On stability of linear hybrid mechanical systems with distributed components
We present a new approach to the solution of problems of stability of hybrid systems based on the constructive determination of elements of a matrix-valued functional.
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| Datum: | 2008 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2008
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| author | Martynyuk, A. A. Slyn'ko, V. I. Мартынюк, А. А. Слынько, В. И. Мартынюк, А. А. Слынько, В. И. |
| author_facet | Martynyuk, A. A. Slyn'ko, V. I. Мартынюк, А. А. Слынько, В. И. Мартынюк, А. А. Слынько, В. И. |
| author_sort | Martynyuk, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:46:54Z |
| description | We present a new approach to the solution of problems of stability of hybrid systems based on the constructive determination of elements of a matrix-valued functional. |
| first_indexed | 2026-03-24T02:37:09Z |
| format | Article |
| fulltext |
UDK 531.36
A. A. Mart¥ngk, V. Y. Sl¥n\ko (Yn-t mexanyky NAN Ukrayn¥, Kyev)
OB USTOJÇYVOSTY LYNEJNÁX
HYBRYDNÁX MEXANYÇESKYX SYSTEM
S RASPREDELENNÁM ZVENOM
We present a new scheme of solving the problem of stability of a hybrid system based on the
constructive creation of elements of a matrix-valued functional.
Navedeno novyj pidxid do rozv’qzannq zadaçi pro stijkist\ hibrydno] systemy, wo ©runtu[t\sq na
konstruktyvnij pobudovi elementiv matryçnoznaçnoho funkcionala.
1. Vvedenye. SloΩn¥e system¥ s raznorodn¥my podsystemamy prynqto naz¥-
vat\ hybrydn¥my. Mnohye system¥ bol\ßoj razmernosty so sloΩnoj struktu-
roj sostoqt yz podsystem men\ßej razmernosty, kotor¥e obæedynqgtsq v
sloΩnug krupnomasßtabnug systemu. Obwye zadaçy kaçestvennoho analyza
hybrydn¥x system udobno rassmatryvat\ v banaxovom prostranstve. Uslovyq
suwestvovanyq reßenyj system, sostoqwyx yz dvux komponent, osnov¥vagtsq
na teoreme Banaxa o nepodvyΩnoj toçke sΩymagweho operatora y yntehral\-
n¥x predstavlenyqx reßenyj uravnenyq syl\no parabolyçeskoho typa. V rabo-
te [1] uslovyq ustojçyvosty hybrydnoj system¥ poluçen¥ na osnove vektornoj
funkcyy Lqpunova y pryncypa sravnenyq. V rabotax [2, 3] nameçen obwyj pod-
xod prymenenyq matryçnoznaçnoho funkcyonala Lqpunova pry yssledovanyy
ustojçyvosty po Lqpunovu y praktyçeskoj ustojçyvosty hybrydnoj system¥.
Osnovnoj trudnost\g prymenenyq poluçenn¥x rezul\tatov qvlqetsq otsutst-
vye konstruktyvnoho podxoda k postroenyg vektornoho yly matryçnoznaçnoho
funkcyonala Lqpunova dlq rassmatryvaemoj system¥. Reßenye zadaçy ob
ustojçyvosty uprowaetsq, esly nezavysym¥e podsystem¥ ymegt svojstvo πks-
ponencyal\noj ustojçyvosty, no pry πtom rol\ funkcyj svqzy predpolahaetsq
destabylyzyrugwej. V πtom sluçae dostatoçn¥e uslovyq ustojçyvosty ves\ma
hrub¥e y mohut prymenqt\sq lyß\ kak oryentyrugwye v dannom napravlenyy.
Takym obrazom, postroenye kaçestvennoj teoryy hybrydn¥x system qvlqetsq
odnoj yz aktual\n¥x problem mexanyky y prykladnoj matematyky.
V dannoj stat\e pryveden nov¥j podxod k reßenyg zadaçy ob ustojçyvosty
hybrydnoj system¥, osnovann¥j na konstruktyvnom postroenyy πlementov mat-
ryçnoznaçnoho funkcyonala. PredloΩenn¥j podxod pozvolqet bolee polno
uçyt¥vat\ dynamyçeskye svojstva reßenyj obosoblenn¥x podsystem hybrydnoj
system¥.
2. Postanovka zadaçy. Budem rassmatryvat\ hranyçnug zadaçu dlq syste-
m¥ vyda
∂
∂
u
t
= Lu B x yT+ ( ) ,
dy
dt
= Cy D x u t x dx+ ∫ ( ) ( , )
Ω
, (1)
u ∂Ω = 0,
hde u ∈ R , x ∈ R
n
, y ∈ R
m
, Ω ⊂ R
n
— otkr¥taq ohranyçennaq odnosvqznaq ob-
last\ s hranycej klassa C2+α , B C m∈ +2 α( ),Ω R , D C m∈ +2 α( ),Ω R , C — posto-
qnnaq ( m × m ) -matryca, L — dyfferencyal\noe v¥raΩenye vtoroho porqdka.
Yzvestno, çto suwestvuet nev¥roΩdennaq ( m × m ) -matryca W takaq, çto
© A. A. MARTÁNGK, V. Y. SLÁN|KO, 2008
204 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
OB USTOJÇYVOSTY LYNEJNÁX HYBRYDNÁX MEXANYÇESKYX SYSTEM … 205
systema (1) preobrazovanyem z = Wy pryvodytsq k vydu
∂
∂
u
t
= Lu B x z B x zT T+ +1 1 2 2( ) ( ) ,
dz
dt
1 = C z D x u t x dx1 1 1+ ∫ ( ) ( , )
Ω
,
(2)
dz
dt
2 = C z D x u t x dx2 2 2+ ∫ ( ) ( , )
Ω
,
u ∂Ω = 0,
hde z m
1
1∈R , z m
2
2∈R , m m1 2+ = m, B x1( ) — m1-vektor-funkcyq, B x2( ) —
m2-vektor-funkcyq, a matryc¥ C1 , C2 ymegt svojstva
max ( )�λi C1 < 0,
min ( )�λi C2 ≥ 0,
hde C1 — postoqnnaq ( )m m1 1× -matryca, C2 — postoqnnaq ( )m m2 2× -matry-
ca.
Poskol\ku v dal\nejßem yssleduetsq systema (1), pryvedennaq k vydu (2),
budem opuskat\ çertu nad Di , Bi, i = 1, 2.
Otnosytel\no prav¥x çastej system¥ (1) sdelaem sledugwye dopolnytel\-
n¥e predpoloΩenyq.
PredpoloΩenye/1. Systema (1) takova, çto:
a) operator L ymeet vyd
Lu = ∂
∂
( ) ∂
∂
+ ( ) ∂
∂
+
= =
∑ ∑x
x u
x
x u
x
x u
i
ij
ji j
m
i
ii j
m
α β γ
, ,
( )
1 1
;
b) suwestvugt postoqnn¥e µi , i = 1, 2, 3, 4, 5, takye, çto
µ ξ1
2
1
i
i
m
=
∑ ≤ α ξ ξij i j
i
m
x( )
=
∑
1
≤ µ ξ2
2
1
i
i
m
=
∑ , µ1, µ2 > 0,
βi
i
m
x2
1
( )
=
∑ ≤ µ3,
µ4 ≤ γ ( x ) ≤ µ5 , x ∈Ω .
Zametym, çto pry v¥polnenyy uslovyq b) nezavysymaq podsystema
∂
∂
u
t
=
= Lu qvlqetsq syl\no parabolyçeskoj (sm. [4]).
Zadaçy ob ustojçyvosty nulevoho reßenyq u = 0, y = 0 budem rassmat-
ryvat\ otnosytel\no dvux mer
ρ0 = ρ = u dx z z2
1
2
2
2
1 2
Ω
∫ + +
/
.
Pryvedem sootvetstvugwye opredelenyq, prynqv vo vnymanye rezul\tat¥
stat\y [5].
Opredelenye/1. Reßenye u = 0, ( ),z zT T T
1 2 = 0 system¥ (2) naz¥vaetsq:
1) ustojçyv¥m po meram ( ρ0 , ρ ) , esly dlq lgb¥x napered zadann¥x t0N∈
∈ R + y ε > 0 suwestvuet δ = δ ( t0 , ε ) takoe, çto dlq vsex naçal\n¥x uslo-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
206 A. A. MARTÁNGK, V. Y. SLÁN|KO
vyj u ( t0 , x ) = ϕ ( x ) , x ∈ Ω , z0 ∈ R
m, udovletvorqgwyx neravenstvu ρ0 < δ ,
v lgboj moment vremeny t > t0 ymeet mesto neravenstvo ρ ( t, z, u ) < ε ;
2) asymptotyçesky ustojçyv¥m po meram ( ρ0 , ρ) , esly ono ustojçyvo po
meram ( ρ0 , ρ ) y ρ ( t, z, u ) → 0 pry t → ∞ ;
3) πksponencyal\no ustojçyv¥m po meram ( ρ0 , ρ) , esly suwestvugt polo-
Ωytel\n¥e postoqnn¥e λ, M takye, çto
ρ ( t, z, u ) ≤ M t z e t tρ ϕ λ
0 0 0
0( , , ) ( )− − , t ≥ t0 .
3. Dostatoçn¥e uslovyq ustojçyvosty. Yssledovanye ustojçyvosty nu-
levoho reßenyq system¥ (2) budem provodyt\ s pomow\g matryçnoznaçnoho
funkcyonala U ( u, z ) =
vij i j
( )
,
⋅[ ] =1
3
. Proyzvodnaq πtoho funkcyonala v¥çyslq-
etsq pokomponentno. Dyahonal\n¥e komponent¥ matryçnoho funkcyonala es-
testvenno prynqt\ v vyde
v11( )u = u dx2
Ω
∫ , v22 1( )z = z PzT
1 1, v33 2( )z = z zT
2 2 , (3)
hde P — poloΩytel\no opredelennaq symmetryçnaq matryca, udovletvorqg-
waq matryçnomu uravnenyg Lqpunova
C P PCT
1 1+ = – Q,
a Q — poloΩytel\no opredelennaq symmetryçnaq matryca.
Vnedyahonal\n¥e πlement¥ matryçnoznaçnoho funkcyonala U u z z( , , )1 2 bu-
dem yskat\ v vyde
v12 1( , )u z = z P x u t x dxT
1 12( ) ( , )
Ω
∫ , v13 2( , )u z = z P x u t x dxT
2 13( ) ( , )
Ω
∫ ,
(4)
v23 1 2( , )z z = z P zT
1 23 2,
hde P x C m
12
2 1( ) ,( )∈ Ω R , P x C m
13
2 2( ) ,( )∈ Ω R , P23 — postoqnnaq matryca.
Dlq η > 0 opredelym skalqrn¥j funkcyonal v( , , , )u z z1 2 η = η ηTU u z z( , , )1 2 ,
dlq kotoroho spravedlyvo sledugwee utverΩdenye.
PredloΩenye/1. Pust\ πlement¥ vij t( , )⋅ funkcyonala U u z( , ) v¥çysle-
n¥ sohlasno (3), (4). Tohda dlq skalqrnoho funkcyonala v( , , )u y η spravedlyva
dvustoronnqq ocenka
λ ρm
TH CH( ) ≤ v( , , , )u z z1 2 η ≤ λ ρM
TH C H( ) ,
hde
C =
1
1
12 13
12 23
13 23
2 2
2
2
− −
− −
− −
P x P x
P x P P
P x P
L L
L m
L
( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
Ω Ω
Ω
Ω
λ ,
C =
1
1
12 13
12 23
13 23
2 2
2
2
P x P x
P x P P
P x P
L L
L M
L
( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
Ω Ω
Ω
Ω
λ
,
H = diag( , , )η η η1 2 3 , P x L( ) ( )2 Ω =
i
n
iP x dx
=
∑∫
1
2
1 2
( )
/
Ω
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
OB USTOJÇYVOSTY LYNEJNÁX HYBRYDNÁX MEXANYÇESKYX SYSTEM … 207
Dokazatel\stvo. Yspol\zovav neravenstvo Koßy – Bunqkovskoho, ocenym
vnedyahonal\ne πlement¥ v1j ju z( , ) , j = 1, 2, matryçnoznaçnoho funkcyonala
–
i
n
i jP x dx u x dx
=
∑∫ ∫
1
1
2
1 2
2
1 2
( ) ( )
/ /
Ω Ω
≤ v1j ju z( , ) ≤
≤
i
n
i jP x dx u x dx
=
∑∫ ∫
1
1
2
1 2
2
1 2
( ) ( )
/ /
Ω Ω
, j = 1, 2.
Yz πtoj ocenky y ocenok dlq v2 1j u z( , ), j = 2, 3, xaraktern¥x dlq kvadratyç-
n¥x form, utverΩdenye sleduet oçevydn¥m obrazom.
Dlq reßenyq zadaçy postroenyq vnedyahonal\n¥x πlementov vij ( )⋅ , i, j = 1,
2, 3, i ≠ j, potrebuetsq sledugwee predpoloΩenye.
PredpoloΩenye/2. Hranyçnaq zadaça dlq system¥ lynejn¥x dyfferency-
al\n¥x uravnenyj v çastn¥x proyzvodn¥x
M P x C P x PD x B x P D xT
12 1 12
2
1
1
1
2
1
3
1
23 2( ) ( ) ( ) ( ) ( )+ + + +η
η
η
η
η
η
= 0,
M P x C P x D x B x P D xT T
13 2 13
3
1
2
1
3
2
2
1
23 1( ) ( ) ( ) ( ) ( )+ + + +η
η
η
η
η
η
= 0, x ∈ Ω ,
(5)
( ) ( ) ( ) ( ) ( )C P P C P x B x dx B x P x dxT T T
1 23 23 2 2 3 1 2 12 2 1 3 1 13+ + +∫ ∫η η η η η η
Ω Ω
= 0,
P12 ∂Ω = 0,
P13 ∂Ω = 0,
hde M — formal\no soprqΩennoe dyfferencyal\noe v¥raΩenye k L, ymeet
reßenye P x12( ), P x C13
2( ) ( )∈ Ω takoe, çto
i
n
jiP x dx
=
∑∫
1
1
2 ( )
Ω
< ∞ , j = 2, 3.
Uslovyq ustojçyvosty nulevoho reßenyq u = 0, ( ),z zT T T
1 2 = 0 system¥ (1)
soderΩatsq v sledugwem utverΩdenyy.
Teorema/1. PredpoloΩym, çto lynejnaq hybrydnaq systema (1) takova,
çto suwestvugt reßenye P x C n( ) ( , )∈ 2 Ω R , udovletvorqgwee uslovyqm
predpoloΩenyqN 2, vektor η ∈ +R
2
y postoqnn¥e ν > 0, c1 > 0 takye, çto
v¥polnqetsq neravenstvo
2 21
2
1 2 1 12η η η
Ω Ω Ω
∫ ∫ ∫+u udx D x udx P x udxTL ( ) ( ) +
+ 2 1 3 2 13η η
Ω Ω
∫ ∫D x udx P x udxT ( ) ( ) ≤ – c u
L2
2
( )Ω
y matryc¥
η η η1 2 12 1 1 12 2
2
Ω
∫ +( ) −P x B x B x P x dx QT T( ) ( ) ( ) ( ) ,
η η η1 3 13 2 2 13 3
2
2 2
Ω
∫ +( ) + +P x B x B x P x dx C CT T T( ) ( ) ( ) ( ) ( )
otrycatel\no poluopredelenn¥e (otrycatel\no opredelenn¥e).
Tohda sostoqnye ravnovesyq u = 0, ( ),z zT T T
1 2 = 0 ustojçyvo (πksponen-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
208 A. A. MARTÁNGK, V. Y. SLÁN|KO
cyal\no ustojçyvo) po meram ( ρ0 , ρ ) .
Dokazatel\stvo. V sylu predloΩenyqN1 skalqrn¥j funkcyonal v( , , )u z η
poloΩytel\no opredelenn¥j po mere ρ y dopuskaet beskoneçno mal¥j v¥sßyj
predel po mere ρ0 . Ocenym polnug proyzvodnug πtoho funkcyonala vdol\ re-
ßenyj system¥ (2):
d u z z
dt
v( , , , )
( )
1 2
2
η
=
η1
2
1 1 2 22
Ω
∫ + +( )u u B x z B x z dxT TL ( ) ( ) +
+ 2 1 2 1 1 1 12η η C z D x udx P x udx
T
+
∫ ∫
Ω Ω
( ) ( ) +
+
2 1 2 1 12 1 1 2 2η η z P x u B x z B x z dxT T T
Ω
∫ + +( )( ) ( ) ( )L + 2 2
2
1 1 1 1η z P C z D x u dxT +
∫
Ω
( ) +
+ 2 23
2
2 2 2 2 2 3 1 1 1 23 2η η ηz C z D x udx C z D x udx P zT
T
+
+ +
∫ ∫
Ω Ω
( ) ( ) +
+ 2 2 3 1 23 2 2 2η η z P C z D x u dxT +
∫
Ω
( ) + 2 1 3 2 2 2 13η η C z D x udx P x udx
T
+
∫ ∫
Ω Ω
( ) ( ) +
+
2 1 3 2 13 1 1 2 2η η z P x u B x z B x z dxT T T
Ω
∫ + +( )( ) ( ) ( )L =
2 1
2η
Ω
∫ u u dxL +
+ 2 21 2 1 12 1 3 2 13η η η η
Ω Ω Ω Ω
∫ ∫ ∫ ∫+D x udx P x udx D x udx P x udxT T( ) ( ) ( ) ( ) +
+ z P x B x B x P x dx Q zT T T
1 1 2 12 1 1 12 2
2
1η η η
Ω
∫ +( ) −
( ) ( ) ( ) ( ) +
+ z P x B x B x P x dx C C zT T T T
2 1 3 13 2 2 13 3
2
2 2 2η η η
Ω
∫ +( ) + +
( ) ( ) ( ) ( ) ( ) +
+ 2 1 1 2 12 1 12 2
2
1 1
2
1 3 2 23 2z MP x C P x PD x B x P D x u dxT T
Ω
∫ +( ) + + +( )η η η η η η( ) ( ) ( ) ( ) ( ) +
+ 2 2 1 3 13 2 13 3
2
2 1
2
2 2 3 23 1z P x C P x D x B x P D x u dxT T T
Ω
∫ +( ) + + +( )η η η η η ηM ( ) ( ) ( ) ( ) ( ) +
+ 2 1 1 23 23 2 2 3 1 2 12 2 1 3 1 13 2z C P P C P x B x dx B x P x dx zT T T T+( ) + +
∫ ∫η η η η η η
Ω Ω
( ) ( ) ( ) ( ) ≤
≤ − + +( ) −
∫c u P x B x B x P x dx Q z
L M
T T
2
2
1 2 12 1 1 12 2
2
1
2
( )
( ) ( ) ( ) ( )
Ω
Ω
λ η η η +
+ λ η η ηM
T T TP x B x B x P x dx C C z1 3 13 2 2 13 3
2
2 2 2
2
Ω
∫ +( ) + +
( ) ( ) ( ) ( ) ( ) .
Otmetym, çto pry ocenke polnoj proyzvodnoj funkcyonala v( , , )u y η pry-
menena formula yntehryrovanyq po çastqm
Ω
∫ P x udx( )L =
Ω
∫ M P x udx( ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
OB USTOJÇYVOSTY LYNEJNÁX HYBRYDNÁX MEXANYÇESKYX SYSTEM … 209
DokaΩem teper\ ustojçyvost\ sostoqnyq ravnovesyq u = 0, ( ),z zT T T
1 2 = 0 sys-
tem¥ (1) po meram ( ρ0 , ρ ) . Dlq zadann¥x t0 ≥ 0 y ε > 0 v¥berem δ ( ε ) =
=
λ
λ
εm
T
M
T
H CH
H CH
( )
( )
. Pust\ u ( t0 , x ) = ϕ ( x ) , ( )( ), ( )z t z tT T T
1 0 2 0 = z0 y
ρ0 =
Ω
∫ + +
ϕ2
1
2
2
2
1 2
( )
/
x dx z z < δ ( ε ) .
Tohda
ρ λ( ) ( )t H CHm
T ≤ v( ( ), ( ), ( ), )u t z t z t1 2 η ≤
≤ v( , , )ϕ ηz0 ≤ ρ λ0 M
TH CH( ) < λ εm
TH CH( ) ,
poπtomu ρ ( t, z, u ) < ε, çto dokaz¥vaet ustojçyvost\ po meram ( ρ0 , ρ ) .
DokaΩem πksponencyal\nug ustojçyvost\ sostoqnyq ravnovesyq u = 0,
( ),z zT T T
1 2 = 0 system¥ (1). V sylu uslovyj teorem¥ y ocenky proyzvodnoj
funkcyonala v( , , , )u z z1 2 η vdol\ reßenyj system¥ (1) dlq funkcyy
γ ( t ) = v u t t z z t t z z t t z( ; , , ), ( ; , , ), ( ; , , )0 0 1 0 0 2 0 0ϕ ϕ ϕ( )
spravedlyva cepoçka neravenstv
d t
dt
γ ( )
≤ – βρ0 ≤ –
β
λ
γ
M
TH CH
t
( )
( ),
hde β — poloΩytel\naq postoqnnaq, otkuda naxodym
ρ ( t ) ≤
λ
λ
ρ β
λ
m
T
M
T
M
T
H CH
H CH H CH
t t
( )
( ) ( )
exp ( )0 0− −
.
Teorema dokazana.
4. O konstruktyvnom reßenyy system¥ (5). Dalee rassmotrym vopros o
reßenyy system¥ yntehro-dyfferencyal\n¥x uravnenyj (5) dlq opredelenyq
vnedyahonal\n¥x πlementov matryçnoznaçnoho funkcyonala Lqpunova. Vvedem
lynejn¥j operator F na prostranstve E ( )m m1 2× -matryc po formule F X =
= C X XCT
1 2+ . Esly λ λi jC C( ) ( )1 2+ ≠ 0 pry vsex i, j = 1, 2, … , m, hde λi( )⋅ —
sobstvenn¥e znaçenyq matryc¥ ( )⋅ , to operator F obratym. Yz posledneho
uravnenyq system¥ (5) naxodym
P23 = –
η
η
η
η
1
3
1
12 2
1
2
1
1 13F F− −∫ ∫−( ) ( )( ) ( ) ( ) ( )P x B x dx B x P x dxT T
Ω Ω
.
Podstavlqq πto v¥raΩenye v pervoe y vtoroe uravnenyq system¥ (5), poluçaem
uravnenyq
MP C P PD x B xT
12 1 12
2
1
1
1
2
1+ + +η
η
η
η
( ) ( ) –
– F F− −∫ ∫−1
12 2 2
3
1
1
1 13 2( ) ( )( ) ( ) ( ) ( ) ( ) ( )P B D x d B P D x dT Tξ ξ ξ η
η
ξ ξ ξ
Ω Ω
= 0,
MP C P D x B xT
13 2 13
3
1
2
1
3
2+ + +η
η
η
η
( ) ( ) –
–
η
η
ξ ξ ξ ξ ξ ξ2
1
1
12 2 1
1
1 13 1F F− −∫ ∫−( ) ( )( ) ( ) ( ) ( ) ( ) ( )P B D x d B P D x dT T T T
Ω Ω
= 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
210 A. A. MARTÁNGK, V. Y. SLÁN|KO
Vvedem oboznaçenyq
P ( x ) = ( )( ), ( )P x P xT T T
12 13 , C ( x ) = diag [ ],C CT T
1 2 ,
G ( x ) = –
η
η
η
η
η
η
η
η
2
1
1
1
2
1
3
1
2
1
3
2PD x B x D x B xT
T
( ) ( ), ( ) ( )+ +
y operator E ( x, ξ ) , dejstvugwyj po pravylu
E ( x, ξ ) P ( ξ ) =
− −
− −
− −
− −
F F
F F
1
12 2 2
1
1 13 2
1
12 2 1
1
1 13 1
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
P B D x B P D x
P B D x B P D x
T T
T T T T
ξ ξ ξ ξ
ξ ξ ξ ξ
.
Tohda systema uravnenyj (5) svodytsq k hranyçnoj zadaçe vyda
M P ( x ) + C P ( x ) = G ( x ) +
E x d( , ) ( )ξ ξ ξP
Ω
∫ , x ∈ Ω ,
(6)
P ( x ) = 0, x ∈ ∂ Ω .
Rassmotrym odnorodnug hranyçnug zadaçu dlq system¥ dyfferencyal\n¥x
uravnenyj v çastn¥x proyzvodn¥x
M P ( x ) + C P ( x ) = 0, x ∈ Ω ,
(7)
P ( x ) = 0, x ∈ ∂ Ω ,
y predpoloΩym, çto dlq zadaçy (7) suwestvuet funkcyq Hryna Γ ( x, ξ ) . Pry
πtom hranyçnaq zadaça dlq uravnenyq (6) svodytsq k yntehral\nomu uravnenyg
P ( x ) =
Γ
ΩΩ
( , ) ( ) ( , ) ( )x E d dξ ξ ξ τ τ τ ξG P+
∫∫ . (8)
Posle nesloΩn¥x preobrazovanyj uravnenyq (8) poluçym yntehral\noe uravne-
nye Fredhol\ma vtoroho roda
P ( x ) =
˜ ( ) ˜ ( , ) ( )G Px x d+ ∫ Γ
Ω
τ τ τ, (9)
hde
˜ ( )G x = Γ
Ω
( , ) ( )x dξ ξ ξG∫ , ˜ ( , )Γ x τ = Γ
Ω
( , ) ( , )x E dξ ξ τ ξ∫ .
PredpoloΩym, çto
˜ ( ) ( )G x L∈ 2 Ω , a qdro
˜ ( , )Γ x τ qvlqetsq qdrom so slaboj
osobennost\g. V πtom sluçae uravnenye (9) pryvodytsq k uravnenyg
P ( x ) = ˜ ( ) ˜ ( , ) ( )G Pr rx x d+ ∫ Γ
Ω
τ τ τ , (10)
hde
˜ ( ) ( )Gr x L∈ 2 Ω , Γ Ω Ωr x L L( , ) ( ) ( )τ ∈ ×2 2 .
Pust\ ψk x( ) , k = 1, 2, … , k, — zamknutaq ortonormyrovannaq systema
funkcyj yz L2( )Ω . Tohda funkcyy ψ ψi jx s( ) ( ), i, j = 1, 2, … , obrazugt zamk-
nutug systemu funkcyj yz L2( )Ω Ω× . Poskol\ku
˜ ( , ) ( )Γ Ω Ωr x s L∈ ×2 , qdro
˜ ( , )Γr x τ moΩno razloΩyt\ v matryçn¥j rqd, sxodqwyjsq v srednekvadratyçnom:
˜ ( , )Γr x s =
i j
ij
r
i jx s
=
∞
=
∞
∑ ∑
1 1
Γ ψ ψ( ) ( ) , (11)
hde Γij
r
— postoqnn¥e matryc¥. Uravnenye zamknutosty dlq πtoho rqda ymeet
vyd
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
OB USTOJÇYVOSTY LYNEJNÁX HYBRYDNÁX MEXANYÇESKYX SYSTEM … 211
Ω Ω
Γ Γ∫ ∫ r
T
rx s x s dxds( , ) ( , ) =
i j
ij
r T
ij
r
=
∞
=
∞
∑ ∑
1 1
( )Γ Γ . (12)
Dalee dlq matryc budem yspol\zovat\ normu A E = tr ( )A AT
. Tohda yz
ravenstva (12) sleduet
Ω Ω
Γ∫ ∫ r Ex s dxds( , ) 2 =
i j
ij
r
E
=
∞
=
∞
∑ ∑
1 1
2
Γ . (13)
Poskol\ku rqd v pravoj çasty (13) sxodytsq, suwestvuet natural\noe çyslo
N takoe, çto
Ω Ω
Γ Γ∫ ∫ ∑ ∑−
= =
r
i
N
j
N
ij
r
i j
E
x s x s dxds( , ) ( ) ( )
1 1
2
ψ ψ <
1
4
. (14)
Oboznaçym
˜ ′Γr =
i
N
j
N
ij
r
i jx s
= =
∑ ∑
1 1
Γ ψ ψ( ) ( ) , ˜ ′′Γr = ˜ ˜Γ Γr r− ′ .
Tohda yntehral\noe uravnenye (10) moΩno pryvesty k vydu
P P( ) ˜ ( , ) ( )x x dr− ′′∫
Ω
Γ τ τ τ = ˜ ( ) ˜ ( , ) ( )G Pr rx x d+ ′∫
Ω
Γ τ τ τ .
Esly
˜ ′′Γrm — m -yteryrovannoe qdro, to
P ( x ) =
˜ ˜( ) ˜ ( , ) ( ) ˜ ( , ) ( ) ˜ ( , ) ( )G P G Pr r
m
rm r rx x d x s s s d ds+ ′ + ′′ + ′
∫ ∑ ∫ ∫
=
∞
Ω Ω Ω
Γ Γ Γτ τ τ τ τ τ
0
.
(15)
V dal\nejßem takΩe celesoobrazno rassmatryvat\ „ukoroçennoe” yntehral\noe
uravnenye
PM ( x ) =
˜ ˜( ) ˜ ( , ) ( ) ˜ ( , ) ( ) ˜ ( , ) ( )G P G Pr r
m
M
rm r r Mx x d x s s s d ds+ ′ + ′′ + ′
∫ ∑ ∫ ∫
=Ω Ω Ω
Γ Γ Γτ τ τ τ τ τ
0
.
(16)
Otmetym, çto uravnenyq (15) y (16) qvlqgtsq uravnenyqmy s v¥roΩdenn¥my
qdramy. Dalee opyßem proceduru reßenyq πtyx uravnenyj y ukaΩem ocenku
pohreßnosty
P P− M L2
2
( )Ω =
Ω
∫ −P P( ) ( )x x dxM
2 .
Uravnenye (15) perepyßem tak:
P ( x ) =
˜ ˜( ) ˜ ( , ) ( )G Gr
m
rm rx x s s ds+ ′′
=
∞
∑ ∫
0 Ω
Γ +
+
Ω Ω
Γ Γ Γ∫ ∑ ∑ ∑ ∫
= = =
∞
+ ′′
i
N
j
N
ij
r
i
m
rm ij
r
i jx x s ds x d
1 1 0
ψ ψ ψ τ τ τ( ) ˜ ( , ) ( ) ( ) ( )P ,
a uravnenye (16) — v vyde
PM ( x ) = ˜ ˜( ) ˜ ( , ) ( )G Gr
m
M
rm rx x s s ds+ ′′
=
∑ ∫
0 Ω
Γ +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
212 A. A. MARTÁNGK, V. Y. SLÁN|KO
+
Ω Ω
Γ Γ Γ∫ ∑ ∑ ∑ ∫
= = =
+ ′′
j
N
i
N
ij
r
i
m
M
rm ij
r
i j Mx x s ds x d
1 1 0
ψ ψ ψ τ τ τ( ) ˜ ( , ) ( ) ( ) ( )P .
Vvedem oboznaçenyq
Xk =
Ω
∫ ψ τ τ τk d( ) ( )P ,
Akj =
Ω Ω
Γ Γ∫ ∑ ∑ ∫
= =
∞
+ ′′
i
N
m
rm ij
r
i kI x s ds x x dx
1 0
˜ ( , ) ( ) ( )ψ ψ ,
Bk =
Ω Ω
Γ∫ ∑ ∫+ ′′
=
∞
ψk r
m
rm rx x x s s ds dx( ) ( ) ˜ ( , ) ( )˜ ˜G G
0
,
Xk
M( ) =
Ω
∫ ψ τ τ τk M d( ) ( )P ,
Akj
M( ) =
Ω Ω
Γ Γ∫ ∑ ∑ ∫
= =
+ ′′
i
N
m
M
rm ij
r
i kI x s ds x x dx
1 0
˜ ( , ) ( ) ( )ψ ψ ,
Bk
M( ) =
Ω Ω
Γ∫ ∑ ∫+ ′′
=
∞
ψk r
m
rm rx x x s s ds dx( ) ( ) ˜ ( , ) ( )˜ ˜G G
0
.
Yntehral\n¥e uravnenyq (15) y (16) πkvyvalentn¥ systemam lynejn¥x al-
hebrayçeskyx uravnenyj
Xk =
j
N
kj j kX
=
∑ +
1
A B ,
(17)
Xk
M( ) =
j
N
kj
M
j
M
k
MX
=
∑ +
1
A B( ) ( ) ( ).
Oboznaçym
X = X XT
N
T
1 , ,…[ ], X M( ) = ( ) ( )( ) ( ), ,X XM T
N
M T T
1 …[ ] ,
A = A ij i j
N[ ] =, 1
, A ( )M = A ij
M
i j
N( )
,[ ] =1
,
B =
B B1
T
N
T T
, ,…[ ] , B ( )M =
( ) ( )( ) ( ), ,B B1
T M
N
T M T
…[ ] .
Dal\nejßej cel\g qvlqetsq ustanovlenye ocenky pohreßnostej εk =
= X Xk k
M− ( ) . Oçevydno, çto
X Xk k
M− ( ) =
i
N
ij ij
M
j
M
i
N
ij j j
M
k k
MX X X
= =
∑ ∑− + − + −
1 1
( ) ( )( ) ( ) ( ) ( )A A A B B .
Otsgda sleduet
( )( )( )I − −A X X M = ( )( ) ( ) ( )A A B B− + −M M MX .
Poskol\ku matryca ( )I − A obratyma, to
εk ≤ X X M− ( ) ≤
( ) ( ) ( ) ( )I − − + −( )−A A A B B1
E
M
E
M M
E
X .
Ocenym norm¥ A A− ( )M
E
y B B− ( )M
E
. Ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
OB USTOJÇYVOSTY LYNEJNÁX HYBRYDNÁX MEXANYÇESKYX SYSTEM … 213
A Akj kj
M
E
− ( ) =
i
N
m M
rm ij
r
i k
E
x s x x dx ds
= = +
∞
∑ ∑ ∫ ∫ ′′
1 1Ω Ω
Γ Γ˜ ( , ) ( ) ( )ψ ψ ≤
≤
i
N
m M
rm E ij
r
E i kx s x x dx ds
= = +
∞
∑ ∑ ∫ ∫ ′′
1 1Ω Ω
Γ Γ˜ ( , ) ( ) ( )ψ ψ .
Oboznaçym γ ki = ψ ψi jx x dx2 2( ) ( )
Ω∫ y, yspol\zovav neravenstvo Koßy – Bunq-
kovskoho, poluçym
A Akj kj
M
E
− ( ) ≤
i
N
m M
rm L kj ij
r
E
= = +
∞
×∑ ∑ ′′
1 1
1 2
2
˜ ( )
( )
/Γ Ω Γ
Ω Ω
γ µ ,
hde
˜
( )
′′
×
Γ
Ω Ωrm L2
2
=
Ω Ω
Γ∫ ∫ ′′˜ ( , )rm E
x s dx ds
2
≤
1
22m ,
µ ( )Ω — lebehova mera mnoΩestva Ω . Otsgda sleduet ocenka
A A− ( )M
E
2
=
k
N
j
N
kj kj
M
E
= =
∑ ∑ −
1 1
2
A A ( ) ≤
≤
k
N
j
N
i
N
M kj ij
r
E
= = =
∑ ∑ ∑
1 1 1
1 2
2
1
2
γ µ / ( )Ω Γ =
µ γ( )Ω Γ
22
1 1 1
2
M
k
N
j
N
i
N
kj ij
r
E
= = =
∑ ∑ ∑
.
Analohyçno ustanavlyvaetsq ocenka
B B− ( )M
E
2
≤
N
M r L
2
2
2
2 2
˜
( )
G
Ω
.
Takym obrazom,
εk ≤
( ) ˜( )
/
( )
( )
I −
+
−
= = =
∑ ∑ ∑A G1
1 1 1
2 1 2
2
2
E
k
N
j
N
i
N
kj ij
r
E
M
E r L
M
X Nµ γΩ Γ
Ω
.
Ocenym normu
( )I − −A 1
, yspol\zovav predstavlenye
I − A = I − + −A A A( ) ( )M M = ( )( ( ) ( ))( ) ( ) ( )I I I− − − −−A A A AM M M1 .
Poskol\ku norma matryc¥ A A− ( )M
mala pry bol\ßyx M , to
( ( ) ( ))( ) ( )I I− − −− −A A AM M1 1 = I +
+ ( ) ( ) ( ) ( )( ) ( ) ( ) ( )I I− − + − −− −A A A A A AM M M M1 1 2 + … .
Poπtomu spravedlyva ocenka
( )I − −A 1
E
≤
( )
( )
( )
( ) ( )
I
I
−
− − −
−
−
A
A A A
M
E
M
E
M
E
1
11
.
Okonçatel\no poluçym
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
214 A. A. MARTÁNGK, V. Y. SLÁN|KO
εk ≤
( ) ˜
( )
( )
/
( )
( )
( )
( )
( )
I
I
−
+
−
−
−
= = =
−
= = =
∑ ∑ ∑
∑ ∑ ∑
A G
A
M
E
k
N
j
N
i
N
kj ij
r
E
M
E r L
M
M
E
M
k
N
j
N
i
N
kj ij
X N1
1 1 1
2 1 2
1
1 1 1
2
2 1
2
µ γ
γ µ
Ω Γ
Ω Γ
Ω
rr
E
2 1 2/ .
(18)
Dalee rassmotrym raznost\ P P( ) ( )x xM− y ocenym ee po norme L2( )Ω :
P P( ) ( )x xM− =
m M
rm rx s s ds
= +
∞
∑ ∫ ′′
1 Ω
Γ̃ ( , ) ( )G̃ +
+
i
N
j
N
ij
r
i
m
rm ij
r
i j j
Mx x s ds x X X
= = =
∞
∑ ∑ ∑ ∫+ ′′
−
1 1 0
Γ Γ Γ
Ω
ψ ψ( ) ˜ ( , ) ( ) ( )( ) +
+
i
N
j
N
m M
rm ij
r
i j
Mx s ds x X
= = = +
∞
∑ ∑ ∑ ∫ ′′
1 1 1 Ω
Γ Γ˜ ( , ) ( ) ( )ψ .
Otsgda naxodym
P P( ) ( ) ( )x xM L− 2 Ω ≤ 1
2
1 22
1 1
2
M r L
i
N
j
N
ij
r
E
˜
( )
( )( ( ))G
Ω
Ω Ω Γ+ +
= =
∗∑ ∑ µ µ ε +
+ 1
2 1 1
3
M
i
N
j
N
ij
r
E j
M
E
X
= =
∑ ∑ Γ Ωµ ( ) ( ) ,
hde çerez ε∗
oboznaçena pravaq çast\ neravenstva (18). YzloΩenn¥e ocenky
pozvolqgt sformulyrovat\ uslovyq asymptotyçeskoj ustojçyvosty hybrydnoj
system¥ (1). Vvedem oboznaçenyq
PM x( ) = ( ) ( )( ) ( ),P PM T M T T
12 13( ) ,
P M
23
( ) = –
η
η
η
η
1
3
1
12 2
1
2
1
1 13F F− −∫ ∫−( ) ( )( ) ( )( ) ( ) ( )( ( ))P x B x dx B x P x dxM T M T
Ω Ω
,
∆ = 1
2
1 22
1 1
2
M r L
i
N
j
N
ij
r
E
˜
( )
( )( ( ))G
Ω
Ω Ω Γ+ +
= =
∗∑ ∑ µ µ ε +
+ 1
2 1 1
3
M
i
N
j
N
ij
r
E j
M
E
X
= =
∑ ∑ Γ Ωµ ( ) ( ) ,
∆∗ =
F − +
1 1
3
2
1
2
12 2
E L LB B∆ Ω Ω
η
η
η
η( ) ( ) .
Sledstvye/1. Pust\ systema uravnenyj (1) takova, çto pry dostatoçno
bol\ßom natural\nom M v¥polnqgtsq sledugwye uslovyq:
1) v¥polnqetsq neravenstvo λ λi jC C( ) ( )1 2+ ≠ 0;
2) systema lynejn¥x alhebrayçeskyx uravnenyj ymeet edynstvennoe reße-
nye Xj
M( ), j = 1, … , N ;
3) matryc¥
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
OB USTOJÇYVOSTY LYNEJNÁX HYBRYDNÁX MEXANYÇESKYX SYSTEM … 215
C ∗ =
1
1
12 13
12 23
13 23
2 2
2
2
P P
P P P
P P
M
L
M
L
M
L m
M
E
M
L
M
E
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
Ω Ω
Ω
Ω
∆ ∆
∆ ∆
∆ ∆
+ +
+ +
+ +
∗
∗
λ ,
C∗ =
1
1
12 13
12 23
13 23
2 2
2
2
− − − −
− − − −
− − − −
∗
∗
P P
P P P
P P
M
L
M
L
M
L m
M
E
M
L
M
E
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
Ω Ω
Ω
Ω
∆ ∆
∆ ∆
∆ ∆
λ
poloΩytel\no opredelen¥;
4) suwestvuet poloΩytel\naq postoqnnaq c takaq, çto v¥polnqgtsq
neravenstva
2 21
2
1 2 1 12η η η
Ω Ω Ω
∫ ∫ ∫+u udx D x udx P x udxT ML ( ) ( )( ) +
+ 2 1 3 2 13η η
Ω Ω
∫ ∫D x udx P x udxM( ) ( )( ) +
+ 2 1 2 1 1 3 2
2
2 2 2∆ Ω Ω Ω
η η η ηD D uL L L( ) ( ) ( )
+( ) ≤ – c u
L2
2
( )Ω
;
5) matryc¥
η η η η η1 2 12 1 1 12 1 2 1 2
22 2
Ω
Ω ∆∫ +( ) + −P x B x B x P x dx B I QM T M T
L
( ) ( )
( )( ) ( ) ( ) ( )( ) ,
η η η η η1 3 13 2 2 13 1 3 2 3
2
2 22 2
Ω
Ω ∆∫ +( ) + + +P x B x B x P x dx B I C CM T M T
L
T( ) ( )
( )( ) ( ) ( ) ( ) ( )( )
otrycatel\no poluopredelenn¥e (otrycatel\no opredelenn¥e).
Tohda sostoqnye ravnovesyq u = 0, ( ),z zT T T
1 2 = 0 system¥ (1) ustojçyvo
(πksponencyal\no ustojçyvo) po meram ( ρ0 , ρ ) .
Prymer. Yssleduem ustojçyvost\ sostoqnyq ravnovesyq lynejnoj hybryd-
noj system¥, sostoqwej yz dvux uravnenyj
∂
∂
u
t
= a u
x
b x y t2
2
2
∂
∂
+ ( ) ( ), u ( t0
, x ) = ϕ ( x ) ,
dy
dt
= c y d x u t x dx
l
2
0
+ ∫ ( ) ( , ) , y ( t0 ) = y0 ,
u ( t, 0 ) = u ( t, l ) = 0,
hde y0 ∈ R , ϕ ( x ) ∈ C [ 0, l ] , ϕ ( 0 ) = ϕ ( l ) = 0. Pust\ η1 = η2 = 1. Postroym
matryçnoznaçn¥j funkcyonal U ( u, y ) s πlementamy
v11( )u = u t x dx
l
2
0
( , )∫ , v22( )y = y
2
, v12( , )u y = y P x u t x dx
l
( ) ( , )
0
∫ ,
hde P ( x ) ∈ C
2
[ 0, l ] — reßenye hranyçnoj zadaçy
a
d P
dx
c P x b x d x2
2
2
2+ + +( ) ( ) ( ) = 0,
P ( 0 ) = P ( l ) = 0.
Netrudno pokazat\, çto esly cl / π a ne qvlqetsq cel¥m çyslom, to
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
216 A. A. MARTÁNGK, V. Y. SLÁN|KO
P ( x ) =
sin( ) sin( ( ) )( ( ) ( ))
sin( )
/ /
/
cx a c l s a b s d s ds
a cl a
l
− +∫0
2 –
sin( ( ) )
( ( ) ( ))/c x s a
a
b s d s ds
x −
+∫ 2
0
.
S uçetom yzvestnoho neravenstva Frydryxsa
u t x dx
l
2
0
( , )∫ ≤ l u
x
dx
l2
2
2
0
π
∂
∂
∫
uslovyq πksponencyal\noj ustojçyvosty po meram ( ρ0 , ρ ) prynymagt vyd
–
a
l
d x dx P x dx
l lπ
+
∫ ∫
2
2
0
1 2
2
0
1 2
( ) ( )
/ /
< 0,
c P x b x dx
l
2
0
+ ∫ ( ) ( ) < 0.
PokaΩem, çto v systeme (1) vozmoΩna stabylyzacyq sostoqnyq ravnovesyq pod
vlyqnyem svqzej meΩdu podsystemamy. Pust\ b ( x ) = b0 , d ( x ) = d0 , hde b0 ,
d0 — postoqnn¥e, ( c l ) / a = ( k + 1 / 2 ) π , k — neotrycatel\noe celoe çyslo.
Tohda
P ( x ) =
b d
ac
cx a cx ak0 0 11 1
+ + − −[ ]+( ) sin( ) cos( )/ / .
Uslovyq asymptotyçeskoj ustojçyvosty prynymagt vyd
–
a
l
d b d
ac
l
al
c
kπ
+ + + −
2
0 0 0 2
1 2
2 1
5
( )
/
< 0,
c
b b d
ac
l
a
c
k2 0 0 0 1
2+ + + −
( )
( ) < 0.
∏ty uslovyq v¥polnqgtsq, naprymer, pry sledugwyx znaçenyqx parametrov:
a = c = 1, l = π / 2 , b0 = 1, d0 = – 1,3. V πtom sluçae svqzy meΩdu podsyste-
mamy qvlqgtsq stabylyzyrugwymy.
V zaklgçenye otmetym, çto predloΩenn¥j podxod analyza ustojçyvosty
hybrydn¥x system qvlqetsq razvytyem yssledovanyj [6 – 9], svqzann¥x s razra-
botkoj metoda matryçn¥x funkcyj Lqpunova v teoryy ustojçyvosty dvyΩenyq.
1. Bart¥ßev A. V. Prymenenye vektor-funkcyy Lqpunova dlq yssledovanyq ustojçyvosty
dvuxkomponentn¥x system // Vektor-funkcyy Lqpunova y yx postroenye. – Novosybyrsk:
Nauka, 1980. – S.N237 – 257.
2. Vujyçyç V. A., Mart¥ngk A. A. Nekotor¥e zadaçy mexanyky neavtonomn¥x system. – Bel-
hrad: Mat. yn-t SANU, 1991. – 109Ns.
3. Mart¥ngk A. A. O praktyçeskoj ustojçyvosty hybrydn¥x system // Prykl. mexanyka. –
1989. – 25, # 2. – S.N101 – 107.
4. Lad¥Ωenskaq O. A. Kraev¥e zadaçy matematyçeskoj fyzyky. – M.: Nauka, 1973. – 408 s.
5. Martynyuk A. A. The Lyapunov matrix function and stability of hybrid systems // Nonlinear Anal.
– 1986. – 10, # 12. – P. 1449 – 1457.
6. Sl¥n\ko V. Y. Ob uslovyqx suwestvovanyq reßenyj odnoho klassa lynejn¥x hybrydn¥x
system // Dop. NAN Ukra]ny. – 2006. – # 5. – S.N53 – 58.
7. Sl¥n\ko V. Y. Ob uslovyqx suwestvovanyq slab¥x reßenyj odnoho klassa lynejn¥x hyb-
rydn¥x system // Tam Ωe. – # 9. – S.N56 – 62.
8. Mart¥ngk A. A. K teoryy prqmoho metoda Lqpunova // Dokl. Akademyy nauk. – 2006. – 408,
# 3. – S.N309 – 312.
9. Slyn’ko V. I. Stability condition for linear impulsive systems with delay // Int. Appl. Mech. – 2005.
– 41, # 6. – P. 697 – 704.
Poluçeno 13.11.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
|
| id | umjimathkievua-article-3150 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:37:09Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3c/da5c7f05d61ea25400cbd5ff00d3eb3c.pdf |
| spelling | umjimathkievua-article-31502020-03-18T19:46:54Z On stability of linear hybrid mechanical systems with distributed components Об устойчивости линейных гибридных механических систем с распределенным звеном Martynyuk, A. A. Slyn'ko, V. I. Мартынюк, А. А. Слынько, В. И. Мартынюк, А. А. Слынько, В. И. We present a new approach to the solution of problems of stability of hybrid systems based on the constructive determination of elements of a matrix-valued functional. Наведено новий підхід до розв'язання задачі про стійкість гібридної системи, що грунтується на конструктивній побудові елементів матричнозначного функціонала. Institute of Mathematics, NAS of Ukraine 2008-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3150 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 2 (2008); 204–216 Український математичний журнал; Том 60 № 2 (2008); 204–216 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3150/3048 https://umj.imath.kiev.ua/index.php/umj/article/view/3150/3049 Copyright (c) 2008 Martynyuk A. A.; Slyn'ko V. I. |
| spellingShingle | Martynyuk, A. A. Slyn'ko, V. I. Мартынюк, А. А. Слынько, В. И. Мартынюк, А. А. Слынько, В. И. On stability of linear hybrid mechanical systems with distributed components |
| title | On stability of linear hybrid mechanical systems with distributed components |
| title_alt | Об устойчивости линейных гибридных механических систем с распределенным звеном |
| title_full | On stability of linear hybrid mechanical systems with distributed components |
| title_fullStr | On stability of linear hybrid mechanical systems with distributed components |
| title_full_unstemmed | On stability of linear hybrid mechanical systems with distributed components |
| title_short | On stability of linear hybrid mechanical systems with distributed components |
| title_sort | on stability of linear hybrid mechanical systems with distributed components |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3150 |
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