Stabilization of the Cauchy problem for integro-differential equations
In the present paper, we obtain a criterion for the stabilization of the Cauchy problem for an integro-differential equation in the class of functions of polynomial growth γ ≥ 0.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509839738798080 |
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| author | Kengne, E. Tayou, Simo J. Кенне, Е. Таю, Сімо Ж. |
| author_facet | Kengne, E. Tayou, Simo J. Кенне, Е. Таю, Сімо Ж. |
| author_sort | Kengne, E. |
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| description | In the present paper, we obtain a criterion for the stabilization of the Cauchy problem for an integro-differential equation in the class of functions of polynomial growth γ ≥ 0. |
| first_indexed | 2026-03-24T02:47:29Z |
| format | Article |
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UDC 517.9
E. Kengne (Univ. Dschang, Cameroon),
J. Tayou Simo (Univ. Yaoundé, Cameroon)
STABILIZATION OF CAUCHY PROBLEM
FOR INTEGRO-DIFFERENTIAL EQUATIONS
STABILIZACIQ ZADAÇI KOÍI
DLQ INTEHRO-DYFERENCIAL|NYX RIVNQN|
In the present paper, we obtain the criterion of stabilization of Cauchy problem for an integro-differential
equation in the class of functions of polynomial growth γ ≥ 0.
OderΩano kryterij stabilizaci] zadaçi Koßi dlq intehro-dyferencial\noho rivnqnnq u klasi
funkcij z polinomial\nym zrostannqm γ ≥ 0.
1. Introduction. In the present paper, we consider the integro-differential equation
∂
∂
u x t
t
( , )
= P
x
u x t Q
x
u x d
t∂
∂
+ ∂
∂
∫( , ) ( , )
0
τ τ, ( x, t ) ∈ Π∞ = Rn × [0, + ∞),
(1.1)
under the initial condition
u ( x, 0 ) = u0 ( x ) , x ∈ Rn, (1.2)
where P ( σ ) and Q ( σ ) are arbitrary polynomials with complex constant coefficients
(σ ∈ Rn ); here u : Π∞ → C is the unknown function; u0 : R
n → C is a given
function;
∂
∂x
= ∂
∂
∂
∂
… ∂
∂( )x x xn1 2
, , , . We study problem (1.1), (1.2) under the condi-
tion Q ( σ ) ≠ 0 (∀ σ ∈ Rn). Here
0
t
u x d∫ ( , )τ τ is a control (the system input).
Introduce the following Banach space of functions of some polynomial growth
γ ≥ 0:
Hm, γ = f C f
f x
x
xm n
m
m n
∈ = ∂
∂
+( ) < + ∞
≤
−( ) : max sup
( )
,R
R
γ α
α
α
γ1 ,
where α = (α1, α2, … , αn ) is a multiindex α = α1 + α2 + … + αn and
∂
∂
α
αx
=
=
∂
∂
… ∂
∂
α
α
α
α
1
1
1x x
n
n
n
, , .
Definition 1.1. We say that problem (1.1), (1.2) is stable in the class of functions
of polynomial growth γ ≥ 0 if for every nonnegative integer m there exists a
nonnegative integer l, so that for every initial function u0 ( x ) of space Hl, γ
, each
solution u ( x, t ) of problem (1.1), (1.2) belongs to the space Hm , γ for each t ∈
∈ [ 0, T ] , and
∂ ⋅
∂
j
j
m
u t
t
(, )
, γ
→ 0, t → + ∞, j = 0, 1. (1.3)
If we consider problem (1.1), (1.2) in the space S (where S is the Schwartz space
and S′ is the dual space of tempered distribution [1]) and apply the Fourier transform,
we obtain
© E. KENGNE, J. TAYOU SIMO, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11 1571
1572 E. KENGNE, J. TAYOU SIMO
∂
∂
v( , )σ t
t
=
P i t Q i d
t
( ) ( , ) ( ) ( , )σ σ σ σ τ τv v+ ∫
0
(in S′), (1.4)
v ( σ, 0 ) = v0 ( σ ) (in S′), (1.5)
where v ( σ , t ) and v 0 ( σ ) are the Fourier transforms of u ( x, t ) and u 0 ( x )
respectively:
v ( ⋅, t ) = Fx
{u ( ⋅, t )}, v0 = Fx
{u0}
(Fx is the operator of Fourier transform with respect to x).
If we introduce the vector function
v ( σ, t ) =
v
v
,
d
dt
T
,
it is easily seen from (1.4) and (1.5) that v ( σ, t ) is a solution of the following Cauchy
problem:
d t
dt
v( , )σ
= A ( σ ) v, v ( σ, 0 ) = v0 ( σ ) (in S′), (1.6)
where
A ( σ ) =
0 1
Q i P i( ) ( )σ σ
and v0 ( σ ) = v0 1( )( , ( ))σ σP i T .
In Section 2, we prove some auxiliary lemmas. The criterion of stabilization of
problem (1.1), (1.2) in the class of functions of polynomial growth is established in
Section 3.
2. Preliminaries. Let λ1 ( σ ) and λ2 ( σ ) be the eigenvalues of matrix A ( σ ) and
let
Λ ( σ ) = max Re ( ), Re ( )λ σ λ σ1 2{ };
here Re z is the real part of the complex z . Because Q ( i σ ) ≠ 0 for every σ ∈ Rn,
we conclude that λ1 ( σ ) λ2 ( σ ) ≠ 0 (∀ σ ∈ Rn ).
Lemma 2.1. Let the function Λ ( σ ) satisfy the condition
Λ ( σ ) < 0 (∀ σ ∈ Rn ). (2.1)
Then there exist constants β < 0 and q ∈ Q such that
Λ ( σ ) < β σ1 2+( )q
(∀ σ ∈ Rn ). (2.2)
Proof. Let δ ( r ) be a real function defined as
δ ( r ) = sup ( )
:σ σ
σ
∈ =
{ }
R
n r
Λ .
It is obvious that δ ( r ) is defined on [0, + ∞) . It follows from (2.1) that δ ( r ) < 0 for
all r ≥ 0. By applying the results of [2] (Appendix A) to δ ( r ) , we find that δ ( r ) is
piecewise continuous on [0, + ∞) and for some constants M < 0 and q ∈ Q,
δ ( r ) = M r oq 1 1+( )( ) (r → + ∞) ;
therefore there exists β < 0 such that δ ( r ) ≤ β 1 2+( )r
q
for all r ≥ 0, which
implies the estimate (2.2) and Lemma 2.1 is proved.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
STABILIZATION OF CAUCHY PROBLEM FOR INTEGRO-DIFFERENTIAL EQUATIONS 1573
Lemma 2.2. Let the function Λ ( σ ) satisfy condition (2.1). Then
R ( σ, t ) =
1 4 0
1 1 2
2 1 2
1 2
1 2
1 2 1 2
2
1
1 1 1
2
2
1 2 2 1
1 2 2 1
1
λ λ
λ λ
λ λ λ λ
λ
λ λ λ
λ λ λ λ
λ λ λ λ
λ
−
− −
− −
+ ≠
+
+ +
+
P e e e e
P e e e e
if P Q
t
t
e if P
t t t t
t t t t
t
( )
( ) ( )
, ,
( )
( )
, 44 0Q =
(2.3)
is a multiplicator in S (here λj = λj ( σ ) , P = P ( i σ ) and Q = Q ( i σ )).
Proof. By using the estimate of a matrix exponential in [3] (Chap. 1, Sect. 6) (see
also [4]) and estimate (2.2), we obtain
R( , )σ t ≤ C ed t q
1 1+( ) +( )σ β σ (∀ σ ∈ Rn, ∀ t ≥ 0),
where C > 0, and d = max (deg P, deg Q). Therefore
∂
∂
α
α
σ
σ
R( , )t
≤ C ed t q
α
α α β σσ1 1 1+( ) +( ) − +( ) (∀ σ ∈ Rn, ∀ t ≥ 0) (2.4)
for any miltiindex α and some Cα > 0. Hence, R ( σ, t ) is a multiplicator in S.
Corollary 2.1. If conditions (2.1) is satisfied, then the solution of Cauchy
problem (1.6) in S ′ reads
v ( σ, t ) = R( , )( , ) ( )σ σt T1 1 0v (in S′) (t ≥ 0) . (2.5)
In fact, if conditions (2.1) is valid, then function R ( σ, t ) given by (2.3) will be a
multiplicator in S, and (2.5) follows from estimate (2.4).
3. Criterion of the stabilization of problem (1.1), (1.2).
Theorem 3.1. In order that the Cauchy problem (1.1), (1.2) should be stable in
the space of functions of polynomial growth γ ≥ 0, it is necessary and sufficient that
condition (2.1) should be valid.
Proof. Necessity. Let problem (1.1), (1.2) be stable in the space of functions of
polynomial growth γ ≥ 0. Assume on contrary that condition (2.1) is violated. Then
for some σ0 ∈ Rn we have Λ ( σ0 ) > 0. Without loss of generality, suppose that
Re λ1 ( σ0 ) = Λ ( σ0 ) ≥ 0 and Re λ1 ( σ0 ) ≥ Re λ2 ( σ0 ). Further we find the solution of
the Cauchy problem for equation (1.1) with the initial condition
u ( x, 0 ) =
λ σ λ σ
λ σ
λ σ λ σ
λ σ λ σ
σ
σ
1 0 2 0
1 0
1 0 2 0
1 0 2 0
0
0
( ) ( )
( )
, ( ) ( ),
, ( ) ( );
.
.
− ≠
=
e
e
ix
ix
if
if
here
x . σ0 =
i
n
i ix
=
∑
1
0σ ,
if x = ( x1, … , xn ) , σ0 = ( σ01, … , σ0n ) . Obviously, the solution of this problem reads
u ( x, t ) =
e e
t e
t ix t ix
t ix
λ σ σ λ σ σ
λ σ σ
λ σ
λ σ
λ σ λ σ
λ σ λ σ λ σ
1 0 0 2 0 0
1 0 0
2 0
1 0
1 0 2 0
1 0 1 0 2 01
( ) . ( ) .
( ) .
( )
( )
, ( ) ( ),
( ) , ( ) ( ).
+ +
+
− ≠
+( ) =
if
if
If λ1 ( σ0 ) = λ2 ( σ0 ) then
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1574 E. KENGNE, J. TAYOU SIMO
| u ( x, t ) | = 1 1 0
0+ λ σ σ( ) ( )t et∆
and we have
lim ( , )
t
u x t
→ + ∞
> 0,
which contradicts the hypothesis that problem (1.1), (1.2) is stable in the space of
functions of polynomial growth γ ≥ 0.
If λ1 ( σ0 ) ≠ λ2 ( σ0 ) then
| u ( x, t ) | = e et tΛ( ) ( ( ) ( ))( )
( )
σ λ σ λ σλ σ
λ σ
0 2 0 1 01 2 0
1 0
− − ≥
≥ e et tΛ( ) (Re ( ) Re ( ))( )
( )
σ λ σ λ σλ σ
λ σ
0 2 0 1 01 2 0
1 0
− − > 0,
and we have
lim ( , )
t
u x t
→ + ∞
> 0,
which contradicts the hypothesis that problem (1.1), (1.2) is stable in the space of
functions of polynomial growth γ ≥ 0.
Sufficiency. Consider for equation (1.1) the Cauchy problem with the initial
condition
u ( x, 0 ) = u0
( x ) , x ∈ Rn. (3.1)
Because of the fulfillment of condition (2.1), the solution of Cauchy problem (1.6)
associated to the Cauchy problem (1.1), (3.1) is given by (2.5) and the first component
of vector v ( σ, t ) is the solution (in S′) of the Cauchy problem for equation (1.4)
with the initial condition v ( σ, 0 ) = v0
( σ ) = F ux{ }0 :
v ( σ, t ) =
1
1 1 2
1 2
2 1
0
1 2
1
0
1 2
1 2
1
λ σ λ σ
σ λ σ λ σ σ σ
λ σ λ σ
λ σ σ
λ σ λ σ
λ σ λ σ
λ σ
( ) ( )
( ) ( ) ( ) ( ) ( ),
( ) ( ),
( )( ) ( ),
( ) ( ).
( ) ( )
( )
−
−( ) + −( )[ ]
≠
+ +[ ]
≠
P i e P i
t e
t t
t
v
v
if
if
Therefore the function
u ( x, t ) = F tσ σ− { }1 v( , ) (σ ∈ Rn )
is the unique solution of Cauchy problem (1.1), (3.1) in S′ (see [3, 5 – 6]). Let m ∈
∈ N0 = N ∪ {0} and show that for some large l ∈ N the function u ( x, t ) belongs
(with respect to x) to the class Hm, γ (for every t ≥ 0) and satisfies condition (1.3) as
soon as u0 ∈ Hl, γ . Let e ( x ) be a compactly supported infinitely differentiable
function on Rn satisfying the condition
j n
e x j
∈
∑ −
Z
( ) ≡ 1
and whose support lies in x xn∈ ≤{ }R : 1 (see [7 – 11]).
Let u xj
0( ) = e x u x j( ) ( )0 + and v j
0( )σ = F ux j{ }0 . Then the function
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
STABILIZATION OF CAUCHY PROBLEM FOR INTEGRO-DIFFERENTIAL EQUATIONS 1575
vj ( σ, t ) =
1
1 1 2
1 2
2 1
0
1 2
1
0
1 2
1 2
1
λ σ λ σ
σ λ σ λ σ σ σ
λ σ λ σ
λ σ σ
λ σ λ σ
λ σ λ σ
λ σ
( ) ( )
( ) ( ) ( ) ( ) ( ),
( ) ( ),
( )( ) ( ),
( ) ( ),
( ) ( )
( )
−
−( ) + −( )[ ]
≠
+ +[ ]
≠
P i e P i
t e
t t
j
t
j
v
v
if
if
is the solution of the Cauchy problem (1.4), (1.5) in which the initial function v0 ( σ ) is
replaced by v j
0( )σ . Therefore uj ( x, t ) = F tjσ
− ⋅1{ ( , )}v is the solution of problem (1.1),
(3.1) with u0
( x ) replaced by u xj
0( ) ; here j ∈ Z
n. Because u xj
0( ) ≡ e x u x j( ) ( )0 + ,
it is evident that for some M > 0 that does not depend on j ∈ Z
n, we have
uj l
0
,γ
≤ M u j
l
0 1
,γ
γ+( ) .
From estimate (2.4) and estimate
σ
σ
σ σλ
α
α
ν∂
∂
( )v j
0( ) ≤ C u j
lα ν λ γ
γ
, , ,
0 1 +( ) ,
σ ∈ R
n, α = (α1, … , αn ) an arbitrary multiindex, | ν | + | λ | ≤ l, it follows that
∂
∂
( )
α
α
ν
σ
σ σvj( , )t ≤ M u e j
l
d t q
1
0 1 11 1( , , ) ,α ν λ γ
α α λ β σ γσ+( ) +( )+( ) − − +( ) ,
where | ν | + | λ | ≤ l and α is arbitrary. If we choose λ from the condition
| λ | = α α+( ) − + +1 1d n ,
then we obtain
x
x
u x tj
α
α
α
∂
∂
( , ) ≤ M t u j
l2
0 1( , ) ,
( )α ν γ
γρ +( ) ,
where α is arbitrary,
| ν | < l d n− +( ) + − −α α1 1,
and
ρ ( t ) =
( ) ,
exp( ) .
/1 0
0
1+ <
≥
t q
t q
q for
forβ
Because 1 +( )j γ ≤ 1 1+ +( ) +( )x j xγ γ , if we choose an α from the condition
| α | = n – E ( – γ ) + 1 (here E ( – γ ) stands for the integer part of –γ), we obtain
∂
∂
α
αx
u x tj ( , ) ≤ M t u x j x
l
n
ν γ
γρ( )
,
0 11 1+ +( ) +( )− − , (3.2)
where ν ≤ m, l ≥ m n E d E+ − +( ) + −(– ) ( )γ γ2 ; consequently
u ( x, t ) =
j
ju x j t
∈
∑ −
Z
( , ) (3.3)
is solution of the Cauchy problem (1.1), (3.1), belongs to Hm, γ for all t ≥ 0 and
satisfies the condition
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1576 E. KENGNE, J. TAYOU SIMO
u t m( , )⋅ , γ ≤ M t um l
ρ
γ
( ) 0
,
(∀ t ≥ 0) . (3.4)
Because ρ ( t ) → 0 as t → + ∞, we conclude from (3.2) – (3.4) that u t( , )⋅ ∈ Hm, γ
(t ≥ 0).
By analogy, we prove that
∂
∂
⋅u
t
t( , ) ≤ ′M t um l
ρ
γ
( ) 0
,
(∀ t ≥ 0) . (3.5)
It is sufficient to notice that the Cauchy problem (1.6) is equivalent to the Cauchy
problem
d t
dt
2
2
v( , )σ
=
P i
d t
dt
Q i t( )
( , )
( ) ( , )σ σ σ σv
v+ ,
v ( σ, 0 ) = v0 ( σ ) , ′vt ( , )σ 0 = P i( ) ( )σ σv0 .
It follows from (3.4) and (3.5) that u ( x, t ) satisfies the condition (1.3). Hence Cauchy
problem (1.1), (1.2) is stable in the class of functions of some polynomial growth γ ≥ 0
and Theorem 3.1 is proved.
Example 3.1. Consider the heat conduction equation
∂
∂
u
t
= ∂
∂
− ∫
2
2
0
4u
t
u x d
t
( , )τ τ, x ∈ R, t ≥ 0.
For this equation, P ( i σ ) = – σ2
, Q ( i σ ) = – 4, and
Λ ( σ ) =
− + − ∈ −∞ − + ∞
− ∈ −
σ σ σ
σ σ
2 4
2
16 2 2
2 2
for
for
( , ] [ , ),
( , ).
∪
Therefore Λ ( σ ) < 0 for every σ ∈ R , and by Theorem 3.1, the Cauchy problem for
this equation is stable in the classes of polynomial growth.
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M., 1958.
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P. 117 – 130.
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S.61 – 72.
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– S. 169 – 18 4.
Received 22.01.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
|
| id | umjimathkievua-article-3709 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:47:29Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6e/2f220ddac816a9cf3169fff4f921306e.pdf |
| spelling | umjimathkievua-article-37092020-03-18T20:02:37Z Stabilization of the Cauchy problem for integro-differential equations Стабілізація задачі Коші для інтегро-диференціальних рівнянь Kengne, E. Tayou, Simo J. Кенне, Е. Таю, Сімо Ж. In the present paper, we obtain a criterion for the stabilization of the Cauchy problem for an integro-differential equation in the class of functions of polynomial growth γ ≥ 0. Одержано критерій стабілізації задачі Коші для інтегро-диференціального рівняння у класі функцій з поліноміальним зростанням &gamma; &ge; 0. Institute of Mathematics, NAS of Ukraine 2005-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3709 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 11 (2005); 1571–1576 Український математичний журнал; Том 57 № 11 (2005); 1571–1576 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3709/4143 https://umj.imath.kiev.ua/index.php/umj/article/view/3709/4144 Copyright (c) 2005 Kengne E.; Tayou Simo J. |
| spellingShingle | Kengne, E. Tayou, Simo J. Кенне, Е. Таю, Сімо Ж. Stabilization of the Cauchy problem for integro-differential equations |
| title | Stabilization of the Cauchy problem for integro-differential equations |
| title_alt | Стабілізація задачі Коші для інтегро-диференціальних рівнянь |
| title_full | Stabilization of the Cauchy problem for integro-differential equations |
| title_fullStr | Stabilization of the Cauchy problem for integro-differential equations |
| title_full_unstemmed | Stabilization of the Cauchy problem for integro-differential equations |
| title_short | Stabilization of the Cauchy problem for integro-differential equations |
| title_sort | stabilization of the cauchy problem for integro-differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3709 |
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