On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations
A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negat...
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| Date: | 2010 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2010
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2970 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508976616046592 |
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| 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. |
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| datestamp_date | 2020-03-18T19:41:38Z |
| description | A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action. |
| first_indexed | 2026-03-24T02:33:46Z |
| format | Article |
| fulltext |
UDK 517.925
A. O. Yhnat\ev (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
O SUWESTVOVANYY FUNKCYY LQPUNOVA V VYDE
KVADRATYÇNOJ FORMÁ DLQ SYSTEM
LYNEJNÁX DYFFERENCYAL|NÁX URAVNENYJ
S YMPUL|SNÁM VOZDEJSTVYEM
A system of linear differential equations with impulse effect at fixed times is considered. Sufficient
conditions for the existence of a positive definite quadratic form are obtained. This form is such that its
derivative along the solutions of differential equations and its variations at points of impulse effect are
negative definite quadratic forms regardless of the moments of impulse effects.
Rozhlqnuto systemu linijnyx dyferencial\nyx rivnqn\ z impul\snog di[g u fiksovani momenty
çasu. Otrymano dostatni umovy isnuvannq dodatno oznaçeno] kvadratyçno] formy tako], wo ]]
poxidna çynnosti dyferencial\nyx rivnqn\ ta ]] zminy v toçkax imul\snoho vplyvu [ nehatyvno
oznaçenymy kvadratyçnymy formamy nezaleΩno vid momentiv impul\sno] di].
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” xa-
rakter. Takaq ydealyzacyq pryvodyt k neobxodymosty yssledovat\ dynamyçe-
skye system¥ s razr¥vn¥my traektoryqmy yly, ynaçe, dyfferencyal\n¥e urav-
nenyq s ympul\sn¥m vozdejstvyem. Sejças teoryq dyfferencyal\n¥x uravne-
nyj s ympul\sn¥m vozdejstvyem predstavlqet soboj yntensyvno razvyvagweesq
napravlenye matematyky, razlyçn¥e aspekt¥ kotoroho yzloΩen¥ v monohrafy-
qx [1, 2]. V poslednye hod¥ opublykovan¥ sotny prykladn¥x rabot, v kaçestve
matematyçeskyx modelej kotor¥x yspol\zovan¥ dyfferencyal\n¥e uravnenyq
s ympul\sn¥m vozdejstvyem. Vsledstvye πtoho zametno uvelyçylos\ çyslo ma-
tematyçeskyx rabot po yssledovanyg razlyçn¥x aspektov teoryy ympul\sn¥x
system [3 – 14]. Nastoqwaq stat\q posvqwena yzuçenyg ustojçyvosty reßenyj
system s ympul\sn¥m vozdejstvyem.
2. Osnovn¥e opredelenyq y postanovka zadaçy. Rassmotrym systemu
ob¥knovenn¥x dyfferencyal\n¥x uravnenyj s ympul\sn¥m vozdejstvyem
dx
dt
f t x= ( , ) , t i≠ τ , i = 1, 2, … , (1)
∆ x J xt i
i= =τ ( ) , 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)
opys¥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¥ poluçagt skaçkoobrazn¥e
yzmenenyq. Oboznaçym
BH : = x x x x Hn
n∈ = + … + ≤{ }R : 1
2 2 ,
Gi : = ( , ) : ,t x t x Bn
i i H∈ < < ∈{ }+
−R 1
1τ τ , G Gi
i
: =
=
∞
1
∪ .
Sformulyruem hypotez¥, kotor¥m moΩet udovletvorqt\ systema (1), (2):
H1 . Funkcyq f = ( , , )f fn1 … : G → Rn
ravnomerno neprer¥vna v R+ × BH ;
© A. O. YHNAT|EV, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11 1451
1452 A. O. YHNAT|EV
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∈ .
H2 . Funkcyy Ji : BH → Rn , i ∈N , neprer¥vn¥ y udovletvorqgt uslovyg
Lypßyca s konstantoj L v BH y Ji ( )0 = 0 pry i ∈N .
H3 . Suwestvuet konstanta h H∈( , )0 takaq, çto esly x Bh∈ , to x +
+ J xi ( ) ∈ BH pry i ∈N .
H4 . Konstant¥ τi udovletvorqgt uslovyqm
0 = τ0 < τ1 < τ2 < … , lim
i
i→∞
= ∞τ .
Budem oboznaçat\ çerez x t t x( , , )0 0 pry t t> 0 reßenye system¥ (1), (2),
udovletvorqgwee uslovyg x t t x( , , )0 0 0 = 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 x( , , )0 0 budem ponymat\ x t t, 0( + 0, x0 + J xi ( )0 ) (pry t t> 0 ). Znaçenye
πtoho reßenyq v moment t takΩe budem oboznaçat\ x t t x( , , )0 0 . ∏to reßenye
budem predpolahat\ neprer¥vno dyfferencyruem¥m po t na lgbom yz mno-
Ωestv Gi y neprer¥vn¥m sleva v toçkax razr¥va: x t xi( , , )τ 0 0 = x i(τ − 0 ,
t0 , x0 ) .
Pry v¥polnenyy hypotez H1 – H3 systema (1), (2) dopuskaet tryvyal\noe
reßenye
x ≡ 0. (3)
Sformulyruem ponqtyq ustojçyvosty y prytqΩenyq tryvyal\noho (nule-
voho) 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 . Esly pry πtom δ
moΩno v¥brat\ ne zavysqwym ot t0 , to reßenye (3) naz¥vaetsq ravnomerno
ustojçyv¥m.
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;
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 x( , , )0 0 ∈ 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)
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;
ravnomerno asymptotyçesky ustojçyv¥m, esly ono ravnomerno ustojçyvo y
ravnomerno prytqhyvagwee.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
O SUWESTVOVANYY FUNKCYY LQPUNOVA V VYDE KVADRATYÇNOJ FORMÁ… 1453
Opredelenye 4. Tryvyal\noe reßenye system¥ (1), (2) naz¥vaetsq πkspo-
nencyal\no ustojçyv¥m, esly suwestvugt poloΩytel\n¥e konstant¥ γ y α
takye, çto reßenyq system¥ (1), (2) obladagt svojstvom x t t x( , , )0 0 ≤
≤ γ αx e t
0
−
.
Oçevydno, çto esly reßenye (3) system¥ (1), (2) πksponencyal\no ustojçyvo,
to ono takΩe ravnomerno asymptotyçesky ustojçyvo.
Dlq yssledovanyq ustojçyvosty reßenyq (3) S.=Y.=Hurhuloj y N.=A.=Peres-
tgkom [15] predloΩeno yspol\zovat\ metod funkcyj Lqpunova, kotor¥j pred-
polahaet suwestvovanye poloΩytel\no opredelennoj funkcyy V t x( , ) , proyz-
vodnaq kotoroj vdol\ neprer¥vnoj system¥ (1) y varyacyq kotoroj v sylu
dyskretnoj system¥ (2) nepoloΩytel\n¥. V rabotax [3, 5, 10] pry predpoloΩe-
nyy v¥polnenyq hypotez H1 – H4 ukazan¥ uslovyq, pry kotor¥x teorema Hur-
hul¥ – Perestgka ob asymptotyçeskoj ustojçyvosty obratyma.
Pry yssledovanyy nekotor¥x processov, proysxodqwyx v real\nom myre y
opys¥vaem¥x dyfferencyal\n¥my uravnenyqmy s ympul\sn¥m vozdejstvyem
(1), (2), vaΩn¥m qvlqetsq postroenye (yly xotq b¥ dokazatel\stvo suwestvova-
nyq) poloΩytel\no opredelennoj funkcyy Lqpunova takoj, çto
dV
dt
=
∂
∂=
∑ V
x
f
ii
n
i
1
+
∂
∂
V
t
, t i≠ τ , x BH∈ ,
y
∆ V t i=τ = V xi iτ τ+ +( )0 0, ( ) – V xi( , )τ , i = 1, 2, … ,
odnovremenno qvlqgtsq otrycatel\no opredelenn¥my funkcyqmy [16 – 19]. V
nastoqwej stat\e rassmatryvaetsq lynejnaq systema s postoqnn¥my koπffy-
cyentamy
dx
dt
Ax= , t i≠ τ , i = 1, 2, … , (5)
∆ x Bxt i= =τ , i = 1, 2, … , (6)
hde A y B — kvadratn¥e nev¥roΩdenn¥e matryc¥, πlementamy kotor¥x qv-
lqgtsq dejstvytel\n¥e çysla.
Stavytsq zadaça: najty uslovyq, pry kotor¥x suwestvuet poloΩytel\no op-
redelennaq kvadratyçnaq forma V x( ) = x PxT
takaq, çto ee proyzvodnaq
vdol\ reßenyj system¥ (5) y ee varyacyq na skaçkax system¥, v¥çyslennaq v
sylu (6), qvlqgtsq otrycatel\no opredelenn¥my otnosytel\no x . Zdes\ y v
dal\nejßem dlq matryc¥ K lgboj razmernosty K T
oboznaçaet transponyro-
vannug matrycu.
Dyskretnug systemu (6) zapyßem v vyde
x i( )τ + 0 = ( ) ( )E B x i+ τ , i = 1, 2, … ,
hde E — edynyçnaq (n × n)-matryca.
Yntuytyvno ponqtno, çto esly τi udovletvorqgt uslovyg τi+1 – τi < θ1 ,
hde θ1 > 0 dostatoçno malo, to suwestvenn¥j vklad v dynamyku system¥ (5),
(6) vnosqt dyskretn¥e uravnenyq (6) y, sledovatel\no, nuΩno potrebovat\, çto-
b¥ sobstvenn¥e çysla matryc¥ E + B leΩaly vnutry edynyçnoho kruha komp-
leksnoj ploskosty. Esly Ωe τi udovletvorqgt uslovyg τi+1 – τi > θ2 , hde
θ2 > 0 — dostatoçno bol\ßoe çyslo, to dlq asymptotyçeskoj ustojçyvosty nu-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1454 A. O. YHNAT|EV
levoho reßenyq system¥ (5), (6) neobxodymo, çtob¥ matryca A b¥la hurvyce-
voj. Poπtomu predstavlqetsq estestvenn¥m sledugwee predpoloΩenye.
PredpoloΩenye 1. Sobstvenn¥e çysla matryc¥ A raspoloΩen¥ v levoj
otkr¥toj poluploskosty kompleksnoj ploskosty, a sobstvenn¥e çysla mat-
ryc¥ B — vnutry kruha edynyçnoho radyusa s koordynatamy ( , )−1 0 na komp-
leksnoj ploskosty.
Rassmotrym teper\ te sluçay, hde postavlennaq zadaça moΩet b¥t\ reßena.
3. Osnovn¥e rezul\tat¥. 3.1. Matryc¥ A y B kommutyrugt. Mat-
rycu P budem naz¥vat\ poloΩytel\no opredelennoj y oboznaçat\ P > 0, esly
kvadratyçnaq forma x PxT
poloΩytel\no opredelena.
V rassmatryvaemom sluçae nuΩno pokazat\, çto suwestvuet symmetryçnaq
poloΩytel\no opredelennaq matryca P takaq, çto
x A P PA xT T( )+ < 0, (7)
x E B P E B P xT T( ) ( )+ + − < 0. (8)
Teorema 1. Esly matryc¥ A y B kommutyrugt (AB = BA) y v¥polnen¥
uslovyq predpoloΩenyq=1, to pry lgb¥x τi , udovletvorqgwyx hypoteze H4 ,
y pry lgboj poloΩytel\no opredelennoj matryce R funkcyq
V x x PxT( ) = , (9)
qvlqgwaqsq reßenyem matryçn¥x uravnenyj Lqpunova
A Q QA RT + = − , (10)
( ) ( )E B P E B P QT+ + − = − , (11)
est\ funkcyq Lqpunova, udovletvorqgwaq uslovyqm (7) y (8).
Dokazatel\stvo. Vnaçale pokaΩem, çto yzmenenye ∆ V funkcyy V pry
t i= τ qvlqetsq otrycatel\no opredelennoj kvadratyçnoj formoj otnosytel\-
no x i( )τ . Dejstvytel\no,
∆ V x x x i
( ) ( )= τ = V x i( )τ +( )0 – V x i( )τ( ) = x PxT
i i( ) ( )τ τ+ +0 0 –
– x PxT
i i( ) ( )τ τ = x E B P E B xT
i
T
i( ) ( ) ( ) ( )τ τ+ + – x PxT
i i( ) ( )τ τ =
= x E B P E B P xT
i
T
i( ) ( ) ( ) ( )τ τ+ + − = −x QxT
i i( ) ( )τ τ .
Poskol\ku sobstvenn¥e çysla matryc¥ A ymegt otrycatel\n¥e vewestvenn¥e
çasty, a matryca R poloΩytel\no opredelena, matryca Q, qvlqgwaqsq reße-
nyem matryçnoho uravnenyq Lqpunova (10), takΩe poloΩytel\no opredelena,
sledovatel\no,
∆ V x x x i
( ) ( )= τ = −x QxT
i i( ) ( )τ τ < 0.
Teper\ pokaΩem, çto
�V = x A P PA xT T( )+ predstavlqet soboj otrycatel\-
no opredelennug kvadratyçnug formu yly, druhymy slovamy, A PT + PA < 0.
Dlq πtoho najdem Q yz (11) y podstavym v (10). V rezul\tate poluçym
A P E B P E BT T− + + ( ) ( ) + P E B P E B AT− + + ( ) ( ) = – R. (12)
Tak kak matryc¥ A y B kommutyrugt, to
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
O SUWESTVOVANYY FUNKCYY LQPUNOVA V VYDE KVADRATYÇNOJ FORMÁ… 1455
( )E B A+ = A E B( )+ , A E BT T( )+ = ( )E B AT T+ . (13)
Perepyßem ravenstvo (12), yspol\zovav (13):
( )A P PAT + – ( ) ( ) ( )E B A P PA E BT T+ + + = – R .
Poskol\ku po predpoloΩenyg sobstvenn¥e çysla matryc¥ E + B raspoloΩen¥
vnutry edynyçnoho kruha kompleksnoj ploskosty, a R poloΩytel\no oprede-
lena, matryca A PT + PA qvlqetsq otrycatel\no opredelennoj [20, s. 215].
Sledstvye 1. Pry v¥polnenyy uslovyj teorem¥=1 nulevoe reßenye syste-
m¥ (5), (6) πksponencyal\no ustojçyvo.
3.2. Matryc¥ A y B symmetryçn¥. Rassmotrym sluçaj, kohda mat-
ryc¥ A y B symmetryçn¥, t.=e. A = AT , B = BT
.
Teorema 2. Esly matryc¥ A y B symmetryçn¥ y v¥polnen¥ uslovyq
predpoloΩenyq=1, to pry lgb¥x τi , udovletvorqgwyx hypoteze H4 , nulevoe
reßenye system¥ (5), (6) πksponencyal\no ustojçyvo y suwestvuet poloΩy-
tel\no opredelennaq kvadratyçnaq forma (9) takaq, çto v¥polnqgtsq uslo-
vyq (7) y (8).
Dokazatel\stvo. V sylu predpoloΩenyq=1 y symmetryçnosty matryc A y
B suwestvugt dejstvytel\n¥e çysla λ1 y λ2 takye, çto λ1 > 0, λ2 < 1,
( )A E+ λ1 < 0,
( )B E E+ −2
2
2λ < 0. (14)
Rassmotrym kvadratyçnug formu
V x( ) = x xT = x1
2 + … + xn
2 . (15)
Ocenym
�V . Na lgbom yntervale neprer¥vnosty ( , )τ τi i+1
�V = 2 x AxT ≤ −2 1λ x xT = −2 1λ V . (16)
V toçkax razr¥va, uçyt¥vaq symmetryçnost\ matryc¥ B, ymeem
∆ V x t i
( ) =τ = V x V xi i( ) ( )τ τ+( ) − ( )0 = x E B E xT
i i( ) ( ) ( )τ τ+ −
2 ,
otkuda s uçetom uslovyq (14) poluçaem
∆ V x t i
( ) =τ ≤ − −( ) ( )1 2
2λ τV x i( ) . (17)
Ne narußaq obwnosty predpoloΩym, çto naçal\n¥j moment vremeny t0 pry-
nadleΩyt yntervalu ( , )0 1τ . Yz ocenok (16) y (17) pry t ∈ τ τk k, +( ]1 naxodym
V x t t x( , , )0 0( ) ≤ e V xt t k− −2
2
2
0
1 0λ λ( ) ( ) .
Uçyt¥vaq, çto λ2 < 1, zaklgçaem, çto nulevoe reßenye system¥ (5), (6) πks-
ponencyal\no ustojçyvo, y v kaçestve funkcyy Lqpunova moΩno v¥brat\ kvad-
ratyçnug formu (15), çto y trebovalos\ dokazat\.
3.3. Dvumern¥j sluçaj. Oboznaçym
C = E + B. (18)
Dlq proyzvol\noho (n-mernoho) sluçaq postavlennaq zadaça svodytsq k sle-
dugwej: najty uslovyq, pry kotor¥x suwestvugt symmetryçn¥e poloΩytel\-
no opredelenn¥e matryc¥ P, Q1 , Q2 takye, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1456 A. O. YHNAT|EV
C PCT – P + Q1 = 0 (19)
y
A PT + PA + Q2 = 0. (20)
UmnoΩaq obe çasty uravnenyq (19) na 2 y preobrazuq levug çast\, poluçaem
( ) ( )C E P C ET − + + ( ) ( )C E P C ET + − + 2 1Q = 0. (21)
Poskol\ku edynyca ne qvlqetsq sobstvenn¥m çyslom matryc¥ C, matryc¥ C –
– E y CT – E qvlqgtsq nev¥roΩdenn¥my, sledovatel\no, suwestvugt (C –
– E)−1
y ( )C ET − −1
, pryçem oçevydno
( )C ET − −1 = ( )C E
T
−( )−1 . (22)
UmnoΩym obe çasty matryçnoho ravenstva (21) sleva na ( )C ET − −1
y sprava na
( )C E− −1
. V rezul\tate budem ymet\
P C E C E( ) ( )+ − −1 + ( ) ( )C E C E PT T− +−1 +
+= 2 1
1
1( ) ( )C E Q C ET − −− − = 0.
Oboznaçym D = (C + E) ( )C E− −1 , Q∗ = 2 1 1( ) ( )C E Q C E
T
−( ) −− −
. Zametym
[21, s. 268, 618], çto matryca D qvlqetsq hurvycevoj. Uçyt¥vaq ravenstvo (22),
ubeΩdaemsq, çto C ymeet sobstvenn¥e çysla, leΩawye v otkr¥tom edynyçnom
kruhe kompleksnoj ploskosty, tohda y tol\ko tohda, kohda v¥polnqetsq mat-
ryçnoe uravnenye Lqpunova
PD + D PT + Q∗ = 0, (23)
hde D — hurvyceva matryca. S uçetom toho, çto matryca C ymeet vyd (18), v¥-
razym matryc¥ D y Q∗ neposredstvenno çerez B :
D = D B( ) = ( )2 1E B B+ − , Q∗ = Q B Q∗( , )1 = 2 1
1
1( )B Q BT− − .
Teper\ perejdem k yzuçenyg sluçaq, kohda v systeme (5), (6) x ∈R2 , A ∈ ×R2 2 ,
B ∈ ×R2 2
. Pryvedem dva opredelenyq yz [22] y dve lemm¥, dokazann¥e v [22].
Opredelenye 5. Puçkom σα A D,[ ] matryc A y D nazovem odnopara-
metryçeskoe semejstvo matryc α A + ( )1 − α D , hde α — dejstvytel\n¥j
parametr yz otrezka [0, 1].
Opredelenye 6. Puçok σα A D,[ ] naz¥vaetsq hurvycev¥m, esly matryca
α A + ( )1 − α D qvlqetsq hurvycevoj pry lgbom α ∈ [0, 1].
Lemma 1. Dlq toho çtob¥ suwestvovaly symmetryçn¥e poloΩytel\no op-
redelenn¥e matryc¥ P , Q1 , Q2 takye, çto odnovremenno v¥polnqgtsq mat-
ryçn¥e ravenstva (20) y (23), neobxodymo y dostatoçno, çtob¥ puçky
σα A D,[ ] y σα A D, −
1
b¥ly hurvycev¥my.
Lemma 2. Dlq toho çtob¥ suwestvovaly symmetryçn¥e poloΩytel\no op-
redelenn¥e matryc¥ P , Q1 , Q2 takye , çto odnovremenno v¥polnqgtsq
matryçn¥e ravenstva (20) y (23), neobxodymo y dostatoçno, çtob¥ matryc¥
AD y AD−1
ne ymely dejstvytel\n¥x otrycatel\n¥x sobstvenn¥x znaçenyj.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
O SUWESTVOVANYY FUNKCYY LQPUNOVA V VYDE KVADRATYÇNOJ FORMÁ… 1457
Yspol\zuq lemm¥ 1 y 2, poluçaem sledugwye teorem¥.
Teorema 3. Esly v systeme dyfferencyal\n¥x uravnenyj s ympul\sn¥m voz-
dejstvyem (5), (6) matryc¥ A y B takov¥, çto puçky σα A D,[ ] y σα A ,
D−
1
, hde D = ( )2 1E B B+ −
, qvlqgtsq hurvycev¥my, to nulevoe reßenye
system¥ (5), (6) πksponencyal\no ustojçyvo pry lgbom v¥bore τi , i = 1, 2, … .
Teorema 4. Esly v systeme dyfferencyal\n¥x uravnenyj s ympul\sn¥m voz-
dejstvyem (5), (6) A y B takov¥ , çto matryc¥ A D y AD−1
, hde D =
= ( )2 1E B B+ −
, ne ymegt dejstvytel\n¥x otrycatel\n¥x sobstvenn¥x zna-
çenyj, to nulevoe reßenye system¥ (5), (6) πksponencyal\no ustojçyvo pry
lgbom v¥bore τi , i = 1, 2, … .
4. Prymer. Dlq yllgstracyy poluçenn¥x rezul\tatov rassmotrym v ka-
çestve prymera systemu (5), (6), v kotoroj n = 2,
A =
−
− −
1 1
1 3
, B =
1
3
2
3
2
2− −
.
PokaΩem, çto suwestvugt poloΩytel\no opredelenn¥e kvadratyçn¥e form¥
P, Q y R takye, çto v¥polnqgtsq uslovyq (10), (11), y nulevoe reßenye system¥
(5), (6) πksponencyal\no ustojçyvo nezavysymo ot v¥bora τi i = 1, 2, … .
Perv¥j sposob. Najdem proyzvedenyq matryc AB y BA:
AB =
−
− −
− −
1 1
1 3
1
3
2
3
2
2
=
− −
5
2
7
2
7
2
9
2
,
BA =
1
3
2
3
2
2
1 1
1 3
− −
−
− −
=
− −
5
2
7
2
7
2
9
2
.
Poskol\ku AB = BA, na osnovanyy teorem¥=1 poluçaem, çto dejstvytel\no su-
westvugt poloΩytel\no opredelenn¥e kvadratyçn¥e form¥ P, Q y R takye,
çto v¥polnqgtsq uslovyq (10), (11), y nulevoe reßenye system¥ (5), (6) πkspo-
nencyal\no ustojçyvo nezavysymo ot v¥bora τi , i = 1, 2, … .
Vtoroj sposob. Tak kak n = 2, moΩno vospol\zovat\sq teoremoj=4. Dlq
πtoho dostatoçno pokazat\, çto matryc¥ AD y AD−1
ne ymegt dejstvytel\-
n¥x otrycatel\n¥x sobstvenn¥x znaçenyj, hde D = ( )2 1E B B+ −
. Proverym
πto. Dlq πtoho posledovatel\no naxodym
2E + B =
3
3
2
3
2
0−
, B−1 =
− −
8 6
6 4
,
D = ( )2 1E B B+ − =
− −
15 12
12 9
, AD =
27 21
21 15− −
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1458 A. O. YHNAT|EV
D−1 =
1
3
3 4
4 5− −
, AD−1 =
− −
7
3
3
3
11
3
.
Matryc¥ AD y AD−1
ymegt kratn¥e sobstvenn¥e znaçenyq (sootvetstvenno
6 y 2 3/ ), kotor¥e ne qvlqgtsq otrycatel\n¥my, çto y sledovalo dokazat\.
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Poluçeno 28.06.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
|
| id | umjimathkievua-article-2970 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:33:46Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e0/4cf9468f642c9d22ea82b3951b532de0.pdf |
| spelling | umjimathkievua-article-29702020-03-18T19:41:38Z On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations О существовании функции Ляпунова в виде квадратичной формы для систем линейных дифференциальных уравнений с импульсным воздействием Ignat'ev, A. O. Игнатьев, А. О. Игнатьев, А. О. A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action. Розглянуто систему лінійних диференціальних рівнянь з імпульсною дією у фіксовані моменти часу. Отримано достатні умови існування додатно означеної квадратичної форми такої, що її похідна чшшосі і диференціальних рівнянь та її зміни в точках імульсного впливу є негативно означеними квадратичними формами незалежно від моментів імпульсної дії. Institute of Mathematics, NAS of Ukraine 2010-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2970 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 11 (2010); 1451–1458 Український математичний журнал; Том 62 № 11 (2010); 1451–1458 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2970/2690 https://umj.imath.kiev.ua/index.php/umj/article/view/2970/2691 Copyright (c) 2010 Ignat'ev A. O. |
| spellingShingle | Ignat'ev, A. O. Игнатьев, А. О. Игнатьев, А. О. On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| title | On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| title_alt | О существовании функции Ляпунова в виде квадратичной формы для систем линейных дифференциальных уравнений с импульсным воздействием |
| title_full | On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| title_fullStr | On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| title_full_unstemmed | On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| title_short | On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| title_sort | on the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2970 |
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