

(18.1) eouncm8enno u ono

ua06pa~aemcR,

8e1fi,-

mop-rjJyn1fi,Lfueu

UO

(t)

(18.9), nenpepbl8nou

npu

8cex

anaTJ,enuR,X

tot

-<

t

-<

tj3.

(/)yn1fi,LfuR,

UO

(t)

nponoPLfUOnaJ1,bna

MUnlM£aJ1,bnOU

rjJynl'Lfuu

hO

(t)

u aa8ucum

J1,uneuno

om

1fi,oopounam

xf

u

x~.

PaccMoTpHM

rrpocToH

rrpHMep.

IIpH

8TOM

oYAeM

rrpeCJIe,IJ;oBaTb

,IJ;Be

n;eJIH.

Bo-rrepBhIx,

BhIJlCHHM

Ha

aTOM

rrpHMepe,

KaK

B ,IJ;eHCTBHTeJIbHO-

CTH orrpe,IJ;eJIJleTCJI

MHHHMaJIbHaJl

4!YHKIJ;HJI

hO

{'t), a

BO-BTOPhIX,

3,IJ;eCb

oY,IJ;eT

HaiiAeHo

pemenHe

,IJ;JIJI

aa,IJ;alIH,

KOTOpyIO

MhI

perrmM

B rrOCJIe,IJ;y-

IOm;HX

rraparpa4!ax

rrpH

APyrHx

YCJIOBHJlX

OrrTHMaJIbHOCTH.

TeMCaMhIM

oopaayeTcJI

HarJIJI,IJ;HhIH

MaTepHaJI,

nOKaahIBaIOm;HH

BJIHJlHHe

BhIoopa

MHHHMH3HpyeMoH

HHTeHCHBHOCTH

x

[u]

Ha

xapaKTep

IIOJIYlIaIOm;erOCJl

OIITHMaJIbHOrO

yrrpaBJIeHHJI

H

cooTBeTcTByIOm;ero

eMy

OrrTHMaJIbHOrO

,IJ;BHmeHHJI

XO

(t).

IIpHMep

18.1.

PaccMoTpHM

aa,n;a'IY

06

yrrpaBJleBlHl

MaTep.l1aJIbHOii

TO'lKoii,

.,n;mImym;eiicH B

BepnmaJlbHOH

rrJlOCKOCTliI

rro,n;

,n;eHcTBliIeM peaKTliIBHoH CliIJlhI liI

C.I1J1hI

THmeCTH. By,n;eM C'IliITaTb peaKT.I1BHYIO CHJIy yrrpaBJIHIOD:J;liIM Boa,n;eHcTBHeM.

IIeperrliIlIIeM

lIa

§ 3

(rr.3°)

ypaBHeHHH

,ll;BliImeHliIH

HameH

TO'lKH

Xl

=

X2,

x

2

= U

l

,

z3

=

Xi'

z4

=

~

-

g.

(18.12)

IIyCTb

t<1.

= 0,

t{3

= 1.

Ha'IaJlbHoe

H IWHe'IHOe

COCTOHHliIH

<paaoBoro BeKTopa

X'=

{Xl'

X

2

, X

3

,

X,}

BhI6epeM

TaKHM.I1

1

):

x<1.

=

{-

1,0,0,0},

x(3

=

{O,

0, 0,

OJ.

1)

MhI

He 06CYln,n;aeM

MacmTa6bI

paCCMaTpHBaeMLIX BeJlH'IHH, rrOCKOJIbI<y

rrpHMep HOCIIT HJIJIIOCTpaTIiBHhIH

xapaKTep.

AHaJIOrH'IlIOC 3aM:e'lanlie

CJIe,n;yeT

UMeTh U lln,n;y

llCIO,n;y

B ,n;aJU>lleiimeM.

c.

§ 18]

YIIPABJIEHHEC

lIUIHUMAJIbHOfI

::HIEPrI-IE:l1 121

Mu

6YlleM

MHHHMHaHpOBaTb

BeJIll'HlHy

t(j

'(,

X

[It]

=

[~

[ui

('r) +

u~

(T)] dTJ

ta.

He B,n;aBalICb B «pHaU'IecKKiI:

CMhICJI

nora

yCJIOBUH. <DYHIOIliH h (T) =

B'

(T)

X

xS

[1', tpl·t, cpe,n;n KOTOphIX Ha,n;JIemHT

a,n;eCb

HCKaTb MHHliIMaJIbHYIO

«PYHIO~BIO

h

O

(T),

IIOJIy'IHM,

ecmr

BOCrrOJIbayeMcH peayJIbTaTaMH 1I3 §

13

(CM.

crp.

81),

rAe

6blJI0 nOKa3aHO, 'ITO

tp,

- T

o

1

o

II

[t(:l'

T]

= X [t(:l.

T]

B

(T)

=

S'

[T, t(:l] B

(T)

=

o

t{3

-T

o

1

H,

CJIe,n;OBaTeJIbHO,

[II

(t(:l

-

T)

+1 J

2

h

(T)

=

H'

[t(:l' 1'] I =

B'

(T)

S [T, t(:l] .

1=

1

Up,

_ 1') +

I,

.

3

"

HaM

IIoTpe6yeTclIem;e

BeKTop

c (15.2). B

,n;aHHOM

CJIy'Iae,

CHOBa

comaCHO

MaTepHaJIy Ha §

13,

HMeeM:

1

t(:l-

T 0

o

1 0

o

X [t(:l'

T]

=

o 0 1

t{3-1'

o 0 0

1

H II03TOMY

t(:l-

ta.

0

o

1 0

o

c-

-0

-

[~'J

0

~

o 1

t(:l-ta.

rn-

o 0

1

tp,

- T 0

t(:llio

1 0

000

l

dT

=

(t(:l-

ta.)2

o I

°l

f 1

o

1

t{3

-

TOg,

2

~

~a."0

o

Ollf-g

g(I,-I.)

TaKUM

06paaoM,

aap;a'Ia (18.4),

eCJIH

y'IeCTb,

'ITO

ta.

= 0, t(:l = 1,

CBOAJ1TCH

a,n;eCb

K rrp06JIeMe: HaHTH

1

(pO)2

=

min

~

{[II

(1

-

T)

+1

2

F+

[L3

(1

- 1') +

!4F}

dT

=

/ °

.

(1.

• 1 2

2)

=

Ill/Ill

3 1

1

-1

1

/

2

1,1

2

+ 3 1

3

-I-

1

3

!"

-I-

1

4

(18.13)

c

.>

······

(

128

CB01iCTBA

OllTI1MA.TIbHOrO

~7llPAB.TIEHI11I

[rJl.

H

upn

ff

II

+

Tla+gl4=

1.

(18.14)

,qJIll

OnpeAeJIeR.II.1l

::ITOrO

MHIlllMyMa

BOCrrOJIb3yeMcH

MeTO.n;OM

MHOJKHTeJIen

JIarpaHlKa

(CM.

[23*],

T.

1, § 212). B

COOTBeTCTBHH

C

8THM

MeTo.n;OM

COCTaBHM

BbIpaJKeHHe

1

2

1

2

2.

g ,

r

=Tll

+lll2+l2+

Tla+lal4+l4+/.,(Ll+TLa+gl4-1),

ll;HtP¢epeH~HpYH

KOTOpOe

'laCTHbIM

06pa30M

nOCJIe.n;OBaTeJIbHO

no

L

I

, 1

2

,

la,

H 1

4

,

llOJIy'IHM

.n;JIH

orrpe.n;eJIeHHJI

BeJIH'lHH

l?

cHcTeMy

JIHHenHbIX

3JIre6paH'IeCKHX

ypaBHeHHii

2

Tll

+

l2

+

/.,

= 0,

II

+

21

2

= 0

(f

T

2

l

a+

l

4+

TA=O,

la

+

21

1

+

gA

=

O.

8Ta

CHCTeMa

BMeCTe

C

yCJIOBHeM

(18.14) .n;OCTaTO'IHa .n;JIHj

BbI'lHCJIeIlllH

HeH3BeCT-

HbIX

l?

H

A.

ITpO));eJIaB

Heo6-

oX

J

XO));HMbIe

BbIKJIa)J;KH,

Haii.n;eM:

,

o

12

° 6

II

=g2+

12'

l2=-g2

+12,

l

o 0

lO-

__

ri_

3

=.

4 -

g2

+12 .

o

x,

CJIeJ\OBaTl:IJIbHO,

B

AaHHOMCJIy'Iae

1

[6

-

121:J

hO(1:)=

g2

+12 g .

PHC.

18.3.

H

(pO)2

=~p2

[hO

(1:)]=1/(g2 +12).

06pa~aHCb

Terrepb

K

paBeHcTBy

(18.9),

lIOJIyqaeM

HBHblii

BR));

OrrTHMaJIbHOrO

yrrpaBJIeHHH]

j

u~

(t)]

[6

-

12tj

UO(t)

= = .

(18.15)

[

u~

(t) g

ECJIH

HaHJ\eHHoe

BbIpaJKeHHe

));JIH

¢YHK~Rii

u~

(t)

H

u~

(t)

rrO));CTaBHTb B

ypaB-

HeHHH

(18.12) H

rrpoHHTerpHpoBaTb

::ITH

ypaBHeIlllH

lIpH

Ha'IaJIbHOM

yCJIOBHH

x (0) = xa. =

{-1,

0,

O,O}, TO

rrOJIy'IHM

OrrTHMaJIbHOe

p;BHJKeHHe

XO

(t),

Tpa-

eKTopHH

KOToporo

B

lIJIOCKOCTH

(Xl'

Xa)

BbIrJIH));HT

BeCbMa

IIp

OCTO

(CM.

pHC.

18.3,

rp;e

H306paJKeHbI

TaHJKe

BeKTopbI

yrrpaBJIeHHH

uO (t),

cooTBeTcTBYIO~He

pa3JIH'I-

HbIMTO'IKaM

lIYTH).

TaKoii

BH)];

p;BHJKeHHH

XO

(t)

061>HCHHeTcH

TeM,

'ITO

B

p;aHHOM

CJIy'Iae

POJIb

ylIpaBJIeHHH

U~

(t),

KaK

::ITO BH));HO

H3

(18.15), CBOP;HTCH

rrpocTo

K

KOMrreHcaU;HH

CHJIbI

TJUKeCTH.

3aKOIPUIB

paccMoTpeHHe

aToro

npHMepa,

no,n;qepKHeM

em;e

paa,

qTO

BCJI

npon;e,n;ypa,

BKJIIOqaJl

Bl>IqJICJIeHHe

MHHHMaJIbHOH

.pyHKn;HH

hO

('r),

6l>IJIa

rrpo,n;eJIaHa

JIUmb

C

TOH

n;eJIbIO,

qT06bI

Ha

KOHKpeTHoM

MaTepHaJIe

,',

~

....

.::::)

C

§ 18]

ynPABJIEHHE

C

MHHHMAJIbH01i

8HEPrIfE:14

129

npOHJIJIIOCTpHpOllaTb

06m;He

co06pameHHJI,

npHBe,n;eHHl>Ie

B

HaCTOJl-

m;eM

naparpa.pe.

PaayMeeTCJI,

nOCKOJIbKY

MI>I

06JIa,n;aeM

.popMyJIaMH

(18.10),

onpe,n;eJIJlIOm;HMH

a,IleCb

OnTHMaJIbHOe

ynpaBJI6HHe

UO

(t) 6

R6-

UOU

8a.M.,.uymoii

¢OPMe,

npoxo,n;HTb

aTOT

nyn

Kam,n;I>IH

pas

npH

peme-

HUH

KOHKpeTHl>IX

aaJJ;aq

HaJIHmHe.

BOJIee

Toro,

aa,n;aqy

4.2

06

onTH-

MaJIbHOM

ynpaBJIeHH.H

npH

x

[u]

BH,n;a (18.1)

Boo6m;e

uette.t£ecoo6pa8uo

pemaTb

onHCbIBaeMI>IM

HaMH

nyTeM,

TaK

KaK

OHa

B8CbMa

npOCTO

pe-

maeTCJI

cpaay,

HCXO,n;JI

H3

KJIaCCHqecKHX

MeTOJJ;OB

BapHan;HOHHoro

HCqHCJIeHUJI.

ITOJIb3YJlCb

aTHM

06CTOJlTeJIbCTBOM,

MI>I

paCCMOTpHM

ee

em;e

paa

a,n;ecb

,lJ;J1H

llJIJIIOCTpan;HH

CBJlaH

3TH

X MeTo,n;OB C

np06JIeMOH

MO-

MeHTOB.

JITaK,

onHmeM

pemeHUe

aa,n;aqH

(15.1), (18.1),

paCCMaTpHBaH

ee

He

KaK

np06JIeMy

MoMeHToB,

a

KaK

uaonepHM.eTpHqecKyIO

aa,n;aqy

Ba-

pHan;HOHHoro

HCqHCJIeHUJI

(CM.

CTp.

124).

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eM

, KorJJ;a

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B COOTBeTCTBHH C

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(CM.

CTp.

124),

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(18.8) F = u

2

('r),

G

j

=

h(j)

('t) U ('t), HaH,n;eM:

n

UO

('t) = -

~

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('t).

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::ho

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('t) rro,n;CTaBHM B

paBeHCTBa

(15.1);

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1

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H

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me

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5 H.

H.

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6

130

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B

rrpOu;ecce

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a

)

= XiX, X (til) = X

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(18.8),

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113

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23,

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[179]

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x

[u]

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1;

= A

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X +b (t)

u,

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=

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=

~

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(18.17)

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lIa

B03MomHblX.

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1 [u,

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1

[u

('r), X ('r)]

(18.17),

a

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x =

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e.

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t

f

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t.

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0

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0

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ex

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x

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t

x (t) =

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lex

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~

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11

2

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~

X ['t,

ftlb(ft)U(ft)dftI12]dt}'

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lex

(18.19)

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1{3

x~

=

~

h(i)

(t)u('t)dt,

(18.20)

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t

ex

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2

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(18.21)

lex

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n

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lex lex

lex

i=l

.

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Teneph

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B

aTOM

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(18.21)

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n T

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06pa30M:

1{3 1{3

1{3

%[u]

=f~

~

K(t,

ft)u(t)u(ft)d'tdft+'~

U

2

(ft)dft}'/z.

(18.22)

,

'ex

lex'

lex

IIOJIOmHM,

'ITO

B

Ka'IeCTBe

npOCTpaHCTBa

fJi3*

{u

('t)}

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meCTBO

{u

('t)} q,YHKIJ;HB: U ('t) C

lIHTerpnpyeMbIM

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(-r)

onpe,n;eJIHM

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He

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(18.1), a

paBeHcTBoM

p*

[u]

= % [uJ,

rp;e

%

[u]

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JIOB: (18.22).

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npOCTpaH-

CTBO,

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(T);

t

ex

-<

T

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til}.

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(T);

t

ex

<' T <'

t~}

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(CM.

IIpMJIOmeHMe,

§ 56,

CTp.

421)

co

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npou88eiJenueM

(u

l

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2

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q,YHKIJ;HB: U

l

(t)

H U (t),

3a-

2

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~

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(UI,

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~ ~

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(T,

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dt

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~

III

(-r)

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z

(-r)

d-r. (18.23)

lex

I"

--~~~~-'--~~"C>."''--'-''-'-''

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B § 56

(CM.

CTp.

421). B -qaCTHOCTH,

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1

,

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npoH8Be,ll;eHHeM

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CJIe,ll;OBaTeJII>HO,

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p*

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=

x.

[ul

(18.22), oOJla,ll;aeT

BceMH

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HOpMhI.

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MhI

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(18.22).

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([14*

J,

CTp.

198; CM.

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§ 59,

CTp.

435).

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{u

(-r)}

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("t)}

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(18.20)

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1

~

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(18.24)

tot

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h

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12.1.

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(CM. § 59,

CTp.

435),

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1/1

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("t)

II

U

("t),

T.

e.

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(18.23)

BhIpameHHeM

til

(h

(or),

u

(or))

=

~ ~

K

(or,

'it)

h

(or)

u ('it)

dor

d'it +

~

h

(or)

u

(or)

dor.

(18.25)

ta.

tot

ta.

IIocTapaeMcH

noaToMy

npeoopa30BaTI>

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(18.24)

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qTOOhI

ero

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ObIJIO npe,ll;CTaBHTI> B

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HymHoro

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til

~

K

(or,

'it)

P ('it)

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+p

(or)

= h

(or).

(18.26)

tot

YpaBHeBHe

(18.26)

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npH

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('it)

113

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¥Ra3aBHLIH

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135

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([18*1,

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19).

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BO

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(18.26)

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npll

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h

("t)

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me

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h

(-r)

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0,

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(18.26)

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(18.26) B

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(18.24),

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ta.

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=

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+

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u

(1')).

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ta.

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(18.24)

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IJ;HIO

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u,

HO

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h

("t),

a Ha,ll;

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(T).

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fi

)

(T)

B

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(18.26), BaH,ll;eM,

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(T)

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p(i}

(T)

_

perneBHe

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(18.26).

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(18.19), (18.20)

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Tenepl>

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(i

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...

, n)

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liTO

p*

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%

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= P

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=

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l'p

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l'c

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,

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(18.4)

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l

H

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18.1.

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fl

x =

O.

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4,2 C

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XIX

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i=1

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x

2

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A,

(18.27)

O",

'",'

G

§ 18]

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C

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137

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u

2

)

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a cPYHKn;HH S

(T)

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. 1

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-

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+

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(18.28)

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Красовский Н.Н. Теория управления движением

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