2. Abstract Formulation of the Problem

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−Δ^1/2ubΔv1ωf1t, xin0, τ×Ω,vtcΔu−dΔv− β−Δ^1/2v1ωf2t, xin0, τ×Ω,u v 0, on0, T×∂Ω,u0, x u0x,v0, x v0x, x∈Ω, whereΩis a bounded domain inR^NN ≥ 1,u0, 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: ifλ^1/2₁ ρ1β >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−Δ^1/2ubΔv1ωf1t, x in0, τ×Ω, v_tcΔu−dΔv−β−Δ^1/2v1_ωf₂t, x in0, τ×Ω,

uv0, on0, τ×∂Ω, u0, x u0x, v0, x v0x, x∈Ω,

1.1

whereΩis a bounded domain inR^NN≥1,u0, v0∈L²Ω, the 2×2 diffusion matrix

D a b

c d

1.2

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: ifλ^1/2₁ ρ₁β >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:

u_ta∂²u

∂x² β∂u

∂xb∂²v

∂x² 1_ωf₁t, 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∈XC_UBR,whereC_UBis 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 t∈ 0, t1.

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−βI_2×2−Δ^1/2z1ωft, x, in0, τ×Ω, z0, on0, τ×∂Ω,

z0, x z₀x, x∈Ω,

1.8

wherez u, v^T ∈R², the distributed controlsf f1, f2^T ∈L² 0, τ;L²Ω;R², andI_2×2 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 0 on 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

e^αjtβ_i,j 0, ∀t∈ 0, t1, i1,2, . . . , m 1.9

if and only if

β_i,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 enjoy the interior controllability with a minor effort.

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 λ1 < 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

E_nξ

γj

k1

ξ, φ_j,k φ_j,k. 2.2

So,{Ej}is a family of complete orthogonal projections inHandξ_∞

j1E_jξ,ξ∈H.

iii Δgenerates an analytic semigroup{T_Δt}given by

T_Δtξ^∞

j1

e^−λjtE_jξ. 2.3

Now, we denote byZthe Hilbert spaceH² L²Ω;R²and define the following operator:

A:DA⊂Z−→Z, Aψ−DΔψβI_2×2−Δ^1/2ψ 2.4

withDA H²Ω,R²∩H₀¹Ω,R². Therefore, for allz∈DA, we obtain

Az^∞

j1

λ^1/2_j

λ^1/2_j DβI_2×2

P_jz, 2.5

z^∞

j1

P_jz, z^{2 ∞}

j1

P_jz², z∈Z, 2.6

where

P_j E_j 0

0 E_j

2.7 is a family of complete orthogonal projections inZ.

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

e^AjtP_jz, z∈Z, t≥0. 2.10

Then the following hold.

aTt is a linear and bounded operator if e^Ajt ≤ 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

A_jP_jz, z∈DA 2.11

with

DA

⎧⎨

⎩z∈Z: ∞ j1

AjPjz² <∞

⎫⎬

⎭. 2.12

cThe spectrumσAofAis given by

σA ^∞

j1

σ Aj

, 2.13

whereA_j A_jP_j:RPj → RPj.

Theorem 2.2. The operator−Adefine by2.5is the infinitesimal generator of a strongly continuous semigroup{Tt}_t≥0given by:

Ttz^∞

j1

e^AjtP_jz, z∈Z, t≥0, 2.14

wherePj diag Ej, EjandAj RjPjwith

Rj

−aλj −bλj

−cλj −dλj

−β

⎡

⎣λ^1/2_j 0 0 λ^1/2_j

⎤

⎦. 2.15

Moreover, ifλ^1/2₁ ρ1β >0, then there existsM >0 such that

Tt ≤Mexp

−λ^1/2₁

λ^1/2₁ ρ1β 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

A_j−λ^1/2_j

λ^1/2_j DβI_2×2

P_j, P_jdiag E_j, E_j

. 2.18

Therefore,A_jR_jP_jwith

R_j

−aλj −bλj

−cλj −dλj

−β

⎡

⎣λ^1/2_j 0 0 λ^1/2_j

⎤

⎦, A_jP_jP_jA_j. 2.19

Clearly thatAj is a bounded linear operatorlinear and continuous. That is, there exists M_j>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

e^Dte^ρ1tQ₁e^ρ2tQ₂. 2.21

Hence,

e^Rjte^Γ1jtQ₁e^Γ2jtQ₂, with Γjs−λ^1/2_j

λ^1/2_j ρ_sβ

, s1,2. 2.22

This implies the existence of positive numbersα, Msuch that

e^Ajt≤Me^αt, j1,2, . . . . 2.23

Therefore,−Agenerates a strongly continuous semigroup{Tt}_t≥0given by2.14.

Finally, ifλ^1/2₁ ρ₁β >0, then

−λ^1/2₁

λ^1/2₁ ρ1β

≥ −λ^1/2_j

λ^1/2_j ρiβ

, 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 Now, one is ready to formulate and prove the main theorem of this work.

Theorem 3.3main theorem. Ifλ^1/2₁ ρ1β >0, then for allτ >0 and 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

B_ω^∗ B_ω, T^∗tz^∞

j1

e^R∗jtP_j^∗z, z∈Z, t≥0. 3.6

On the other hand,

R_j−λ^1/2_j

λ^1/2_j a b

c d

β 1 0

0 1

−λ^1/2_j

λ^1/2_j DβI_2×2

. 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

e^Dte^ρ1tQ₁e^ρ2tQ₂. 3.8

Hence,

e^Rjte^Γj1tQ₁e^Γj2tQ₂, withΓjs−λ^1/2_j

λ^1/2_j ρ_sβ

, s1,2. 3.9

Therefore,

B_ω^∗T^∗tz^∞

j1

B^∗_ωe^Rj∗tP_j^∗z^∞

j1

2 s1

e^ΓjstB^∗_ωP_s,j^∗ z, 3.10

whereP_s,jQ_sP_jP_jQ_s.

Now, suppose forz∈ZthatB^∗_ωT^∗tz0, for allt∈ 0, τ. Then,

B^∗_ωT^∗tz^∞

j1

B^∗_ωe^R∗jtP_j^∗z^∞

j1

2 s1

e^ΓjstB_ω^∗P_s,j^∗ z0

⇐⇒^∞

j1

2 s1

e^Γjst B_ω^∗P_s,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

B_ω^∗P_s,j^∗ z

x Q^∗_s

⎡

⎢⎢

⎣

γj

k1

z1, φj,k 1ωφj,kx

γj

k1

z₂, φ_j,k 1_ωφ_j,kx

⎤

⎥⎥

⎦ 0

0

, j 1,2,3,4, . . .; s1,2. 3.12

SinceQ1Q2I_R², we get that

⎡

⎢⎢

⎣

γj

k1

z1, φj,k φj,kx

γj

k1

z₂, φ_j,k φ_j,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 ofE_jz_iγj

k1< z_i,φ_j,k > φ_j,k. Then, fromTheorem 1.2we get that

⎡

⎢⎢

⎣

γj

k1

z₁, φ_j,k φ_j,kx

γj

k1

z₂, φ_j,k φ_j,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.

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