Open Access

Interior Controllability of a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq1_HTML.gif Reaction-Diffusion System with Cross-Diffusion Matrix

Boundary Value Problems20092009:560407

DOI: 10.1155/2009/560407

Received: 30 December 2008

Accepted: 27 May 2009

Published: 11 June 2009

Abstract

We prove the interior approximate controllability for the following https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq2_HTML.gif reaction-diffusion system with cross-diffusion matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq3_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq5_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq7_HTML.gif , on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq11_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq12_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq14_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq15_HTML.gif diffusion matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq16_HTML.gif has semisimple and positive eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq18_HTML.gif is an arbitrary constant, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq19_HTML.gif is an open nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq21_HTML.gif denotes the characteristic function of the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq22_HTML.gif , and the distributed controls https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq23_HTML.gif . Specifically, we prove the following statement: if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq24_HTML.gif (where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq25_HTML.gif is the first eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq26_HTML.gif ), then for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq27_HTML.gif and all open nonempty subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq28_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq29_HTML.gif the system is approximately controllable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq30_HTML.gif .

1. Introduction

In this paper we prove the interior approximate controllability for the following https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq31_HTML.gif reaction-diffusion system with cross-diffusion matrix
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq32_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq33_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq34_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq35_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq36_HTML.gif diffusion matrix
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ2_HTML.gif
(1.2)

has semisimple and positive eigenvalues, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq37_HTML.gif is an arbitrary constant, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq38_HTML.gif is an open nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq40_HTML.gif denotes the characteristic function of the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq41_HTML.gif , and the distributed controls https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq42_HTML.gif . Specifically, we prove the following statement: if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq43_HTML.gif (the first eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq44_HTML.gif ), then for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq45_HTML.gif and all open nonempty subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq46_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq47_HTML.gif , the system is approximately controllable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq48_HTML.gif .

When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq49_HTML.gif this system takes the following particular form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ3_HTML.gif
(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 system (1.3) on the unbounded domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq50_HTML.gif . That is to say, he studies the system:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ4_HTML.gif
(1.4)
supplemented with the initial conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ5_HTML.gif
(1.5)

where the diffusion coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq52_HTML.gif are assumed positive constants, while the diffusion coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq54_HTML.gif and the coefficient https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq55_HTML.gif are arbitrary constants. He assume also the following three conditions:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq58_HTML.gif ,

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq59_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq60_HTML.gif is the space of bounded and uniformly continuous real-valued functions,

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq62_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq64_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq66_HTML.gif are locally Lipshitz; namely, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq67_HTML.gif and all constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq68_HTML.gif , there exist a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq69_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ6_HTML.gif
(1.6)

is verified for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq71_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq72_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq74_HTML.gif .

We note that the hypothesis (H1) implies that the eigenvalues of the matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq75_HTML.gif are simple and positive. But, this condition is not necessary for the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq76_HTML.gif to be positive, in fact we can find matrices https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq77_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq79_HTML.gif been negative and having positive eigenvalues. For example, one can consider the following matrix:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ7_HTML.gif
(1.7)

whose eigenvalues are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq81_HTML.gif .

The system (1.1) can be written in the following matrix form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ8_HTML.gif
(1.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq82_HTML.gif , the distributed controls https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq83_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq84_HTML.gif is the identity matrix of dimension https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq85_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq86_HTML.gif with Dirichlet boundary condition are real analytic functions.

Theorem 1.2 (see [2, Theorem  1.23, page 20]).

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq87_HTML.gif is open, nonempty, and connected set, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq88_HTML.gif is real analytic function in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq89_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq90_HTML.gif on a nonempty open subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq91_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq92_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq93_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq94_HTML.gif .

Lemma 1.3 (see [3, Lemma  3.14, page 62]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq96_HTML.gif be two sequences of real numbers such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq97_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ9_HTML.gif
(1.9)
if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ10_HTML.gif
(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 (1.8) can be written as an abstract differential equation; to this end, we consider the following notations:

Let us consider the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq99_HTML.gif the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq100_HTML.gif , each one with finite multiplicity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq101_HTML.gif equal to the dimension of the corresponding eigenspace. Then, we have the following well-known properties (see [3, pages 45-46]).

(i)There exists a complete orthonormal set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq102_HTML.gif of eigenvectors of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq103_HTML.gif .

(ii)For all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq104_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ11_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq105_HTML.gif is the inner product in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq106_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ12_HTML.gif
(2.2)

So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq107_HTML.gif is a family of complete orthogonal projections in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq110_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq111_HTML.gif generates an analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq112_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ13_HTML.gif
(2.3)
Now, we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq113_HTML.gif the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq114_HTML.gif and define the following operator:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ14_HTML.gif
(2.4)
with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq115_HTML.gif . Therefore, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq116_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ15_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ16_HTML.gif
(2.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ17_HTML.gif
(2.7)

is a family of complete orthogonal projections in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq117_HTML.gif .

Consequently, system (1.8) can be written as an abstract differential equation in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq118_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ18_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq120_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq122_HTML.gif is a bounded linear operator.

Now, we will use the following Lemma from [4] to prove the following theorem.

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq123_HTML.gif be a Hilbert separable space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq125_HTML.gif two families of bounded linear operator in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq126_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq127_HTML.gif a family of complete orthogonal projection such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ19_HTML.gif
(2.9)
Define the following family of linear operators:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ20_HTML.gif
(2.10)

Then the following hold.

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq128_HTML.gif is a linear and bounded operator if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq130_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq131_HTML.gif , continuous for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq132_HTML.gif .

(b)Under the above (a), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq133_HTML.gif is a strongly continuous semigroup in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq134_HTML.gif , whose infinitesimal generator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq135_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ21_HTML.gif
(2.11)
with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ22_HTML.gif
(2.12)
(c)The spectrum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq136_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq137_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ23_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq138_HTML.gif .

Theorem 2.2.

The operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq139_HTML.gif define by (2.5) is the infinitesimal generator of a strongly continuous semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq140_HTML.gif given by:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ24_HTML.gif
(2.14)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq142_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ25_HTML.gif
(2.15)
Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq143_HTML.gif , then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq144_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ26_HTML.gif
(2.16)

Proof.

In order to apply the foregoing Lemma, we observe that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq145_HTML.gif can be written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ27_HTML.gif
(2.17)
with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ28_HTML.gif
(2.18)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq146_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ29_HTML.gif
(2.19)
Clearly that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq147_HTML.gif is a bounded linear operator (linear and continuous). That is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq148_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ30_HTML.gif
(2.20)

In fact, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq149_HTML.gif .

Now, we have to verify condition (a) of Lemma 2.1. To this end, without loss of generality, we will suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq150_HTML.gif . Then, there exists a set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq151_HTML.gif of complementary projections on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq152_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ31_HTML.gif
(2.21)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ32_HTML.gif
(2.22)
This implies the existence of positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq153_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ33_HTML.gif
(2.23)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq154_HTML.gif generates a strongly continuous semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq155_HTML.gif given by (2.14).

Finally, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq156_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ34_HTML.gif
(2.24)

and using (2.14) we obtain (2.16).

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 all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq158_HTML.gif , the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ35_HTML.gif
(3.1)
where the control function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq159_HTML.gif belonging to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq160_HTML.gif admits only one mild solution given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ36_HTML.gif
(3.2)

Definition 3.1 (approximate controllability).

The system (2.8) is said to be approximately controllable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq161_HTML.gif if for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq162_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq163_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq164_HTML.gif such that the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq165_HTML.gif of (3.2) corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq166_HTML.gif verifies:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ37_HTML.gif
(3.3)
The following result can be found in [5] for the general evolution equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ38_HTML.gif
(3.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq168_HTML.gif are Hilbert spaces, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq169_HTML.gif is the infinitesimal generator of strongly continuous semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq170_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq171_HTML.gif , the control function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq172_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq173_HTML.gif .

Theorem 3.2.

System (3.4) is approximately controllable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq174_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ39_HTML.gif
(3.5)

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

Theorem 3.3 (main theorem).

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq175_HTML.gif , then for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq176_HTML.gif and all open nonempty subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq177_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq178_HTML.gif the system, (2.8) is approximately controllable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq179_HTML.gif .

Proof.

We will apply Theorem 3.2 to prove the approximate controllability of system (2.8). With this purpose, we observe that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ40_HTML.gif
(3.6)
On the other hand,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ41_HTML.gif
(3.7)
Without lose of generality, we will suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq180_HTML.gif . Then, there exists a set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq181_HTML.gif of complementary projections on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq182_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ42_HTML.gif
(3.8)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ43_HTML.gif
(3.9)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ44_HTML.gif
(3.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq183_HTML.gif .

Now, suppose for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq184_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq185_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq186_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ45_HTML.gif
(3.11)
Clearly that, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq187_HTML.gif is a decreasing sequence. Then, from Lemma 1.3, we obtain for all x https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq188_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ46_HTML.gif
(3.12)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq189_HTML.gif , we get that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ47_HTML.gif
(3.13)
On the other hand, from Theorem 1.1 we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq190_HTML.gif are analytic functions, which implies the analyticity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq191_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq192_HTML.gif . Then, from Theorem 1.2 we get that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ48_HTML.gif
(3.14)

Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq194_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq195_HTML.gif . This completes the proof of the main theorem.

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad de Los Andes

References

  1. Badraoui S: Asymptotic behavior of solutions to a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq196_HTML.gif reaction-diffusion system with a cross diffusion matrix on unbounded domains. Electronic Journal of Differential Equations 2006, 2006(61):1–13.MathSciNet
  2. Axler S, Bourdon P, Ramey W: Harmonic Function Theory, Graduate Texts in Mathematics. Volume 137. Springer, New York, NY, USA; 1992:xii+231.View Article
  3. Curtain RF, Pritchard AJ: Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences. Volume 8. Springer, Berlin, Germany; 1978:vii+297.View Article
  4. Leiva H: A lemma on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq197_HTML.gif -semigroups and applications PDEs systems. Quaestiones Mathematicae 2003, 26(3):247–265. 10.2989/16073600309486057MATHMathSciNetView Article
  5. Curtain RF, Zwart H: An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics. Volume 21. Springer, New York, NY, USA; 1995:xviii+698.View Article

Copyright

© H. Larez and H. Leiva. 2009

This article is published under license to BioMed Central Ltd. 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.