Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation
We obtain sufficient conditions for the stability of a dynamical system in a semi-Markov medium under the conditions of diffusion approximation by using asymptotic properties of the compensation operator for a semi-Markov process and properties of the Lyapunov function for an averaged system.
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3388 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509468032237568 |
|---|---|
| author | Chabanyuk, Ya. M. Чабанюк, Я. М. |
| author_facet | Chabanyuk, Ya. M. Чабанюк, Я. М. |
| author_sort | Chabanyuk, Ya. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:51Z |
| description | We obtain sufficient conditions for the stability of a dynamical system in a semi-Markov medium under the conditions of diffusion approximation by using asymptotic properties of the compensation operator for a semi-Markov process and properties of the Lyapunov function for an averaged system. |
| first_indexed | 2026-03-24T02:41:35Z |
| format | Article |
| fulltext |
UDK 519.21+62
Q. M. Çabangk (Nac. un-t „L\viv. politexnika”)
STIJKIST| DYNAMIÇNO} SYSTEMY
Z NAPIVMARKOVS|KYMY PEREMYKANNQMY
V UMOVAX DYFUZIJNO} APROKSYMACI}
We obtain sufficient conditions of the stability of a dynamical system in the semi-Markov space under
the conditions of the diffusion approximation by using asymptotic properties of the compensation
operator for the semi-Markov process and properties of the Lyapunov function for an averaged system.
Poluçen¥ dostatoçn¥e uslovyq ustojçyvosty dynamyçeskoj system¥ v polumarkovskoj srede v
uslovyqx dyffuzyonnoj approksymacyy s yspol\zovanyem asymptotyçeskyx svojstv kompensy-
rugweho operatora dlq polumarkovskoho processa, a takΩe svojstv funkcyy Lqpunova dlq us-
rednennoj system¥.
1. Vstup. Vyvçennq stijkosti dynamiçnyx system u vypadkovomu seredovywi po-
çalosq z rozvytku teori] vypadkovyx evolgcij [1, 2]. V umovax dyfuzijno] ap-
roksymaci] dynamiçno] systemy z markovs\kym zburennqm problemu stijkosti
vperße bulo rozv’qzano v roboti [3] z vykorystannqm martynhal\no] xaraktery-
zaci] vidpovidnoho markovs\koho procesu, a takoΩ v robotax V.7S. Korolgka
(dyv., napryklad, [4]).
Stijkist\ dynamiçno] systemy z napivmarkovs\kym peremykannqm v umovax
userednennq ta dyfuzijno] aproksymaci] vyvçalas\ u robotax A.7V. Sviwuka
(dyv. [5]) zvedennqm do markovs\koho procesu. Pry c\omu vykorystovuvalas\
martynhal\na xarakteryzaciq vidpovidnoho markovs\koho procesu z dodatkovog
komponentog linijnoho procesu.
V danij roboti analiz stijkosti dynamiçno] systemy z napivmarkovs\kymy pe-
remykannqmy budemo rozhlqdaty u bil\ß zahal\nij formi i realizuvaty z vy-
korystannqm kompensugçoho operatora dlq napivmarkovs\koho procesu, vvede-
noho v roboti [6]. Asymptotyçne zobraΩennq kompensugçoho operatora, wo po-
budovane v danij roboti, faktyçno zvodyt\ problemu stijkosti systemy z napiv-
markovs\kymy peremykannqmy do analohiçno] problemy z markovs\kymy peremy-
kannqmy.
2. Postanovka zadaçi. Dynamiçna systema v napivmarkovs\komu seredovywi
v umovax dyfuzijno] aproksymaci] zada[t\sq evolgcijnym dyferencial\nym riv-
nqnnqm
du t
dt
C u t x t C u t x t
ε
ε εε ε ε( ) ( ), ( / ) ( ), ( / )= ( ) + ( )−1
1
2
0
2
, (1)
u uε ( )0 0= ,
de ε > 0 — malyj parametr, a uε
( t ) = u t k dk
ε( ), ,=( )1 .
Ívydkosti Ck ( u, x ) = C u x i dki( , ); ,=( )1 , k = 1, 0, u ∈ R
d, x ∈ X, zadovol\nq-
gt\ umovy, wo zabezpeçugt\ isnuvannq hlobal\nyx rozv’qzkiv determinovanyx
system pry koΩnomu ε > 0:
du t
dt
C u t x C u t xx
x x
ε
ε εε( )
( ), ( ),= ( ) + ( )−1
1 0 , x X∈ . (2)
Tut x ( t ) , t ≥ 0, — napivmarkovs\kyj proces (NMP) u standartnomu fazovomu
prostori staniv ( X, � ) , wo porodΩu[t\sq procesom markovs\koho vidnovlennq
x n , τ n , n ≥ 0, qkyj zada[t\sq napivmarkovs\kym qdrom [7]
Q ( t, x, B ) = P ( x, B ) Gx ( t ) ,
de stoxastyçne qdro
© Q. M. ÇABANGK, 2007
1290 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
STIJKIST| DYNAMIÇNO} SYSTEMY Z NAPIVMARKOVS|KYMY … 1291
P ( x, B ) : = P x B x xn n+ ∈ ={ }1 , B ∈� ,
vyznaça[ vkladenyj lancgh Markova (VLM) x n = x ( τn ) v momenty vidnovlennq
τ θn k
k
n
=
=
∑
1
, n ≥ 0, τ0 0= ,
çerez intervaly θ k + 1 = τ k + 1 – τ k
miΩ momentamy vidnovlennq. Pry c\omu θ n
vyznaçagt\sq funkciqmy rozpodilu
G t t x x tx n n x( ) := ≤ ={ } = ≤{ }+P Pθ θ1 .
Napivmarkovs\kyj proces zada[t\sq spivvidnoßennqm
x t x t( ) ( )= ν , t ≥ 0,
de liçyl\nyj proces ν (t ) vyznaça[t\sq formulog
ν τ( ) : max :t n tn= ≤{ }, t ≥ 0 .
NMP x ( t ), t ≥ 0, rozhlqda[mo rehulqrnyj ta rivnomirno erhodyçnyj [8] zi
stacionarnym rozpodilom π ( B ), B ∈ �, qkyj zadovol\nq[ spivvidnoßennq [9]
π ( dx ) = ρ ( dx ) m ( x ) / m,
de
m x G t dtx x( ) E ( )= =
∞
∫θ
0
, G t G tx x( ) ( )= −1 ,
m dx m x
X
= ∫ ρ( ) ( ) ,
a ρ(B) — stacionarnyj rozpodil VLM xn
, n ≥ 0:
ρ ρ( ) ( ) ( , )B dx P x B
X
= ∫ , ρ( )X = 1.
Dlq intensyvnosti çasu perebuvannq vvedemo poznaçennq
q x m x( ) ( )= −1 , q m= −1.
Pry m (x ) = 0 poklademo q ( x ) = ∞.
Dali budemo vykorystovuvaty takoΩ suprovodΩugçyj markovs\kyj proces
x
0 ( t ) , t ≥ 0, wo zada[t\sq heneratorom
Q x q x P x dy y x
X
ϕ ϕ ϕ( ) ( ) ( , ) ( ) ( )= −[ ]∫ , (3)
na test-funkciqx ϕ ( x ) banaxovoho prostoru � ( X ) dijsnoznaçnyx funkcij z
supremum-normog ϕ( )x : = sup ( )
x X
x
∈
ϕ .
Stacionarnyj rozpodil π ( B ) , B ∈ �, porodΩu[ proektor P v � ( X ) , wo
zada[t\sq rivnistg [ 9 ]
Πϕ ϕ( ) ˆ ( )x x= 1 , ˆ : ( ) ( )ϕ π ϕ= ∫ dx x
X
, 1( )X ≡ 1.
Budemo vykorystovuvaty takoΩ potencial\nyj operator (potencial) R 0
[ 9 ],
wo vyznaça[t\sq spivvidnoßennqm QR 0 = R 0
Q = I – P.
Stijkist\ stoxastyçno] systemy ( 1 ) rozhlqda[t\sq v umovax stijkosti usered-
neno] systemy, qka pry umovi balansu
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1292 Q. M. ÇABANGK
π( ) ( , )dx C u x
X
1 0∫ ≡ , u Rd∈ ,
vyznaça[t\sq rozv’qzkom stoxastyçno] systemy [ 10 ]
du t C u t dt d t( ) ( ) ( )= ( ) +0 ζ ,
(4)
d t a u t dt u t dw tζ σ( ) ( ) ( ) ( )= ( ) + ( ) .
Userednennq zdijsng[t\sq za stacionarnym rozpodilom π ( dx ):
C u dx C u x
X
0 0( ) : ( ) ( , )= ∫ π .
Dyfuzijnyj proces ζ ( t ) , t ≥ 0, vyznaça[t\sq vektor-funkci[g zsuvu
a u a u a u( ) ( ) ( )= +1 2 ,
de
a u dx C u x R C u x
X
1 1 0 1( ) ( ) ( , ) ( , )= − ′∫ π ,
(5)
a u q dx x C u x C u x
X
2 1 1
1
2
( ) ( ) ( , ) ( , )= ( ) ′∫ ρ µ ,
ta matryceg dyspersi] σ ( u ) , wo vyznaça[t\sq spivvidnoßennqm
B ( u ) = σ ( u ) σ∗ ( u ),
pry umovi pozytyvno] vyznaçenosti matryci
B ( u ) = B0 ( u ) + B 1 ( u ), (6)
de
B u dx C u x R C u x
X
0 1 0 12( ) ( ) ( , ) ( , )= − ∫ π ,
(7)
B u q dx x C u x
X
1 1
2( ) ( ) ( ) ( , )= ∫ ρ µ .
V (5) ta (7)
µ( ) ( ) ( )x m x m x= −2
22 ,
m x G s dsx
2 2
0
( ) ( )( )=
∞
∫ , a G t G s dsx x
t
( )( ) : ( )2 =
∞
∫ .
ZauvaΩennq 1. Dlq pokaznykovyx funkcij rozpodilu z intensyvnistg
q ( x )7= m
–
1
( x ) ma[mo µ ( x ) = 0. OtΩe, çleny a 2 ( u ) zsuvu ta B 2 ( u ) dyspersi]
xarakteryzugt\ nemarkovist\ peremykagçoho procesu.
ZauvaΩennq 2. Henerator stoxastyçno] systemy (4) vyznaça[t\sq na test-
funkciqx ϕ ( u ) ∈ C
2
( R
d
) spivvidnoßennqm
Lϕ ϕ ϕ( ) ( ) ( ) Tr ( ) ( )u C u u B u u= ′ + ′′[ ]1
2
, (8)
de
C u C u a u( ) ( ) ( )= +0 . (9)
Zadaça polqha[ v tomu, wob za umov zbiΩnosti stoxastyçno] systemy (1) do
useredneno] systemy (4) pry ε → 0 vstanovyty dodatkovi umovy, wo zabezpe-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
STIJKIST| DYNAMIÇNO} SYSTEMY Z NAPIVMARKOVS|KYMY … 1293
çugt\ stijkist\ poçatkovo] systemy (1) pry vsix ε ≤ ε 0 (ε 0 — dostatn\o male
çyslo).
Stijkist\ systemy (1) rozhlqda[t\sq v umovax eksponencial\no] stijkosti
useredneno] systemy [4]
du t
dt
C u t
˜( ) ˜( )= ( )0 .
3. Formulgvannq rezul\tatu.
Teorema. Nexaj dlq useredneno] stoxastyçno] systemy (4), wo vyznaça-
[t\sq heneratorom (8), isnu[ funkciq Lqpunova V ( u ), u ∈ R
d
, dlq qko] vyko-
nugt\sq umova eksponencial\no] stijkosti:
C 1 ) LV u C V u( ) ( )≤ − 0 , C 0
> 0,
a takoΩ nastupni dodatkovi umovy pry k, r, l = 0, 1:
C 2
) C u x V u C V uk ( , ) ( ) ( )′ ≤ 1 , C 1
> 0,
C u x R C u x V u C V uk r( , ) ( , ) ( ) ( )0 2
′[ ]′ ≤ , C 2
> 0,
C u x R C u x C u v V u C V uk r( , ) ( , ) ( , ) ( ) ( )0 1 3
′[ ]′
′
≤ , C 3
> 0;
C3 ) funkci] rozpodilu G x ( t ), t ≥ 0, x ∈ X, zadovol\nqgt\ umovu Kramera
rivnomirno po x ∈ X :
sup ( )
x X
ht
xe G t dt H
∈
∞
∫ ≤ < + ∞
0
, h > 0,
a takoΩ magt\ misce ocinky
0 < ≤ ≤ < + ∞m m x m( ) .
Todi dlq vsix ε ≤ ε 0 rozv’qzok evolgcijnoho rivnqnnq (1) pry vsix poçat-
kovyx umovax u
ε
( 0 ) ≤ u 0 (u 0 — dostatn\o male) [ asymptotyçno stijkym
z imovirnistg 1:
Ρ lim ( )
t
u t
→∞
={ } =ε 0 1.
4. Kompensugçyj operator. Rozßyrenyj proces markovs\koho vidnovlennq
(RPMV) zada[t\sq poslidovnistg
u un n
ε ε ετ= ( ) , x xn n
ε ε ετ= ( ) , τ ε τε
n n= 2
, n ≥ 0 . (10)
Oznaçennq [ 5 ]. Kompensugçyj operator (KO) RPMV ( 1 0) vyznaça[t\sq
spivvidnoßennqm
Lεϕ( , , )u x t =
= ε ϕ τ τ ϕε ε ε ε ε ε−
+ + + = = ={ } −[ ]2
1 1 1q x u x u u x x t u x tn n n n n n( ) , , ( , , )E ( , , ) . (11)
Rozhlqnemo sukupnist\ napivhrup C xt
ε( ) , t ≥ 0 , x X∈ , wo porodΩu[t\sq
suprovodΩugçog systemog (2) ta vyznaça[t\sq heneratorom
C Cε εϕ ϕ( ) ( ) ( , ) ( )x u u x u= ′ , (12)
de
Cε ε( , ) ( , ) ( , )u x C u x C u x= +−1
1 0 ,
a takoΩ operator Cε( )x , wo ma[ vyhlqd
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1294 Q. M. ÇABANGK
C C C Cε
εε ε( ) : ( ) ( ) ( )x x x x= = +1 0 , (13)
skladovi qkoho vyznaçagt\sq za formulamy
C1 1( ) ( ) : ( , ) ( )x u C u x uϕ ϕ= ′ , C0 0( ) ( ) ( , ) ( )x u C u x uϕ ϕ= ′ .
Lema 1. KO (11) dlq RPMV (10) na test-funkciqx ϕ ( u, x ) ma[ vyhlqd
Lε
ε
εϕ ε ϕ ϕ( , ) ( ) ( ) ( ) ( , ) ( , ) ( , )u x q x G ds C x P x dy u y u xx s
X
= −
−
∞
∫∫2
0
2 . (14)
Dovedennq. Oskil\ky
E Eϕ ϕ ϕε ε
θ
ε ε
ε
ε( , ) ( ) ( , ) ( ) ( ) ( , ) ( , )u x C x u x G ds C x P x dy u y
x x s
X
1 1 1
0
2= =
∞
∫ ∫ ,
to z (11) ma[mo (14).
Lema 2. KO (14) na test-funkciqx ϕ ( u, ⋅ ) ∈ C3 ( R
d ) dopuska[ asympto-
tyçni zobraΩennq
Lε εϕ ε ϕ ε θ ϕ( , ) ( , ) ( ) ( , )u x Q u x x u x= +− −2 1
0 =
= ε ϕ ε ϕ θ ϕε− −+ +2 1
1 1Q u x Q x u x x u x( , ) ( ) ( , ) ( ) ( , ) =
= ε ϕ ε ϕ ϕ εθ ϕε− −+ + +2 1
1 2 2Q u x Q x u x Q x u x x u x( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ), (15)
de operator Q vyznaçeno v (3), a operatory Q x1( ) i Q x2( ) vyznaçagt\sq
spivvidnoßennqmy
Q x u x x u x1 1( ) ( , ) ( ) ( , )ϕ ϕ= C P ,
Q x u x x x x u x2 0 2 1
2( ) ( , ) ( ) ( ) ( ) ( , )ϕ µ ϕ= +[ ]C C P ,
µ2
2
2
( )
( )
( )
x
m x
m x
= ,
a zalyßkovi çleny magt\ vyhlqd
θ ϕ ϕε
ε
ε
0 1( ) ( , ) ( ) ( ) ( , )x u x q x x x u x= ( )C G P , (16)
θ ϕ ϕε
ε
ε
1
2
2 0( ) ( , ) ( ) ( ) ( ) ( ) ( , )x u x q x x x x u x= [ ]C G + C P , (17)
θ ϕ ϕε
ε
ε ε
2
3
3
2
22
( ) ( , ) ( ) ( ) ( )
( )
( ) ( , )x u x q x x x
m x
x u x=
C G + C P . (18)
Tut operatory Gk xε( ), k = 1 3, , vyznaçagt\sq rekursi[g
Gk x
k
s
x G s ds C xε
ε
ε( ) ( ) ( )( )=
∞
∫ 2
0
,
de G sx
( )( )1
= G sx( ), a C C C C2 0 1 02ε ε( ) ( ) ( ) ( )x x x x= +[ ].
Dovedennq. Spoçatku vykorysta[mo oçevydne zobraΩennq
L Lε εε ε= +− −2 2
1Q x( ) , (19)
de
L G P1
ε
ε( ) : ( ) ( )x q x x I= −[ ] , (20)
ta v pravij çastyni (20)
Gε ε
ε( ) : ( ) ( )x G ds C xx s
=
∞
∫
0
2 . (21)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
STIJKIST| DYNAMIÇNO} SYSTEMY Z NAPIVMARKOVS|KYMY … 1295
Dali, intehrugçy çastynamy ta vykorystovugçy rivnqnnq dlq napivhrupy
dC x x C x ds
s sε
ε ε
ε
εε2 2
2( ) ( ) ( )= C ,
z uraxuvannqm umovy7 C3 teoremy otrymu[mo dlq (21) take zobraΩennq:
G C Gε
ε εε( ) ( ) ( )x I x x− = 2
1 , (22)
de
G1
0
2
ε
ε
ε( ) ( ) ( )x G s dsC xx s
=
∞
∫ . (23)
Analohiçno z (23) ma[mo
G C G1
2
2
ε ε εε( ) ( ) ( ) ( )x m x I x x= + , (24)
de
G2
2
0
2
ε
ε
ε( ) : ( ) ( )( )x G s ds C xx s
=
∞
∫ , (25)
a takoΩ z (25) znaxodymo
G C G2
2 2
32
ε ε εε( )
( )
( ) ( )x
m x
I x x= + , (26)
de
G3
3
0
2
ε
ε
ε( ) : ( ) ( )( )x G s dsC xx s
=
∞
∫ ,
i
G s G s dsx x
s
( ) ( )( ) : ( )3 2=
∞
∫ .
Ob’[dnugçy (22), (24) ta (26), oderΩu[mo
G C C C Gε
ε ε ε εε ε ε( ) ( ) ( ) ( ) ( ) ( ) ( )x I m x x m x x x x− = + [ ] + [ ]2 4
2
2 6 3
3 . (27)
Teper, vraxovugçy (12) ta (13), ostatoçno z (27) ma[mo
G C C Cε
εε ε ε θ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x I m x x m x x m x x x− = + +[ ] +1
2
0 2 1
2 3
3 , (28)
de zalyßkovyj çlen
θ εε
ε
ε
3 2 1 0 0
2 3
32( ) ( ) ( ) ( ) ( ) ( ) ( )x m x x x x x x= +( ) +C C C C G . (29)
Ob’[dnugçy (19), (20), (28) i (29), otrymu[mo asymptotyçni zobraΩennq (15).
5. Zburena funkciq Lqpunova. Dovedennq teoremy bazu[t\sq na vykorys-
tanni rozv’qzku problemy synhulqrnoho zburennq (RPSZ) [4] dlq kompensugço-
ho operatora, podanoho v asymptotyçnomu zobraΩenni (15).
Vvedemo zburenu funkcig Lqpunova
V u x V u V u x V u xε ε ε( , ) ( ) ( , ) ( , )= + +1
2
2 , (30)
de V u( ) — funkciq Lqpunova dlq hranyçno] dyfuzi] (4).
Lema 3. Na zburenij funkci] Lqpunova (30) KO (14) dopuska[ zobraΩennq
L Lε ε εεθV u x V u x V uL( , ) ( ) ( ) ( )= + ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1296 Q. M. ÇABANGK
de L — henerator hranyçno] dyfuzi] (4), a zalyßkovyj operator θε
L x( ) vyz-
naça[t\sq spivvidnoßennqm
θ θ θ θε ε ε ε
L x V u x V u x V u x x V u x( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( , )= + +0 1 1 2 2P P . (31)
Zburennq funkci] Lqpunova magt\ zobraΩennq
V u x R Q x V u1 0 1( , ) ( ) ( )= − , (32)
V u x R x V u2 0( , ) ˜( ) ( )= L . (33)
Tut
˜( ) : ( )L L Lx x= − , (34)
L P( ) : ( ) ( ) ( )x Q x Q x R Q x= −2 1 0 1 .
Dovedennq lemy 3 [ bezposerednim vysnovkom RPSZ (dyv. [9, s. 52], lema 3.3)
ta asymptotyçnyx zobraΩen\ (15). Pry c\omu hranyçnyj operator L obçyslg-
[t\sq za formulog
LΠ Π Π Π Π= −Q x Q x R Q x2 1 0 1( ) ( ) ( ) ,
a zalyßkovyj operator θε
L x( ) vyznaça[t\sq ob’[dnannqm çleniv pry odnakovyx
stepenqx ε u rozkladi
L P P Pε ε εε ε εθV u x Q Q x Q x x V u( , ) ( ) ( ) ( ) ( )= + + +[ ]− −2 1
1 2 2 +
+ ε εθε− + +[ ]1
1 1 1Q Q x x V u x( ) ( ) ( , )P P + Q x V u x+ ( )[ ]εθε
0 2P ( , ) ,
wo pryvodyt\ do formul (31) – (33).
Vysnovok 1. Hranyçnyj operator, wo vyznaça[ userednenu dyfuzig (4), za-
da[t\sq rivnistg (8), de C ( u ) i B ( u ) obçyslggt\sq za formulamy vidpovidno
(9) i (6).
Vraxovugçy zobraΩennq (32), (33) zburggçyx funkcij Vk ( u, x ), k = 1, 2, ta
vyraz (31) dlq zalyßkovoho operatora θε
L x( ) , otrymu[mo nastupne zobraΩennq
zalyßkovoho operatora na funkciqx Lqpunova V ( u ) :
θ θ θ θε ε ε ε
L x V u x R Q x R x V u( ) ( ) ( ) ( ) ˜( ) ( )= − +[ ]2 1 0 1 0 0P P L .
Na pidstavi vyraziv (16) – (18) dlq zalyßkovyx operatoriv θε
k x( ) , k = 0, 1, 2,
ta zobraΩennq (34) operatora
˜( )L x robymo vysnovok, wo u zalyßkovoho ope-
ratora θε
L x( ) digt\ operatory dyferencigvannq po zminnij u ∈ Rd
ne vywe
tret\oho porqdku. Krim toho, z umov teoremy vyplyva[, wo operatory Gk xε( ),
k = 0, 1, 2, 3, ta potencial R 0 [ obmeΩenymy u prostori funkcij V ( u ) ∈
∈ C3( R
3
) . OtΩe, ma[ misce takyj vysnovok.
Vysnovok 2. V umovax teoremy ma[ misce ocinka
θε
L Lx V u c V u( ) ( ) ( )≤ .
Z ohlqdu na umovu C1 teoremy ta asymptotyçne zobraΩennq (15) moΩemo
sformulgvaty takyj vysnovok.
Vysnovok 3. V umovax C1 – C3 teoremy pry vsix ε ≤ ε0 (ε0 — dostatn\o ma-
le, ε0 ≤ c / cL ) ma[ misce klgçova nerivnist\
Lε εV u x cV u( , ) ( )≤ − , c > 0. (35)
6. Dovedennq teoremy. Zaverßennq dovedennq teoremy realizu[t\sq za
sxemog roboty [10]. Iz zobraΩen\ (32), (33) funkcij zburennq Vk ( u , x ) , k = 1, 2,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
STIJKIST| DYNAMIÇNO} SYSTEMY Z NAPIVMARKOVS|KYMY … 1297
ta umovy C2 teoremy ma[mo nastupnu dvostoronng ocinku dlq zbureno] funkci]
Lqpunova V u xε( , ):
0 1 1< − ≤ ≤ +( ) ( ) ( , ) ( ) ( )ε εεc V u V u x c V u . (36)
Martynhal\na xarakteryzaciq procesu ηε( )t :7= V uε ε ετ( )( , x t( / )ε2 ) [ 6] ta
klgçova nerivnist\ (35) xarakteryzugt\ proces ηε( )t , qk nevid’[mnyj super-
martynhal [10]. OtΩe, isnu[ z imovirnistg odynycq nevid’[mna hranycq vε
:
Ρ lim ( ( ), ( / ))
t
V u t x t v
→∞
=
=ε ε εε2 1.
Pry c\omu vypadkova velyçyna vε
ma[ skinçenne matematyçne spodivannq, os-
kil\ky
EV u t x t V u x c V uε ε εε ε( ), ( / ) ( , ) ( ) ( )2 1( ) ≤ ≤ + .
Vraxovugçy dodatkovu vlastyvist\ funkci] Lqpunova
V u( ) → ∞ , u → ∞ , (37)
robymo vysnovok, wo Ρ vε < ∞{ } = 1. Znovu Ω taky z klgçovo] nerivnosti (35) ta
ocinky (36) ma[mo
Ρ lim ( )
t
V u t
→∞
( ) ={ } =ε 0 1,
tobto
Ρ lim ( )u
n
n
ε τ
→∞
={ } =0 1.
Nasamkinec\, pozytyvnist\ funkci] Lqpunova V( u ) > 0 pry u ≠ 0, vlas-
tyvist\ (37) ta rehulqrnist\ napivmarkovs\koho procesu x ( t ) , t ≥ 0, pryvodqt\
do tverdΩennq teoremy.
Avtor vyslovlg[ podqku akademiku NAN Ukra]ny V.7S.7Korolgku za uvahu
do vykladenoho materialu.
1. Hersh T., Griego R. Random evolution — theory and applications // Univ. New Mexico. Techn.
Repts. – 1969. – 180. – P. 15 – 38.
2. Pinsky M. Random evolution // Springer Lect. Notes Math. – 1975. – 451. – P. 89 – 100.
3. Blankenship G. L., Papanicolaou G. C. Stability and control of stochastic systems with wide band
noise disturbances // SIAM J. Appl. Math. – 1978. – 34. – P. 437 – 476.
4. Korolgk V. S. Stijkist\ stoxastyçnyx system u sxemi dyfuzijno] aproksymaci] // Ukr. mat.
Ωurn. – 1998. – 50, #71. – S. 36 – 47.
5. Swishchuk A. V. Stability of semi-Markov evolutionary stochastic systems in averaging and
diffusion approximation schemes // Asymptotyçnyj analiz vypadkovyx evolgcij. – Ky]v : In-t
matematyky NAN Ukra]ny, 1994. – S. 255 – 269.
6. Svyrydenko M. N. Martynhal\naq xarakteryzacyq predel\n¥x raspredelenyj v prostran-
stve funkcyj bez razr¥vov vtoroho roda // Mat. zametky. – 1998. – 43, #75. – S. 398 – 402.
7. Korolyuk V. S., Swishchuk A. V. Evolution of systems in random media. – CRC Press, 1995. –
352 p.
8. Korolgk V. S., Svywuk A. V. Polumarkovskye sluçajn¥e πvolgcyy. – Kyev : Nauk. dumka,
1992. – 246 s.
9. Korolyuk V. S., Korolyuk V. V. Stochastic models of systems. – Netherland: Kluwer, 1999. –
185 p.
10. Korolgk V. S., Çabangk Q. M. Stijkist\ dynamiçno] systemy z napivmarkovs\kymy peremy-
kannqmy za umov stijkosti useredneno] systemy // Ukr. mat. Ωurn. – 2002. – 54, #72. –
S.71957–7204.
OderΩano 10.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
|
| id | umjimathkievua-article-3388 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:35Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/80/eb8e436b174d7aefcce526869578d380.pdf |
| spelling | umjimathkievua-article-33882020-03-18T19:52:51Z Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation Стійкість динамічної системи з напівмарковськими перемиканнями в умовах дифузійної апроксимації Chabanyuk, Ya. M. Чабанюк, Я. М. We obtain sufficient conditions for the stability of a dynamical system in a semi-Markov medium under the conditions of diffusion approximation by using asymptotic properties of the compensation operator for a semi-Markov process and properties of the Lyapunov function for an averaged system. Получены достаточные условия устойчивости динамической системы в полумарковской среде в условиях диффузионной аппроксимации с использованием асимптотических свойств компенсирующего оператора для полумарковского процесса, а также свойств функции Ляпунова для усредненной системы. Institute of Mathematics, NAS of Ukraine 2007-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3388 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 9 (2007); 1290–1296 Український математичний журнал; Том 59 № 9 (2007); 1290–1296 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3388/3519 https://umj.imath.kiev.ua/index.php/umj/article/view/3388/3520 Copyright (c) 2007 Chabanyuk Ya. M. |
| spellingShingle | Chabanyuk, Ya. M. Чабанюк, Я. М. Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation |
| title | Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation |
| title_alt | Стійкість динамічної системи з напівмарковськими перемиканнями в умовах дифузійної апроксимації |
| title_full | Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation |
| title_fullStr | Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation |
| title_full_unstemmed | Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation |
| title_short | Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation |
| title_sort | stability of a dynamical system with semi-markov switchings under conditions of diffusion approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3388 |
| work_keys_str_mv | AT chabanyukyam stabilityofadynamicalsystemwithsemimarkovswitchingsunderconditionsofdiffusionapproximation AT čabanûkâm stabilityofadynamicalsystemwithsemimarkovswitchingsunderconditionsofdiffusionapproximation AT chabanyukyam stíjkístʹdinamíčnoísistemiznapívmarkovsʹkimiperemikannâmivumovahdifuzíjnoíaproksimacíí AT čabanûkâm stíjkístʹdinamíčnoísistemiznapívmarkovsʹkimiperemikannâmivumovahdifuzíjnoíaproksimacíí |