Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function
By the method of combining the matrix-valued Lyapunov functional and comparison theorem, connected Lyapunov stability and practical stability of large scale delay system are studied deeply. A series of new sufficient conditions are proposed. These results are not only of theoretical but also of prac...
Збережено в:
| Опубліковано в: : | Прикладная механика |
|---|---|
| Дата: | 2013 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут механіки ім. С.П. Тимошенка НАН України
2013
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/87804 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function / J.F. Sun, X.L. Wang // Прикладная механика. — 2013. — Т. 49, № 5. — С. 139-144. — Бібліогр.: 6 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859707548509143040 |
|---|---|
| author | Sun, J.F. Wang, X.L. |
| author_facet | Sun, J.F. Wang, X.L. |
| citation_txt | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function / J.F. Sun, X.L. Wang // Прикладная механика. — 2013. — Т. 49, № 5. — С. 139-144. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Прикладная механика |
| description | By the method of combining the matrix-valued Lyapunov functional and comparison theorem, connected Lyapunov stability and practical stability of large scale delay system are studied deeply. A series of new sufficient conditions are proposed. These results are not only of theoretical but also of practical value.
Методом об‘єднання матрично-значних функціоналів Ляпунова і теореми порівняння досліджено зв‘язну стійкість за Ляпуновим і практичну стійкість великих систем з запізненням. Запропоновано ряд нових достатніх умов. Результати мають не лише теоретичний сенс, але також практичне значення.
|
| first_indexed | 2025-12-01T04:00:02Z |
| format | Article |
| fulltext |
2013 ПРИКЛАДНАЯ МЕХАНИКА Том 49, № 5
ISSN0032–8243. Прикл. механика, 2013, 49, № 5 139
J . F . S u n , X . L . W a n g
CONNECTED STABILITY ANALYSIS OF DELAY SYSTEMS VIA
THE MATRIX-VALUED LYAPUNOV FUNCTION
Harbin Institute of Technology (Weihai),
Shangdong province 2, Wenhua Xi Road Weihai, China;
e-mail: sunif_70@126.com
Abstract. By the method of combining the matrix-valued Lyapunov functional and
comparison theorem, connected Lyapunov stability and practical stability of large scale de-
lay system are studied deeply. A series of new sufficient conditions are proposed. These
results are not only of theoretical but also of practical value.
Key words: delay system, connected Lyapunov stability, practical stability.
1. Designations.
Let 0 0([ , 0], ), [ , ), 0nC C R J t t . For any C the norm
0
sup ( )
s
s
is
used. For nx R , max rx x , 1, 2,...,r n . If 0([ , ), )nx C t R , then tx C is de-
termined as ( ) ( )tx s x t s , 0s . We designate :H
nC C H , where
0H or H .
2. Description of the system and decomposition.
Consider the large scale system modeled by functional differential equation
0 0( , ), ,H
t t n
dx
f t x x C
dt
(1)
where : H n
nf J C R . Provided that the vector-function f maps the bounded sets into
the bounded sets, for each 0t J and 0
H
nC there exists a unique solution
00( , )( )tx t x t determined on some interval 0[ , ]t t , 0 , and if 1H H is such that
0 0 1( , ( )x t t H , then .
The system (1) is decomposed into m interconnected subsystems
1( )
( , ) ( , , , ),
i
i m
i t i t t
dx t
f t x g t x x
dt
(2)
where 1, 2,...,mi I m , ( , )i i
i
H n
i nf J C R , ( , )inH
i ng C J C R and i
i
n n . We
assume that the functions ( , )i tg t x depend on m m -matrix of interconnections [ ]ij
t tE e ,
1 1 2 2( , ) ( , , , , ),i i im m
i t i t t t t t tg t x g t e x e x e x (3)
140
when mi I , where the elements ([ ,0],[0,1])ij
te C depend in general case on the delay
( ) ( ), [ ,0].ij ij i
t te x e t x t
We designate by tE the fundamental matrix of interactions with the elements 1
ij
te if
jx is contained in ( , )i tg t x ; 0
ij
te if jx is not contained in ( , )i tg t x .
For 0tE we get from system (2) the independent subsystems of functional differen-
tial equations of smaller dimensions
0 0
( )
( , ), ,
i
i i i
i t t
dx t
f t x x C
dt
(4)
where in
ix R and i i
i
H n
i nf J C R . Moreover, we assume ( ,0) 0if t and for subsys-
tems (2) ( ,0) ( ,0) 0i if t g t for all t J and mi I , i.e. the state 1 ... 0mx x x is the
unique equilibrium state of system (1) and subsystems (2).
For subsystem (2), whose functions ( , ), 1, 2,...,i tg t x i m , depend on the matrix of in-
teractions tE , the problem on practical stability of motion is reduced to the establishment of
conditions under which the solution
00( , ) ( )tx t x t of system (2) possesses certain qualitative
properties for given estimates of initial and subsequent deviations on the infinite interval.
3. Matrix-valued functional.
For system (2) we construct the matrix-valued functional
( , ) [ ( , )], ,ij mU t t i j I (5)
with the elements satisfying the following conditions.
1H . The elements ( , ) ( , )i i
i
H mi
ii nt C J C R , 1 i im n , ( ,0) 0,ii t are locally
Lipschitz in i ;
2H .The elements ( , , ) ( , ),i jI I
i i
m mH Hi j
ij n nt C J C C R are locally Lipschitz in i
and j for all ( ) mi j I .
By means of the real vector , 0mR , we construct the functional
( , , ) ( , ) ,TV t U t (6)
which is continuous and definite on the set H
nJ C by conditions 1 2H H . The upper de-
rivative of functional (6) along solutions of system (2) is determined by the formula
( , , ) ( , ) ,TD V t D U t (7)
where ( , )D U t
0
1
lim sup ( , ( , )) ( , ) .tU t x t U t
Note that ( , )D U t is com-
puted element-wise.
4. Definitions of connected stability of system (2).
Taking into account the results of paper [3] we shall cite the definitions of stability no-
tion incorporated in this paper.
Definition 1. The equilibrium state 0x of system (1) is called
a) connectedly stable if for every 0 and 0 0t there exists 0( , ),t such that
0( , )( )x t t whenever 0[ , ]nC t t for all t tE E ;
b) uniformly connectedly stable if in definition (a) the value does not depend on 0t ;
141
c) asymptotically connectedly stable if it is connectedly stable and for any 0 0t there
exists 0 such that 0( , ( ) 0,x t t as ,t whenever ,nC for all t tE E ;
d) uniformly asymptotically connectedly stable if it is uniformly connectedly stable and
there exists some 0 and for every 0 there exists 0 such that 0( , ( ) ,x t t
whenever 0 0[ , ]nC t t for all t tE E .
5. Conditions of connected stability of system (2).
Using matrix-valued functional (5) and its derivative (7) and applying the theorems of
comparison principle for functional-differential equations (see [1]) we shall set out a series
of sufficient conditions for connected stability of the equilibrium state 0x of system (1).
Theorem 1. Let system of functional-differential equations (1) be such that
1) there exists the matrix-valued functional ( , ) ( , )H m m
nU t C J C R , ( ,0) 0U t for
all t J and ( , )U t is locally Lipschitz in for every t J ;
2) there exist m m constant matrices 1( )A and 1( )B , real vector mR , 0
and comparison functions 1 2( (0) ), ( ),i i
i i mu u i I , of Hahn class K so that
1 1 1( (0) ) ( ) ( (0) )Tu A u
, 1
( , )
m
i j ij
i j
u t
2 1 2( ) ( ) ( )Tu B u for all t J and H
nC ;
3) there exists the comparison function ( , )W C J R R such that
( , , ) ( , ( , , ))D V t W t V t (8)
for all ( , ) H
nt J C and all matrices of interaction t tE E . Then the certain type of sta-
bility of zero solution to the comparison equation
0 0( , ), ( ) 0
du
W t u u t u
dt
(9)
and the restrictions on the matrices 1 1( ), ( )A B imply the corresponding type of connected
stability of the equilibrium state of system (1) with decomposition (2).
Proof. Provided that the matrices 1( )A and 1( )B are positive definite, functional (6)
is positive definite and decreasing. Further, we apply Theorem 4.4.3 from [1] and determine
certain type of connected stability of system (1).
Corollary 1. Let
1) conditions (1) and (2) of Theorem 1 be satisfied;
2) the matrix 1( )A be positive definite, the matrix 1( ) 0B and the comparison func-
tion ( , ( , , )) 0W t V t .
Then the equilibrium state 0x of system (1) with decomposition (2) is connectedly
stable.
Corollary 2. Let
1) conditions (1) and (2) of Theorem 1 be satisfied;
2) the matrices 1( )A and 1( )B be positive definite and the comparison function
( , ( , , )) 0W t V t .
Then the equilibrium state 0x of system (1) with decomposition (2) is uniformly
connectedly stable.
Corollary 3. Let
1) conditions (1) and (2) of Theorem 1 be satisfied;
2) the matrices 1( )A and 1( )B be positive definite;
3) the zero solution of comparison equation (9) be uniformly asymptotically stable.
142
Then the equilibrium state 0x of system (1) with decomposition (2) is uniformly as-
ymptotically connectedly stable.
Theorem 2. Let system of functional differential equations (1) be such that
1) conditions (1) and (2) of Theorem 1 are satisfied;
2) there exist a constant m m matrix 1( ), , 0mC R and functions 3 ( )i
i tu x ,
3iu is of class K for all mi I , such that 3 1 3( , , ) ( ) ( ) ( )T
t tD V t u x C u x for any
( , ) H
nt J C and any matrices of interactions t tE E , where 3 ( )T
tu x =
31 3( ( ),..., ( ))l m
t m tu x u x ;
3) the matrices 1( )A and 1( )B are positive definite and the matrix 1( )C is negative
definite.
Then the equilibrium state 0x of system (1) with decomposition (2) is uniformly as-
ymptotically connectedly stable.
Theorem 3. Let in system of equations (1) the vector function ( , )f t be bounded in
and
1) conditions (1) and (2) of Theorem 1 are satisfied;
2) there exist a constant m m matrix 2 ( ), , 0mC R and functions 4 ( )i
i tu x of
class K for all mi I such that 4 2 4( , , ) ( ) ( ) ( )T
t tD V t u x C u x for all ( , ) H
nt J C
and any matrices of interconnections t tE E ;
3) the matrices 1( )A and 1( )B are positive definite and the matrix 2 ( )C is negative
definite.
Then the equilibrium state 0x of system (1) with decomposition (2) is uniformly as-
ymptotically connectedly stable.
6. Matrix-valued function on space product.
For system (4) we construct the matrix-valued function
( , , ) [ ( , , )], , 1, 2,...,t ij tU t x x v t x x i j m , (10)
with the elements satisfying the following conditions.
3H . The elements ( , ), ( , 0, 0) 0i
i
H
ii iinv C J C C R v t are locally Lipschitz in ix ;
4H . The elements ( , ),ji
i j
HH
ij n nv C J C C C C R ( , 0, 0, 0) 0ijv t are locally
Lipschitz in ,i jx x for all ( ) mi j I .
By means of the real vector , 0mR , we construct the function
( , , , ) ( , , ) ,T
t tV t x x U t x x (11)
which is definite on the space product nR C and locally Lipschitz in x , providing condi-
tions of assumptions 3H and 4H are satisfied. Further we define
( , , , ) ( , , ) ,T
t tD V t x x D U t x x (12)
where
( , , ) lim sup ( ,tD U t x x U t 1( , ), ( )) ( , , )] :t t h tx f t x x U t x x
0 . (13)
Note that when formula (12) is properly applied, ( , , )tD U t x x is computed element-wise.
7. Conditions of connected practical stability of system (2).
In view of the results from [1, 4] we shall formulate the following definitions.
Definition 2. System (2) is called
143
a) connectedly practically stable, if given estimates of ( , ), 0A A , the condition
0 nC implies 0 0( , )( )x t t A for all 0t t and all t tE E ;
b) connectedly asymptotically practically stable, if conditions of definition (a) are satis-
fied and 0 0lim ( , )( ) 0
t
x t t
.
The other definitions of connected practical stability can be formulated in terms of
Definition 2.
Theorem 4. Let system of functional differentional equations (1) be such that
1) there exists a matrix-valued function ( , )H m m
nU C J C C R , ( ,0,0) 0U t for all
t J and ( , , )tU t x x is locally Lipschitz in x for ( , , ) ( ) ( )tt x x J S A C A ;
2) there exist a real vector , 0R , constant m m matrices ( )A and ( )B and a
comparison function 1 2( ), ( ( ) ), 1, 2,...,i i tu x u x i m , 1 2,i iu u K , such that 1 1( ) ( ) ( )Tu x A u x
, 1
( , , )
m
i j ij t
i j
v t x x
2 2( ) ( ) ( )T
t tu x B u x for all ( , , ) ( ) ( )tt x x J S A C A ;
3) there exists a comparison function ( , )W C J R R such that ( , , , )tD V t x x
( , ( , , , ))tW t V t x x for all ( , , ) ( ) ( )tt x x J S A C A and all matrices of interactions
t tE E ;
4) the matrices A and B are positive definite and ( ) ( ) ( ) ( )M mB a A b where
( )m A is the minimal and ( )M B is the maximal eigenvalues of the matrices A and B re-
spectively and ,a b are of class K .
Then the certain type of practical stability of zero solution to the equation
0 0( , ), ( ) 0
du
W t u u t u
dt
(14)
implies the certain type of connected practical stability of system (2).
Proof. Note first that under conditions (1) and (2) of Theorem 4 for the function
( , , )tV t x x determined by (11) the estimate
( ) ( ) ( , , ) ( ) ( ( ) )m t M tA b x V t x x B a x (15)
is true. This follows from the fact that for function 1 2, , 1, 2,...,i iu u K i m , there exist
functions ( ( ) )ta x and ( )b x of class K such that 1 1( ) ( ) ( )Tb x u x u x and
2 2( ( ) ) ( ( ) ) ( ( ) ).T
t t ta x u x u x Further we have from condition (3) of Theorem 4 for the
function
0 00 0( ) ( , ( , )( ), ( , ))t tm t V t x t x t x t x ( ) ( , ( ))D m t W t m t which together with the
condition
00 0( , , )tV t x x u yield the estimate
0 00 0 0 0( , ( , )( ), ( , )) ( , , ),t t tV t x t x t x t x r t t u 0t t (16)
according to the comparison principle (see[1] Theorem 4.1.1). Let the zero solution of equa-
tion (14) be practically stable. Given ( ( ) ( ), ( ) ( ))M mB a A b A , we have
0 0( , , ) ( ) ( )mu t t u A b A , (17)
provided that
0 ( ) ( )Mu B a . (18)
144
Let
0x and
0
( )tx . (19)
We shall demonstrate that
00( , )( )tx t x t A for all 0 .t t
Assume that this is not true and that there exists 1 0t t such that for the solution
00( , ) ( )tx t x t with initial condition (19) the correlations
00 1( , ) ( )tx t x t A and
00( , ) ( )tx t x t A hold for 0 1t t t .
Estimate (15) yields
0 1 01 0 1 0( , ( , )( ), ( , )) ( ) ( )t t t mV t x t x t x t x A b A (20)
Let
0 0 00 0 0 0 0( , ( , )( ), ( , ))t t tu V t x t x t x t x . Then for all 0 1t t t , estimate (16) is valid,
where 0 0( , , )r t t u is the maximal solution of equation (14). Since 0 2( ) T
Mu B u
0 02( ( ) ) ( ( ) )t tx u x ( ) ( ),M B a we find by the comparison principle and inequalities (15).
1 0 1 0( ) ( ) ( ) ( ) ( )T
m mA b A A u x u x
0 1 01 0 1 0( , ( , )( ), ( , ))t t tV t x t x t x t x 1 0 0( , , ) ( ) ( )mr t t u A b A .
(21)
The obtained contradiction shows that 1t J and therefore system (2) is connectedly practi-
cally stable.
Р Е ЗЮМ Е . Методом об‘єднання матрично-значних функціоналів Ляпунова і теореми порів-
няння досліджено зв‘язну стійкість за Ляпуновим і практичну стійкість великих систем з запізнен-
ням. Запропоновано ряд нових достатніх умов. Результати мають не лише теоретичний сенс, але
також практичне значення.
1. Lakshimikantham V., Leela S., Martynyuk A.A. Practical Stability of Nonlinear Systems. – Singapore:
World Scientific, 1990. – 215 p.
2. Lakshmikantham V.,Leela S., Sivasundaram S. Lyapunov functions on product spaces and theory of delay
differential equation // J. Math. Anal. and Appl. – 1991. – 154 – P. 391 – 402.
3. Martynyuk A.A., Sun Z.Q. A matrix-valued Lyapunov functional and stability of systems with delay
// Dokl. Akad. Nauk. – 1998. – 359, N 2. –P.165 – 167.
4. Martynyuk A.A., Sun Z.Q. On Connected Practical Stability of Motion of Systems with Delay// Int. Appl.
Mech. – 1999. – 35, N 1. – P. 87 – 92.
5. .Martynyuk A.A., Sun Z.Q. Stability analysis for nonlinear system with small parameter. – Beijin: Science
Publishing House, 2006. – P. 212 – 214.
6. Martynyuk. A.A., Martynyuk-Chernienko Yu.A. Uncertain Dynamical Systems: Stability and Motion Con-
trol. – Boca – Raton: CRC Press, 2012. – 237 p.
From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 10.09.2012 Утверждена в печать 26.06.2013
|
| id | nasplib_isofts_kiev_ua-123456789-87804 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0032-8243 |
| language | English |
| last_indexed | 2025-12-01T04:00:02Z |
| publishDate | 2013 |
| publisher | Інститут механіки ім. С.П. Тимошенка НАН України |
| record_format | dspace |
| spelling | Sun, J.F. Wang, X.L. 2015-10-25T18:43:15Z 2015-10-25T18:43:15Z 2013 Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function / J.F. Sun, X.L. Wang // Прикладная механика. — 2013. — Т. 49, № 5. — С. 139-144. — Бібліогр.: 6 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/87804 By the method of combining the matrix-valued Lyapunov functional and comparison theorem, connected Lyapunov stability and practical stability of large scale delay system are studied deeply. A series of new sufficient conditions are proposed. These results are not only of theoretical but also of practical value. Методом об‘єднання матрично-значних функціоналів Ляпунова і теореми порівняння досліджено зв‘язну стійкість за Ляпуновим і практичну стійкість великих систем з запізненням. Запропоновано ряд нових достатніх умов. Результати мають не лише теоретичний сенс, але також практичне значення. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function Анализ связной устойчивости систем с задержкой с помощью матрично-значной функции Ляпунова Article published earlier |
| spellingShingle | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function Sun, J.F. Wang, X.L. |
| title | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function |
| title_alt | Анализ связной устойчивости систем с задержкой с помощью матрично-значной функции Ляпунова |
| title_full | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function |
| title_fullStr | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function |
| title_full_unstemmed | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function |
| title_short | Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function |
| title_sort | connected stability analysis of delay systems via the matrix-valued lyapunov function |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/87804 |
| work_keys_str_mv | AT sunjf connectedstabilityanalysisofdelaysystemsviathematrixvaluedlyapunovfunction AT wangxl connectedstabilityanalysisofdelaysystemsviathematrixvaluedlyapunovfunction AT sunjf analizsvâznoiustoičivostisistemszaderžkoispomoŝʹûmatričnoznačnoifunkciilâpunova AT wangxl analizsvâznoiustoičivostisistemszaderžkoispomoŝʹûmatričnoznačnoifunkciilâpunova |