Controllability of Matrix Sylvester System and Sylvester Integro-differential System
In the present article, Sylvester matrix first order differential system and matrix first order integro-differential system are studied. A set of sufficient conditions for controllability and complete controllability of the system is presented. As a necessary tool, a variation of parameter formula i...
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nasplib_isofts_kiev_ua-123456789-618642025-02-09T15:21:16Z Controllability of Matrix Sylvester System and Sylvester Integro-differential System Murty, K.N. Krishna, M.V. Ramesh, P. Математические методы и модели In the present article, Sylvester matrix first order differential system and matrix first order integro-differential system are studied. A set of sufficient conditions for controllability and complete controllability of the system is presented. As a necessary tool, a variation of parameter formula is developed for the non-linear Sylvester system. Исследованы дифференциальные и интегро-дифференциальные матричные системы Сильвестра первого порядка. Представлен набор достаточных условий управляемости и полной управляемости систем. Как необходимый инструмент для нелинейной системы Сильвестра получена разновидность параметрической формулы. Досліджено диференціальні та інтегро-диференціальні матричні системи Сильвестра першого порядку. Наведено набір достатніх умов керованості та повної керованості систем. Як необхідний інструмент для нелінійної системи Сильвестра отримано різновид параметричної формули. 2013 Article Controllability of Matrix Sylvester System and Sylvester Integro-differential System / K.N. Murty, M.V. Krishna, P. Ramesh // Электронное моделирование. — 2013 — Т. 35, № 1. — С. 43-55. — Бібліогр.: 5 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/61864 en Электронное моделирование application/pdf Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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Математические методы и модели Математические методы и модели |
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Математические методы и модели Математические методы и модели Murty, K.N. Krishna, M.V. Ramesh, P. Controllability of Matrix Sylvester System and Sylvester Integro-differential System Электронное моделирование |
| description |
In the present article, Sylvester matrix first order differential system and matrix first order integro-differential system are studied. A set of sufficient conditions for controllability and complete controllability of the system is presented. As a necessary tool, a variation of parameter formula is developed for the non-linear Sylvester system. |
| format |
Article |
| author |
Murty, K.N. Krishna, M.V. Ramesh, P. |
| author_facet |
Murty, K.N. Krishna, M.V. Ramesh, P. |
| author_sort |
Murty, K.N. |
| title |
Controllability of Matrix Sylvester System and Sylvester Integro-differential System |
| title_short |
Controllability of Matrix Sylvester System and Sylvester Integro-differential System |
| title_full |
Controllability of Matrix Sylvester System and Sylvester Integro-differential System |
| title_fullStr |
Controllability of Matrix Sylvester System and Sylvester Integro-differential System |
| title_full_unstemmed |
Controllability of Matrix Sylvester System and Sylvester Integro-differential System |
| title_sort |
controllability of matrix sylvester system and sylvester integro-differential system |
| publisher |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| publishDate |
2013 |
| topic_facet |
Математические методы и модели |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/61864 |
| citation_txt |
Controllability of Matrix Sylvester System and Sylvester Integro-differential System / K.N. Murty, M.V. Krishna, P. Ramesh // Электронное моделирование. — 2013 — Т. 35, № 1. — С. 43-55. — Бібліогр.: 5 назв. — англ. |
| series |
Электронное моделирование |
| work_keys_str_mv |
AT murtykn controllabilityofmatrixsylvestersystemandsylvesterintegrodifferentialsystem AT krishnamv controllabilityofmatrixsylvestersystemandsylvesterintegrodifferentialsystem AT rameshp controllabilityofmatrixsylvestersystemandsylvesterintegrodifferentialsystem |
| first_indexed |
2025-11-27T09:07:05Z |
| last_indexed |
2025-11-27T09:07:05Z |
| _version_ |
1849933878773415936 |
| fulltext |
K.N. Murty
1
, M.V. Krishna, P. Ramesh
Department of Science and Humanities,
Sreenidhi Institute of Science and Technology
(Yamnampet, Hyderabad — 501301 (A.P.) India,
nkanuri@hotmail.com)
Controllability of Matrix Sylvester System
and Sylvester Integro-differential System
In the present article, Sylvester matrix first order differential system and matrix first order
integro-differential system are studied. A set of sufficient conditions for controllability and com-
plete controllability of the system is presented. As a necessary tool, a variation of parameter for-
mula is developed for the non-linear Sylvester system.
Èññëåäîâàíû äèôôåðåíöèàëüíûå è èíòåãðî-äèôôåðåíöèàëüíûå ìàòðè÷íûå ñèñòåìû
Ñèëüâåñòðà ïåðâîãî ïîðÿäêà. Ïðåäñòàâëåí íàáîð äîñòàòî÷íûõ óñëîâèé óïðàâëÿåìîñòè è
ïîëíîé óïðàâëÿåìîñòè ñèñòåì. Êàê íåîáõîäèìûé èíñòðóìåíò äëÿ íåëèíåéíîé ñèñòåìû
Ñèëüâåñòðà ïîëó÷åíà ðàçíîâèäíîñòü ïàðàìåòðè÷åñêîé ôîðìóëû.
K e y w o r d s: control function, resolvent matrix, sylvester system, integro-differential equa-
tion, Fubini’s theorem, Volterra integro-differential equation, nonlinear systems.
1. Introduction. In this paper, we shall be concerned with the general first order
Sylvester system
� � � �T t A t T t T t B t C t U t D t( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), (1)
where A, B are continuous (n � n) matrices on J = [t0, t1] and C is an (n �m) con-
tinuous matrix, U is an (m � m) continuous matrix on J, and D is an (m � n) ma-
trix. The Sylvester system (1) arises in a number of areas of control engineering,
feedback systems, dynamical systems and its general form solution in terms of
solution of fundamental matrix is given by Murty, Howell and Sivasundaram in
1992 [1]. Most of the results presented on control systems are new and include
general first order vector system as a particular case. More specifically, the pa-
per is organized as follows.
Section 2, presents general solution of (1) in-terms of two fundamental ma-
trix solutions of � �T A t T( ) and � �T B t T* ( ) and then present a set of sufficient
conditions for the controllability of (1).
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 43
1 2010 Mathematics Subject Classification primary: 37N35, 93Cxx
� K.N. Murty, M.V. Krishna, P. Ramesh, 2013
In section 3, we consider the general first order integro-differential equation
of the form
� � � � ��T t A t T t T t B t K t s T s T s K t
t
t
( ) ( ) ( ) ( ) ( ) [ ( , ) ( ) ( ) ( ,
0
1 2 s ds F t)] ( )� , (2)
satisfying T t T( )0 0� , where K t s1( , ) and K t s2( , ) are continuous square matrices
of the order (n � n) and( , )t s R� �
2 and F C R R n n� �
�[ , ]. We first present the varia-
tion of parameters formula for (1) and then present sufficient conditions for the
controllability of the integro-differential system (2). Section 4 deals with nonlin-
ear control system. In this section, we consider the matrix integro-differential
system of the form
� � � � �T t A t T t U t T t T t B t T t U t K
t
t
( ) ( , ( ), ( )) ( ) ( ) ( , ( ), ( ))
0
1( , , ( ), ( )) ( )t s T s U s T s �
� �T s K t s T s U s ds C t T t U t U t D t T2 2( ) ( , , ( ), ( )) ( , ( ), ( )) ( ) ( ,* ( ), ( ))t U t (3)
and present a set of sufficient conditions for its controllability, and then present
sufficient conditions for complete controllability of (3). Note that the matrices
involved are of appropriate dimension and are all continuous (n � n) matrices in
their arguments. Here the nonlinear matrix A is such that the elements aij �
� a t T t V tij ( , ( ), ( )) are scalars. Some remarks are applicable in respect of the
matrices B, K1, K2, C and D [2, 3].
In [1] the general solution of the homogeneous system � � �T t A t T t( ) ( ) ( )
�T t B t( ) ( ) is given by T t Y t CZ t( ) ( ) ( )*� , where Y (t) is a fundamental matrix
solution of � �T t A t T t( ) ( ) ( ) and Z (t) is a fundamental matrix solution of � �T t( )
�B t T t* ( ) ( ). In this paper, we seek a particular solution of the nonhomogeneous
system.
� � � �T t A t T t T t B t F t( ) ( ) ( ) ( ) ( ) ( ) (4)
in the form Y t( )C t( ) Z t* ( ). It is easy to verify that
C t Y s F s Z s ds
t
t
( ) ( ) ( ) ( ) .*�
�
1 1
0
Thus a particular solution of (4) is given by
T t Y t Y s F s Z s ds Z tp
t
t
( ) ( ) ( ) ( ) ( ) ( )* *�
�
�
�
�
�
�
�
1 1
0
,
K.N. Murty, M.V. Krishna, P. Ramesh
44 ISSN 0204–3572. Electronic Modeling. 2013. V. 35. ¹ 1
and any solution of (4) is of the form
T t Y t CZ t Y t Y s F s Z s ds
t
t
( ) ( ) ( ) ( ) ( ) ( ) ( )* *� �
�
�
�
�
�
�
1 1
0
�
Z t* ( ).
Using this general solution, we can at once write the general solution of the con-
trol system (1) as
T t Y t CZ t Y t Y s C s U s D s Z s ds
t
t
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* *� � 1 1
0
�
�
�
�
�
�
�
Z t* ( ).
2. Controllability of the Sylvester system. In this section, we shall be con-
cerned with establishing a necessary and sufficient condition for the controll-
ability and observability of the nonhomogeneous system (1) under smooth con-
ditions. The Sylvester system (1) is said to be time invariant, if all the coefficient
matrices are constant, otherwise it is called time varying.
Definition1. The linear time varying system (1) is said to be controllable on
[ , ]t t0 1 , if for any initial time t0 and any initial state T t T( )0 0� there exists a con-
tinuous input signal U(t) such that the corresponding solution of (1) satisfies
T t T( )1 1� .
The time varying system (1) is said to be completely controllable on [ , ]t t0 1 ,
if it is controllable for all t t t�[ , ]0 1 . We use the following notation Y t Y t( ) ( ) �1
0
�� ( , )t t0 and Z t Z t t t( ) ( ) ( , ) �1
0 0� .
Theorem 1. The system (1) is completely controllable, if, and only if, the
( )n n� symmetric matrix
V t t t s C s C s t s dsT T
t
t
( , ) ( , ) ( ) ( ) ( , )0 1 0 0
0
1
� � � � ,
(5)
where � ( , )t t0 is the fundamental matrix solution � �T A t T( ) satisfying
� ( , )t t I0 0 � is non-singular, and that for some positive constant C, det
( ( , ))UC t t C0 1 � . The control function U defined by
U t C t t t V t t T t t T t tT T( ) ( ) ( , ) ( , ) ( , ) ( ,� � �0
1
0 1 0 1 0 1 1 0��� �) ( , )�� t t0 (6)
transfer T (t0) = T0 into T (t1) = T1.
P r o o f. First, we suppose thatV t t( , )0 1 given by (5) is non-singular. Then
U (t) given by (6) is well defined. Further, any solution T (t) of (1) satisfying
T t T( )0 0� is given by
T t t t T t t( ) ( , ) ( , )� �� 0 0 0��
Controllability of Matrix Sylvester System and Sylvester Integro-differential System
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 45
�
�
�
�
�
�
�
�� �( , ) ( , ) ( ) ( ) ( , ) (t t t s C s U s t s ds t
t
t
0 0 0 0
0
� �� � , )t .
Substitute the general form of the control function U(t) given by (6) in the above
equation, we get
T t t t T t t t t t s C s C sT( ) ( , ) ( , ) ( , ) ( , ) ( ) ( )1 1 0 0 0 1 1 0 0� � � ��� �T
t
t
t s( , )0
0
� �
� � V t t T t t T t t ds t t1
0 1 0 1 0 1 1 0 0 1( , )[ ( , ) ( , )] ( , )� � �� �
� � � � ��( , ) ( , ) ( , ) ( , ) ( , )t t T t t t t V t t V t t1 0 0 0 1 0 1 0 1
1
0 1
� �� �T t t T t t t t0 1 0 1 1 0 0 1� � �( , ) ( , ) ( , )� �
� �� � � � � �� �( , ) ( , ) ( , ) ( , ) ( , )t t T t t t t T t t t t1 0 0 0 1 1 0 0 0 1 0 1 ( , )t t1 0 �
� � �T t t t t t t T t t T1 1 0 0 1 0 0 1 1 1 1� � � ��� �( , ) ( , ) ( , ) ( , ) .
Thus, the control function U transforms T t T( )0 0� into T t T( )1 1� and hence the
Sylvester system is controllable. This is true for all t t t�[ , ]0 1 , it follows that the
Sylvester system is completely controllable.
Conversely, suppose that the system (1) is completely controllable. Let� be
an arbitrary constant matrix of the order (n n� ). Since V is symmetric, we have
� � � �* * * *( , ) ( ) ( ) ( , )V t s C s C s t s ds
t
t
� �� � �0 0
0
1
0 .
Let � ��� * ( , ) ( )� t s C s0 . Then
� � � � �* * ( , ) ( , ) | |V t s t s ds ds
t
t
t
t
� � �� �0 0
2
0
1
0
1
0 . (7)
HenceV t t( , )0 1 is positive semi-definite. Suppose that there exists an ~� � 0 such
that ~ ~� �TV �0. Then in view of (7),
~
( , )� t s ds
t
t
0
2
0
1
0� � ,
K.N. Murty, M.V. Krishna, P. Ramesh
46 ISSN 0204–3572. Electronic Modeling. 2013. V. 35. ¹ 1
which means
~
� �0on [ , ]t t0 1 . Since the system is completely controllable, there
exists a control V such that T t( )1 0� if T t( ) ~
0 ��. Hence
T t t t s C s u s s t ds
t
t
1 0 1 0 0
0
1
� �
�
�
�
�� � �( , ) ~ ( , ) ( , ( ), ( , ))� �
�
�
�
��� ( , )t t0 1 0,
which implies
~ ( , ) ( , ( ), ( , ))� �� � � �t s C s u s s t ds
t
t
0 0
0
1
,
and consequently |~| ~ ~� � � � �2 � V , which is a contradiction. Since ~� �� implies
|~|� �2 � . Hence the claim.
3. Integro-differential system. In this section we develop the method of
variation of parameters formula for integro-differential equations of the Volter-
ra type in terms of resolvent kernels and then offer sufficient conditions for the
controllability of the Sylvester matrix integro-differential system [2]. More spe-
cifically, we consider the matrix linear integro-differential equation of the form
� � � ��T t A t T t K t s T s ds F t
t
t
( ) ( ) ( ) ( , ) ( ) ( )1
0
, T t T( )0 0� , (8)
where A (t), K t s1( , ) are (n n� ) continuous matrices for t R� � and ( , )t s R� �
2
and � �F C R R n n� �
�, and T is a square matrix of the order n. We get for
t s t0 � � � �
� ( , ) ( ) ( , )t s A t K t d
s
t
� �� 1 � �,
R t s I R t s d
s
t
1 1( , ) ( , ) ( , )� �� � � �� , (9)
where I is the identity matrix of the order n and K t s t s R t s1 1 0( , ) ( , ) ( , )� � �� for
t t s0 � � . We shall state the following result relative to the linear integro-differ-
ential equation (8).
Theorem 2. Assume that A (t), K t s1( , ) are (n n� ) continuous matrices for
t R� � , ( , )t s R� �
2 and � �F C R R n n� �
�, . Then the solution of (8) is given by
T t R t t T R t s F s ds
t
t
( ) ( , ) ( , ) ( )� ��1 0 0 1
0
, T t T( )0 0� ,
Controllability of Matrix Sylvester System and Sylvester Integro-differential System
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 47
where R t s( , ) is the unique solution of the partial differential equation
�
�
� � �
R
s
t s R t s A s R t K s d
s
t
1
1 1 1 0( , ) ( , ) ( ) ( , ) ( , )� � �� (10)
with R t t I1( , ) � .
P r o o f. Since � is continuous, it follows that R1 in (9) exists and subse-
quently
�
�
R t s
s
1( , )
exists and satisfies (10). Let T (t) be a solution of (8) for t t� 0.
Then, if we set p s R t s T s( ) ( , ) ( )� 1 , we have
� � � �p s
R t s
s
T s R t s A s T s K s u T u du( )
( , )
( ) ( , ) ( ) ( ) ( , ) ( )
�
�
1
1 1
t
s
F s
0
� �
�
�
�
�
�
�
( ) .
Integrating in between the limits t0 to t yields,
p t p t
R t s
s
T s R t s A s T s R t
t
t
( ) ( )
( , )
( ) ( , ) ( ) ( ) ( � � ��0
1
1 1
0
�
�
, ) ( )s F s ds
��
��
�
�
�
�
�
�
�
�
� �
t
t
t
s
R t s K s u T u du ds
0 0
1 1( , ) ( , ) ( ) .
Applying Fubini’s theorem [4], we get
p t R t t T
R t s
s
R t s A s
t
t
( ) ( , )
( , )
( , ) ( ) � �
��
��1 0 0
1
1
0
�
�
�
�
�
�
�� �
s
t
t
t
R t u K s u du T s ds R t s F s ds1 1 1
0
( , ) ( , ) ( ) ( , ) ( ) .
Using (10), we get
p t R t t T R t s F s ds
t
t
( ) ( , ) ( , ) ( )� ��1 0 0 1
0
.
Since p t R t t T t( ) ( , ) ( )� 1 and R t t I1( , ) � , we have
T t R t t T R t s F s ds
t
t
( ) ( , ) ( , ) ( )� ��1 0 0 1
0
.
Now, to prove that T is a solution of (8), let T (t) be the solution of (10) satis-
fying T t T( )0 0� existing for t t0 � � �. Then
K.N. Murty, M.V. Krishna, P. Ramesh
48 ISSN 0204–3572. Electronic Modeling. 2013. V. 35. ¹ 1
t
t
t
t
R t s T s ds R t t T t R t t T
R
0 0
1 1 1 0 0
1� �� � ( , ) ( ) ( , ) ( ) ( , )
(� t s
s
T s ds
, )
( )
�
�
� � �
t
t
t
t
R t s F s ds
R t s
s
T s ds
0 0
1
1( , ) ( )
( , )
( )
�
�
.
Using (10) and Fubini’s theorem, we get
t
t
t
s
R t s T s A s T s K s u T u du F s
0 0
1 1� ��
�
( , ) ( ) ( ) ( ) ( , ) ( ) ( )�
�
�
�
�
�ds 0 .
Since R t s1( , ) is a non-zero continuous matrix for t s t0 � � � �, we have
� ��T s A s T s K s u T u du F s
t
s
( ) ( ) ( ) ( , ) ( ) ( )1
0
0 .
Therefore, T is a solution of
� � � �T t A t T t K s u F u du
t
s
( ) ( ) ( ) ( , ) ( )1
0
.
Thus the proof of the theorem is complete.
Theorem 3. The matrix integro-differential system
� � � ��T t A t T t K t s T s ds C t U t
t
t
( ) ( ) ( ) ( , ) ( ) ( ) ( )1
0
, t J� , (11)
T t T( )0 0�
is completely controllable, if, and only if, the controllability matrix
� ( , ) ( , ) ( ) ( ) ( , )* *t t R t C s C s R t s ds
t
t
0 1 1 0 1 0
0
1
� � �
is non-singular, where R t s1( , ) is the resolvent matrix. The control function
U t C t R t t t t T R t t T( ) ( ) ( , ) ( , )[ ( , ) ]*�
1 0
1
0 1 0 0 1 1�
defined for t t t0 1� � transfers T t T( )0 0� to T t T( )1 1� .
Controllability of Matrix Sylvester System and Sylvester Integro-differential System
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 49
P r o o f. Any solution T (t) of (11) is given by
T t R t t T R t C U d
t
t
( ) ( , ) ( , ) ( ) ( )*� ��0 0 1
0
� � � �
and hence
T t R t t T R t C U d
t
t
( ) ( , ) ( , ) ( ) ( )*
1 1 0 0 1 0
0
1
� �
�
�
�
�
�
�
�� � � � � R t t1 1 0( , )�
� � T R t C C R t t t T R t0 0 0
1
0 1 0 1 0( , ) ( )( ( ) ( , ) ( , ))( (* *� � � � � , ) )t T d
t
t
1 1
0
1
��
�
�
�
�
�
�
�
� � R t t T t t t t T R t t T T1 1 0 0 0 1
1
0 1 0 0 1 1( , )[ ( , ) ( , )( ( , ) )]� � 1.
Converse is similar to the proof of Theorem 1.
Theorem 4. Assume that B (t) and K t s2( , ) are continuous (n n� ) matrices
for t R� � , ( , )t s R� �
2 and � �F C R R n n� �
�, . Then the solution of
� � � ��T t T t B t T s K t s ds F t
t
t
( ) ( ) ( ) ( ) ( , ) ( )2
0
, (12)
T t T( )0 0�
is given by
T t T R t t F R t d
t
t
( ) ( , ) ( ) ( , )* *� ��0 2 0 2
0
� � �,
where R2 (t, s) is the resolvent kernel and is the unique solution of
�
�
� � �
s
R t s B s R t s K s R t d
s
t
2 2 2 2 0* * *( , ) ( ) ( , ) ( , ) ( , )� � �� (13)
with R t t I2( , ) � .
P r o o f. The proof is similar to the proof of Theorem 2.
Theorem 5. The matrix integro-differential system (12) is completely con-
trollable, if, and only if, the controllability matrix
� � � � � �( , ) ( , ) ( ) ( ) ( , )* *t t R t D D R t d
t
t
0 1 2 0 2 0
0
1
� �
K.N. Murty, M.V. Krishna, P. Ramesh
50 ISSN 0204–3572. Electronic Modeling. 2013. V. 35. ¹ 1
is non-singular, where R t s2( , ) is the resolvent matrix. The control function
U (t) given by
� �U t D T R t t t t T R t t T( ) ( ) ( , ) ( , ) ( , )* * *�
2 0
1
0 0 2 0 1 1�
defined for t t t0 1� � transfers T t T( )0 0� to T t T( )1 1� .
P r o o f. The proof is similar to Theorem 1.
We shall now consider the superposition of these two systems and present a
set of sufficient conditions for the complete controllability of the Sylvester
integro-differential system
� � � � ��T t A t T t T t B t K t s T s
t
t
( ) ( ) ( ) ( ) ( ) [ ( , ) ( )1
0
� �T s K t s ds C t U t D t( ) ( , )] ( ) ( ) ( )*
2 , (14)
T t T( )0 0� .
Theorem 6. The matrix Sylvester integro-differential system (14) satisfy-
ing T t T( )0 0� has a unique solution given by
T t R t t T R t t R t C U D t R( ) ( , ) ( , ) ( , ) ( ) ( ) ( , )* * *� �
1 0 0 2 0 1 2� � � � ( )t d
t
t
�
0
� ,
where R1 and R2 are the solutions of the partial differential equations (10) and
(13), respectively.
P r o o f. The proof is similar to Theorem 2.
Theorem 7. The matrix integro-differential system (13) is completely con-
trollable, if, and only if, the
� ( , ) ( , ) ( ) ( ) ( , )* *t t R t s C s C s R t s ds
t
t
0 1 1 0 1 0
0
1
� �
and
� ( , ) ( , ) ( ) ( ) ( , )* *t t R t s D s D s R t s ds
t
t
0 1 2 0 2 0
0
1
� �
are non-singular, where R1(t, s) and R2(t, s) are resolvent matrices. The control
function U (t) given by
U t C t R t t t t T R t t T R t( ) ( ) ( , ) ( , ) ( , ) ( ,* *�
1 0
1
0 1 0 1 0 1 1 2 1� � �t0 ) �
� � 1
0 1 2 0 1( , ) ( , ) ( )*t t R t t D t
is defined for t t t0 1� � transfers T t T( )0 0� to T t T( )1 1� .
Controllability of Matrix Sylvester System and Sylvester Integro-differential System
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 51
P r o o f. Any solution T of the matrix Sylvester integro-differential system
is given by
T t R t t T R t t R t C U D R t( ) ( , ) ( , ) ( , ) ( ) ( ) ( ) (* * *� �
1 0 0 2 0 1 2� � � � , )� �d
t
t
0
� . (15)
Substituting the general form of the control U (t) in (15), we get
T t R t t T R t t( ) ( , ) ( , )*
1 1 1 0 0 2 1 0� �
� �R t t R t C
t
t
1 1 0 1 0
0
1
( , ) ( , ) ( )� � � C R t t t T( ) ( , ) ( , )[*� �1 0
1
0 1 0�
R t t T R t t t t R t t D1 0 1 1 2 1 0
1
0 1 2 1( , ) ( , )] ( , ) ( , ) ( )*� � � D R t d* *( ) ( , )� � � �2 0
� R t t T R t t R t t t t t t
1 1 0 0 2 1 0 1 1 0 0 1
1
0 1( , ) ( , ) ( , ) ( , ) ( , )* � � [T0
R t t T R t t t t t t R t t1 0 1 1 2 1 0 0 1
1
0 1 2 0 1( , ) ( , )] ( , ) ( , ) ( , )*� � �
� �R t t T R t t R t t T R t t
1 1 0 0 2 1 0 1 1 0 0 2 0 1( , ) ( , ) ( , ) ( , )* *
� �R t t R t t T R t t R t t R t t1 1 0 0 1 1 2 1 0 2 0 1 1 1 1( , ) ( , ) ( , ) ( , ) ( , )* * T R t t T1 2 1 1 1
* ( , ) � .
Conversely, suppose the system (14) is completely controllable. Then it can
easily be proved as in Theorem 1, that� ( , )t t0 1 and� ( , )t t0 1 are positive definite
matrices.
4. Nonlinear control system. Observe that the system (4) is nonlinear. We
aim at finding the existence of controllability conditions so that it be completely
controllable. For this purpose, let
~
( )T t and
~
( )U t be two given n n� matrices con-
tinuous in t J� . The associated linear matrix integro-differential equation with
(4) is given by
� � � �T t A t T t U t T t T t B t T t U t( ) ( ,
~
( ),
~
( )) ( ) ( ) ( ,
~
( ),
~
( ))
� �K t s T s U s T s T s K t s T s U s ds1 2( , ,
~
( ),
~
( )) ( ) ( ) ( , ,
~
( ),
~
( ))
t
t
0
� �
�C t T t U t U t D t T t U t( ,
~
( ),
~
( )) ( ) ( ,
~
( ),
~
( ))* , (16)
T t T( )0 0� ,
where T0 is a given (n n� ) matrix. Note that (16) is similar to the system (13).
Since (16) is linear, Theorems 2 and 4 are applicable. Hence the resolvent matri-
K.N. Murty, M.V. Krishna, P. Ramesh
52 ISSN 0204–3572. Electronic Modeling. 2013. V. 35. ¹ 1
ces R t s T s U s1( , ,
~
( ),
~
( )), R t s T s U s2( , ,
~
( ),
~
( )) exist for t J� and yield the solution
of the initial value problem (16) and is given by
T t R t t T t U t T R t t T t U t R( ) ( , ,
~
( ),
~
( )) ( , ,
~
( ),
~
( ))*� �0 0 2 0 1
( , ,
~
( ),
~
( ))t s T s U s
t
t
��
0
�C s T s U s U s D t s T s U s R t s T* * *( ,
~
( ),
~
( )) ( ) ( , ,
~
( ),
~
( ) ( , ,
~
2 ( ),
~
( )))s U s ds. (17)
The representation of the solution given in (17) is a consequence of the su-
perposition of conclusions obtained in Theorems 2, 4 and 7. Hence we have the
following theorem.
Theorem 8. The matrix integro-differential system (17) is completely con-
trollable, if, and only if, the control matrices
� ( , ,
~
( ),
~
( )) ( , ,
~
( ),
~
( ))t t T t U t R t s T s U s
t
t
0 1 1 1 1 0
0
1
� ��
�C s T s U s C s T s U s R t s T s* *( ,
~
( ),
~
( )) ( ,
~
( ),
~
( )) ( , ,
~
( ),
~
1 0 U s ds( )) (18)
and
� ( , ,
~
( ),
~
( )) ( , ,
~
( ),
~
( ))*t t T t U t R t s T s U s
t
t
0 1 1 1 2 0
0
1
� ��
�D s T s U s D s T s U s R t s T s* *( ,
~
( ),
~
( )) ( ,
~
( ),
~
( )) ( , ,
~
( ),
~
2 0 U s ds( )) (19)
are non-singular. The control function U (t) is given by
U t C t T t U t R t t T t U t t( ) ( ,
~
( ),
~
( )) ( , ,
~
( ),
~
( )) ( ,*�
1 0
1
0� t T t U t,
~
( ),
~
( ))1 1 �
� [ ( , ,
~
( ),
~
( )), ( , ,
~
( ),
~*T R t t T t U t T R t t T t U0 1 0 1 1 1 1 2 1 0 1 ( ))]t1 �
� � 1
0 1 1 1 2 1 1 1( , ,
~
( ),
~
( )) ( , ,
~
( ),
~
( )) (*t t T t U t R t t T t U t D t T t U t,
~
( ),
~
( ))1 . (20)
Here the matrices� and� as given in (18), (19) are assumed to be non-singular.
P r o o f. Substitute (20) in (17) and prove that T (t1) = T1. Conversely, as-
sume that the Sylvester system (16) is completely controllable. Then by follow-
ing the proof of Theorem 1, we show that the control matrices� and � are posi-
tively definite. The details are omitted.
We now turn our attention to the nonlinear system given in (4). Taking the
relations (17) and (20) in consideration, we let
F C J C J C J C Jn n n n n n n n: ( ) ( ) ( ) ( )� � � �� �
Controllability of Matrix Sylvester System and Sylvester Integro-differential System
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 53
such that
F T t U t T t U t(
~
( ),
~
( )) ( ( ), ( ))� .
The operator F is continuous on C J C Jn n n n� ��( ) ( ). Let there exist a closed
bounded, convex set! in C J C Jn n n n� ��( ) ( ) such that (T, U) = F T U(
~
,
~
) for any
~
,
~
T U in !, then from the relations (17), (20), F (!) is bounded. We apply
Schauder’s fixed point theorem and conclude that there exists at least one fixed
point of F. Hence, we have the Theorem 8.
We consider a more complicated system and using Kronecker product of
matrices, we obtain sufficient conditions for the system controllability.
� � � ��T t A t T t B t K t s T s K t s ds F t
t
t
( ) ( ) ( ) ( ) ( , ) ( ) ( , ) ( )1 2
0
is equivalent to
V T t A B V T t( ( )) ( ) ( ( ))*� � " �
� " � "� [ ( , ) ( , )] ( ( )) ( ( ) )*K t s K t s V T s ds F t I
t
t
n1 2
0
, (21)
whereV T t( ( ))� is an n2 1� vector, ( )*A B" is an ( )n n2 2� matrix and K t s1( , )"
"K t s2
* ( , ) are ( )n n2 2� continuous matrices and" denotes the Kronecker product.
The following theorem is a simple consequence of Theorem 1 in [5].
Theorem 9. Assume that ( )*A B" and K K1 2" * are continuous ( )n n2 2�
matrices for t R� � and ( , )t s R� �
2 and F t I n( )" is an ( )n n2 2� continuous ma-
trix. Then the solution of (21) is given by
V T t R t t T R t s F s I dsn
t
t
( ( )) ( , ) ( , )( ( ) )� � "�1 0 0 1
0
, T t T( )0 0� ,
where R t s1( , ) is the unique solution of
�
�s
R t s R t s A s B s1 1( , ) ( , )( ( ) ( ))*� " �
� " "�R t K s K s F s I d
s
t
n1 1 2( , )( ( , ) ( , ))( ( ) )*� � � �
R1 being ( )n n2 2� matrix and the symbol " means Kronecker product.
K.N. Murty, M.V. Krishna, P. Ramesh
54 ISSN 0204–3572. Electronic Modeling. 2013. V. 35. ¹ 1
The other results like Theorem 3, 4 and 5 follow. We can consider more
general systems like
� � �T t F A t T t g B t f K t s T s g K t s d( ) ( ( )) ( ) ( ( )) ( ( , )) ( ) ( ( , ))1 2 s F t
t
t
0
� � ( ),
which can be converted into a vector system by using Kroneker product of matri-
ces as
V T t f A t g B t V T t( ( )) [ ( ( )) ( ( ))] ( ( ))*� � " �
� " � "� [ ( ( , )) ( ( , ))] ( ( )) ( ( ) )*f K t s g K t s V T s ds F t I
t
t
n1 2
0
.
The results on controllability and observability criteria can be discussed for
the above system as in linear systems [5]. In order to avoid monotony, we even
omit starting those results.
Äîñë³äæåíî äèôåðåíö³àëüí³ òà ³íòåãðî-äèôåðåíö³àëüí³ ìàòðè÷í³ ñèñòåìè Ñèëüâåñòðà ïåð-
øîãî ïîðÿäêó. Íàâåäåíî íàá³ð äîñòàòí³õ óìîâ êåðîâàíîñò³ òà ïîâíî¿ êåðîâàíîñò³ ñèñòåì.
ßê íåîáõ³äíèé ³íñòðóìåíò äëÿ íåë³í³éíî¿ ñèñòåìè Ñèëüâåñòðà îòðèìàíî ð³çíîâèä ïàðà-
ìåòðè÷íî¿ ôîðìóëè.
REFERENCES
1. Murty K.N., Howell G., Sivasundaram S. Two multi-point nonlinear Lyapunov systems.
Existence and uniqueness // J. Math. Anal. Appl. — 1992. —167. — P. 505—512.
2. Lakshmikantham V., Deo S.G. Method of Variation of Parameters for Dynamic Systems.—
Vol. 1. — Gordon and Breach Science Publishers, 1998.
3. Yamamoto Y. Controllability of nonlinear systems// J. of Optim. Theory and Appl. — 1977. —
22. — P. 41—49.
4. Miller R.K. On the linearization of Volterra integral equations// J. Math. Anal. Appl. —
1968. — 23. — P. 198—206.
5. Barnett S. Matrix differential equations and Kronecker product//SIAM Appl. Math. —
1973. — 14. — P. 1—5.
Submitted 29.10.12
Controllability of Matrix Sylvester System and Sylvester Integro-differential System
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2013. Ò. 35. ¹ 1 55
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/KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020ace0d488c9c80020c2dcd5d80020c778c1c4c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
/NOR <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>
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/SVE <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>
/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|