Some Results on Asymptotic Stability of Order α
Quasi-equiasymptotic stability of order $α (α ∈ ℝ_{+} * )$ with respect to a part of variables is considered. Some sufficient conditions, a converse theorem, and a theorem on multistability are proved.
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| Дата: | 2005 |
|---|---|
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2005
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509707140071424 |
|---|---|
| author | Vu, Thi Thu Huong Vu, Tuan Ву, Тхі Тху Хуонг Ву, Туан |
| author_facet | Vu, Thi Thu Huong Vu, Tuan Ву, Тхі Тху Хуонг Ву, Туан |
| author_sort | Vu, Thi Thu Huong |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | Quasi-equiasymptotic stability of order $α (α ∈ ℝ_{+} * )$ with respect to a part of variables is considered. Some sufficient conditions, a converse theorem, and a theorem on multistability are proved. |
| first_indexed | 2026-03-24T02:45:23Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDC 517.9
Vu Tuan, Vu Thi Thu Houng (Hanoi Univ. Education, Vietnam)
SOME RESULTS
ON THE ASYMPTOTIC STABILITY OF ORDER αααα
DEQKI REZUL|TATY
PRO ASYMPTOTYÇNU STIJKIST| PORQDKU αααα
The quasi-equiasymptotic stability of order α α ∈ +( )R
* with respect to a part of variables is
considered. Some sufficient conditions, a converse theorem and a theorem for multistability are proved.
Rozhlqnuto problemu kvazirivnomirno asymptotyçno] stijkosti porqdku α α ∈ +( )R
*
vidnosno
çastyny zminnyx. Dovedeno deqki dostatni umovy, obernenu teoremu ta teoremu pro mul\ty-
stijkist\.
1. Introduction. Consider the differential system of the form
dx
dt
X t x= ( , ), (1)
X t( , )0 0≡ for all t I a∈ = ∞[ ) ⊂ +, R .
Denoting
x x x y y z z
y
zn
T
m p
T= …( ) = … …( ) =
1 1 1, , , , , , , ,
X X X Y Y Z Z
Y
Zn
T
m p
T= …( ) = … …( ) =
1 1 1, , , , , , , ,
n = m + p, p ≥ 0, m ≥ 0,
D t x t I y H z= ∈ < < + ∞{ }( , ): , , , H > 0,
we assume that
X D n: → R ,
( , ) ( , )t x X t x�
is continuous and satisfies some uniqueness condition of solutions in D (see [1]). In
the paper, we introduce the notion of quasi-equiasymptotic stability of order α
α ∈( )+R
* , establish some sufficient conditions for this stability, prove a converse
theorem and a theorem for the x-uniform stability and (at the same time) y-quasi-
equiasymptotic stability of order α (one case of multistability) [2]. First, we give
some definitions.
© VU TUAN, VU THI THU HOUNG, 2005
250 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SOME RESULTS ON THE ASYMPTOTIC STABILITY OF ORDER α 251
Definition 1. The trivial solution x = 0 of (1) is said to be:
i) x- (or y-, respectively) quasi-equiasymptotically stable of order α α ∈( )+R
* ,
if given any ε > 0 and any t0 ∈ I, there exist δ = δ ε( , )t0 and T = T t( , )0 ε
such that if x0 < δ, then
x t t x t t( ; , ) ( )0 0 0< − −ε α
(or y t t x t t( ; , ) ( )0 0 0< − −ε α , respectively)
for all t ≥ t0 + T;
ii) x- (or y-, respectively) equiasymptotically stable of order α α ∈( )+R
* , if it
is stable in the sense of Liapunov and (at the same time) x- (or y -, respectively)
quasi-equiasymptotically stable of order α ;
iii) stable (uniformly stable, respectively) in the sense of Liapunov and (at the
same time) y-quasi-equiasymptotically stable of order α if given any ε > 0 and
any t0 ∈ I, there exist δ = δ ε( , )t0 > 0 (δ = δ ε( ) > 0, respectively) and T =
= T t( , )0 ε such that if x0 < δ, then x t t x( ; , )0 0 < ε for all t ≥ t0 a n d
y t t x( ; , )0 0 < ε α( )t t− −
0 for all t ≥ t0 + T.
2. Asymptotic stability of order αααα. 2.1. Sufficient conditions.
Theorem 1. Suppose that there exists a Liapunov function V t x( , ) defined on
D such that
i) V t( , )0 0≡ ;
ii) x V t x≤ ( , );
iii) ˙ ( , ) ( , )( )V t x
t
V t x1 ≤ − α for α > 0, t ∈ I.
Then the trivial solution x = 0 of (1) is equiasymptotically stable of order β
( 0 < β < α ) .
The proof of this theorem is similar to the proof of Theorem 1 in the paper [2].
Theorem 2. Suppose that there exists a Liapunov function V t x( , ) defined on
D such that
i) V t( , )0 0≡ ;
ii) y V t x≤ ( , ) ;
iii) ˙ ( , ) ( , )( )V t x
t
V t x1 ≤ − α for α > 0, t ∈ I.
Then the trivial solution x = 0 of (1) is y-equiasymptotically stable of order β
( 0 < β < α ) .
Proof. Given any ε > 0 ( ε < H ) and any t ∈ I = a, +∞[ ) for any x satisfying
the condition y = ε, inequality ii) implies V t x( , ) ≥ ε. Because of the continuity of
V t x( , ) and V t( , )0 0 0≡ , given t0 ∈ I, there exists δ = δ ε( , )t0 > 0 such that if
x0 < δ, then V t x( , )0 0 < ε. Assume that there exists a solution x t t x( ; , )0 0 of (1)
such that x0 < δ and y t t x( ; , )1 0 0 = ε at t1 ∈ I. From iii) it follows that
V t x t t x1 1 0 0, ( ; , )( ) ≤ V t x( , )0 0 and then
ε ε≤ ( ) ≤ <V t x t t x V t x1 1 0 0 0 0, ( ; , ) ( , ) .
This contradiction shows that if x0 < δ then y t t x( ; , )0 0 < ε for all t ≥ t0
, i.e.,
the trivial solution x = 0 is y-stable in the sense of Liapunov. Given γ > 0, we
denote by x t t x( ; , )0 0 the solution of (1) satisfying the condition x0 < γ. By virtue
of iii), we have
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
252 VU TUAN, VU THI THU HOUNG
V t x t t x V t x t
t
t V t x t t, ( ; , ) ( , ) ( , )( )0 0 0 0
0
0 0 0 0( ) ≤
≤ −
−
−
α
α α (2)
for all t sufficiently large. Let M t( , )0 γ = max x0 = γ V t x( , )0 0 , T T t= ( , , )0 ε γ such
that
0 0
0 0
≤
−
<−
M t
t t t
( , )
( )
γ ε
α β α
for 0 < β < α and for all t ≥ t0 + T. From (2) it follows that for all t ≥ t0 + T we
have
y t t x( ; , )0 0 ≤ V t x t t x, ( ; , )0 0( ) ≤ t V t x t t0 0 0 0
α α( , )( )− − ≤
≤ t
M t
t t
0
0
0
α
α β
γ( , )
( )− − ( )t t− −
0
β < t0
α ε
αt0
( )t t− −
0
β = ε β( )t t− −
0
which proves that the trivial solution x = 0 of (1) is y-quasiasymptotically stable of
order β.
The theorem is proved.
2.2. Converse theorems for linear system. We shall study now converse
theorems on the quasi-equiasymptotic stability. We consider the linear system
dx
dt
A t x= ( ) , (3)
where A t( ) is a continuous n × n matrix on I (see [3, 4]).
Theorem 3. Suppose that there exist M > 1 and α > 0 such that
x t t x( ; , )0 0 ≤ M x
t
t0
0
α
for all t ≥ t0, (4)
where x t t x( ; , )0 0 is a solution of (3). Then there exists a Liapunov function
V t x( , ), which satisfies the following conditions:
i) x V t x M x≤ ≤( , ) ;
ii) V t x V t x M x x( , ) ( , )− ′ ≤ − ′ ;
iii) ˙ ( , ) ( , )( )V t x
t
V t x3 ≤ − α .
Proof. Let V t x( , ) be defined by
V t x( , ) = sup ( ; , )
τ
α
τ τ
≥
+ +
0
x t t x
t
t
.
It is clear that x V t x≤ ( , ). On the other hand, from (4) we have
x t t x( ; , )+ τ ≤ M x t
t +
τ
α
,
which implies
V t x( , ) = sup ( ; , )
τ
α
τ τ
≥
+ +
0
x t t x
t
t
≤ sup
τ≥0
M x
t
t
+
τ α t
t +
τ
α
= M x .
Thus, we obtain
x V t x M x≤ ( , ) .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SOME RESULTS ON THE ASYMPTOTIC STABILITY OF ORDER α 253
Since the system (3) is linear, we have
x t x x t x x t x x( ; , ) ( ; , ) ( ; , )τ τ τ− ′ = − ′ . (5)
Hence,
V t x V t x( , ) ( , )− ′ = sup ( ; , ) sup ( ; , )
τ
α
τ
α
τ τ τ τ
≥ ≥
+ +
− + ′ +
0 0
x t t x
t
t
x t t x
t
t
≤
≤ sup
τ≥0
x t t x x t t x( ; , ) ( ; , )+ − + ′{ }τ τ t
t
+
τ α
≤
≤ sup
τ≥0
x t t x x t t x( ; , ) ( ; , )+ − + ′τ τ t
t
+
τ α
= sup
τ≥0
x t t x x( ; , )+ − ′τ t
t
+
τ α
≤
≤ sup
τ≥0
M x x− ′ t
t +
τ
α
t
t
+
τ α
= M x x− ′ .
Thus, the condition ii) is established. Now we shall prove the continuity of V t x( , ).
From i) and ii) it follows that V t x( , ) is continuous at 0. It remains to prove the
continuity of V t x( , ) at x ≠ 0. For δ ≥ 0 we have
V t x V t x( , ) ( , )+ ′ −δ ≤ V t x V t x( , ) ( , )+ ′ − +δ δ +
+ V t x V t x t t x( , ) , ( ; , )+ − + +( )δ δ δ + V t x t t x V t x+ +( ) −δ δ, ( ; , ) ( , ) . (6)
Since V t x( , ) is Lipschitzian in x and x t t x( ; , )+ δ is continuous, the first two terms
are small when x x− ′ and δ are small. Let us consider the third term. Since
x t t x t t x+ + + +( )δ τ δ δ; , ( ; , ) = x t t x( ; , )+ +δ τ ,
we have
V t x t t x V t x+ +( ) −δ δ, ( ; , ) ( , ) =
= sup ; , ( ; , ) sup ( ; , )
τ
α
τ
α
δ τ δ δ δ τ
δ
τ τ
≥ ≥
+ + + +( ) + +
+
− + +
0 0
x t t x t t x
t
t
x t t x
t
t
=
= sup ; , sup ( ; , )
τ
α
τ
α
δ τ δ τ
δ
τ τ
≥ ≥
+ +( ) + +
+
− + +
0 0
x t t x
t
t
x t t x
t
t
=
= sup ; , sup ( ; , )
τ δ
α α
τ
α
τ τ
δ
τ τ
≥ ≥
+( ) +
+
− + +
x t t x
t
t
t
t
x t t x
t
t0
.
Put a( )δ = maxτ δ≥ x t t x( ; , )+ τ t
t
+
τ α
. Then a( )δ is continuous and bounded
because
a( )δ ≤ a( )0 = V t x( , ) ≤ M x < ∞
and a( )δ → a( )0 as δ → 0. Thus,
V t x t t x V t x+ +( ) −δ δ, ( ; , ) ( , ) = a t
t
a( ) ( )δ
δ
α
+
− 0 → 0
as δ → 0. Therefore, V t x( , ) is continuous. Finally, we shall establish condition iii).
We have
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
254 VU TUAN, VU THI THU HOUNG
V t h x t h t x+ +( ), ( ; , ) =
= sup
τ≥0
x t h t h x t h t x+ + + +( )τ; , ( ; , )
t h
t h
+ +
+
τ α
=
= sup ; , ( ; , )
τ
α α
τ τ
≥
+ + + +( ) + +
+
0
x t h t h x t h t x
t h
t
t
t h
=
= sup ; ,
τ
α α
τ τ
≥
+( ) +
+
h
x t t x
t
t
t
t h
≤
≤ sup ; ,
τ
α α
τ τ
≥
+( ) +
+
0
x t t x
t
t
t
t h
= V t x( , ) t
t h+
α
,
which implies
V t h x t h t x V t x
h
V t x
t
t h
h
+ +( ) − ≤ +
−
, ( ; , ) ( , )
( , )
α
1
.
Since
lim
h
t
t h
h→
+
−
0
1
α
= lim
h → 0
α
α
t
t h+
−1 −
+
t
t h( )2 = – α
t
,
we obtain
˙ ( , ) ( , )V t x
t
V t x3 ≤ − α .
This completes the proof.
Theorem 4. Suppose that there exist M > 1 and α > 0 such that
y t t x M x
t
t
( ; , )0 0 0
0≤
α
for all t ≥ t0
, where x t t x( ; , )0 0 = y zT T T,( ) is a solution of (3). Then there exists
a Liapunov function V t x( , ), which satisfies the following:
i) y V t x M x≤ ≤( , ) ;
ii) V t x V t x M x x( , ) ( , )− ′ ≤ − ′ ;
iii) ˙ ( , ) ( , )( )V t x
t
V t x3 ≤ − α .
This theorem can be proved by the same argument used in the proof of Theorem 3.
3. Liapunov stability and y-quasi-equiasymptotic stability of order αααα .
Consider the differential system of the form
˙ ( ) ( ) ( , , )y A t y B t z Y t y z= + + ,
(7)
˙ ( ) ( ) ( , , )z C t y D t z Y t y z= + +
and the linear system relatively
˙ ( ) ( )y A t y B t z= + ,
(8)
˙ ( ) ( )z C t y D t z= + .
Assume that the following conditions are valid:
Y t Y t z( , , ) ( , , )0 0 0 0≡ ≡ , Z t Z t z( , , ) ( , , )0 0 0 0≡ ≡ ,
(9)
t Y t y z Z t y z
y
( , , ) ( , , )+( ) → 0
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SOME RESULTS ON THE ASYMPTOTIC STABILITY OF ORDER α 255
as y + z → 0. We shall prove a theorem, generalized Theorem 1 in [5], on case
of multistability [6 – 8].
Theorem 5. Suppose that the trivial solution x = 0 of the linear system (8) is
uniformly stable and there exist M > 1, α > 0 such that
y t t x M x
t
t
( ; , )0 0 0
0≤
α
for all t ≥ t0
,
where x t t x( ; , )0 0 = y zT T T,( ) is a solution of (8). Then the trivial solution y =
= z = 0 of (7), for which condition (9) holds, is x-stable in the sense of Liapunov
and (at the same time) is y-quasi-equiasymptotically stable of order α1 (0 < α1 <
< α ).
Proof. Since y t t x M x
t
t
( ; , )0 0 0
0≤
α
or all t ≥ t0
, by virtue of Theorem 4,
there exists a Liapunov function V t x( , ) satisfying the following conditions:
y V t x M x≤ ≤( , ) , (10)
V t x V t x M x x( , ) ( , )− ′ ≤ − ′ ,
˙ ( , ) ( , )( )V t x
t
V t x8 ≤ − α . (11)
Differentiating the function V along the system (7), we have
˙ ( , ) ˙ ( , ) ( , )( ) ( )V t x V t x R t x7 8= + ,
where
R t x V
x
X t x( , ) , ( , )*= ∂
∂
,
X Y ZT T T* ,= ( ) ,
〈 ⋅ 〉 is inner product. By virtue of (9) – (11), the following conditions hold in the
domain { t ≥ 0, x ≤ h } :
R t x
M
t
y
M
t
V t x( , ) ( , )≤ ≤ε ε
,
where ε → 0 as x → 0. Consequently, there exists β (0 < β < h) such that, in the
domain { t ≥ t0
, x ≤ β }, we have
˙ ( , ) ( , ) ( , ) ( , )( )V t x
t
V t x
M
t
V t x
t
V t x7
1≤ − + ≤ −α ε α
, (12)
where α1 = α – εM, 0 < α1 < α, for sufficiently small ε > 0. We consider a solution
x t t x( ; , )0 0 of (7), where t a0 ≥ , x0 ≤ δ ( 0 < δ < β ), for which the inequality
x t t x( ; , )0 0 ≤ β holds at least in an interval T t t= ( )0, * . Therefore, by virtue of (12)
and (10), we have
y t t x( ; , )0 0 ≤ V t x t t x, ( ; , )0 0( ) ≤ M x
t
t0
0
1
α
⇒
⇒ y t t x( ; , )0 0 ≤ M x t t t0 0 0
1 1α α−( )− (13)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
256 VU TUAN, VU THI THU HOUNG
for t T∈ * . It follows from (9) and (13) that
X t x t t x, ( ; , )0 0( ) ≤ ε α α
1 0 0
11 1M x t t− +( )
( t ∈ T; α2 = ε1
M → 0 as x → 0 ). It is clear that the solution of the nonlinear
system (7) is of the form
x t t x( ; , )0 0 = K t t x( ; )0 0 +
t
t
K t X x t x d
0
0 0∫ ( , ) ( , ( ; , ))τ τ τ τ , (14)
where U t( ) is a fundamental matrix of the linear system (8) and K t t( , )0 =
= U t U t( ) ( )−1
0 . Because of the uniform stability in the sense of Liapunov of the trivial
solution of (8), there exists N = const ≥ 1 such that K t t N( , )0 ≤ for t ≥ t0 ≥ a (see
[9]). Then (14) implies
x t t x( ; , )0 0 ≤ N x0 +
t
t
N x t d
0
1 1
2 0 0
1∫ − +α τ τα α( ) ≤
≤ N x t t0 2 1
1
2 1
1
01 1 1+ −( )− − −α α α α α α ≤ N x0 2 1
11 +( )−α α . (15)
Given ε > 0 ( ε < β ) we choose δ = δ ε( , )t0 > 0 and x0 < δ such that
δ < min ;M t N− − − −
+( )[ ]{ }1
0 2 1
1 1
1 1α α α ε .
Then x t t x( ; , )0 0 < ε, t T∈ . By virtue of (13), we have
y t t x( ; , )0 0 < ε αt t−( )−
0
1 , t T∈ .
Thus, for all t satisfying the condition
x t t x( ; , )0 0 ≤ β,
the inequality
x t t x( ; , )0 0 < ε
is valid. Hence, ε < β, the inequality
x t t x( ; , )0 0 < ε
is satisfied for all t ≥ t0
, and
y t t x( ; , )0 0 < ε αt t−( )−
0
1 for all t ≥ t0
,
that is the trivial solution x = 0 of (7) is Liapunov stable and (at the same time) is y-
quasi-equiasymptotically stable of order α.
The theorem is proved.
Example.
dx
dt
x
t
y t
t
= − +2 2 cos
,
t ≥ 1. (16)
dy
dt
xt t y t
x t
t
= − + + +2
2 2 1
sin sin
cos( )
,
First, it is easy to see that the general solution of the linear system
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
SOME RESULTS ON THE ASYMPTOTIC STABILITY OF ORDER α 257
dx
dt
x
t
= − 2 ,
dy
dt
xt t y t= − +2 sin sin
is
x
C
t
= 1
2
,
t ≥ 1.
y C C e t= + −
1 2
cos ,
Hence, it is clear that the trivial solution x = y = 0 of (7) is uniformly stable in the
sense of Liapunov. On the other hand, the zero solution is x-quasi-equiasymptotically
stable of order 2. Since the nonlinear part of (16) satisfies condition (9), by virtue of
Theorem 5 the zero solution of (16) is uniformly stable and, at the same time, is x-
quasi-equiasymptotically stable of order 2.
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Received 12.12.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
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| id | umjimathkievua-article-3592 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:45:23Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/29/46fce74ba7a5634d4506caf11b656529.pdf |
| spelling | umjimathkievua-article-35922020-03-18T19:59:22Z Some Results on Asymptotic Stability of Order α Деякі результати про асимптотичну стійкість порядку α Vu, Thi Thu Huong Vu, Tuan Ву, Тхі Тху Хуонг Ву, Туан Quasi-equiasymptotic stability of order $α (α ∈ ℝ_{+} * )$ with respect to a part of variables is considered. Some sufficient conditions, a converse theorem, and a theorem on multistability are proved. Розглянуто проблему квазірівномірно асимптотичної стійкості порядку &alpha; відносно частини змінних. Доведено деякі достатні умови, обернену теорему та теорему про мультистійкість. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3592 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 250–257 Український математичний журнал; Том 57 № 2 (2005); 250–257 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3592/3916 https://umj.imath.kiev.ua/index.php/umj/article/view/3592/3917 Copyright (c) 2005 Vu Thi Thu Huong; Vu Tuan |
| spellingShingle | Vu, Thi Thu Huong Vu, Tuan Ву, Тхі Тху Хуонг Ву, Туан Some Results on Asymptotic Stability of Order α |
| title | Some Results on Asymptotic Stability of Order α |
| title_alt | Деякі результати про асимптотичну стійкість порядку α |
| title_full | Some Results on Asymptotic Stability of Order α |
| title_fullStr | Some Results on Asymptotic Stability of Order α |
| title_full_unstemmed | Some Results on Asymptotic Stability of Order α |
| title_short | Some Results on Asymptotic Stability of Order α |
| title_sort | some results on asymptotic stability of order α |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3592 |
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