Interior Controllability of a http://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

  • Hanzel Larez1 and

    Affiliated with

    • Hugo Leiva1Email author

      Affiliated with

      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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq5_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq7_HTML.gif , on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq11_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq12_HTML.gif is a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq14_HTML.gif , the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq15_HTML.gif diffusion matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq16_HTML.gif has semisimple and positive eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq18_HTML.gif is an arbitrary constant, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq19_HTML.gif is an open nonempty subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq20_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq22_HTML.gif , and the distributed controls http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq24_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq25_HTML.gif is the first eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq26_HTML.gif ), then for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq27_HTML.gif and all open nonempty subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq28_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq29_HTML.gif the system is approximately controllable on http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ1_HTML.gif
      (1.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq32_HTML.gif is a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq33_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq34_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq35_HTML.gif , the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq36_HTML.gif diffusion matrix
      http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq37_HTML.gif is an arbitrary constant, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq38_HTML.gif is an open nonempty subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq39_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq41_HTML.gif , and the distributed controls http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq43_HTML.gif (the first eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq44_HTML.gif ), then for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq45_HTML.gif and all open nonempty subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq46_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq47_HTML.gif , the system is approximately controllable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq48_HTML.gif .

      When http://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:
      http://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 http://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:
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ5_HTML.gif
      (1.5)

      where the diffusion coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq51_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq54_HTML.gif and the coefficient http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq58_HTML.gif ,

      (H2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq59_HTML.gif where http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq62_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq64_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq66_HTML.gif are locally Lipshitz; namely, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq67_HTML.gif and all constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq68_HTML.gif , there exist a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq69_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ6_HTML.gif
      (1.6)

      is verified for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq71_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq73_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq77_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq78_HTML.gif and http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ7_HTML.gif
      (1.7)

      whose eigenvalues are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq80_HTML.gif and http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ8_HTML.gif
      (1.8)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq82_HTML.gif , the distributed controls http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq83_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq84_HTML.gif is the identity matrix of dimension http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq88_HTML.gif is real analytic function in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq89_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq90_HTML.gif on a nonempty open subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq91_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq92_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq93_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq95_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq97_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ9_HTML.gif
      (1.9)
      if and only if
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq99_HTML.gif the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq100_HTML.gif , each one with finite multiplicity http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq102_HTML.gif of eigenvectors of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq103_HTML.gif .

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

      So, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq109_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq110_HTML.gif

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq111_HTML.gif generates an analytic semigroup http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq112_HTML.gif given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ13_HTML.gif
      (2.3)
      Now, we denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq113_HTML.gif the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq114_HTML.gif and define the following operator:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ14_HTML.gif
      (2.4)
      with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq115_HTML.gif . Therefore, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq116_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ15_HTML.gif
      (2.5)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ16_HTML.gif
      (2.6)
      where
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq118_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ18_HTML.gif
      (2.8)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq120_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq121_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq123_HTML.gif be a Hilbert separable space and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq124_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq126_HTML.gif , with http://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
      http://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:
      http://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) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq130_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq131_HTML.gif , continuous for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq132_HTML.gif .

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

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

      Theorem 2.2.

      The operator http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq140_HTML.gif given by:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ24_HTML.gif
      (2.14)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq142_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ25_HTML.gif
      (2.15)
      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq143_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq144_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq145_HTML.gif can be written as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ27_HTML.gif
      (2.17)
      with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ28_HTML.gif
      (2.18)
      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq146_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ29_HTML.gif
      (2.19)
      Clearly that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq148_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ30_HTML.gif
      (2.20)

      In fact, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq150_HTML.gif . Then, there exists a set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq151_HTML.gif of complementary projections on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq152_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ31_HTML.gif
      (2.21)
      Hence,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq153_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ33_HTML.gif
      (2.23)

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

      Finally, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq156_HTML.gif , then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq158_HTML.gif , the initial value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ35_HTML.gif
      (3.1)
      where the control function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq159_HTML.gif belonging to http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq161_HTML.gif if for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq163_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq164_HTML.gif such that the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq165_HTML.gif of (3.2) corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq166_HTML.gif verifies:
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ38_HTML.gif
      (3.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq167_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq168_HTML.gif are Hilbert spaces, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq170_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq171_HTML.gif , the control function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq172_HTML.gif belongs to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq174_HTML.gif if and only if
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq175_HTML.gif , then for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq176_HTML.gif and all open nonempty subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq177_HTML.gif of http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ40_HTML.gif
      (3.6)
      On the other hand,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq180_HTML.gif . Then, there exists a set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq181_HTML.gif of complementary projections on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq182_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ42_HTML.gif
      (3.8)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ43_HTML.gif
      (3.9)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ44_HTML.gif
      (3.10)

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

      Now, suppose for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq184_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq185_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq186_HTML.gif . Then,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ45_HTML.gif
      (3.11)
      Clearly that, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq188_HTML.gif that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ46_HTML.gif
      (3.12)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq189_HTML.gif , we get that
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq191_HTML.gif , http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_Equ48_HTML.gif
      (3.14)

      Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F560407/MediaObjects/13661_2008_Article_858_IEq194_HTML.gif , which implies that http://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

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      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.