Null controllability of a cascade model in population dynamics

0.26 MB·

by  Bedr'Eddine Ainseba

NULL CONTROLLABILITY OF A CASCADE MODEL IN
POPULATION DYNAMICS
BEDR’EDDINE AINSEBA, YOUNES ECHARROUDI AND LAHCEN MANIAR
Abstract. In this paper, we are concerned with the null controllability of a linear
7
1 population dynamics cascade systems (or the so-called prey-predator models) with two
0 different dispersion coefficients which degenerate in the boundary and with one control
2 force. We develop first a Carleman type inequality for its adjoint system, and then an

n observability inequality which allows us to deduce the existence of a control acting on a
a subset of the space domain which steers both populations of a certain age to extinction
J
in a finite time.

5
1

]
C
O
1. Introduction
.
h
t
a
We consider the coupled population cascade system
m
[ ∂y ∂y
+ (k (x)y ) +µ (t,a,x)y = ϑχ in Q, (1.1)
1 ∂t ∂a − 1 x x 1 ω
v ∂p ∂p
3 + (k (x)p ) +µ (t,a,x)p+µ (t,a,x)y = 0 in Q,
8 ∂t ∂a − 2 x x 2 3
0 y(t,a,1) = y(t,a,0) = p(t,a,1) = p(t,a,0) = 0 on (0,A) (0,T),
4 ×
0 y(0,a,x) = y0(a,x);p(0,a,x) = p0(a,x) in QA,
.
1 A
0 y(t,0,x) = β (t,a,x)y(t,a,x)da in Q ,
1 T
7
Z0
1 A
v: p(t,0,x) = β2(t,a,x)p(t,a,x)da in QT,
Xi Z0
where Q = (0,T) (0,A) (0,1), Q = (0,A) (0,1), Q = (0,T) (0,1) and we will
r A T
a denote q = (0,T)× (0,A)× ω. The system (1.1×) models the dispersi×on of a gene in two
× ×
given populations which are in interaction. In this case, x represents the gene type and
y(t,a,x) and p(t,a,x) as the distributions of individuals of age a at time t and of gene
type x of both populations. The parameters β (t,a,x) (respectively β (t,a,x)), µ (t,a,x)
1 2 1
2000 Mathematics Subject Classification. 35K65, 92D25, 93B05,93B07.
Key words and phrases. Degeneratepopulationdynamicsmodel,cascadesystems,Carlemanestimate,
observability inequality, null controllability.
Institut de Math´ematiques de Bordeaux,UMR-CNRS 5251,Universit´eBordeauxSegalen, 3 Placede
la Victoire, 33076 Bordeaux Cedex, France, e-mail: [email protected].
Private university of Marrakesh,Km 13 Route d’Amizmiz, Marrakesh,Morocco,
e-mail: [email protected],
D´epartementdeMath´ematiques,Facult´edesSciencesSemlalia,LaboratoireLMDP,UMMISCO(IRD-
UPMC), B. P. 2390 Marrakech40000,Maroc, e-mail: [email protected].
1
2 BEDR’EDDINEAINSEBA,YOUNESECHARROUDIANDLAHCEN MANIAR
(respectively µ (t,a,x)) are respectively the natural fertility and mortality rates of indi-
2
viduals of age a at time t and of gene type x of the population whose distribution is y
(respectively p), µ can be interpreted as the interaction coefficient between two popula-
3
tions (cancer cells and healthy cells for instance) which depends on x, t and a, the subset
ω is the region where a control ϑ is acting. Such a control corresponds to an external
A
supply or to removal of individuals on the subdomain ω. Finally, β (t,a,x)y(t,a,x)da
0 1
A
and β (t,a,x)p(t,a,x)da are the distributions of the newborns of the two populations
0 2 R
that are of gene type x at time t.
R
The control problems of (1.1) or in general of coupled systems take an intense inter-
est and are widely investigated in many papers, among them we find [3], [7], [17] and the
referencestherein. Infact, in[3]theauthorsstudiedacoupledreaction-diffusionequations
describing interaction between a prey population and predator population. The goal of
this work was to look for a suitable control supported on a small spatial subdomain which
guarantees the stabilization of the predator population to zero. In [17], the objective was
different. More precisely, the authors considered an age-dependent prey-predator system
and they proved the existence and uniqueness for an optimal control (called also ”optimal
effort”) which gives the maximal harvest via the study of the optimal harvesting problem
associated to their coupled model.
However, the previous results were found in the case when the diffusion coefficients are
constants. This leads Ait Ben Hassi et al. in [7] to generalize the model of [3] and in-
vestigate a semilinear parabolic cascade systems with two different diffusion coefficients
allowed to depend on the space variable and degenerate at the left boundary of the space
domain. Moreover, the purpose of this paper was to show the null controllability via a
Carleman type inequality of the adjoint problem of the associated linearized system using
the results of [8] (or [12]) and with the help of the Schauder fixed point theorem. On
the other hand, a massive interest was given to the question of null controllability of the
population dynamics models in the case of one equation both in the case without diffu-
sion (see for example [9]) and with diffusion (see for instance [1, 2, 4, 5, 15] in the case
of a constant diffusion coefficient). Recently, a more general case was investigated by B.
Ainseba and al. in [6] and [13]. Indeed, in [6] the authors allowed the dispersion coeffi-
cient to depend on the variable x and verifies k(0) = 0 (i.e, the coefficient of dispersion k
degenerates at 0) and they tried to obtain the null controllability in such a situation with
β L∞ basing on the work done in [8] for the degenerate heat equation to establish a
∈
new Carleman estimate for a suitable full adjoint system and afterwards his observability
inequality. However, the main controllability result of [6] was shown under the condition
T A (as in [9]) and this constitutes a restrictiveness on the ”optimality” of the control
≥
time T since it means, for example, that for a pest population whose the maximal age A
may equal to a many days (may be many months or years) we need much time to bring
the population to the zero equilibrium. In the same trend and to overcome the condition
T A, L. Maniar et al in [13] suggested the fixed point technique implemented in [15]
≥
and which requires that the fertility rate must belong to C2(Q) and consists briefly to
demonstrate in a first time the null controllability for an intermediate system with a fer-
tility function b L2(Q ) instead of Aβ(t,a,x)y(t,a,x)da and to achieve the task via a
∈ T 0
Leray-Schauder theorem.
R
But up now, little is known about the null controllability question of population dynamics
NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 3
cascade systems both in degenerate and nondegenerate cases to our knowledge and the
work done in this paper will address to such a control problem and it will be a general-
ization of the results established in [6] and [13]. More precisely, following the strategy of
[7] we expect in this contribution to prove the null controllability of system (1.1) when
T (0,δ) where δ (0,A) small enough in the case of one control force. That is, we show
∈ ∈
that for all y ,p L2(Q ) and δ (0,A) small enough, there exists a control ϑ L2(q)
0 0 A
∈ ∈ ∈
such that the associated solution of (1.1) verifies
y(T,a,x) = 0, a.e. in (δ,A) (0,1),
× (1.2)
p(T,a,x) = 0, a.e in (δ,A) (0,1).
(
×
SucharesultisgottenundertheconditionsthatallthenaturalratespossessanL∞ regularity
−
(see (2.4) beneath) and the dispersion coefficients are different and depend on the gene
type with a degeneracy in the left hand side of its domain, i.e k (0) = 0;i = 1,2 (e.g
i
k = xα, α > 0). In this case, we say that (1.1) is a degenerate population dynamics
i
cascade system. Genetically speaking, such a property is natural since it means that if
each population is not of a gene type, it can not be transmitted to its offspring.
The remainder of this paper is organized as follows: in Section 2, we give the well-
posedness result of system (1.1) and we bring out a Carleman inequality for an intermedi-
ate trivial adjoint system which helps us to prove the main Carleman estimate for the full
adjoint model. With the aid of this inequality, we establish in Section 3 the observability
inequality and show the main result of the null controllability of (1.1). The last section
takes the form of an appendix wherein we will give the proofs of some basic tools.
2. Well-posedness and Carleman estimates
2.1. Well-posedness result. For this section and for the sequel, we assume that the
dispersion coefficients k ,i = 1,2 satisfy the hypotheses
i
k C([0,1]) C1((0,1]), k > 0 in (0,1] and k (0) = 0,
i ∈ ∩ i i (2.3)
γ [0,1) : xk′(x) γk (x), x [0,1].
(cid:26) ∃ ∈ i ≤ i ∈
The last hypothesis on k means in the case of k(x) = xαi that 0 α < 1. Similarly,
i i
≤
all results of this paper can be obtained also in the case of 1 α < 2 taking, instead
i
≤
of Dirichlet condition on x = 0, the Newmann condition (k (x)u )(0) = 0. On the other
i x
hand, we assume that the rates µ , µ , µ , β and β verify
1 2 3 1 2
µ ,µ ,µ ,β ,β L∞(Q), µ ,µ ,µ ,β ,β 0 a.e in Q,
1 2 3 1 2 1 2 3 1 2
∈ ≥ (2.4)
β (.,0,.) 0 a.e. in (0,T) (0,1), for i = 1,2.
( i
≡ ×
The third assumption in(2.4) on thefertility rates β andβ is natural since the newborns
1 2
are not fertile.
As in [13], we discuss the well-posedness of (1.1) by introducing the weighted spaces
H1(0,1) and H2(0,1) defined by
ki ki
H1(0,1) := u L2(0,1) : u is abs. cont. in [0,1] : √k u L2(0,1),u(1) = u(0) = 0 ,
ki { ∈ i x ∈ }
H2(0,1) := u H1(0,1) : k (x)u H1(0,1) ,
( ki ∈ k i x ∈
n o
4 BEDR’EDDINEAINSEBA,YOUNESECHARROUDIANDLAHCEN MANIAR
endowed respectively with the norms
u 2 := u 2 + √k u 2 , u H1(0,1),
k kHk1i(0,1) k kL2(0,1) k i xkL2(0,1) ∈ ki
u 2 := u 2 + (k (x)u ) 2 , u H2(0,1),
( k kHk2i k kHk1i(0,1) k i x xkL2(0,1) ∈ ki
with i = 1,2 (see [7], [8], [12] or the references therein for the properties of such a spaces).
We recall from [11, 12] that the operators C u := (k (x)u ) , u D(C ) = H2(0,1),i =
i i x x ∈ i ki
1,2 are closed self-adjoint and negative with dense domains in L2(0,1).
On the other hand, in the Hilbert space H = (L2((0,A) (0,1)))2, the system (1.1) can
×
be rewritten abstractly as an inhomogeneous Cauchy problem in the following way
X′(t) = AX(t)+B(t)X(t)+f(t),
y (2.5)
X(0) = 0 ,
 p0 !
y(t)  0
where X(t) = ,A = A1 ; D(A) = D( ) D( ),
p(t) 0 A1 × A2
2
(cid:18) (cid:19) (cid:18) A (cid:19)
ϑ(t,., )χ (.) M 0
f(t) = · ω , B(t) = µ1(t) , where M w = µ (t)w, the
(cid:18) 0 (cid:19) (cid:18) Mµ3(t) Mµ2(t) (cid:19) µj(t) − j
operators : L2((0,A) (0,1)) L2((0,A) (0,1)) and : L2((0,A) (0,1))
1 2
A × → × A × →
L2((0,A) (0,1)) are defined respectively by:
×
θ(a,x) = ∂θ +(k (x)θ ) , θ D( ),
A1 −∂a 1 x x ∀ ∈ A1
D( ) = θ(a,x) : θ, θ L2((0,A) (0,1)),θ(a,0) = θ(a,1) = 0,θ(0,x) = Aβ (a,x)θ(a,x)da ,
( A1 { A1 ∈ × 0 1 }
(2.6)
R
and
θ(a,x) = ∂θ +(k (x)θ ) , θ D( ),
A2 −∂a 2 x x ∀ ∈ A2
D( ) = θ(a,x) : θ, θ L2((0,A) (0,1)),θ(a,0) = θ(a,1) = 0,θ(0,x) = Aβ (a,x)θ(a,x)da .
( A2 { A2 ∈ × 0 2 }
(2.7)
R
It iswell-known, from[16]andthe references therein that theoperators and defined
1 2
above generate a C semigroups. On the other hand, one can see thAat the oAperator A
0
−
is diagonal and B(t) is a bounded perturbation. Therefore, the following well-posedness
result holds (see for instance [7] for a similar result of cascade parabolic equations).
Theorem 2.1. i) The operator A generates a C semigroup.
0
−
ii) Under the assumptions (2.3) and (2.4) and for all ϑ L2(Q) and (y ,p ) (L2(Q ))2,
0 0 A
∈ ∈
the system (1.1)admits a uniquesolution (y,p). Thissolution belongsto E := C([0,T],(L2((0,A)
×
(0,1)))2) C([0,A],(L2((0,T) (0,1)))2) L2((0,T) (0,A),H1 (0,1) H1 (0,1)). More-
∩ × ∩ × k1 × k2
over, the solution of (1.1) satisfies the following
inequality
sup (y(t),p(t)) 2 + sup (y(a),p(a)) 2
k kL2(QA)×L2(QA) k kL2(QT)×L2(QT)
t∈[0,T] a∈[0,A]
1 A T
+ (( k y )2 +( k p )2)dtdadx
1 x 2 x
Z0 Z0 Z0
p p
C ϑ2dtdadx+ (y ,p ) 2 . (2.8)
≤ k 0 0 kL2(QA)×L2(QA)
(cid:18)Zq (cid:19)
NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 5
2.2. Carleman inequality results. In this paragraph, we show a Carleman type in-
equality for the following adjoint system of (1.1)
∂u ∂u
+ +(k (x)u ) µ (t,a,x)u µ (t,a,x)v = β (t,a,x)u(t,0,x) in Q, (2.9)
∂t ∂a 1 x x − 1 − 3 − 1
∂v ∂v
+ +(k (x)v ) µ (t,a,x)v = β (t,a,x)v(t,0,x) in Q,
∂t ∂a 2 x x − 2 − 2
u(t,a,1) = u(t,a,0) = v(t,a,1) = v(t,a,0) = 0 on (0,T) (0,A),
×
u(T,a,x) = u (a,x) in Q ,
T A
v(T,a,x) = v (a,x) in Q ,
T A
u(t,A,x) = v(t,A,x) = 0 in Q .
T
To do this, we prove firstly the Carleman estimate for the following intermediate system
∂u ∂u
+ +(k (x)u ) µ (t,a,x)u µ (t,a,x)v = h in Q, (2.10)
∂t ∂a 1 x x − 1 − 3 1
∂v ∂v
+ +(k (x)v ) µ (t,a,x)v = h in Q,
2 x x 2 2
∂t ∂a −
u(t,a,1) = u(t,a,0) = v(t,a,1) = v(t,a,0) = 0 on (0,T) (0,A),
×
u(T,a,x) = u (a,x) in Q ,
T A
v(T,a,x) = v (a,x) in Q ,
T A
u(t,A,x) = v(t,A,x) = 0 in Q ,
T
with (u ,v ) (L2(Q ))2 and h ,h L2(Q). Such a system can be rewritten in the
T T A 1 2
∈ ∈
following way
∂u ∂u
+ +(k (x)u ) µ (t,a,x)u = h +µ (t,a,x)v in Q, (2.11)
∂t ∂a 1 x x − 1 1 3
u(t,a,1) = u(t,a,0) = 0 on (0,T) (0,A),
×
u(T,a,x) = u (a,x) in Q ,
T A
u(t,A,x) = 0 in Q ,
T
where v is the solution of
∂v ∂v
+ +(k (x)u ) µ (t,a,x)v = h in Q, (2.12)
∂t ∂a 2 x x − 2 2
v(t,a,1) = v(t,a,0) = 0 on (0,T) (0,A),
×
v(T,a,x) = v (a,x) in Q ,
T A
v(t,A,x) = 0 in Q .
T
Classically, the proof of such a kind of estimates is based tightly on the choice of the
so-called weight functions. In our case, these functions are set in the following way
ϕ (t,a,x) := Θ(t,a)ψ (x),i = 1,2,
i i
1
 Θ(t,a) := ,
(t(T t))4a4
 (2.13)
 ψi(x) := λi 0x−kir(r)dr −di ,
φ(t,a,x) =(cid:16)Θ(a,t)eκσ(x),Φ((cid:17)t,a,x) = Θ(a,t)Ψ(x),Ψ(x) = eκσ(x) e2κkσk∞,
R
 −





6 BEDR’EDDINEAINSEBA,YOUNESECHARROUDIANDLAHCEN MANIAR
where σ is the function given by
σ C2([0,1]),σ(x) > 0 in (0,1),σ(0) = σ(1) = 0,
∈ (2.14)
σ (x) = 0 in [0,1] ω ,
x 0
(cid:26) 6 \
ω ⋐ ω is an open subset. The existence of this function is proved in [14, Lemma 1.1]. λ ,
0 i
d for i = 1,2 and κ are supposed to verify following assumptions
i
d > 1 , λ1 d2 ,
1 k1(1)(2−γ) λ2 ≥ d1−R01 k1r(r)dr (2.15)
( κ ≥ k4σlkn∞2,d2 ≥ k2(1)5(2−γ),
with λ I = [k2(1)(2−γ)(e2κkσk∞−1), 4(e2κkσk∞−eκkσk∞)) which can be shown not empty (see
2 ∈ d2k2(1)(2−γ)−1 3d2
Lemma 4.3 in the appendix). On other hand, in the light of the first and the fourth
conditions in (2.15) on d and d , one can observe that ψ (x) < 0 for all x [0,1], and
1 2 i
∈
Θ(t,a) + as t 0+,T− and a 0+.
→ ∞ → →
Now, westatethefirstresult ofthissectionwhichistheintermediateCarlemanestimate
satisfied by solution of system (2.10).
Theorem 2.2. Assume that k satisfy the hypotheses (2.3) and let A > 0 and T > 0 be
i
given. Then, there exist two positive constants C and s , such that every solution (u,v)
0
of (2.10) satisfies, for all s s , the following inequality
0
≥
x2 x2
s3Θ3 u2 +sΘk (x)u2 e2sϕ1dtdadx+ s3Θ3 v2 +sΘk (x)v2 e2sϕ2dtdadx
k (x) 1 x k (x) 2 x
ZQ(cid:18) 1 (cid:19) ZQ(cid:18) 2 (cid:19)
C (h2 +h2)e2sΦdtdadx+ s3Θ3(u2 +v2)e2sΦdtdadx . (2.16)
≤ 1 2
(cid:18)ZQ Zq (cid:19)
The proof of Theorem 2.2 needs two basic results. These results are concerned with
Carleman type inequalities in both cases degenerate and nondegenerate. The first one is
stated in the following proposition
Proposition 2.3. Consider the following system with h L2(Q), µ L∞(Q) and k
∈ ∈
verifies the hypotheses (2.3)
∂u ∂u
+ +(k(x)u ) µ(t,a,x)u = h, (2.17)
x x
∂t ∂a −
u(t,a,1) = u(t,a,0) = 0,
u(T,a,x) = u (a,x),
T
u(t,A,x) = 0.
Then, there exist two positive constants C and s , such that every solution of (2.17)
0
satisfies, for all s s , the following inequality
0
≥
x2
s3 Θ3 u2e2sϕdtdadx+s Θk(x)u2e2sϕdtdadx (2.18)
k(x) x
ZQ ZQ
A T
C h 2 e2sϕdtdadx+sk(1) Θu2(a,t,1)e2sϕ(a,t,1)dtda ,
≤ | | x
(cid:18)ZQ Z0 Z0 (cid:19)
NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 7
where ϕ and Θ are the weight functions defined by
ϕ(t,a,x) := Θ(t,a)ψ(x) with :
Θ(t,a) := 1 , (2.19)
 ψ(x) := c ((t(Tx−t)r)4da4r c ).
 1 0 k(r) − 2
with c2 > k(1)(12−γ), c1 > 0 and γ is the parRameter defined by (2.3).
For the proof of this proposition, we refer the reader to [13, Proposition 3.1]. The
second result is the following
Proposition 2.4. Let us consider the following system
∂z ∂z
+ +(k(x)z ) c(t,a,x)z = h in Q , (2.20)
x x b
∂t ∂a −
z(t,a,b ) = z(t,a,b ) = 0 on (0,T) (0,A),
1 2
×
where Q := (0,T) (0,A) (b ,b ), (b ,b ) [0,1], h L2(Q ), k C1([0,1]) is a
b 1 2 1 2 b
× × ⊂ ∈ ∈
strictly positive function and c L∞(Q ). Then, there exist two positive constants C and
b
∈
s , such that for any s s , z verifies the following estimate
0 0
≥
(s3φ3z2 +sφz2)e2sΦdtdadx
x
ZQb
A T
C h2e2sΦdtdadx+ s3φ3z2e2sΦdtdadx , (2.21)
≤
(cid:18)ZQb ZωZ0 Z0 (cid:19)
where φ, Θ and Φ are defined by (2.13) and σ by (2.14).
For the proof of Proposition 2.4, a careful computations allow us to adapt the same
procedure of [2, Lemma 2.1] to show (2.21) in case where k is a positive general nonde-
generate coefficient, with our weight function Θ(t,a) = 1 and the source term h.
t4(T−t)4a4
Besides the two Propositions 2.3 and 2.4, we must bring out another important result
Lemma 2.5. Under assumptions (2.15), the functions ϕ , ϕ and Φ defined by (2.13)
1 2
satisfy the following inequalities
ϕ ϕ ,
1 ≤ 2 (2.22)
4Φ < ϕ Φ.
(cid:26) 3 2 ≤
Proof. By the definitions of ϕ , ϕ and Φ and taking into account that Θ is positive,
1 2
showing the results of (2.22) is equivalent to show
ψ ψ ,
1 ≤ 2 (2.23)
4Ψ < ψ Ψ.
(cid:26) 3 2 ≤
Thefirstinequalityin(2.23)isassuredbythesecondassumptionin(2.15)whilethesecond
one is deduced from λ I = [k2(1)(2−γ)(e2κkσk∞−1), 4(e2κkσk∞−eκkσk∞)) and this achieves the
2 ∈ d2k2(1)(2−γ)−1 3d2
proof. (cid:3)
Now, we can address the proof of Theorem 2.2.
8 BEDR’EDDINEAINSEBA,YOUNESECHARROUDIANDLAHCEN MANIAR
Proof. Let us introduce the smooth cut-off function ξ : R R defined as follows
→
0 ξ(x) 1, x R,
≤ ≤ ∈
ξ(x) = 1, x [0, 2x1+x2], (2.24)

∈ 3
ξ(x) = 0, x [x1+2x2,1].
∈ 3
Let u and v be respectively the solutions of (3.75) and (3.76). Set w := ξu, z := ξv and

put ω′ = (2x1+x2, x1+2x2). Then, (w,z) satisfies the following system
3 3
∂w ∂w
+ +(k (x)w ) µ (t,a,x)w = µ (t,a,x)z +ξh +(k ξ u) +ξ k u in Q, (2.25)
∂t ∂a 1 x x − 1 3 1 1 x x x 1 x
∂z ∂z
+ +(k (x)z ) µ (t,a,x)z = ξh +(k ξ v) +ξ k v in Q,
2 x x 2 2 2 x x x 2 x
∂t ∂a −
w(t,a,1) = w(t,a,0) = z(t,a,1) = z(t,a,0) = 0 on (0,T) (0,A),
×
w(T,a,x) = w (a,x) in Q ,
T A
z(T,a,x) = z (a,x) in Q ,
T A
w(t,A,x) = z(t,A,x) = 0 in Q .
T
Using Proposition2.3 for the inhomogeneous term ξ(h + µ v) + (k ξ u) + ξ k u , the
1 3 1 x x x 1 x
definition of ξ and Young inequality, we get the following inequality
x2
(sΘk w2 +s3Θ3 w2)e2sϕ1dtdadx
1 x k
ZQ 1
C( [ξ2(h +µ v)2 +((k ξ u) +ξ k u )2]e2sϕ1dtdadx
1 3 1 x x x 1 x
≤
ZQ
A T
+sk (1) Θw2(t,a,1)e2sϕ1(t,a,1)dtda)
1 x
Z0 Z0
C [µ2z2 +ξ2h2 +((k ξ u) +ξ k u )2]e2sϕ1dtdadx. (2.26)
≤ 3 1 1 x x x 1 x
ZQ
Thanks again to the definition of ξ, we have
1
((k ξ u) +ξ k u )2e2sϕ1dx
1 x x x 1 x
Z0
(8(k ξ )2u2 +2((k ξ ) )2u2)e2sϕ1dx
≤ Zω′ 1 x x 1 x x
C (u2 +u2)e2sϕ1dx. (2.27)
≤ Zω′ x
On the other hand, since x2 is non-decreasing, with the help of Hardy-Poincar´e inequal-
k2(x)
ity stated in [8] and since ϕ ϕ we get
1 2
≤
1 µ 2 1 k (x) µ 2 1
µ2z2e2sϕ1dx k 3k∞ 2 (zesϕ2)2dx Ck 3k∞ k (x)((zesϕ2) )2dx.
3 ≤ k (1) x2 ≤ k (1) 2 x
Z0 2 Z0 2 Z0
Thus, from the definition of ψ , we obtain
2
1 1 1 x2
µ2z2e2sϕ1dx C k (x)z2e2sϕ2dx+C s2Θ2 z2e2sϕ2dx.
3 ≤ 2 x k (x)
Z0 Z0 Z0 2
NULL CONTROLLABILITY OF A CASCADE MODEL IN POPULATION DYNAMICS 9
Hence, for s quite large we get
1 1 1 1 1 x2
µ2z2e2sϕ1dx sΘk (x)z2e2sϕ2dx+ s3Θ3 z2e2sϕ2dx. (2.28)
3 ≤ 2 2 x 2 k (x)
Z0 Z0 Z0 2
Combining (2.26), (2.27) and (2.28), for s quite large the following inequality holds
x2
(sΘk w2 +s3Θ3 w2)e2sϕ1dtdadx (2.29)
1 x k
ZQ 1
1 x2
C h2e2sϕ1dtdxda+ (sΘk (x)z2 +s3Θ3 z2)e2sϕ2dtdadx
≤ 1 2 2 x k (x)
ZQ ZQ 2
A T
+C (u2 +u2)e2sϕ1dtdadx.
1 x
Zω′ Z0 Z0
Applying the same way with ξh +(k ξ v) +ξ k v we obtain
2 2 x x x 2 x
x2
(sΘk z2 +s3Θ3 z2)e2sϕ2dtdadx
2 x k
ZQ 2
A T
C h2e2sϕ2dtdxda+C (v2 +v2)e2sϕ2dtdadx. (2.30)
≤ 2ZQ 2 3Zω′ Z0 Z0 x
Therefore, forsquitelargeweconcludebyinequalities(2.29)and(2.30)andagainϕ ϕ
1 2
≤
that
x2 x2
(sΘk w2 +s3Θ3 w2)e2sϕ1dtdadx+ (sΘk z2 +s3Θ3 z2)e2sϕ2dtdadx
1 x k 2 x k
ZQ 1 ZQ 2
A T
C (h2 +h2)e2sϕ2dtdadx+C (u2 +v2 +u2 +v2)e2sϕ2dtdadx.
≤ 4ZQ 1 2 5Zω′ Z0 Z0 x x
Using Caccioppoli’s inequality (4.87), the last inquality becomes
x2 x2
(sΘk w2 +s3Θ3 w2)e2sϕ1dtdadx+ (sΘk z2 +s3Θ3 z2)e2sϕ2dtdadx
1 x k 2 x k
ZQ 1 ZQ 2
C (h2 +h2)e2sϕ2dtdadx+C s2Θ2(u2 +v2)e2sϕ2dtdadx. (2.31)
≤ 6 1 2 7
ZQ Zq
Now, let W := ηu and Z := ηv with η = 1 ξ. Then W and Z are supported in (x ,1)
1
−
and verify the following system
∂W ∂W
+ +(k (x)W ) µ (t,a,x)W = µ (t,a,x)Z +ηh +(k η u) +η k u in Q ,
∂t ∂a 1 x x − 1 3 1 1 x x x 1 x x1
(2.32)
∂Z ∂Z
+ +(k (x)Z ) µ (t,a,x)Z = ηh +(k η v) +η k v in Q ,
∂t ∂a 2 x x − 2 2 2 x x x 2 x x1
W(t,a,1) = W(t,a,x ) = Z(t,a,1) = Z(t,a,x ) = 0 on (0,T) (0,A),
1 1
×
W(t,a,x) = W (a,x) in Q ,
T A
Z(t,a,x) = Z (a,x) in Q ,
T A
W(t,A,x) = Z(t,A,x) = 0 in Q ,
T
where, Q := (0,T) (0,A) (x ,1). Then, the system satisfied by W and Z is non-
x1 × × 1
degenerate. Hence, applying Proposition 2.4 on the first equation of (2.32) for b = x ,
1 1
10 BEDR’EDDINEAINSEBA,YOUNESECHARROUDIANDLAHCEN MANIAR
b = 1 and h := η(h +µ v)+(k η u) +η k u , with the aid of Caccioppoli’s inequality
2 1 3 1 x x x 1 x
stated in [13, Lemma 5.1], thanks to the definition of η and Young inequality and taking
s quite large we obtain the following estimate
(s3φ3W2 +sφW2)e2sΦdtdadx
x
ZQ
A T
C (η(h +µ v)+(kη u) +kη u )2e2sΦdtdadx+ s3Θ3u2e2sΦdtdadx
1 3 x x x x
≤
(cid:18)ZQ ZωZ0 Z0 (cid:19)
A T
C η2(h +µ v)2e2sΦ +((kη u) +kη u )2e2sΦdtdadx+ s3Θ3u2e2sΦdtdadx
1 3 x x x x
≤
(cid:18)ZQ ZωZ0 Z0 (cid:19)
A T
e
C( η2(h +µ v)2e2sΦdtdadx+ (8(kη )2u2 +2((kη ) )2u2)e2sΦdtdadx
≤ ZQ 1 3 Zω′ Z0 Z0 x x x x
A T
e
+ s3Θ3u2e2sΦdtdadx)
ZωZ0 Z0
A T A T
C η2(h +µ v)2e2sΦdtdadx+ (u2 +u2)e2sΦdtdadx+ s3Θ3u2e2sΦdtdadx
≤ 1(cid:18)ZQ 1 3 Zω′ Z0 Z0 x ZωZ0 Z0 (cid:19)
A T
e
C η2(h +µ v)2e2sΦdtdadx+ s3Θ3u2e2sΦdtdadx
2 1 3
≤
(cid:18)ZQ ZωZ0 Z0 (cid:19)
Ce (h2 +µ2Z2)e2sΦdtdadx+ s3Θ3u2e2sΦdtdadx , (2.33)
≤ 3 1 3
(cid:18)ZQ Zq (cid:19)
e
with Φ and φ are defined in (2.13) and ω′ is defined in the beginning of the proof. On the
other hand, using the fact that x x2 is non-decreasing, Hardy-Poincar´e inequality
7→ k2(x)
for the function ZesΦ and the definition of ψ we have for s quite large the following
2
inequality
x2
µ2Z2e2sΦdx c k (x)Z2e2sΦdtdadx+ s2Θ2 Z2e2sΦdtdadx
3 ≤ 2 x k (x)
ZQ (cid:18)ZQ ZQ 2 (cid:19)
1
(s3φ3Z2 +sφZ2)e2sΦdtdadx. (2.34)
≤ 2 x
ZQ
Therefore, injecting (2.34) in (2.33) we get
(s3φ3W2 +sφW2)e2sΦdtdadx (2.35)
x
ZQ
1
C h2e2sΦdtdadx+ s3Θ3u2e2sΦdtdadx + (s3φ3Z2 +sφZ2)e2sΦdtdadx.
≤ 1 2 x
(cid:18)ZQ Zq (cid:19) ZQ
Replying the same argument for the source term h := ηh + (k η v) + η k v we infer
2 2 x x x 2 x
that
(s3φ3Z2 +sφZ2)e2sΦdtdadx C h2e2sΦdtdadx+ s3Θ3v2e2sΦdtdadx .(2.36)
x ≤ 8 2
ZQ (cid:18)ZQ Zq (cid:19)