Exponential stability of a program manifold of indirect control systems
We establish sufficient conditions for the exponential stability of a program manifold of indirect control systems and conditions for the fast operation of a regulator, overcontrol, and monotone damping of a transient process in the neighborhood of the program manifold.
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| Дата: | 2010 |
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| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508904699461632 |
|---|---|
| author | Zhumatov, S. S. Жуматов, С. С. Жуматов, С. С. |
| author_facet | Zhumatov, S. S. Жуматов, С. С. Жуматов, С. С. |
| author_sort | Zhumatov, S. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:12Z |
| description | We establish sufficient conditions for the exponential stability of a program manifold of indirect control systems and conditions for the fast operation of a regulator, overcontrol, and monotone damping of a transient process in the neighborhood of the program manifold. |
| first_indexed | 2026-03-24T02:32:37Z |
| format | Article |
| fulltext |
UDK 517.925:62.50
S. S. Ûumatov
(Yn-t matematyky M-va obrazovanyq y nauky Respublyky Kazaxstan, Almat¥)
∏KSPONENCYAL|NAQ USTOJÇYVOST|
PROHRAMMNOHO MNOHOOBRAZYQ
SYSTEM NEPRQMOHO UPRAVLENYQ
We obtain sufficient conditions of exponential stability of program manifold of indirect control systems.
We establish conditions of regulator high speed, the reregulation, and of the monotonic attenuation of a
transition process in the neighborhood of program manifold.
Vstanovleno dostatni umovy eksponencial\no] stijkosti prohramnoho mnohovydu system neprq-
moho keruvannq, a takoΩ umovy ßvydkodi] rehulqtora, pererehulgvannq, monotonnoho zhasannq
perexidnoho procesu v okoli prohramnoho mnohovydu.
Zadaça postroenyq vseho mnoΩestva system dyfferencyal\n¥x uravnenyj,
ymegwyx zadannug yntehral\nug kryvug, b¥la sformulyrovana v rabote [1],
hde pryveden y metod ee reßenyq. ∏ta zadaça poluçyla dal\nejßee razvytye
kak zadaça postroenyq system dyfferencyal\n¥x uravnenyj po zadannomu
yntehral\nomu mnohoobrazyg, reßenyq razlyçn¥x obratn¥x zadaç dynamyky,
postroenyq system prohrammnoho dvyΩenyq. Sleduet otmetyt\, çto v processe
reßenyq πtyx zadaç postroenye ustojçyv¥x system, qvlqqs\ odnoj yz osnovn¥x
zadaç teoryy ustojçyvosty, prevratylos\ v samostoqtel\nug teoryg. Podrob-
n¥j obzor πtyx rabot pryveden v [2]. Postroenyg system avtomatyçeskoho up-
ravlenyq po zadannomu mnohoobrazyg posvqwen¥ rabot¥ [3 – 5]. V nyx system¥
upravlenyq b¥ly postroen¥ dlq sluçaq, kohda nelynejnaq funkcyq ϕ σ( ) qv-
lqetsq skalqrnoj. Ustanovlen¥ dostatoçn¥e uslovyq absolgtnoj ustojçy-
vosty. V [6, 7] reßen¥ zadaçy postroenyq system avtomatyçeskoho upravlenyq,
kohda nelynejnaq funkcyq qvlqetsq vektornoj y udovletvorqet uslovyqm
lokal\noj kvadratyçnoj svqzy. Yssledovanyg voprosov ob πksponencyal\noj
ustojçyvosty tryvyal\noho reßenyq posvqwena rabota [8]. V [9, 10] ustanov-
len¥ uslovyq πksponencyal\noj ustojçyvosty dlq system avtomatyçeskoho
upravlenyq opredelennoho klassa. Zadaçy synteza asymptotyçesky ustojçyv¥x
system, obladagwyx zadann¥m kaçestvom, sformulyrovan¥ v rabote [11], hde
dan metod synteza zakonov obratnoj svqzy. V [12 – 14] ustanovlen¥ uslovyq
pryvodymosty k kanonyçeskoj forme y uslovyq razreßymosty zadaçy Koßy, a
takΩe yssledovan¥ vopros¥ suwestvovanyq peryodyçeskyx reßenyj dlq urav-
nenyj, ne razreßenn¥x otnosytel\no starßej proyzvodnoj. V rabote [15] polu-
çen¥ dostatoçn¥e uslovyq asymptotyçeskoj ustojçyvosty prohrammnoho mno-
hoobrazyq v¥roΩdenn¥x system avtomatyçeskoho upravlenyq.
V nastoqwej rabote reßagtsq zadaçy ustanovlenyq uslovyj πksponencyal\-
noj ustojçyvosty prohrammnoho mnohoobrazyq y synteza system, obladagwyx
napered zadann¥my svojstvamy v vyde nekotoroho mnohoobrazyq.
Postanovka zadaçy. Rassmotrym zadaçu postroenyq ustojçyvoj system¥
upravlenyq sledugwej struktur¥:
�x = f t x B( , ) − ξ , �ξ = ϕ σ( ) , σ = P RTω ξ− 1 , (1)
po zadannomu ( )n s− -mernomu hladkomu yntehral\nomu mnohoobrazyg Ω( )t ,
opredelqemomu vektorn¥m uravnenyem
ω ( , )x t = 0, (2)
hde ω — ( )s n≤ -mern¥j vektor, t I∈ = ∞[ , [0 , pry uslovyy R1 0> , f t x( , ) —
nekotoraq n-mernaq vektor-funkcyq, B, P y R — postoqnn¥e matryc¥ raz-
© S. S. ÛUMATOV, 2010
784 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
∏KSPONENCYAL|NAQ USTOJÇYVOST| PROHRAMMNOHO MNOHOOBRAZYQ … 785
mernostej n r× , s r× y r r× sootvetstvenno, x — n-mern¥j, a ξ , σ, ϕ — r-
mern¥e vektor¥, nelynejnaq vektor-funkcyq upravlenyq ϕ σ( ) po otklone-
nyg ot zadannoj prohramm¥ udovletvorqet uslovyqm
ϕ ( )0 = 0, 0 < σ ϕ σT ( ) ≤ σ σT K ∀ ≠σ 0 . (3)
V prostranstve Rn
v¥delym oblast\ G R( ) :
G R( ) = ( , ) : ( , )t x t t x≥ ∧ ≤ < ∞{ }0 ω ρ . (4)
Uçyt¥vaq neobxodymoe y dostatoçnoe uslovye toho, çto mnohoobrazye Ω budet
yntehral\n¥m dlq system¥ (1), ymeem
�ω = F t x H B( , , )ω ξ− , �ξ = ϕ σ( ) , σ = P RTω ξ− 1 , H =
∂
∂
ω
x
, (5)
hde
∂
∂
+
∂
∂
ω ω
t x
f t x( , ) = F t x( , , )ω , F t x( , , )ω — funkcyq Eruhyna [2], udovlet-
vorqgwaq uslovyg F t x( , , )0 ≡ 0. Pry F = F t t( , , ( , ))ω ξ ω systema (5) naz¥va-
etsq zamknutoj; ξ = ξ ω( , )t — mnoΩestvo zakonov obratnoj svqzy [11]. Pola-
haq F = –A ω , hde A — hurvyceva matryca razmernosty s s× , yz (5) poluçaem
�ω = – A ω – H B ξ , �ξ = ϕ σ( ) , σ = P RTω ξ− 1 . (6)
Sleduet otmetyt\, çto pry postroenyy ustojçyv¥x system avtomatyçeskoho
upravlenyq na prohrammnoe mnohoobrazye (2) takΩe naklad¥vaetsq dopolny-
tel\noe trebovanye ustojçyvosty.
Opredelenye,1. Prohrammnoe mnohoobrazye Ω( )t naz¥vaetsq πksponen-
cyal\no ustojçyv¥m pry t → ∞ otnosytel\no vektor-funkcyy ω , esly v
oblasty (4) suwestvugt N > 0 , α > 0 takye, çto v¥polnqetsq
z t( ) ≤ N z t t t( ) exp ( )[ ]0 0− −α , t ≥ t0
, (7)
dlq lgboj funkcyy ω ( , )t x0 0 y ϕ σ( ) , udovletvorqgwej uslovyqm (3), z
2
=
= ω ξ2 2+ .
Zadaça,1. Ustanovym dostatoçn¥e uslovyq πksponencyal\noj ustojçyvos-
ty prohrammnoho mnohoobrazyq Ω( )t system upravlenyq otnosytel\no vek-
tor-funkcyy ω.
Voz\mem dve sfer¥ ω0 = R , ω0 = ω ( )t0 , ω ( )t0
∗ = ε , R >> ε . Ras-
smotrym vse mnoΩestvo reßenyj uravnenyq (6), naçynagwyxsq na sfere R y
naz¥vaem¥x R-reßenyqmy. Dlq asymptotyçesky ustojçyv¥x system dlq lgboho
ω0 na sfere R suwestvuet t0
∗
takoe, pry kotorom v¥polnqgtsq uslovyq
ω ω( , , )t t0 0 0
∗ = ε, ω ω( , , )t t0 0 < ε ∀ > ∗t t0 . (8)
Pust\ t∗ = sup
ω0
0t
∗ .
Opredelenye,2. Ynterval t t∗ − 0 naz¥vaetsq vremenem rehulyrovanyq v
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
786 S. S. ÛUMATOV
zamknutoj systeme, esly lgboe R-reßenye v¥xodyt na sferu ε pry t t0
∗ ∗≤
y ostaetsq vnutry nee pry t t> ∗
.
Zadaça,2. Dano mnoΩestvo M zakonov obratnoj svqzy. Trebuetsq opre-
delyt\ eho podmnoΩestvo M1 , na kotorom v¥polnqetsq uslovye t R∗( , , )ε ξ ≤
≤ t
z
( t
z
— zadannoe vremq).
Zadaça reßaetsq dlq system, asymptotyçesky ustojçyv¥x otnosytel\no vek-
tor-funkcyy ω.
∏ksponencyal\naq ustojçyvost\ prohrammnoho mnohoobrazyq.
Teorema,1. Pust\ ϕ σ( ) udovletvorqet uslovyqm (3) y suwestvuet po-
loΩytel\no opredelennaq funkcyq V ( , )ω ξ > 0, proyzvodnaq kotoroj v sylu
system¥ (6) qvlqetsq otrycatel\no opredelennoj – �V W= >( , )ω ξ 0 . Tohda
prohrammnoe mnohoobrazye Ω( )t πksponencyal\no ustojçyvo otnosytel\no
vektor-funkcyy ω.
Dokazatel\stvo. Dlq system¥ (6) stroym poloΩytel\no opredelennug
funkcyg Lqpunova
V = ω ω ω ξ ξ ξ ϕ β σ
σ
T T T TL L L d0 1 2
0
2+ + + ∫ , L = LT > 0, (9)
hde L =
L L
L LT
0 1
1 2
> 0, β = diag β β1, ,… r > 0. Proyzvodnaq funkcyy (9)
po vremeny t v sylu system¥ (6) prymet vyd
– �V ≡≡≡≡ W = ω ω ω ξ ξ ξ ω ϕ ξ ϕ ϕ ρϕT T T T T TG G g G G0 1 2 12 2 2+ + + + + > 0,
(10)
G0 = A L L AT
0 0+ , G1 = L B A LT
0 1+ , g = BL L BT
1 0+ ,
G2 = – L A PT
1
1
2
+ β , G3 = – L B PT
2 + β , ρ = βR1 ,
G =
G G G
G g G
G G
T
T T
0 1 2
1 3
2 3 ρ
> 0.
Na osnovanyy svojstva (3) y struktur¥ obratnoj svqzy σ spravedlyv¥ sledug-
wye ocenky:
0 < ϕ σ β σ
σ
T d( )
0
∫ <
β
σ1 1 2
2
k
, 0 < ϕ 2
< k1
2 2σ ,
(11)
ρ ω ν ξ1
2
1
2+ ≤ σ 2
≤ ρ ω ν ξ2
2
2
2+ ,
hde k1 = min ki{ } , β1 = βi{ } , i = 1, … , r, ki — sobstvenn¥e çysla matryc¥
K, a ρ1, ρ2 y ν1 , ν2 opredelqgtsq tak:
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
∏KSPONENCYAL|NAQ USTOJÇYVOST| PROHRAMMNOHO MNOHOOBRAZYQ … 787
ρ1 = min
ω
ω ω
ω ω≠0
T T
T
PP
, ρ2 = max
ω
ω ω
ω ω≠0
T T
T
PP
,
ν1 = min
ξ
ξ ξ
ξ ξ≠0
1 1
T T
T
R R
, ν2 = max
ξ
ξ ξ
ξ ξ≠0
1 1
T T
T
R R
.
V sylu poloΩytel\noj opredelennosty funkcyy V (9) y ee proyzvodnoj – �V
(10) s uçetom ocenok (11) ymegt mesto sootnoßenyq
γ ω ξ1
2 2+( ) ≤ V ≤ γ ω ξ2
2 2+( ) , (12)
g1
2 2ω ξ+( ) ≤ – �V ≤ g2
2 2ω ξ+( ) . (13)
Zdes\
γ1 = min ;l
k
l
k
1
1 1
1 1
1 1
12 2
+ +
β
ρ
β
ν ,
γ 2 = max ;l
k
l
k
2
1 1
2 2
1 1
22 2
+ +
β
ρ
β
ν ,
g1 = min ;g k k0 1
2
1 1
2
11 1+ +{ }ρ ν , g2 = max ;g k ks 1 11
2
2 1
2
2+ +{ }ρ ν ,
l1, l2 , g0 , gs — naymen\ßye y naybol\ßye znaçenyq sobstvenn¥x çysel L y G.
Pust\ z
2
= ω ξ2 2+ . Tohda, prynymaq vo vnymanye sootnoßenyq (12),
(13), poluçaem ocenky
γ α2
1
0 1 0
− −V t texp ( ) ≤ z
2
≤ γ α1
1
0 2 0
− −V t texp ( ) , (14)
hde
V0 = z Lz dT T
0 0
0
0
+ ∫ ϕ σ β σ
σ
( ) , z0 = z t( )0 , σ0 = σ ( )t0 ,
α1 = –
g2
1γ
, α2 = –
g1
2γ
.
Otsgda v sylu neravenstva (12) ymeem
z t( )
2
≤ γ γ α2 1
1
0
2
02− −z t t t( ) exp ( ) , t t≥ 0 , α
α
= 2
2
.
Takym obrazom, pry t t≥ 0 naxodym
z t( ) ≤ N z t t t( ) exp ( )0 0α − , t t≥ 0 ,
hde N = −γ γ2 1
1 , a z t( )0 dostatoçno mala.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
788 S. S. ÛUMATOV
Uslovye b¥strodejstvyq rehulqtora. Na osnovanyy ocenok (13), (14) na
sfere R v¥polnqetsq neravenstvo
z
2
≤ γ γ α1
1
2
2
1 0
− −R t texp ( ) . (15)
Pry t t= ∗
0 yz (15) poluçaem
γ γ α1
1
2
2
1 0 0
− ∗ −R t texp ( ) = ε2 ,
otkuda sleduet
t t0 0
∗ − = α
ε γ
γ
2
1
2
1
2
2
− ln
R
.
Uslovyq b¥strodejstvyq rehulqtora ymegt vyd
–α
γ
ε γ
2
1
2
2
2
1
− sup ln
R
R
= t
z
,
hde t
z
— zadannoe vremq.
Zadaça pererehulyrovanyq. Rassmotrym lgboe R-reßenye zamknutoj sys-
tem¥ (6), opredelennoe dlq kakoho-lybo zakona obratnoj svqzy ξ = ξ ω( , )t , y
proyzvol\nug poloΩytel\nug funkcyg Φ( )ω .
Otnoßenye
Π = sup
( )
t
S
S
Φ ω −
, S > 0 , (16)
naz¥vaetsq pererehulyrovanyem [11], S = ω ( )∞ — ustanovyvßeesq znaçenye
rehulyruemoj velyçyn¥ posle zaverßenyq perexodnoho processa.
Zadaça,3. Dano mnoΩestvo M zakonov obratnoj svqzy. Trebuetsq opre-
delyt\ takoe eho podmnoΩestvo M2 , na kotorom v¥polnqetsq neravenstvo
Π = Π( , , )R S ξ ≤ Π
z
( Π
z
— zadannoe çyslo).
Dlq reßenyq zadaçy pererehulyrovanyq poloΩym Φ( , )ω ξ = V , hde V
opredelqetsq sootnoßenyem (9). Tohda na osnovanyy (15) pererehulyrovanye P
budet ymet\ vyd
Π = sup
exp ( )
t
t t s
s
γ γ α1
1
2 2 0
− − −
, s = z( )∞ > 0.
Otsgda, prynymaq vo vnymanye, çto α2 0< , s uçetom (14) naxodym
Π =
γ γ1
1
2
2− −R s
s
> 0
pry uslovyy R s> −γ γ2
1
1 .
Esly Π
z
— zadannoe çyslo, to uslovye pererehulyrovanyq poluçym v vyde
R2 ≤ s( )Π
z
+ −1 1 2
1γ γ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
∏KSPONENCYAL|NAQ USTOJÇYVOST| PROHRAMMNOHO MNOHOOBRAZYQ … 789
Zadaça o monotonnom zatuxanyy. Perexodn¥j process v zamknutoj syste-
me naz¥vaetsq monotonno zatuxagwym [11], esly suwestvuet takoj zakon obrat-
noj svqzy ξ = ξ ω( , )t , çto lgboe R -reßenye uravnenyq (6) udovletvorqet
uslovyg
�ω ω( , , )t t0 0 ≤ C t texp ( )[ ]− −α 0 , C > 0 , α > 0 . (17)
Zadaça,4. Dano mnoΩestvo M zakonov obratnoj svqzy. Trebuetsq opre-
delyt\ takoe eho podmnoΩestvo M
z
, na kotorom lgboe R-reßenye uravne-
nyq (6) zatuxaet monotonno.
Dlq poluçenyq uslovyq monotonnoho zatuxanyq dopolnytel\no trebuetsq
dyfferencyruemost\ vektor-funkcyy ϕ σ( ) po σ y çtob¥ çastnaq proyzvod-
naq udovletvorqla uslovyg
K1 ≤
∂
∂
ϕ
σ
≤ K2 , K1 0> , K2 0> .
Dyfferencyruq systemu (6) po vremeny t , naxodym
��ω = – A N� �ω ξ− ,
��ξ =
∂
∂
ϕ
σ
σ� , (18)
�σ = P RT � �ω ξ− .
Stroym funkcyg Lqpunova
V = � � � �ω ω ξ ξT TL L0 1+ , L0 = LT
0 > 0, L1 = LT
1 > 0, (19)
proyzvodnaq kotoroj v sylu system¥ (18) ymeet vyd
– �V = � � � � � �ω ω ω ξ ξ ξT T TG G G0 1 22+ + . (20)
Zdes\
G0 = A L L AT
0 0+ , G1 = L N P L
T
0 1−
∂
∂
ϕ
σ
,
G2 = M L L MT
1 1+ , M =
∂
∂
ϕ
σ
R ,
G =
G G
G GT
0 1
1 2
> 0.
Yz sootnoßenyj (19), (20) ymeem
l1
2 2
� �ω ξ+
≤ V ≤ l2
2 2
� �ω ξ+
, (21)
g t1
2 2
( ) � �ω ξ+
≤ – �V ≤ g t2
2 2
( ) � �ω ξ+
, (22)
hde l1 = min ,{ }( ) ( )l li i
0 1 , l2 = max ,{ }( ) ( )l li i
0 1 , g t1( ) = min ( ){ }g ti , g t2( ) =
= max ( ){ }g ti ; li
( )0 , li
( )1 , g ti ( ) — sobstvenn¥e çysla matryc L0 , L1 , G.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
790 S. S. ÛUMATOV
Na osnovanyy ocenok (21), (22) poluçaem
V t t0 1 0exp ( )α − ≤ V ≤ V t t0 2 0exp ( )α − , (23)
α1 = – l g1 2/ , α2 = – l g2 1/ , g1 = inf ( ){ }
t
g t1 , g2 = sup ( ){ }
t
g t2 .
Polahaq �z
2
= � �ω ξ2 2
+ , v sylu (21), (23) ymeem
�z
2
≤ �R t t0
2
2 0exp ( )α − ,
�R0
2 = � �ω ξ( ) ( )t t0
2
0
2
+ .
Otsgda naxodym uslovye monotonnosty zatuxanyq perexodnoho processa syste-
m¥ (18)
�z ≤ �R t t0 0exp ( )α − ,
hde α = α2 2/ .
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Poluçeno 02.09.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
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| id | umjimathkievua-article-2909 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:32:37Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/52/82a09e04d4d0db99e7fad16118030b52.pdf |
| spelling | umjimathkievua-article-29092020-03-18T19:40:12Z Exponential stability of a program manifold of indirect control systems Экспоненциальная устойчивость программного многообразия систем непрямого управления Zhumatov, S. S. Жуматов, С. С. Жуматов, С. С. We establish sufficient conditions for the exponential stability of a program manifold of indirect control systems and conditions for the fast operation of a regulator, overcontrol, and monotone damping of a transient process in the neighborhood of the program manifold. Встановлено достатні умови експоненціальної стійкості програмного миоговиду систем непрямого керування, а також умови швидкодії регулятора, перерегулювання, монотонного згасання перехідного процесу в околі програмного миоговиду Institute of Mathematics, NAS of Ukraine 2010-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2909 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 6 (2010); 784–790 Український математичний журнал; Том 62 № 6 (2010); 784–790 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2909/2569 https://umj.imath.kiev.ua/index.php/umj/article/view/2909/2570 Copyright (c) 2010 Zhumatov S. S. |
| spellingShingle | Zhumatov, S. S. Жуматов, С. С. Жуматов, С. С. Exponential stability of a program manifold of indirect control systems |
| title | Exponential stability of a program manifold of indirect control systems |
| title_alt | Экспоненциальная устойчивость программного многообразия систем непрямого управления |
| title_full | Exponential stability of a program manifold of indirect control systems |
| title_fullStr | Exponential stability of a program manifold of indirect control systems |
| title_full_unstemmed | Exponential stability of a program manifold of indirect control systems |
| title_short | Exponential stability of a program manifold of indirect control systems |
| title_sort | exponential stability of a program manifold of indirect control systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2909 |
| work_keys_str_mv | AT zhumatovss exponentialstabilityofaprogrammanifoldofindirectcontrolsystems AT žumatovss exponentialstabilityofaprogrammanifoldofindirectcontrolsystems AT žumatovss exponentialstabilityofaprogrammanifoldofindirectcontrolsystems AT zhumatovss éksponencialʹnaâustojčivostʹprogrammnogomnogoobraziâsistemneprâmogoupravleniâ AT žumatovss éksponencialʹnaâustojčivostʹprogrammnogomnogoobraziâsistemneprâmogoupravleniâ AT žumatovss éksponencialʹnaâustojčivostʹprogrammnogomnogoobraziâsistemneprâmogoupravleniâ |