Mechanical analogs of linear impulsive systems
The linear system of differential equations with pulse influence is considered for which the condition of construction of its mechanical analogs is obtained.
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2011
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| author | Pryz, A. M. Приз, А. М. |
| author_facet | Pryz, A. M. Приз, А. М. |
| author_sort | Pryz, A. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:34:07Z |
| description | The linear system of differential equations with pulse influence is considered for which the condition of construction of its mechanical analogs is obtained. |
| first_indexed | 2026-03-24T02:28:40Z |
| format | Article |
| fulltext |
UDK 531.36
A. M. Pryz (In-t matematyky NAN Ukra]ny, Ky]v)
MEXANIÇNI ANALOHI} LINIJNYX IMPUL|SNYX SYSTEM
The linear system of differential equations with pulse influence is considered for which the condition of
construction of its mechanical analogs is obtained.
Rassmotrena lynejnaq systema dyfferencyal\n¥x uravnenyj s ympul\sn¥m vozdejstvyem, dlq
kotoroj poluçeno uslovye postroenyq ee mexanyçeskyx analohyj.
Pobudovu zahal\no] teori] system z impul\snog di[g ta ]x qkisnyj analiz vykla-
deno v monohrafiqx [1, 2] ta in. Dyferencial\ni rivnqnnq z impul\snog di[g vy-
nykagt\ pry matematyçnomu modelgvanni real\nyx procesiv z korotkoçasnymy
zburennqmy [3 – 7]. U statti [8] rozv'qzano zadaçu znaxoΩdennq mexaniçno] ana-
lohi] dlq systemy dyferencial\nyx rivnqn\ u formi Koßi bez impul\sno] di],
pry c\omu vidpovidna zadaça pry naqvnosti impul\siv zalyßalasq vidkrytog. U
danij roboti otrymano umovu isnuvannq j alhorytm pobudovy mexaniçnyx analo-
hij dlq linijnyx system dyferencial\nyx rivnqn\ z impul\snog di[g.
Budemo doslidΩuvaty linijni systemy dyferencial\nyx rivnqn\ z impul\s-
nog di[g vyhlqdu [1]
� �x t Ax t( ) ( )= , x t x( )0 0= , t k≠ τ ,
(1)
x t Px t( ) ( )+ = � , t k= τ , k = 1, 2, … .
Tut x n∈R2 ,
�A , �P n n∈ ×R2 2 — stali matryci, a momenty impul\sno] di] zado-
vol\nqgt\ dvostoronng ocinku 0 < θ1 ≤ τk+1 – τk ≤ θ2 < ∞.
Zapyßemo systemu (1) u vyhlqdi
�
�
x t
x t
1
2
( )
( )
=
� �
� �
A A
A A
x t
x t
11 12
21 22
1
2
( )
( )
, x t y( )0 0= , t k≠ τ ,
(2)
x t
x t
1
2
( )
( )
+
+
=
� �
� �
P P
P P
x t
x t
11 12
21 22
1
2
( )
( )
, t k= τ , k = 1, 2, … ,
de vsi matryci magt\ odnakovyj porqdok. Same taka bloçna forma zapysu vlas-
tyva zadaçi pro mexaniçni analohi].
Dlq perßoho rivnqnnq systemy (1) dovedeno nastupnu teoremu [8].
Teorema 1. Stacionarna linijna systema
�
�
x
x
1
2
=
� �
� �
A A
A A
x
x
11 12
21 22
1
2
, x t x( )0 0= ,
de xi n∈R ,
�Aij n n∈ ×R , i, j = 1, 2, linijnym stacionarnym peretvorennqm vek-
tora stanu
y = Tx, T =
T T
T A T A T A T A
11 12
11 11 12 21 11 12 12 22
� � � �+ +
, (3)
zvodyt\sq do vyhlqdu
© A. M. PRYZ, 2011
140 ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
MEXANIÇNI ANALOHI} LINIJNYX IMPUL|SNYX SYSTEM 141
�
�
y
y
1
2
=
O I
C B
y
y− −
1
2
, y t y( )0 0=
( yi n∈R , I n n∈ ×R — odynyçna matrycq), todi i til\ky todi, koly dlq vsix
vlasnyx znaçen\ λi matryci �A vykonugt\sq umovy
rank ( )�A Ii− λ ≥ n, i = 1 2, n ,
de I n n∈ ×R2 2 , a matryci B i C znaxodqt\sq analityçno z rivnosti C B =
= −[ ] −T T A T11 12
2 1�
.
U teoremiF1 i skriz\ dali çerez matrycg I poznaça[mo odynyçnu matrycg. }]
porqdok zavΩdy dorivng[ n abo 2n, wo bude lehko vydno z vyraziv, do skladu
qkyx vona vxodyt\.
Pobudu[mo dlq (1) mexaniçnu analohig [8]. Budemo vvaΩaty, wo dlq perßo-
ho rivnqnnq systemy (1), (2) ma[ misce teoremaF1. Todi, zastosovugçy (3) do (2), v
zahal\nomu vypadku otrymu[mo
�y t Ay t( ) ( )= , y t y( )0 0= , t k≠ τ ,
(4)
y t Py t( ) ( )+ = , t k= τ , k = 1, 2, … ,
de A = TAT� −1
, P = TPT� −1
. Zvidsy vyplyva[ ekvivalentnist\ matryc\ A ta
�A i
P ta
�P . Zapyßemo (4) u bloçnij formi
�
�
y t
y t
1
2
( )
( )
=
O I
C B
y t
y t− −
1
2
( )
( )
, y t( )0 = y0 , t k≠ τ ,
y t
y t
1
2
( )
( )
+
+
=
P P
P P
y t
y t
11 12
21 22
1
2
( )
( )
, t k= τ , k = 1, 2, … .
U zahal\nomu vypadku matrycq P ne ma[ vyznaçeno] kanoniçno] formy. Ale
nas bude cikavyty ]] special\nyj vyhlqd
P =
I O
M I
, M ≠ 0. (5)
Bil\ß detal\no kanoniçna forma (5) ta vlastyvosti matryci
�P budut\ doslid-
Ωeni nyΩçe.
Oskil\ky perße rivnqnnq v systemi (4) [ mexaniçnog analohi[g [8] perßoho
rivnqnnq z (2), to zupynymosq na druhyx rivnqnnqx obox system ta znajdemo
umovy, pry vykonanni qkyx rivnqnnq
y t( )+ = Py t( ) , t k= τ , k = 1, 2, … ,
bude mexaniçnog analohi[g dlq
x t( )+ = �Px t( ) , t k= τ , k = 1, 2, … ,
a matrycq M bude matryceg uzahal\nenyx impul\siv udarnyx syl.
Dlq c\oho navedemo neobxidni teoretyçni vidomosti [9]. U vypadku, koly sy-
ly digt\ na dostatn\o korotkomu promiΩku çasu (napryklad, pry zitknenni dvox
til), ]x nazyvagt\ impul\snymy [9]. Pid impul\som syly F rozumigt\ intehral
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
142 A. M. PRYZ
F dt
t∆
∫ ,
de ∆ t — neskinçenno malyj promiΩok çasu, pid ças qkoho di[ cq syla. Pry na-
qvnosti impul\snyx syl rivnqnnq LahranΩa u vvedenyx poznaçennqx moΩna za-
pysaty u vyhlqdi
∂
∂
+
+
L t
q tj
( )
( )�
–
∂
∂
L t
q tj
( )
( )�
= S j , j = 1, n , (6)
de koordynaty q j systemy do di] impul\su poznaçeno qk q tj ( ) , a pislq di] —
qk q tj ( )+
, S j = d qij ii
n
=∑ 1
— uzahal\nenyj impul\s udarno] syly, L — lahran-
Ωian systemy.
Oskil\ky L = T – Π, T =
1
2 1
a q qij i ji j
n � �
, =∑ i Π =
1
2 1
c q qij i ji j
n
, =∑ , to
∂
∂
L
q j�
=
∂
∂
T
q j�
–
∂
∂
Π
�q j
= a qij i
i
n
=
∑
1
.
Pry c\omu rivnqnnq (6) moΩna zapysaty v takij formi:
a q t q tij i i
i j
n
� �( ) ( )
,
+
=
−( )∑
1
= d qij j
i
n
=
∑
1
. (7)
Perejdemo v rivnqnni (7) vid koordynatno] formy do matryçno], poznaçyvßy
A = ( ) ,a i j
n
=1 > 0, D = ( ) ,d i j
n
=1 . Rozhlqnemo vypadok, koly matrycq D [ nevyrod-
Ωenog. Todi rivnqnnq (7) zapyßemo u vyhlqdi
Aq t�( )+ = Aq t�( ) + Dq t( ) .
Vykona[mo zaminu y = A q, pislq çoho otryma[mo
�y t( )+ = �y t( ) + My t( ) , M DA= −1.
Vidomo, wo pid di[g impul\snyx syl koordynaty systemy ne zminggt\sq. Cq
fizyçna vlastyvist\ impul\no] systemy zada[t\sq umovog y t( )+ = y t( ) , qka ra-
zom z ostannim rivnqnnqm pislq zaminy y1 = y, y2 = �y utvorg[ druhu çastynu
systemy (4):
y t y t1 1( ) ( )+ = ,
y t2( )+ = My t1( ) + y t2( ) ,
abo
y t( )+ =
I O
M I
y t
( ) .
Nastupna lema mistyt\ umovy zvedennq matryci
�P do P vyhlqdu (5).
Lema 1. Qkwo dlq matryci �P n n∈ ×R2 2 vykonugt\sq umovy:
1) vsi vlasni znaçennq matryci �P dorivnggt\ odynyci;
2) forma Ûordana matryci �P ne mistyt\ klityn porqdku vywe druhoho,
do toho Ω kil\kist\ takyx klityn dorivng[ n;
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
MEXANIÇNI ANALOHI} LINIJNYX IMPUL|SNYX SYSTEM 143
3) spravdΩu[t\sq totoΩnist\
� � �P P I P12 22
2
21( )− − = – I, de
�Pij — kvadrat-
ni matryci porqdku n,
to vona podibna do matryci P vyhlqdu
P =
I O
M I
, M ≠ 0.
Dovedennq. Z rivnosti PT = TP� otrymu[mo
( )P I T− =
O O
M O
T T
T T
11 12
21 22
=
=
T T
T T
P I P
P P I
11 12
21 22
11 12
21 22
−
−
� �
� �
= T P I( )� − , (8)
de
�P – I n n∈ ×R2 2 — matrycq ranhu n. Budemo vvaΩaty, wo matrycq
�P I22 −
nevyrodΩena, todi spravdΩu[t\sq rivnist\ [10] �P I11 − = � � �P P I P12 22
1
21( )− −
, pi-
slq pidstanovky qko] v (8) oderΩu[mo
O O
M O
T T
T T
11 12
21 22
=
T T
T T
P P I P P11 12
21 22
12 22
1
21 12
− −� � � �
�
( )
PP P I21 22
� −
abo
T P P I P11 12 22
1
21
� � �( )− − + T P12 21
� = 0,
T P11 12
� + T P I12 22( )� − = 0,
T P P I P21 12 22
1
21
� � �( )− − + T P22 21
� = MT11 ,
T P21 12
� + T P I22 22( )� − = MT12 .
Dlq zruçnosti ostanng systemu zapyßemo takym çynom:
T P T P I P I P11 12 12 22 22
1
21
� � � �+ −( ) − −( ) ( ) = 0,
T P11 12
� + T P I12 22( )� − = 0,
T P T P I P I P21 12 22 22 22
1
21
� � � �+ −( ) − −( ) ( ) = MT11 ,
T P21 12
� + T P I22 22( )� − = MT12 ,
zvidky otryma[mo umovu zvidnosti matryci
�P I− do P – I abo
�P do P :
� � �P P I P12 22
2
21( )− − = – I. (9)
NevyrodΩenist\ M vyplyva[ z toho, wo dlq matryc\ P vyhlqdu (5) porqdok
klityn Ûordana ne perevywu[ 2 ta kil\kist\ takyx klityn dorivng[ ranhu M.
Lemu dovedeno.
Vstanovymo teper umovy, qki povynni zadovol\nqty
�A i
�P dlq odnoçasno-
ho ]x zvedennq do A ta P pretvorennqm (3). Tym samym vydilymo klas impul\s-
nyx system, dlq qkyx isnu[ ]x mexaniçna analohiq (2), (5).
Teorema 2. Nexaj dlq matryc\ impul\sno] systemy (1), (2) spravdΩugt\sq
teoremaF1, lemaF1 i umova
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
144 A. M. PRYZ
det ( )I P P I
A A
A A
− −
−� �
� �
� �12 22
1 11 12
21 22
−
−� �P P I
I
12 22
1( )
≠ 0.
Todi isnu[ peretvorennq y = Tx vyhlqdu
T =
T O
O T
I P P I
A P P
11
11
12 22
1
11 12
− −
−
−� �
� �
( )
( 222
1
21 12 12 22
1
22− − −
− −I A A P P I A) ( )� �
,
z dopomohog qkoho z systemy (1), (2) moΩna otrymaty ]] mexaniçnu analohig
vyhlqdu
�
�
y t
y t
1
2
( )
( )
=
O I
C B
y t
y t− −
1
2
( )
( )
, y t( )0 = y0 , t k≠ τ ,
(10)
y t
y t
1
2
( )
( )
+
+
=
I O
M I
y t
y t
1
2
( )
( )
, t k= τ , k = 1, 2, … .
Dovedennq. Iz systemy
T12 = − − −T P P I11 12 22
1� �( ) ,
(11)
MT11 – T22 – T P P21 12 22
� � = 0
z uraxuvannqm (3) ta (9) otrymu[mo matrycg T, qka odnoçasno zvodyt\
�A do A
i
�P do P :
T =
T O
O T
I P P I
A P P
11
11
12 22
1
11 12
− −
−
−� �
� �
( )
( 222
1
21 12 12 22
1
22− − −
− −I A A P P I A) ( )� �
. (12)
Pidstavymo komponenty matryci T iz (12) u druhe rivnqnnq systemy (11) ta
oderΩymo vyraz dlq matryci M v qvnomu vyhlqdi
M = T I P P I
A A
A A
11 12 22
1 11 12
21 22
− −
−� �
� �
� �
( )
−
−
−
� �P P I
I
T
12 22
1
11
1( )
. (13)
Dovedemo odnoçasnu nevyrodΩenist\ matryc\ T i M ta vstanovymo vidpo-
vidnu zaleΩnist\ miΩ
�A i
�P . Proanalizu[mo vyrazy (3), (12) ta (13) dlq mat-
ryc\ T i M vidpovidno. Matrycq T iz (3) povynna buty nevyrodΩenog. Vyny-
ka[ pytannq pro nevyrodΩenist\ matryci T iz (12), de dlq ]] „Fdobudovy” vyko-
rystano vidpovidni bloky matryci
�P – I iz (8). Vyxodqçy z formuly dlq vy-
znaçnyka bloçno] matryci [10]
∆ =
A B
C D
= AD CB− ,
de vraxovano, wo AB = BA, AC = CA, umovu nevyrodΩenosti T iz (12) zapysu[mo
u vyhlqdi
det ( )I P P I
A A
A A
− −
−� �
� �
� �12 22
1 11 12
21 22
−
−� �P P I
I
12 22
1( )
≠ 0. (14)
ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 1
Lehko baçyty, wo umova (14) ta nevyrodΩenist\ matryci M iz (13) ta T iz
(12) vykonugt\sq todi i til\ky todi, koly stovpci matryci
� �P P I I
T
12 22
1( )−
−
ne budut\ mistyty Ωodnoho vlasnoho vektora matryci
�A , qkyj vidpovida[ ]] nu-
l\ovomu vlasnomu znaçenng.
Teoremu dovedeno.
Zaznaçymo, wo oskil\ky vybir matryci T11 v teoremiF2 ne [ odnoznaçnym, ma[
misce mnoΩyna mexaniçnyx analohij dlq odni[] impul\sno] systemy. Taka ne-
odnoznaçnist\ dozvolq[ za dopomohog vil\nyx parametriv matryci T11 vykonaty
pevni sprowennq struktury matryc\ B i C iz (10) abo zvesty matrycg M do
formy Ûordana.
1. Samojlenko A. M., Perestgk N. A. Dyfferencyal\n¥e uravnenyq s ympul\sn¥m vozdejst-
vyem. – Kyev: Vywa ßk., 1987. – 288 s.
2. C¥pkyn Q. Z. Teoryq lynejn¥x ympul\sn¥x system. – M.: Fyzmathyz, 1958. – 724 s.
3. Martynyuk A. A., Shen J. N., Stavroulakis I. P. Stability theorems in impulsive equations with
infinite delay // Advances in Stability Theory at the End of the 20 th Century / Ed. A. A. Martynyuk.
– London; New York: Taylor and Francis, 2003. – P. 153 – 175.
4. Laryn V. B. Upravlenye ßahagwymy apparatamy. – Kyev: Nauk. dumka, 1980. – 168 s.
5. Laryn V. B. K voprosu postroenyq modely ßahagweho apparata // Prykl. mexanyka. – 2003.
– 39, # 4. – S. 122 – 132.
6. Sl¥n\ko V. Y. Lynejn¥e matryçn¥e neravenstva y ustojçyvost\ dvyΩenyq ympul\sn¥x
system // Dop. NAN Ukra]ny. – 2008. – # 4. – S. 68 – 71.
7. Denysenko V. S., Sl¥n\ko V. Y. Ympul\snaq stabylyzacyq mexanyçeskyx system v modelqx
Takahy – Suheno // Int. Appl. Mech. – 2009. – 45, # 10.
8. Novyc\kyj V. V., Petryßyna L. V. Dekompozyciq ta mexaniçni analohi]. 1. Linijni stacionar-
ni systemy // Vopros¥ analytyçeskoj mexanyky y ee prymenenyj: Praci In-tu matematyky
NAN Ukra]ny. – 1999. – 26. – S. 251 – 256.
9. Holdstejn H. Klassyçeskaq mexanyka. – 2-e yzd. – M.: Nauka, 1975. – 416 s.
10. Hantmaxer F. R. Teoryq matryc. – M.: Nauka, 1967. – 576 s.
OderΩano 23.03.10
|
| id | umjimathkievua-article-2705 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:28:40Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/80f6e4723d2f62c62cb51a3384232279.pdf |
| spelling | umjimathkievua-article-27052020-03-18T19:34:07Z Mechanical analogs of linear impulsive systems Механічні аналогії лінійних імпульсних систем Pryz, A. M. Приз, А. М. The linear system of differential equations with pulse influence is considered for which the condition of construction of its mechanical analogs is obtained. Рассмотрена линейная система дифференциальных уравнений с импульсным воздействием, для которой получено условие построения ее механических аналогий. Institute of Mathematics, NAS of Ukraine 2011-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2705 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 1 (2011); 140-145 Український математичний журнал; Том 63 № 1 (2011); 140-145 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2705/2166 https://umj.imath.kiev.ua/index.php/umj/article/view/2705/2167 Copyright (c) 2011 Pryz A. M. |
| spellingShingle | Pryz, A. M. Приз, А. М. Mechanical analogs of linear impulsive systems |
| title | Mechanical analogs of linear impulsive systems |
| title_alt | Механічні аналогії лінійних імпульсних систем |
| title_full | Mechanical analogs of linear impulsive systems |
| title_fullStr | Mechanical analogs of linear impulsive systems |
| title_full_unstemmed | Mechanical analogs of linear impulsive systems |
| title_short | Mechanical analogs of linear impulsive systems |
| title_sort | mechanical analogs of linear impulsive systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2705 |
| work_keys_str_mv | AT pryzam mechanicalanalogsoflinearimpulsivesystems AT prizam mechanicalanalogsoflinearimpulsivesystems AT pryzam mehaníčníanalogíílíníjnihímpulʹsnihsistem AT prizam mehaníčníanalogíílíníjnihímpulʹsnihsistem |