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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions

NI Mahmudov* and S Zorlu

Author Affiliations

Eastern Mediterranean University, via Mersin 10, Gazimagusa, T.R. North Cyprus, Turkey

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Boundary Value Problems 2013, 2013:118  doi:10.1186/1687-2770-2013-118

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/118


Received:12 December 2012
Accepted:22 March 2013
Published:8 May 2013

© 2013 Mahmudov and Zorlu; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss the approximate controllability of nonlinear fractional integro-differential system under the assumptions that the corresponding linear system is approximately controllable. Using the fixed-point technique, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional integro-differential equations are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

1 Introduction

Controllability is one of the fundamental concepts in mathematical control theory, which plays an important role in control systems. The controllability of nonlinear systems represented by evolution equations or inclusions in abstract spaces and qualitative theory of fractional differential equations has been extensively studied by several authors. An extensive list of these publications can be found in [1-44] and the references therein. Recently, the approximate controllability for various kinds of (fractional) differential equations has generated considerable interest. A pioneering work on the approximate controllability of deterministic and stochastic systems has been reported by Bashirov and Mahmudov [5], Dauer and Mahmudov [8] and Mahmudov [10]. Sakthivel et al.[28] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. On the other hand, the fractional differential equation has gained more attention due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Yan [45] derived a set of sufficient conditions for the controllability of fractional-order partial neutral functional integro-differential inclusions with infinite delay in Banach spaces. Debbouche and Baleanu [1] established the controllability result for a class of fractional evolution nonlocal impulsive quasi-linear delay integro-differential systems in a Banach space using the theory of fractional calculus and fixed point technique. However, there exists only a limited number of papers on the approximate controllability of the fractional nonlinear evolution systems. Sakthivel et al.[28] studied the approximate controllability of deterministic semilinear fractional differential equations in Hilbert spaces. Wang [40] investigated the nonlocal controllability of fractional evolution systems. Surendra Kumar and Sukavanam [33] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order using the contraction principle and the Schauder fixed-point theorem. More recently, Sakthivel et al.[27] derived a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations.

In this paper, we discuss the approximate controllability of nonlinear fractional integro-differential system under the assumption that the corresponding linear system is approximately controllable. We consider the following fractional integro-differential control system involving nonlocal conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M1">View MathML</a>

(1)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M4">View MathML</a>, stands for the Caputo fractional derivative of order β, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M8">View MathML</a> are given functions to be specified later. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M9">View MathML</a> is the infinitesimal generator of a compact analytic semigroup of bounded linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11">View MathML</a>, on a real Hilbert space X. B is a linear bounded operator from a real Hilbert space U to X.

The rest of this paper is organized as follows. In Section 2, we give some preliminary results on the fractional powers of the generator of an analytic compact semigroup and introduce the mild solution of system (1). In Section 3, we study the existence of mild solutions for system (1) under the feedback control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M12">View MathML</a> defined in (5). We show that the control system (1) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M13">View MathML</a> provided that the corresponding linear system is approximately controllable. Finally, an example is given to demonstrate the applicability of our result.

2 Preliminaries

In this section, we introduce some facts about the fractional powers of the generator of a compact analytic semigroup, the Caputo fractional derivative that are used throughout this paper.

We assume that X is a Hilbert space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M15">View MathML</a> be the Banach space of continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16">View MathML</a> into X with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M17">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M18">View MathML</a>. In this paper, we also assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M19">View MathML</a> is the infinitesimal generator of a compact analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>, of uniformly bounded linear operator in X, that is, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M22">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M23">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11">View MathML</a>. Without loss of generality, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M25">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M26">View MathML</a> is the resolvent set of A. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M27">View MathML</a>, we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M28">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M29">View MathML</a>

It follows that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M28">View MathML</a> is an injective continuous endomorphism of X. Hence we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M31">View MathML</a>, which is a closed bijective linear operator in X. It can be shown that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32">View MathML</a> has dense domain and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M33">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M34">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M35">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M37">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M38">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M39">View MathML</a>, I is the identity in X. (For proofs of these facts, we refer to the literature [15,20,22].)

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a> the Hilbert space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M41">View MathML</a> equipped with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M42">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M43">View MathML</a>, which is equivalent to the graph norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M45">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M34">View MathML</a> (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M47">View MathML</a> ) and the embedding is continuous. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32">View MathML</a> has the following basic properties.

Lemma 1[42]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M10">View MathML</a>have the following properties.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M51">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M53">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M54">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11">View MathML</a>.

(iii) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M58">View MathML</a>is bounded inXand there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M59">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M60">View MathML</a>

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M28">View MathML</a>is a bounded linear operator for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M62">View MathML</a>.

Let us recall the following known definitions of fractional calculus. For more details, see [43,44].

Definition 2 The fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M27">View MathML</a> with the lower limit 0 for a function f is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M64">View MathML</a>

provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M65">View MathML</a>, where Γ is the gamma function.

Definition 3 The Caputo derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M27">View MathML</a> with the lower limit 0 for a function f can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M67">View MathML</a>

The Caputo derivative of a constant is equal to zero. If f is an abstract function with values in X then the integrals which appear in Definitions 2 and 3 are taken in Bochner’s sense.

According to Definitions 2 and 3, it is suitable to rewrite the problem (1) in the equivalent integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M68">View MathML</a>

(2)

provided that the integral in (2) exists. Applying the Laplace transform

to (2) and using the method similar to that used in [38] we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M70">View MathML</a>

where

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M72">View MathML</a> is a probability density function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M73">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M76">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77">View MathML</a>, we define two families <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M79">View MathML</a> of operators by

respectively.

The following lemma follows from the results given in [37-39].

Lemma 4The operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M82">View MathML</a>have the following properties.

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11">View MathML</a>, and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M84">View MathML</a>, we have the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M86">View MathML</a>are linear and bounded operators, i.e. for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M88">View MathML</a>

(ii) The operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M90">View MathML</a>are strongly continuous for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M11">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93">View MathML</a>are norm continuous inXfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>.

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93">View MathML</a>are compact operators inXfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>.

(v) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>, the restriction of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>and the restriction of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>are norm continuous.

(vi) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a>, the restriction of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M85">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>and the restriction of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M93">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>are compact operators in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>.

(vii) For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M110">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M111">View MathML</a>

In this paper, we adopt the following definition of mild solution of equation (1).

Definition 5 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M112">View MathML</a> is said to be a mild solution of (1) if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M113">View MathML</a> the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M114">View MathML</a>

(3)

is satisfied.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M115">View MathML</a> is bounded if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M116">View MathML</a>. In what follows, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M117">View MathML</a>.

3 Approximate controllability

In this section, we state and prove conditions for the approximate controllability of semilinear fractional control integro-differential systems. To do this, we first prove the existence of a fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M118">View MathML</a> defined below using Krasnoselskii’s fixed-point theorem. Secondly, in Theorem 11, we show that under the uniform boundedness of f and g the approximate controllability of fractional systems (1) is implied by the approximate controllability of the corresponding linear system (4).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M119">View MathML</a> be the state value of (1) at terminal time T corresponding to the control u and the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M120">View MathML</a>. Introduce the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M121">View MathML</a>, which is called the reachable set of system (1) at terminal time T, its closure in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M123">View MathML</a>.

Definition 6 The system (1) is said to be approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M124">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M125">View MathML</a>, that is, given an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126">View MathML</a> it is possible to steer from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M120">View MathML</a> to within a distance ε from all points in the state space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M128">View MathML</a> at time T.

Consider the following linear fractional differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M129">View MathML</a>

(4)

The approximate controllability for linear fractional system (4) is a natural generalization of approximate controllability of linear first order control system [9,10,12]. It is convenient at this point to introduce the controllability and resolvent operators associated with (4) as

respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M131">View MathML</a> denotes the adjoint of B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M132">View MathML</a> is the adjoint of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M86">View MathML</a>. It is straightforward that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M134">View MathML</a> is a linear bounded operator.

Theorem 7[10]

LetZbe a separable reflexive Banach space and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M135">View MathML</a>stands for its dual space. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M136">View MathML</a>is symmetric. Then the following two conditions are equivalent:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M137">View MathML</a>is positive, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M138">View MathML</a>for all nonzero<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M139">View MathML</a>.

2. For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M140">View MathML</a>strongly converges to zero as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141">View MathML</a>. Here, Jis the duality mapping ofZinto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M135">View MathML</a>.

Lemma 8The linear fractional control system (4) is approximately controllable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M144">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141">View MathML</a>in the strong operator topology.

Proof The lemma is a straightforward consequence of Theorem 7. Indeed, the system (4) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M146">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M147">View MathML</a> for all nonzero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M77">View MathML</a>, see [7]. By Theorem 7, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M149">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M151">View MathML</a>. □

Remark 9 Notice that positivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M134">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M153">View MathML</a>. In other words, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M154">View MathML</a>, approximate controllability of the linear system (4) is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M156">View MathML</a>.

Before proving the main results, let us first introduce our basic assumptions.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M157">View MathML</a> are continuous and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M158">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M159">View MathML</a> and functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M161">View MathML</a> such that

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M163">View MathML</a> is a Lipschitz function with Lipschitz constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M164">View MathML</a>.

(Hc) The linear system (4) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M124">View MathML</a>.

Using the hypothesis (Hc), for an arbitrary function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M166">View MathML</a>, we choose the feedback control function as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M167">View MathML</a>

(5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M168">View MathML</a>, where r is a positive constant. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M169">View MathML</a> is clearly a bounded closed and convex subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M170">View MathML</a>. We will show that when using the above control the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M171">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M172">View MathML</a>

where

has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M174">View MathML</a>.

Theorem 10Let the assumptions (H1) and (H2) be satisfied. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M175">View MathML</a>, the fractional Cauchy problem (1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M176">View MathML</a>has at least one mild solution provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M177">View MathML</a>

(6)

where

Proof It is easy to see that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126">View MathML</a> the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M118">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M181">View MathML</a> into itself.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M183">View MathML</a>. Using assumption (H1) yield the following estimations,

and

(7)

From (6) and the assumption (H2), it follows that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M187">View MathML</a> such that

(8)

Therefore, from (7) and (8), it follows that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M187">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M191">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M192">View MathML</a>. Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M126">View MathML</a> the fractional Cauchy problem (1) with the control (5) has a mild solution if and only if the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M194">View MathML</a> has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M195">View MathML</a>.

In what follows, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M197">View MathML</a> satisfy the conditions of Krasnoselskii’s fixed-point theorem. From (H2) and (6), we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M196">View MathML</a> is a contraction. Next, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199">View MathML</a> is completely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M200">View MathML</a>.

Step 1: We first prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M195">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M203">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M204">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M205">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M170">View MathML</a>. Therefore, it follows from the continuity of f, g and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M207">View MathML</a> that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208">View MathML</a>,

Also, by (H1), we see that

Since

using the Lebesgue dominated convergence theorem that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208">View MathML</a>, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M213">View MathML</a>

implying that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M214">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M205">View MathML</a>. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M200">View MathML</a>.

Step 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199">View MathML</a> is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M219">View MathML</a>.

For the sake of brevity, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M220">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208">View MathML</a> be fixed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M222">View MathML</a> be small enough. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M223">View MathML</a>, we define the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M224">View MathML</a>

Therefore, from Lemma 4, we see that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M225">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M226">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>. Since

approaches to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M229">View MathML</a>, using the total boundedness, we conclude that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M208">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M231">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>.

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M233">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M234">View MathML</a> small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M235">View MathML</a>

where

Therefore, it follows from (H1) and Lemma 4 that

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M238">View MathML</a>

from which it is easy to see that all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M240">View MathML</a>, tend to zero independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M241">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M243">View MathML</a>. Thus, we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M244">View MathML</a>

and the limit is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M245">View MathML</a>. The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M246">View MathML</a> is trivial. Consequently, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M247">View MathML</a> is equicontinuous. Now applying the Arzela-Ascoli theorem, it results that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M199">View MathML</a> is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M195">View MathML</a>.

Therefore, applying Krasnoselskii’s fixed-point theorem, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M118">View MathML</a> has a fixed point, which gives rise to a mild solution of Cauchy problem (1) with control given in (5). This completes the proof. □

Theorem 11Let the assumptions (H1), (H2) and (Hc) be satisfied. Moreover, assume the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M251">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M163">View MathML</a>are bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M253">View MathML</a>. Then the semilinear fractional system (3) is approximately controllable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M254">View MathML</a>.

Proof It is clear that all assumptions of Theorem 10 are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M255">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M256">View MathML</a> be a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M257">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M169">View MathML</a>. Any fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M257">View MathML</a> is a mild solution of (3) under the control

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M260">View MathML</a>

and satisfies the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M261">View MathML</a>

(9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M262">View MathML</a>

Moreover, by the boundedness of the functions f and g and Dunford-Pettis theorem, we have that the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M264">View MathML</a> are weakly compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M265">View MathML</a>, so there are subsequences still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M266">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M267">View MathML</a>, that weakly converge to, say, f and g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M265">View MathML</a>. On the other hand, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M269">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M270">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M271">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M2">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M273">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M274">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141">View MathML</a> because of compactness of the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M276">View MathML</a>

Then from (9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M277">View MathML</a>

(10)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M141">View MathML</a>. This proves the approximate controllability of (1). □

4 Applications

Example 1 As an application to Theorem 11, we study the following simple example. Consider a control system governed by the fractional partial differential equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M279">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M282">View MathML</a>.

Let us take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M283">View MathML</a> and define the operator A by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M284">View MathML</a> with the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M285">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M286">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M289">View MathML</a> . Clearly −A generates a compact analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M21">View MathML</a> in X and it is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M292">View MathML</a>

Clearly, the assumption (H1) is satisfied. On the other hand, it can be easily seen that the deterministic linear system corresponding to (11) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16">View MathML</a>; see [12].

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M294">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M295">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M296">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M297">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M298">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M299">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M300">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M301">View MathML</a> satisfies the following conditions:

1. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M303">View MathML</a> are continuous and uniformly bounded.

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M304">View MathML</a>.

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M281">View MathML</a> is continuously differentiable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M306">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M307">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M308">View MathML</a>, the Mittag-Leffler special function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M309">View MathML</a>

Therefore,

Define

Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M312">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M313">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M314">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M315">View MathML</a> is bounded and Lipschitz continuous. On the other hand, it is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M316">View MathML</a> are continuous.

Next, we show that the linear system corresponding to (11) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M318">View MathML</a> is defined as follows:

By Remark 9, the linear system corresponding to (11) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M321">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M322">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M323">View MathML</a>. This follows from the representation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M324">View MathML</a>.

Now, we note that the problem (11) can be reformulated as the abstract problem. Thus, by Theorem 11, the system (11) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M16">View MathML</a>, provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/118/mathml/M326">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the reviewers for their valuable comments and helpful suggestions that improved the note’s quality.

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