EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 1277-1296
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Boundary sentinel with given sensitivity in population
dynamics problem and parameters identification

Mifiamba Soma1, Somdouda Sawadogo2,∗

1 Département de Mathématiques et Informatique, Université Joseph Ki-ZERBO,
Ouagadougou, Burkina Faso.
2 Département de Mathématiques, Institut des Sciences, Ouagadougou, Burkina Faso.

Abstract. The notion of sentinels with given sensitivity was introduced by J.L.Lions [11] in order
to identify parameters in the problem of pollution ruled by a parabolic equation. He proves that
the existence of such sentinels is reduced to the solution of exact controllability problem with
constraints on the state. In population dynamics model, we reconsider this notion of sentinels in a
more general framework. We prove the existence of the boundary sentinels by solving a boundary
null-controllability problem with constraint on the control. Our results use Carleman inequality
which is adapted to the constraint.

2010 Mathematics Subject Classifications: 49J20, 93B05, 92D25, 35Q92, 35Q93

Key Words and Phrases: Population dynamics, optimal control, controllability, sentinels, Car-
leman inequality

1. Introduction

The notion of sentinel was introduced by J.L.Lions to study systems with incomplete
data [11]. The notion permits us to distinguish and to analyse two types of incomplete
data: the so-called pollution terms at which we look for information, independently of
the other type of incomplete data which is the missing terms and that we do not want to
identify.

Typically, the Lions’s sentinel is a functional defined on an open set O where we
consider three functions: the ”observation” yobs corresponding to measurements, a given
”mean” function h0, and a control function w to be determined.

Let us remind that Lions’s sentinel theory [11] relies on the following three features:
the state equation y which is governed by a partial differential equation, the observation
system and some particular evaluation function: the sentinel itself. More precisely, we
consider a linear model (1) describing the dynamics of population with age dependence,
spatial structure with incomplete data.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3479

Email addresses: mifiambasoma@yahoo.fr (M. Soma), sawasom@yahoo.fr (S. Sawadogo)

http://www.ejpam.com 1277 c© 2019 EJPAM All rights reserved.



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1278

Let Ω be an open and bounded domain of RN , N ∈ {1, 2, 3}, with boundary Γ
of C∞. For the time T > 0 and the life expectancy of an individual A > 0, we set
U = (0, T ) × (0, A), Q = U × Ω, QA = (0, A) × Ω, QT = (0, T ) × Ω, Σ = U × Γ,
Σ1 = U × Γ1, where Γ1 is a non-empty open subset of Γ. Then consider the following two
stroke problem:

∂y
∂t + ∂y

∂a −∆y + µy = 0 in Q
y(0, a, x) = y0 + τ ŷ0 in QA

y(t, 0, x) =

∫ A

0
β(t, a, x)y(t, a, x)da in QT

y =

 ξ +

M∑
i=1

λiξ̂i on Σ1

0 on Σ \ Σ1

(1)

where :

- y(t, a, x) is the distribution of a-year old individuals at time t at the point x ∈ Ω.

- β(t, a, x) ≥ 0 and µ(t, a, x) ≥ 0 are respectively the natural fertility and the natural
death rate of age a at time t and position x ∈ Ω.

- Thus, the formula

∫ A

0
β(t, a, x)y(t, a, x)da denotes the distribution of newborn individ-

uals at time t and location x.

- The boundary condition is unknown on a part Σ1 of the boundary and represents a

pollution with a structure of the form ξ +

M∑
i=1

λiξ̂i . In this structure, the functions

ξ and ξ̂i, i = 1, . . .M are known whereas the real λi, i = 1, . . .M are unknown.

- The initial distribution of individuals is unknown and its structure is of the form y0+τ ŷ0

where the function y0 is known and the term τ ŷ0 is unknown.

System (1) is a system with incomplete data because the information on the boundary
condition as well as on the initial condition are partially or completely unknown. Here,
the pollution is isolated on the boundary Γ\Γ1 . The missing term is located in the initial
conditions. In what follows, we assume as in [8] that:

(H1) :


β ∈ L∞(Q), β(t, a, x) ≥ 0 a.e. in Q;

sup
(t,x)∈]0,T [×Ω

∫
]0,A[

(‖β2(t, a, x)‖+ ‖∇β‖2(t, a, x)da);

∃ δ ∈ (0, A) s.t. β(a, ., ) = 0 for a ∈ (δ, A);

(H2) : µ ∈ C([0, T ]× [0, A]× Ω̄), µ(t, a, x) ≥ 0 a.e in Q



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1279

(H3) :



∀t, 0 < t < A, ∀x ∈ Ω, lim
∫ A

0 µ(ι, a− t+ ι, x)dι = +∞
a−→A

;

∀t, A < t < T, ∀x ∈ Ω, lim
∫ a

0 µ(t− a+ α, α, x)dα = +∞
a−→A

;

∇µ ∈ [L∞(Q)]n.

We also assume that:

- y0 and ŷ0 belong to L2(QA), ξ and ξ̂i belong to L2(Σ),

- the reals τ , λi 1 ≤ i ≤ M are sufficiently small and ‖ŷ0‖L2(QA) ≤ 1, and we set
λ = (λ1, . . . , λM ) .

Under the above assumptions on the data, one can prove as in [17] that problem (1) has
a unique solution in L2(Q). For the sake of simplicity, we denote

y(t, a, x;λ, τ) (2)

the unique solution of (1). Therefore, the map

(λ, τ) 7→ y(λ, τ) is in C1(R× R;L2(Q)). (3)

For more literature on the model describing the dynamics of population with age depen-
dence and spatial structure as well as for some existence results on such problem, we refer
for instance to [1, 3, 8, 17] and the reference therein. Recently S. Sawadogo [16] use the
sentinel method to control the migration of a single species population subjected to a
migratory phenomenon.

For the model (1), we are interested in identifying the parameters λi without any at-
tempt at computing τ ŷ0.

To identify these parameters, we use the theory of sentinel in a general framework.
More precisely, Let O be a nonempty open subset of Γ\Γ1 and let y = y(t, a, x;λ, τ) =
y(λ, τ) be the solution of (1). Then for any non-empty open subset γ of Γ\Γ1 such that
O ∩ γ 6= ∅, we look for a function S(λ, τ) solution to the following problem : given
h0 ∈ L2(U ×O), find w ∈ L2(U × γ) such that

i) the function S defined by

S(λ, τ) =

∫
U

∫
O
h0
∂y

∂ν
(λ, τ)dtdadΓ +

∫
U

∫
γ
w
∂y

∂ν
(λ, τ)dtdadΓ, (4)

satisfies :



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1280

- S is stationary to the first order with respect to missing term τ ŷ0

∂S

∂τ
(0, 0) = 0 ∀ ŷ0 (5)

- S is sensitive to the first order with respect to pollution terms λiξ̂i:

∂S

∂λi
(0, 0) = ci 1 ≤ i ≤M, (6)

where ci, 1 ≤ i ≤M, are given constants not all identically zero.

ii) The control w is of minimal norm in L2(U × γ) among ” the admissible controls”, i.e.

‖w‖2L2(U×γ) = min
w̄∈E
‖w̃‖2L2(U×γ), (7)

where
E =

{
w̃ ∈ L2(U × γ), such that (w̃, S(w̃)) satisfies (4)− (7)

}
. (8)

Remark 1. J.L.Lions refers to the function S as a sentinel with given sensitivity ci. In
(6), the ci are chosen according to the importance which is conferred to the component ξi
of the pollution.

Remark 2. Notice that for the J.L.Lions’s sentinels defined by (4)-(7), the observatory
O ⊂ (Γ \ Γ1) is also the support of the control function w.

For more information on the theory of sentinel, we refer to [9–11, 14, 15, 20] and the
reference therein. We set y0 = y(0, 0) ∈ L2(Q), the solution of (1) when λ = 0 and τ = 0
and we denote respectively by yτ and yλi , the derivatives of y at (0, 0) with respect to τ
and λi, i.e. :

yτ = lim
τ→0

y(0, τ)− y(0, 0)

τ

and

yλi = lim
λi→0

y(λi, 0)− y(0, 0)

λ
.

Then yτ and yλi are respectively solutions of

∂yτ
∂t + ∂yτ

∂a −4yτ + µyτ = 0 in Q,

yτ (0, a, x) = ŷ0 in QA,

yτ (t, 0, x) =

∫ A

0
β(t, a, x)yτ (t, a, x)da in QT ,

yτ = 0 on Σ,

(9)

and



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1281



∂yλi
∂t +

∂yλi
∂a −4yλi + µyλi = 0 in Q,

yλi(0, a, x) = 0 in QA,

yλi(t, 0, x) =

∫ A

0
βyλi(t, a, x)da in QT ,

yλi = ξ̂iχΣ1 on Σ,

(10)

where χX denote now and in the sequel, the characteristic function of the set X. Under the
assumptions (H1)−(H4), the systems (9) and (10) have respectively a unique solution yτ ∈
L2(Q) and yλi ∈ L2(Q) (see [8, 17]). From now on, we make the following assumptions:

- The functions
ξ̂i.χΣ1

, 1 ≤ i ≤M are linearly independent (11)

- Any function ρ such that
∂ρ
∂t + ∂ρ

∂a −∆ρ+ µρ = 0 in Q,

ρ = 0 in U × γ,
∂ρ
∂ν = 0 on U × γ,

(12)

is identically zero.

and we set

Y = Span{∂yλ1

∂ν
χγ , ...,

∂yλM
∂ν

χγ}. (13)

The vector subspace of L2(U × γ) generated by M functions {∂yλi∂ν χγ}
M
i=1.

Yθ =
1

θ
Y

The vector subspace of L2(U × γ) generated by M functions {1
θ

∂yλi
∂ν χγ}

M
i=1,

where θ is the positive function precisely defined later on by (31).

Remark 3. We will prove in Lemma 1 that the function {∂yλi∂ν χγ}
M
i=1 and {1

θ

∂yλi
∂ν χγ}

M
i=1

are linearly independent.

We now consider the following boundary null-controllability problem :

given h0 ∈ L2(U ×O), w0 ∈ Yθ, find v ∈ L2(U × γ) such that

v ∈ Y ⊥, (14)

and if q = q(t, a, x; v) is solution of



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1282


−∂q
∂t −

∂q
∂a −4q + µq = βq(t, 0, x) in Q,

q = h0χO + (w0 − v)χγ on Σ,

q(T, a, x) = 0 in QA,

q(t, A, x) = 0 in QT ,

(15)

q satisfy
q(0, a, x; v) = 0 in QA. (16)

Remark 4. Let us notice that if v exists, the set

E = {v ∈ Y ⊥such that (v, q = q(t, a, x; v)) satisfies (15)− (16)} (17)

is a non-empty closed, and convex set in L2(U×γ). Therefore there exists v ∈ E of minimal
norm.

The problem (14) − (16) is a null boundary controllability problem with constraint
on the control. When Y ⊥ = L2(U × γ), this problem becomes a null controllability
problem without constraint on the control. This kind of problem has been studied by
many authors with various methods [2, 5]. In this paper we solve the boundary null
controllability problem with constraint on the control (14)− (16), this allows us to prove
the existence of the sentinel with given sensitivity (4) − (7). More precisely, we have the
following results:

Theorem 1. Let Ω be a bounded open subset of RN with boundary Γ of class C∞. Let Γ1

be a non-empty open subset of Γ. Let also O and γ be two non empty subsets of Γ\Γ1, such
that O ∩ γ 6= 0. Assume that the assumptions of the data of the system (1) are satisfied.
Assume also that (11) and (12) holds. Then the existence of sentinel (4)− (7) holds if and
only if, the boundary null-controllability problem with constraints on the control (14)−(16)
has a solution.

To prove the boundary null-controllability problem with constraints on the control
(14)− (16), we use an inequality of Carleman adapted to the constraint that we establish
by means of a global Carleman inequality. More precisely we prove the following results.

Theorem 2. Assume that the hypotheses of Theorem 1 are satisfied. Then there exists a
positive real weight function θ (a precise definition of θ will be given later on (31)) such
that, for any function h0 ∈ L2(U ×O) with θh0 ∈ L2(U ×O) there exists a unique control
v̂ ∈ L2(U × γ) such that (v̂, q̂) with q̂ = q(v̂) is solution of null boundary controllability
problem with constraint on the control (14)− (16) and provides a control ŵ = w0χγ − v̂ of
the sentinel problem satisfying (7) . Moreover, the control ŵ is given by

ŵ = P (w0) + (I − P )(
∂ρ̂

∂ν
χγ), (18)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1283

where P is the orthogonal projection operator from L2(U × γ) into Y, w0 ∈ Yθ depends on
h0 and ci, i ∈ {1, . . . ,M}, and will be precisely determined in (27) and ρ̂ satisfies

∂ρ̂
∂t + ∂ρ̂

∂a −4ρ̂+ µρ̂ = 0 in Q,

ρ̂ = 0 on Σ

ρ̂(t, 0, x) =

∫ A

0
β(t, a, x)ρ̂(t, a, x)da in QT ,

ρ̂(0, ., .) = 0 in QA,

(19)

The rest of the paper is organized as follows: section 2 is devoted to the equivalence be-
tween the sentinel problem and the null boundary controllability problem with constraint
on the control. In this section we give the proof of Theorem 1. In section 3, we establish
Carleman inequalities necessary to solve the boundary null-controllability problem with
constraint on the control (14) − (16). In subsection 3.2 we give the proof of Theorem 2.
In section 4, we formulate the sentinel and we identify the parameters.

2. Equivalence between the sentinel problem and the null boundary
controllability problem with constraint on the control

In this subsection we prove Theorem 1. But before going further, we need the
following result:

Lemma 1. Assume that (11) and (12) holds. Then the functions
∂yλi
∂ν χγ , 1 ≤ i ≤ M

are linearly independent. Moreover the functions 1
θ

∂yλi
∂ν χγ , 1 ≤ i ≤ M , are also linearly

independent.

Proof. Let αi ∈ R, 1 ≤ i ≤ M be such that
M∑
i

αi
∂yλi
∂ν

χγ = 0 . Set k =
M∑
i

αiyλi ,

using (10) k is solution of

∂k
∂t + ∂k

∂a −∆k + µk = 0 in Q,

k(0, a, x) = 0 in QA,

k(t, 0, x) =

∫ A

0
βk(t, a, x)da in QT ,

k =

M∑
i=1

αiξ̂i.χΣ1 on Σ,

∂k
∂ν = 0 on U × γ.

(20)

Assumption (12) allows us to say that k = 0 inQ. Therefore, we deduce that

M∑
i=1

αiξ̂i.χΣ1 =

0 on Σ. Then it follows from (11) that αi = 0 for 1 ≤ i ≤M . The second assertion of the
lemma follows immediately.



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1284

Now, let us prove Theorem 1. To this end, we interpret (5) and (6). Actually, in view
of (4), the stationary condition (5) and respectively the sensitivity conditions (6) hold if
and only if∫

U

∫
O
h0
∂yτ
∂ν

dtdadΓ +

∫
U

∫
γ
w
∂yτ
∂ν

dtdadΓ = 0.∀ŷ0, ‖ŷ0‖L2(QA) ≤ 1 (21)

and ∫
U

∫
O
h0
∂yλi
∂ν

dtdadΓ +

∫
U

∫
γ
w
∂yλi
∂ν

dtdadΓ = ci, 1 ≤ i ≤M. (22)

Therefore, in order to transform equation (21), we consider the following adjoint equation
−∂q
∂t −

∂q
∂a −4q + µq = βq(t, 0, x) in Q,

q = h0χO + wχγ on Σ,

q(T, a, x) = 0 in QA,

q(t, A, x) = 0 in QT ,

(23)

Since h0χO + wχγ ∈ L2(Σ), the assumptions (H1) − (H2) ensure that that (23) has a
unique solution q ∈ L2(Q). Now multiplying both sides of the differential equation in (23)
by yτ solution of (9) and integrating by parts in Q, we get∫

U

∫
O
h0
∂yτ
∂ν

dtdadΓ+

∫
U

∫
γ
w
∂yτ
∂ν

dtdadΓ =

∫ A

0

∫
γ
q(0, a, x)ŷ0dadx ∀ ŷ0 ∈ L2(QA) (24)

Thus, the condition (5) or (21) holds if and only if

q(0, a, x; v) = 0 in QA. (25)

Then, multiplying both sides of the differential equation in (23) by yλi solution of (10)
and integrating by parts in Q, we have

∫
U

∫
O
h0
∂yλi
∂ν

dtdadΓ +

∫
U

∫
γ
w
∂yλi
∂ν

dtdadΓ =

∫
Σ1

∂q

∂ν
ξ̂i.χΓ1dtda, 1 ≤ i ≤M.

Thus, the condition the condition (6) or (22) is equivalent to∫
Σ1

∂q

∂ν
ξ̂i.χΓ1dtda = ci, 1 ≤ i ≤M. (26)

Now, consider the matrix(∫ T

0

∫ A

0

∫
γ

1

θ

∂yλj
∂ν

∂yλi
∂ν

dtdadΓ

)
1≤i,j≤M

.



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1285

Since this matrix is symmetric positive definite therefore, there exists a unique w0 ∈ Yθ
such that

ci −
∫
U

∫
O
h0
∂yλi
∂ν

dtdadΓ =

∫
U

∫
γ
w0
∂yλi
∂ν

dtdadΓ. 1 ≤ i ≤M (27)

Consequently, combining (27) with (22), we observe that condition (6) ( or the constraints
(26)) holds if and only if

w − w0 = −v ∈ Y ⊥,

where Y is given by (13). Replacing w by w0 − υ in the second expression of (23), we
obtain (15). We just have proved that the sentinel problem (4) − (7) hold if and only if
null controllability problem with constraint on the control (14)− (16) has a solution.

Remark 5. If E is the set of admissible control v ∈ L2(U × γ) such that (14) − (16) is
satisfied, then E is a closed convex subset of L2(U × γ). Since w0 − E is also a closed
convex subset of L2(U × γ), we can obtain w to be of minimum norm in L2(U × γ) by
minimizing the norm of w0 − v when v ∈ E. Then the pair (v, q(v)) satisfying (14)− (16)
necessarily provides a control w satisfying (7)

3. Study of the boundary null-controllability problem with constraint
on the control

In this section, we prove existence of the solution of the boundary null controllability
problem (14)−(16) and of course uniqueness if we want the control to be of minimal norm
among admissible controls. The main tool we use is an observability inequality adapted
to the constraint (14) which itself is a consequence of a global Carleman inequality.

3.1. An adapted Carleman inequality

The observability inequality we are looking for is a consequence of the global Car-
leman’s inequality. We consider an auxiliary function an auxillary function ψ ∈ C2(Ω)
which satisfies the following conditions :

ψ(x) > 0, ∀x ∈ Ω,

∇ψ > α,∀x ∈ Ω,
ψ(x) = 0, ∀x ∈ Γ \ γ,
∂ψ

∂ν
< 0, ∀x ∈ Γ \ γ.

(28)

Such a function exists according to A. Fursikov and O. Yu. Imanuvilov [7]. For any
positive parameter value λ we define the following weight functions :

ϕ (t, a, x) =
eλ(m|ψ|∞+ψ(x))

at (A− a) (T − t)
, (29)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1286

η(t, a, x) =
e2λm|ψ|∞ − eλ(m|ψ|∞+ψ(x))

at (A− a) (T − t)
, (30)

with m ≥ 1. Since ϕ does not vanish in Q, for all s > 0 and λ > 0, we set

1

θ2
= min

[
e−2sη

(
ϕ−1, ϕ, ϕ3, ϕ, |∂ψ

∂ν
|
)]

(31)

and we adopt the following notations :
L =

∂

∂t
+

∂

∂a
−∆ + µI

L∗ = − ∂

∂t
− ∂

∂a
−∆ + µI

V =
{
ρ ∈ C∞

(
Q
)
, ρ = 0 on Σ

} (32)

Using the notations given by (32) and the definition of θ given by (31), we have the
following boundary Carleman inequality:

Proposition 1. [Global Carleman inequality] Let ψ,ϕ and η be defined respectively
by (28) − (30). Then, there exists numbers λ0 = λ0 (γ, µ) > 1, s0 = s0 (γ, µ, T ) > 1, C0 =
C0 (γ, µ) > 0 and C1 = C1 (γ, µ) > 0 such that for any λ ≥ λ0, for any s ≥ s0, for any
ρ ∈ ν, the following estimate holds :∫

Q

e−2sη

sϕ

(
|ρt + ρa|2 + |∆ρ|2

)
dadtdx+

∫
Q
e−2sη

(
sλ2ϕ|∇ρ|2 + s3λ4ϕ3|ρ|2

)
dtdadx

+C0

∫ T

0

∫ A

0

∫
Γ\γ

se−2sηϕ(−∂ψ
∂ν

)|∂ρ
∂ν
|2dtdadΓ

≤ C1

[∫
Q
e−2η|Lρ|2dtdadx+

∫ T

0

∫ A

0

∫
γ
se−2ηϕ|∂ρ

∂ν
|2dtdadΓ

]
. (33)

Proof. See [18]

As ψ belong to C2(Ω) and ϕe−2sη is bounded, then 1
θ is also bounded in Q. Hence,

from Proposition 1, we have this other inequality :

Proposition 2. Let θ be defined by (31). Then, there exists numbers λ0 = λ0(Ω, γ, µ) > 1,
s0 = s0(Ω, γ, µ, T ) > 1, C0 = C0(Ω, γ, µ) > 0, and C1 = C1(Ω, γ, µ) > 0 such that, for
any λ ≥ λ0, for any s ≥ s0, and for any ρ ∈ V,

∫
U

∫
Ω

1

θ2

(
|∂ρ
∂ν
|2 + |∆ρ|2 + |∇ρ|2 + |ρ|2

)
dtdadΓ+C0

∫
U

∫
Γ

1

θ2
|∂ρ
∂ν
|2dtdadΓ

≤ C1

[∫
Q
|Lρ|2dtdadx+

∫
U

∫
γ
|∂ρ
∂ν
|2dtdadΓ

]
. (34)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1287

Lemma 2. Under the assumptions of Lemma 1. Let Y be the real vector subspace of
L2(U × γ) of finite dimension defined in (13). Then any function ρ such that

∂ρ
∂t + ∂ρ

∂a −∆ρ+ µρ = 0 in Q,

ρ(0, ., .) = 0 in QA,

ρ = 0 on Σ \ Σ1,
∂ρ
∂ν |γ ∈ Y,

(35)

is identically zero.

Proof. For any ρ verifying (35) there exists αi ∈ R, 1≤ i ≤ M , such that ∂ρ
∂ν =

M∑
i=1

αi
∂yi
∂ν

. We set z = ρ−
M∑
i=1

αiyi. Using (10), we have
∂z
∂t + ∂z

∂a −∆z + µz = 0 in Q,

z(0, ., .) = 0 in QA,

z = 0 on Σ \ Σ1,
∂z
∂ν = 0 on U × γ.

(36)

As γ ⊂ Γ \ Γ1, we have z = 0 and ∂z
∂ν = 0 in U × γ. Then it follows from (12) that

z = 0 in Q. Consequently, we deduce on the one hand that ρ =
M∑
i=1

αiyi and on the other

hand that
M∑
i=1

αiξ̂i = 0 on Σ1. Hence, it follows from assumption (11) that αi = 0 for

1 ≤ i ≤M .Thus, ρ = 0 in Q.

Proposition 3 (Adapted Carleman inequality). Under the Assumption of Lemma1
. Let Y be the real vector subspace of L2(U × γ) of finite dimension defined in (13) and
P be the orthogonal projection operator from L2(U × γ) into Y . Let also θ be the function
defined by (31). Then, there exists numbers λ0 = λ0(Ω, γ, µ) > 1, s0 = s0(Ω, γ, µ, T ) > 1,
C0 = C0(Ω, γ, µ) > 0 and C1 = C1(Ω, γ, µ) > 0 such that, for any λ ≥ λ0, for any s ≥ s0,
and for any ρ ∈ V,

∫
U

∫
Ω

1

θ2
|∂ρ
∂ν
|2dtdadΓ ≤ C1

[∫
Q
|Lρ|2dtdadx+

∫
U

∫
γ
|P ∂ρ
∂ν
− ∂ρ

∂ν
|2dtdadΓ

]
. (37)

Proof. As in [9], we use a well known compactness-uniqueness argument and the
inequality (34). Indeed, suppose that (37) does not hold. Then for any j ∈ N, there exists
ρj ∈ V such that ∫

U

∫
Ω
|Lρj |2dtdadx ≤

1

j
, (38)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1288

∫
U

∫
γ
|P ∂ρj

∂ν
− ∂ρj
∂ν

χγ |2dtdadΓ ≤ 1

j
, (39)

∫
U

∫
Γ

1

θ2
|∂ρj
∂ν
|2dtdadΓ = 1. (40)

In what follows, we prove in three steps that (38)− (40) yields contradiction.

Step 1. We have∫
U

∫
γ

1

θ2
|P ∂ρj

∂ν
|2dtdadΓ ≤ 2

∫
U

∫
γ

1

θ2
|P ∂ρj

∂ν
|2dtdadΓ

+ 2

∫
U

∫
γ

1

θ2
|P ∂ρj

∂ν
− ∂ρj
∂ν

χγ |2dtdadΓ.

(41)

Since 1
θ2 is bounded, using (38) and (39), it follows that there exists a positive constant

C such that

∀j ∈ N,
∫
U

∫
γ
|P ∂ρj

∂ν
|2dtdadΓ ≤ C. (42)

As ∂ρ
∂νχγ = P ∂ρ

∂νχγ + ( ∂ρ∂νχγ − P
∂ρ
∂νχγ), using (40) and (42), we obtain

‖∂ρj
∂ν
‖2L2(U×γ) ≤ C. (43)

Step 2. Let L2
(

1
θ , U × γ

)
= {ρ ∈ L2(U × Ω);

∫
U

∫
Γ

1

θ2

∣∣∣∣∂ρj∂ν
∣∣∣∣2 dtdadΓ <∞}.

Then in view of (40) and (43), we deduce from (34) that ,
(
∂ρ
∂t + ∂ρ

∂a

)
, (

∂ρj
∂ν ), (∇ρj), (ρj)

and (∆ρj) are bounded in L2
(

1
θ , U × γ

)
. Let us the take a subsequence still denoted by

(ρj) such that

ρj ⇀ ρ weakly in L2(
1

θ
, U × γ), (44)

∂ρj
∂ν

⇀
∂ρ

∂ν
weakly in L2(

1

θ
, U × γ). (45)

Then follows from (28) − (30) and the definition of 1
θ given by (31) that (ρj) and (∆ρj)

are bounded in L2(]β, T − β[×]α, A− α[×Ω) for any β > 0 and any α > 0. In particular,
for all β > 0 and any α > 0, we have

ρj ⇀ ρ weakly in L2(]β, T − β[×]α, A− α[×Ω)

∂ρj
∂ν

⇀
∂ρ

∂ν
weakly in L2(]β, T − β[×]α, A− α[×Σ).

which implies that



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1289

ρj ⇀ ρ weakly in D′(Q)

∂ρj
∂ν

⇀
∂ρ

∂ν
weakly in D′(Σ).

Therefore, we get from (38) and (43) that

Lρj −→ Lρ = 0 strongly in L2(U × Ω), (46)

∂ρj
∂ν

⇀
∂ρ

∂ν
strongly in L2(U × γ). (47)

And, since P is a compact operator, we deduce from (47) that

P
∂ρj
∂ν
−→ P

∂ρ

∂ν
strongly in L2(U × γ). (48)

In view of (39), we also have

∂ρj
∂ν
− P ∂ρj

∂ν
−→ 0 strongly in L2(U × γ). (49)

Thus combining (48) and (49), we get

P
∂ρj
∂ν
−→ ∂ρj

∂ν
strongly in L2(U × γ). (50)

Thanks to the uniqueness of the limit in L2(U × γ) , the convergence relations (48)-(49)
and (50) imply that P ∂ρ

∂ν = ∂ρ
∂νχγ . This means that ∂ρ

∂νχγ ∈ Y . We thus have proved
that ρ verifies (35). Hence thanks to Lemma 2, ρ is identically zero.

Therefore, (50) becomes

∂ρj
∂ν
−→ 0 strongly in L2(U × γ). (51)

Step 3. Since ρj ∈ ν, it follows from the observability inequality (34) that∫
U

∫
Ω

1

θ2
|∂ρj
∂ν
|2dtdadΓ ≤ C1

[∫
Q
|Lρj |2 dtdadx+

∫
U

∫
γ
|∂ρj
∂ν
|2dtdadΓ

]
.

Therefore passing this latter inequality to the limit while using (46) and (51), we obtain

lim
j−→∞

∫
U

∫
Ω
|∂ρj
∂ν
|2dtdadx = 0.

The contradiction occurs with (40).



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1290

3.2. Proof of Theorem 2

In this subsection, we are concerned with the proof of Theorem 2. That is, the optimal-
ity system for the control v̂ such that the pair (v̂; q̂) verifies (14)− (16). Since a classical
way to derive this optimality system is the method of penalization due to J.L.Lions [11],
here we use this method.

Step 1. Let w0 be defined by (27). If v ∈ Y ⊥ and q is solution of (15) then q(0, ., .) ∈
L2(QA) and we can define the functional

Jε(v) =
1

2
‖w0 − v‖2L2(U×γ) +

1

2ε
‖q(0, ., .)‖2L2(QA). (52)

We consider the optimal control problem: Find vε ∈ Y ⊥ such that

Jε(vε) = min
v∈Y ⊥

Jε(v) (53)

Since Y ⊥ is a closed and convex subset of L2(U × γ), it is classical to prove that there
exists a unique solution to (53). If we write qε the solution of (15) corresponding to vε
using an adjoint state ρε , we have that the triplet
(qε , ρε vε ) is solution of the first order optimality system:

L∗qε = βqε(t, 0, x) in Q,

qε(T, a, x) = 0 in QA,

qε(t, A, x) = 0 in QT ,

qε = h0χO + (w0 − vε)χγ on Σ,

(54)



Lρε = 0 in Q,

ρε(0, a, x) = 1
ε qε(0, a, x) in QA,

ρε(t, 0, x) =

∫ A

0
β(t, a, x)ρε(t, a, x)da in QT ,

ρε = 0 on Σ,

(55)

vε = (w0χγ −
∂ρε
∂ν

χγ)− P (w0χγ −
∂ρε
∂ν

χγ) ∈ Y ⊥. (56)

Step 2.

Multiplying the state equation (54) by ρε and integrating by parts over Q, we get

1

ε
‖qε(0, ., .)‖2L2(QA) =

∫
U

∫
O
h0
∂ρε
∂ν

dtdΓ +

∫
U

∫
γ
(w0 − vε)

∂ρε
∂ν

dtdΓ.



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1291

Which in view of (56) and the fact that vε ∈ Y ⊥ give

1

ε
‖qε(0, ., .)‖2L2(QA) =

∫
U

∫
O
h0
∂ρε
∂ν

dtdΓ

+

∫
U

∫
γ
(w0 − vε)(w0 − vε − P (w0χγ −

∂ρε
∂ν

χγ))dtdΓ.

=

∫
U

∫
O
h0
∂ρε
∂ν

dtdΓ

− ‖w0 − vε‖L2(U×γ) + ‖Pw0χγ‖L2(U×γ) +

∫
U

∫
γ
w0
∂ρε
∂ν

dtdΓ.

As on U × γ
w0 − vε = Pw0χγ + (I − P )

∂ρε
∂ν

χγ .

We have that

‖w0 − vε‖L2(U×γ) = ‖(I − P )
∂ρε
∂ν

χγ‖2L2(U×γ) + ‖Pw0χγ‖2L2(U×γ)

so that

1

ε
‖qε(0, ., .)‖2L2(QA) + ‖(I − P )

∂ρε
∂ν

χγ‖2L2(U×γ) =

∫
U

∫
O
h0
∂ρε
∂ν

dtdΓ

+

∫
U

∫
γ
w0
∂ρε
∂ν

dtdΓ.

This implies that

1

ε
‖qε(0, ., .)‖2L2(QA) + ‖(I − P )

∂ρε
∂ν

χγ‖2L2(U×γ) ≤ (

∫
U

∫
O

(θh0)2dtdΓ)
1
2 (

∫
U

∫
γ

1

θ2

∂ρε
∂ν

2

dtdΓ)
1
2

+ (

∫
U

∫
O

(θw0)2dtdΓ)
1
2 (

∫
U

∫
γ

1

θ2

∂ρε
∂ν

2

dtdΓ)
1
2 .

(57)
If we apply the adapted Carleman inequality (37) to ρε we obtain∫

U

∫
Γ

1

θ2
|∂ρε
∂ν
|2dtdΓ ≤ C

∫
U

∫
γ
|(I − P )

∂ρε
∂ν

χγ |2dtdaΓ, (58)

where C > 0 is independent of ε. From (57), the choice of w0 ∈ Yθ and the hypothesis on
h0, we deduce that

1
ε‖qε(0, ., .)‖

2
L2(QA) + 1

2‖(I − P )∂ρε∂ν χγ‖
2
L2(U×γ)

≤ C(

∫
U

∫
ω
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2

(59)

and then

‖vε‖2L2(U×ω) ≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2 (60)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1292

‖qεχω‖2L2(U×γ) ≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2 (61)

In view of (58) and (59), we get

‖1

θ

∂ρε
∂ν
‖L2(Σ) ≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2 (62)

and using (59) and the fact that 1
θ is bounded, we have

‖1

θ
P
∂ρε
∂ν
‖L2(U×γ) ≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2

Therefore, Y being a finite dimensional vector subspace of L2(U × γ),we deduce that

‖P ∂ρε
∂ν
‖L2(U×γ) ≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2 (63)

from which we deduce by using (59) that

‖∂ρε
∂ν
‖L2(U×γ) ≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2 (64)

Using Proposition 2 , we have that∫
U

∫
Ω

1

θ2
(|∂ρε
∂ν
|2 + |∆ρε |2 + |∇ρε |2 + |ρε |2)dtdadΓ

≤ C(

∫
U

∫
γ
θ2|w0|2dtdadΓ +

∫
U

∫
O
θ2|h0|2dtdadΓ)

1
2

(65)

Step 3.

We prove the convergence of (vε, qε)ε and ρε towards v̂, q̂ and ρ as ε −→ 0. According
to (60), (61) and (62) we can extract subsequences of (vε, qε)ε ( still called (vε, qε)ε
) such that

vε ⇀ ṽ weakly in L2(U × γ), (66)

qε ⇀ q̃ weakly in L2(U ;H1
0 (Ω)), (67)

1

θ
ρε ⇀ ρ̃ weakly in L2(

1

θ
,Q). (68)

As vε belong to Y ⊥ which is closed vector subspace of L2(U × γ), we have

ṽ ∈ Y ⊥. (69)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1293

The traces (q̃(0, ., .), q̃(., 0, .)), (q̃(T, ., .), q̃(., A, .)) and ∂q̃
∂ν exists and belong respec-

tively to (L2(QA))2 × (L2(QT ))2 and L2(Σ)(see [8]).

So, using (66) and (67) while passing (54) to the limit as ε −→ 0, we can prove
that

q̃ is solution of
L∗q̃ = βq̃(t, 0, x) in Q,

q̃(T, a, x) = 0 in QA,

q̃(t, A, x) = 0 in QT ,

q̃ = h0χO + (w0 − ṽ)χγ on Σ,

(70)

and it follows from (59) that

qε(0, ., .) ⇀ q̃(0, ., .) = 0 weakly in L2(Q). (71)

In view of (69), (70) and (71), (υ̃, q̃) verifies the null controllability (14)− (16) and
there exists a solution to the boundary null controllability problem. Moreover, it is
clear from (68) that ρ̃ satisfies

Lρ̃ = 0 in Q

ρ̃(t, 0, x) =

∫ A

0
β(t, a, x)ρ̃(t, a, x)da in QT ,

ρ̃ = 0 on Σ

From (64)
∂ρε
∂ν

⇀
∂ρ̃

∂ν
weakly in L2(U × γ) (72)

We know on the one hand that (ṽ, q̃) is solution to null controllability (14)− (16) ,
and on the other other hand that, there exists a unique v̂ ∈ ε such that (w0 − v̂) is
of minimal norm in L2(U ×γ). If we denote by q̂ the corresponding solution to (15),
we have q̂(0, ., .) = 0 and, as ṽ ∈ E ,

1

2
‖w0 − vε‖2L2(U×γ) ≤ Jε(vε) ≤ Jε(v̂) =

1

2
‖w0 − v̂‖2L2(U×γ)

and

1

2
‖w0 − v̂‖2L2(U×γ) ≤

1

2
‖w0 − vε‖2L2(U×γ)

Using (66)

lim inf
ε−→0

1

2
‖w0 − vε‖2L2(U×γ) ≥

1

2
‖w0 − v̂‖2L2(U×γ)



M.Soma , S. Sawadogo / Eur. J. Pure Appl. Math, 12 (3) (2019), 1277-1296 1294

Hence,
ṽ = v̂

and

vε ⇀ ṽ strongly in L2(U × γ).

Writing ρ̃ = ρ̂ , we obtain

v̂ = (I − P )

(
w0χγ −

∂ρ̂

∂ν
χγ

)
.

4. Formulation of the sentinel with given sensitivity and identification
of parameters λi

According to Theorem 2, if we replace in (4) w by

ŵ = P (w0) + (I − P )

(
∂ρ̂

∂ν
χγ

)
,

the function S defined by

S(λ, τ) =

∫
U

∫
O
h0
∂y

∂ν
(λ, τ)dtdadΓ +

∫
U

∫
γ
(P (w0) + (I − P )(

∂ρ̂

∂ν
χγ))

∂y

∂ν
(λ, τ)dtdadΓ,

is such that (ŵ, S(ŵ)) verified the sentinel problem (4)− (7). To estimate the parameters
λi , one proceeds as follows: assume that the solution of (1) when λ = 0 and τ = 0 is
known. Then, one has the following information

S(λ, τ)− S(0, 0) ≈
M∑
i=1

λi
∂S

∂λi
(0, 0).

Therefore, fixing i ∈ {1, . . . ,M} and choosing

∂S

∂λj
(0, 0) = 0 for j 6= i and

∂S

∂λi
(0, 0) = ci,

one obtains the following estimate of the parameter λi:

λi ≈
1

ci
(S(λ, τ)− S(0, 0)) ,

we deduce that

λi ≈
1

ci

{∫
U

∫
O
h0

(
m0 −

∂y0

∂ν
dtdadΓ

)}
+

1

ci

{∫
U

∫
γ

(
P (w0) + (I − P )

(
∂ρ̂

∂ν
χγ

))(
m0 −

∂y0

∂ν

)
dtdadΓ

}
,

where m0 is a measure of the flux of the population taken on the observatory O ∪ γ and
y0 is solution of (1) when λ = 0 and τ = 0.



REFERENCES 1295

References

[1] M.Langlais, Age-Dependent Population Diffusion with External Constraint.
J.Math.Biology 14(1982):77-94.

[2] B.Ainseba, Exact and approximate controllability of age and space population dy-
namics structured model.J.Math.Anal.App.(2002).

[3] B.Ainseba et M.Langlais, On a population dynamics control problem with age de-
pendence and spatial structure.Journal of Mathematical Analysis and Applications
248(2000),455–474.

[4] B. Ainseba et M.Langlais, Sur un probléme de contrle d’une population structurée en
ge et en espace.C.R.Acad.Sci.Paris,t.323(1996 ),serie I, P.269–274.

[5] B.Ainseba and S.Anita, Local exact controllability of the age-dependent population
dynamics with diffusion. Abstract Appl.Anal.6(2001) 357-368.

[6] H.Brezis, Analyse fonctionnelle.Théorie et application. Masson(1983)

[7] A.Fursikov,O.Imanualov, Controllability of evolution equation.Lecture Notes Series
34, RIM-GARC, Seoul National University,1996.

[8] M.Giovanna and M.Langlais, Age-dependent population diffusion with external con-
straint. J.Math.Biology(1982) 14:77-94.

[9] G.M.Mophou and O.Nakoulima, Sentinels with given sensitivity. European Journal
of Applied Mathematics, vol.19(2008), pp.21-40.

[10] G.M.Mophou and J.P.Puel, Boundary sentinels with given sensitiv-
ity.Rev.Mat.Complut. vol.22(2009), no.1,165-185.

[11] J.L.Lions, Sentinelles pour les systemes distribués données incompltes.Recherches en
Mathématiques Appliques, vol.21, Masson, Paris,1992.

[12] J.L.Lions and E.Magenes, Problémes aux limites non homogénes et applications.
Paris, Dunod,1968,Vol. 1et 2

[13] O. Nakoulima, Contrôlabilité zéro avec contraintes sur le contrle. C.R.Acad.Sci. Paris,
Ser.I 339/6 (2004) 405-410.

[14] O. Nakoulima, A revision of J.-L. Lions’ notion of sentinels. Portugal. Math. (N.S)
vol.65, Fasc. 1(2008), 1-22.

[15] O.Nakoulima and S.Sawadogo, Internal pollution and discriminating sentinel in pop-
ulation dynamics problem. International Journal of Evolution Equations, vol 2,
n01,pp.29-46, 2007.



REFERENCES 1296

[16] S.Sawadogo, Control of a migration problem of a population by the sentinel method,
Journal of Nonlinear Evolution Equation and Application, ID JNEEA-1810291 (Ac-
cepted for publication)

[17] A. Ouédraogo, O. Traoré, sur un probléme de dynamique des populations, Imhotep.
Vol 4 n01(2003).

[18] O. Traoré, Null controllability of a nonlinear population dynamics problem, e Interna-
tional Journal of Mathematics and Mathematical Sciences, Vol 26, Article ID 49279,
pp. 1-20.

[19] J. C. Saut and B. Scheure, Unique continuation for some evolution equation,
J.Differential Equation, 66 (1987) n0.1, 118-139.

[20] J.-P. Kernévez, the sentinel method and its application to environmental pollution
problem, CRC Mathematical modelling Series, CRC Press, Boca Raton, FL, (1997).

[21] Y. Simporé and O. Traoré, Null Controllability of a nonlinear dissipative system and
application to the detection of the Incomplete parameter for a Nonlinear Population
Dynamics Model, International Journal of Mathematics and Mathematical Sciences
Article ID 2820613, (2016).

[22] Lions, J.L. and Magenes, E. Nonhomogenous Boundary Value Problem and Applica-
tions, Grundlehren B, 181(1972), Springer Verlag.