Stability analysis of large-scale functional differential systems
The present paper is focused on a new method for analysis of stability of solutions of a large-scale functional differential system via matrix-valued Lyapunov-Krasovskii functionals. The stability conditions are based on information about the dynamical behavior of subsystems of the general system an...
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| author | Martynyuk, A. A. Мартинюк, А. А. |
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| description | The present paper is focused on a new method for analysis of stability of solutions of a large-scale functional differential system via matrix-valued Lyapunov-Krasovskii functionals. The stability conditions are based on information about the dynamical behavior of subsystems of the general system and properties of the functions of interconnection between them. |
| first_indexed | 2026-03-24T02:40:12Z |
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UDC 517.9
A. A. Martynyuk (Inst. Mech. Nat. Acad. Sci. Ukraine, Kyiv)
STABILITY ANALYSIS OF LARGE-SCALE
FUNCTIONAL DIFFERENTIAL SYSTEMS
ANALIZ STIJKOSTI VELYKOMASÍTABNYX
FUNKCIONAL|NO-DYFERENCIAL|NYX SYSTEM
The present paper is focused on the new method of analysis of stability of solutions of large-scale
functional differential system via matrix-valued functional Liapunov – Krasovskii. The stability
conditions are based on information about dynamical behaviour of subsystems of the general system and
properties of the functions of interconnection between them.
Zaproponovano odyn novyj metod analizu stijkosti rozv’qzkiv velykomasßtabno] funkcional\-
no-dyferencial\no] systemy na osnovi matryçnoznaçnoho funkcionala Lqpunova – Krasovs\ko-
ho. Umovy stijkosti ©runtugt\sq na dynamiçnij povedinci pidsystem zahal\no] systemy ta vlas-
tyvostqx funkcij zv’qzku miΩ nymy.
1. Introduction. Stability of continuous systems of ordinary differential equations has
been studied by many authors for more that 100 years while stability investigation of
solutions to functional differential equations has been undertaken quite recently. It is
known (see, for example, [2, 12, 23]) that there are two global approaches of
qualitative investigation of the dynamical behavior of solutions to systems of this class.
One is the method of Liapunov – Razumikhin functions and the other is the method of
Liapunov – Krasovskii functionals. Each approach is being intensively developed and
new interesting results are obtained in both directions (see [1, 3 – 11, 13, 14, 22 – 24,
26]).
The method of matrix-valued Liapunov functions worked out for the last years in
qualitative stability theory of nonlinear systems (see [16, 17]) allows generalization
investigation of solutions to functional differential equations in the framework of both
of the approaches mentioned above.
In this paper, stability analysis of large-scale systems of functional differential
equations with finite delay via the matrix-valued functionals of Liapunov – Krasovskii
are considered. A new theorem on uniform asymptotic stability is established. These
results generalize some theorems in the stability theory of functional differential
equations via scalar Liapunov functionals obtained recently.
The results of this paper are arranged as follows.
We provide a statement the problem in Section 2 and provide necessary information
on functional differential equations and matrix-valued Liapunov – Krasovskii
functionals in Section 3.
In Section 4 we elaborate on the various types of qualitative properties (i.e.,
stability properties) of large-scale functional differential equations that we will
consider. In Section 5 we specialize the results of Section 4 for a class of functional
differential equations and proposes new forms of decomposition-aggregation of large-
scale systems of functional differential equations via the application of matrix-valued
functionals. This follows us to establish new sufficient stability conditions for the
systems of equations under consideration in terms of special matrices property of fixed
sign.
2. Statement the problem. For x ∈ R
n, | ⋅ | denotes the Euclidean norm of x .
Let C = C ( [ – τ, 0 ], R
n
) be the space of continuous functions which map [ – τ, 0 ] into
R
n and for ϕ ∈ C, || ϕ || ≤ sup ( )− ≤ ≤τ θ ϕ θ0 , CH is a set ϕ ∈ C for which || ϕ || < H,
and xt
, as an element of C, is defined by correlation xt
(
θ
) = x
(
t + θ
), – τ ≤ θ ≤ 0.
Throughout this paper we will use a seminorm | ⋅ | τ on C with | ϕ | τ ≤ || ϕ || for
all ϕ ∈ C.
We consider the large-scale systems described by functional differential equations
© A. A. MARTYNYUK, 2007
382 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
STABILITY ANALYSIS OF LARGE-SCALE FUNCTIONAL DIFFERENTIAL SYSTEMS 383
d x
dt
= f (
t, xt ), xt0
= ϕ0 ∈ C, t0 ≥ 0, (2.1)
where f ∈ C
(
R
+ × C, R
n
), x ∈ C
(
[ t0 – τ, ∞ ], R
n
), and xt ∈ C . In (2.1), d
x / d
t
denotes the right-hand derivative of x at x ∈ R
+. We suppose that for every ϕ ∈ C
and for every t0 ≥ 0, system (2.1) possesses a unique solution xt (
t0
, ϕ
) with xt0
= ϕ
and we denote by x
(
t
) = x
(
t; t0
, ϕ
) the value of xt
(
t0
, ϕ
) at t (for details see [2]).
Let the system (2.1) be decomposed into m interconnected subsystems described
by the equations
d x
dt
i
= f t xi t
i,( ) + g t x xi t t
m, , ,1 …( ), (2.2)
where i ∈ Im = { 1, 2,…, m },
i
m
in=∑ 1
= n , f i ∈
C R RH
n
i
i
+ ×( )C , , and g i ∈
∈
C R RH
ni
+ ×( )C , , CH =
C H1
× … ×
C Hm
, i.e.,
C Hi
is a set of ϕi ∈ C for which
|| ϕ
i
|| < Hi
, i ∈ Im
. When gi
(
t, xt ) ≡ 0 for all i ∈ Im
, from (2.2) we obtain the
isolated subsystems
d x
dt
i
= f t xi t
i,( ), xt
i
0
∈ ϕ0
i ∈ C Hi
, t0 ≥ 0 . (2.3)
We also assume that f ( t, 0, … , 0 ) = 0, fi ( t, 0 ) = 0, and gi ( t, 0, … , 0 ) = 0 for all
i ∈ Im
, so that x = 0 ( x
i = 0 ) are the solutions of (2.1) or (2.) respectively.
3. Auxiliary results. We will need some notions which we present now.
Definition 3.1. The functional U
(
t, ϕ
) = vij t( , )⋅[ ], i, j = 1, 2, … , m,
continuous and defined on R
+ × CH which together with the upper right Dini
derivative of V
(
t, ϕ, η
) = η
T U
(
t, ϕ
) η, η ∈ Rm
+ , defined by
D V t+ ( , , ) ( . )ϕ η 2 1 =
= limsup ( , ( , ), ) ( , ( , ), ) :V t h x t V t x t h ht h t+ −[ ] →{ }+
− +
0 0
1 0ϕ η ϕ η , (3.1)
solves the problem of stability of solution x = 0, is called Liapunov matrix-valued
functional.
Definition 3.2. The function w : [ 0, ∞ ) → [ 0, ∞ ), w ( 0 ) = 0, w ( r ) strictly
increasing with w( 0 ) = 0 and w ( r ) → ∞ for r → + ∞, is called a wedge
function (for short we will use w ∈ W ).
Definition 3.3. The zero solution of (2.1) is
i) stable if for each ε > 0 there exists δ = δ
(
t0
, ε
) such that [( , )t0 ϕ ∈
∈ R CH+ × , ϕ δ< ≥ ], t t0 implies | x
(
t; t0
, ϕ
) | < ε for all t ≥ t0
;
ii) uniformly stable if it is stable and if in the Definition 2.3(a) the value δ does
not depend on t0
;
iii) asymptotically stable (AS) if it is stable and for any t0 ≥ 0 there exists ∆ >
> 0 such that [ ∈ ×+( , )t R CHϕ , ϕ < ≥ ]∆, t t0 implies | x
(
t; t0
, ϕ
) | → 0 for
t → + ∞;
iv) uniformly asymptotically stable (UAS) if it is uniformly stable and there exists
δ
* > 0 such that for each ε > 0 there exists T > 0 such that [ ∈ ×+( , )t R CH0 ϕ ,
ϕ δ< ≥ ]*, t0 0 implies | x
(
t; t0
, ϕ
) | < ε for all t ≥ t0 + T.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
384 A. A. MARTYNYUK
4. Stability of large-scale system. It is known that the problem of constructing
appropriate Liapunov – Krasovskii functionals which solve the stability problem of
solution x = 0 of system (2.1) is a central one in the direct Liapunov method for
functional differential equations (2.1). The application of matrix-valued functionals
allows one to extend the class of admissible components for construction of the
appropriate Liapunov – Krasovskii functional.
Further we shall investigate systems of connected equations (2.2) and independent
subsystems (2.3) in the framework of general methodology of qualitative analysis of
large-scale systems motion.
4.1. Approach A1
. This approach is based on the system of conditions below.
Assumption 4.1. There exist functionals
vii
i
Ht R C R
i
( , ):ϕ + +× → and
vij
i j
H Ht R C C R
i j
( , , ):ϕ ϕ + × × → ,
wedge functions w̃1, w̃2, w̃3 ∈ W-class and constants ãij , b̃ij , c̃ij , i, j = 1, 2, …
… , m, and 0 < Ki < H such that
1) ˜ ˜ ( )a wii i
i
1
2 0ϕ( ) ≤ vi i ( t, ϕ
i
) ≤ ˜ ˜b wii i
i
2
2 ϕ
τ( ) + ˜ ˜c wii i
i
3
2 ϕ( ) for all ( t,
ϕ
i
) ∈ R+ × CKi
, i = 1, 2, … , m;
2) ˜ ˜ ( ) ˜ ( )a w wij i
i
j
j
1 10 0ϕ ϕ( ) ( ) ≤ vi j ( t, ϕ
i, ϕ
j
) ≤ ˜ ˜ ˜b w wij i
i
j
j
2 2ϕ ϕ
τ τ( ) ( ) +
+ ˜ ˜ ˜c w wij i
i
j
j
3 3ϕ ϕ( ) ( ) for all ( t, ϕ
i
, ϕ
j) ∈ R+ × CKi
× CK j
, i ≠ j ∈ Im .
The following assertion holds true.
Proposition 4.1. If all conditions of Assumption 4.1 are satisfied, then in the
domain of values ( t, ϕ ) ∈ R+ × CK for the functional
V
(
t, ϕ, η
) = η
T U
(
t, ϕ
) η, η ∈ Rm
+ , η > 0, (4.1)
the bilateral inequality
˜ ( ) ˜ ˜ ( )w H A H wT T
1 10 0ϕ ϕ( ) ( ) ≤ V
(
t, ϕ, η
) ≤
≤ ˜ ˜ ˜w H B H wT T
2 2ϕ ϕτ τ( ) ( ) + ˜ ˜ ˜w H C H wT T
3 3ϕ ϕ( ) ( ) (4.2)
is fulfilled for all ( t, ϕ ) ∈ R+ × CK
, where
˜ ( )wT
1 0ϕ( ) = ˜ ( ) , , ˜ ( )w wm
m
11
1
10 0ϕ ϕ( ) … ( )( ),
w̃T
2 ϕ τ( ) = ˜ , , ˜w wm
m
12
1
2ϕ ϕ
τ τ( ) … ( )( ) ,
w̃T
3 ϕ( ) = ˜ , , ˜w wm
m
13
1
3ϕ ϕ( ) … ( )( ) ,
à = ãij[ ], ãij = ã ji , B̃ = b̃ij[ ], b̃ij = b̃ji ,
C̃ = c̃ij[ ], c̃ij = c̃ ji , H = diag [ η1, … , ηm ] ,
CK = CK1
× CK2
× … × CKm
.
Proposition 4.1 is proved by direct substitution by estimates 1), 2) of Assumption
4.1 in the expression of functional
V
(
t, ϕ, η
) =
i j
m
i j ij t
,
( , )
=
∑ ⋅
1
η η v
with subsequent simple transformations.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
STABILITY ANALYSIS OF LARGE-SCALE FUNCTIONAL DIFFERENTIAL SYSTEMS 385
Assumption 4.2. There exist functionals vi i ( t, ϕ
i
), i = 1, 2, … , m, and v i j ( t,
ϕ
i, ϕ
j
), i ≠ j ∈ Im , mentioned in Assumption 4.1 which are locally Lipschitzian in
ϕ
i and ( ϕ
i, ϕ
j
) respectively, wedge functions w̃i4, i = 1, 2, … , m , and
continuous positive function dk i ( t ), dk i j ( t ) : R+ → R+, k = 1, 2, 3, i, j = 1, 2, … , m,
i ≠ j, such that
1) D tii
i+ v ( , ) ( . )ϕ 2 3 ≤ d wi i
i
1 4
2˜ ϕ
τ( ) for all ( t, ϕ
i ) ∈ R+ × CKi
;
2) D tii
i+ v ( , ) ( . )ϕ 2 2 – D tii
i+ v ( , ) ( . )ϕ 2 3 ≤
i
i j
m
ij i
i
j
jd w w
=
≠
∑ ( ) ( )
1
1 4 4˜ ˜ϕ ϕ
τ τ
for all ( t, ϕ
i, ϕ
j
) ∈ R+ × CKi
× CK j
, i ≠ j;
3) D tij
i j+ v ( , , ) ( . )ϕ ϕ 2 2 ≤ d wi i
i
2 4
2˜ ϕ
τ( ) ≤
≤
i
i j
m
ij i
i
j
jd w w
=
≠
∑ ( ) ( )
1
2 4 4˜ ˜ϕ ϕ
τ τ
+
i
i j
m
ij i
i
j
jd w w
=
≠
∑ ( ) ( )
1
3 4 4˜ ˜ϕ ϕ
τ τ
+ d wi j
j
3 4
2˜ ϕ
τ( )
for all ( t, ϕ
i, ϕ
j
) ∈ R+ × CKi
× CK j
, i ≠ j.
Proposition 4.2. If all conditions of Assumption 4.2 are satisfied, then
D V t+ ( , , ) ( . )ϕ η 2 2 ≤ ˜ ˜w DwT
4 4ϕ ϕτ τ( ) ( ) (4.3)
for all ( t, ϕ ) ∈ R+ × CK
, CK = CK1
× CK2
× … × CKm
, where
w̃4 ϕ τ( ) = ˜ , , ˜w wm
m T
14
1
4ϕ ϕ
τ τ( ) … ( )( ) ,
D ( t ) = d tij ( )[ ], di j = dj i
, i, j = 1, 2, … , m,
di i ( t ) = ηi id2
1 + 2
2
2 3
j
j i
m
i j i jd d
=
>
∑ +η η ( ), i = 1, 2, … , m,
di j ( t ) =
1
2
2
1 1ηi ij jid t d t( ) ( )+( ) + η ηi j
s
s j
m
sj
s
s j
m
sjd t d t
=
≠
=
≠
∑ ∑+
1
2
1
3( ) ( ) , (i ≠ j) ∈ Im
.
Proof. Estimate (4.3) is obtained by direct substitution by estimates 1) – 3) from
Assumption 4.2 in the expression
D V t+ ( , , )ϕ η = η ϕ ηT D U t+ ( , ) , η ∈ Rm
+ ,
where D U t+ ( , )ϕ is computed component-wise along solutions of subsystems (2.3)
and (2.2) respectively.
Estimates (4.2), (4.3) and Theorem 4.1 of the paper [15] enable us to establish new
stability conditions for the solution x = 0 of system (2.1) as follows.
Theorem 4.1. Assume that for system (2.1) the functional U ( t, ϕ ), is
constructed for the components of which all estimates of Assumption 4.1 are fulfilled
and for the upper right derivatives D
+
vi j ( t, ⋅ ) estimates 1) – 3) of Assumption 4.2
are fulfilled, and moreover
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
386 A. A. MARTYNYUK
a) in estimates (4.2) matrices A = H A HT ˜ , B = H B HT ˜ and C = H C HT ˜
are positive definite;
b) there exist r0 ≤ mini iK such that λm ( A ) w1 ( r ) – λM ( C ) w3 ( r ) > 0 for all
r ∈ ( 0, r0
), where w1 , w2 are the wedge functions such that w1 ( r ) ≤ ˜ ( ) ˜ ( )w r w rT
1 1 ,
w3 ( r ) ≥ ˜ ( ) ˜ ( )w r w rT
3 3 for any r ∈ ( 0, r0
).
Then the zero solution of system (2.1) is
a) uniformly stable, if the constant matrix D M ≥
1
2
D t D tT ( ) ( )+( ) in inequality
(4.3) is negative semidefinite;
b) uniformly asymptotically stable if the matrix DM mentioned in condition a)
is negative definite.
Proof. Under condition a) of Theorem 4.1 estimate (4.2) implies that the
functional (4.1) is positive definite and decreasing and the upper right derivative of the
functional V ( t, ϕ, η ) satisfies the condition
D V t+ ( , , ) ( . )ϕ η 2 1 ≤ 0 for all ( t, ϕ ) ∈ R+ × CK .
Therefore all conditions of Theorem 4.1(2) of the paper [15] are satisfied and the zero
solution of system (2.1) is uniformly stable. Assertion b) of this theorem is proved in
the same way in view of Theorem 4.1(3) of the same paper [15].
4.2. Approach A2
. System of conditions of Assumption 4.2 outlines a general
approach to stability analysis of zero solution of system (2.1). The essence of this
approach is that stability conditions of zero solution of system (2.1) are established
based on the analysis of dynamical properties of independent subsystems (4.3) and
qualitative estimates of interconnection functions between them. Below the other
system of conditions is presented under which stability of system (2.1) can be studied
in the framework of this approach.
Assumption 4.3. There exist functionals vi i ( t, ϕ
i
) and v i j ( t, ϕ
i, ϕ
j
),
mentioned in Assumption 4.1 wedge functions w̃i4, i = 1, 2, … , m, and constants
dk i , di j , k = 1, 2, i, j = 1, 2, … , m, such that
1) D tii
i+ v ( , ) ( . )ϕ 2 3 ≤ d wi i
i
1 4
2˜ ϕ
τ( ) for all ( t, ϕ
i
) ∈ R+ × CKi
;
2)
i
m
i ii
i
ii
iD t D t
=
+ +∑ −
1
2 2 2 3η ϕ ϕv v( , ) ( , )( . ) ( . ) +
+
j
j i
m
j ij
i jD t
=
≠
+∑ ( )
1
2 2η ϕ ϕv ( , , ) ( . ) ≤
i
m
i i
id w
=
∑ ( )
1
2 4
2˜ ϕ
τ
+
+
j
j i
m
ij i
i
j
jd w w
=
≠
∑ ( ) ( )
1
4 4˜ ˜ϕ ϕ
τ τ
for all ( t, ϕ
i, ϕ
j
) ∈ R+ × CKi
× CK j
.
Remark 4.1. In some cases condition 2) from Assumption 4.3 allows one to
estimate more precisely the effect of the interconnection functions g t x xi t t
m, , ,1 …( ), i =
= 1, 2, … , m, on the whole system dynamics.
Proposition 4.3. If all conditions of Assumption 4.3 are satisfied, then
D V t+ ( , , ) ( . )ϕ η 2 2 ≤ ˜ ˜w E wT
4 4ϕ ϕτ τ( ) ( ) (4.4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
STABILITY ANALYSIS OF LARGE-SCALE FUNCTIONAL DIFFERENTIAL SYSTEMS 387
for all ( t, ϕ ) ∈ R+ × CK , where w̃4 ϕ τ( ) is determined as in Proposition 4.2
E = [
εi j
], εi j = εj i
, i, j = 1, 2, … , m,
εi i = d1 i + d2 i
, i = 1, 2, … , m,
εi j =
1
2
dij , i ≠ j, (i, j) ∈ Im
.
This proposition is proved in the same way as Proposition 4.2.
Theorem 4.2. Assume that for system (2.2) the matrix-valued functional U ( t, ϕ )
is constructed for which all estimates of Assumption 4.1 are fulfilled and for the
upper right derivatives D
+
vi j ( t, ⋅ ) estimates of Assumption 4.3 and conditions a),
b) of Theorem 4.1 are satisfied. Then the zero solution of system (2.1) is
a) uniformly stable, if the matrix E in estimate (4.4) is negative semidefinite;
b) uniformly asymptotically stable, if the matrix E in estimate (4.4) is negative
definite.
The proof o f Theorem 4.2 is omitted, since it is similar to the proof of
Theorem 4.1.
4.3. Approach B1
. In distinction to the system of conditions from Assumptions
4.2 and 4.3, which is the basis for Theorem 4.2 we shall indicate the approach of
establishing stability conditions for zero solution of system (2.1), in which the
interacting subsystem is not divided into free subsystem and interconnection functions.
Besides some restrictions are imposed simultaneously on all interacting sub-
systems (2.1).
Assumption 4.4. There exist functionals vi i ( t, ϕ
i
) and v i j ( t, ϕ
i, ϕ
j
),
mentioned in Assumption 4.1, wedge functions w̃i4 and bounded on any finite
interval functions β1 i ( t ), β2 i ( t ), β3 i ( t ), βk i j ( t ), k = 1, 2, 3, ( i ≠ j ) ∈ Im
, such
that
1) D tii
i+ v ( , ) ( . )ϕ 2 3 ≤ β ϕ
τ1 4
2
i i
it w( ) ˜ ( ) +
j
j i
m
ij i
i
j
jt w w
=
≠
∑ ( ) ( )
1
1 4 4β ϕ ϕ
τ τ
( ) ˜ ˜
for all ( t, ϕ
i, ϕ
j
) ∈ R+ × CKi
× CK j
, i = 1, 2, … , m;
2) D tij
i j+ v ( , , ) ( . )ϕ ϕ 2 2 ≤ β ϕ
τ2 4
2
i i
it w( ) ˜ ( ) +
+
j
i j
m
ij i
i
j
jt w w
=
≠
∑ ( ) ( )
1
2 4 4β ϕ ϕ
τ τ
( ) ˜ ˜ +
+
j
i j
m
ij i
i
j
jt w w
=
≠
∑ ( ) ( )
1
3 4 4β ϕ ϕ
τ τ
( ) ˜ ˜ + β ϕ
τ3 4
2
i i
it w( ) ˜ ( ) for all ( t, ϕ
i, ϕ
j
) ∈ R+ ×
× CKi
× CK j
, ( i ≠ j ) ∈ Im
.
Proposition 4.4. If all conditions of Assumption 4.4 are satisfied, then
D V t+ ( , , ) ( . )ϕ η 2 2 ≤ ˜ ( ) ˜w t wT
4 4ϕ θ ϕτ τ( ) ( ) (4.5)
for all ( t, ϕ ) ∈ R+ × CK . Here the elements of matrix θ ( t ) = [
θi j ( t )
], θ i j = θj i for
all ( i, j ) ∈ Im
, are determined as:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
388 A. A. MARTYNYUK
θi i ( t ) = η βi i t2
1 ( ) + 2
2
2 3
j
j i
m
i j i jt t
=
>
∑ +( )η η β β( ) ( ) , i = 1, 2, … , m,
θi j ( t ) =
1
2
2
1η βi ij t( ) + η η β βi j
i
i j
m
ij
j
j i
m
ijt t
=
≠
=
≠
∑ ∑+
1
2
1
3( ) ( ) , ( i ≠ j ) ∈ Im
.
The proof of this assertion is similar to those of Propositions 4.2 and 4.3 and so are
omitted here.
Theorem 4.3. Assume that for system (2.1) the matrix-valued functional U ( t,
ϕ ), is constructed for components of which all conditions of Assumption 4.1 are
fulfilled and for the upper derivatives D
+
vi j ( t, ⋅ ) estimates of Assumption 4.4 are
satisfied as well as conditions a) and b) from Theorem 4.1.
Then the zero solution of system (2.1):
a) uniformly stable, if the constant m × m -matrix θM ≥
1
2
θ θT t t( ) ( )+( ) in
estimate (4.5) is negative semidefinite;
b) uniformly asymptotically stable, if the matrix θM mentioned in condition a)
is negative definite.
The proof of this theorem is similar to that of Theorem 4.2.
4.4. Approach B2
. This approach is based on the following system of
conditions.
Assumption 4.5. There exist functionals vi i ( t, ϕ
i
) and vi j ( t, ϕ
i, ϕ
j
) for ( i ≠
≠ j ) ∈ Im
, mentioned in Assumption 4.1, wedge functions w̃i4 and constants β i i
,
i = 1, 2, … , m, βi j
, ( i ≠ j ) ∈ Im
, such that
i
m
i ii
iD t
=
+∑ ( )
1
2
2 2η ϕv ( , ) ( . ) +
2
1
1
2
2 2
i
m
j
j i
m
i j ij
i jD t
=
−
=
>
+∑ ∑ ( )η η ϕ ϕv ( , , ) ( . ) ≤
≤
i
m
ii i
iw
=
∑ ( )
1
4
2β ϕ
τ
˜ +
i
m
j
j i
m
ij i
i
j
jw w
=
−
=
>
∑ ∑ ( ) ( )
1
1
2
4 4β ϕ ϕ
τ τ
˜ ˜
for all ( t, ϕ
i, ϕ
j
) ∈ R+ × CKi
× CK j
.
Similarly to the above proposition the following assertion takes place.
Proposition 4.5. If all conditions of Assumption 4.5 are satisfied, then
D V t+ ( , , ) ( . )ϕ η 2 2 ≤ ˜ ˜w PwT
4 4ϕ ϕτ τ( ) ( ) (4.6)
for all ( t, ϕ ) ∈ R+ × CK , where the matrix P = [
ρi j
], ρ i j = ρj i for all ( i, j ) ∈ Im
,
has the elements
ρi j = βi i
, i = 1, 2, … , m; ρi j =
1
2
βij , ( i ≠ j ) ∈ Im
.
The following result establishes stability conditions for zero solution of (2.1).
Theorem 4.4. Assume that for interacting subsystems (2.2) the matrix-valued
functional U ( t, ϕ ), is constructed, for the components of which all conditions of
Assumption 4.1 are satisfied and for the upper derivatives D
+
vi j ( t, ⋅ ) the estimates
of Assumption 4.5 are fulfilled as well as conditions a) and b) from Theorem 4.1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
STABILITY ANALYSIS OF LARGE-SCALE FUNCTIONAL DIFFERENTIAL SYSTEMS 389
Then the zero solution of system (2.1) is
a) uniformly stable if the matrix P in the inequality (4.6) is negative
semidefinite;
b) uniformly asymptotically stable if the matrix P in the inequality (4.6) is
negative definite.
The proof of this theorem is similar to that of Theorem 3.1.
Remark 4.2. Theorems 4.1 – 4.4 given a series of corollaries. Below a corollary
of Theorem 4.1, is presented being of importance in the next section.
Corollary 4.1. Assume that for system (2.2) the matrix-valued functional U ( t,
ϕ ) be constructed such that the functional U ( t, ϕ, η ) is continuous on R+ × CK ,
and locally Lipschitzian in ϕ and
i) ˜ ( ) ˜ ( )w AwT
1 10 0ϕ ϕ( ) ( ) ≤ V ( t, ϕ, η ) ≤ ˜ ˜w BwT
2 2ϕ ϕ( ) ( ) , where A, B
are constant m × m matrices and w̃1, w̃2 are wedge functions;
ii) D V t+ ( , , ) ( . )ϕ η 2 2 ≤ – ˜ ( ) ˜ ( )w DwT
3 30 0ϕ ϕ( ) ( ) for all ( t, ϕ ) ∈ R+ × C K ,
where D̃ is a constant m × m matrix;
iii) λm ( A ) > 0, λm ( B ) > 0, λM D̃( ) > 0.
Then the zero solution of system (2.2) is uniformly asymptotically stable.
5. Applications. Consider a linear delay system consisting of two interconnected
subsystems. To analyse stability of its zero solution we shall apply Approach A1
. Let
the system be of the form
ẋ1 = A1 x1 ( t ) + B1 x2 ( t ) + C1 x1 ( t – τ ) + D1 x2 ( t – τ ),
(5.1)
ẋ2 = A2 x1 ( t ) + B2 x2 ( t ) + C2 x1 ( t – τ ) + D2 x2 ( t – τ ),
where τ > 0, x1 ∈ Rm1 , x2 ∈ Rm2 , m 1 + m2 = n, Ai , Bi , Ci , Di , i = 1, 2, are
constant matrices of the corresponding dimensions.
Independent subsystems of system (5.1) are
ẋ1 = A1 x1 ( t ) + C1 x1 ( t – τ ),
(5.2)
ẋ2 = B2 x2 ( t ) + D2 x2 ( t – τ ).
For system (5.2) we construct the matrix-valued functional U ( t, ϕ ) with the
elements
v11 = x t P x tT
1 11 1( ) ( ) +
−
∫ + +
τ
0
1 11 1x t s P x t s dsT ( ) ( ) ,
v22 = x t P x tT
2 22 2( ) ( ) +
−
∫ + +
τ
0
2 22 2x t s P x t s dsT ( ) ( ) , (5.3)
v12 = x t P x tT
1 12 2( ) ( ) +
−
∫ + +
τ
0
1 12 2x t s P x t s dsT ( ) ( ) ,
which satisfy estimates characteristic for the quadratic forms
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
390 A. A. MARTYNYUK
λm P x t( ) ( )11 1
2 +
−
∫ +
τ
λ
0
11 1
2
m P x t s ds( ) ( ) ≤ v11 ( t, ϕ1 ) ≤
≤ λm P x t( ) ( )11 1
2 +
−
∫ +
r
M P x t s ds
0
11 1
2λ ( ) ( ) ,
λm P x t( ) ( )22 2
2 +
−
∫ +
τ
λ
0
22 2
2
m P x t s ds( ) ( ) ≤ v22 ( t, ϕ2 ) ≤
≤ λM P x t( ) ( )22 2
2 +
−
∫ +
τ
λ
0
22 2
2
M P x t s ds( ) ( ) , (5.4)
– λM
TP P x t x t1 2
12 12 1 2
/ ( ) ( ) ( ) –
−
∫ / ( ) + +
τ
λ
0
1 2
12 12 1 2M
TP P x t s x t s ds( ) ( ) ≤
≤ v12 ( t, ϕ1 , ϕ2 ) ≤ λM
TP P x t x t12 12 1 2( ) ( ) ( ) +
+
−
∫ ( ) + +
τ
λ
0
12 12 1 2M
TP P x t s x t s ds( ) ( ) .
Here P11 , P12 are symmetric positive definite matrices and P12 is a constant matrix.
It is easy to verify that for the functional
V ( t, x, η ) = η T U ( t, x ( s ) ) η, η = (1, 1) T, (5.5)
the bilateral estimate
u t H A H u tT T
0 0( ) ( ) +
−
∫ + +
τ
0
0 0u t s H A H u t s dsT T( ) ( ) ≤ V ( t, x, η ) ≤
≤ u t H B H u tT T
0 0( ) ( ) +
−
∫ + +
τ
0
0 0u t s H B H u t s dsT T( ) ( ) (5.6)
is valid, where
u tT
0 ( ) = x t x tT T
1 2( ), ( )( ), u t sT
0 ( )+ = x t s x t sT T
1 2( ), ( )+ +( ),
H T = H = diag (1,1),
A =
λ λ
λ λ
m M
T
M
T
m
P P P
P P P
( )
( )
11
1 2
12 12
1 2
12 12 22
− ( )
− ( )
/
/
,
B =
λ λ
λ λ
M M
T
M
T
M
P P P
P P P
( )
( )
11
1 2
12 12
1 2
12 12 22
/
/
( )
( )
.
It follows from estimate (5.6) that for functional (5.5) to be positive definite it is
sufficient that the matrix H T A H to be positive definite.
For the upper right derivative of functional (5.5) along solutions of system (5.1) one
can easily obtain the estimate
D V t x s+ ( ), ( )
( . )5 1
≤ u T G u (5.7)
where the following notations are used
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
STABILITY ANALYSIS OF LARGE-SCALE FUNCTIONAL DIFFERENTIAL SYSTEMS 391
u T = u t u t1 2( , ) , ( , )⋅ ⋅( ),
G =
λ λ λ
λ λ λ
M M M
T
M
T
M M
C F K K
K K D L
( ) ( ) ( )
( ) ( ) ( )
+
+
/
/
2
2
1 2
1 2
,
where
C =
λ λ
λ λ
M
T
M
T
M
T
M
A P P A P P C P C
P C P C P
1 11 11 1 11
1 2
11 1 11 1
1 2
11 1 11 1 11
+ +( ) [ ]
[ ] −
/
/
( )( )
( )( ) ( )
,
D =
λ λ
λ λ
M
T
M
T
M
T
M
B P P B P P D P D
P D P D P
2 22 22 2 22
1 2
22 2 22 2
1 2
22 2 22 2 22
+ +( ) [ ]
[ ] −
/
/
( )( )
( )( ) ( )
,
F =
λ λ
λ
M M
T
M
T
P A P C P C
P C P C
( ) ( )( )
( )( )
12 2
1 2
12 2 12 2
1 2
12 2 12 2
1
2
1
2
0
/
/
( )
( )
,
K =
λ λ
λ
M
T
M
T
M
T
QQ YY
W W
1 2 1 2
1 2 0
/ /
/
( ) ( )
( )
,
L =
λ λ
λ
M
T
M
T T T
M
T T T
B P P D P D
P D P D
1 12
1 2
12 1 12 1
1 2
12 1 12 1 0
( ) ( )( )( )
( )( )( )
/
/
,
Q = P11
B1 + A PT
2 22 + A PT
1 12 + P12
B2
,
Y = P11
D1 + P12
D2
,
W = C PT
2 22 + C PT
1 12.
Estimates (5.6) and (5.7) yield stability criterion for the state x = 0 of system (5.1)
formulated below.
Theorem 5.1. Assume that for system (5.1) the matrix-valued functional U ( t, ϕ )
is constructed with components (5.3). If in estimate (5.6) the matrices A and B
are positive definite and in estimate (5.7) the matrix G is negative definite then the
zero solution of system (5.1) is uniformly asymptotically stable.
The proof of this theorem follows from Theorem 4.1.
Remark 5.1. It is easy to show that conditions of Theorem 5.1 are fulfilled
provided that
a) λM ( P11 ) λM ( P22 ) > λM
TP P12 12( ),
b) λM ( C ) + 2 λM ( F ) < 0,
c) λM ( D ) + 2 λM ( L ) < 0,
d) ( λM ( C ) + 2 λM ( F ) ) ( λM ( D ) + 2 λM ( L ) ) > λM ( K K
T
).
Further, to study system (5.1) we apply Approach B2 and the functional U ( ϕ )
with the elements (5.3). Let η ∈ R+
2 , η = (1, 1) T. Then, in the framework of this
approach one need to study sign-definiteness of the upper right derivative
D V x+ ( , ) ( . )η 5 1 = D x+ v11 1 5 2( ) ( . ) + D x+ v22 2 5 2( ) ( . ) +
+ 2 12 1 2 5 1D x x+ v ( , ) ( . ) . (5.8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
392 A. A. MARTYNYUK
Having accomplished simple transformations in the expression (5.8) we get the
estimate
D V x+ ( , ) ( . )η 5 1 ≤ u T S u, (5.9)
where u T = u u1 2,( ) and
uT
1 = x t x t1 1( ) , ( )−( )τ , uT
2 = x t x t2 2( ) , ( )−( )τ ,
S =
λ λ
λ λ
M M
T
M
T
M
C K K
K K D
1 2 1 2
1 2 1 2
/ /
/ /
( ) ( )
( ) ( )
,
C =
c c
c c
11 12
21 22
, D =
d d
d d
11 12
21 22
, K =
k k
k k
11 12
21 22
,
c11 = λM
T TA P P A P A A P P1 11 11 1 12 2 2 12 11+ + + +( ),
c21 = c12 = λM
TP C P P P C P C1 2
11 1 12 2 11 1 12 2
/ + +( )( )( ) ,
c22 = λM ( – P11 );
d11 = λM
T T TB P P B B P P B P2 22 22 2 1 12 12 1 22+ + + +( ) ,
d21 = d12 = λM
TP D P D P D P D1 2
22 2 12 1 22 2 12 1
/ + +( )( )( ) ,
d22 = λM ( – P22 );
k11 = λM
T TP B A P A P P B P1 2
11 1 2 22 1 12 12 2 12
1
2
/ + + + +( )
×
× P B A P A P P B PT T
T
11 1 2 22 1 12 12 2 12
1
2
+ + + +( )
,
k12 = λM
TP D P D P D P D1 2
11 1 12 2 11 1 12 2
/ + +( )( )( ) ,
k21 = λM
TP C P C P C P C1 2
22 2 12 1 22 2 12 1
/ + +( )( )( ) ,
k22 =
1
2
1 2
12 12λM
TP P/ ( ).
Estimates (4.6) and (5.9) provide the following result.
Theorem 5.2. Assume that for system (5.1) the matrix-valued functional U ( ϕ )
is constructed with components (5.3). If in estimate (5.4) the matrices A and B
are positive definite and in estimate (5.9) the matrix S is negative definite, then the
zero solution of system (5.1) is uniformly asymptotically stable.
Remark 5.2. It is easy to verify that conditions of Theorem 5.2 are satisfied
provided that
a) λM ( P11 ) λM ( P22 ) > λM
TP P12 12( ),
b) λM ( C ) < 0,
c) λM ( D ) < 0,
d) λM ( C ) λM ( D ) > λM ( K K
T
).
6. Bibliographical comments. The intensive investigation of functional
differential equations is motivated by many problem from mechanics, biology,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
STABILITY ANALYSIS OF LARGE-SCALE FUNCTIONAL DIFFERENTIAL SYSTEMS 393
elasticity, replicator dynamics, viscoelasticity, electricity, reactor dynamics, heat flow,
chemical oscillations, and neural networks. The books [1, 2, 11, 21] and papers [3, 13,
20, 22] are wonderful source of the problems in the direction.
In Section 2 we present some auxiliary results. It is based on some known results
[2, 12, 27].
In the Section 3 we used some results from [15, 18, 19].
Many authors try to find effective approaches to construction of Liapunov
functionals (see [4 – 7, 9, 24, 25]). However the problem is, in general, open.
The approaches presented in this paper are aimed at solving the problem of stability
of the zero solution of large-scale systems functional differential equations. These
approaches have a considerable potential for further development and applications.
First, we note that insignificant modification of the conditions of Theorem 4.1
allows us to establish a new boundedness conditions for motions in functional
differential systems with finite delay by further development of results of the papers [8,
10]. Also note that the proposed approach enables us to determine the stabilizing
(destabilizing) role of delay, since it admits the existence of both stable and unstable
subsystems in the initial system.
Second, the approach developed in this paper make it possible to apply efficiently
some general results (see, for example, [3, 9, 12]) on stability of solutions to large-scale
functional differential equations which contain functionals satisfying estimates of the
type
w1 0ϕ( )( ) ≤ v ( t, ϕ ) ≤ w2 ϕ( ) (6.1)
or its generalizations, for example, in the form
w1 0ϕ( )( ) ≤ v ( t, ϕ ) ≤ w2 ϕ( ) + w L3 2
ϕ ⋅( ) , (6.2)
where ⋅ L2
is the norm in space L2
, and wi ( ⋅ ) is the comparison function of class
K ( K R ). We weaken conditions (6.1) or (6.2) by expansion of the set of components
vi j , appropriate for construction of suitable functional. Some possibilities of the
proposed technique of stability analysis is applicable to quasilinear equations which
remain an urgent object of investigations, including estimates of stability domains in
parameter space.
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matrix functions. – New York: Marcel Dekker, 2002. – 301 p.
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Received 26.09.2006
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|
| id | umjimathkievua-article-3314 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:12Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/63/f7bbc993376adb8475fae0053b6c2a63.pdf |
| spelling | umjimathkievua-article-33142020-03-18T19:51:00Z Stability analysis of large-scale functional differential systems Аналіз стійкості великомасштабних функціонально-диференціальних систем Martynyuk, A. A. Мартинюк, А. А. The present paper is focused on a new method for analysis of stability of solutions of a large-scale functional differential system via matrix-valued Lyapunov-Krasovskii functionals. The stability conditions are based on information about the dynamical behavior of subsystems of the general system and properties of the functions of interconnection between them. Запропоновано один новий метод аналізу стійкості розв'язків великомасштабної функціонально-диференціальної системи на основі матричнозначного функціонала Ляпунова - Красовського. Умови стійкості ґрунтуються на динамічній поведінці підсистем загальної системи та властивостях функцій зв'язку між ними. Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3314 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 382–394 Український математичний журнал; Том 59 № 3 (2007); 382–394 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3314/3373 https://umj.imath.kiev.ua/index.php/umj/article/view/3314/3374 Copyright (c) 2007 Martynyuk A. A. |
| spellingShingle | Martynyuk, A. A. Мартинюк, А. А. Stability analysis of large-scale functional differential systems |
| title | Stability analysis of large-scale functional differential systems |
| title_alt | Аналіз стійкості великомасштабних функціонально-диференціальних систем |
| title_full | Stability analysis of large-scale functional differential systems |
| title_fullStr | Stability analysis of large-scale functional differential systems |
| title_full_unstemmed | Stability analysis of large-scale functional differential systems |
| title_short | Stability analysis of large-scale functional differential systems |
| title_sort | stability analysis of large-scale functional differential systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3314 |
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