Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea

I?/eo? Aieiaeeie? Aieiaeeie?iae/

OAeE 517.977.5; 519.863

IIOEI?CAOe?ss NO?OEOO?

A AeEIAI?*IEO NENOAIAO IA INIIA?

OCAAAEUeIAIIAI I?EIOeEIO AAEEIAIA

01.05.04-nenoaiiee aiae?c ?

oai??y iioeiaeueieo ??oaiue

Aaoi?aoa?ao

aeena?oaoe?? ia caeiaoooy a/aiiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a-1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia ia eaoaae?? iiaeaethaaiiy neeaaeieo nenoai Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea.

Iaoeiaee ea??aiee:

aeieoi? oaoi?/ieo iaoe, i?ioani? Aa?auaiei Oaae?? Aai?a?eiae/
(Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, i?ioani?)

Io?oe?ei? iiiiaioe: 1. aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani? Aie/oe
Ieaenaiae? Aiae??eiae/ (?inoeooo iaoaiaoeee IAI Oe?a?ie, i?ia?aeiee
iaoeiaee ni?a?ia?oiee)

2. eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe, aeioeaio Iaoa??iei Aieiaeeie?
Oeoiiiae/ (Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, aeioeaio)

I?ia?aeia onoaiiaa:

?inoeooo e?aa?iaoeee ?iai? A.I. Aeooeiaa IAI Oe?a?ie, a?aeae?e
iiaeaethaaiiy ?ioi?iaoe?eii-ooieoe?iiaeueieo nenoai, i. Ee?a.

Caoeno a?aeaoaeaoueny «22» ea?oiy 1999?. ia can?aeaii? niaoe?ae?ciaaii?
a/aii? ?aaee Ae 26.001.09 Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana
Oaa/aiea, i. Ee?a, i?. Aeaaeai?ea Aeooeiaa, 6, ei?i. 2, o-o e?aa?iaoeee,
aoae. 40 i 14 aiaeei?. (Oae. 266-10-58. Oaen 266-12-49. E-mail:
[email protected]).

C aeena?oaoe??th iiaeia iciaeiieoenue o Iaoeia?e a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea, i. Ee?a, aoe. Aieiaeeie?nueea, 58.

Aaoi?aoa?ao ?ic?neaiee «18» aa?aciy 1999?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee

A.I. Oaa/aiei

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. I?e iaoaiaoe/iiio iiaeaethaaii? yaeu oa i?ioean?a
aeey ?o aaeaeaaoiiai iieno iaiao?aeii a?aoiaoaaoe cia/io e?euee?noue
ia?aiao??a. Oea oneeaaeith? ieoaiiy ?aae?caoe?? iiaeae? ia i?aeoeoe?.
Oiio iaoaiaoe/io iiaeaeue aeioe?eueii ?icaeyaeaoe a oae?e oi?i?, yea a
a?aoiaoaaea noae? aca?iica’ycee i?ae i?aenenoaiaie, caeaaeiino? i?ae
ia?aiao?aie nenoaie, a iiiaeeie, a yeeo a?aeaoaa?oueny i?ioean,
iaeaaeaee a aei no?oeoo?ii caaeaieo eean?a. Oaeee i?aeo?ae aeicaiey? ia
o?eueee ciaioeoe e?euee?noue ci?iieo aaee/ei a iiaeae? aac ao?aoe
aeinoaoiueiai ??aiy ?? aaeaeaaoiino? oa ni?inoeoe ?ic?aooiie
iioeiaeueieo oa?aeoa?enoee, aea ? eiino?othaaoe nenoaio a
aei/ii-no?oeoo?i?e oi?i? c iiaeaeueoith iioei?caoe??th ia?aiao??a o
no?oeoo?ao, ye? i?ino?oa ?aae?coaaoe oaoi?/ii. Oaea iaoiaeeea o?nii
iia’ycaia c iaoaiaoe/ieie i?iaeaiaie, ye? aeieeathoue i?e aeine?aeaeaii?
aeeiai?/ieo nenoai c? ci?iiith no?oeoo?ith. sseui iioei?caoe?y
ia?aiao??a nenoaie i?iaiaeeoueny a o?eniaaiiio no?oeoo?iiio aeaeyae?, oi
aaaeeeai anoaiiaeoe no?oeoo?o, yea a caaeaaaea iioeiaeueiee ?aaeei
ooieoe?iioaaiiy ia’?eoo. Oea icia/a?, ui i?aenenoaie iathoue
eiiiiioaaoenue oae, uia aoaeue-yea aeiionoeia ci?ia aca?iica’yceo i?ae
ieie ia iie?auoaaea e?eoa??e yeino? an??? eiino?oeoe??.

A aaaaoueio i?eeeaaeieo caaea/ao aaaeeeai aecia/eoe iaeanoue an?o
ii/aoeiaeo aeaieo, aeey yeeo a?aeiia?aei? ?ica’ycee ia ii?oooaaee a
caaeaieo oaciaeo iaiaaeaiue. I?e canoinoaaii? no?oeoo?iiai i?aeoiaeo
iioeiaeuei? iiiaeeie ioe?iththoueny aaiiao?e/iith oi?iith c o?eniaaiiai
eeano. Oae? ioe?iee iie?auothoueny ca ?aooiie iaeaaeiiai aeai?o
ia?aiao??a nenoaie, i?e/iio a aeayeeo aeiaaeeao iioei?caoe?y ioe?iee
caiaeeoueny aei ?ica’ycoaaiiy a?aeiia?aeii? caaea/? iioeiaeueiiai
ea?oaaiiy iao?e/ieie aeeoa?aioe?aeueieie ??aiyiiyie. Oae? caaea/?
aeine?aeaeoaaeeny a i?aoeyo Auaieiaa E.O., Aa?auaiea O.A., ?iaeueyiiaa
N.A., Ee?e/aiea I.O., Ieoaeaae/a A.N., Iiiaaeeioey A.?., Ooe?ia A.?. c
aeei?enoaiiyi i?eioeeio iaeneioio Iiio?ya?ia, ?acoeueoao?a oai???
no?eeino? oa /ooeeaino?. Iaeiae ne?ae a?aecia/eoe, ui oaeeie i?aeoiaeaie
iniaeeaino? aeacaieo i?iaeai ioiieththoueny ia iiaiith i??ith.
Iai?eeeaae, ieoaiiy canoiniaiino? i?eioeeio iaeneioio aei iioei?caoe??
?aeaeieo nenoai oneeaaeith?oueny, yeui aaiaeyoueny iaiaaeaiiy ia oi/ee
ia?aeeth/aiiy.

O i?iiiiiaai?e aeena?oaoe?ei?e ?iaio? aeine?aeaeothoueny caaea/? aeai?o
iioeiaeueii? no?oeoo?e aeeiai?/ieo nenoai ia iniia? iaoiaeo aeeiai?/iiai
i?ia?aioaaiiy. I?eioeei Aaeeiaia ? iaeiei c ooiaeaiaioaeueieo o oai???
iioeiaeueiiai ea?oaaiiy. Aea i?e aea/aii? ??cieo i?eeeaaeieo i?iaeai
iinoa? ieoaiiy i?i canoiniai?noue i?eioeeio iioeiaeueiino? aeey
?ica’ycoaaiiy ?ioeo eean?a caaea/ iioei?caoe??. Oiio aeine?aeaeaiiy
caaaeueieo aeanoeainoae iaoiaeo aeeiai?/iiai i?ia?aioaaiiy ? aeooaeueiei
ye c iaoaiaoe/ii?, oae ? c i?eeeaaeii? oi/ee ci?o.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. ?iaioa aeeiiaia
a?aeiia?aeii aei ieaio iaoeiaeo aeine?aeaeaiue eaoaae?e iiaeaethaaiiy
neeaaeieo nenoai oaeoeueoaoo e?aa?iaoeee Ee?anueeiai oi?aa?neoaoo ?iai?
Oa?ana Oaa/aiea, a oaeiae iia’ycaia c iaoeiaeie a?aioaie ?97544 oa
?97508 I?i?noa?noaa iaoee ? oaoiieia?? Oe?a?ie c ooiaeaiaioaeueieo oa
i?eeeaaeieo aeine?aeaeaiue.

Iaoa ?iaioe:

1) iioe?eoe i?eioeei Aaeeiaia ia caaea/? iioei?caoe?? aaeeoeaieo
ooieoe?e iiiaeei ia no?oeoo?ii caaeaieo eeanao;

2) canoinoaaoe ocaaaeueiaiee i?eioeei Aaeeiaia aeey ?ica’ycoaaiiy caaea/
aeai?o iioeiaeueii? no?oeoo?e aeeiai?/ii? nenoaie, aeayeeo caaea/
iioeiaeueiiai ea?oaaiiy io/eii o?a?eoi??e oa iioei?caoe?? nenoai c
aeaioiiceoe?eiei ea?oaaiiyi;

3) io?eiaoe aeeoa?aioe?aeueia ??aiyiiy Aaeeiaia aeey caaea/?
iioeiaeueiiai ea?oaaiiy iao?e/iei aeeoa?aioe?aeueiei ??aiyiiyi ? ia eiai
iniia? iiaoaeoaaoe /enaeuei? iaoiaee ?ica’ycoaaiiy aeayeeo caaea/
iioei?caoe?? ioe?iie a caaea/ao i?aeoe/ii? no?eeino?;

4) iiaoaeoaaoe /enaeuei? aeai?eoie iioei?caoe?? no?oeoo? aeeiai?/ieo
nenoai ia aac? io?eiaieo oai?aoe/ieo ?acoeueoao?a;

5) ai?iaoaaoe ?ic?iaeai? aeai?eoie aeey ?ic?aooieo iioeiaeueii?
no?oeoo?e a aeayeeo nenoaiao i?enei?aiiy oa oieonoaaiiy.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a iieyaa? a oiio, ui a
aeena?oaoe?? aia?oa:

( noi?ioeueiaaii ? aeiaaaeaii i?eioeei iioeiaeueiino? Aaeeiaia aeey
caaea/? iioei?caoe?? aaeeoeaii? ooieoe?? iiiaeei ia no?oeoo?ii caaeaieo
eeanao;

( ia iniia? ocaaaeueiaiiai i?eioeeio Aaeeiaia noi?ioeueiaaii ?
aeiaaaeaii i?eioeei iioeiaeueiino? aeey caaea/ aeai?o iioeiaeueii?
no?oeoo?e aeeiai?/ii? nenoaie, aeaaaeaii ??aiyiiy Aaeeiaia a
?ioaa?aeuei?e oi?i? ?, o aeiaaeeo c iao?eniaaieie oi/eaie ia?aeeth/aiiy,
a aeeoa?aioe?aeuei?e oi?i?;

( aeey caaea/? iioei?caoe?? nenoai c aeaioiiceoe?eiei ea?oaaiiyi ? aeey
aeayeeo caaea/ iioeiaeueiiai ea?oaaiiy io/eii o?a?eoi??e c ?ioaa?aeueiei
e?eoa???i yeino?, aacoth/enue ia ocaaaeueiaiiio i?eioeei? Aaeeiaia,
noi?ioeueiaaii ? aeiaaaeaii i?eioeei iioeiaeueiino? oa io?eiaii ??aiyiiy
Aaeeiaia a ?ioaa?aeuei?e oi?i?;

( c aeei?enoaiiyi ocaaaeueiaiiai i?eioeeio Aaeeiaia noi?ioeueiaaii ?
aeiaaaeaii i?eioeei iioeiaeueiino? aeey caaea/? iioeiaeueiiai ea?oaaiiy
iao?e/iei aeeoa?aioe?aeueiei ??aiyiiyi c ?ioaa?aeueiei e?eoa???i yeino?
?, ye iane?aeie, aeaaaeaii ??aiyiiy Aaeeiaia a aeeoa?aioe?aeuei?e oi?i?;

( iiaoaeiaaii aeai?eoie aeey ?ica’ycoaaiiy caaea/ aeai?o iioeiaeueii?
no?oeoo?e oa iioei?caoe?? iaeano? ii/aoeiaeo oiia i?aeoe/ii? no?eeino?
aeeiai?/ii? nenoaie ia iniia? i?eioeeio Aaeeiaia;

( ?ic?iaeaii /enaeuei? iaoiaee aeey ?ica’ycoaaiiy caaea/ aeai?o
iioeiaeueii? no?oeoo?e a nenoaiao i?enei?aiiy oa oieonoaaiiy. I?iaaaeaii
a?aeiia?aeiee ia/enethaaeueiee aenia?eiaio.

Iaoiaee aeine?aeaeaiiy. A ?iaio? aeei?enoaii iaoiaee iaoaiaoe/iiai
aiae?co, oai??? iioeiaeueiiai ea?oaaiiy, iioei?caoe?? oa oai???
no?eeino?, /enaeuei? iaoiaee.

Oai?aoe/ia oa i?aeoe/ia oe?ii?noue. ?acoeueoaoe aeena?oaoe?? iiaeooue
aooe aeei?enoaieie aeey ?ica’ycoaaiiy caaea/ aeai?o iioeiaeueii?
no?oeoo?e a aeeiai?/ieo nenoaiao, i?e aeine?aeaeaii? caaea/ i?aeoe/ii?
no?eeino?, a oaeiae i?e iioei?caoe?? nenoai i?enei?aiiy oa oieonoaaiiy
oa ?ioeo oaoi?/ieo nenoai.

Iniaenoee aianie aaoi?a. A ?iaioao, ui aeeiiai? c? ni?aaaoi?aie,
iniaenoee aianie aaoi?a iieyaaa a iaaiai?aii? iinoaiiaie caaea/,
aeeiiaii? an?o iniiaieo aeiaaaeaiue, ?ic?aooie?a ? a oi?ioethaaii?
aeniiae?a.

Ai?iaaoe?y ?iaioe. Iaoa??aee aeena?oaoe?eii? ?iaioe aeiiia?aeaeenue ia
iaoeiaeo nai?ia?ao eaoaae?e iiaeaethaaiiy neeaaeieo nenoai oaeoeueoaoo
e?aa?iaoeee Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea, a oaeiae ia
eiioa?aioe?yo:

( Oe?a?inuee?e eiioa?aioe?? «Iiaeaethaaiiy ? aeine?aeaeaiiy no?eeino?
nenoai»(20-24 o?aaiy 1996?., Ee?a);

( I?aeia?iaei?e eiioa?aioe?? «Modeling and investigation of systems
stability»(19-23 o?aaiy 1997?., Ee?a).

Ioae?eaoe??. Ca oaiith aeena?oaoe?? aeeiiaii 7 ?ia?o, 6 c yeeo
iioae?eiaaii o aeaeyae? noaoae, oac aeiiia?aeae eiioa?aioe?e. Iaeia
?iaioa aeaiiiiaaia a AeIOA Oe?a?ie. Iniiai? ?acoeueoaoe iioae?eiaaii a
noaooyo [1-3]. Iiaiee nienie ioae?eaoe?e i?eaiaeeoueny a aeena?oaoe?ei?e
?iaio?.

No?oeoo?a oa ianya aeena?oaoe??. Aeena?oaoe?eia ?iaioa aeeeaaeaia ia 120
noi??ieao iaoeiiieniiai oaenoo, neeaaea?oueny ?c anooio, /ioe?ueio
?icae?e?a, aeniiae?a, nieneo aeei?enoaieo aeaea?ae c 102 iaeiaioaaiue,
aeiaeaoeo.

CI?NO ?IAIOE

O anooi? i?eaiaeeoueny caaaeueia oa?aeoa?enoeea aeena?oaoe?eii? ?iaioe:
iaa?oioiao?oueny aeooaeuei?noue aea?aii? oaie, aiiinothoueny iniiai?
io?eiai? ?acoeueoaoe oa aena?oeth?oueny ?o i?aeoe/ia cia/aiiy.

Ia?oee ?icae?e aeena?oaoe?? i?enay/aiee ocaaaeueiaiith iaoiaeo
aeeiai?/iiai i?ia?aioaaiiy ia eean caaea/ iioei?caoe?? aaeeoeaieo
ooieoe?e iiiaeei oa canoinoaaiith io?eiaieo ?acoeueoao?a i?e
aeine?aeaeaii? aeayeeo iioei?caoe?eieo caaea/.

I?ae?icae?e 1.1 ia? iaeyaeiaee oa?aeoa?. A iueiio ia?aaeyiooi e?oa?aoo?o
ia oaio aeena?oaoe?eii? ?iaioe oa iaaaaeaii iinoaiiaee caaea/, ye?
aeine?aeaeothoueny a aeena?oaoe??.

((E)=((B(). I?eionoeii, ui ((A()>0, ((B()>0.

??aia ioeth.

((, a(A}.

m(L)=m(I() i?e oiia? (L(d)=(G((d), L((D, (L((, (G(((.

): a(D}.

A i?ae?icae?e? 1.2 ?iaioe ?icaeyaeathoueny caaea/? aeai?o iioeiaeueii?
no?oeoo?e aeeiai?/ii? nenoaie. Iaoae O(Rn– oaciaee i?ino?? ? aeey
eiaeiiai t([t0,T] aecia/aii iiiaeeio ((t)(X. ?icaeyiaii nenoaio
aeeoa?aioe?aeueieo ??aiyiue aeaeyaeo

(x,t), t(((i-1,(i) (1)

c oiiaaie

x(t0)=x0, (2)

(x((i-0),(i), i=1,2,…,p. (3)

} — iiiaeeia i?aaeo /anoei nenoaie (1.1), ui a?aeiia?aea? iiiaeei?
{(i}.

}), yea a i?i?i?coaaea e?eoa??e yeino?

(x,t)dt+O(x(T)). (4)

Ooo f0(x,t) — ?ioaa?iaaia ca Eaaaaii ia ?ica’yceao nenoaie (1) ooieoe?y
i?e t([t0,T], O(x) — iaia?a?aia ooieoe?y.

})() –o?a?eoi??y (1), ui a?aeiia?aea? iioeiaeuei?e no?oeoo??.

(x,t)dt+O(x(T)) ia o?a?eoi??yo nenoaie (1), ui ?icaeyaea?oueny ia
i?ii?aeeo t((s,T] c oiiaaie x(s+0)=x((s+0) oa (3). Ia? i?noea ianooiiee
i?eioeei iioeiaeueiino?.

})(( ia i?ii?aeeo t((s,T] .

Inoaoi/ii io?eia?ii ??aiyiiy Aaeeiaia a ?ioaa?aeuei?e oi?i? aeey
caaea/? 1.1:

(x(s+(s-0),s+(s),s+(s+0)}.

Ia? i?noea oaeiae ??aiyiiy Aaeeiaia a aeeoa?aioe?aeuei?e oi?i?:

(x,s+0)}=0,

(((s-0).

},{(x((i)}) oa aaeoi?ii x0(((t0), yeee aecia/a? ii/aoeiao oiiao aeey
nenoaie (1). Ooo (x((i)=x((i+0)-x((i-0). I?eionoeii O(x)(0. Caaea/a
iieyaa? a oiio, uia ciaeoe oaeo no?oeoo?o nenoaie (1), yea i?i?i?co?
e?eoa??e yeino? (4).

(x,t)dt.

},{(x((i)})(( ia a?ae??ceo [s1,s2].

}() – o?a?eoi??y nenoaie (1), ui a?aeiia?aea? ?? iioeiaeuei?e
no?oeoo??.

?icaeyiaii aeiiii?aeio caaea/o. Cao?eno?ii oi/eo s((t0,T). Iaiao?aeii
i?i?i?coaaoe e?eoa??e yeino? (4) ia o?a?eoi??yo nenoaie (1) i?e t((s,T]
c oiiaaie x(s+0)=x((s+0) oa (3).

}((.

sseui B(x,s) ? ooieoe??th Aaeeiaia caaea/? 1.3, o((((i-1), oi ia? i?noea
??aiyiiy Aaeeiaia caaea/? 1.3 a ?ioaa?aeuei?e oi?i?

(x((i-0),(i),(i+0)}.

Aeae? ?icaeyiooi caaea/o aeai?o iioeiaeueii? no?oeoo?e aeeiai?/ii?
nenoaie, ui iieno? io/ie o?a?eoi??e. Io?eiaii oaa?aeaeaiiy, aiaeia?/i?
oai?aiai 1.4 oa 1.6.

A i?ae?icae?e? 1.4 ia iniia? iiiaeeiiiai i?eioeeio iioeiaeueiino?
aeiaaaeaii i?eioeei iioeiaeueiino? Aaeeiaia, a oaeiae ??aiyiiy Aaeeiaia
a ?ioaa?aeuei?e oa aeeoa?aioe?aeuei?e oi?iao aeey caaea/? iioeiaeueiiai
ea?oaaiiy aeeiai?/iith nenoaiith c aeaioiiceoe?eiei ea?oaaiiyi.

A i?ae?icae?e? 1.5 ?icaeyiooi aeaye? caaea/? iioeiaeueiiai ea?oaaiiy
io/eii o?a?eoi??e. Iaoae X(Rn — iiiaeeia iaiaaeaiue ia aaeoi? oaciaeo
eii?aeeiao, U(Rm – iiiaeeia iaiaaeaiue ia aaeoi? ea?oaaiiy. ?icaeyiaii
ea?iaaio nenoaio

=f(x,u,t), t([t0,T] (5)

c ii/aoeiaith oiiaith

x(t0)(M0, (6)

aea x(t)(X – aaeoi? oaciaeo eii?aeeiao ?ici??iino? n, u(t)(U— aaeoi?
ea?oaaiiy ?ici??iino? m, f(x,u,t)— n-aei??ia aaeoi?-ooieoe?y, yea
caaeiaieueiy? oiiae oai?aie ?nioaaiiy oa ?aeeiino? ?ica’yceo caaea/?
Eio?, I0(O– eiiiaeoia iiiaeeia. Iicia/eii M(t,u)={x(X:x=x(t,u,x0),
x0(M0}— ia?aoei io/ea o?a?eoi??e nenoaie (5) a iiiaio t([t0,T].

Caaea/a iioeiaeueiiai ea?oaaiiy io/eii o?a?eoi??e iieyaa? a oiio, uia
ciaeoe ea?oaaiiy u(=u((t) ? a?aeiia?aeiee eiio io/ie o?a?eoi??e
X(={x(t):x(t)=x(t,u(,x0), x0(M0, t([t0,T]}, ye? i?i?i?cothoue
ooieoe?iiae

(x,T)dx. (7)

Ooo ((x,u,t), g(x,T) — ?ioaa?iaai? ca Eaaaaii ia ia?aoeiao io/ea
ooieoe??. ?ica’yceii caaea/? (5)-(7) iacaaii ia?o (u(,X().

?icaeyiaii aeiiii?aeio caaea/o. Iaoae s((t0,T)– o?eniaaiee iiiaio /ano.
Iaiao?aeii i?i?i?coaaoe ooieoe?iiae

(x,T)dx

) — ?ica’ycie aeiiii?aeii? caaea/?.

).

C oai?aie 1.10 aeieeaa?, ui aeeiio?oueny ??aiyiiy Aaeeiaia a
?ioaa?aeuei?e oi?i?

(x,u,t)dxdt+B(M(s+(s,u),s+(s)}.

), i=0,1,…,N-1 — caaeai? m-aei??i? iaia?a?ai? ii noeoiiino? ci?iieo
aaeoi?-ooieoe??, aij — /eneia? ia?aiao?e, j=1,2,…,ki, i=0,1,…,N-1.

) ia a?ae??ceo t([s,T].

Ia? i?noea ??aiyiiy Aaeeiaia

(x,(i),t)dxdt+B(M(ti+1,(i),ti+1)},

)(U, t([ti,ti+1), j=1,2,…,ki, i=0,1,…,N-1. Aeae? a i?ae?icae?e?
1.5 ?icaeyaeathoueny aiaeia?/i? ?acoeueoaoe aeey caaea/? iioeiaeueiiai
ea?oaaiiy io/eii o?a?eoi??e c aeaioiiceoe?eiei ea?oaaiiyi.

?icae?e 2 aeena?oaoe?eii? ?iaioe i?enay/aiee aeine?aeaeaiith caaea/
iioei?caoe?? ioe?iie i?aeoe/ii? no?eeino? aeeiai?/ieo nenoai ia iniia?
iaoiaeo Aaeeiaia. A i?ae?icae?e? 2.1 ?icaeyaeathoueny iinoaiiaee caaea/
oa aeyaey?oueny ?o ca’ycie c caaea/aie iioeiaeueiiai ea?oaaiiy
iao?e/ieie aeeoa?aioe?aeueieie ??aiyiiyie. I?e oeueiio canoinoaaiiy
iaoiaeo aeeiai?/iiai i?ia?aioaaiiy i?eaiaeeoue aei iaiao?aeiino?
ciaoiaeaeaiiy iio?aeii? a?ae ooieoe?? iao?e/iiai a?aoiaioo. Oiio a
i?ae?icae?e? 2.2 iaaaaeaii iaiao?aei? oaa?aeaeaiiy ii
aeeoa?aioe?thaaiith ooieoe?e oeio ne?aeo. A i?ae?icae?e? 2.3 ?iaioe
?icaeyaea?oueny caaea/a iioeiaeueiiai ea?oaaiiy iao?e/iei
aeeoa?aioe?aeueiei ??aiyiiyi

=F(X,U,t). (8)

, X(t0)=X0, X0(Y – a?aeiia iao?eoey, Y(Mm(n – iiiaeeia iaiaaeaiue ia
o?a?eoi??? iao?e/iiai aeeoa?aioe?aeueiiai ??aiyiiy, V(Mp(q – iiiaeeia
iaiaaeaiue ia iao?eoeth ea?oaaiiy, Mm(n – iiiaeeia iao?eoeue ?ici??iino?
m(n.

Caaea/a iioeiaeueiiai ea?oaaiiy iao?e/iei ??aiyiiyi (8) iieyaa? a oiio,
uia ciaeoe iao?eoeth ea?oaaiiy Uiio=Uiio(t)(V oaeo, ui i?i?i?co?
ooieoe?iiae

+O(X(T)), (9)

aea f0(X,U,t)– ?ioaa?iaaia ia [t0,T] ooieoe?y, O(X(T))– iaia?a?aia
ooieoe?y. Aeey caaea/? (8), (9) ia? i?noea i?eioeei iioeiaeueiino?
Aaeeiaia, yeee a ?iaio? aeiaaaeaii ia iniia? ocaaaeueiaiiai i?eioeeio
Aaeeiaia. sseui ooieoe?y Aaeeiaia B(X,s) caaea/? (8), (9) ? iaia?a?aii
aeeoa?aioe?eiaaiith ii X oa ii s, oi ??aiyiiy Aaeeiaia aeey caaea/? (8),
(9) ia? aeaeyae

=0, (10)

B(X,T)=O(X). (11)

A i?ae?icae?e? 2.4 ?ica’yco?oueny aaaeeeaa caaea/a i?i?i?caoe??
ooieoe?iiaeo

+tr(SoX(T)S)

ia ?ica’yceao iao?e/iiai aeeoa?aioe?aeueiiai ??aiyiiy oeio Eyioiiaa

=(D(t)+H(t)U(t))X(t)+X(t)(D(t)+H(t)U(t))o+R(t).

Ho(t)((t)H(t), I(Uiio)=tr(((t0)X0)+tr((t0), iao?eoey ((t) caaeiaieueiy?
??aiyiith

((s)H(s)Ho(s)((s)=0, s([t0,T], ((T)=SSo,

a iao?eoey ((t) caaeiaieueiy? ??aiyiith

=((s)R(s), s([t0,T], ((T)=0.

=A(t,u)x, i?e oiia?, ui iaiaaeaiiy At={x(Rn: (lso(t)x((1,s=1,2,…,N}
caaeai? o?eueee a iiiaio /ano t=T i N=1 caiaeeoueny aei ciaoiaeaeaiiy
?ica’yceo nenoaie

=0, i=1,2,…,r. (12)

Ooo x(Rn — aaeoi? noaio nenoaie, u(Rr — aaeoi? ea?oaaiiy, A(Mn(n —
iao?eoey c iaia?a?aieie eiiiiiaioaie, iao?eoey ? aecia/a?oueny c
??aiyiiy

+PA(+A(oP+(A(oA(=0, P(T)=l1(T)l1o(T),

aea A(=A(s,uiio), uiio — ?ica’ycie nenoaie (12), s([t0,T], (>0. Aeae?
?icaeyaeathoueny ?io? caaea/? iioei?caoe?? ioe?iie i?aeoe/ii? no?eeino?
e?i?eii? nenoaie ? io?eiaii ni?aa?aeiioaiiy aiaeia?/i? (12).

Ieoaiiy iiaoaeiae aeai?eoi?a ia iniia? oai?aoe/ieo ?acoeueoao?a
?icae?e?a 1, 2 aena?oeth?oueny a ?icae?e? 3 aeena?oaoe?eii? ?iaioe.
Oae, a i?ae?icae?e? 3.1 iaaiaeyoueny /enaeuei? iaoiaee ?ica’ycoaaiiy
caaea/ aeai?o iioeiaeueii? no?oeoo?e aeeiai?/ii? nenoaie. Aiie
aacothoueny ia io?eiaieo a i?ae?icae?e? 1.3 i?eioeeiao iioeiaeueiino? oa
??aiyiiyo Aaeeiaia. Ia iniia? oai?aoe/iiai iaoa??aeo i?ae?icae?eo 1.4 a
i?ae?icae?e? 3.2 aoaeo?oueny i?ioeaaeo?a ciaoiaeaeaiiy iioeiaeueieo
oi/ie ia?aeeth/aiiy a caaea/? iioei?caoe?? nenoaie c aeaioiiceoe?eiei
ea?oaaiiyi. I?ae?icae?e 3.3 i?enay/aii canoinoaaiith ?acoeueoao?a c
i?ae?icae?eo 2.5 aei iiaoaeiae /enaeueieo iaoiae?a iioei?caoe?? ioe?iie
a caaea/ao i?aeoe/ii? no?eeino?. Iieacaii, ui ni?aa?aeiioaiiy (12) ?
aiaeia?/i? eiio nenoaie iiaeia ?ica’ycoaaoe ca aeiiiiiaith ?oa?aoe?eieo
i?ioeaaeo? oeio a?aae??ioiiai nioneo. A i?ae?icae?e? 3.4 iaaaaeaii
aeai?eoie aeey ?ica’ycoaaiiy aeayeeo caaea/ iioeiaeueiiai ea?oaaiiy
io/eii o?a?eoi??e ia iniia? ??aiyiue Aaeeiaia, io?eiaieo a i?ae?icae?e?
1.5.

A ?icae?e? 4 ?icaeyaea?oueny ieoaiiy iiaeaethaaiiy iioeiaeueii?
aeeiai?ee ca?yaeaeaieo io/e?a ia iniia? ?acoeueoao?a iiia?aaei?o
?icae?e?a ?iaioe. A i?ae?icae?e? 4.1 iaaiaeyoueny ??aiyiiy ?ooo /anoeiie
a aeaeo?iiaai?oieo iieyo oa iinoaiiaee caaea/. A i?ae?icae?e? 4.2 ?iaioe
?icaeyaeathoueny aeai?eoie ciaoiaeaeaiiy iioeiaeueii? no?oeoo?e a
nenoaiao i?enei?aiiy oa oieonoaaiiy. Iaeei c i?aeoiae?a iniiaaiee ia
aeai?? aaciaeo aeaiaio?a nenoaie. Oae, ca aacia? aeaiaioe ia?aca?yaeiiai
i?enei?thaa/a iiaeia aea?aoe i?enei?thth/o o?oaeo, eaaae?oiieueio e?ico,
o?ueioaeaeo?iaeio e?ico, ?ia?n?eio e?ico, iiai?ioiee iaai?o. Iiaoaeiaai?
aeai?eoie aacothoueny ia ?acoeueoaoao i?ae?icae?eo 3.1. A i?ae?icae?e?
4.3 iaaiaeyoueny ?acoeueoaoe ia/enethaaeueiiai aenia?eiaioo.
A?aeiia?aeia i?ia?aia ianyaii 33KB ?ic?iaeaia ia iia? nenoaie
Mathematica 3.0 a na?aaeiaeu? Windows 95 aeey IAII PC Pentium.

O aeniiaeao noi?ioeueiaaii iniiai? ?acoeueoaoe aeena?oaoe?eii? ?iaioe.
Aeiaeaoie i?noeoue iien i?ia?aie.

AENIIAEE

A aeena?oaoe?? iaea?aeai? iia? iaoeiai iaa?oioiaai? ?acoeueoaoe a aaeoc?
no?oeoo?ii? iioei?caoe??, iioeiaeueiiai ea?oaaiiy oa i?aeoe/ii?
no?eeino? aeeiai?/ieo nenoai. Aiie iiaeooue aooe aeei?enoai? i?e
?ica’ycoaaii? caaea/ iioei?caoe?? oa iioeiaeueiiai ea?oaaiiy
aeeiai?/ieie nenoaiaie, i?e no?oeoo?i?e iioei?caoe?? oaoi?/ieo nenoai,
cie?aia, i?e aeai?? iioeiaeueii? no?oeoo?e a nenoaiao i?enei?aiiy oa
oieonoaaiiy.

Iniiaieie ?acoeueoaoaie i?iiiiiaaii? aeena?oaoe?? ?:

1. Aeey caaea/ iioei?caoe?? ooieoe?? iiiaeei ia no?oeoo?ii caaeaieo
eeanao noi?ioeueiaaii ? aeiaaaeaii oai?aie, ye? ? ocaaaeueiaiiyi
i?eioeeio iioeiaeueiino? Aaeeiaia.

2. Aeey caaea/ aeai?o iioeiaeueii? no?oeoo?e aeeiai?/ii? nenoaie ia
iniia? ocaaaeueiaiiai i?eioeeio Aaeeiaia noi?ioeueiaaii ? aeiaaaeaii
i?eioeei iioeiaeueiino?, aeaaaeaii ??aiyiiy Aaeeiaia a ?ioaa?aeuei?e
oi?i? ?, o aeiaaeeo c iao?eniaaieie oi/eaie ia?aeeth/aiiy, oaeiae o
aeeoa?aioe?aeuei?e oi?i?.

Aeoiaey/e c ocaaaeueiaiiai i?eioeeio Aaeeiaia, io?eiaii a?aeiia?aeia
??aiyiiy a ?ioaa?aeuei?e oi?i? aeey caaea/? iioei?caoe?? nenoai c
aeaioiiceoe?eiei ea?oaaiiyi ? aeey aeayeeo caaea/ iioeiaeueiiai
ea?oaaiiy io/eii o?a?eoi??e c ?ioaa?aeueiei e?eoa???i yeino?.

Io?eiaii ??aiyiiy Aaeeiaia a aeeoa?aioe?aeuei?e oi?i? aeey caaea/?
iioeiaeueiiai ea?oaaiiy iao?e/iei aeeoa?aioe?aeueiei ??aiyiiyi c
?ioaa?aeueiei e?eoa???i yeino?. Oea ??aiyiiy canoiniaaia aeey
ciaoiaeaeaiiy ?ica’yceo caaea/? iioeiaeueiiai ea?oaaiiy iao?e/iei
aeeoa?aioe?aeueiei ??aiyiiyi oeio Eyioiiaa.

Ia iniia? iaoiaeo Aaeeiaia iiaoaeiaaii /enaeuei? aeai?eoie aeey
?ica’ycoaaiiy caaea/ aeai?o iioeiaeueii? no?oeoo?e aeeiai?/ii? nenoaie,
iioei?caoe?? nenoai c aeaioiiceoe?eiei ea?oaaiiyi, iioeiaeueiiai
ea?oaaiiy io/eii o?a?eoi??e, iioei?caoe?? a no?oeoo?ieo oi?iao iaeano?
ii/aoeiaeo oiia i?aeoe/ii? no?eeino? aeeiai?/ieo nenoai.

6. ?ic?iaeaii /enaeuei? iaoiaee aeey ?ica’ycoaaiiy caaea/ aeai?o
iioeiaeueii? no?oeoo?e a nenoaiao i?enei?aiiy ? oieonoaaiiy oa
i?iaaaeaii a?aeiia?aeiee ia/enethaaeueiee aenia?eiaio.

NIENIE IIOAE?EIAAIEO I?AOeUe

CA OAIITH AeENA?OAOe??

1. Aa?auaiei O.A., Ie/eo? A.A. No?oeoo?iay iioeiecaoeey aeeiaie/aneeo
nenoai ia iniiaa iaiauaiiiai i?eioeeia Aaeeiaia //I?iaeaiu oi?aaeaiey e
eioi?iaoeee. -1997. -?6. -n. 6-13.

2. Aa?auaiei O.A., Aaoiyeia A.I., Ie/eo? A.A. I iaaeeooa?aioee?oaiuo
caaea/ao iioeiaeueiiai oi?aaeaiey iao?e/iuie aeeooa?aioeeaeueiuie
o?aaiaieyie //Eeaa?iaoeea e nenoaiiue aiaeec. -1998. -?5. -n. 92-101.

3. Aa?auaiei O.A., Ie/eo? A.A. Ia iioeiaeueiuo iiiaeanoaao i?aeoe/aneie
onoie/eainoe//I?iaeaiu oi?aaeaiey e eioi?iaoeee. -1998. -?5. -n. 5-18.

A ?iaioao, ui aeeiioaaeenue c? ni?aaaoi?aie, iaoeiaiio ea??aieeo
i?ioani?o Aa?auaieo O.A. iaeaaeeoue iinoaiiaea caaea/ oa ?aeiiaiaeaoe??
uiaei iaoiae?a ?o ?ica’ycaiiy, Aaoiyeiao I.I. — o/anoue a iaaiai?aii?
?acoeueoao?a oa oaa?aeaeaiiy, iia’ycai? c iaiao?aeieie oiiaaie
iioeiaeueiino?.

I?/eo? A.A. Iioei?caoe?y no?oeoo? a aeeiai?/ieo nenoaiao ia iniia?
ocaaaeueiaiiai i?eioeeio Aaeeiaia. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.05.04 — nenoaiiee aiae?c
? oai??y iioeiaeueieo ??oaiue. — Ee?anueeee oi?aa?neoao ?iai? Oa?ana
Oaa/aiea, Ee?a, 1999 ?.

Aeena?oaoe?th i?enay/aii ieoaiith no?oeoo?ii? iioei?caoe?? aeeiai?/ieo
nenoai iaoiaeii aeeiai?/iiai i?ia?aioaaiiy. Ia iniia? io?eiaieo
?acoeueoao?a ii iioei?caoe?? ooieoe?? iiiaeei aeine?aeaeaii caaea/?
aeai?o iioeiaeueii? no?oeoo?e aeeiai?/ii? nenoaie, iioei?caoe?? nenoai c
aeaioiiceoe?eiei ea?oaaiiyi, iioeiaeueiiai ea?oaaiiy io/eii o?a?eoi??e,
iioei?caoe?? ioe?iie a caaea/ao i?aeoe/ii? no?eeino? ? iiaoaeiaaii
aeai?eoie aeey ?o ?ica’ycoaaiiy. ?ic?iaeaio iaoiaeeeo canoiniaaii aei
iiaeaethaaiiy iioeiaeueii? aeeiai?ee ca?yaeaeaieo io/e?a.

Eeth/ia? neiaa: no?oeoo?ia iioei?caoe?y, iioeiaeueia ea?oaaiiy, i?eioeei
Aaeeiaia, ooieoe?y iiiaeei, iao?e/ia aeeoa?aioe?aeueia ??aiyiiy, io/ie.

Ie/eo? A.A. Iioeiecaoeey no?oeoo? a aeeiaie/aneeo nenoaiao ia iniiaa
iaiauaiiiai i?eioeeia Aaeeiaia. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.05.04 — nenoaiiue
aiaeec e oai?ey iioeiaeueiuo ?aoaiee. — Eeaaneee oieaa?neoao eiaie
Oa?ana Oaa/aiei, Eeaa, 1999 a.

Aeenna?oaoeey iinayuaia aii?ino no?oeoo?iie iioeiecaoeee aeeiaie/aneeo
nenoai iaoiaeii aeeiaie/aneiai i?ia?aiie?iaaiey. A ia?aii ?acaeaea
?aaiou i?eioeei Aaeeiaia iaiauaaony ia eeann caaea/ iioeiecaoeee
aaeaeeoeaiuo ooieoeee iiiaeanoa. Ia iniiaa iieo/aiiuo ?acoeueoaoia
aeieacuaathony oai?aiu oeia i?eioeeia Aaeeiaia aeey caaea/ auai?a
iioeiaeueiie no?oeoo?u aeeiaie/aneie nenoaiu, iioeiaeueiiai oi?aaeaiey
io/eii o?aaeoi?ee, iioeiecaoeee aeeiaie/aneie nenoaiu n
aeaooiiceoeeiiiui oi?aaeaieai a oaeaea auaiaeyony o?aaiaiey Aaeeiaia a
eioaa?aeueiie, a a iaeioi?uo neo/ayo e a aeeooa?aioeeaeueiie, oi?iao.

Aoi?ie ?acaeae aeenna?oaoeee iinayuai iioeiecaoeee ioeaiie i?aeoe/aneie
onoie/eainoe iaoiaeii aeeiaie/aneiai i?ia?aiie?iaaiey. ?anniao?eaathony
niioaaonoaothuea iinoaiiaee caaea/ e iaia?oaeeaaaony eo naycue n
caaea/aie iioeiaeueiiai oi?aaeaiey iao?e/iuie aeeooa?aioeeaeueiuie
o?aaiaieyie. I?e yoii i?eiaiaiea iaoiaea aeeiaie/aneiai i?ia?aiie?iaaiey
i?eaiaeeo e iaiaoiaeeiinoe iaoiaeaeaiey i?iecaiaeiie io ooieoeee
iao?e/iiai a?aoiaioa. Iiyoiio ?anniao?eaathony iaiaoiaeeiua
ooaa?aeaeaiey ii aeeooa?aioee?iaaieth ooieoeee oeia neaaea. Aeey caaea/e
iioeiaeueiiai oi?aaeaiey iao?e/iuie aeeooa?aioeeaeueiuie o?aaiaieyie n
iiiiuueth iaiauaiiiai i?eioeeia Aaeeiaia aeieacuaaaony niioaaonoaothuay
oai?aia e auaiaeeony o?aaiaiea Aaeeiaia a aeeooa?aioeeaeueiie oi?ia.
Iieo/aiiua ?acoeueoaou i?eiaiythony aeey iaoiaeaeaiey iioeiaeueiiai
oi?aaeaiey iao?e/iui o?aaiaieai oeia Eyioiiaa n eaaae?aoe/iui e?eoa?eai
ea/anoaa. Iieacuaaaony, /oi caaea/e iioeiecaoeee ioeaiie i?aeoe/aneie
onoie/eainoe eeiaeiuo aeeiaie/aneeo nenoai naiaeyony e ?aoaieth nenoai
iao?e/iuo o?aaiaiee.

Aii?in iino?iaiey aeai?eoiia ia iniiaa oai?aoe/aneeo ?acoeueoaoia
?acaeaeia 1, 2 inaauaaony a ?acaeaea 3 aeenna?oaoeeiiiie ?aaiou.
I?eaiaeyony /eneaiiua iaoiaeu ?aoaiey caaea/ auai?a iioeiaeueiie
no?oeoo?u aeeiaie/aneie nenoaiu, i?ioeaaeo?a iaoiaeaeaiey iioeiaeueiuo
oi/ae ia?aeeth/aiey a caaea/a iioeiecaoeee nenoai n aeaooiiceoeeiiiui
oi?aaeaieai, aeai?eoiu ?aoaiey caaea/ iioeiaeueiiai oi?aaeaiey io/eii
o?aaeoi?ee. Iie aace?othony ia iieo/aiiuo a ?acaeaea 1 i?eioeeiao
iioeiaeueiinoe e o?aaiaieyo Aaeeiaia. Ia iniiaa ?acoeueoaoia aoi?iai
?acaeaea no?iyony /eneaiiua iaoiaeu iioeiecaoeee ioeaiie a caaea/ao
i?aeoe/aneie onoie/eainoe. Iieacaii, /oi eo ?aoaiea iiaeii inouanoaeyoue
n iiiiuueth eoa?aoeeiiiuo i?ioeaaeo? oeia a?aaeeaioiiai nionea.

A ?acaeaea 4 ?anniao?eaaaony aii?in iiaeaee?iaaiey iioeiaeueiie
aeeiaieee ca?yaeaiiuo io/eia. I?eaiaeyony o?aaiaiey aeaeaeaiey /anoeoe a
yeaeo?iiaaieoiuo iieyo e iinoaiiaee caaea/. ?anniao?eaathony aeai?eoiu
iaoiaeaeaiey iioeiaeueiie no?oeoo?u a nenoaiao onei?aiey e oieone?iaee.
Iaeei ec iiaeoiaeia iniiauaaaony ia auai?a aaciauo yeaiaioia nenoaiu.
Iai?eia?, aaciauie yeaiaioaie ia?aca?yaeiiai onei?eoaey iiaeii aua?aoue
onei?eoaeueioth o?oaeo, eaaae?oiieueioth eeico, o?aoyeaeo?iaeioth eeico,
eiia?neiiioth eeico, iiai?ioiue iaaieo. Iino?iaiiua /eneaiiua iaoiaeu
iniiauaathony ia aeai?eoiao, iieo/aiiuo a ?acaeaea 3 ?aaiou.
Aiaeece?othony ?acoeueoaou au/eneeoaeueiiai yenia?eiaioa.

Eeth/aaua neiaa: no?oeoo?iay iioeiecaoeey, iioeiaeueiia oi?aaeaiea,
i?eioeei Aaeeiaia, ooieoeey iiiaeanoa, iao?e/iia aeeooa?aioeeaeueiia
o?aaiaiea, io/ie.

Pichkur V.V. Structures optimization in dynamical system on the basis of
the generalized Bellman’s principle. — Manuscript.

Thesis for degree of Candidate of Science (Ph.D) in Physics and
Mathematics, the speciality 01.05.04 — system analysis and theory of
optimal solutions. — Kyiv Shevchenko University, Kyiv, 1999.

The dissertation is devoted to the question of the dynamical system
structural optimization by Bellman’s method. On the basis of obtained
theoretical results some problems of optimal structure choice of
dynamical systems, systems optimization with two-position control,
optimal control of trajectories bunch, optimization of practical
stability estimates are investigated and corresponding algorithms are
built. The elaborated methods are applied for modeling of optimal
dynamics of charge beams.

Key words: structural optimization, optimal control, Bellman’s
principle, sets function, matrix differential equation, beam.

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