On equiasymptotic stability of solutions of doubly-periodic impulsive systems
A system of ordinary differential equations with impulse effects at fixed moments of time is considered. This system admits the zero solution. Sufficient conditions of the equiasymptotic stability of the zero solution are obtained.
Збережено в:
| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3246 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509298940968960 |
|---|---|
| author | Ignat'ev, A. O. Игнатьев, А. О. Игнатьев, А. О. |
| author_facet | Ignat'ev, A. O. Игнатьев, А. О. Игнатьев, А. О. |
| author_sort | Ignat'ev, A. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:15Z |
| description | A system of ordinary differential equations with impulse effects at fixed moments of time is considered.
This system admits the zero solution. Sufficient conditions of the equiasymptotic stability of the zero solution are obtained. |
| first_indexed | 2026-03-24T02:38:53Z |
| format | Article |
| fulltext |
UDK 517.925
A. O. Yhnat\ev (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
OB ∏KVYASYMPTOTYÇESKOJ USTOJÇYVOSTY
REÍENYJ DVOQKOPERYODYÇESKYX SYSTEM
S YMPUL|SNÁM VOZDEJSTVYEM
A system of ordinary differential equations with impulse effects at fixed moments of time is considered.
This system admits the zero solution. Sufficient conditions of the equiasymptotic stability of the zero
solution are obtained.
Rozhlqnuto systemu zvyçajnyx dyferencial\nyx rivnqn\ z impul\snym vplyvom u fiksovani mo-
menty çasu, qka dopuska[ nul\ovyj rozv’qzok. Vstanovleno dostatni umovy ekviasymptotyçno]
stijkosti nul\ovoho rozv’qzku.
1. Vvedenye. Pry matematyçeskom opysanyy πvolgcyy real\n¥x processov s
kratkovremenn¥my vozmuwenyqmy vo mnohyx sluçaqx dlytel\nost\g vozmuwe-
nyj udobno prenebreç\ y sçytat\, çto πty vozmuwenyq ymegt „mhnovenn¥j”
xarakter. Takaq ydealyzacyq pryvodyt k neobxodymosty yssledovat\ dynamy-
çeskye system¥ s razr¥vn¥my traektoryqmy yly, ynaçe, dyfferencyal\n¥e
uravnenyq s ympul\sn¥m vozdejstvyem. Sejças teoryq dyfferencyal\n¥x
uravnenyj s ympul\sn¥m vozdejstvyem predstavlqet soboj yntensyvno razvyva-
gweesq napravlenye matematyky, razlyçn¥e aspekt¥ kotoroho yzloΩen¥ v mo-
nohrafyqx [1 – 5]. V poslednye hod¥ opublykovano mnoho prykladn¥x rabot, v
kotor¥x v kaçestve matematyçeskyx modelej yspol\zovan¥ dyfferencyal\n¥e
uravnenyq s ympul\sn¥m vozdejstvyem. Vsledstvye πtoho uvelyçylos\ koly-
çestvo rabot po yssledovanyg razlyçn¥x aspektov teoryy ympul\sn¥x system
[6 – 13]. Nastoqwaq stat\q posvqwena yzuçenyg ustojçyvosty reßenyj system
s ympul\sn¥m vozdejstvyem. Ona qvlqetsq prodolΩenyem y razvytyem rabo-
t¥=[8].
2. Osnovn¥e opredelenyq. Rassmotrym systemu ob¥knovenn¥x dyffe-
rencyal\n¥x uravnenyj s ympul\sn¥m vozdejstvyem
dx
dt
= f t x( , ), t i≠ τ , i = 1, 2,=… , (1)
∆ x t i= τ = J xi( ) , i = 1, 2,=… , (2)
hde t ∈ +R : = 0, ∞[ ) — vremq, i ∈N (N — mnoΩestvo natural\n¥x çysel), τi
— konstant¥, x
n∈R , f : Rn +1 → Rn
, Ji : Rn → Rn
. Uravnenyq (1), (2) opy-
s¥vagt dynamyku system¥, sostoqwej yz dvux çastej: neprer¥vnoj (pry t i≠ τ ),
opys¥vaemoj ob¥knovenn¥my dyfferencyal\n¥my uravnenyqmy, y dyskretnoj
(v moment¥ τi), kohda reßenyq system¥ skaçkoobrazno yzmenqgtsq. Obozna-
çym
B x x x x HH
n
n= ∈ = + …+ ≤{ }R : 1
2 2
,
Gi : = ( , ) : ,–t x t x Bn
i i H∈ < < ∈{ }+R 1
1τ τ , G : =
i
iG
=
∞
1
∪ .
Sformulyruem hypotez¥ H
1
– H
5
, kotor¥m moΩet udovletvorqt\ syste-
ma=(1), (2).
H
1
. Funkcyq f = ( f1, … , fn) : G → Rn
ravnomerno neprer¥vna v R+ × BH ;
f t( , )0 ≡ 0, y suwestvuet konstanta L > 0 takaq, çto f t x( , ) – f t y( , ) ≤
≤ L x y– pry ( , )t x ∈ G, ( , )t y ∈ G, x BH∈ , y BH∈ .
H
2
. Funkcyy Ji : BH → Rn
, i ∈N , neprer¥vn¥ y udovletvorqgt uslovyg
© A. O. YHNAT|EV, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10 1317
1318 A. O. YHNAT|EV
Lypßyca s konstantoj L v BH , Ji( )0 = 0 pry i ∈N .
H
3
. Suwestvuet konstanta h ∈ ( , )0 H takaq, çto esly x Bh∈ , to x +
+ J xi( ) ∈ BH pry i ∈N .
H
4
. Konstant¥ τi udovletvorqgt uslovyqm
0 = τ τ τ0 1 2< < < … , lim
i
i
→∞
τ = ∞.
H
5
. Konstant¥ τi udovletvorqgt uslovyg: dlq lgb¥x T > 0, t > 0 otre-
zok t t T, +[ ] soderΩyt ne bolee p konstant τi , pryçem çyslo p zavysyt
tol\ko ot T y ne zavysyt ot t.
Oboznaçym çerez x t t( , 0 , x0) pry t > t0 reßenye system¥ (1), (2), udovletvo-
rqgwee uslovyg x t t( ,0 0 , x0) = x0 v sluçae, kohda t i0 ≠ τ , i ∈N . Esly Ωe
t i0 = τ pry kakom-lybo natural\nom i, to pod v¥raΩenyem x t t( , 0 , x0) budem
ponymat\ x t t, 0( + 0, x0 + J xi( )0 ) (pry t > t0). Znaçenye πtoho reßenyq v mo-
ment t takΩe oboznaçym çerez x t t( , 0 , x0). ∏to reßenye budem predpolahat\
neprer¥vno dyfferencyruem¥m po t na lgbom yz mnoΩestv Gi y neprer¥v-
n¥m sleva v toçkax razr¥va: x ti( ,τ 0 , x0) = x i(τ – 0, t0, x0).
Pry v¥polnenyy hypotez H
1
– H
3
systema (1), (2) dopuskaet tryvyal\noe
reßenye
x ≡ 0 . (3)
Sformulyruem ponqtyq ustojçyvosty y prytqΩenyq tryvyal\noho (nulevo-
ho) reßenyq system¥ (1), (2).
Opredelenye 1. Tryvyal\noe reßenye system¥ (1), (2) naz¥vaetsq ustoj-
çyv¥m, esly dlq lgb¥x ε > 0, t0 ∈ +R moΩno ukazat\ δ = δ ε( , )t0 > 0 takoe,
çto esly x0 ≤ δ, to x t t x( , , )0 0 ≤ ε pry t > t0.
Opredelenye 2. Reßenye (3) system¥ (1), (2) naz¥vaetsq:
prytqhyvagwym, esly dlq lgboho t0 ∈ +R suwestvuet λ = λ( )t0 > 0 y
dlq lgb¥x ε > 0 y x B0 ∈ λ suwestvuet σ = σ ε( , , )t x0 0 > 0 takoe, çto
x t t x( , , )0 0 ≤ ε dlq vsex t ≥ t0 + σ;
πkvyprytqhyvagwym, esly dlq lgboho t0 ∈ +R najdetsq λ = λ( )t0 > 0
takoe, çto dlq lgboho ε > 0 suwestvuet σ = σ ε( , )t0 > 0 takoe, çto dlq
lgb¥x x B0 ∈ λ , t ≥ t0 + σ v¥polnqetsq neravenstvo x t t x( , , )0 0 ≤ ε;
ravnomerno prytqhyvagwym, esly ymeetsq takoe λ > 0, çto dlq lgboho
ε > 0 najdetsq σ = σ ε( ) > 0 takoe, çto dlq lgb¥x t0 ∈ +R , x B0 ∈ λ , t ≥
≥ t0 + σ spravedlyvo x t t( , 0 , x0) ∈ Bε .
Yn¥my slovamy, reßenye (3) system¥ (1), (2) naz¥vaetsq:
prytqhyvagwym, esly dlq lgb¥x t0 ∈ +R , x B0 ∈ λ spravedlyvo predel\-
noe sootnoßenye
lim ( , , )
t
x t t x
→∞
0 0 = 0; (4)
πkvyprytqhyvagwym, esly sootnoßenye (4) v¥polnqetsq ravnomerno po
x B0 ∈ λ ;
ravnomerno prytqhyvagwym, esly predel\noe sootnoßenye (4) v¥polnqetsq
ravnomerno po x B0 ∈ λ , t0 ∈ +R .
Opredelenye 3. Tryvyal\noe reßenye system¥ (1), (2) naz¥vaetsq:
asymptotyçesky ustojçyv¥m, esly ono ustojçyvo y prytqhyvagwee;
πkvyasymptotyçesky ustojçyv¥m, esly ono ustojçyvo y πkvyprytqhyva-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
OB ∏KVYASYMPTOTYÇESKOJ USTOJÇYVOSTY REÍENYJ … 1319
gwee;
ravnomerno asymptotyçesky ustojçyv¥m, esly ono ravnomerno ustojçyvo
y ravnomerno prytqhyvagwee.
Opredelenye 4. Budem hovoryt\, çto funkcyq ω : R+ → R+ prynadle-
Ωyt klassu K K( )ω ∈ , esly ona neprer¥vna, stroho vozrastaet y ω( )0 = 0.
Vvedem sledugwye opredelenyq.
Opredelenye 5. Budem hovoryt\, çto funkcyq V : R+ × BH → R pry-
nadleΩyt klassu V V0 0( )V ∈ , esly V ravnomerno neprer¥vna na R+ × BH .
Budem hovoryt\, çto funkcyq V ∈V0 prynadleΩyt klassu V V1 1( )V ∈ ,
esly V qvlqetsq neprer¥vno dyfferencyruemoj na R+ × BH y ee proyzvod-
naq v¥çyslqetsq po formule
dV
dx
V
t
= ∂
∂
+
i
n
i
i
V
x
f t x
=
∑
1
∂
∂
( , ) .
Analohyçno rabote [8] vvedem sledugwee opredelenye.
Opredelenye 6. Funkcyq g : R+ → Rm
, m ∈N , naz¥vaetsq fynal\no
nenulevoj, esly dlq lgboho M > 0 suwestvuet t > M takoe, çto g t( ) ≠ 0.
Çyslovaq posledovatel\nost\ uk k{ } =
∞
1 naz¥vaetsq fynal\no nenulevoj,
esly dlq lgboho natural\noho çysla M suwestvuet k > M takoe, çto uk ≠
≠ 0.
3. Osnovn¥e rezul\tat¥. Rassmotrym systemu s ympul\sn¥m vozdejstvy-
em (1), (2) v predpoloΩenyy, çto v¥polnqgtsq hypotez¥ H
1
– H
5
y suwestvugt
ω1 0> , ω2 0> , q ∈N takye, çto
f t x( , )+ ω1 ≡ f t x( , ), τk q+ = ω τ2 + k ,
(5)
J xk q+ ( ) ≡ J xk ( ) ∀ ∈x BH ∀ ∈ +t R ∀ ∈k N .
Esly ω ω1 2/ — racyonal\noe çyslo, to systema (1), (2) qvlqetsq peryody-
çeskoj. Yzuçenyg ustojçyvosty reßenyq (3) system¥ (1), (2) v sluçae ee peryo-
dyçnosty posvqwena stat\q [8]. V nej pokazano, çto asymptotyçesky ustojçy-
v¥e reßenyq peryodyçeskyx system qvlqgtsq ravnomerno asymptotyçesky us-
tojçyv¥my, y poluçen¥ dostatoçn¥e uslovyq yx asymptotyçeskoj ustojçyvos-
ty. V dal\nejßem budem rassmatryvat\ systemu (1), (2) v predpoloΩenyqx (5) y
yrracyonal\nosty çysla ω ω1 2/ . V çastnom sluçae J xi( ) ≡ 0, i ∈N , systema
(1), (2) qvlqetsq peryodyçeskoj s peryodom ω1, a v sluçae f t x( , ) ≡ 0 — peryo-
dyçeskoj s peryodom ω2. V obwem sluçae f t x( , ) � 0, J xi( ) � 0 systemu (1),
(2) budem naz¥vat\ dvoqkoperyodyçeskoj.
Perejdem k yssledovanyg ustojçyvosty nulevoho reßenyq dvoqkoperyo-
dyçeskoj system¥ (1), (2). Vnaçale pryvedem yzvestn¥j rezul\tat Kronekera
[14, s. 9].
Lemma 1. Pust\ ω1 y ω2 — proyzvol\n¥e dejstvytel\n¥e çysla. Kako-
vo b¥ ny b¥lo ε > 0, moΩno ukazat\ L = L( )ε > 0 takoe, çto v kaΩdom yn-
tervale dlyn¥ L najdetsq, po krajnej mere, odno çyslo T, udovletvorqg-
wee systeme neravenstv
T s– ω1 < ε, T m– ω2 < ε,
hde s y m — nekotor¥e cel¥e çysla.
Rassmotrym monotonno stremqwugsq k nulg posledovatel\nost\ poloΩy-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1320 A. O. YHNAT|EV
tel\n¥x çysel εi i{ } =
∞
1. V sootvetstvyy s lemmoj=1 dlq kaΩdoho εi suwestvuet
posledovatel\nost\ çysel Ti k,{ } takaq, çto Ti k, < Ti k, +1 , Ti k, → ∞ pry k →
→ ∞ y v¥polnqgtsq neravenstva Ti k, – si k, ω1 < εi , Ti k, – mi k, ω1 < εi , hde
si k, , mi k, — cel¥e çysla. Ne narußaq obwnosty, budem sçytat\ Ti k, < Ti k+1, ,
i ∈N , k ∈N . Oboznaçym Ti i, = Ti , si i, = si , mi i, = mi y rassmotrym „dyaho-
nal\nug” posledovatel\nost\ Ti{ }. Dlq nee poluçaem
T si i– ω1 < εi , T mi i– ω2 < εi . (6)
V sootvetstvyy s (6) ymeem
Ti = si ω1 + δi1 = mi ω2 + δi2 , hde δi1 < εi , δi2 < εi . (7)
Oboznaçym xk = x t( 0 + mk ω2 , t0 , x0). Pust\ t t∗ > 0 — fyksyrovann¥j mo-
ment vremeny. PokaΩem, çto
lim ( , , ) – ( , , )
k
k kx t t x x t m t x
→∞
∗ ∗ +0 2 0 0ω = 0. (8)
Traektoryy I y II system¥ (1), (2), naçynagwyesq v moment¥ vremeny t0 y t0 +
+ mk ω2 v toçke xk , za vremq ∆ t = t∗ – t0 perexodqt sootvetstvenno v toçky
x t( ∗, t0 , xk ) y x t( ∗ + m t mk kω ω2 0 2, + , xk ) = x t( ∗ + m t xkω2 0 0, , ) . Traektoryg II
system¥ (1), (2), proxodqwug çerez toçku xk pry t = t0 + mk ω2 , moΩno trak-
tovat\ kak traektoryg system¥
dx
dt
= f t m xk( , )+ ω2 , t i≠ τ , (9)
∆ x = J xi m qk+ ( ), t i= τ , (10)
s toj Ωe naçal\noj toçkoj xk y naçal\n¥m momentom vremeny t0 . Uçyt¥vaq,
çto J xi m qk+ ( ) ≡ J xi( ) , x BH∈ , v sylu (7) y ω1-peryodyçnosty po t funkcyy f,
f t m xk( , )+ ω2 ≡ f t s xk k k( – , )+ +ω δ δ1 1 2 ≡
≡ f t xk k( – , )+ δ δ1 2 , x BH∈ , t ∈R,
poluçaem, çto dyskretnaq systema (10) sovpadaet s systemoj (2), a prav¥e çasty
neprer¥vn¥x system (9) y (1) ymegt svojstvo
lim ( – , ) – ( , )
k
k kf t x f t x
→∞
+ δ δ1 2 = 0
ravnomerno po t ∈R, x BH∈ v sylu ravnomernoj neprer¥vnosty funkcyy
f t x( , ). Poskol\ku kolyçestvo toçek ympul\sn¥x vozdejstvyj τi na otrezke
t t0, ∗[ ] koneçno y reßenye system¥ s ympul\sn¥m vozdejstvyem neprer¥vno za-
vysyt ot prav¥x çastej (sm., naprymer, [1], teorema=2.5), otsgda sleduet spra-
vedlyvost\ predel\noho sootnoßenyq (8).
Teorema 1. Pust\ dlq dvoqkoperyodyçeskoj system¥ dyfferencyal\n¥x
uravnenyj s ympul\sn¥m vozdejstvyem (1), (2) suwestvuet funkcyq V t x( , ) =
= V t x1( , ) + V t x2( , ), V1 1∈V , V2 1∈V , takaq, çto
V t x1 1( , )+ ω ≡ V t x1( , ) , V t x2 2( , )+ ω ≡ V t x2( , ), t ∈ +R , x BH∈ , (11)
V t x( , ) ≥ a x( ) , a ∈K , V t( , )0 ≡ 0, (12)
dV
dt
≤ 0 pry ( , )t x G∈ , (13)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
OB ∏KVYASYMPTOTYÇESKOJ USTOJÇYVOSTY REÍENYJ … 1321
V x J xi iτ + +( )0, ( ) – V xi( , )τ ≤ 0, i ∈N , x BH∈ . (14)
Esly vdol\ lgboho fynal\no nenulevoho reßenyq system¥ (1), (2) v¥pol-
nqetsq xotq b¥ odno yz uslovyj:
dV
dt
— fynal\no nenulevaq funkcyq,
V iτ({ + 0, x + J xi( )) – V xi( , )τ } — fynal\no nenulevaq posledovatel\nost\,
to reßenye (3) system¥ (1), (2) πkvyasymptotyçesky ustojçyvo.
Dokazatel\stvo. Ustojçyvost\ nulevoho reßenyq system¥ (1), (2) sledu-
et yz yzvestnoj teorem¥ Hurhul¥ – Perestgka [15]. ∏to znaçyt, çto dlq proyz-
vol\n¥x ε > 0, t0 ∈ +R suwestvuet takoe δ = δ ε( , )t0 > 0, çto dlq lgboho
x B0 ∈ δ v¥polnqetsq neravenstvo x t t x( , , )0 0 ≤ ε. PokaΩem, çto dlq lgboho
x B0 ∈ δ v¥polnqetsq predel\noe sootnoßenye
lim , ( , , )
t
V t x t t x
→∞
( )0 0 = 0.
PredpoloΩym protyvnoe: pust\ suwestvugt x B0 ∈ δ y η > 0 takye, çto
V t x,( ( , , )t t x0 0 ) > η > 0 pry t ≥ t0 . Funkcyq V t x,( ( , , )t t x0 0 ) v sylu uslovyj
(12) – (14) neotrycatel\na y qvlqetsq monotonno nevozrastagwej funkcyej
vremeny, poπtomu suwestvuet predel
lim , ( , , )
t
V t x t t x
→∞
( )0 0 = V0 ≥ η > 0 .
Rassmotrym monotonno stremqwugsq k nulg posledovatel\nost\ poloΩytel\-
n¥x çysel εi i{ } =
∞
1 y sootvetstvugwye posledovatel\nosty natural\n¥x çysel
mi i{ } =
∞
1 y si i{ } =
∞
1, udovletvorqgwye sootnoßenyqm (6) y (7). Oboznaçym xk =
= x t( 0 + m t xkω2 0 0, , ) . Poskol\ku reßenye (3) system¥ (1), (2) ustojçyvo, po-
sledovatel\nost\ xk{ } ohranyçena y ymeet predel\nug toçku x∗
. Ne narußaq
obwnosty, budem sçytat\, çto sama posledovatel\nost\ xk{ } sxodytsq k x∗
. V
sylu ravnomernoj neprer¥vnosty funkcyj V1, V2 y svojstva (11) ymeem
V t xk2 0( , ) = V t m xk k2 0 2( , )+ ω ,
V t xk1 0( , ) = V t s xi k1 0 1( , )+ ω = V t m xi i i k1 0 2 1( – , )+ + δ δ ,
lim ( – , )
i
i i i kV t m x
→∞
+ +1 0 2 2 1ω δ δ = V t xk1 0( , ),
V t x( , )0
∗ = lim ( , )
k
kV t x
→∞
0 = lim ( , ) ( , )
k
k kV t x V t x
→∞
+[ ]1 0 2 0 =
= lim lim ( – , )
k i
i i i kV t m x
→∞ →∞
+ +1 0 2 2 1ω δ δ + lim ( , )
k
k kV t m x
→∞
+2 0 2ω =
= lim ( – , ) ( , )
k
k k k k k kV t m x V t m x
→∞
+ + + +[ ]1 0 2 2 1 2 0 2ω δ δ ω =
= lim ( , ) ( , )
k
k k k kV t m x V t m x
→∞
+ + +[ ]1 0 2 2 0 2ω ω =
= lim ( , )
k
k kV t m x
→∞
+0 2ω = lim , ( , , )
k
k kV t m x t m t x
→∞
+ +( )0 2 0 2 0 0ω ω = V0.
Rassmotrym teper\ traektoryg x t t x( , , )0
∗
pry t0 < t < ∞ . PredpoloΩym,
çto πta traektoryq qvlqetsq fynal\no nenulevoj. V πtom sluçae na nej
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1322 A. O. YHNAT|EV
suwestvugt toçky, hde lybo
˙ , ( , , )V t x t t x( )∗
0 < 0, lybo V iτ( + 0, x J xi+ )( ) –
– V xi( , )τ < 0, t.=e. moΩno ukazat\ moment vremeny t t∗ > 0 , v kotor¥j v¥pol-
nqetsq uslovye
V t x t t x∗ ∗ ∗( ), ( , , )0 = V V1 0
∗ < . (15)
Poskol\ku posledovatel\nost\ xk{ } sxodytsq k toçke x∗
, vsledstvye ne-
prer¥vnoj zavysymosty reßenyj ot naçal\n¥x dann¥x [1, s. 21] moΩno zapysat\
x t t x( , , )∗ ∗
0 = lim ( , , )
k
kx t t x
→∞
∗
0 , otkuda
lim , ( , , )
k
kV t x t t x
→∞
∗ ∗( )0 = V1
∗
. (16)
Uçyt¥vaq, çto pry v¥polnenyy uslovyj teorem¥ spravedlyvo predel\noe soot-
noßenye (8), poluçaem neravenstvo
x t t x x t m t xk k( , , ) ( , , )∗ ∗− +0 2 0 0ω ≤ γ k ,
hde lim
k
k
→∞
γ = 0. V sylu toho, çto V = V1 + V2 y funkcyq V2 peryodyçna po t s
peryodom ω2, ymeem
V t x V t m xk( , ) – ( , )∗ ∗ + ω2 =
= V t x V t m xk1 1 2( , ) – ( , )∗ ∗ + ω < M k( )ε ∀ ∈x BH , (17)
hde M k( )ε → 0 pry k → ∞, a yz uslovyq (16) sleduet sootnoßenye
V t x t m t x Vk
∗ ∗ ∗+( ), ( , , ) –ω2 0 0 1 < ηk , (18)
hde lim
k
k
→∞
η = 0. Yz (17) poluçaem neravenstvo
V t x t m t x V t m x t m t xk k k
∗ ∗ ∗ ∗+( ) + +( ), ( , , ) – , ( , , )ω ω ω2 0 0 2 2 0 0 < M k( )ε , (19)
a yz neravenstv (18), (19) sleduet, çto
V t m x t m t x Vk k
∗ ∗ ∗+ +( )ω ω2 2 0 0 1, ( , , ) – < ηk + M k( )ε , (20)
hde ηk + M k( )ε → 0 pry k → ∞. V to Ωe vremq
lim , ( , , )
k
k kV t m x t m t x
→∞
∗ ∗+ +( )ω ω2 2 0 0 = V0. (21)
Sootnoßenyq Ωe (20) y (21) protyvoreçat neravenstvu V1
∗ < V0, t.=e. predpola-
haq, çto traektoryq x t t x( , , )0
∗
fynal\no nenulevaq, poluçaem protyvoreçye.
V sluçae, kohda x t t x( , , )0
∗
ne qvlqetsq fynal\no nenulevoj, suwestvuet T∗ >
> 0 takoe, çto x t t x( , , )0
∗ ≡ 0 pry t T≥ ∗, otkuda v sylu svojstv funkcyy V
ymeem V t( , x t t x( , , )0
∗ ) = 0 pry t T≥ ∗. ∏to oznaçaet, çto v πtom sluçae takΩe
suwestvuet takoe t t∗ > 0 , çto v¥polnqetsq uslovye (15), y m¥ snova pryxodym
k protyvoreçyg. Poluçennoe protyvoreçye pokaz¥vaet, çto lgboe reßenye s
x0 ≤ δ ymeet svojstvo
lim , ( , , )
k
V t x t t x
→∞
( )0 0 = 0. (22)
PokaΩem, çto predel\noe sootnoßenye (22) v¥polnqetsq ravnomerno po
x B0 ∈ δ . Yz (22) sleduet, çto dlq lgb¥x ε > 0, t0 ∈ +R y x B0 ∈ δ moΩno
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
OB ∏KVYASYMPTOTYÇESKOJ USTOJÇYVOSTY REÍENYJ … 1323
ukazat\ σ ε( , , )t x0 0 > 0, pry kotorom V t0 +( σ , x t t x( , , )0 0 0+ )σ ≤ a( )/ε 2 . V sylu
neprer¥vnosty funkcyy V t x( , ) ot x y neprer¥vnoj zavysymosty reßenyj ot
naçal\n¥x uslovyj, v nekotoroj okrestnosty Q x( )0 toçky x0 v¥polnqetsq
neravenstvo
V t x t t x0 0 0 0+ + ′( )σ σ, ( , , ) ≤ a( )ε pry ′ ∈x Q x0 0( ). (23)
Vsledstvye monotonnoho vozrastanyq funkcyy V t x( , ) vdol\ reßenyj yz
(23) sleduet, çto V t( , x t t x( , , )0 0′ ) ≤ a( )ε pry t ≥ t0 + σ ε( , , )t x0 0 , ′ ∈x Q x0 ( ) .
Kompaktnaq oblast\ Bδ okaz¥vaetsq pokr¥toj systemoj okrestnostej
Q x( )0{ }, yz kotoroj po lemme Hejne – Borelq [16, s. 49] moΩno v¥delyt\ ko-
neçnoe podpokr¥tye Q1, … , Qj s sootvetstvugwymy çyslamy σ1, … , σ j . Po-
loΩym σ ε( , )t0 = max σ1{ , … , σ j}. Tohda
V t x t t x, ( , , )0 0( ) ≤ a( )ε dlq lgb¥x x B0 ∈ δ , t ≥ t0 + σ ε( , )t0 . (24)
Yz ocenok (12) y (24) poluçaem x t t x( , , )0 0 ≤ ε pry x B0 ∈ δ , t ≥ t0 + σ ε( , )t0 , a
πto dokaz¥vaet, çto nulevoe reßenye system¥ (1), (2) πkvyasymptotyçesky
ustojçyvo.
Teorema 2. Pust\ dlq dvoqkoperyodyçeskoj system¥ dyfferencyal\n¥x
uravnenyj s ympul\sn¥m vozdejstvyem (1), (2) suwestvuet funkcyq V t x( , ) =
= V t x1( , ) + V t x2( , ), V1 1∈V , V2 1∈V , udovletvorqgwaq uslovyqm (11),
V t x( , ) ≤ b x( ), b ∈K , (25)
dV
dt
≥ 0 ( , )t x G∈ , (26)
V x J xi iτ + +( )0, ( ) – V xi( , )τ ≥ 0, i ∈N . (27)
Pust\, krome toho, vdol\ lgboho fynal\no nenulevoho reßenyq system¥
uravnenyj (1), (2) v¥polnqetsq xotq b¥ odno yz uslovyj:
dV dt/ — fynal\no nenulevaq funkcyq;
V iτ +({ 0 , x J xi+ )( ) – V xi( , )τ } — fynal\no nenulevaq posledovatel\nost\.
Tohda esly v lgboj skol\ uhodno maloj okrestnosty naçala koordynat pry
lgbom t > 0 najdetsq toçka x takaq, çto V t x( , ) > 0, to reßenye (3)
system¥ (1), (2) neustojçyvo.
Dokazatel\stvo. Pust\ ε < H — nekotoroe poloΩytel\noe çyslo. V¥be-
rem proyzvol\noe t0 ∈ +R y skol\ uhodno maloe δ > 0. PokaΩem, çto suwest-
vugt x B0 ∈ δ y t > t0 takye, çto x t t x( , , )0 0 > ε. Dlq πtoho v¥berem x B0 ∈ δ
tak, çto V t x( , )0 0 = V0 > 0. PredpoloΩym protyvnoe:
x t t x( , , )0 0 ≤ ε (28)
pry vsex t > t0. Yz uslovyq (25) poluçaem
V t x( , ) < V0 pry x < b V–1
0( ) = η, t ∈ +R ,
hde b–1
— funkcyq, obratnaq k funkcyy b. Uçyt¥vaq predpoloΩenyq (26) –
(28), zaklgçaem, çto polutraektoryq x t( ) = x t t x( , , )0 0 udovletvorqet uslo-
vyqm
η ≤ x t t x( , , )0 0 ≤ ε.
Rassmotrym monotonno stremqwugsq k nulg posledovatel\nost\ poloΩy-
tel\n¥x çysel εi i{ } =
∞
1, sootvetstvugwye posledovatel\nosty natural\n¥x çy-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1324 A. O. YHNAT|EV
sel mi i{ } =
∞
1 y si i{ } =
∞
1, udovletvorqgwye sootnoßenyqm (6), (7), a takΩe po-
sledovatel\nost\ toçek x j{ } , hde x j = x t( 0 + mjω2, t0 , x0), j = 1, 2, … . Uçy-
t¥vaq, çto πta posledovatel\nost\ raspoloΩena v kompaktnom mnoΩestve, yz
nee moΩno v¥delyt\ sxodqwugsq k toçke x∗ podposledovatel\nost\, pryçem
x∗ udovletvorqet uslovyg η ≤ x∗ ≤ ε. Ne narußaq obwnosty, budem sçytat\,
çto sama posledovatel\nost\ x j{ } sxodytsq k toçke x∗ .
Funkcyq v( )t = V t( , x t t x( , , )0 0 ) qvlqetsq monotonno neub¥vagwej y ohra-
nyçennoj sverxu çyslom b( )ε , sledovatel\no, suwestvuet predel
lim ( )
t
t
→∞
v = lim , ( , , )
t
V t x t t x
→∞
( )0 0 = v0 = V t x( , )0 ∗
(sm. dokazatel\stvo teorem¥ 1), pryçem
V t x t t x, ( , , )0 0( ) ≤ v0 . (29)
Rassmotrym teper\ polutraektoryg x t( , t x0, )∗ pry t > t0. V sylu pred-
poloΩenyj teorem¥ ymeetsq lybo moment vremeny t1 takoj, çto dV t1( ,
x t t x( , , )1 0 ∗ ) / dt > 0, lybo moment ympul\snoho vozdejstvyq τk takoj, çto V kτ( =+
+ 0, x t xk( , , )τ 0 ∗ + J xk ( ( , , )τk t x0 ∗ ) – V kτ( , x t xk( , , )τ 0 ∗ ) > 0. ∏to oznaçaet, çto
suwestvuet moment vremeny t∗ > t0 takoj, çto V t∗( , x t t x( , , )∗ ∗ )0 = v1 > v0 .
Poskol\ku posledovatel\nost\ x j{ } sxodytsq k toçke x∗ , vsledstvye nepre-
r¥vnoj zavysymosty reßenyj ot naçal\n¥x dann¥x poluçaem neravenstvo
x t t x x t t x j( , , ) – ( , , )∗ ∗ ∗0 0 < γ
pry vsex j > N( )γ , kakovo b¥ ny b¥lo napered zadannoe çyslo γ > 0. Sledova-
tel\no,
lim , ( , , )
j
jV t x t t x
→∞
∗ ∗( )0 = v1 . (30)
Kak y pry dokazatel\stve pred¥duwej teorem¥, moΩno pokazat\, çto
lim , ( , , )
j
j jV t m x t m t x
→∞
∗ ∗+ +( )ω ω2 2 0 0 = v0 (31)
y
V t m x t m t xj j∗ ∗+ +( )ω ω2 2 0 0 1, ( , , ) – v < η j + M j( )ε , (32)
hde η j + M j( )ε → 0 pry j → ∞ . Sootnoßenyq Ωe (30) – (32) pryvodqt k pro-
tyvoreçyg, tak kak v1 > v0 . Poluçennoe protyvoreçye dokaz¥vaet, çto pred-
poloΩenye (28) neverno, t.=e. reßenye (3) system¥ (1), (2) qvlqetsq neustojçy-
v¥m.
4. Prymer. V kaçestve yllgstracyy rassmotrym systemu dyfferencyal\-
n¥x uravnenyj s ympul\sn¥m vozdejstvyem
dx
dt
y t xy t x x y
dy
dt
x t x y t x y
= + +
=
– cos sin –
cos – sin –
α α
α α
2 3 2
2 2
2
pry t ≠ πk, k ∈N ,
∆ x = – 2x, ∆ y = – y pry t = πk, k ∈N ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
OB ∏KVYASYMPTOTYÇESKOJ USTOJÇYVOSTY REÍENYJ … 1325
hde α — yrracyonal\noe çyslo. V¥berem v kaçestve funkcyy Lqpunova V =
= 1
2
2 2x y+( ) . Tohda
dV
dt
= –x x y4 32+ – x y2 2 = – ( – )x x y2 2 ≤ 0,
∆V t k= π = 1
2
2 2 2 2x x y y x y+( ) + +( )[ ]∆ ∆ – – = –1
2
2y ≤ 0.
V sluçae, kohda pry skol\ uhodno bol\ßyx t vdol\ traektoryy system¥ v¥pol-
nqgtsq neravenstva x t( ) ≠ 0, x t( ) ≠ y t( ), funkcyq
dV
dt
vdol\ πtoj traektoryy
qvlqetsq fynal\no nenulevoj. V sluçaqx x t( ) = 0 yly x t( ) = y t( ) poluçaem,
çto ∆V qvlqetsq fynal\no nenulev¥m vdol\ lgboj fynal\no nenulevoj
traektoryy. Sledovatel\no, v sylu teorem¥=1 nenulevoe reßenye rassmatry-
vaemoj system¥ qvlqetsq πkvyasymptotyçesky ustojçyv¥m.
1. Samojlenko A. M., Perestgk N. A. Dyfferencyal\n¥e uravnenyq s ympul\sn¥m vozdejst-
vyem. – Kyev: Vywa ßk., 1987. – 288 s.
2. Bainov D. D., Simeonov P. S. Systems with impulse effect: stability, theory and applications. –
New York etc.: Halsted Press, 1989. – 256 p.
3. Lakshmikantham V., Bainov D. D., Simeonov P. S. Theory of impulsive differential equations. –
Singapure etc.: World Sci., 1989.
4. Haddad W. M., Chellaboina V., Nersesov S. G. Impulsive and hybrid dynamical systems: stability,
dissipativity, and control. – Princeton: Princeton Univ. Press, 2006. – 520 p.
5. Li Z., Soh Y., Wen C. Switched and impulsive systems: analysis, design, and applications. – Berlin
etc.: Springer, 2005. – 274 p.
6. Perestgk M. O., Çernikova O. S. Do pytannq pro stijkist\ intehral\nyx mnoΩyn system
impul\snyx dyferencial\nyx rivnqn\ // Ukr. mat. Ωurn. – 2002. – 54, # 2. – S. 249 – 257.
7. Hladylyna R. Y., Yhnat\ev A. O. O neobxodym¥x y dostatoçn¥x uslovyqx asymptotyçeskoj
ustojçyvosty dlq ympul\sn¥x system // Tam Ωe. – 2003. – 55, # 8. – S. 1035 – 1043.
8. Hladylyna R. Y., Yhnat\ev A. O. Ob ustojçyvosty peryodyçeskyx system s ympul\sn¥m
vozdejstvyem // Mat. zametky. – 2004. – 76, v¥p. 1. – S. 44 – 51.
9. Yhnat\ev A. O. Metod funkcyj Lqpunova v zadaçax ustojçyvosty reßenyj system dyffe-
rencyal\n¥x uravnenyj s ympul\sn¥m vozdejstvyem // Mat. sb. – 2003. – 194, # 10. – S.=117
– 132.
10. Borysenko S. D., Iovane G., Giordano P. Investigations of the properties motion for essential non-
linear systems perturbed by impulses on some hypersurfaces // Nonlinear Anal. – 2005. – 62. – P.
345 – 363.
11. Bojçuk A. A., Perestgk N. A., Samojlenko A. M. Peryodyçeskye reßenyq ympul\sn¥x dyf-
ferencyal\n¥x system v krytyçeskyx sluçaqx // Dyfferenc. uravnenyq. – 1991. – 27, # 9.
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linear Oscillations. – 2004. – 7, # 1. – P. 78 – 82.
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Poluçeno 06.03.07,
posle dorabotky — 11.06.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
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| id | umjimathkievua-article-3246 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:53Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ba/93a0ceb29000f5e0c38e7d8c6f815fba.pdf |
| spelling | umjimathkievua-article-32462020-03-18T19:49:15Z On equiasymptotic stability of solutions of doubly-periodic impulsive systems Об эквиасимптотической устойчивости решений двоякопериодических систем с импульсным воздействием Ignat'ev, A. O. Игнатьев, А. О. Игнатьев, А. О. A system of ordinary differential equations with impulse effects at fixed moments of time is considered. This system admits the zero solution. Sufficient conditions of the equiasymptotic stability of the zero solution are obtained. Розглянуто систему звичайних диференціальних рівнянь з імпульсним впливом у фіксовані моменти часу, яка допускає нульовий розв'язок. Встановлено достатні умови еквіасимптотичної стійкості нульового розв'язку. Institute of Mathematics, NAS of Ukraine 2008-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3246 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 10 (2008); 1317–1325 Український математичний журнал; Том 60 № 10 (2008); 1317–1325 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3246/3238 https://umj.imath.kiev.ua/index.php/umj/article/view/3246/3239 Copyright (c) 2008 Ignat'ev A. O. |
| spellingShingle | Ignat'ev, A. O. Игнатьев, А. О. Игнатьев, А. О. On equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| title | On equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| title_alt | Об эквиасимптотической устойчивости решений двоякопериодических систем с импульсным воздействием |
| title_full | On equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| title_fullStr | On equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| title_full_unstemmed | On equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| title_short | On equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| title_sort | on equiasymptotic stability of solutions of doubly-periodic impulsive systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3246 |
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