Stability of a program manifold of control systems with locally quadratic relations
We establish sufficient conditions for the absolute stability of a program manifold of control systems. In the case where the Jacobi matrix is degenerate, sufficient conditions for the absolute stability of a program manifold is obtained by reduction to the central canonical form.
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| Datum: | 2009 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509049012879360 |
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| 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. |
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| collection | OJS |
| datestamp_date | 2020-03-18T19:43:35Z |
| description | We establish sufficient conditions for the absolute stability of a program manifold of control systems. In the case where the Jacobi matrix is degenerate, sufficient conditions for the absolute stability of a program manifold is obtained by reduction to the central canonical form. |
| first_indexed | 2026-03-24T02:34:55Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.925:62.50
S. S. Ûumatov
(Yn-t matematyky M-va obrazovanyq y nauky Respublyky Kazaxstan, Almat¥)
USTOJÇYVOST| PROHRAMMNOHO
MNOHOOBRAZYQ SYSTEM UPRAVLENYQ
S LOKAL|NO KVADRATYÇNÁMY SVQZQMY
Sufficient conditions of the absolute stability of program manifold of control systems are obtained. For
the case where the Jacobi matrix is degenerate, by the reduction to the central canonical form, sufficient
conditions of the absolute stability of program manifold are found.
Vstanovleno dostatni umovy absolgtno] stijkosti prohramnoho mnohovydu system upravlinnq.
Dlq vypadku, koly matrycq Qkobi [ vyrodΩenog, ßlqxom zvedennq do central\no] kanoniçno]
formy otrymano dostatni umovy absolgtno] stijkosti prohramnoho mnohovydu.
V rabote N. P. Eruhyna [1] postroeno vse mnoΩestvo system dyfferencyal\n¥x
uravnenyj, ymegwyx zadannug yntehral\nug kryvug, sformulyrovana zadaça y
dan metod ee reßenyq. Dal\nejßee razvytye πta zadaça poluçyla v rabotax
A. S. Halyullyna, Y. A. Muxametzqnova, R. H. Muxarlqmova y yx uçenykov [2 –
– 4] kak zadaça postroenyq system dyfferencyal\n¥x uravnenyj, v kotor¥x
hladkoe mnohoobrazye, opredelqemoe pereseçenyem hyperpoverxnostej, qvlqet-
sq yntehral\n¥m.
Rassmotrym materyal\nug systemu, ymegwug (n – s)-mernoe yntehral\noe
mnohoobrazye Ω( )t ≡ ω( , )t x = 0, dvyΩenyq kotoroj opys¥vagtsq ob¥knoven-
n¥my dyfferencyal\n¥my uravnenyqmy
ẋ = f (t, x) – Bξ , ξ̇ = ϕ σ( ), σ = P RTω ξ– , (1)
hde B, P, R � sootvetstvenno ( )n r× -, ( )s r× - y ( )r r× -matryc¥, x — n-mer-
n¥j vektor sostoqnyq obæekta, f � n-mernaq vektor-funkcyq, ω � s ≤ n-mer-
n¥j vektor, ξ � r-mern¥j vektor upravlenyq po otklonenyg ot zadannoj proh-
ramm¥, udovletvorqgwyj uslovyqm lokal\noj kvadratyçnoj svqzy
ϕ θ σ ϕT K– –1( ) > 0, θ = diag θ θ1, ,… r , K = KT > 0. (2)
Poskol\ku mnohoobrazye Ω( )t qvlqetsq yntehral\n¥m dlq system¥ (1), ymeet
mesto ω̇ = ∂ω
∂t
+ H f (t, x, u) = F (t, ω, u), H = ∂ω
∂x
� matryca Qkoby, F (t, 0, u) ≡ 0
� nekotoraq s-mernaq vektor-funkcyq; pry F = t( , ω, ξ ω( , )t ) systema naz¥-
vaetsq zamknutoj, ξ = ξ ω( , )t � mnoΩestvo zakonov obratnoj svqzy [5].
Zadannaq prohramma Ω( )t toçno v¥polnqetsq lyß\ pry uslovyy, kohda
naçal\n¥e znaçenyq vektora sostoqnyq system¥ udovletvorqgt uslovyqm ω(t0 ,
x0) = 0. No πty uslovyq ne vsehda mohut b¥t\ toçno v¥polnen¥. Poπtomu pry
postroenyy system prohrammnoho dvyΩenyq sleduet ymet\ v vydu ewe y trebo-
vanyq ustojçyvosty prohrammnoho mnohoobrazyq Ω( )t otnosytel\no vektor-
funkcyy ω.
Postroym systemu dyfferencyal\n¥x uravnenyj (1), yntehral\noe mnoho-
obrazye Ω( )t kotoroj ymelo b¥ svojstvo ustojçyvosty.
Pust\ F = – Aω , A � hurvyceva ( )s s× -matryca. Dyfferencyruq mnohoob-
razye Ω( )t po vremeny t v sylu system¥ (1), poluçaem
© S. S. ÛUMATOV, 2009
418 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
USTOJÇYVOST| PROHRAMMNOHO MNOHOOBRAZYQ SYSTEM UPRAVLENYQ … 419
ω̇ = – Aω – HBξ, ξ̇ = ϕ σ( ), σ = P RTω ξ– . (3)
Opredelenye. Prohrammnoe mnohoobrazye Ω( )t naz¥vaetsq absolgtno
ustojçyv¥m otnosytel\no vektor-funkcyy ω, esly ono ustojçyvo v celom na
reßenyqx uravnenyq (1) pry lgboj ω(t0 , x0) y ϕ σ( ), udovletvorqgwej us-
lovyqm (2).
Stavytsq zadaça: poluçyt\ uslovyq absolgtnoj ustojçyvosty prohrammnoho
mnohoobrazyq Ω( )t otnosytel\no vektor-funkcyy ω.
Rassmotrym neavtonomnug systemu prohrammnoho mnohoobrazyq
ω̇ = F t( , )ω . (4)
PredpoloΩym, çto: a) F(t, 0) = 0 y F t( , )ω udovletvorqet uslovyqm su-
westvovanyq y edynstvennosty nulevoho reßenyq system¥, b) suwestvuet neko-
toraq neotrycatel\naq lokal\naq kvadratyçnaq svqz\ S [6]
S = S t( , )ω ≥ 0 ∧ S t( , )0 = 0. (5)
Teorema 1. Esly suwestvugt poloΩytel\no opredelennaq funkcyq
V = V t( , )ω > 0 (6)
y neotrycatel\noe çyslo α takye, çto
M tω ω, ( )[ ] = V t( , )ω + α τ ω τ τ
t
t
S d
0
∫ [ ], ( ) > 0, (7)
hde ω( )t � lgboe reßenye s uslovyem (5),
– ˙
( )
M
4
= W tω( )( ) > 0, (8)
to prohrammnoe mnohoobrazye Ω( )t asymptotyçesky ustojçyvo pry v¥polne-
nyy uslovyj (5) otnosytel\no vektor-funkcyy ω.
Dokazatel\stvo. Dopustym, çto [7]
lim ( )
t
M t
→∞
[ ]ω = α0 > 0, (9)
hde α0 � nekotoroe çyslo. DokaΩem, çto α0 = 0. V sylu (8) ymeem
dM
dt
≤ –α1 α1 0>( ) . (10)
Tohda dlq lgboho t > t0 s uçetom neravenstva (10) naxodym
M tω( )[ ] = M tω( )0[ ] +
t
t
dM
dt
d
0
∫ τ ≤ M tω( )0[ ] – α0 0( – )t t . (11)
Yz (11) pry dostatoçno bol\ßom t sleduet, çto M tω( )[ ] stanovytsq otryca-
tel\noj, çto protyvoreçyt neravenstvu (9). Znaçyt, ymeet mesto uslovye
lim ( )
t
M t
→∞
[ ]ω = 0. (12)
V sylu predpoloΩenyq (5) neravenstvo (12) v¥polnqetsq tohda y tol\ko tohda,
kohda
lim ( )
t
t
→∞
ω = 0. (13)
Analohyçno dokaz¥vaetsq sledugwaq teorema.
Teorema 2. Esly suwestvugt funkcyq V = V( )ω > 0, obladagwaq svojst-
vom
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
420 S. S. ÛUMATOV
V = V( )ω → ∞ pry ω( )t → ∞, (14)
y neotrycatel\noe çyslo α takye, çto ymegt mesto sootnoßenyq (6) y (7),
to prohrammnoe mnohoobrazye Ω( )t asymptotyçesky ustojçyvo v celom pry
uslovyqx (5) otnosytel\no vektor-funkcyy ω.
Teper\ rassmotrym systemu (3) v sluçae, kohda upravlenye qvlqetsq prqm¥m
y skalqrn¥m:
ω̇ = – Aω – bξ , ξ = ϕ σ( ), σ = cTω . (15)
Na osnovanyy teorem¥ 2 dostatoçn¥e uslovyq asymptotyçeskoj ustojçyvosty v
celom prohrammnoho mnohoobrazyq Ω( )t moΩno poluçyt\ pry v¥polnenyy ne-
ravenstv
ϕ( )0 = 0 ∨ k1
2σ < σ ϕ σ( ) < k2
2σ , (16)
hde k1, k2 � nekotor¥e postoqnn¥e, a takΩe neravenstv (5) � (7) y uslovyq (8)
pry
V( )ω = ω ωT L + β ϕ σ σ
σ
0
∫ ( )d > 0, (17)
hde β � neotrycatel\noe çyslo.
Druhymy slovamy, spravedlyva sledugwaq teorema.
Teorema 3. Esly suwestvugt vewestvennaq matryca L = LT > 0 y neot-
rycatel\n¥e çysla α y β takye, çto v¥polnqetsq odno yz uslovyj
F =
2G g
gT ρ
> 0, (18)
2ρG – ggT > 0, (19)
2G > 0 ∧ ρ – 2 1 1− g G gT – > 0, (20)
to prohrammnoe mnohoobrazye Ω( )t asymptotyçesky ustojçyvo v celom ot-
nosytel\no vektor-funkcyy ω pry uslovyqx
S( )ω = σ ϕ σ ϕ σ σ– ( ) ( ) ––k k2
1
1( )( ) ≥ 0, (21)
hde k ≥ 0, k2 ≤ ∞,
A LT + L A + k ccT
1 α = 2G, (22)
g = L b + 2 11
1 2
1– ––β αA c k k cT +( ) , (23)
ρ = αk2
1– + βc bT . (24)
Zameçanye 1. Esly nelynejnaq funkcyq ϕ σ( ) dyfferencyruema y razla-
haetsq v rqd Maklorena ϕ σ( ) = h0σ + ψ σ( ) , to vmesto (15) poluçaem systemu
ω̇ = – Ãω – bξ , ξ = ψ σ( ), σ = cTω . (25)
Zdes\ Ã zavysyt ot h0, b y c:
à = A + h bcT
0 ,
a funkcyq ψ udovletvorqet uslovyqm
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
USTOJÇYVOST| PROHRAMMNOHO MNOHOOBRAZYQ SYSTEM UPRAVLENYQ … 421
ψ( )0 = 0 ∨ k̃1
2σ < σ ψ σ( ) < k̃2
2σ ∀σ ≠ 0, (26)
hde k̃1 = k1 – h0, k̃2 = k2 – h0.
Tohda prohrammnoe mnohoobrazye Ω( )t asymptotyçesky ustojçyvo v celom
pry uslovyy (26), esly v¥polnqgtsq sootnoßenyq
à LT + Là + k̃ ccT
1α = 2G,
(27)
ρ = L b + 2 1– ˜ –β αA c cT( ), ρ = α ˜ –k2
1 + βc bT .
Yspol\zuq ravenstva [8]
P HT + HP = U = uij
s
1
, (28)
P = diag ρ ρ1, ,… n , H =
uij
i j
s
ρ ρ+
1
, (29)
moΩno sformulyrovat\ sledugwug teoremu.
Teorema 4. Dlq absolgtnoj ustojçyvosty prohrammnoho mnohoobrazyq
Ω( )t v uhle k k1 2,] [ otnosytel\no vektor-funkcyy ω dostatoçno v¥polne-
nyq odnoho yz neravenstv (18) � (20), hde 2G, g y ρ opredelqgtsq tak:
U + αccT = 2G, ρ = αk2
1– + p dT ,
(30)
g = L d + 2 11
1 2
1– ––β αPp k k p+( )[ ],
yly
U > 0 ∧ ρ > 0, (31)
L b = 2 11
1 2
1– – –α β+( )[ ]k k p Pp . (32)
Uravnenye (32) est\ obobwennoe razreßagwee uravnenye Lur\e [9]. Zdes\
uslovye 2G > 0 yz (23) zameneno uslovyem U > 0, tak kak α ≥ 0.
Zameçanye 2. Dlq v¥polnenyq sootnoßenyj (24), (32) neobxodymo, çtob¥
v¥polnqlys\ sootvetstvenno neravenstva
α 1 1 2
1+( )k k b cT– > βb A cT T , (33)
α 1 1 2
1+( )k k d pT– > βd AP pT –1 , (34)
esly L = LT > 0.
Teorema 5. Dlq asymptotyçeskoj ustojçyvosty v celom prohrammnoho
mnohoobrazyq Ω( )t pry uslovyy (21) otnosytel\no vektor-funkcyy ω dosta-
toçno, çtob¥
G = Q > 0 ∧ L = LT > 0 ∧ α ≥ 0, (35)
a takΩe lybo
γ1 ≥ 0 ∧ γ 2 < 0 ∧ γ 3 ≥ 0 ∧ γ 2
2 – γ γ1 3 > 0, (36)
lybo
γ1 ≥ 0 ∧ γ 3 < 0, (37)
hde matryca 2Q opredelqetsq formuloj (5.18) [4, s. 115], a γ s opredelq-
etsq tak:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
422 S. S. ÛUMATOV
γ1 = ( ) –γ γ5
1
5
T G , γ 5 = A cT , γ 2 = γ γ4
1
5
T G– – c bT ,
(38)
γ 4 = L b – α
2
1 1 2
1+( )k k c– , γ 3 = γ γ4
1
4
T G– – 2 2
1αk– .
Dokazatel\stvo. Pust\ matryca L = LT > 0 opredelena yz uravnenyq (22),
α � zadannoe neotrycatel\noe çyslo y 2Q > 0. Tohda yz uslovyj ustojçyvosty
(20) ostaetsq lyß\ poslednee:
2ρ – g G gT –1 > 0, (39)
kotoroe svodytsq k neravenstvu
γ β1
2 + 2 2γ β + γ 3 < 0. (40)
Poskol\ku G > 0, vsehda ymeet mesto neravenstvo γ1 > 0, esly γ s ≠ 0. Poπto-
mu dlq suwestvovanyq neotrycatel\noho parametra β dostatoçno v¥polnenyq
uslovyj (36) lybo (37). Sledovatel\no, na osnovanyy teorem¥ 3 verna teo-
rema 5.
Zameçanye 3. Funkcyq vyda
M = ω ωT L + β ϕ σ σ
σ
0
∫ ( )d + α ω δ δ
0
t
S d∫ ( )( ) (41)
otlyçaetsq ot funkcyy Lqpunova typa Lur\e (α = 0) nalyçyem posledneho
slahaemoho. Preymuwestvom uslovyj (18) yly (19), yly (20) qvlqetsq to, çto
ony neobxodym¥ y dostatoçn¥ dlq poloΩytel\noj opredelennosty –Ṁ = W.
Krome toho, yz nyx moΩno takΩe poluçyt\, kak sledstvyq, yzvestn¥e uslovyq
absolgtnoj ustojçyvosty dlq razlyçn¥x uhlov pry k1 = 0, k2 = k, k2 = ∞.
Teper\ rassmotrym sluçaj, kohda xarakterystyçeskoe uravnenye matryc¥
H t( ) pry s = n ymeet r nulev¥x kornej, a mnohoobrazye Ω( )t zadano v lynej-
nom vyde
ω( , )t x ≡ H t x1( ) = 0, (42)
hde H t1( ) ∈ Rs n× � zadannaq neprer¥vnaq matryca.
Naßej cel\g qvlqetsq pryvedenye poluçennoj system¥ k central\noj ka-
nonyçeskoj forme [10 – 12] y ustanovlenye uslovyj ustojçyvosty prohrammno-
ho mnohoobrazyq. V¥berem funkcyg Eruhyna sledugwym obrazom:
F (t, x, ω) = – ( )A t2 ω . (43)
Tohda v¥raΩenye ∂ω
∂t
+ H t( ) ẋ = F (t, x, ω) zapyßetsq v vyde
H t x( ) ˙ = A t2( )ω – ∂ω
∂t
. (44)
Prynymaq vo vnymanye sootnoßenye (42), ymeem
H t x( ) ˙ = – ( )A t ω – q t( ), (45)
hde
H t( ) ∈ Rs s× , A t( ) = A t H t2 1( ) ( ) , q t( ) = ∂ω
∂t
.
Rassmotrym operator L t( ) = – ( )A t – H t( )
d
dt
.
Teorema 6 [11]. Pust\ A t( ), H t( ) ∈ C m2 (α, β), rank H t( ) = s – r y mat-
ryca H t( ) ymeet v yntervale (α, β ) poln¥j Ωordanov nabor otnosytel\no
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
USTOJÇYVOST| PROHRAMMNOHO MNOHOOBRAZYQ SYSTEM UPRAVLENYQ … 423
operatora L t( ) , kotor¥j sklad¥vaetsq v r kletok porqdka l 1 , l2 , … , lr
pry max
i
il = m. Tohda suwestvugt neosob¥e pry vsex t ∈ (α, β ) ( s × s )-mat-
ryc¥ ˜ ( )M t , G t1( ) ∈ C1(α, β) takye, çto umnoΩenyem na ˜ ( )M t y zamenoj z =
= G t y( ) systema (44) pryvodytsq k central\noj kanonyçeskoj forme
E
J
y
s r–
˙
0
0
=
M t
E
y
l
( ) 0
0
+ ˜ ( ) ( )M t q t ,
hde l = l1 + … + lr , J = diag (J1, … , Jr ), J j � Ωordanov¥e kletky porqdka lj ,
j = 1, r .
V¥berem funkcyg Eruhyna v vyde
F (t, x, ω) = – ( )A t2 ω – B t( )ξ , ξ̇ = ϕ σ( ), σ = PTω , (46)
hde A t2( ) ∈ Rs s× � hurvyceva, B t( ) ∈ Rs k× � neprer¥vnaq, P ∈ Rk k× � po-
stoqnnaq matryc¥, ϕ, σ ∈Rk � vektor¥, a nelynejnaq vektor-funkcyq ϕ σ( )
udovletvorqet lokal\n¥m uslovyqm kvadratyçnoj svqzy y dyfferencyruema
po σ:
ϕ( , )t 0 ≡ 0 ∧ 0 < σ ϕ σT K t1 ( , ) < σ σT K t( ) ∀ ≠σ 0 , (47)
K2 <
∂ϕ
∂σ
< K3. (48)
Zdes\ K, Ki , i = 1, 2, 3, � dyahonal\n¥e matryc¥.
S uçetom (42) y (46) poluçym systemu (44) v vyde
H t x( ) ˙ = – ( )A t2 ω – q(t) – B t( )ξ , ξ̇ = ϕ σ( ), σ = PTω , (49)
hde ϕ σ( ) udovletvorqet uslovyqm (47), (48).
Na osnovanyy teorem¥ 6 systema (49) pryvodytsq k central\noj kanonyçes-
koj forme
u̇ = – ( )M t u – M t q1 1( ) – M t B t1 1 1 1( ) ( ) ( )ϕ σ ,
ν̇ = – ν – M t q2 2( ) – M t B t2 2 2 2( ) ( ) ( )ϕ σ ,
(50)
σ1 = Q uT
1 , σ2 = Q uT
2 ,
σ = σ1 + σ2, y = uT T T
ν( ) .
Zdes\ q t1( ), q t2( ) yhragt rol\ postoqnno dejstvugwyx vozmuwenyj. Dlq sys-
tem¥ (50) bez vozmuwenyj stroym funkcyg Lqpunova vyda
V = u LuT + ν νT JNJ +
0
1 1 1
1σ
ϕ σ β σ∫ T d( ) +
0
2 2 2
2σ
ϕ σ β σ∫ T Jd( ) , (51)
hde L = LT > 0, N = NT > 0, a β1, β2 � poloΩytel\n¥e dyahonal\n¥e mat-
ryc¥.
Dyfferencyruq funkcyg (51) po vremeny t, v sylu system¥ (50) poluçaem
–V̇ = u G uT
0 + 2 1 1 1u GT ϕ σ( ) + ϕ σ ϕ σ1 1 2 1 1
T G( ) ( ) +
+ ν νT G3 + 2 4 2 2ν ϕ σT G ( ) + ϕ σ ϕ σ2 2 5 2 2
T G( ) ( ) > 0,
esly v¥polnqgtsq obobwenn¥e uslovyq Syl\vestra
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
424 S. S. ÛUMATOV
G G
G G
G G
G G
T
T
0 1
1 2
3 4
4 5
0 0
0 0
0 0
0 0
> 0, (52)
G0 = L M + M LT , G1 = LM B1 1 + 1
2 1 1M QT β , G2 = 2 JNJ,
G3 = β1 1 1 1Q M BT , G4 = JNJVM2B2 + 1
2 2 2QTβ , G5 = β2 2 2 2Q M BT .
Teorema 7. Pust\ nelynejnost\ ϕ σ( ) udovletvorqet uslovyqm (47), (48)
y suwestvugt poloΩytel\no opredelenn¥e matryc¥ L , N , β . Tohda dlq ab-
solgtnoj ustojçyvosty prohrammnoho mnohoobrazyq Ω( )t otnosytel\no vek-
tor-funkcyy ω bez vozmuwenyj dostatoçno v¥polnenyq uslovyq (52).
Yz ustojçyvosty system¥ (50) bez vozmuwenyj sleduet, çto
q t1( ) ≤ δ ε( ) , q t2( ) ≤ δ ε( ) pry u ≤ ε, ν ≤ ε. (53)
Teorema 8. Pust\ v¥polnqgtsq vse uslovyq teorem¥ 7. Tohda dlq abso-
lgtnoj ustojçyvosty prohrammnoho mnohoobrazyq Ω( )t otnosytel\no vek-
tor-funkcyy ω dostatoçno v¥polnenyq uslovyj (53).
1. Eruhyn N. P. Postroenye vseho mnoΩestva system dyfferencyal\n¥x uravnenyj, ymegwyx
zadannug yntehral\nug kryvug // Prykl. matematyka y mexanyka. � 1952. � 16, v¥p. 6. �
S. 653 � 670.
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Poluçeno 06.10.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
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| id | umjimathkievua-article-3030 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:55Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0a/0efbbffee62eb2b203976193744b6e0a.pdf |
| spelling | umjimathkievua-article-30302020-03-18T19:43:35Z Stability of a program manifold of control systems with locally quadratic relations Устойчивость программного многообразия систем управления с локально квадратичными связями Zhumatov, S. S. Жуматов, С. С. Жуматов, С. С. We establish sufficient conditions for the absolute stability of a program manifold of control systems. In the case where the Jacobi matrix is degenerate, sufficient conditions for the absolute stability of a program manifold is obtained by reduction to the central canonical form. Встановлено достатні умови абсолютної стійкості програмного многовиду систем управління. Для випадку, коли матриця Якобі є виродженою, шляхом зведення до центральної канонічної форми отримано достатні умови абсолютної стійкості програмного многовиду. Institute of Mathematics, NAS of Ukraine 2009-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3030 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 3 (2009); 418-424 Український математичний журнал; Том 61 № 3 (2009); 418-424 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3030/2809 https://umj.imath.kiev.ua/index.php/umj/article/view/3030/2810 Copyright (c) 2009 Zhumatov S. S. |
| spellingShingle | Zhumatov, S. S. Жуматов, С. С. Жуматов, С. С. Stability of a program manifold of control systems with locally quadratic relations |
| title | Stability of a program manifold of control systems with locally quadratic relations |
| title_alt | Устойчивость программного многообразия систем управления с локально квадратичными связями |
| title_full | Stability of a program manifold of control systems with locally quadratic relations |
| title_fullStr | Stability of a program manifold of control systems with locally quadratic relations |
| title_full_unstemmed | Stability of a program manifold of control systems with locally quadratic relations |
| title_short | Stability of a program manifold of control systems with locally quadratic relations |
| title_sort | stability of a program manifold of control systems with locally quadratic relations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3030 |
| work_keys_str_mv | AT zhumatovss stabilityofaprogrammanifoldofcontrolsystemswithlocallyquadraticrelations AT žumatovss stabilityofaprogrammanifoldofcontrolsystemswithlocallyquadraticrelations AT žumatovss stabilityofaprogrammanifoldofcontrolsystemswithlocallyquadraticrelations AT zhumatovss ustojčivostʹprogrammnogomnogoobraziâsistemupravleniâslokalʹnokvadratičnymisvâzâmi AT žumatovss ustojčivostʹprogrammnogomnogoobraziâsistemupravleniâslokalʹnokvadratičnymisvâzâmi AT žumatovss ustojčivostʹprogrammnogomnogoobraziâsistemupravleniâslokalʹnokvadratičnymisvâzâmi |