Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains

Boundary Value Problems20102010:281238

DOI: 10.1155/2010/281238

Received: 11 July 2009

Accepted: 5 January 2010

Published: 19 January 2010

Abstract

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq1_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq5_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq8_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq9_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq10_HTML.gif is a smooth bounded domain, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq11_HTML.gif , the diffusion matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq12_HTML.gif has semisimple and positive eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq15_HTML.gif is an open nonempty set, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq16_HTML.gif is the characteristic function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq17_HTML.gif . Specifically, we prove that under some conditions over the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq18_HTML.gif , the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq19_HTML.gif the system is approximately controllable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq20_HTML.gif .

1. Introduction

In this paper we prove controllability for the following reaction-diffusion system with cross diffusion matrix:

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq21_HTML.gif is an open nonempty set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq23_HTML.gif is the characteristic function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq24_HTML.gif .

We assume the following assumptions.

(H1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq25_HTML.gif is a smooth bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq26_HTML.gif .

(H2) The diffusion matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq27_HTML.gif has semisimple and positive eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq28_HTML.gif

(H3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq29_HTML.gif are real constants, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq30_HTML.gif are real constants belonging to the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq31_HTML.gif

(H4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq32_HTML.gif

(H5) The distributed controls http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq33_HTML.gif .

Specifically, we prove the following statements.

(i) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq35_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq36_HTML.gif is the first eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq37_HTML.gif with Dirichlet condition, or if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq38_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq39_HTML.gif then, under the hypotheses (H1)–(H3), the semigroup generated by the linear operator of the system is exponentially stable.

(ii) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq40_HTML.gif and under the hypotheses (H1)–(H5), then, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq41_HTML.gif and all open nonempty subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq42_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq43_HTML.gif the system is approximately controllable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq44_HTML.gif

This paper has been motivated by the work done in [1] and the work done by H. Larez and H. Leiva in [2]. In the work [1], the auther studies the asymptotic behavior of the solution of the system

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ2_HTML.gif
(1.2)

supplemeted with the initial conditions

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ3_HTML.gif
(1.3)

The author proved that in the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq45_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq46_HTML.gif is the space of bounded uniformly continuous real valued functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq47_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq49_HTML.gif are locally Lipshitz and under some conditions over the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq50_HTML.gif , and if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq51_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq52_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq53_HTML.gif Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq55_HTML.gif satisfy the system of ordinary differential equations

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ4_HTML.gif
(1.4)

with the initial data

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ5_HTML.gif
(1.5)

The same result holds for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq56_HTML.gif

In the work done in [2], the authers studied the system (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq57_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq58_HTML.gif They proved that if the diffusion matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq59_HTML.gif has semi-simple and positive eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq61_HTML.gif then if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq62_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq63_HTML.gif is the first eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq64_HTML.gif ), the system is approximately controllable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq65_HTML.gif for all open nonempty subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq66_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq67_HTML.gif

2. Notations and Preliminaries

In the following we denote by

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq69_HTML.gif the set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq70_HTML.gif matrices with entries from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq71_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq73_HTML.gif the set of all measurable functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq74_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq75_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq77_HTML.gif the set of all the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq78_HTML.gif that have generalized derivatives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq79_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq80_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq82_HTML.gif the closure of the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq83_HTML.gif in the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq84_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq86_HTML.gif the set of all the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq87_HTML.gif that have generalized derivatives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq88_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq89_HTML.gif .

We will use the following results.

Theorem 2.1 (cf. [3]).

Let us consider the following classical boundary-eigenvalue problem for the laplacien:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ6_HTML.gif
(2.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq90_HTML.gif is a nonempty bounded open set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq92_HTML.gif .

This problem has a countable system of eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq94_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq95_HTML.gif .

(i) All the eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq96_HTML.gif have finite multiplicity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq97_HTML.gif equal to the dimension of the corresponding eigenspace http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq98_HTML.gif .

(ii) Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq99_HTML.gif be a basis of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq100_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq101_HTML.gif then the eigenvectors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq102_HTML.gif form a complete orthonormal system in the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq103_HTML.gif Hence for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq104_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq105_HTML.gif If we put http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq106_HTML.gif then we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq107_HTML.gif .

(iii) Also, the eigenfunctions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq108_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq109_HTML.gif is the space of infinitely continuously differentiable functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq110_HTML.gif and compactly supported in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq111_HTML.gif .

(iv) For all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq112_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq113_HTML.gif .

(v) The operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq114_HTML.gif generates an analytic semigroup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq115_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq116_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ7_HTML.gif
(2.2)

Definition 2.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq117_HTML.gif a real number, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq118_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ8_HTML.gif
(2.3)

In particular, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq120_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq121_HTML.gif form a complete orthonormal system in the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq122_HTML.gif then it is dense in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq123_HTML.gif , and hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq124_HTML.gif is dense in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq125_HTML.gif .

Proposition 2.3 (cf. [4]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq126_HTML.gif be a Hilbert separable space and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq128_HTML.gif two families of bounded linear operators in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq129_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq130_HTML.gif a family of complete orthogonal projections such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq131_HTML.gif

Define the following family of linear operators http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq132_HTML.gif Then

(a) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq133_HTML.gif is a linear and bounded operator if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq134_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq135_HTML.gif continiuous for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq136_HTML.gif

(b) under the above condition (a), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq137_HTML.gif is a strongly continiuous semigroup in the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq138_HTML.gif whose infinitesimal generator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq139_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ9_HTML.gif
(2.4)

Theorem 2.4 (cf. [5]).

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq140_HTML.gif is connected, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq141_HTML.gif is a real function in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq142_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq143_HTML.gif on a nonempty open subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq144_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq145_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq146_HTML.gif .

3. Abstract Formulation of the Problem

In this section we consider the following notations.
  1. (i)

    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq147_HTML.gif is a Hilbert space with the inner product

     
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ10_HTML.gif
(3.1)

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq148_HTML.gif We define

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ11_HTML.gif
(3.2)
  1. (iii)

    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq149_HTML.gif then we can define the linear operator

     
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ12_HTML.gif
(3.3)

where

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ13_HTML.gif
(3.4)

Therefore, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq150_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ14_HTML.gif
(3.5)

If we put

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ15_HTML.gif
(3.6)

then (3.3) can be written as

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ16_HTML.gif
(3.7)

and we have for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq151_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ17_HTML.gif
(3.8)

Consequently, system (1.1) can be written as an abstract differential equation in the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq152_HTML.gif in the following form:

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ18_HTML.gif
(3.9)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq154_HTML.gif is a bounded linear operator from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq155_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq156_HTML.gif .

4. Main Results

4.1. Generation of a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq157_HTML.gif -Semigroup

Theorem 4.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq158_HTML.gif , then, under hypotheses (H1)–(H3), the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq159_HTML.gif defined by (3.3) is the infinitesimal generator of strongly continuous semigroup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq160_HTML.gif given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ19_HTML.gif
(4.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ20_HTML.gif
(4.2)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ21_HTML.gif
(4.3)
Moreover, if
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ22_HTML.gif
(4.4)
then the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq161_HTML.gif -semigoup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq162_HTML.gif is exponentially stable, that is, there exist two positives constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq163_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ23_HTML.gif
(4.5)

Proof.

In order to apply the Proposition 2.3, we observe that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq164_HTML.gif can be written as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ24_HTML.gif
(4.6)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ25_HTML.gif
(4.7)

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq166_HTML.gif

Now, we have to verify condition (a) of the Proposition 2.3. We shall suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq167_HTML.gif Then, there exists a set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq168_HTML.gif of complementary projections on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq169_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ26_HTML.gif
(4.8)
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq170_HTML.gif is the matrix passage from the canonical basis of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq171_HTML.gif to the basis composed with the eigenvectors of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq172_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ27_HTML.gif
(4.9)
Hence,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ28_HTML.gif
(4.10)
We have also
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ29_HTML.gif
(4.11)
From (4.10)-(4.11) into (4.7) we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ30_HTML.gif
(4.12)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ31_HTML.gif
(4.13)
As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq173_HTML.gif we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ32_HTML.gif
(4.14)
As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq174_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq175_HTML.gif then this implies the existence of a positive number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq176_HTML.gif and a real number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq177_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq178_HTML.gif    for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq179_HTML.gif Therefore http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq180_HTML.gif is a strongly continious semigroup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq181_HTML.gif given by (4.1). We can even estimate the constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq183_HTML.gif as follows.
  1. (i)
    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq184_HTML.gif As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq185_HTML.gif , then there exist constants
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ33_HTML.gif
    (4.15)
     
hence, if we put
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ34_HTML.gif
(4.16)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ35_HTML.gif
(4.17)
we easily obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ36_HTML.gif
(4.18)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq186_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq187_HTML.gif . If we put
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ37_HTML.gif
(4.19)
then we find that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ38_HTML.gif
(4.20)

Therefore, the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq188_HTML.gif generates a strongly continuous semigroup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq189_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq190_HTML.gif given by expression (4.1).

Finally, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq191_HTML.gif we have already proved (4.20). Using (4.20) into (4.1) we get that the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq192_HTML.gif -semigoup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq193_HTML.gif is exponentially stable. The expression (4.5) is verfied with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq195_HTML.gif is defined by (4.19).

Theorem 4.2.

If
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ39_HTML.gif
(4.21)
then, under the hypotheses (H1)–(H3), the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq196_HTML.gif defined by (3.3) is the infinitesimal generator of strongly continuous semigroup exponentially stable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq197_HTML.gif defined by (4.1). Specially, there exist two positives constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq198_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ40_HTML.gif
(4.22)

To prove this result, we need the following lemma.

Lemma 4.3.

For every two real positives constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq200_HTML.gif , one has for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq201_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ41_HTML.gif
(4.23)
and for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq202_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ42_HTML.gif
(4.24)

Proof of Lemma 4.3.

It is easy to verify that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq203_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq204_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq205_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq206_HTML.gif , then we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ43_HTML.gif
(4.25)

Hence, we get (4.23).

Also, it is easy to verify that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq207_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq208_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq209_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq210_HTML.gif , then we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ44_HTML.gif
(4.26)

Hence, from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq211_HTML.gif (4.26) we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq212_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq214_HTML.gif , which gives (4.24).

With the same manner we can prove that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq215_HTML.gif and every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq216_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ45_HTML.gif
(4.27)
and consequently, for every two real positives constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq217_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq218_HTML.gif and every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq219_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ46_HTML.gif
(4.28)

Now, we are ready to prove Theorem 4.2.

Proof of Theorem 4.2.

By applying Proposition 2.3 we start from formula (4.12) and we put
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ47_HTML.gif
(4.29)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ48_HTML.gif
(4.30)
To estimate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq220_HTML.gif we have in taking into account http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq221_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ49_HTML.gif
(4.31)
and applying the Lemma 4.3 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq222_HTML.gif we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ50_HTML.gif
(4.32)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq224_HTML.gif . But we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq225_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq226_HTML.gif Then we get for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq227_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ51_HTML.gif
(4.33)
From (4.31)-(4.33) we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ52_HTML.gif
(4.34)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ53_HTML.gif
(4.35)

and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq228_HTML.gif

Applying Lemma 4.3 and taking into account (4.21) we get with the same manner that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq229_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ54_HTML.gif
(4.36)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ55_HTML.gif
(4.37)
and or every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq230_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ56_HTML.gif
(4.38)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ57_HTML.gif
(4.39)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ58_HTML.gif
(4.40)
From (4.34)-(4.40) into (4.12) we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ59_HTML.gif
(4.41)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq231_HTML.gif is defined by (4.17) and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ60_HTML.gif
(4.42)

Using (4.41) into (4.1) we get that the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq232_HTML.gif -semigoup http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq233_HTML.gif generated by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq234_HTML.gif is exponentially stable. Expression (4.22) is verfied with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq235_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq236_HTML.gif is defined by (4.42).

4.2. Approximate Controllability

Befor giving the definition of the approximate controllabiliy for the sytem (3.9), we have the following known result: for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq237_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq238_HTML.gif the initial value problem (3.9) admits a unique mild solution given by

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ61_HTML.gif
(4.43)

This solution is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq239_HTML.gif

Definition 4.4.

System (3.9) is said to be approximately controllable at time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq240_HTML.gif whenever the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq241_HTML.gif is densely embedded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq242_HTML.gif ; that is,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ62_HTML.gif
(4.44)

The following criteria for approximate controllability can be found in [6].

Criteria 1.

System (3.9) is approximately controllable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq243_HTML.gif if and only if
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ63_HTML.gif
(4.45)

Now, we are ready to formulate the third main result of this work.

Theorem 4.5.

If the following condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ64_HTML.gif
(4.46)

is satisfied; then, under hypotheses (H1)–(H5), for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq244_HTML.gif and all open subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq245_HTML.gif system (3.9) is approximately controllable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq246_HTML.gif .

Proof.

The proof of this theorem relies on the Criteria 1 and the following lemma.

Lemma 4.6.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq247_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq248_HTML.gif be sequences of real numbers such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq249_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq251_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq252_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq253_HTML.gif one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ65_HTML.gif
(4.47)

Proof of Lemma 4.6.

By analyticity we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq254_HTML.gif and from this we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq255_HTML.gif . Under the assumptions of the lemma we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq256_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq257_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq258_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq259_HTML.gif , we divide http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq260_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq261_HTML.gif and we pass http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq262_HTML.gif we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq263_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq264_HTML.gif we divide http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq265_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq266_HTML.gif and we pass http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq267_HTML.gif and get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq268_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq269_HTML.gif , we divide http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq270_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq271_HTML.gif and we pass http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq272_HTML.gif and get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq273_HTML.gif But in this we case we can integrate under the symbol of sommation over the intervall http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq274_HTML.gif and we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq275_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq276_HTML.gif . Continuing this way we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq277_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq278_HTML.gif

We are now ready to prove Theorem 4.5. For this purpose, we observe that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ66_HTML.gif
(4.48)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq279_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq280_HTML.gif -semigroup generated by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq281_HTML.gif .

Without lose of generality, we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq282_HTML.gif Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ67_HTML.gif
(4.49)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq283_HTML.gif

Now, suppose for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq284_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq285_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq286_HTML.gif Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ68_HTML.gif
(4.50)
If (4.46) is satisfied, then (4.50) take the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ69_HTML.gif
(4.51)
Then, from lemma 4.6 we obtain that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq287_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq288_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ70_HTML.gif
(4.52)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq289_HTML.gif we get that all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq290_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ71_HTML.gif
(4.53)
On the other hand, from Theorem 2.4 we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq291_HTML.gif are analytic functions, which implies the analticity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq292_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq293_HTML.gif Then we can conclude that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq294_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq295_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ72_HTML.gif
(4.54)

Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq296_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq297_HTML.gif which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq298_HTML.gif This completes the proof of Theorem 4.5.

Authors’ Affiliations

(1)
Laboratoire LAIG, Université du 08 Mai 1945

References

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Copyright

© Salah Badraoui. 2010

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.