Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 18, pp. 1–22.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

NULL CONTROLLABILITY OF COUPLED SYSTEMS OF

DEGENERATE PARABOLIC INTEGRO-DIFFERENTIAL

EQUATIONS

BRAHIM ALLAL, GENNI FRAGNELLI, JAWAD SALHI

Abstract. This article concerns the null controllability of a coupled system

of two degenerate parabolic integro-differential equations with one locally dis-

tributed control force. Since the memory terms do not allow applying the
standards Carleman estimates directly, we start by proving a null control-

lability result for an associated nonhomogeneous degenerate coupled system
employing new Carleman estimates with appropriate weight functions. As a

consequence, we deduce the null controllability result for the initial memory

system by using the Kakutani’s fixed point Theorem.

1. Introduction

This article studies the null controllability of a coupled system of two degener-
ate parabolic equations involving memory terms, by means of a single distributed
control force. More precisely, we consider the system

y1t − (a(x)y1x)x + b11y1 + b12y2 = H1(t, y1) + 1ωu, (t, x) ∈ Q,
y2t − (a(x)y2x)x + b21y1 + b22y2 = H2(t, y2), (t, x) ∈ Q,

y1(t, 1) = y2(t, 1) = 0, t ∈ (0, T ),{
y1(t, 0) = y2(t, 0) = 0, if a is weakly degenerate,

(ay1x)(t, 0) = (ay2x)(t, 0) = 0, if a is strongly degenerate,
t ∈ (0, T ),

y1(0, x) = y0
1(x), y2(0, x) = y0

2(x), x ∈ (0, 1),

(1.1)

where Q = (0, T )×(0, 1), ω b (0, 1) is a non-empty open set, 1ω is the corresponding
characteristic function, bij := bij(t, x) ∈ L∞(Q) and u = u(t, x) is the distributed
control function. By Hk(t, yk) we denote the following quantity

Hk(t, yk) =

∫ t

0

hk(t, r, x)yk(r, x) dr, k = 1, 2, (1.2)

where hk = hk(t, r, x) ∈ L∞((0, T ) × Q), k = 1, 2, are memory kernels. Moreover,
the diffusion coefficient a degenerates at x = 0 and we say that

2020 Mathematics Subject Classification. 35K65, 45K05, 93C05, 93B05.
Key words and phrases. Carleman estimates; parabolic systems involving memory terms;

observability inequality; null controllability.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted October 9, 2021. Published February 22, 2023.

1



2 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

• a is weakly degenerate (WD) if a ∈ C[0, 1] ∩C1(0, 1] is such that a(0) = 0,
a > 0 on (0, 1] and there exists α ∈ [0, 1), such that xa′(x) ≤ αa(x) for all
x ∈ [0, 1].
• a is strongly degenerate (SD) if a ∈ C1[0, 1] is such that a(0) = 0, a > 0 on

(0, 1] and there exists α ∈ [1, 2), such that xa′(x) ≤ αa(x) for all x ∈ [0, 1];
moreover,

∃β ∈ (1, α], x 7→ a(x)

xβ
is nondecreasing near 0, if α > 1,

∃β ∈ (0, 1), x 7→ a(x)

xβ
is nondecreasing near 0, if α = 1.

The study of controllability properties for (1.1) is motivated by numerous real
world applications. Indeed, degenerate partial differential equations play a major
role in modeling many processes coming from physics, biology and finance. How-
ever, in several complex problems, the history of the phenomena under investigation
is of relevance and must be incorporated in the mathematical model. As it is by
now classical, standard PDEs models cannot provide a good description of such
processes. For this reason, PDEs have been replaced by partial integro-differential
equations that take into account this memory effect, and that have been largely
investigated in previous decades.

Up to now, the controllability of degenerate parabolic equations with distributed
controls has been largely developed in several recent papers, see [2, 7, 8, 11] and
the references therein. Moreover, in the last recent years an increasing interest
has been devoted to the study of controllability properties for parabolic equations
involving memory terms, see [9, 15, 16, 19, 20, 21]. But, very little is known for
the controllability analysis of parabolic equations that couple a degenerate diffusion
coefficient with a nonlocal reaction term. We refer to [4, 6, 22] for some related
results. See also [5] for a similar work on this theme.

In this work, we aim to extend those known results to coupled systems of kind
(1.1). More precisely, we seek for suitable conditions on the kernels h1 and h2 so
that the coupled system (1.1) is null controllable, that is to say, for any initial data
(y0

1 , y
0
2), there exists a control function u such that the associated solution to (1.1)

vanishes at the end of the time horizon [0, T ]. To our knowledge, this is the first
paper dealing with a coupled system of degenerate parabolic equations in presence
of memory terms.

The starting point for proving the null controllability for the integro-differential
system (1.1) is to show the null controllability for the nonhomogeneous degenerate
parabolic system without memory

y1t − (a(x)y1x)x + b11y1 + b12y2 = F1 + 1ωu, (t, x) ∈ Q,
y2t − (a(x)y2x)x + b21y1 + b22y2 = F2, (t, x) ∈ Q,

y1(t, 1) = y2(t, 1) = 0, t ∈ (0, T ),{
y1(t, 0) = y2(t, 0) = 0, (WD),

(ay1x)(t, 0) = (ay2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

y1(0, x) = y0
1(x), y2(0, x) = y0

2(x), x ∈ (0, 1)

(1.3)

for arbitrary functions F1, F2 ∈ L2(Q).



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 3

The proof of this result relies on a new modified Carleman inequality for the
associated adjoint problem with some weight functions that blow up as t→ T . The
new Carleman inequality is the key point to derive the null controllability result
for an intermediate problem similar to the integro-differential system (1.1). At the
end, we deduce the desired controllability result for the original problem using a
classical fixed point argument.

This article is organized in the following way: in Section 2, we first consider the
nonhomogeneous degenerate system (1.3) studying its well posedness, the Carleman
estimates for the associated adjoint problem and, finally, its null controllability. As
a consequence, in Section 3, by means of Kakutani’s fixed point Theorem, we prove
that system (1.1) is null contrallable under a decaying condition on the kernels h1

and h2 only at t = T . In the last section, we show the same controllability result
for kernels vanishing in a neighborhood of the initial time.

2. Null controllability of a nonhomogeneous degenerate system

As stated in the introduction, we first study system (1.3).

2.1. Well-posedness. To study the well-posedness of the degenerate system (1.3),
we first recall the following weighted Sobolev spaces (in the sequel, a.c. means
absolutely continuous):

In the (WD) case we use

H1
a(0, 1) :=

{
y ∈ L2(0, 1) : y a.c. in [0, 1],

√
ayx ∈ L2(0, 1) and y(1) = y(0) = 0

}
,

H2
a(0, 1) :=

{
y ∈ H1

a(0, 1) : ayx ∈ H1(0, 1)
}
.

In the (SD) case we use

H1
a(0, 1) :=

{
y ∈ L2(0, 1) : y locally a.c. in (0, 1],

√
ayx ∈ L2(0, 1) and y(1) = 0

}
,

H2
a(0, 1) :=

{
y ∈ H1

a(0, 1) : ayx ∈ H1(0, 1)
}

=
{
y ∈ L2(0, 1) : y locally a.c. in (0, 1], ay ∈ H1

0 (0, 1),

ayx ∈ H1(0, 1) and (ayx)(0) = 0
}
.

In both cases, the norms are defined as

‖y‖2H1
a

:= ‖y‖2L2(0,1) + ‖
√
ayx‖2L2(0,1), ‖y‖2H2

a
:= ‖y‖2H1

a
+ ‖(ayx)x‖2L2(0,1).

Now we recall a well-posedness result for system (1.3) (see, for instance, [1]).

Proposition 2.1. Assume that (y0
1 , y

0
2) ∈ L2(0, 1)2, (F1, F2) ∈ L2(Q)2, and u ∈

L2(Q). Then, system (1.3) admits a unique weak solution

(y1, y2) ∈WT := L2(0, T ;H1
a(0, 1)2) ∩ C([0, T ];L2(0, 1)2) (2.1)

such that

‖(y1, y2)‖L2(0,T ;H1
a(0,1)2) + sup

t∈[0,T ]

‖(y1(t), y2(t))‖L2(0,1)2

≤ C
(
‖(y0

1 , y
0
2)‖L2(0,1)2 + ‖(F1, F2)‖L2(Q)2 + ‖1ωu‖L2(Q)2

)
,

(2.2)

for some positive constant C. Moreover, if (y0
1 , y

0
2) ∈ H1

a(0, 1)2, then

(y1, y2) ∈ ZT := L2(0, T ;H2
a(0, 1)2) ∩H1(0, T ;L2(0, 1)2)



4 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

and
‖(y1, y2)‖L2(0,T ;H2

a(0,1)2) + ‖(y1, y2)‖H1(0,T ;L2(0,1)2)

≤ C
(
‖(y0

1 , y
0
2)‖H1

a(0,1)2 + ‖(F1, F2)‖L2(Q)2 + ‖1ωu‖L2(Q)2

)
,

(2.3)

for some positive constant C.

2.2. Carleman estimates. In this subsection, we establish a Carleman type esti-
mate for the nonhomogeneous adjoint system

−v1t − (a(x)v1x)x + b11v1 + b21v2 = g1, (t, x) ∈ Q,
−v2t − (a(x)v2x)x + b12v1 + b22v2 = g2, (t, x) ∈ Q,

v1(t, 1) = v2(t, 1) = 0, t ∈ (0, T ),{
v1(t, 0) = v2(t, 0) = 0, (WD),

(av1x)(t, 0) = (av2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

v1(T, x) = vT1 (x), v2(T, x) = vT2 (x), x ∈ (0, 1),

(2.4)

where vT1 , v
T
2 ∈ L2(0, 1) and g1, g2 ∈ L2(Q).

To develop a Carleman estimate for (2.4), some suitable weight functions are
needed. As in [2], we introduce the weight functions

ψ(x) := γ
(∫ x

0

y

a(y)
dy − d

)
, θ(t) :=

1(
t(T − t)

)4 , ϕ(t, x) := θ(t)ψ(x). (2.5)

Now, let ω̃ be an arbitrary open subset of ω and ρ ∈ C2([0, 1]) be such that

ρ > 0, in (0, 1), ρ(0) = ρ(1) = 0, and ρx 6= 0, in [0, 1]\ω̃,

and define

Ψ(x) := eλρ(x) − e2λ‖ρ‖∞ , Φ(t, x) := θ(t)Ψ(x). (2.6)

We also define

σ := 4Φ− 3ϕ and σ1 := 2Φ− ϕ. (2.7)

By taking the parameters λ, d such that

d > 4d? := 4

∫ 1

0

y

a(y)
dy and λ >

1

‖ρ‖∞
ln
(4(d− d∗)
d− 4d∗

)
, (2.8)

one can show that the interval
(
e2λ‖ρ‖∞

d−d? , 4(e2λ‖ρ‖∞−eλ‖ρ‖∞ )
3d

)
is nonempty. This

permits to choose the constant γ (see (2.5)) in such a way that

e2λ‖ρ‖∞

d− d∗
< γ <

4
(
e2λ‖ρ‖∞ − eλ‖ρ‖∞

)
3d

. (2.9)

With this choice of the parameters d, λ and γ one can readily show that the
above weight functions satisfy the following inequalities which will play a crucial
role in the sequel.

Lemma 2.2. (1) maxx∈[0,1] ψ(x) ≤ minx∈[0,1] Ψ(x);

(2) 4
3 maxx∈[0,1] Ψ(x) ≤ minx∈[0,1] ψ(x);

(3) 4
3Φ(t, x) ≤ ϕ(t, x) ≤ Φ(t, x), for all (t, x) ∈ Q;

(4) ϕ(t, x) ≤ Φ(t, x) ≤ σ1(t, x) ≤ σ(t, x) < 0, for all (t, x) ∈ Q.



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 5

From the definition of the function θ, we observe that

|θ′(t)| ≤ Cθ3/2(t), ∀ t ∈ [0, T ], and θ(t)→ +∞ as t→ 0−, T+. (2.10)

Then, the next Carleman estimate holds (see [3, Theorem 3.3]).

Theorem 2.3. Assume that a is (WD) or (SD) and let T > 0. Then, there
exist two positive constants C and s0, such that the solution (v1, v2) ∈ ZT of (2.4)
satisfies∫∫

Q

(
sθa(x)(v2

1x + v2
2x) + s3θ3 x2

a(x)
(v2

1 + v2
2)
)
e2sϕ dt dx

≤ C
(∫∫

Q

(g2
1 + g2

2)e2sΦ dt dx+

∫∫
Qω

s3θ3(v2
1 + v2

2)e2sΦ dtdx
)
,

(2.11)

for all s ≥ s0. Here Qω = (0, T )× ω.

To obtain the controllability for the degenerate nonlocal system (1.1) with only
one control force, we need to show the following Carleman estimate with a single
locally distributed observation.

Theorem 2.4. Assume that a is (WD) or (SD) and let T > 0. Suppose that for
some open subset ω̂ b ω

b21 ≥ b0 > 0, in (0, T )× ω̂. (2.12)

Then, there exist two positive constants C and s0, such that the solution (v1, v2) ∈
ZT of (2.4) satisfies∫∫

Q

(
sθa(x)(v2

1x + v2
2x) + s3θ3 x2

a(x)
(v2

1 + v2
2)
)
e2sϕ dt dx

≤ C
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx
)
,

(2.13)

for all s ≥ s0.

Proof. Let us consider a nonnegative smooth cut-off function ζ ∈ C∞([0, 1]) such
that

0 ≤ ζ(x) ≤ 1, ζ(x) =

{
1, x ∈ ω̂,
0, x ∈ (0, 1) \ ω.

(2.14)

Multiplying the first equation in (2.4) by s3θ3ζe2sΦv2 and integrating on Q, we
have∫∫

Q

ζb21s
3θ3e2sΦv2

2 dt dx =

∫∫
Q

ζs3θ3e2sΦ (v2 (av1x)x + v2v1t) dt dx

−
∫∫

Q

ζb11s
3θ3e2sΦv2v1 dt dx

+

∫∫
Q

ζs3θ3e2sΦv2g1 dt dx.

(2.15)



6 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

Integrating by parts and using the second equation of (2.4), we obtain∫∫
Q

ζs3θ3e2sΦv2 (av1x)x dt dx = −
∫∫

Q

ζas3θ3e2sΦv1xv2x dt dx

+

∫∫
Q

s3θ3a
(
ζe2sΦ

)
x
v1v2x dt dx

+

∫∫
Q

s3θ3
(
a
(
ζe2sΦ

)
x

)
x
v1v2 dt dx

(2.16)

and ∫∫
Q

ζs3θ3e2sΦv2v1t dt dx

= −
∫∫

Q

ζas3θ3e2sΦv1xv2x dt dx−
∫∫

Q

ζb12s
3θ3e2sΦv2

1 dt dx

−
∫∫

Q

as3θ3
(
ζe2sΦ

)
x
v1v2x dt dx−

∫∫
Q

ζb22s
3θ3e2sΦv1v2 dt dx

−
∫∫

Q

ζs3
(
θ3e2sΦ

)
t
v1v2 dt dx+

∫∫
Q

ζs3θ3e2sΦv1g2 dt dx.

(2.17)

Combining the identities (2.15)-(2.17), it follows that∫∫
Q

ζb21s
3θ3e2sΦv2

2 dt dx

= −

I1︷ ︸︸ ︷
2

∫∫
Q

ζas3θ3e2sΦv1xv2x dt dx−

I2︷ ︸︸ ︷∫∫
Q

ζb12s
3θ3e2sΦv2

1 dt dx

+

I3︷ ︸︸ ︷∫∫
Q

(
s3θ3

(
a
(
ζe2sΦ

)
x

)
x
− ζ(b11 + b22)s3θ3e2sΦ − ζs3

(
θ3e2sΦ

)
t

)
v2v1 dt dx

+

I4︷ ︸︸ ︷∫∫
Q

ζs3θ3e2sΦv1g2 dt dx+

I5︷ ︸︸ ︷∫∫
Q

ζs3θ3e2sΦv2g1 dt dx .

(2.18)
Now, we estimate the integrals I1, I2, I3, I4 and I5. Applying the Young’s inequality,
one has

|I1| = |2
∫∫

Q

ζas3θ3e2sΦv1xv2x dt dx|

=
∣∣2 ∫∫

Q

(
s1/2θ1/2a1/2esϕv2x

)(
s

5
2 θ

5
2 ζa1/2es(2Φ−ϕ)v1x

)
dt dx

∣∣
≤ ε

∫∫
Q

sθae2sϕv2
2x dt dx+

1

ε

J︷ ︸︸ ︷∫∫
Q

s5θ5ζ2ae2s(2Φ−ϕ)v2
1x dt dx

(2.19)

for every ε > 0.



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 7

The term J should be estimated by an integral of v2
1 . For this, we multiply the

first equation in (2.4) by s5θ5ζ2e2s(2Φ−ϕ)v1 and we integrate by parts to obtain

J = −

J1︷ ︸︸ ︷
1

2

∫∫
Q

s5ζ2
(
θ5e2s(2Φ−ϕ)

)
t
v2

1 dt dx

+

J2︷ ︸︸ ︷
1

2

∫∫
Q

s5θ5
(
a
(
ζ2e2s(2Φ−ϕ)

)
x

)
x
v2

1 dt dx−

J3︷ ︸︸ ︷∫∫
Q

ζ2b11s
5θ5e2s(2Φ−ϕ)v2

1 dt dx

−

J4︷ ︸︸ ︷∫∫
Q

ζ2b21s
5θ5e2s(2Φ−ϕ)v1v2 dt dx+

J5︷ ︸︸ ︷∫∫
Q

ζ2s5θ5e2s(2Φ−ϕ)g1v1 dt dx .

(2.20)

Since |θ̇| ≤ Cθ2 and supp ζ b ω, we obtain

|Jk| ≤ C
∫∫

Qω

s7θ7e2s(2Φ−ϕ)v2
1 dt dx, k ∈ {1, 2, 3}.

Moreover, using the Young’s inequality, the boundedness of a/x2 in ω and again
the fact that supp ζ b ω, the term J4 can be estimated in the following way

|J4| =
∣∣∣ ∫∫

Q

(
s3/2θ3/2

(x2

a

)1/2
esϕv2

)(
s

7
2 θ

7
2 b21ζ

2
( a
x2

)1/2

es(4Φ−3ϕ)v1

)
dt dx

∣∣∣
≤ ε2

∫∫
Q

s3θ3x
2

a
e2sϕv2

2 dt dx+ Cε

∫∫
Qω

s7θ7e2s(4Φ−3ϕ)v2
1 dt dx.

Similarly,

|J5| ≤ C
∫∫

Qω

s3θ3e2s(2Φ−ϕ)g2
1 dt dx+ C

∫∫
Qω

s7θ7e2s(2Φ−ϕ)v2
1 dt dx.

On the other hand, thanks to Lemma 2.2, one can check that

2Φ− ϕ ≤ 4Φ− 3ϕ. (2.21)

Hence,

|J | ≤ ε2

∫∫
Q

s3θ3x
2

a
e2sϕv2

2 dt dx+ Cε

∫∫
Qω

s7θ7e2s(4Φ−3ϕ)v2
1 dt dx

+ C

∫∫
Qω

s3θ3e2s(2Φ−ϕ)g2
1 dt dx.

(2.22)

Putting together inequalities (2.19) and (2.22), we obtain

|I1| ≤ ε
∫∫

Q

sθae2sϕv2
2x dt dx+ ε

∫∫
Q

s3θ3x
2

a
e2sϕv2

2 dt dx

+ Cε

∫∫
Qω

s7θ7e2s(4Φ−3ϕ)v2
1 dt dx+ C

∫∫
Qω

s3θ3e2s(2Φ−ϕ)g2
1 dt dx.

(2.23)

In view of Lemma 2.2, we also have

|I2| ≤ C
∫∫

Qω

s3θ3e2sΦv2
1 dt dx ≤ C

∫∫
Qω

s3θ3e2s(4Φ−3ϕ)v2
1 dt dx. (2.24)



8 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

Proceeding as before, we obtain

|I3| ≤ C
∫∫

Qω

s5θ5e2sΦv1v2 dt dx

≤ ε
∫∫

Q

s3θ3x
2

a
e2sϕv2

2 dt dx+ Cε

∫∫
Qω

s7θ7e2s(2Φ−ϕ)v2
1 dt dx

(2.25)

and

|I5| ≤ ε
∫∫

Q

s3θ3x
2

a
e2sϕv2

2 dt dx+ Cε

∫∫
Qω

s3θ3e2s(2Φ−ϕ)g2
1 dt dx. (2.26)

Finally, using once again the Young’s inequality and Lemma 2.2, it follows that

|I4| ≤ C
∫∫

Qω

s3θ3e2sΦv2
1 dt dx+ C

∫∫
Qω

s3θ3e2sΦg2
2 dt dx

≤ C
∫∫

Qω

s3θ3e2s(4Φ−3ϕ)v2
1 dt dx+ C

∫∫
Qω

s3θ3e2sΦg2
2 dt dx.

(2.27)

Combining the estimates (2.18), (2.23)-(2.27) together with (2.12) and (2.21), we
obtain

b0

∫∫
Qω

s3θ3e2sΦv2
2 dt dx

≤
∫∫

Q

ζb21s
3θ3e2sΦv2

2 dt dx

≤ 3ε
(∫∫

Q

sθae2sϕv2
2x dt dx+

∫∫
Q

s3θ3x
2

a
e2sϕv2

2 dt dx
)

+ Cε

∫∫
Qω

s7θ7e2s(4Φ−3ϕ)v2
1 dt dx+ Cε

∫∫
Qω

s3θ3e2s(2Φ−ϕ)(g2
1 + g2

2) dt dx.

Hence, using the Carleman estimate (2.11) together with the previous inequality
with ε = b0

6C , where C is the positive constant in (2.11), we readily deduce the
desired result. �

Next, using (2.13), we are going to establish a new Carleman inequality with a
modified weight time function that blows up only as t→ T . This will give the null
controllability result for system (1.1) imposing a decaying condition on the kernels
h1 and h2 only at t = T . Thus, as in [12], we introduce the weight function

β(t) :=

{
θ(T2 ) =

(
2
T

)8
, for t ∈ [0, T2 ],

θ(t), for t ∈ [T2 , T ],

and the associated weight functions

ϕ̃(t, x) = β(t)ψ(x), Φ̃(t, x) := β(t)Ψ(x),

σ̃ = 4Φ̃− 3ϕ̃, σ̃1 = 2Φ̃− ϕ̃.
(2.28)

In what follows we will use the notation

Φ̂(t) := max
x∈[0,1]

Φ(t, x), ϕ̂(t) := max
x∈[0,1]

ϕ(t, x) = γ(d∗ − d)β(t),

ϕ∗(t) := min
x∈[0,1]

ϕ(t, x) = −γdβ(t), Φ∗(t) := min
x∈[0,1]

Φ(t, x).
(2.29)

Using Lemma 2.2, one can easily check that the next inequalities hold.



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 9

Lemma 2.5. (1) 4
3 Φ̂(t) ≤ ϕ∗(t) and ϕ̂(t) ≤ Φ∗(t), for all t ∈ (0, T );

(2) 4
3 Φ̃ ≤ ϕ̃ ≤ Φ̃ in Q;

(3) ϕ̃ ≤ Φ̃ ≤ σ̃1 ≤ σ̃ < 0, in Q.

Now, we are ready to state our main modified Carleman inequality.

Lemma 2.6. Assume that the conditions of Theorem 2.4 hold and let T ∗ ∈ (T2 , T ).
Then, there exist two positive constants C and s0 such that every solution (v1, v2) ∈
ZT of system (2.4) satisfies

e2sϕ̂(0)

∫ 1

0

(
v2

1(0) + v2
2(0)

)
dx+

∫∫
Q

(v2
1 + v2

2)e2sϕ dt dx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx

+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx
)
,

(2.30)

for all s ≥ s0.

Proof. Let us first prove that∫ T

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

≤ C
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx
)
,

(2.31)

for some positive constant C.

Using the monotonicity of x2

a(x) and the Hardy-Poincaré inequality given in [2,

Proposition 2.1], we have∫ 1

0

v2
1e

2sϕ dx ≤ 1

a(1)

∫ 1

0

a(x)

x2
(v1e

sϕ)2 dx ≤ C
∫ 1

0

a(x)(v1e
sϕ)2

x dx. (2.32)

Since ϕx(t, x) = γθ(t) x
a(x) , we have∫ 1

0

v2
1e

2sϕ dx ≤ C
∫ 1

0

(
a(x)v2

1x + s2θ2 x2

a(x)
v2

1

)
e2sϕ dx. (2.33)

Proceeding in a similar way, one can easily obtain∫ 1

0

(v2
1 + v2

2)e2sϕ dx ≤ C
∫ 1

0

(
a(x)(v2

1x + v2
2x) + s2θ2 x2

a(x)
(v2

1 + v2
2)
)
e2sϕ dx. (2.34)

Therefore, observing that ϕ̃ = ϕ in [T2 , T ] and applying the Carleman inequality
(2.13), we obtain∫ T

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

=

∫ T

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ dt dx

≤
∫ T

T
2

∫ 1

0

(
sθa(x)(v2

1x + v2
2x) + s3θ3 x2

a(x)
(v2

1 + v2
2)
)
e2sϕ dt dx



10 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

≤ C
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx
)
,

which gives (2.31).
On the other hand, let ξ ∈ C∞([0, T ]) be a cut-off function such that

0 ≤ ξ ≤ 1, ξ(t) :=

{
1, for t ∈ [0, T/2],

0, for t ∈ [T ∗, T ],
(2.35)

and define wi = ξ̃vi, i = 1, 2, where ξ̃ = ξesϕ̂(0) and (v1, v2) satisfies the adjoint
system (2.4). Thus, (w1, w2) solves

−w1t − (a(x)w1x)x + b11w1 + b21w2 = −ξ̃′v1 + ξ̃g1, (t, x) ∈ Q,

−w2t − (a(x)w2x)x + b12w1 + b22w2 = −ξ̃′v2 + ξ̃g2, (t, x) ∈ Q,
w1(t, 1) = w2(t, 1) = 0, t ∈ (0, T ),{

w1(t, 0) = w2(t, 0) = 0, (WD),

(aw1x)(t, 0) = (aw2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

w1(T, x) = w2(T, x) = 0, x ∈ (0, 1).

(2.36)

Thanks to the energy estimate (2.2), one has

(‖w1(0)‖2L2(0,1) + ‖w2(0)‖2L2(0,1)) + (‖w1‖2L2(Q) + ‖w2‖2L2(Q))

≤ C
∫∫

Q

(
(−ξ̃′v1 + ξ̃g1)2 + (−ξ̃′v2 + ξ̃g2)2

)
dt dx,

which yields

e2sϕ̂(0)
(
‖v1(0)‖2L2(0,1) + ‖v2(0)‖2L2(0,1) + ‖ξv1‖2L2(Q) + ‖ξv2‖2L2(Q)

)
≤ C

∫∫
Q

(
(ξ′)2(v2

1 + v2
2) + (ξ)2(g2

1 + g2
2)
)
e2sϕ̂(0) dt dx

= Ce2sϕ̂(0)

∫ T∗

T
2

∫ 1

0

(ξ′)2(v2
1 + v2

2) dt dx

+ Ce2sϕ̂(0)

∫ T∗

0

∫ 1

0

(ξ)2(g2
1 + g2

2) dt dx.

(2.37)

Using that ξ′(t) = 0 in [0, T2 ], ξ(t) = 0 in [T ∗, T ] and ϕ ≤ ϕ̂(0), one has

e2sϕ̂(0)
(
‖v1(0)‖2L2(0,1) + ‖v2(0)‖2L2(0,1)

)
+

∫ T
2

0

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

≤ e2sϕ̂(0)
(
‖v1(0)‖2L2(0,1) + ‖v2(0)‖2L2(0,1)

)
+

∫∫
Q

ξ2(v2
1 + v2

2)e2sϕ̂(0) dt dx

≤ C
(∫ T∗

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ̂(0) dt dx+

∫ T∗

0

∫ 1

0

(g2
1 + g2

2)e2sϕ̂(0) dt dx
)

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫ T∗

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

+

∫ T∗

0

∫ 1

0

(g2
1 + g2

2)e2sϕ̃ dt dx
)
,

(2.38)



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 11

since ϕ∗(T ∗) ≤ ϕ̃ in (0, T ∗)× (0, 1).
By (2.31), we have∫ T∗

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

≤
∫ T

T
2

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

≤ C
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx
)
.

Plugging the above inequality in (2.38), we obtain

e2sϕ̂(0)
(
‖v1(0)‖2L2(0,1) + ‖v2(0)‖2L2(0,1)

)
+

∫ T
2

0

∫ 1

0

(v2
1 + v2

2)e2sϕ̃ dt dx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx

+

∫ T∗

0

∫ 1

0

(g2
1 + g2

2)e2sϕ̃ dt dx
)
.

(2.39)

Using the definition of the modified weights, in particular the fact that ϕ̃ ≤ σ̃1 in
Q, together with (2.31) and (2.39), it follows that

e2sϕ̂(0)

∫ 1

0

(
v2

1(0) + v2
2(0)

)
dx+

∫∫
Q

(v2
1 + v2

2)e2sϕ̃ dt dx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

s3θ3(g2
1 + g2

2)e2sσ1 dt dx

+

∫∫
Q

(g2
1 + g2

2)e2sσ̃1 dt dx+

∫∫
Qω

s7θ7v2
1e

2sσ dt dx
)
.

(2.40)

Finally, observe that for c > 0 and n ≥ 0, the function x 7→ xne−cx is non-increasing
for x sufficiently large. Thus, using the fact that β(t) ≤ θ(t), one has

(sθ)ne2sσ ≤ (sβ)ne2sσ̃, (sθ)ne2sσ1 ≤ (sβ)ne2sσ̃1

for s large enough. This, together with (2.40), gives the estimate (2.30). This
completes the proof of Lemma 2.6. �

2.3. Null controllability result. In this subsection, as a consequence of Lemma
2.6, we will show the null controllability for the nonhomogeneous system (1.3) with
more regular solution. This result will be the key tool in the proof of the null
controllability for the memory system (1.1). To this purpose, we introduce the
following weighted space where the controllability will be solved:

Es :=
{

(y1, y2) ∈ ZT | (sβ)−
3
2 e−sσ̃1(y1, y2) ∈ L2(Q)2

}
endowed with the associated norm

‖y‖2Es :=

∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx.

Remark 2.7. If (y1, y2) belongs to Es, then (y1, y2) ∈ C([0, T ];L2(0, 1)2) and∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx < +∞.



12 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

Since σ̃1 < 0, one has

y1(T, ·) = y2(T, ·) = 0 in (0, 1).

From the modified Carleman inequality, we can obtain the following null con-
trollability result for (1.3).

Theorem 2.8. Assume that the conditions of Theorem 2.4 hold. Let T > 0,
T ∗ ∈ (T2 , T ) and suppose that e−sϕ̃(F1, F2) ∈ L2(Q)2 with s ≥ s0. Then, for any

(y0
1 , y

0
2) ∈ H1

a(0, 1)2, there exists u ∈ L2(Q) such that the associated solution (y1, y2)
of system (1.3) belongs to Es.

Moreover, there exists a positive constant C such that∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

e−2sϕ̃(F 2
1 + F 2

2 ) dt dx

+ e−2sϕ̂(0)(‖y0
1‖2L2(0,1) + ‖y0

2‖2L2(0,1))
)
.

(2.41)

Proof. Let us introduce the functional

J(y1, y2, u) =

∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx, (2.42)

where u ∈ L2(Q) and (y1, y2) satisfies the system

y1t − (a(x)y1x)x + b11y1 + b12y2 = F1 + 1ωu, (t, x) ∈ Q,
y2t − (a(x)y2x)x + b21y1 + b22y2 = F2, (t, x) ∈ Q,

y1(t, 1) = y2(t, 1) = 0, t ∈ (0, T ),{
y1(t, 0) = y2(t, 0) = 0, (WD),

(ay1x)(t, 0) = (ay2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

y1(0, x) = y0
1(x), y2(0, x) = y0

2(x), x ∈ (0, 1)

y1(T, x) = y2(T, x) = 0, x ∈ (0, 1).

(2.43)

By standard arguments (see for instance [17]), J attains its minimum at a unique
point (ȳ1, ȳ2, ū).

We are going to prove the existence of a dual variable z̄ = (z̄1, z̄2) such that

(ȳ1, ȳ2) = (sβ)3e2sσ̃1L∗(z̄1, z̄2), in Q,

ū = −1ω(sβ)7e2sσ̃ z̄1, in Q,

where L∗z̄ = −z̄t − (a(x)z̄x)x +B∗z̄, with B = (bij)1≤i,j≤2 such that

z̄(·, 1) = 0 and

{
z̄(·, 0) = 0, (WD)

(az̄x)(·, 0) = 0, (SD)
on (0, T ). (2.44)

Let us define the linear space

Xa =
{
w ∈ C∞(Q)2 : w satisfies (2.44)

}
.

In addition, we set

β(z, w) =

∫∫
Q

(sβ)3e2sσ̃1(L∗z · L∗w) dt dx+

∫∫
Qω

(sβ)7e2sσ̃z1w1 dtdx, (2.45)



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 13

for all z, w ∈ Xa, and

`(w) =

∫∫
Q

F · w dt dx+

∫ 1

0

y0 · w(0)dx, ∀w ∈ Xa, (2.46)

where F = (F1, F2) and y0 = (y0
1 , y

0
2) are the functions in (1.3).

Observe that the Carleman inequality (2.30) holds for all w ∈ Xa. Notably, we
have

e2sϕ̂(0)

∫ 1

0

(
w2

1(0) + w2
2(0)

)
dx+

∫∫
Q

(w2
1 + w2

2)e2sϕ̃ dt dx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]β(w,w),

for all w ∈ Xa.
Now, let us denote by X̃a the completion of Xa with the norm ‖w‖X̃a =

(β(w,w))1/2. Thus, X̃a is a Hilbert space with this norm.

Clearly, β is a strictly positive, symmetric and continuous bilinear form in X̃a.
Moreover, in view of the above inequality, one can see that the linear form ` is

continuous in X̃a. Indeed, employing the Cauchy-Schwarz inequality, one has

|`(w)| =
∫∫

Q

(F · w) dt dx+

∫ 1

0

y0 · w(0)dx

≤ Ces[ϕ̂(0)−ϕ∗(T∗)]
((∫∫

Q

e−2sϕ̃(F 2
1 + F 2

2 ) dt dx
)1/2

+ e−sϕ̂(0)(‖y0
1‖L2(0,1) + ‖y0

2‖L2(0,1))
)
‖w‖X̃a ,

(2.47)

for all w ∈ X̃a.
Hence, in view of Lax-Milgram’s Lemma, there exists one and only one z̄ ∈ X̃a

satisfying

β(z̄, w) = `(w), ∀w ∈ X̃a. (2.48)

Moreover,

‖z̄‖X̃a ≤ Ce
s[ϕ̂(0)−ϕ∗(T∗)]

((∫∫
Q

e−2sϕ̃(F 2
1 + F 2

2 ) dt dx
)1/2

+ e−sϕ̂(0)(‖y0
1‖L2(0,1) + ‖y0

2‖L2(0,1))
)
.

(2.49)

Let us set

(ȳ1, ȳ2) = (sβ)3e2sσ̃1L∗(z̄1, z̄2) and ū = −1ω(sβ)7e2sσ̃ z̄1. (2.50)

Using these definitions together with (2.49), it is not difficult to check that (ȳ1, ȳ2)
and ū satisfy∫∫

Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

e−2sϕ̃(F 2
1 + F 2

2 ) dt dx

+ e−2sϕ̂(0)(‖y0
1‖2L2(0,1) + ‖y0

2‖2L2(0,1))
) (2.51)

which yields (2.41).



14 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

To complete the proof, it suffices to check that (ȳ1, ȳ2, ū) satisfies the system
(2.43). First of all, notice that (ȳ1, ȳ2) ∈ Es and ū ∈ L2(Q). Denote by (ỹ1, ỹ2) the
(weak) solution of (1.3) associated to the control function u = ū. Then ỹ = (ỹ1, ỹ2)
is also the unique solution of (1.3) defined by transposition. Therefore, ỹ is the
unique function in L2(Q)2 satisfying∫∫

Q

ỹ ·Gdt dx =

∫∫
Q

1ωūz1 dt dx+

∫∫
Q

F · z dt dx+

∫ 1

0

y0 · z(0) dx, (2.52)

for all G = (G1, G2) ∈ L2(Q)2, where z := (z1, z2) solves

−z1t − (a(x)z1x)x + b11z1 + b21z2 = G1, (t, x) ∈ Q,
−z2t − (a(x)z2x)x + b12z1 + b22z2 = G2, (t, x) ∈ Q,

z1(t, 1) = z2(t, 1) = 0, t ∈ (0, T ),{
z1(t, 0) = z2(t, 0) = 0, (WD),

(az1x)(t, 0) = (az2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

z1(T, x) = z2(T, x) = 0, x ∈ (0, 1).

Now, using the expressions of (ȳ1, ȳ2) and ū (see (2.50)) in (2.52), we easily obtain∫∫
Q

ȳ ·Gdt dx

=

∫∫
Q

1ωūz1 dt dx+

∫∫
Q

F · z dt dx+

∫ 1

0

y0 · z(0) dx, ∀G ∈ L2(Q)2.

This together with (2.52), implies that ȳ = ỹ. Thus, the control ū ∈ L2(ω× (0, T ))
drives the state (ȳ1, ȳ2) ∈ Es to zero at time T . �

3. Null controllability for the integro-differential system

In this section, we establish our main null controllability result for the integro-
differential system (1.1).

At first, we recall that proceeding as in [14], thanks to a fixed point argument
and invoking Proposition 2.1, one can show that the following well-posedness result
holds.

Proposition 3.1. Assume that (y0
1 , y

0
2) ∈ L2(0, 1)2 and u ∈ L2(Q). Then system

(1.1) admits a unique solution (y1, y2) ∈WT .

Before presenting our main result, in what follows we start by proving some
technical results.

Lemma 3.2. Let ϕ̃ be the function in (2.28). Then

− ϕ̃(t, x) ≤ γd

(T/4)4(T − t)4
, ∀ (t, x) ∈ Q, (3.1)

where γ and d are the constants in (2.5).

Proof. By the definition of ϕ̃, we see that

− ϕ̃(t, x) ≤ γdβ(t), ∀ (t, x) ∈ Q. (3.2)



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 15

We next observe that, when t ∈ (0, T2 ), we have

1

T 4
≤ 1

(T − t)4
,

which yields

β(t) =
( 4

T 2

)4

≤
( 4

T

)4 1

(T − t)4
, ∀ t ∈

(
0,
T

2

)
. (3.3)

On the other hand,

β(t) ≤
( 2

T

)4 1

(T − t)4
, ∀ t ∈

(T
2
, T
)
.

This together with (3.3) gives

β(t) ≤
( 4

T

)4 1

(T − t)4
, ∀ t ∈ (0, T ). (3.4)

Then, putting (3.4) in (3.2), we finally deduce (3.1). �

Lemma 3.3. Let T ∗ = (1 + ε)T/2. Assume that d > 5d∗ and

ε ∈
(

0,

√
1− 4

√
4

5
(

d

d− d∗
)
)
.

Then
5

2
ϕ̂(0)− 2ϕ∗(T ∗) < 0. (3.5)

Proof. From the definitions of ϕ̂ and ϕ∗, one has

5

2
ϕ̂(0)− 2ϕ∗(T ∗) =

5

2
γ(d∗ − d)β(0) + 2γdβ(T ∗)

= γ
( 2

T

)8[5

2
(d∗ − d) +

2d

(1− ε2)4

]
=
dγ

2

( 2

T

)8[
− 5

(d− d∗)
d

+
4

(1− ε2)4

]
.

(3.6)

On the other hand, using the fact that d > 5d∗, we immediately have

4

5

d

(d− d∗)
< 1.

Hence, taking ε ∈
(

0,

√
1− 4

√
4
5 ( d
d−d∗ )

)
, it results

ε2 < 1− 4

√
4

5

( d

d− d∗
)

and, in particular,

(1− ε2)4 >
4

5

d

(d− d∗)
.

This is equivalent to

−5(d− d∗)
d

+
4

(1− ε2)4
< 0

and, by (3.6), the claim follows. �

Next, we make the following assumption on the kernels h1 and h2.



16 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

Hypothesis 3.4. Assume that a is (WD) or (SD) and h1, h2 satisfy

e
c0s

(T−t)4 hk ∈ L∞((0, T )×Q), k = 1, 2, (3.7)

with c0 := γd
(

4
T

)4
. Fix s ≥ s0 such that

2Ces(
5
2 ϕ̂(0)−2ϕ∗(T∗)) < 1, (3.8)

where C and s0 are the constants in (2.41) and Theorem 2.8, respectively.

Thanks to the previous hypothesis, we are able to prove the main result of this
paper.

Theorem 3.5. Assume the conditions of Theorem 2.4 and Hypothesis 3.4. Then
for any (y0

1 , y
0
2) ∈ H1

a(0, 1)2, there exists a control function u ∈ L2(Q) such that the
associated solution (y1, y2) ∈ ZT of (1.1) satisfies

y1(T, ·) = y2(T, ·) = 0 in (0, 1). (3.9)

The proof of this theorem is based on the following generalized version of Kaku-
tani’s fixed point Theorem, due to Glicksberg [13].

Theorem 3.6. Let B be a non-empty convex, compact subset of a locally convex
topological vector space X. If Λ : B → B is a convex set-valued mapping with closed
graph and Λ(B) is closed, then Λ has a fixed point.

Proof of Theorem 3.5. To prove the desired result, we begin by showing the null
controllability for the system

y1t − (a(x)y1x)x + b11y1 + b12y2 =

∫ t

0

h1(t, r, x)w1(r, x) dr + 1ωu, (t, x) ∈ Q,

y2t − (a(x)y2x)x + b21y1 + b22y2 =

∫ t

0

h2(t, r, x)w2(r, x) dr, (t, x) ∈ Q,

y1(t, 1) = y2(t, 1) = 0, t ∈ (0, T ),{
y1(t, 0) = y2(t, 0) = 0, (WD),

(ay1x)(t, 0) = (ay2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

y1(0, x) = y0
1(x), y2(0, x) = y0

2(x), x ∈ (0, 1),

(3.10)
for each (w1, w2) ∈ Es,M =

{
(w1, w2) ∈ Es : ‖(sβ)−3/2e−sσ̃1(w1, w2)‖L2(Q) ≤ M

}
,

where M and s are two arbitrary positive constants to be fixed later. More precisely,
as a first step we prove that this system is null controllable under Hypothesis 3.4.
As consequence, we obtain the null controllability result for the original memory
system through a fixed point technique.

Notice that Es,M is a non empty, bounded, closed, and convex subset of L2(Q)2.
Now, let (w1, w2) ∈ Es,M . By (3.1), we obtain∫∫

Q

e−2sϕ̃
(∫ t

0

hk(t, r, x)wk(r, x) dr
)2

dt dx

≤ T
∫∫

Q

∫ t

0

e−2sϕ̃h2
k(t, r, x)w2

k(r, x) dr dt dx

(by (3.1))



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 17

≤ T
∫∫

Q

∫ t

0

e
2sγd

(T/4)4(T−t)4 h2
k(t, r, x)w2

k(r, x) dr dt dx, k = 1, 2.

Next, using the condition (3.7), it follows that∫∫
Q

e−2sϕ̃
(∫ t

0

hk(t, r, x)wk(r, x) dr
)2

dt dx ≤ CT
∫∫

Q

w2
k dt dx, (3.11)

for k = 1, 2 and some positive constant C.
Applying Hölder’s inequality and using that sup(t,x)∈Q

(
(sβ(t))3e2sσ̃1(t,x)

)
< +∞

and (w1, w2) ∈ Es,M , we deduce that∫∫
Q

e−2sϕ̃
((∫ t

0

h1(t, r, x)w1(r, x) dr
)2

+
(∫ t

0

h2(t, r, x)w2(r, x) dr
)2)

dt dx

≤ CT sup
(t,x)∈Q

(
(sβ(t))3e2sσ̃1(t,x)

)∫∫
Q

(sβ)−3e−2sσ̃1(w2
1 + w2

2) dt dx

≤ CTM2 < +∞.

Therefore, setting Fk :=
∫ t

0
hk(t, r, x)wk(r, x) dr, k = 1, 2, we have e−sϕ̃(F1, F2) ∈

L2(Q)2. It follows from Theorem 2.8 that the system (3.10) is null controllable,
that is, for any (y0

1 , y
0
2) ∈ H1

a(0, 1)2 and (w1, w2) ∈ Es,M , there exists a control
function u ∈ L2(Q) such that the solution of (3.10) fulfills y1(T, ·) = y2(T, ·) = 0 in
(0, 1). Furthermore, in this case, the control u satisfies the estimate∫∫

Qω

(sβ)−7e−2sσ̃u2 dtdx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(
M2 + e−2sϕ̂(0)(‖y0

1‖2L2(0,1) + ‖y0
2‖2L2(0,1))

)
.

(3.12)

In the following, we extend this controllability result to the memory system (1.1).
First, we introduce the mapping Λ : Es,M → 2Es defined by

Λ(w1, w2) =
{

(y1, y2) ∈ Es : (y1, y2) is a solution of (3.10), such that

y1(T, ·) = y2(T, ·) = 0, for a control u ∈ L2(Q) satisfying (3.12)
}
.

Here, X = L2(Q)2 and B = Es,M .
Clearly, Λ(w1, w2) is a convex set of L2(Q)2. Moreover, thanks to the null

controllability of the system (3.10), Λ(w1, w2) is non empty. Let us now prove that
Λ is compact and has closed graph. This will be done in the next few steps.
• Λ(Es,M ) ⊂ Es,M for a sufficiently large M . Indeed, using the inequality (2.41),

condition (3.7) and proceeding as in (3.11), we obtain∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

e−2sϕ̃
(∫ t

0

h1(t, r, x)w1(r, x) dr
)2

dt dx

+

∫∫
Q

e−2sϕ̃
(∫ t

0

h2(t, r, x)w2(r, x) dr
)2

dt dx+ e−2sϕ̂(0)(‖y0
1‖2L2(0,1)

+ ‖y0
2‖2L2(0,1))

)



18 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(∫∫

Q

(w2
1 + w2

2) dt dx+ e−2sϕ̂(0)(‖y0
1‖2L2(0,1) + ‖y0

2‖2L2(0,1))
)

≤ Ce2s[ϕ̂(0)−ϕ∗(T∗)]
(

sup
(t,x)∈Q

(
(sβ(t))3e2sσ̃1(t,x)

)∫∫
Q

(sβ)−3e−2sσ̃1(w2
1 + w2

2) dt dx

+ e−2sϕ̂(0)(‖y0
1‖2L2(0,1) + ‖y0

2‖2L2(0,1))
)
.

Using that sup(t,x)∈Q(sβ(t))3esσ̃1(t,x) < +∞ and Lemma 2.5, it is not difficult to

show that

sup
(t,x)∈Q

esσ̃1(t,x) ≤ es(2Φ̂(0)−ϕ∗(0)) ≤ e s2ϕ
∗(0) ≤ e s2 ϕ̂(0). (3.13)

The above estimate together with the fact that (w1, w2) ∈ Es,M implies that∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx

≤ CM2es[
5
2 ϕ̂(0)−2ϕ∗(T∗)] + Ce−2sϕ∗(T∗)

(
‖y0

1‖2L2(0,1) + ‖y0
2‖2L2(0,1)

)
.

On the other hand, from (3.5) and (3.8), we obtain

Ces[
5
2 ϕ̂(0)−2ϕ∗(T∗)] ≤ 1

2
. (3.14)

Hence, for M sufficiently large, we deduce that∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx

≤ M2

2
+ Ce−2sϕ̂(0)(‖y0

1‖2L2(0,1) + ‖y0
2‖2L2(0,1)) ≤M

2,

(3.15)

which yields ∫∫
Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx ≤M2. (3.16)

Thus, Λ maps Es,M into itself, i.e., Λ(Es,M ) ⊂ Es,M .
• Λ(w1, w2) is a closed subset of L2(Q)2. Let (w1, w2) fixed and (yn1 , y

n
2 ) ∈

Λ(w1, w2) such that (yn1 , y
n
2 )→ (y1, y2). Let us show that (y1, y2) ∈ Λ(w1, wn). In

fact, by definition we have that (yn1 , y
n
2 ) is, together with a control function un the

solution of the system

yn1t − (a(x)yn1x)x + b11y
n
1 + b12y

n
2 =

∫ t

0

h1(t, r, x)w1(r, x) dr + 1ωun, (t, x) ∈ Q,

yn2t − (a(x)yn2x)x + b21y
n
1 + b22y

n
2 =

∫ t

0

h2(t, r, x)w2(r, x) dr, (t, x) ∈ Q,

yn1 (t, 1) = yn2 (t, 1) = 0, t ∈ (0, T ),{
yn1 (t, 0) = yn2 (t, 0) = 0, (WD),

(ayn1x)(t, 0) = (ayn2x)(t, 0) = 0, (SD),
t ∈ (0, T ),

yn1 (0, x) = y0
1(x), yn2 (0, x) = y0

2(x), x ∈ (0, 1),

(3.17)
with ∫∫

Q

(sβ)−3e−2sσ̃1(y2
1 + y2

2) dt dx+

∫∫
Qω

(sβ)−7e−2sσ̃u2 dtdx ≤M2. (3.18)



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 19

Furthermore, in view of Proposition 2.1, the solution (yn1 , y
n
2 ) is bounded in ZT .

Thus, thanks to the Aubin-Lions Theorem, this implies that Λ(Es,M ) is relatively
compact in L2(Q)2.

Hence, by Proposition 2.1 and (3.18), we infer that, on a subsequence (denoted
by the same index n) we have the convergences:

1ωun → 1ωu weakly in L2(Q),

(yn1 , y
n
2 )→ (y1, y2) weakly in ZT ,

(yn1 , y
n
2 )→ (y1, y2) strongly in C(0, T ;L2(0, 1)2).

By passing to the limit in (3.17), it follows that (y1, y2) is a controlled solution of
(3.10) associated to the control u. Consequently, (y1, y2) ∈ Λ(w1, w2) and Λ(Es,M )
is closed and compact of L2(Q)2.
• Λ(w1, w2) has closed graph in L2(Q)2. We need to prove that if (wn1 , w

n
2 ) →

(w1, w2) and (yn1 , y
n
2 )→ (y1, y2) with (yn1 , y

n
2 ) ∈ Λ(w1, w2), then (y1, y2) ∈ Λ(w1, w2).

Using the last two steps, one can easily prove that (y1, y2) ∈ Λ(w1, w2). Therefore,
we can apply the fixed point theorem (see Theorem 3.6) in the L2(Q)2 topology
for the mapping Λ to conclude that there is at least one (y1, y2) ∈ Es,M such that
(y1, y2) ∈ Λ(w1, w2). This completes the proof. �

As a consequence of Theorem 3.5 and arguing as in scalar case (see [4]), one can
show the following result.

Theorem 3.7. Assume the conditions of Theorem 2.4 and Hypothesis 3.4. Then
for any (y0

1 , y
0
2) ∈ L2(0, 1)2, there exists a control function u ∈ L2(Q) such that the

associated solution (y1, y2) ∈WT of (1.1) satisfies

y1(T, ·) = y2(T, ·) = 0 in (0, 1).

4. Concluding remarks

We are interested in proving that assumption (3.7) on the decay in time of
the kernels h1 and h2 as t approaches T− can be substituted by the following
assumption:

Hypothesis 4.1. Assume that a is (WD) or (SD) and suppose that there exists
t0 ∈ (0, T ) such that

supphk(t, ·, x) b (t0, T ), k = 1, 2, ∀ (t, x) ∈ Q. (4.1)

Observe that in this case we do not require condition (3.8). Then the following
null controllability result holds.

Theorem 4.2. Assume Hypothesis 4.1. Then for any (y0
1 , y

0
2) ∈ L2(0, 1)2, there

exists a control function u ∈ L2(Q) such that the associated solution (y1, y2) ∈WT

of (1.1) satisfies

y1(T, ·) = y2(T, ·) = 0 in (0, 1).

Moreover,

‖u‖L2(Q) ≤ Ct0‖y0‖L2(0,1)2 ,

for some positive constant Ct0 depending on t0.



20 B. ALLAL, G. FRAGNELLI, J. SALHI EJDE-2023/18

Proof. Consider the controlled parabolic system

w1t − (a(x)w1x)x + b11w1 + b12w2 = 1ωv, (t, x) ∈ (0, t0)× (0, 1),

w2t − (a(x)w2x)x + b21w1 + b22w2 = 0, (t, x) ∈ (0, t0)× (0, 1),

w1(t, 1) = w2(t, 1) = 0, t ∈ (0, t0),{
w1(t, 0) = w2(t, 0) = 0, (WD),

(aw1x)(t, 0) = (aw2x)(t, 0) = 0, (SD),
t ∈ (0, t0),

w1(0, x) = y0
1(x), w2(0, x) = y0

2(x), x ∈ (0, 1),

(4.2)

where (y0
1 , y

0
2) is the initial condition in (1.1).

Thanks to [1, Theorem 4.2] (see also [10, Theorem 3.10]), there exists v ∈
L2((0, t0)×(0, 1)) such that the associated solution (w1, w2) ∈ L2

(
0, t0;H1

a(0, 1)2
)
∩

C
(
[0, t0];L2(0, 1)2

)
satisfies

w1(t0, ·) = w2(t0, ·) = 0 in (0, 1).

Moreover, there exists a positive constant Ct0 depending on t0 such that

‖v‖L2((0,t0)×(0,1)) ≤ Ct0‖y0‖L2(0,1)2 . (4.3)

Now, we consider the uncontrolled integro-differential system

z1t − (a(x)z1x)x + b11z1 + b12z2

=

∫ t

t0

h1(t, r, x)z1(r, x) dr, (t, x) ∈ (t0, T )× (0, 1),

z2t − (a(x)z2x)x + b21z1 + b22z2

=

∫ t

t0

h2(t, r, x)z2(r, x) dr, (t, x) ∈ (t0, T )× (0, 1),

z1(t, 1) = z2(t, 1) = 0, t ∈ (t0, T ),{
z1(t, 0) = z2(t, 0) = 0, (WD),

(az1x)(t, 0) = (az2x)(t, 0) = 0, (SD),
t ∈ (t0, T ),

z1(t0, x) = w1(t0, x) = 0, z2(t0, x) = w2(t0, x) = 0, x ∈ (0, 1).

(4.4)

Using Proposition 3.1, we infer that (z1, z2) = (0, 0) is the unique solution of (4.4).
Finally, we set

(y1, y2) :=

{
(w1, w2), in [0, t0],

(z1, z2), in [t0, T ]
and u :=

{
v, in [0, t0],

0, in [t0, T ].

Note that, according to Hypothesis 4.1 and the previous definition, one has∫ t

0

hk(t, r, x)yk(r, x) dr =

∫ t

t0

hk(t, r, x)zk(r, x) dr = 0, (4.5)

for k = 1, 2, and t ∈ (t0, T ).
We can readily show that (y1, y2) ∈ L2

(
0, T ;H1

a(0, 1)2
)
∩ C

(
[0, T ];L2(0, 1)2

)
solves the system (1.1) associated to u and is such that

y1(T, ·) = y2(T, ·) = 0 in (0, 1).

Furthermore, using (4.3), we have that u satisfies the estimate

‖u‖L2(Q) = ‖v‖L2((0,t0)×(0,1)) ≤ Ct0‖y0‖L2(0,1)2 .



EJDE-2023/18 NULL CONTROLLABILITY OF INTEGRO-DIFFERENTIAL SYSTEMS 21

This completes the proof. �

Observe that with this technique we obtain also an estimate on the control
function through the norm of the initial data and thus we can estimate the cost for
controlling the solution of the system to zero.

Acknowledgements. G. Fragnelli was supported by the FFABR Fondo per il fi-
nanziamento delle attività base di ricerca 2017, by the INdAM - GNAMPA Project
2020 Problemi inversi e di controllo per equazioni di evoluzione e loro applicazioni,
by PRIN 2017-2019 Qualitative and quantitative aspects of nonlinear PDEs, and
by the DEB.HORIZON EU DM737 project 2022 Controllability of PDEs in the
Applied Sciences (COPS). G. Fragnelli is member of the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Isti-
tuto Nazionale di Alta Matematica (INdAM) and a member of UMI “Modellistica
Socio-Epidemiologica (MSE)”.

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Brahim Allal

Ibn Zohr University, Faculty of Applied Sciences, IMIS Laboratory, 86153 Aẗ-Melloul,
Morocco

Email address: b.allal@uiz.ac.ma

Genni Fragnelli
Department of Ecological and Biological Sciences, Tuscia University,

Largo dell’Università, 01100 Viterbo, Italy

Email address: genni.fragnelli@unitus.it

Jawad Salhi

Moulay Ismail University of Meknes, FST Errachidia, MAIS Laboratory, MAMCS Group,
P.O. Box 509 Boutalamine, 52000 Errachidia, Morocco

Email address: j.salhi@umi.ac.ma


	1. Introduction
	2. Null controllability of a nonhomogeneous degenerate system
	2.1. Well-posedness
	2.2. Carleman estimates
	2.3. Null controllability result

	3. Null controllability for the integro-differential system
	4. Concluding remarks
	Acknowledgements

	References