Volume 2009, Article ID 560407,9pages doi:10.1155/2009/560407
Research Article
Interior Controllability of a 2
×
2 Reaction-Diffusion
System with Cross-Diffusion Matrix
Hanzel Larez and Hugo Leiva
Departamento de Matem´aticas, Universidad de Los Andes, M´erida 5101, Venezuela
Correspondence should be addressed to Hugo Leiva,[email protected]
Received 30 December 2008; Accepted 27 May 2009
Recommended by Gary Lieberman
We prove the interior approximate controllability for the following 2×2 reaction-diffusion system with cross-diffusion matrixutaΔu−β−Δ1/2ubΔv1ωf1t, xin0, τ×Ω,vtcΔu−dΔv− β−Δ1/2v1
ωf2t, xin0, τ×Ω,u v 0, on0, T×∂Ω,u0, x u0x,v0, x v0x,
x∈Ω, whereΩis a bounded domain inRNN ≥ 1,u
0, v0∈L2Ω, the 2×2 diffusion matrix
Da b c d
has semisimple and positive eigenvalues 0< ρ1≤ρ2,βis an arbitrary constant,ωis an
open nonempty subset ofΩ, 1ωdenotes the characteristic function of the setω, and the distributed controlsf1, f2∈L2 0, τ;L2Ω. Specifically, we prove the following statement: ifλ11/2ρ1β >0
whereλ1is the first eigenvalue of−Δ, then for allτ >0 and all open nonempty subsetωofΩthe
system is approximately controllable on 0, τ.
Copyrightq2009 H. Larez and H. Leiva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper we prove the interior approximate controllability for the following 2×2
reaction-diffusion system with cross-diffusion matrix
utaΔu−β−Δ1/2ubΔv1ωf1t, x in0, τ×Ω,
vtcΔu−dΔv−β−Δ1/2v1ωf2t, x in0, τ×Ω,
uv0, on0, τ×∂Ω,
u0, x u0x, v0, x v0x, x∈Ω,
1.1
whereΩis a bounded domain inRNN≥1,u0, v0∈L2Ω, the 2×2 diffusion matrix
D
a b c d
has semisimple and positive eigenvalues,βis an arbitrary constant,ωis an open nonempty
subset ofΩ, 1ωdenotes the characteristic function of the setω, and the distributed controls
f1, f2∈L2 0, τ;L2Ω. Specifically, we prove the following statement: ifλ11/2ρ1β >0the
first eigenvalue of−Δ, then for allτ >0 and all open nonempty subsetωofΩ, the system is
approximately controllable on 0, τ.
WhenΩ 0,1this system takes the following particular form:
uta ∂2u
∂x2 β
∂u ∂xb
∂2v
∂x2 1ωf1t, x in0, τ×0,1,
vtc ∂2u
∂x2 d
∂2v
∂x2 β
∂v
∂x 1ωf2t, x in0, τ×0,1, ut,0 vt,0 ut,1 vt,1 0, t∈0, τ,
u0, x u0x, v0, x v0x, x∈0,1.
1.3
This paper has been motivated by the work done Badraoui in 1, where author studies the
asymptotic behavior of the solutions for the system1.3on the unbounded domainΩ R.
That is to say, he studies the system:
uta ∂2u
∂x2 β
∂u ∂xb
∂2v
∂x2 ft, u, v, x∈R, t >0,
vtc∂
2u
∂x2 d
∂2v ∂x2 β
∂v
∂xgt, u, v, x∈R, t >0,
1.4
supplemented with the initial conditions:
ux,0 u0x, vx,0 v0x, x∈R, 1.5
where the diffusion coefficientsaanddare assumed positive constants, while the diffusion
coefficientsb,c and the coefficientβ are arbitrary constants. He assume also the following
three conditions:
H1 a−d24bc >0,cd /0 andad > bc,
H2u0, v0∈XCUBR,whereCUBis the space of bounded and uniformly continuous
real-valued functions,
H3ft, u, v and gt, u, v ∈ X, for all t > 0 and u, v ∈ X. Moreover, f and g are
locally Lipshitz; namely, for allt1 ≥0 and all constantk >0, there exist a constant
LLk, t1>0 such that
ft, w1−ft, w2≤L|w1−w2|, 1.6
is verified for allw1 u1, v1,w2 u2, v2 ∈R×Rwith|w1| ≤k,|w2| ≤ kand
We note that the hypothesisH1implies that the eigenvalues of the matrix D are simple
and positive. But, this condition is not necessary for the eigenvalues ofD to be positive, in
fact we can find matricesDwithaanddbeen negative and having positive eigenvalues. For
example, one can consider the following matrix:
D
5 −6
2 −2
, 1.7
whose eigenvalues areρ11 andρ2 2.
The system1.1can be written in the following matrix form:
ztDΔz−βI2×2−Δ1/2z1ωft, x, in0, τ×Ω,
z0, on0, τ×∂Ω,
z0, x z0x, x∈Ω,
1.8
wherez u, vT ∈R2, the distributed controlsf f
1, f2T ∈L2 0, τ;L2Ω;R2, andI2×2
is the identity matrix of dimension 2×2.
Our technique is simple and elegant from mathematical point of view, it rests on the shoulders of the following fundamental results.
Theorem 1.1. The eigenfunctions of −Δ with Dirichlet boundary condition are real analytic
functions.
Theorem 1.2see 2, Theorem 1.23, page 20. SupposeΩ⊂Rnis open, nonempty, and connected
set, andfis real analytic function inΩwithf 0on a nonempty open subsetωofΩ. Then,f 0
inΩ.
Lemma 1.3 see 3, Lemma 3.14, page 62. Let {αj}j≥1 and {βi,j :i1,2, . . . , m}j≥1 be two
sequences of real numbers such thatα1> α2> α3· · ·. Then
∞
j1
eαjtβ
i,j 0, ∀t∈ 0, t1, i1,2, . . . , m 1.9
if and only if
βi,j 0, i1,2, . . . , m; j1,2, . . . ,∞. 1.10
Finally, with this technique those young mathematicians who live in remote and inhospitable places, far from major research centers in the world, can also understand and
2. Abstract Formulation of the Problem
In this section we choose a Hilbert space where system 1.8can be written as an abstract
differential equation; to this end, we consider the following notations:
Let us consider the Hilbert spaceH L2Ω,R and 0 λ
1 < λ2 < · · · < λj → ∞
the eigenvalues of −Δ, each one with finite multiplicity γj equal to the dimension of the
corresponding eigenspace. Then, we have the following well-known propertiessee 3, pages
45-46.
iThere exists a complete orthonormal set{φj,k}of eigenvectors of−Δ.
iiFor allξ∈D−Δ, we have
−Δξ
∞
j1
λj γj
k1
ξ, φj,k φj,k
∞
j1
λjEjξ, 2.1
where ·,·is the inner product inHand
Enξ γj
k1
ξ, φj,k φj,k. 2.2
So,{Ej}is a family of complete orthogonal projections inHandξ
∞
j1Ejξ,ξ∈H.
iii Δgenerates an analytic semigroup{TΔt}given by
TΔtξ
∞
j1
e−λjtE
jξ. 2.3
Now, we denote byZthe Hilbert spaceH2 L2Ω;R2and define the following
operator:
A:DA⊂Z−→Z, Aψ−DΔψβI2×2−Δ1/2ψ 2.4
withDA H2Ω,R2∩H1
0Ω,R2. Therefore, for allz∈DA, we obtain
Az
∞
j1
λj1/2λj1/2DβI2×2
Pjz, 2.5
z
∞
j1
Pjz, z2
∞
j1
Pjz2, z∈Z, 2.6
where
Pj
Ej 0
0 Ej
2.7
Consequently, system1.8can be written as an abstract differential equation inZ:
z−AzBωf, z∈Z, t≥0, 2.8
wheref∈L2 0, τ;U,UZ, andBω:U → Z,Bωf1ωfis a bounded linear operator.
Now, we will use the following Lemma from 4to prove the following theorem.
Lemma 2.1. LetZbe a Hilbert separable space and{Aj}j≥1,{Pj}j≥1two families of bounded linear
operator inZ, with{Pj}j≥1a family of complete orthogonal projection such that
AjPjPjAj, j≥1. 2.9
Define the following family of linear operators:
Ttz
∞
j1
eAjtP
jz, z∈Z, t≥0. 2.10
Then the following hold.
aTt is a linear and bounded operator if eAjt ≤ gt, j 1,2, . . ., with gt ≥ 0,
continuous fort≥0.
bUnder the above (a),{Tt}t≥0is a strongly continuous semigroup in the Hilbert spaceZ, whose infinitesimal generatorAis given by
Az
∞
j1
AjPjz, z∈DA 2.11
with
DA ⎧ ⎨ ⎩z∈Z:
∞
j1
AjPjz2
<∞ ⎫ ⎬
⎭. 2.12
cThe spectrumσAofAis given by
σA
∞
j1
σAj
, 2.13
whereAj AjPj:RPj → RPj.
Theorem 2.2. The operator−Adefine by2.5is the infinitesimal generator of a strongly continuous semigroup{Tt}t≥0given by:
Ttz
∞
j1
eAjtP
wherePj diag Ej, EjandAj RjPjwith
Rj
−aλj −bλj −cλj −dλj
−β ⎡ ⎣λ
1/2
j 0
0 λ1j/2 ⎤
⎦. 2.15
Moreover, ifλ11/2ρ1β >0, then there existsM >0such that
Tt ≤Mexp−λ11/2λ11/2ρ1β
t, t≥0. 2.16
Proof. In order to apply the foregoing Lemma, we observe that−Acan be written as follows:
−Az∞
j1
AjPjz, z∈DA 2.17
with
Aj−λ1j/2
λ1j/2DβI2×2
Pj, Pjdiag
Ej, Ej
. 2.18
Therefore,AjRjPjwith
Rj
−aλj −bλj −cλj −dλj
−β ⎡ ⎣λ
1/2
j 0
0 λ1j/2 ⎤
⎦, AjPjPjAj. 2.19
Clearly thatAj is a bounded linear operatorlinear and continuous. That is, there exists
Mj>0 such that
Ajz≤Mjz, ∀z∈Z. 2.20
In fact,AjzRjPjz ≤ RjPjz ≤ Rjz.
Now, we have to verify condition a of Lemma 2.1. To this end, without loss
of generality, we will suppose that 0 < ρ1 < ρ2. Then, there exists a set {Q1, Q2} of
complementary projections onR2such that
eDteρ1tQ
1eρ2tQ2. 2.21
Hence,
eRjteΓ1jtQ
1eΓ2jtQ2, with Γjs−λ1j/2
λ1j/2ρsβ
This implies the existence of positive numbersα, Msuch that
eAjt≤Meαt, j1,2, . . . . 2.23
Therefore,−Agenerates a strongly continuous semigroup{Tt}t≥0given by2.14.
Finally, ifλ11/2ρ1β >0, then
−λ1/2 1
λ11/2ρ1β
≥ −λ1/2
j
λ1j/2ρiβ
, j1,2,3, . . .; i1,2, 2.24
and using2.14we obtain2.16.
3. Proof of the Main Theorem
In this section we will prove the main result of this paper on the controllability of the linear
system 2.8. But, before we will give the definition of approximate controllability for this
system. To this end, for allz0∈Zandf∈L20, τ;U, the initial value problem
z−AzBωft, z∈Z,
z0 z0,
3.1
where the control functionfbelonging toL20, τ;Uadmits only one mild solution given by
zt Ttz0
t
0
Tt−sBωfsds, t∈ 0, τ. 3.2
Definition 3.1 approximate controllability. The system 2.8 is said to be approximately
controllable on 0, τif for every z0, z1 ∈ Z, ε > 0, there exists u ∈ L20, τ;Usuch that
the solutionztof3.2corresponding touverifies:
zτ−z1< ε. 3.3
The following result can be found in 5for the general evolution equation:
zAzBft, z∈Z, u∈U, 3.4
where Z, U are Hilbert spaces, A : DA ⊂ Z → Z is the infinitesimal generator of
strongly continuous semigroup {Tt}t≥0 inZ, B ∈ LU, Z, the control function f belongs
toL20, τ;U.
Theorem 3.2. System3.4is approximately controllable on 0, τif and only if
B∗T∗tz0, ∀t∈ 0, τ ⇒z0. 3.5
Theorem 3.3main theorem. Ifλ11/2ρ1β >0, then for allτ >0and all open nonempty subsetω ofΩthe system,2.8is approximately controllable on 0, τ.
Proof. We will applyTheorem 3.2to prove the approximate controllability of system2.8. With this purpose, we observe that
Bω∗ Bω, T∗tz
∞
j1
eR∗jtP∗
jz, z∈Z, t≥0. 3.6
On the other hand,
Rj−λ1j/2
λ1j/2 a b c d β 1 0 0 1
−λ1/2
j
λ1j/2DβI2×2
. 3.7
Without lose of generality, we will suppose that 0< ρ1< ρ2. Then, there exists a set{Q1, Q2}
of complementary projections onR2such that
eDteρ1tQ
1eρ2tQ2. 3.8
Hence,
eRjteΓj1tQ
1eΓj2tQ2, withΓjs−λ1j/2
λ1j/2ρsβ
, s1,2. 3.9
Therefore,
Bω∗T∗tz
∞
j1
B∗ωeRj∗tP∗ jz
∞
j1 2
s1
eΓjstB∗
ωPs,j∗ z, 3.10
wherePs,jQsPjPjQs.
Now, suppose forz∈ZthatB∗ωT∗tz0, for allt∈ 0, τ. Then,
B∗ωT∗tz
∞
j1
B∗ωeR∗jtP∗ jz
∞
j1 2
s1
eΓjstB∗
ωPs,j∗ z0
⇐⇒∞ j1
2
s1
eΓjst
Bω∗Ps,j∗ zx 0, ∀x∈Ω.
3.11
Clearly that,{Γjs}is a decreasing sequence. Then, fromLemma 1.3, we obtain for all x∈ Ω
that
Bω∗Ps,j∗ zx Q∗s ⎡ ⎢ ⎢ ⎣ γj k1
z1, φj,k 1ωφj,kx γj
k1
z2, φj,k 1ωφj,kx ⎤ ⎥ ⎥ ⎦ 0 0
SinceQ1Q2IR2, we get that ⎡
⎢ ⎢ ⎣ γj k1
z1, φj,k φj,kx γj
k1
z2, φj,k φj,kx ⎤ ⎥ ⎥ ⎦
0
0
, j 1,2,3,4, . . .; s1,2, ∀x∈ω. 3.13
On the other hand, fromTheorem 1.1we know thatφn,kare analytic functions, which implies
the analyticity ofEjzi
γj
k1< zi,φj,k > φj,k. Then, fromTheorem 1.2we get that
⎡ ⎢ ⎢ ⎣ γj k1
z1, φj,k φj,kx γj
k1
z2, φj,k φj,kx ⎤ ⎥ ⎥ ⎦
0
0
, j1,2,3,4, . . . , ∀x∈Ω, s1,2. 3.14
HencePjz0,j1,2,3,4, . . ., which implies thatz0. This completes the proof of the main
theorem.
Acknowledgment
This work was supported by the CDHT-ULA-project: 1546-08-05-B.
References
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