9_371_gopal.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 2, 2010, 235-253

ISSN 1307-5543 – www.ejpam.com

Identification of the Memory Kernels and Controllability for

Parabolic Equations

R. Lavanya

Adithya Institute of Technology, Coimbatore, India

Abstract. This paper deals with the controllability and observability properties of the mathematical

models (describing systems with thermal memory) consisting of boundary value problems of parabolic

type, where the differential equation contains additional integral expressions including “memory func-

tions” which describe the memory property of the material. The proof of controllability relies on a

Carleman type estimate and duality arguments.

2000 Mathematics Subject Classifications: 93B05, 93C20, 45K05, 35K50.

Key Words and Phrases: Controllability, Observability, Memory Kernels, Carleman Estimate.

1. Introduction

In many of the applications [4] we begin with a partial differential equation and, through

simplifying assumptions, arrive at an integral or integrodifferential equation which takes the

whole history into account. Lunardi [11] and Unger et al [14], for example, studied the

problem concerned with materials with memory having the property that the mathematical-

physical description of their state at a given point of time includes such states in which the

materials have been at earlier points of time. In the linear theory of heat flow in a rigid ho-

mogeneous isotropic body consisting of material with thermal memory, the following system

of constitutive relationships hold (see [9,14])

e(t, x) = β y(t, x) +

∫ t

−∞
n(t,τ)y(τ, x)dτ, (1)

s(t, x) = −ζ∇y(t, x)−
∫ t

−∞
m(t,τ)∇y(τ, x)dτ, (2)

together with the heat-balance equation:

et(t, x) + divs(t, x) = f (t, x), (3)

Email address: lavnya.gopal�gmail.
om
http://www.ejpam.com 235 c© 2010 EJPAM All rights reserved.



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 236

where e(t, x) is the internal energy, s(t, x) is the heat flux, y(t, x) is the body temperature

with time t ∈ [0, T] for fixed T , x ∈ Ω where Ω ⊂ R3 is an open bounded domain with a

smooth boundary ∂Ω of class C1, f (t, x) is the given heat source, β = cρ (c is the specific

caloric constant; ρ is the density) and ζ is the heat-conduction coefficient.

If we assume that y(t, x) ≡ 0 for −∞ < t < 0, it can be immediately seen that the

relations (1)-(3) lead to the system of the form,

β yt(t, x) − ζ∆y(t, x)−
∫ t

0

m(t,τ)∆y(τ, x)dτ

+
∂

∂ t

�∫ t

0

n(t,τ)y(τ, x)dτ
�
= f (t, x) in (0, T )×Ω,

y(0, x) = y0(x) in Ω,

y(t, x) = 0 on (0, T )× ∂Ω,

where y0(x) is the given initial temperature distribution. The memory kernels n and m are

sufficiently smooth and have support in (t0, t1), where 0 < t0 < t1 < T satisfying m(t, t) =

n(t, t) = 0 and represent the derivatives of the relaxation function of internal energy and

heat flux respectively. Hereafter, for our convenience, assume that β = 1 and ζ = 1 and set

Q = (0, T )×Ω and Σ = (0, T )× ∂Ω.

We now consider the corresponding controlled parabolic system with memory kernels

yt −∆y(t, x)−M t
0 ∗∆y(t) + (N t

0 ∗ y(t))t = f (t, x) +χωu(t, x) in Q

y(0, x) = y0(x) in Ω

y(t, x) = 0 on Σ,



 (4)

where χω is the characteristic function of the open set ω ⊂ Ω, u = u(t, x) is the control

function to be determined which acts on the system through ω while f ∈ L2(Q) is the given

source term. The notations M t
0 ∗∆y and N t

0 ∗ y respectively stand for memory integrals from

0 to t, that is,

M t
0 ∗∆y(t) =

∫ t

0

m(t,τ)∆y(τ)dτ,

N t
0 ∗ y(t) =

∫ t

0

n(t,τ)y(τ)dτ.

The system is null controllable at time T if, for each y0 ∈ H1
0(Ω), there exists a control u ∈

L2(ω× (0, T )) such that the associated solution satisfies

y(T, x) = 0 a.e. x ∈ Ω.

The null controllability of linear parabolic equations without the memory kernels has been

intensively studied by several authors; for instance see Barbu [2], Fernandez-Cara et al [6],

Fursikov and Imanuvilov [7], Imanuvilov [8] and the references cited therein. Fernandez-

Cara and Zuazua [5] studied the approximate controllability for heat equations and Barbu



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 237

and Iannelli [3] discussed the approximate controllability for the system of the form (4) with

the kernel n(·) = 0. Sakthivel et al [12] obtained the exact null controllability result by estab-

lishing a Carleman type inequality for the linear parabolic equation (taking the history into

account),

yt −∆y +

∫ t

0

a(t −τ)y(τ, x)dτ=u(t, x)+ l(t, x) in Q,

y(0, x) = y0(x) in Ω,

α1

∂ y

∂ ν
+ α2 y = 0 on Σ,

where Ω⊂ Rn is a bounded domain with boundary ∂Ω ∈ C1, the kernel a ∈ C1[0, T], a(0) = 0

and α1 ≥ 0 is a constant and α2 ∈ C1(Σ), α2 ≥ 0, while Fernandez-Cara et al [6] studied the

exact controllability of the parabolic equation of the form,

yt −∆y + B(t, x)∇y + a(t, x)y = v(t, x)χω in Q

with Fourier boundary conditions when the coefficients a, B and α2 satisfy a ∈ L∞(Q), B ∈
L∞(Q), and α2 ∈ L∞(Σ). The problem under consideration is interesting and different from

the previous works (see [9,10]) because the derivation of Carleman estimate containing a

special type of integral term for the backward adjoint problem of (4) stated in (5) require

a careful treatment of the surface integrals to guarantee the existence (ie., to settle the in-

tegral term properly so as to get the same upper bound) of this estimate for the parabolic

integrodifferential equations.

Throughout this paper we shall use the following notations for general function spaces.

For each positive integer m, we denote, by Hm(Ω), the Sobolev spaces of functions in L2(Ω)

whose weak derivatives of order less than or equal to m are also in L2(Ω). We define

L2(0, T ; H1(Ω)), the space of all equivalence classes of square integrable functions from (0, T )

to H1(Ω). The space L2(0, T ; L2(Ω)) is analogously defined. Moreover, we set

H1(0, T ; L2(Ω)) = {y ∈ L2(0, T ; L2(Ω)) :
d y

d t
∈ L2(0, T ; L2(Ω))},

H2,1(Q) = {y ∈ L2(0, T ; H1
0(Ω)∩H2(Ω)),

d y

d t
∈ L2(0, T ; L2(Ω))},

where
d y

d t
is taken in the sense of distributions. For the definition and detailed discussion on

these spaces one can refer [1,13].

The paper is organized as follows: In section 2 we establish a Carleman estimate for the

dual problem stated in (5) and we deduce an observability inequality . In section 3, we prove

the null controllability of the system (4) making use of observability inequality and an a priori

estimate for the solution of the system (4).



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 238

2. Carleman and Observability Inequalities

In this section we shall obtain a Carleman inequality and an observability estimate for the

following adjoint system associated with (4),

qt +∆q+M T
t ∗∆q(t) + N T

t ∗ qt(t) = g in Q

q(T, x) = qT (x) in Ω

q(t, x) = 0 on Σ,



 (5)

where qT ∈ L2(Ω), g ∈ L2(Q) and M T
t ∗∆q, N T

t ∗ qt are the corresponding adjoint integrals,

that is,

M T
t ∗∆q(t) =

∫ T

t

m(τ, t)∆q(τ)dτ,

N T
t ∗ qt(t) =

∫ T

t

n(τ, t)qτ(τ)dτ.

To formulate our results, we give some of the frequently used notations, following the idea

used in [7], which provide a fundamental tool in proving the Carleman type estimates. Let

ω0 ⋐ω be a suitably fixed sub domain. Then there exists a function ψ ∈ C2(Ω) such that

ψ(x)> 0 ∀ x ∈ Ω, ψ|∂Ω = 0, |∇ψ(x)|> 0 ∀ x ∈ Ω\ω0.

We define two weight functions that will be used throughout this paper as follows: For fixed

λ > 0 and the function ψ defined above, we introduce functions φ,α : Q→ R defined by the

formulas

φ(t, x) =
eλψ(x)

ξ(t)
, α(t, x) =

e2λΨ(x) − eλψ

ξ(t)
,

where

ξ(t) = t(T − t) and Ψ = ‖ψ(x)‖C(Ω).
Moreover, in proving the main inequality, we need the following estimates for the functions φ

and α: �� ∂ φ
∂ t

�� = |T−2t|
t2(T−t)2

eλψ ≤ C(Ω,ω)Tφ2

�� ∂ α
∂ t

�� = |T−2t|
t2(T−t)2

|e2λΨ − eλψ| ≤ C(Ω,ω) Teλψ

t2(T−t)2)
≤ C(Ω,ω)Tφ2

�� ∂ 2α

∂ t2

�� = |2T2−6T t+6t2 |
t3(T−t)3

|e2λΨ − eλψ| ≤ C(Ω,ω)T 2φ3



 (6)

where C(Ω,ω) is a generic constant. Throughout the proof of the estimate, we use C(Ω,ω),

the generic constant for all the space derivatives of ψ. One can also easily verify the identities

which will be used in the sequel are

∇φ = λφ∇ψ, ∇α= −λφ∇ψ.



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 239

Now we are ready to state and prove the main estimate of this section. Though the proof of

this estimate follows standard technique for general parabolic equations without memory, we

have to do careful calculations on the memory integrals which involves second derivative in

spatial variable as well as first derivative in time variable.

Theorem 1. For any solution q of the dual problem (5) with the kernels m(·, ·) and n(·, ·) have

support in (t0, t1), where 0 < t0 < t1 < T, there exist λ0, s0 and C, the constant depending on

Ω,ω,λ and T such that for every λ≥ λ0, s ≥ s0 the following inequality holds:

LQ,s,φ(q)≤ C
� ∫∫

(0,T)×ω

e−2sαs3φ3|q|2d xd t +

∫∫

Q

e−2sα|g|2d xd t
�

, (7)

where we used the notation

LQ,s,φ(q) =

∫∫

Q

(sφ)−1
�|qt |2+ |∆q|2+

��M T
t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�e−2sαd xd t

+

∫∫

Q

(s3φ3|q|2+ sφ|∇q|2)e−2sαd xd t.

Moreover, the constants λ0, s0 take the form λ0 = C(Ω,ω)[1 +
p

T + T 2 + T 4] and s0 =

C(Ω,ω)[T + T
p

T + T 2 + T 4].

To prove this theorem we need the following lemma in terms of the new transformed

variable p = e−sαq which essentially completes the first part of Theorem 1.

Lemma 1. Let the kernels m(·, ·) and n(·, ·) have support in (t0, t1), where 0 < t0 < t1 < T

and g ∈ L2(Q) be given. There exist eλ0,es0 and C only depending on Ω,ω and T such that, for

any λ ≥ eλ0 = C(Ω,ω)(1+ T 4), any s ≥ es0 = C(Ω,ω)(T + T 2 + T 4), the weak solution of (5)

satisfies

eLQ,s,λ(p) ≤ C
�‖e−sαg‖2

L2(Q)
+ eLQω0 ,s,λ(p) +MQ,s,λ(m; p) + NQ,s,λ(n; p) (8)

+ MQ,s,λ(mt ; p) + NQ,s,λ(nt; p)
�
,

where

eLQ,s,λ(p) =

∫∫

Q

s3λ4φ3|p|2d xd t +

∫∫

Q

sλ2φ|∇p|2d xd t,

MQ,s,λ(m; p) =

∫∫

Q

e−2sαsλφ
��M T

t ∗∆(esαp)(t)
��2d xd t,

NQ,s,λ(n; p) =

∫∫

Q

e−2sαsλφ
��N T

t ∗ (esαp)t(t)
��2d xd t,



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 240

and the notations MQ,s,λ(mt ; p), NQ,s,λ(nt; p) denote the time derivative of the kernels respectively

in MQ,s,λ(m; p), NQ,s,λ(n; p) and Qω0
= (0, T )×ω0.

The proof of Lemma 1 is quite similar to the detailed proof given in [9],[10]. The explicit

dependence of the constant on time and space is not obtained in [12] and we refer to [10],[6]

where the explicit dependence has been computed. Now we need to estimate the memory

integrals appearing on the right hand side of the estimate (8) and in fact this will complete

the proof of Theorem 1.

Proof. First we write the inequality (8) in terms of the original variable by substituting

p = e−sαq to have

∫∫

Q

e−2sαs3λ4φ3|q|2d xd t +

∫∫

Q

sλ2φ|∇(e−sαq)|2d xd t ≤ C
�∫∫

Q

e−2sα|g|2d xd t

+

∫∫

Qω0

e−2sαs3λ4φ3|q|2d xd t +

∫∫

Qω0

sλ2φ|∇(e−sαq)|2d xd t

+MQ,s,λ(m; q) +MQ,s,λ(mt ; q) + NQ,s,λ(n; q)+ NQ,s,λ(nt ; q)
�

.

Note that ∇(e−sαq) = e−sαsλφ∇ψq+ e−sα∇q and

2

∫∫

Q

e−2sαs2λ3φ2∇ψq∇qd xd t ≥ −ρ
∫∫

Q

e−2sαsλ2φ|∇q|2d xd t

− 1

ρ

∫∫

Q

e−2sαs3λ4φ3|∇ψ|2|q|2d xd t,

where the parameter ρ ∈ (0,1). Choose ‖∇ψ‖C(Ω̄) ≤ ρ to obtain

eLQ,s,λ(q)≤ C
�∫∫

Q

e−2sα|g|2d xd t + eLQω0
,s,λ(q)+MQ,s,λ(m; q) (9)

+MQ,s,λ(mt ; q) + NQ,s,λ(n; q)+ NQ,s,λ(nt ; q)
�

,

since we have redefined the notations eL, M , N as follows:

eLQ,s,λ(q) =

∫∫

Q

e−2sαs3λ4φ3|q|2d xd t +

∫∫

Q

e−2sαsλ2φ|∇q|2d xd t,

MQ,s,λ(m; q) =

∫∫

Q

e−2sαsλφ
��M T

t ∗∆q(t)
��2d xd t,



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 241

NQ,s,λ(n; q) =

∫∫

Q

e−2sαsλφ
��N T

t ∗ qt(t)
��2d xd t.

Next we shall express the term |∇q|2 over Qω0
on the right hand side of (9), in terms of |q|2 in

the larger domain ω(since ω0 ⋐ω⊂ Ω). To attain this, let us introduce a truncating function

θ = θ(x), 0≤ θ ≤ 1 satisfying

θ ∈ C2
0 (ω), θ = 1 in ω̄0 and θ = 0 in Ω\ω.

Multiplying (5) by e−2sαθ sλ2φq and integrating over Q, we obtain that
∫∫

Q

e−2sαθ sλ2φ|∇q|2d xd t ≤ 1

4

∫∫

Q

e−2sα|g|2d xd t +
1

4
MQω ,s,λ(m; q)

+
1

4
NQω,s,λ(n; q)−
∫∫

Q

sλ2∇(e−2sαθφ)q∇qd xd t

+

∫∫

Qω

e−2sα(s2λ4φ2 + 2sλ3φ)|q|2d xd t − 1

2

∫∫

Qω

sλ2(e−2sαφ)t |q|2d xd t =

6∑

i=1

Ii . (10)

Now a simple computation yields the following estimates: The integral I4 can be estimated by

C(Ω,ω)

∫∫

Qω0

e−2sα(s3λ4φ3 + sφ(λ4 +λ2))|q|2d xd t +
1

4

∫∫

Qω0

e−2sαsλ2φ|∇q|2d xd t,

where the first integral can be bounded by
∫∫

Qω

e−2sαs3λ4φ3|q|2d xd t, if λ ≥ C(Ω,ω)T 2, s ≥ 1.

The integral I6 has also the same bound,

I6 ≤ C(Ω,ω)T

∫∫

Qω

e−2sα(s2λ2φ3 + sλ2φ2)|q|2d xd t ≤
∫∫

Qω

e−2sαs3λ4φ3|q|2d xd t

for the choice of λ ≥ 1, s ≥ C(Ω,ω)(T + T 3/2). Thus, combining all the preceding inequality,

we obtain
∫∫

Qω0

e−2sαsλ2φ|∇q|2d xd t ≤ C
�∫∫

Q

e−2sα|g|2d xd t +

∫∫

Qω

e−2sαs3λ4φ3|q|2d xd t (11)

+ MQω ,s,λ(m; q) + NQω,s,λ(n; q)
�

.

Using (11), the inequality (9) can be re-estimated as

eLQ,s,λ(q)≤ C
�∫∫

Q

e−2sα|g|2d xd t +

∫∫

(0,T)×ω

e−2sαs3λ4φ3|q|2d xd t



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 242

+MQ,s,λ(m; q) +MQ,s,λ(mt ; q) + NQ,s,λ(n; q)+ NQ,s,λ(nt ; q)
�

, (12)

for any λ ≥ λ0 = C(Ω,ω)[1+
p

T + T 2 + T 4] and s ≥ s̃0 = C(Ω,ω)[T + T 2 + T
p

T + T 4].

Making use of the assumptions on the kernel, Hölder’s inequality and changing the order of

integration, we have

MQ,s,λ(m; q) =

∫∫

Q

e−2sαsλφ
��M T

t ∗∆q(t)
��2d xd t ≤
∫∫

Q

e−2sαsλφ
��M T

0 ∗∆q(t)
��2d xd t

≤
∫∫

Q

e−2sαλφ
�∫ t1

t0

|m(τ, t)|2e(s
2+2sα)φ(τ)dτ
��∫ t1

t0

e−2sαs−1φ−1(τ)|∆q(τ)|2dτ
�

d xd t

≤ C‖m‖2L∞
∫∫

(t0 ,t1)×Ω

e−2sα(sφ)−1λ|∆q|2
�∫ T

0

e−2sαφ(τ)dτ
�

d xd t

≤ C

∫∫

(t0,t1)×Ω

e−2sαλ(sφ)−1|∆q|2d xd t ≤ C

∫∫

Q

e−2sαλ(sφ)−1|∆q|2d xd t, (13)

where C depends on Ω,ω, t0, t1, T , and m. Similarly, estimating the integral NQ,s,λ(n; q), one

can have

NQ,s,λ(n; q) =

∫∫

Q

e−2sαsλφ
��N T

t ∗ qt(t)
��2d xd t ≤ C

∫∫

Q

e−2sαλ(sφ)−1|qt |2d xd t, (14)

where C depends on Ω,ω, t0, t1, T , and n. The similar estimates holds true for MQ,s,λ(mt ; q)

and NQ,s,λ(nt; q).

Indeed one can obtain a sharp estimate for the weight functions (used above) as follows:

Following certain standard analysis used in [5], we obtain

esαφ ≤ C(Ω,ω)(t(t − T ))−1e−sα̃/t(t−T) ≤ 4T−2e−σ(Ω,ω)sT−2

,

where eα= e2λΨ − eλψ and σ = 4 min
x∈Ω eα for s ≥ s0 =max(s̃0, (σ(Ω,ω))−1T 2).

In order to complete the theorem, it remains to obtain an estimate for the terms involving

first order derivative in time and second in space variable. To obtain this, first of all multiply-

ing (5) by e
p−2sαλ
p
(sφ)−1, squaring and then integrating on Q, we get

bLQ,s,λ(q) =

∫∫

Q

e−2sα(sφ)−1λ2|g|2d xd t + 2(D+ E) + 2(F + G) + 2H

−2

∫∫

Q

e−2sα(sφ)−1λ2qt∆qd xd t, (15)



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 243

where

bLQ,s,λ(q) =

∫∫

Q

e−2sα(sφ)−1λ2
�|qt |2 + |∆q|2+

��M T
t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�d xd t,

(D+ E) = −
∫∫

Q

e−2sα(sφ)−1λ2qt

�
M T

t ∗∆q(t) + N T
t ∗ qt(t)
�
d xd t,

(F + G) = −
∫∫

Q

e−2sα(sφ)−1λ2∆q
�

M T
t ∗∆q(t) +N T

t ∗ qt(t)
�
d xd t,

H = −
∫∫

Q

e−2sα(sφ)−1λ2
�

M T
t ∗∆q(t)
��

N T
t ∗ qt(t)
�
d xd t.

Now we have the following estimates by choosing the constants carefully and applying Young’s

inequality followed by Green’s theorem and integration by parts. Integrating by parts with

respect to time in D+ E, we obtain

2(D+ E) = −
∫∫

Q

e−2sαλ2(4αtφ
−1 + 2s−1φ−2φt)q

�
M T

t ∗∆q(t) + N T
t ∗ qt(t)
�
d xd t

+2

∫∫

Q

e−2sαλ2(sφ)−1q
�∫ T

t

mt(τ, t)∆q(τ)dτ+

∫ T

t

nt(τ, t)qτ(τ)dτ
�

d xd t

= D1 + D2, (16)

where we used the assumption m(t, t) = n(t, t) = 0. Since we observe that

D1 ≤
∫∫

Q

e−2sαs3λ4φ3|q|2d xd t

+
1

2

∫∫

Q

e−2sαλ2(sφ)−1
���M T

t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�d xd t

for any λ≥ 1 and s ≥ C(Ω,ω)(T + T
p

T ). The integral D2 can be bounded by

D2 ≤
∫∫

Q

e−2sαs3λ3φ3|q|2d xd t +

∫∫

Q

e−2sαλ(sφ)−1
����
∫ T

t

mt(τ, t)∆q(τ)dτ

���
2

+

���
∫ T

t

nt(τ, t)qτ(τ)dτ

���
2�

d xd t = D21 + D22 (17)



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 244

for s ≥ C(Ω,ω)T 2. Computation similar to (13) gives further that,

D22 ≤
∫∫

Q

e−2sαλ(sφ)−1
h�∫ t1

t0

|mt(τ, t)|2e2sαφdτ
��∫ t1

t0

e−2sαφ−1|∆q(τ)|2dτ
�

+
�∫ t1

t0

|nt(τ, t)|2e2sαφdτ
��∫ t1

t0

e−2sαφ−1|qτ(τ)|2dτ
�i

d xd t

≤ C‖mt‖2L∞
∫∫

Q

e−2sαλ(sφ)−1|∆q|2d xd t + C‖nt‖2L∞
∫∫

Q

e−2sαλ(sφ)−1|qt |2d xd t.

Here we abserve that for any λ ≥ λ0 sufficiently large, the last two integrals can be absorbed

in bLQ,s,λ(q). Now the simple calculation using Green’s formula yields

−2

∫∫

Q

e−2sα(sφ)−1λ2qt∆qd xd t =

∫∫

Q

e−2sαλ3(4− 2(sφ)−1)qt(∇ψ · ∇q)d xd t

+

∫∫

Q

e−2sαλ2(s−1φ−2φt + 2φ−1αt)|∇q|2d xd t

≤ 1

4

∫∫

Q

e−2sα(sφ)−1λ2|qt |2d xd t +

∫∫

Q

e−2sαsλ2φ|∇q|2d xd t, (18)

for any s ≥ C(Ω,ω)(T + T 2 + T
p

T ). Since we have chosen (if necessarily by normalizing)

that ‖∇ψ‖C(Ω̄) ≤ 1/λ and used the fact that α(0) = α(T ) = +∞. Moreover, we have

2H ≤
∫∫

Q

e−2sα(sφ)−1λ2
���M T

0 ∗∆q(t)
��2+
��N T

0 ∗ qt(t)
��2�d xd t, (19)

and

2(F + G) ≤ 1

4

∫∫

Q

e−2sα(sφ)−1λ2|∆q|2d xd t

+8

∫∫

Q

e−2sα(sφ)−1λ2
���M T

0 ∗∆q(t)
��2 +
��N T

0 ∗ qt(t)
��2�d xd t. (20)

Proceeding calculations similar to (13) and (14), we note that the integrals in (19) and the

last integral in (20) can further be estimated as

C‖m‖2L∞
∫∫

Q

e−2sα(sφ)−1λ2|∆q|2d xd t + C‖n‖2L∞
∫∫

Q

e−2sα(sφ)−1λ2|qt |2d xd t. (21)



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 245

Consequently, if C‖m‖2
L∞ ≤ 1

4
and C‖n‖2

L∞ ≤ 1

4
, the estimations (16)-(21) yield

bLQ,s,λ(q)≤ C
�
∫∫

Q

e−2sα|g|2d xd t + eLQ,s,λ(q)
�
. (22)

Eventually, making use of the estimations (13), (14) and choosing λ≥ λ0, s ≥ s0 large enough,

recall that the powers of λ in bLQ,s,λ(q), dominates the powers in (13),(14)), we get

bLQ,s,λ(q)+ eLQ,s,λ(q)

≤ C
�∫∫

Q

e−2sα|g|2d xd t +

∫∫

(0,T)×ω

e−2sα(sφ)3λ4|q|2d xd t
�

. (23)

This completes the proof of the theorem.

Remark 1. Smallness condition on the memory kernels m(·, ·) and n(·, ·) imposed in the esti-

mate (22) is indeed necessary to arrive at such an estimate as the integral involves the second

spatial derivative as the absorption is not possible otherwise. In practice the memory kernels are

exponential functions (with negative exponents, in general) and so the assumption is valid for

appropriate weights.

An important consequence of Theorem 1 is the following observability estimate. The proof

of this estimate is similar to that of the estimate derived for various problems in Fursikov et

al [7] and Fernandez-Cara et al [5]. This estimate essentially gives the unique continuation

property for the solutions of the system (5), precisely, q = 0 in (0, T )×ω implies q ≡ 0 in

(0, T )×Ω; in particular q(0) = 0 in Ω. Now we state the observability inequality for the adjoint

system (5).

Corollary 1. Under the assumptions of theorem 1, there exists a positive constant W depending

on Ω,ω, m, n and T such that

‖q(0)‖2
L2(Ω)
≤W (Ω,ω, T )
� ∫∫

(0,T)×ω

|q|2d xd t +

∫∫

Q

|g|2d xd t
�

(24)

where W (·) = exp
�

C(1+ 1

T
+ 1p

T
+ T + T 2+ T 2(‖m‖2

L∞ + ‖n‖2L∞)
�

and q is the weak solution

of the problem (5).

Proof. Let q be the solution of (5) and g ∈ L2(Q). We shall first prove the variant of the

inequality (24), namely,

‖q(0)‖2
L2(Ω)
≤W ∗(T )
� ∫∫

(T/4,3T/4)×Ω

|q|2d xd t +

∫∫

Q

|g|2d xd t



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 246

+

∫∫

(t0,t1)×Ω

e−2sα(sφ)−1(|∆q|2+ |qt |2)d xd t
�

(25)

where W ∗(·) = exp[C( 1

T
+ T + T 2(‖m‖2

L∞ + ‖n‖2L∞)]. Multiplying (5) by q and integrating on

Ω, we get

−1

2

d

d t

∫

Ω

|q|2d x +

∫

Ω

|∇q|2d x ≤ 3

2

∫

Ω

|q|2d x +
1

2

∫

Ω

�|g|2+
��M T

t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�d x .

It follows that

− d

d t

�
exp[3t]

∫

Ω

|q|2d x
�
≤ exp[3t]

∫

Ω

�|g|2+
��M T

t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�d x . (26)

Integrating (26) with respect to time in 0≤ t ≤ T/4, we have

∫

Ω

|q(0)|2d x ≤ exp[3T/4]

∫

Ω

|q(T/4, x)|2d x

+exp[3T]

∫ T/4

0

∫

Ω

�|g|2+
��M T

t ∗∆q(t)
��2+
��N T

t ∗ qt(t)
��2�d xdτ.

Again integrating (26) from T/4 to t, we get

exp[3T/4]

∫

Ω

|q(T/4, x)|2d x ≤ exp[3T]

∫

Ω

|q|2d x

+exp[3T]

∫ t

T/4

∫

Ω

�|g|2+
��M T

t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�d xdτ

for all t ∈ �T/4,3T/4
�

. Thus we have

∫

Ω

|q(0)|2d x ≤ C
�∫

Ω

|q|2d x +

∫ t

0

∫

Ω

�|g|2+
��M T

t ∗∆q(t)
��2 +
��N T

t ∗ qt(t)
��2�d xdτ
�

, (27)

where C = exp[3T]. By the assumption on the kernel, we have

∫ t

0

∫

Ω

���M T
t ∗∆q(t)

���
2

d xdτ

≤
∫ T

0

∫

Ω

�∫ t1

t0

|m(τ, t)|2es(1+2α(τ))φ(τ)dτ
��∫ t1

t0

e−2sα(τ)s−1φ−1(τ)|∆q(τ)|2dτ
�

d xd t

≤ C T‖m‖2L∞
∫∫

(t0 ,t1)×Ω

e−2sα(sφ)−1|∆q|2d xd t. (28)



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 247

Now estimating the integral

∫ t

0

∫

Ω

��N T
t ∗ qt(t)
��2d xdτ similar to the above and substituting

the preceding estimates into (27) and integrating the resulting inequality with respect to time

in (T/4,3T/4), one can obtain the inequality (25). To complete the proof it suffices to obtain

an estimate for the right hand side integrals of (25) in terms of the L2 integral of q over

(0, T )×ω. From the Carleman estimate for the adjoint system (5), we obtain

LQ,s,1(q) ≤ C
�∫∫

Q

e−2sα|g|2d xd t +

∫∫

(0,T)×ω

e−2sαs3φ3|q|2d xd t
�

. (29)

Here one can easily verify the following weight function estimates using certain standard

analysis (see [5]):

e−2sαφ3 ≤ C(Ω,ω)
1

(t(T − t))3
e2sα̃/t(T−t) ≤ C(Ω,ω)

� 2
T

�6
e−σ(Ω,ω)sT−2 ∀ (t, x) ∈ Q̄,

provided s ≥ s1 = max(s0, 3(σ(Ω,ω))−1T 2), where the constant σ(Ω,ω) = 8 min eα. If we

look at the constants s0 and s1, then we get

s1 ≤ s2 = C(Ω,ω)
�

T + T 2 + T
p

T + T 4
�

.

For s ≥ s2, we have

e−2sαφ3 ≥ C(Ω,ω)
� 16

3T 2

�3
e−C(Ω,ω)sT−2 ∀ (t, x) ∈ [T/4,3T/4]× Ω̄.

Let us fix the constant s = s2 and making use of the above weight function estimates, and

from (29), we deduce the following estimate

∫∫

(T/4,3T/4)×Ω

|q|2d xd t +

∫∫

(t0,t1)×Ω

e−2sα(sφ)−1(|∆q|2+ |qt |2)d xd t

≤fW (·)
� ∫∫

(0,T)×ω

|q|2d xd t +

∫∫

Q

|g|2d xd t
�

,

where fW (·) = exp
�

C(1+ 1

T
+ 1p

T
+ T 2)
�

. Coupling the above estimate with (25), one can

obtain the observability estimate (24).

3. Controllability Results

In this section, we prove a null controllability result for the problem stated in (4). We

shall obtain a solution to the global controllability problem for the equation (4) as a limit of

an approximation process with the aid of certain suitably defined optimal control problem. To



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 248

derive the estimate, we use the maximum principle and the observability inequality which is

derived in the previous section for the dual problem (5). We first obtain an explicit bound for

the weak solution of the system

yt −∆y −M t
0 ∗∆y(t) + (N t

0 ∗ y(t))t = f in Q

y(0, x) = y0(x) in Ω

y(t, x) = 0 on Σ,



 (30)

where f ∈ L2(Q), y0 ∈ H1
0(Ω) are given. The above problem has a unique solution y ∈

L2(0, T ; H1
0(Ω)∩H2(Ω))∩H1([0, T]; L2(Ω)) whenever y0 ∈ H1

0(Ω). The existence and unique-

ness of a solution to this problem is well known, see for example [9]. The following proposi-

tion gives an a priori estimate for the solution of the system (30).

Proposition 1. Let f ∈ L2(Q) and y0 ∈ H1
0(Ω) be given. Then the weak solution y ∈ H2,1(Q) of

the problem (30) satisfies the estimate

‖y‖2
H2,1(Q)

≤ V (·)�‖y0‖2H1
0(Ω)
+ ‖ f ‖2

L2(Q)

�
, (31)

where V (·) = exp[C(1+ T + ‖nt‖L∞ + T 2(‖n‖2
L∞ + ‖nt‖2L∞ + ‖nt t‖2L∞))].

Proof. The proof follows the standard technique. First multiplying (30) by y and integrat-

ing on Ω, we obtain that

1

2

d

d t

∫

Ω

|y|2d x +

∫

Ω

|∇y|2d x ≤ 3

2

∫

Ω

|y|2d x+
1

2

∫

Ω

�| f |2+
��M t

0 ∗∆y(t)
��2+
��(N t

0 ∗ y(t))t
��2�d x .

Applying the differential version of Gronwall’s inequality in the interval 0 to t with 0≤ t ≤ t1

for some fixed t1 ∈ (0, T ), we have

∫∫

(0,t1)×Ω

|∇y|2d xd t +

∫

Ω

|y(t1)|2d x ≤ exp[3t1]
�∫

Ω

|y0|2d x

+

∫∫

(0,t1)×Ω

�| f |2 +
��M t

0 ∗∆y(t)
��2 +
��(N t

0 ∗ y(t))t
��2�d xd t
�

. (32)

Squaring both sides of the equation (30), and integrating on Ω, we obtain

d

d t

∫

Ω

|∇y|2+
∫

Ω

�|yt |2+ |∆y|2+
��M t

0 ∗∆y(t)
��2 +
��(N t

0 ∗ y(t))t
��2�d x = ‖ f ‖2

L2(Ω)
(33)

+2

∫

Ω

∆y
�
(N t

0 ∗ y(t))t −M t
0 ∗∆y(t)
�
d x − 2

∫

Ω

yt

�
(N t

0 ∗ y(t))t −M t
0 ∗∆y(t)
�

d x

+2

∫

Ω

�
M t

0 ∗∆y(t)
�
(N t

0 ∗ y(t))t d x =

4∑

i=1

Ii



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 249

for all t ∈ (0, T ). Using Cauchy inequality one can easily see that

∫ t1

0

I2d t ≤ 1

4

∫∫

(0,t1)×Ω

|∆y|2d xd t + 8

∫∫

(0,t1)×Ω

���M t
0 ∗∆y(t)
��2 +
���
∫ t

0

nt(t,τ)y(τ)dτ

���
2�

d xd t.(34)

Applying Hölder’s inequality, the last integral can further be estimated as

‖m‖2L∞ t2
1

∫∫

(0,t1)×Ω

|∆y|2d xd t + ‖nt‖2L∞ t2
1

∫∫

(0,t1)×Ω

|y|2d xd t.

Integration by parts in time together with the assumptions on the kernel yields,

∫ t1

0

I3d t = 2

∫∫

(0,t1)×Ω

y
�
∫ t

0

nt t(t,τ)y(τ)dτ+ nt(t, t)y(t)
�

d xd t

−2

∫∫

(0,t1)×Ω

y
�
∫ t

0

mt(t,τ)∆y(τ)dτ
�
d xd t = I31 + I32. (35)

Since using Young’s and Hölder’s inequality, we further obtain that

I31 ≤ (1+ 2‖nt‖L∞)
∫∫

(0,t1)×Ω

|y|2d xd t + ‖nt t‖2L∞ t2
1

∫∫

(0,t1)×Ω

|y|2d xd t

and

I32 ≤ ηt2
1‖mt‖2L∞
∫∫

(0,t1)×Ω

|∆y|2d xd t +
1

η

∫∫

(0,t1)×Ω

|y|2d xd t.

Finally, we note that the estimation similar to I32 yields

∫ t1

0

I4d t ≤ ηt2
1‖mt‖2L∞
∫∫

(0,t1)×Ω

|∆y|2d xd t +
1

η
‖n‖2L∞ t2

1

∫∫

(0,t1)×Ω

|y|2d xd t. (36)

If ‖m‖2
L∞(0,T)

t2
1 ≤ 1

16
, integrating (33) in the interval (0,t) and substituting (34)-(36) and

using the Poincaré inequality

∫

Ω

|y|2d x ≤ C(Ω)

∫

Ω

|∇y|2d x , one can have the following

‖y(t1)‖2H1
0(Ω)
+

∫∫

(0,t1)×Ω

�|yt |2 + |∆y|2 +
��M t

0 ∗∆y(t)
��2

+
��(N t

0 ∗ y(t))t
��2�d xd t ≤ M(·)�‖y0‖2H1

0(Ω)
+

∫∫

(0,t1)×Ω

| f |2d xd t
�
, (37)



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 250

where M(·) = exp[C(1+‖nt‖L∞+ t2
1(‖n‖2L∞+‖nt‖2L∞+‖nt t‖2L∞))]. Since we have also chosen

that η ≤ 1

16t2
1‖mt‖2L∞

. With the estimate (37) together with (32) and the Sobolev estimate, one

can conclude the proof.

Now we prove the main result of this work.

Theorem 2. Assume that T > 0 is fixed and y0 ∈ H1
0(Ω) is given and the kernels m(·, ·) and n(·, ·)

have support in (t0, t1) where 0 < t0 < t1 < T. Then there exists a control u ∈ L2(0, T ; L2(ω))

such that the corresponding solution of (4) satisfies

y(T, x) = 0 a.e. x ∈ Ω.

Moreover, the control u can be chosen in such a way that

||u||2
L2(0,T ;L2(ω))

≤W (Ω,ω, T )||y0||2L2(Ω)
,

where the constant W (·) is explicitly given in (24).

Proof. Let us fix T > 0 and y0 ∈ H1
0(Ω). For every ε > 0, let us consider the problem

min
n

Jε(u) : u ∈ L2(0, T ; L2(ω))
o

,

where the functional Jε is defined by

Jε(u) =
1

2

∫∫

(0,T)×ω

|u|2d xd t +
1

2ε

∫

Ω

|y(T, x)|2d x , (38)

where y is the solution of (4) associated with the control u. In order to solve this control

problem, it is enough to prove that the functional Jε has a unique solution (see Fernandez-

Cara et al [5]). Since, Jε is a continuous strictly convex functional in L2(Q) and coercive, that

is,

lim inf
||u||

L2((0,T )×ω→∞
Jε(u) =∞,

Jε has a unique solution (uε, yε) for every ε > 0. Next, we shall obtain the necessary condition

for optimality via maximum principle. We can verify that it is characterized by

uε = −χωqε (39)

where qε is the solution to the adjoint problem

(qε)t +∆qε+M t
0 ∗∆qε(t) + (N

t
0 ∗ qε(t))t = 0 in Q

qε(T, x) = 1

ε
yε(T, x) in Ω

qε(t, x) = 0 on Σ.



 (40)



R. Lavanya / Eur. J. Pure Appl. Math, 3 (2010), 235-253 251

Let us put y = w+ϕ. If y is the solution of (4) associated with u, and w is the weak solution

of the homogeneous problem corresponding to (30), then ϕ satisfies

ϕt −∆ϕ−M t
0 ∗∆φε(t) + (N t

0 ∗φε(t))t = χωu in Q

ϕ(0, x) = 0 in Ω

ϕ(t, x) = 0 on Σ.



 (41)

Now the functional Jε is differentiable at the point u. For u, v ∈ L2(0, T ; L2(ω)), we obtain

〈J ′ε(uε), v〉L2(Q) =

∫∫

(0,T)×ω

uεvd xd t+
1

ε

∫

Ω

y(T )ϕ(T )d x , (42)

where ϕ is the solution to (41) associated with the control v. For the pair (uε, yε) to be a

unique solution of Jε, we must have

〈J ′ε(uε), v〉L2(Q) = 0.

The duality between ϕ and q gives the following

∫∫

(0,T)×ω

qεvd xd t =

∫

Ω

qε(T )ϕ(T )d x =
1

ε

∫

Ω

yε(T )ϕ(T )d x . (43)

In view of (42) and (43), we can identify uε = −χωqε, the optimal control stated in (39).

Next we shall show that (uε, yε) converges along a subsequence of ε in a certain topology. In

order to prove this, we need a suitable estimate for (uε, yε). In particular, we get L2 estimate

for uε. Multiplying (4) by qε(replace y by yε) and (40) by yε and adding and then integrating

on (0, T )×Ω, we have

∫

Ω

qε(T )yε(T )d x =

∫

Ω

y0(x)qε(0)d x +

∫∫

(0,T)×ω

uεqεd xd t.

Making use of the optimality condition qε(T, x) = 1

ε
yε(T, x) and the Young’s inequality, we

obtain

∫∫

(0,T)×ω

|uε|2d xd t +
1

ε

∫

Ω

|yε(T, x)|2d x ≤ η
2
||y0||2L2(Ω)

+
1

2η
||qε(0)||2L2(Ω)

∀ η > 0.

Using Corollary 1, we can choose η appropriately, for instance η =W (Ω,ω, T ); then we have

1

2

∫∫

(0,T)×ω

|uε|2d xd t +
1

ε

∫

Ω

|yε(T, x)|2d x ≤W (Ω,ω, T )||y0||2L2(Ω)
, (44)



REFERENCES 252

where the constant W (·) is given by (24). The Proposition 1 and the estimate (44) allow us

to pass to the weak limit in (4) (after replacing (u, y) by (uε, yε)) as ε→ 0, which gives the

solution of the null controllability problem (4). Since uε is bounded in L2(0, T ; L2(ω)), there

exists a subsequence of ε still indexed by ε such that uε→ u weakly in L2(0, T ; L2(ω)), yε→ y

weakly in H2,1(Q) as ε→ 0. From (44) and Fatou’s lemma for any constant C independent of

ε, we have

||y(T, x)||2
L2(Ω)
≤ lim inf

ε→0

∫

Ω

|yε(T, x)|2d x ≤ lim inf
ε→0

Cε = 0.

It follows that

y(T, x) ≡ 0 a.e. x ∈ Ω.

The estimate for the control u follows from (44) and the proof is thus completed.

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