Controllability for Sobolev type fractional integro-differential systems in a Banach space

Ahmed

Description

In this paper, by using compact semigroups and the Schauder fixed-point theorem, we study the sufficient conditions for controllability of Sobolev type fractional integro-differential systems in a Banach space. An example is provided to illustrate the obtained results. MSC: 26A33, 34G20, 93B05. In this paper, by using compact semigroups and the Schauder fixed-point theorem, we study the sufficient conditions for controllability of Sobolev type fractional integro-differential systems in a Banach space. An example is provided to illustrate the obtained results. MSC: 26A33, 34G20, 93B05.

Sujets

• Fractional calculus
• Controllability

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 15
• Langue: English

Extrait

AhmedAdvances in Difference Equations2012,2012:167
http://www.advancesindifferenceequations.com/content/2012/1/167

R E S E A R C H

Open Access

Controllability for Sobolev type fractional
integro-differential systems in a Banach space

*
Hamdy M Ahmed

*
Correspondence:
[e-mail]
Higher Institute of Engineering,
El-Shorouk Academy, P.O. Box 3
El-Shorouk City, Cairo, Egypt

Abstract
In this paper, by using compact semigroups and the Schauder fixed-point theorem,
we study the sufficient conditions for controllability of Sobolev type fractional
integro-differential systems in a Banach space. An example is provided to illustrate the
obtained results.
MSC:26A33; 34G20; 93B05
Keywords:fractional calculus; Sobolev type fractional integro-differential systems;
controllability; compact semigroup; mild solution; Schauder fixed-point theorem

1 Introduction
A Sobolev-type equation appears in a variety of physical problems such as flow of fluids
through fissured rocks, thermodynamics and propagation of long waves of small
amplitude (see [–]). Recently, there has been an increasing interest in studying the problem of
controllability of Sobolev type integro-differential systems. Balachandran and Dauer []
studied the controllability of Sobolev type integro-differential systems in Banach spaces.
Balachandran and Sakthivel [] studied the controllability of Sobolev type semilinear
integro-differential systems in Banach spaces. Balachandran, Anandhi and Dauer []
studied the boundary controllability of Sobolev type abstract nonlinear integro-differential
systems.
In this paper, we study the controllability of Sobolev type fractional integro-differential
systems in Banach spaces in the following form:

t s

cα
D Ex(t) +Ax(t) =Bu(t) +f t,x(t) +g t,s,x(s),H s,τ,x(τ)dτds,
 
t∈J= [,a],a> ,x() =x, (.)

whereEandAare linear operators with domain contained in a Banach spaceXand ranges

contained in a Banach spaceY. The control functionu(∙) is inL(J,U), a Banach space of
admissible control functions, withUas a Banach space.Bis a bounded linear operator
fromUintoY. The nonlinear operatorsf∈C(J×X,Y),H∈C(J×J×X,X) andg∈C(J×
J×X×X,Y) are all uniformly bounded continuous operators. The fractional derivative
cα
D,  <α<  is understood in the Caputo sense.

2 Preliminaries
In this section, we introduce preliminary facts which are used throughout this paper.

©2012 Ahmed; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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