A Note on the Asymptotic Stability of Fuzzy Differential Equations
We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and the comparison principle for Lyapunov-like functions, we give sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equat...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509776710991872 |
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| author | Le, Van Hien Ле, Ван Хіен |
| author_facet | Le, Van Hien Ле, Ван Хіен |
| author_sort | Le, Van Hien |
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| description | We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and the comparison principle for Lyapunov-like functions, we give sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations. |
| first_indexed | 2026-03-24T02:46:29Z |
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| fulltext |
UDC 517.9
Le Van Hien (Hanoi Univ. Education, Vietnam)
A NOTE ON THE ASYMPTOTIC STABILITY
OF FUZZY DIFFERENTIAL EQUATIONS
PRO ASYMPTOTYÇNU STIJKIST|
NEÇITKYX DYFERENCIAL|NYX RIVNQN|
In this paper, we study the stability of solutions of fuzzy differential equations by Lyapunov’s second
method. By using scale equations and comparison principle for Lyapunov-like functions, we give some
sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations.
Vyvçeno stijkist\ rozv’qzkiv neçitkyx dyferencial\nyx rivnqn\ za dopomohog druhoho metodu
Lqpunova. Za dopomohog masßtabnyx rivnqn\ ta pryncypu porivnqnnq dlq rivnqn\ typu Lq-
punova vstanovleno dostatni umovy stabil\nosti ta asymptotyçno] stabil\nosti rozv’qzkiv neçit-
kyx dyferencial\nyx rivnqn\.
1. Introductions. The investigation of stability of solutions is the most important
problem in the qualitative theory of differential equations. It has been widely applied
in Physics, Mechanics, Control, ... .
Recently, the industrial interest in fuzzy control and logic [1] has dramatically
increased the study of fuzzy systems. The calculus of fuzzy-valued functions has
recently developed [2 – 6] and the investigation of fuzzy differential equations has been
initiated in [7 – 11].
In this paper, we study the stability theory which corresponds to Lyapunov stability
theory for fuzzy differential equations.
2. Preliminaries. Let PK
n
R( ) denote the family of all nonempty compact,
convex subsets of Rn and define the addition and scalar multiplication in PK
n
R( ) as
usual. Let A, B be two nonempty subsets in Rn . The distance between A and B is
defined by Haussdorff metric:
d A B a b a bH
a A b B b B a A
( , ) max sup inf , sup inf= − −
∈ ∈ ∈ ∈
,
where ⋅ denotes a norm in R
n . Then it is clear that P dK
n
HR( )( ), becomes a
metric space. Moreover, the metric space P dK
n
HR( )( ), is complete and separable
(see [12]). Let T = a b;[ ], a ≥ 0, be an interval in R and denote εn = {u : R
n →
→ 0 1;[ ] | u satisfies (i) to (iv) below}:
(i) u is normal, that is, there exists x n
0 ∈R such that u x( )0 = 1;
(ii) u is fuzzy convex, that is, for x, y n∈R and 0 ≤ λ ≤ 1:
u x y u x u yλ λ+ −( ) ≥ [ ]( ) min ( ), ( )1 ;
(iii) u is upper semicontinuous;
(iv) u[ ]0 = x u xn∈ >{ }R : ( ) 0 is a compact subset in Rn .
For 0 < α ≤ 1, we denote u[ ]α = { x n∈R : u x( ) ≥ α}, then from (i) to (iv) it
follows that the α-level u[ ]α ∈ PK
n
R( ) for all α ∈[ ]0 1; . For later purpose, we
define ô ∈ εn as ˆ( )o x = χ{ }( )0 x = 1 if x = 0 and ˆ( )o x = 0 if x ≠ 0. Define a
metric function d : εn × εn → Rn by
d u d uH, sup ,v v[ ] = [ ] [ ]( )
≤ ≤0 1α
α α ;
© LE VAN HIEN, 2005
904 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
A NOTE ON THE ASYMPTOTIC STABILITY OF FUZZY DIFFERENTIAL EQUATIONS 905
then ( , )εn d becomes a complete metric space (see [12]). We list here some
properties of metric d u, v[ ] (see [7, 10, 12]):
(i) d u, v[ ] = d uv,[ ]; d u, v[ ] = 0 ⇔ u = v;
(ii) d u w d u d w, , ,[ ] ≤ [ ] + [ ]v v ;
(iii) d u d uλ λ λ, ,v v[ ] = [ ];
(iv) d u w w+ +[ ], v = d u, v[ ], u, v, w n∈ε , λ ∈R .
For x, y n∈ε , if there exists z n∈ε such that x = y + z, then z is called H-
difference of x and y and is denoted by x – y.
A mapping F : T → εn is differentiable at t0 ∈ T if for small h > 0, there exist
H-differences P (t0 + h ) – F t( )0 ; F t( )0 – P ( t0 – h ) and there exists ′F t( )0 ∈ εn
such that the limits
lim
( ) ( )
h
F t h F t
h→ +
+ −
0
0 0 , lim
( ) ( )
h
F t F t h
h→ +
− −
0
0 0
exist and equal ′F t( )0 .
If F, G are differentiable at t, then ( ) ( )F G t+ ′ = ′F t( ) + ′G t( ) and ( ) ( )λF t′ =
= λ ′F t( ), λ ∈ R (see [6, 7, 10]).
If F : T → εn is strongly measurable and integrable bounded, then it is integrable
on T and
T
F t dt∫ ( ) ∈ εn,
T T
F t dt F t dt∫ ∫
=( ) ( )
α
α , 0 < α ≤ 1, F t F tα
α( ) ( )= [ ] ,
where
T
F t dt∫ α( ) is an Aumann integral. It is well known that
T
F t dt∫[ ]( )
0
=
=
T
F t dt∫ 0( ) (see [7], Remark 4.1). Also the following properties of integral are valid
(see [3, 4, 7, 10]). If F, G: T → εn are integrable on T and λ ∈ R, then:
(i)
T
F G t dt∫ +( )( ) =
T
F t dt∫ ( ) +
T
G t dt∫ ( ) ;
(ii)
T
F t dt∫ ( )( )λ = λ
T
F t dt∫ ( ) ;
(iii) d F G T( ), ( ) :⋅ ⋅[ ] → +R is integrable;
(iv) d F t dt G t dt
T T∫ ∫[ ]( ) , ( ) ≤
T
d F t G t dt∫ [ ]( ), ( ) ;
(v)
a
b
F t dt∫ ( ) =
a
c
F t dt∫ ( ) +
c
b
F t dt∫ ( ) , a ≤ c ≤ b.
If F is continuous, then G t( ) =
a
t
F d∫ ( )τ τ is differentiable on T and ′G t( ) =
= F t( ) ∀ t ∈ T. Moreover, if F is differentiable on T and ′ ⋅F ( ) is integrable on T,
then for all t T∈ we have F t( ) = F t( )0 +
t
t
F d
0
∫ ( )τ τ , a ≤ t0 ≤ t ≤ b . If F is
continuous on T and G t( ) =
a
t
F d∫ ( )τ τ , then for t1 ≤ t2 we have (see [7])
d G t G t t t d F t o
t t
( ), ( ) ( ) sup ( ), ˆ
,
1 2 2 1
1 2
[ ] ≤ − [ ]
[ ]
.
3. Stability. Consider fuzzy differentiable equation:
dx
dt
f t x= ( , ) , x t x( )0 0= , (1)
where f ∈ C[ R+ × S( )ρ , εn], S( )ρ = { x ∈ εn: d x o, ˆ[ ] < ρ }, f t o( , ˆ) ≡ ô .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
906 LE VAN HIEN
In this section, we shall discuss the stability, especially, asymptotically stability of
solutions of Eq. (1) by Lyapunov’s second method. First, we give some notions of
stability which are used in the sequel. Let x t( ) = x t t x( ; , )0 0 be any solution of (1)
existing on t0, ∞[ ]. Denote K = {a ∈ C R R+ +[ ], }, a( )0 = 0, a( )⋅ is increasing}.
Definition 1. The trivial solution x = ô o f (1) is stable if for any ε > 0,
t0 ∈ R+ , there exists δ = δ ε( , )t0 > 0 such that if d x o0, ˆ[ ] < δ, then d x t o( ), ˆ[ ] <
< ε ∀ t ≥ t0 .
Definition 2. The trivial solution x = ô of (1) is uniform-stable if δ in Defi-
nition 1 is independent of t0 , i.e., for any ε > 0, there exists δ = δ ε( ) > 0 such
that if d x o0, ˆ[ ] < δ, t0 ∈ R+ , then d [x ( t; t0 , x0 ), ô] < ε ∀ t ≥ t0 .
Definition 3. The trivial solution x = ô of (1) is asymptotically stable if x = ô
is stable and for any t0 ∈ R+ , there exists ∆ = ∆( )t0 > 0 such that if d x o0, ˆ[ ] <
< ∆, then lim
t→∞
d [x ( t; t0 , x0 ), ô] = 0.
Definition 4. The trivial solution x = ô of (1) is uniform-asymptotically stable
if for any ε > 0, there exist δ0 = δ ε0( ) > 0, T( )ε ≥ 0 such that if d x o0, ˆ[ ] < δ0 ,
t0 ∈ R+ , then d [x ( t; t0 , x0 ), ô] < ε ∀ t ≥ t0 + T( )ε .
Definition 5. The trivial solution x = ô is exponential stable if there exist δ >
> 0, α > 0 such that for any solution x t( ) = x t t x( ; , )0 0 of (1) defines on t0, ∞[ ) :
d x t o d x o e t t( ), ˆ , ˆ ( )[ ] ≤ [ ]( ) − −β α
0
0 , t t≥ 0 ,
where β( )⋅ : 0, δ[ ) → R+ increasing in h ∈ 0, δ[ ).
Before discuss the stability of solutions of (1), we need the following lemma which
corresponds to Comparison Principle (see [10] for detail).
Lemma 1. Suppose that for Eq. (1) there exists a function V ∈ C R+[ × S( )ρ ,
R+] satisfying
V t x V t y Ld x y( , ) ( , ) ,− ≤ [ ] ∀ ( , )t x , ( , ) ( )t y S∈ ×+R ρ ;
D V t x+ ( , ) = lim sup ( , ( , )) ( , ) , ( , )
h h
V t h x h f t x V t x g t V t x
→ +
+ + −[ ] ≤ ( )
0
1 ,
g C( , ) ,⋅ ⋅ ∈ [ ]+R R
2 .
Let r t( ) = r t t w( ; , )0 0 be the maximal solution of equation
′ =w g t w( , ) (2)
existing on t0, ∞[ ) and x t( ) = x t t x( ; , )0 0 be any solution of (1).
Then V t x( , )0 0 ≤ w0 implies V t x t, ( )( ) ≤ r t( ) ∀ t ≥ t0 .
Theorem 1. Suppose that for Eq. (1) there exists a function V ∈ C R+[ × S( )ρ ,
R+] which satisfies the following conditions:
(i) V t x V t y Ld x y( , ) ( , ) ,− ≤ [ ] ∀ ( , )t x , ( , ) ( )t y S∈ ×+R ρ ;
(ii) a d x o V t x, ˆ ( , )[ ]( ) ≤ , V t o( , ˆ) = 0, where α( )⋅ ∈K class;
(iii) D V t x+ ( , ) ≤ g t V t x, ( , )( ) , g C∈ [ ]+R R
2 , , g t( , )0 = 0.
If the solution w = 0 of (2) is stable (asymptotically stable), then the trivial
solution x = ô of (1) is stable (asymptotically stable).
Proof. Let x t( ) = x t t x( ; , )0 0 , t0 ∈ R+ , be any solution of Eq. (1) existing on
t0, ∞[ ) and solution w = 0 of (2) be stable. Then, for any 0 < ε < ρ, there exists
δ0 = δ ε0 0( , )t > 0 such that if 0 ≤ w0 < δ0 , then w t t w( ; , )0 0 < a( )ε ∀ t ≥ t0 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
A NOTE ON THE ASYMPTOTIC STABILITY OF FUZZY DIFFERENTIAL EQUATIONS 907
From (ii), it follows that there exists δ = δ ε( , )t0 > 0 such that V t x( , )0 < δ0 if
d x o, ˆ[ ] < δ. We will show that if d x o, ˆ[ ] < δ, then d x t o( ), ˆ[ ] < ε ∀ t ≥ t0 .
Suppose that d x t o( ), ˆ[ ] ≥ ε for some t
*
> t0 ; then there exists t1 > t0 such that
d x t o( ), ˆ1[ ] = ε; d x t o( ), ˆ[ ] < ε ∀ ∈[ )t t t0 1, .
Let m t( ) = V t x t, ( )( ) , t ≥ t0 , then we have
m t h m t( ) ( )+ − = V t h x t h V t x t+ +( ) − ( ), ( ) , ( ) =
= V t h x t h V t h x t hf t x t+ +( ) − + + ( )( ), ( ) , ( ) , ( ) +
+ V t h x t hf t x t V t x t+ + ( )( ) − ( ), ( ) , ( ) , ( ) ≤
≤ Ld x t h x t hf t x t( ), ( ) , ( )+ + ( )[ ] + V t h x t hf t x t V t x t+ + ( )( ) − ( ), ( ) , ( ) , ( ) .
For small h > 0, H -differences of x t h( )+ and x t( ) are assumed to exist. Let
x t h( )+ = x t( ) + z t( ). Using the properties of metric d x y,[ ], we have
d x t h x t hf t x t( ), ( ) , ( )+ + ( )[ ] = hd
x t h x t
h
f t x t
( ) ( )
, , ( )
+ − ( )
.
Hence,
D m t+ ( ) = lim sup ( ) ( )
h
m t h m t
→ +
+ −[ ]
0
≤
≤ L d
x t h x t
h
f t x t
h
lim sup
( ) ( )
, , ( )
→ +
+ − ( )
0
+
+ lim sup , ( ) , ( ) , ( )
h h
V t h x t hf t x t V t x t
→ +
+ + ( )( ) − ( )[ ]
0
1 =
= Ld x t f t x t D V t x t′ ( )[ ] + ( )+( ), , ( ) , ( ) = D V t x t+ ( ), ( ) =
= D m t+ ( ) ≤ g t m t, ( )( ) , t0 ≤ t ≤ t1.
Applying Lemma 1, m t( ) ≤ r ( t ; t0 , w0 ), w0 = V t x( , )0 0 , t ∈ t t0 1,[ ]. On the other
hand, V t x( , )0 0 < δ0 , so, r ( t ; t0 , w0 ) < a( )ε , t ∈ t t0 1,[ ], and therefore
m t r t t w a( ) ( ; , ) ( )1 1 0 0≤ < ε .
By the choice of t1, we have a( )ε = a d x t o( ), ˆ1[ ]( ) ≤ V t x t1 1, ( )( ) = m t( )1 < a( )ε .
This is a contradiction, whence
d x t o( ), ˆ[ ] < ε ∀ ≥t t0 .
This shows that the trivial solution x = ô of (1) is stable.
If w = 0 of (2) is asymptotically stable, then it’s stable, therefore x = ô of (1) is
stable. For t0 ∈ R+ , there exist δ = δ( )t0 > 0, ∆1 0( )t > 0 such that d x t o( ), ˆ[ ] <
< ρ ∀ t ≥ t0 if d x o0, ˆ[ ] < δ and if 0 ≤ w0 < ∆1 0( )t , then lim
t→∞
w ( t; t0 , w0 ) = 0.
From hypothesises of function V t x( , ) , we can find ∆2 > 0 such that if d x o, ˆ[ ] < ∆2 ,
then V t x( , )0 < ∆1 0( )t . Put ∆ = min ,δ ∆2[ ]. Let x t( ) be any solution of (1),
t0 ∈ R+ , d x o0, ˆ[ ] < ∆. Define m t( ) = V t x t, ( )( ) , t ≥ t0 . By the first part of this
proof we see that D m t+ ( ) ≤ g t m t, ( )( ) . Apply Lemma 1, m t( ) ≤ r ( t ; t0 , w0 ), w0 =
= V t x( , )0 0 , t ≥ t0 . Since w0 = V t x( , )0 0 < ∆1 0( )t , we have lim
t→∞
r ( t; t0 , w0 ) = 0.
From a d x t o( ), ˆ[ ]( ) ≤ V t x t, ( )( ) = m t( ) ≤ r ( t; t0 , w0 ), a( )⋅ ∈ K , it follows that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
908 LE VAN HIEN
lim ( ), ˆ
t
d x t o
→∞
[ ] = 0. This shows that x = ô is asymptotically stable. The proof is
completed.
Theorem 2. Suppose that for Eq. (1) there exists a function V ∈ C R+[ × S( )ρ ,
R+] which satisfies the following conditions:
(i) V t x V t y Ld x y( , ) ( , ) ,− ≤ [ ] ∀ ( , )t x , ( , ) ( )t y S∈ ×+R ρ ;
(ii) a d x o V t x b d x o, ˆ ( , ) , ˆ[ ]( ) ≤ ≤ [ ]( ), a( )⋅ , b( )⋅ ∈K ;
(iii) D V t x+ ( , ) ≤ g t V t x, ( , )( ) , g C∈ [ ]+R R
2 , , g t( , )0 = 0.
If the solution w = 0 of (2) is uniform-stable (uniform-asymptotically stable),
then the trivial solution x = ô o f (1) is uniform-stable (uniform-asymptotically
stable).
Proof. If solution w = 0 of (2) is uniform-stable, then for any ε > 0 there exists
δ0 > 0 such that if t0 ∈ R+ and 0 ≤ w0 < δ0 , then w t t w( ; , )0 0 < a( )ε ∀ t ≥ t0 .
By choosing δ = δ ε( ) > 0 such that b( )δ < a( )δ0 and by the same argument in the
proof of Theorem 1, it can be proved that if d x o, ˆ[ ] < δ, then d [x ( t; t0 , x0 ), ô] <
< ε, t ≥ t0 . This shows that x = ô is uniform-stable.
Now, we assume that w = 0 is uniform-asymptotically stable, then by the first part
of this proof, the trivial solution x = ô is uniform-stable. Hence, there exists δ0 > 0
such that t0 ∈ R+ , d x o0, ˆ[ ] < δ0 implies d [x ( t; t0 , x0 ), ô] < ρ ∀ t ≥ t0 .
Moreover, there exists δ1 > 0 such that for any ε > 0, exists T = T( )ε ≥ 0 such
that if t ≥ 0, 0 ≤ w0 < δ1, then w t t w( ; , )0 0 < a( )ε ∀ t ≥ t0 + T. Put δ =
= min , ( )δ δ0
1
1b−[ ]. By the same argument in the proof of Theorem 1, it can be proved
that if d x o0, ˆ[ ] < δ, then d [x ( t; t0 , x0 ), ô] < ε ∀ t ≥ t0 + T( )ε . This shows that x =
= ô is uniform-asymptotically stable. The proof is completed.
Example 1. Consider a fuzzy-valued function f t x( , ) which satisfies
d f t x o a t d x o( , ), ˆ ( ) , ˆ[ ] ≤ [ ];
0
∞
∫ < ∞a t dt( )
(for example, f t x( , ) = 1
1 2+ t
x , a t( ) = 1
1 2+ t
satisfies all the above conditions).
Then the trivial solution x = ô of (1) is uniform-stable.
Proof. Consider a Lyapunov function V t x( , ) = d x o, ˆ[ ].
Then 1
2
d x o, ˆ[ ] ≤ V t x( , ) ≤ 2 d x o, ˆ[ ] and V t x V t y( , ) ( , )− ≤ d x y,[ ] ∀ ( , )t x ;
(t, y) ∈ R+ × εn. For h > 0, we have
V t h x hf t x+ +( ), ( , ) = d x hf t x o+[ ]( , ), ˆ ≤
≤ d x o, ˆ[ ] + hd f t x o( , ), ˆ[ ] ≤ d x o, ˆ[ ] + ha t d x o( ) , ˆ[ ].
Hence, D V t x+ ( , ) ≤ a t( ) d x o, ˆ[ ] = g t V t x, ( , )( ) , where g t w( , ) = a t w( ) . It’s easy to
show that the solution w = 0 of (2) is uniform-stable, so by Theorem 2, the trivial
solution x = ô of (1) is uniform-stable.
Theorem 3. Suppose that:
(i) f t x( , ) is bounded on R+ × S( )ρ ;
(ii) ∃ V ∈ C SR R+ ×[ ]( ),ρ : V t x V t y( , ) ( , )− ≤ Ld x y,[ ]; a d x o, ˆ[ ]( ) ≤ V t x( , ) ≤
≤ a t d x o0 , , ˆ[ ]( ) , where a( )⋅ ∈ K , a t0( , )⋅ ∈ K ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
A NOTE ON THE ASYMPTOTIC STABILITY OF FUZZY DIFFERENTIAL EQUATIONS 909
(iii) D V t x+ ( , ) + V t x*( , ) ≤ g t V t x, ( , )( ) , where g ∈ C [ R+ × R, R ], g t( , )⋅ is
nondecreasing for each t ∈ R+ a n d V* ∈ C [ R+ × S( )ρ , R+ ], V t x*( , ) ≥
≥ c d x o, ˆ[ ]( ), c( )⋅ ∈ K .
If solution w = 0 of (2) is stable, then the trivial solution x = ô o f (1) is
asymptotically stable.
Proof. By Theorem 1, the trivial solution x = ô is stable. Hence, for t0 ∈ R+ ,
there exists δ1 0( )t such that d x o0, ˆ[ ] < δ1 implies d [x ( t; t0 , x0 ), ô] < ρ ∀ t ≥ t0 .
Moreover, for t0 ∈ R+ , there exists δ2 0( )t > 0 such that if 0 ≤ w0 < δ2 0( )t , then
r ( t; t0 , w0 ) < ρ ∀ t ≥ t0 , where r ( t; t0 , w0 ) is the maximal solution of (2). Since
a t0 0( , )⋅ ∈ K , then there exists δ3 0( )t > 0 such that a t0 0 3( , )δ < δ2 0( )t . Put δ =
= δ( )t0 = min , ,δ δ δ1 2 3{ }. Let x t( ) = x t t x( ; , )0 0 be any solution of (1), d x o0, ˆ[ ] <
< δ. We will show that
lim ( ), ˆ
t
d x t o
→∞
[ ] = 0.
Suppose that lim sup ( ), ˆ
t
d x t o
→∞
[ ] > 0. Then there exists η > 0 and a sequence { }tn →
→ ∞ such that d x t on( ), ˆ[ ] ≥ η, n = 0, 1, 2, … .
By the boundedness of f t x( , ) and by taking a subsequence of tn{ }, we can
assume that there exists M > 0, tn{ } → ∞ such that tn +1 – tn ≥
η
2M
, n ≥ 0.
For t ∈ t t
Mn n, +
η
2
, we have x t( ) = x tn( ) +
t
t
n
f x d∫ ( )τ τ τ, ( ) . Hence
d x t o( ), ˆ[ ] ≥ d x t on( ), ˆ[ ] –
t
t
n
d f x o d∫ ( )[ ]τ τ τ, ( ) , ˆ ≥ η η− M
M2
=
η
2
.
Define m t( ) = V t x t, ( )( ) +
t
t
V x d
0
∫ ( )* , ( )τ τ τ , t t≥ 0 . Then
D m t+ ( ) ≤ D V t x t V t x t+ ( ) + ( ), ( ) , ( )* ≤ g t V t x t, , ( )( )( ) ≤ g t m t, ( )( ) , t t≥ 0 .
Applying Lemma 1, it follows that m t( ) ≤ r ( t; t0 , w0 ), where w0 = V t x( , )0 0 .
Since V t x( , )0 0 ≤ a t d x o0 0 0, , ˆ[ ]( ) < a t0 0( , )δ < δ2 0( )t , we have r t t w( ; , )0 0 < ρ
∀ t ≥ t0 . Therefore,
V t
M
x t
Mn n+ +
η η
2 2
, ≤ r t t w( ; , )0 0 –
k
n
t
t
M
k
k
V x d
=
+
∑ ∫ ( )
0
2
η
τ τ τ* , ( ) ≤
≤ r t t w( ; , )0 0 – c
M
n
η η
2 2
< ρ – c
M
n
η η
2 2
< 0
for n sufficiently large. This is a contradiction and, therefore,
lim ( ), ˆ
t
d x t o
→∞
[ ] = 0.
The proof is completed.
Theorem 4. Let the assumptions (i), (ii) of Theorem 2 hold and
(iii′ ) D V t x V t x+ +( , ) ( , )* ≤ g t V t x, ( , )( ) , g( , )⋅ ⋅ as in Theorem 3, V* ∈ C R+[ ×
× S( )ρ , R+], V t x*( , ) ≥ c d x o, ˆ[ ]( ), c( )⋅ ∈ K .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
910 LE VAN HIEN
If solution w = 0 of (2) is uniform-stable, then the trivial solution x = ô of (1)
is uniform-asymptotically stable.
Proof. By Theorem 2, x = ô of (1) is uniform-stable. Hence, for ε = ρ there
exists δ0 > 0 such that if t0 ∈ R+ , d x o0, ˆ[ ] < δ0 , then
d x t t x o( ; , ), ˆ0 0[ ] < ρ, t ≥ t0 .
We can assume that δ0 satisfies if 0 ≤ w0 < b( )δ0 then r t t w( ; , )0 0 < a( )ρ ∀ t ≥
≥ t0 . By the uniform-stability of x o= ˆ , for any ε > 0, there exists δ > 0 such that if
d x o0, ˆ[ ] < δ, t0 ∈ R+ , then d [x ( t; t0 , x0 ), ô] < ε ∀ t ≥ t0 . Let’s put T = T( )ε =
= 1 +
a
c
( )
( )
ρ
δ
. Let x t( ) = x t t x( ; , )0 0 be any solution of (1), d x o0, ˆ[ ] < δ0 . We will
show that d x o0, ˆ[ ] < δ for some t* ∈ [ t0 , t0 + T( )ε ] . Suppose that d x t o( ), ˆ[ ] ≥ δ
∀ t ∈ [ t0 , t0 + T( )ε ] . Define m t( ) = V t x t, ( )( ) +
t
t
V x d
0
∫ ( )* , ( )τ τ τ , t ≥ t0 . By the
same argument in the proof of Theorem 3, we have m t( ) ≤ r ( t; t0 , w0 ), t ≥ t0 , where
w0 = V t x( , )0 0 and r ( t; t0 , w0 ) is the maximal solution of (2). Therefore,
0 ≤ V t T x t T0 0+ +( ), ( ) ≤
≤ r t T t w V x d
t
t T
( ; , ) , ( )*
0 0 0
0
0
+ − ( )
+
∫ τ τ τ ≤ r t T t w Tc( ; , ) ( )0 0 0+ − δ .
Since V t x( , )0 0 ≤ b d x o, ˆ[ ]( ) < b( )δ0 , we have w0 = V t x( , )0 0 < b( )δ0 and, hence,
r ( t0 + T ; t0 , w0 ) < a( )ρ . Therefore, 0 ≤ V ( t0 + T, x (t0 + T )) < a( )ρ – Tc( )δ < 0.
This contradiction shows that there exists t1 ∈ [ t0 , t0 + T ] such that d x t o( ), ˆ1[ ] <
< δ. On the other hand, x ( t ; t1, x ( t1; t0 , x0 )) = x t t x( ; , )0 0 ∀ t ≥ t1, hence,
d x t o( ), ˆ[ ] < ε ∀ t ≥ t0 + T( )ε .
This shows that the trivial solution x = ô of (1) is uniform-asymptotically stable. The
proof is completed.
Theorem 5. Suppose that for Eq. (1) there exists a function V ∈ C R+[ × S( )ρ ,
R] which satisfies the following conditions:
(i) V t x V t y Ld x y( , ) ( , ) ,− ≤ [ ] ∀ ( , )t x , ( , ) ( )t y S∈ ×+R ρ ;
(ii) λ d x o p, ˆ[ ]( ) ≤ V t x( , ) ≤ Λ d x o p, ˆ[ ]( ) , p > 0, λ, Λ > 0;
(iii) D V t x+ ( , ) ≤ − [ ]( )c d x o p. , ˆ + K e t−α , t ≥ 0, c > 0.
If α > c
Λ
, then the trivial solution x = ô of (1) is exponential stable.
Proof. By Theorem 1, x = ô is uniform-stable. Hence, there exists δ such that
t0 ∈ R+ , d x o0, ˆ[ ] < δ ⇒ d [x ( t; t0 , x0 ), ô] < ρ ∀ t ≥ t0 .
Let’s put M = c
Λ
, m t( ) = V t x t eM t t, ( ) ( )( ) − 0 , t t≥ 0 . We have
D m t+ ( ) ≤ MV t x t eM t t, ( ) ( )( ) − 0 + e D V t x tM t t( ) , ( )− + ( )0 ≤
≤ MV t x t eM t t, ( ) ( )( ) − 0 + e K e c d x oM t t t p( ) , ˆ− − − [ ]( )[ ]0 α ≤
≤ MV t x t eM t t, ( ) ( )( ) − 0 + K e M t t( )( )− −α 0 – c e V t x tM t t
Λ
( ) , ( )− ( )0 =
= K e M t t( )( )− −α 0 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
A NOTE ON THE ASYMPTOTIC STABILITY OF FUZZY DIFFERENTIAL EQUATIONS 911
Apply Lemma 1, m t( ) – m t( )0 ≤ K e d
t
t M t
0
0∫ − −( )( )α τ τ = K
M
e M t t
−
−[ ]− −
α
α( )( )0 1 . By
hypothesises, m t( )0 = V t x( , )0 0 ≤ Λ d x o p
0, ˆ[ ]( ) , we have
m t( ) ≤ K
M
e K
M
d x oM t t p
−
−
−
+ [ ]( )− −
α α
α( )( ) , ˆ0
0Λ .
Put α1 = − −( )M α > 0, then
m t( ) ≤ Λ d x o p
0, ˆ[ ]( ) + K
α1
– K e t t
α
α
1
1 0− −( ) ≤ Λ d x o p
0, ˆ[ ]( ) + K
α1
, t t≥ 0 .
Therefore, V t x t, ( )( ) ≤ β1 0
0d x o e M t t, ˆ ( )[ ]( ) − − , t t≥ 0 , where β1 0d x o, ˆ[ ]( ) =
= Λ d x o p
0, ˆ[ ]( ) + K
α1
. On the other hand, Λ d x t o p( ), ˆ[ ]( ) ≤ V t x t, ( )( ) , t t≥ 0 , so
finally we have
d x t o( ), ˆ[ ] ≤
β
λ
1 0
1
0d x o
e
p
M
p t t, ˆ ( )[ ]( )
− −
, t t≥ 0 .
Denote α = M
p
, β d x o0, ˆ[ ]( ) =
β
λ
1 0
1
d x o p, ˆ[ ]( )
, then
d x t o( ), ˆ[ ] ≤ β αd x o e t t
0
0, ˆ ( )[ ]( ) − − , t t≥ 0 .
This shows that the trivial solution x = ô of (1) is exponential-stable. The proof is
completed.
Acknowledgement. I wish to express my sincere thanks to Professor Vu Tuan for
his encouragement and valuable ideas.
1. Driankov D., Hellendorm H., Rein Frank M. An introduction to fuzzy control. – Berlin: Springer,
1996.
2. Dubois D., Prade H. Towards fuzzy differential calculus. Pt I // Fuzzy Sets and Systems. – 1982. –
8. – P. 1 – 17.
3. Dubois D., Prade H. Towards fuzzy differential calculus. Pt II // Ibid. – P. 105 – 116.
4. Dubois D., Prade H. Towards fuzzy differential calculus. Pt III // Ibid. – P. 225 – 234.
5. Kaleva O. On the calculus of fuzzy valued mapping // Appl. Math. Lett. – 1990. – 3. – P. 55 – 59.
6. Puri M. L., Ralescu D. A. Differential of fuzzy functions // J. Math. Anal. and Appl. – 1983. – 91.
– P. 552 – 558.
7. Kaleva O. Fuzzy differential equations // Fuzzy Sets and Systems. – 1987. – 24. – P. 301 – 317.
8. Kaleva O. The Cauchy problem for fuzzy differential equations // Ibid. – 1990. – 35. – P. 389 –
396.
9. Kloeden P. E. Remark on Peano-like theorems for fuzzy differential equations // Ibid. – 1991. – 44.
– P. 161 – 163.
10. Lakshmikantham V., Mohapatra R. N. Basic properties of solutions of fuzzy differential equations
// Nonlinear Stud. – 2000. – 8. – P. 113 – 124.
11. Nieto J. J. The Cauchy problem for continuous fuzzy differential equations // Fuzzy Sets and
Systems. – 1999. – 102. – P. 259 – 262.
12. Puri M. L., Ralescu D. A. Fuzzy random variables // J. Math. Anal. and Appl. – 1986. – 114. –
P. 409 – 422.
Received 25.10.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
|
| id | umjimathkievua-article-3652 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:46:29Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-36522020-03-18T20:01:15Z A Note on the Asymptotic Stability of Fuzzy Differential Equations Про асимптотичну стійкість нечітких диференціальних рівнянь Le, Van Hien Ле, Ван Хіен We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and the comparison principle for Lyapunov-like functions, we give sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations. Вивчено стійкість розв'язків нечітких диференціальних рівнянь за допомогою другого методу Ляпунова. За допомогою масштабних рівнянь та принципу порівняння для рівнянь типу Ляпунова встановлено достатні умови стабільності та асимптотичної стабільності розв'язків нечітких диференціальних рівнянь. Institute of Mathematics, NAS of Ukraine 2005-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3652 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 7 (2005); 904–911 Український математичний журнал; Том 57 № 7 (2005); 904–911 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3652/4034 https://umj.imath.kiev.ua/index.php/umj/article/view/3652/4035 Copyright (c) 2005 Le Van Hien |
| spellingShingle | Le, Van Hien Ле, Ван Хіен A Note on the Asymptotic Stability of Fuzzy Differential Equations |
| title | A Note on the Asymptotic Stability of Fuzzy Differential Equations |
| title_alt | Про асимптотичну стійкість нечітких диференціальних рівнянь |
| title_full | A Note on the Asymptotic Stability of Fuzzy Differential Equations |
| title_fullStr | A Note on the Asymptotic Stability of Fuzzy Differential Equations |
| title_full_unstemmed | A Note on the Asymptotic Stability of Fuzzy Differential Equations |
| title_short | A Note on the Asymptotic Stability of Fuzzy Differential Equations |
| title_sort | note on the asymptotic stability of fuzzy differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3652 |
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