Báo cáo hóa học: " Research Article Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix" pdf

Introduction In this paper we prove the interior approximate controllability for the following 2×2 reaction-diffusion system with cross-reaction-diffusion matrix 1.1 a b c d 1.2... We no

Volume 2009, Article ID 560407, 9 pages

doi:10.1155/2009/560407

Research Article

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,hleiva@ula.ve

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 matrix u t  aΔu − β−Δ 1/2 u  bΔv  1 ω f1t, x in 0, τ × Ω, v t  cΔu − dΔv −

β−Δ1/2 v 1ω f2t, x in 0, τ × Ω, u  v  0, on 0, T × ∂Ω, u0, x  u0x, v0, x  v0x,

x∈ Ω, where Ω is a bounded domain inRN N ≥ 1, u0, v0∈ L2Ω, the 2 × 2 diffusion matrix

Da b

c d



has semisimple and positive eigenvalues 0 < ρ1≤ ρ2, β is an arbitrary constant, ω is an

open nonempty subset ofΩ, 1ω denotes the characteristic function of the set ω, and the distributed controls f1, f2∈ L2 2Ω Specifically, we prove the following statement: if λ 1/2

1 ρ1 β > 0

where λ1is the first eigenvalue of−Δ, then for all τ > 0 and all open nonempty subset ω of Ω the

system is approximately controllable on

Copyrightq 2009 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-reaction-diffusion matrix

1.1



a b

c d



1.2

has semisimple and positive eigenvalues, β is an arbitrary constant, ω is an open nonempty

approximately controllable on

u t  a ∂2u

∂x2  β ∂u

∂x  b ∂2v

v t  c ∂2u

∂x2  d ∂2v

∂x2  β ∂v

1.3

That is to say, he studies the system:

u t  a ∂2u

∂x2  β ∂u

∂x  b ∂2v

v t  c ∂2u

∂x2  d ∂2v

∂x2  β ∂v

1.4

supplemented with the initial conditions:

where the diffusion coefficients a and d are assumed positive constants, while the diffusion coefficients b, c and the coefficient β are arbitrary constants He assume also the following three conditions:

real-valued functions,

H3 ft, u, v and gt, u, v ∈ X, for all t > 0 and u, v ∈ X Moreover, f and g are

t ∈ 0, t1

We note that the hypothesisH1 implies that the eigenvalues of the matrix D are simple and positive But, this condition is not necessary for the eigenvalues of D to be positive, in fact we can find matrices D with a and d been negative and having positive eigenvalues For

example, one can consider the following matrix:





1.8

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 1.2 see 2, Theorem 1.23, page 20 n is open, nonempty, and connected set, and f is real analytic function in Ω with f  0 on a nonempty open subset ω of Ω Then, f  0

in Ω.

Lemma 1.3 see 3, Lemma 3.14, page 62 j}j≥1 and {β i,j : i  1, 2, , m} j≥1 be two sequences of real numbers such that α1> α2> α3· · · Then

∞



j1

if and only if

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

differential equation; to this end, we consider the following notations:

45-46

ii For all ξ ∈ D−Δ, we have

j1

λ j

γ j



k1



ξ, φ j,k

φ j,k∞

j1

where

E n ξ

γ j



k1



ξ, φ j,k

j1

operator:

Az∞

j1

λ 1/2 j

z∞

j1

P j z, z2 ∞

j1

where

P j



E j 0



2.7

is a family of complete orthogonal projections in Z.

Consequently, system1.8 can be written as an abstract differential equation in Z:

Lemma 2.1 Let Z be a Hilbert separable space and {A j}j≥1 , {P j}j≥1 two families of bounded linear operator in Z, with {P j}j≥1 a family of complete orthogonal projection such that

Define the following family of linear operators:

j1

Then the following hold.

continuous for t ≥ 0.

whose infinitesimal generator A is given by

Az∞

j1

with

⎧

⎨

∞



j1

A j P j z 2 <∞

⎫

⎬

c The spectrum σA of A is given by

j1

σ

A j

where A j  A j P j:RP j  → RP j .

Theorem 2.2 The operator −A define by 2.5 is the infinitesimal generator of a strongly continuous semigroup {Tt} t≥0 given by:

j1

where P j  diag E j , E j j  R j P j with

R j





− β

⎡

1/2

j 0

⎤

Moreover, if λ 1/21 ρ1 β > 0, then there exists M > 0 such that

1

λ 1/21 ρ1 βt

Proof In order to apply the foregoing Lemma, we observe that −A can be written as follows:

j1

with

A j  −λ 1/2

j

R j





− β

⎡

1/2

j 0

⎤

M j > 0 such that

Hence,

e R j t  eΓ1j t Q1 eΓ2j t Q2, with Γjs  −λ 1/2

j



This implies the existence of positive numbers α, M such that

Finally, if λ 1/21 ρ1 β > 0, then

1

λ 1/21 ρ1 β≥ −λ 1/2

j

3 Proof of the Main Theorem

In this section we will prove the main result of this paper on the controllability of the linear

t

0

Definition 3.1 approximate controllability The system 2.8 is said to be approximately

Theorem 3.2 System 3.4

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

Theorem 3.3 main theorem If λ 1/2

Proof We will applyTheorem 3.2to prove the approximate controllability of system2.8 With this purpose, we observe that

j1

On the other hand,

R j  −λ 1/2 j

λ 1/2 j



a b

c d



 β



1 0

0 1



j



Hence,

e R j t  eΓj1 t Q1 eΓj2 t Q2, withΓjs  −λ 1/2

j



Therefore,

B ω∗T∗tz ∞

j1

B∗ω e R j ∗t P j∗z∞

j1

2



s1

ω T∗

B∗ω T∗tz ∞

j1

B∗ω e R∗j t P j∗z∞

j1

2



s1

eΓjs t B ω∗P s,j∗ z 0

j1

2



s1

eΓjs t

B ω∗P s,j∗

3.11

that

B ω∗P s,j∗ z

s

⎡

⎢

⎣

γ j

k1



z1, φ j,k

1ω φ j,k x

γ j

k1



z2, φ j,k

1ω φ j,k x

⎤

⎥

⎦ 

 0 0



Since Q1 Q2 IR 2, we get that

⎡

⎢

⎣

γ j

k1



z1, φ j,k

φ j,k x

γ j

k1



z2, φ j,k

φ j,k x

⎤

⎥

⎦ 

 0 0



k1 < z i , φ j,k > φ j,k Then, fromTheorem 1.2we get that

⎡

⎢

⎣

γ j

k1



z1, φ j,k

φ j,k x

γ j

k1



z2, φ j,k

φ j,k x

⎤

⎥

⎦ 

 0 0



theorem

Acknowledgment

This work was supported by the CDHT-ULA-project: 1546-08-05-B

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j1

where P j  diag E j , E j...

⎡... Q2, with Γjs  −λ 1/2

j



This