Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq3_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq5_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq8_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq9_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq10_HTML.gif is a smooth bounded domain, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq11_HTML.gif , the diffusion matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq12_HTML.gif has semisimple and positive eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq15_HTML.gif is an open nonempty set, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq16_HTML.gif is the characteristic function of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq19_HTML.gif the system is approximately controllable on https://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:

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

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

We assume the following assumptions.

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

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

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

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

(H5) The distributed controls https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq35_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq36_HTML.gif is the first eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq37_HTML.gif with Dirichlet condition, or if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq38_HTML.gif , and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq41_HTML.gif and all open nonempty subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq42_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq43_HTML.gif the system is approximately controllable on https://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

https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq45_HTML.gif where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq47_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq48_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq50_HTML.gif , and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq51_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq52_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq53_HTML.gif Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq54_HTML.gif and https://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

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

with the initial data

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq57_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq59_HTML.gif has semi-simple and positive eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq61_HTML.gif then if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq62_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq63_HTML.gif is the first eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq64_HTML.gif ), the system is approximately controllable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq65_HTML.gif for all open nonempty subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq66_HTML.gif of https://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

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

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq86_HTML.gif the set of all the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq87_HTML.gif that have generalized derivatives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq88_HTML.gif for all https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ6_HTML.gif
(2.1)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq91_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq94_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq95_HTML.gif .

(i) All the eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq96_HTML.gif have finite multiplicity https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq98_HTML.gif .

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

(iii) Also, the eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq108_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq110_HTML.gif and compactly supported in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq111_HTML.gif .

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

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

Definition 2.2.

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

In particular, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq120_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq122_HTML.gif then it is dense in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq123_HTML.gif , and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq124_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq125_HTML.gif .

Proposition 2.3 (cf. [4]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq126_HTML.gif be a Hilbert separable space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq127_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq129_HTML.gif , with https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq132_HTML.gif Then

(a) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq134_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq135_HTML.gif continiuous for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq136_HTML.gif

(b) under the above condition (a), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq138_HTML.gif whose infinitesimal generator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq139_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq140_HTML.gif is connected, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq141_HTML.gif is a real function in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq142_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq143_HTML.gif on a nonempty open subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq144_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq145_HTML.gif in https://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)

    https://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

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

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

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

    Let https://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

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

where

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

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

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

If we put

https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq151_HTML.gif

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq152_HTML.gif in the following form:

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

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq154_HTML.gif is a bounded linear operator from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq155_HTML.gif into https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq157_HTML.gif -Semigroup

Theorem 4.1.

If https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq160_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ19_HTML.gif
(4.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ20_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ21_HTML.gif
(4.3)
Moreover, if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ22_HTML.gif
(4.4)
then the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq161_HTML.gif -semigoup https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq163_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq164_HTML.gif can be written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ24_HTML.gif
(4.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ25_HTML.gif
(4.7)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq165_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq167_HTML.gif Then, there exists a set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq168_HTML.gif of complementary projections on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq169_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ26_HTML.gif
(4.8)
If https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq172_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ27_HTML.gif
(4.9)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ28_HTML.gif
(4.10)
We have also
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ30_HTML.gif
(4.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ31_HTML.gif
(4.13)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq173_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ32_HTML.gif
(4.14)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq174_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq176_HTML.gif and a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq177_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq178_HTML.gif    for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq179_HTML.gif Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq180_HTML.gif is a strongly continious semigroup https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq183_HTML.gif as follows.
  1. (i)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq184_HTML.gif As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq185_HTML.gif , then there exist constants
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ33_HTML.gif
    (4.15)
     
hence, if we put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ34_HTML.gif
(4.16)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ35_HTML.gif
(4.17)
we easily obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ36_HTML.gif
(4.18)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq186_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq187_HTML.gif . If we put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ37_HTML.gif
(4.19)
then we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ38_HTML.gif
(4.20)

Therefore, the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq188_HTML.gif generates a strongly continuous semigroup https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq189_HTML.gif on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq192_HTML.gif -semigoup https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq194_HTML.gif and https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq198_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq200_HTML.gif , one has for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq201_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ41_HTML.gif
(4.23)
and for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq202_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq203_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq204_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq206_HTML.gif , then we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq207_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq208_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq210_HTML.gif , then we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ44_HTML.gif
(4.26)

Hence, from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq211_HTML.gif (4.26) we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq212_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq213_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq215_HTML.gif and every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq216_HTML.gif we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq218_HTML.gif and every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq219_HTML.gif we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ47_HTML.gif
(4.29)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ48_HTML.gif
(4.30)
To estimate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq220_HTML.gif we have in taking into account https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq221_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq222_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ50_HTML.gif
(4.32)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq224_HTML.gif . But we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq225_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq226_HTML.gif Then we get for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq227_HTML.gif that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ52_HTML.gif
(4.34)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ53_HTML.gif
(4.35)

and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq229_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ54_HTML.gif
(4.36)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ55_HTML.gif
(4.37)
and or every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq230_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ56_HTML.gif
(4.38)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ57_HTML.gif
(4.39)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ59_HTML.gif
(4.41)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq231_HTML.gif is defined by (4.17) and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq232_HTML.gif -semigoup https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq233_HTML.gif generated by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq235_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq237_HTML.gif and https://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

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq240_HTML.gif whenever the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq241_HTML.gif is densely embedded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq242_HTML.gif ; that is,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq243_HTML.gif if and only if
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq244_HTML.gif and all open subset https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq247_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq249_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq250_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq251_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq252_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq253_HTML.gif one has
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq254_HTML.gif and from this we get https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq256_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq257_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq258_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq259_HTML.gif , we divide https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq260_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq261_HTML.gif and we pass https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq262_HTML.gif we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq263_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq264_HTML.gif we divide https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq265_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq266_HTML.gif and we pass https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq267_HTML.gif and get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq268_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq269_HTML.gif , we divide https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq270_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq271_HTML.gif and we pass https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq272_HTML.gif and get https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq274_HTML.gif and we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq275_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq276_HTML.gif . Continuing this way we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq277_HTML.gif for all https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ66_HTML.gif
(4.48)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq279_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq280_HTML.gif -semigroup generated by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq282_HTML.gif Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ67_HTML.gif
(4.49)

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

Now, suppose for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq284_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq285_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq286_HTML.gif Then
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq287_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq288_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ70_HTML.gif
(4.52)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq289_HTML.gif we get that all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq290_HTML.gif
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq293_HTML.gif Then we can conclude that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq294_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq295_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_Equ72_HTML.gif
(4.54)

Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq296_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281238/MediaObjects/13661_2009_Article_911_IEq297_HTML.gif which implies that https://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

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Copyright

© Salah Badraoui. 2010

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