Open Access Research

Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions

Yanchao Zang* and Junping Li

Author Affiliations

School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410075, P.R. China

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Boundary Value Problems 2013, 2013:193  doi:10.1186/1687-2770-2013-193


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/193


Received:16 April 2013
Accepted:12 August 2013
Published:28 August 2013

© 2013 Zang and Li; 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

In this paper, the approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions and infinite delay in Hilbert spaces is studied. By using the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result.

MSC: 65C30, 93B05, 34K40, 34K45.

Keywords:
approximate controllability; fixed point principle; fractional impulsive neutral stochastic differential equations; mild solution; nonlocal conditions

1 Introduction

The purpose of this paper is to prove the existence and approximate controllability of mild solutions for a class of fractional impulsive neutral stochastic differential equations with nonlocal conditions described in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M2">View MathML</a> is the Caputo fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M3">View MathML</a>; the state variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M4">View MathML</a> takes values in the real separable Hilbert space H; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M5">View MathML</a> is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M7">View MathML</a>, in the Hilbert space H. The history <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M10">View MathML</a>, belongs to an abstract phase space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11">View MathML</a>. The control function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M12">View MathML</a> is given in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M13">View MathML</a>, U is a Hilbert space; B is a bounded linear operator from U into H. The functions f, h, g, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M14">View MathML</a> are appropriate functions to be specified later. The process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M15">View MathML</a> is a given U-valued Wiener process with a finite trace nuclear covariance operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M16">View MathML</a> defined on a complete probability space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M17">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M21">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M22">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M23">View MathML</a>, respectively. The initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M24">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M25">View MathML</a>-measurable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11">View MathML</a>-valued random variable independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M27">View MathML</a> with finite second moments.

In the past few decades, the theory of fractional differential equations has received a great deal of attention, and they play an important role in many applied fields, including viscoelasticity, electrochemistry, control, porous media, electromagnetic and so on. We refer the reader to the monographs of Kilbas et al.[1], Mill and Ross [2], Podlubny [3] and the references therein. There is also an extensive literature concerned with the fractional differential equations. For example, Benchohra et al. in [4] considered the VIP for a particular class of fractional neutral functional differential equations with infinite delay. Zhou in [5] discussed the existence and uniqueness for fractional neutral differential equations with infinite delay.

In practice, deterministic systems often fluctuate due to environmental noise. So it is important and necessary for us to discuss the stochastic differential systems. On the other hand, the control theory is one of the important topics in mathematics. Roughly speaking, controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. As a result of its widespread use, the controllability of stochastic or deterministic systems all have received extensive attention. Mahmudov [6] investigated the controllability of infinite dimensional linear stochastic systems, and in [7] Dauer and Mahmudov extended the results to semilinear stochastic evolution equations with finite delay. Park, Balasubramaniam and Kumaresan [8] gave the controllability of neutral stochastic functional infinite delay systems. Besides the environmental noise, sometimes, we have to consider the impulsive effects, which exist in many evolution processes, because the impulsive effects may bring an abrupt change at certain moments of time. For the literatures on controllability of stochastic system with impulsive effect, we can see [9-13].

However, to the best of our knowledge, it seems that little is known about approximate controllability of fractional impulsive neutral stochastic differential equations with infinite delay and nonlocal conditions. The aim of this paper is to study this interesting problem. The rest of the paper is organized as follows. In Section 2, we introduce some preliminaries such as definitions of fractional calculus and some useful lemmas. In Section 3, we prove our main results. Finally in Section 4, an example is given to demonstrate the application of our results.

2 Preliminaries

In this section, we introduce some notations and preliminary results, needed to establish our results. Throughout this paper, let U and H be two real separable Hilbert spaces, and we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M28">View MathML</a> the set of all linear bounded operators from U into H. For convenience, we will use the same notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M29">View MathML</a> to denote the norms in U, H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M30">View MathML</a>, and use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M31">View MathML</a> to denote the inner product of U and H without any confusion. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M32">View MathML</a> be a complete probability space with a filtration <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M33">View MathML</a> satisfying the usual conditions (i.e., it is increasing and right continuous, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M34">View MathML</a> contains all P-null sets). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M35">View MathML</a> be a Q-Wiener process defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M32">View MathML</a> with the covariance operator Q, that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M37">View MathML</a>

where Q is a positive, self-adjoint, trace class operator on U. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M38">View MathML</a> be the space of all Q-Hilbert-Schmidt operators from U to H with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M39">View MathML</a>

For the construction of stochastic integral in Hilbert space, see Da Prato and Zabczyk [14]. Let A be the infinitesimal generator of an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M40">View MathML</a> of uniformly bounded linear operators on H, and in this paper, we always assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6">View MathML</a> is compact.

Now, we present the abstract space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M43">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M44">View MathML</a> is a continuous function. The abstract phase space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11">View MathML</a> = {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M47">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M49">View MathML</a> is a bounded and measurable function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M51">View MathML</a>}. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M11">View MathML</a> is endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M53">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M54">View MathML</a> is a Banach space [15,16].

Now, we consider the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M55">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M56">View MathML</a> is the restriction of x to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M58">View MathML</a>. We endow a seminorm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M59">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M60">View MathML</a>, it is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M61">View MathML</a>

Lemma 2.1 (see [17])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M62">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M64">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M65">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M66">View MathML</a>.

Definition 2.1 The fractional integral of order α with the lower limit 0 for a function f is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M67">View MathML</a>

provided the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M68">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M69">View MathML</a> is the gamma function.

Definition 2.2 The Caputo derivative of order α with the lower limit 0 for a function f can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M70">View MathML</a>

Definition 2.3 A stochastic process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M71">View MathML</a> is called a mild solution of the system (1) if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M22">View MathML</a> is measurable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M73">View MathML</a>-adapted, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M74">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M75">View MathML</a> has càdlàg paths on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M76">View MathML</a> a.s., and satisfies the following integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M77">View MathML</a>

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M78">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M79">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M80">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M81">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M82">View MathML</a> is a probability density function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M83">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M84">View MathML</a>

Lemma 2.2[18]

The operatorsandhave the following properties:

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M89">View MathML</a>are linear and bounded operators, i.e., for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M90">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M91">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M93">View MathML</a>are strongly continuous, which means that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M94">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M95">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M96">View MathML</a>

(iii) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M89">View MathML</a>are also compact operators if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6">View MathML</a>is compact for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M97">View MathML</a>.

In order to study the approximate controllability for the fractional control system (1), we introduce the following linear fractional differential system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M102">View MathML</a>

(2)

The controllability operator associated with (2) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M103">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M105">View MathML</a> denote the adjoint of B and , respectively.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M107">View MathML</a> be the state value of (1) at terminal time T, corresponding to the control u and the initial value φ. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M108">View MathML</a> the reachable set of system (1) at terminal time T, its closure in H is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M109">View MathML</a>.

Definition 2.4 The system (1) is said to be approximately controllable on J if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M110">View MathML</a>.

Lemma 2.3[19]

The linear fractional control system (2) is approximately controllable onJif and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M111">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M112">View MathML</a>in the strong operator topology.

Lemma 2.4 ([18] Krasnoselskii’s fixed point theorem)

LetNbe a Banach space, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M113">View MathML</a>be a bounded closed and convex subset ofN, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M115">View MathML</a>be maps of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M113">View MathML</a>intoNsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M117">View MathML</a>for every pair<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M118">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M114">View MathML</a>is a contraction and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M115">View MathML</a>is completely continuous, then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M121">View MathML</a>has a solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M113">View MathML</a>.

3 Main results

In this section, we formulate sufficient conditions for the approximate controllability of system (1). For this purpose, we first prove the existence of solutions for system (1). Second, in Theorem 3.2, we shall prove that system (1) is approximately controllable under certain assumptions. In order to prove our main results, we need the following assumptions.

(H1) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M123">View MathML</a> are continuous, and there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M125">View MathML</a> such that the function satisfies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M126">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M127">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M63">View MathML</a>.

(H2) There exists a positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M131">View MathML</a>

(H3) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M132">View MathML</a> is continuous, and there exists continuous nondecreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M133">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M94">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M135">View MathML</a>

(H4) μ is continuous, and there exists some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M136">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M137">View MathML</a>

(H5) The linear stochastic system (2) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M138">View MathML</a>.

The following lemma is required to define the control function.

Lemma 3.1[6]

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M139">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M140">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M141">View MathML</a>.

Now, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M139">View MathML</a>, we define the control function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M144">View MathML</a>

Theorem 3.1Assume that the assumptions (H1)-(H4) hold. Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142">View MathML</a>, the system (1) has a mild solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M146">View MathML</a>, provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M147">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M148">View MathML</a>

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142">View MathML</a>, define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M150">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M151">View MathML</a>

We shall show that the operator Φ has a fixed point in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M60">View MathML</a>, which is the mild solution of (1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M154">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M155">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M156">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M157">View MathML</a>, and it is clear that x satisfies (1) if and only if z satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M158">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M159">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M160">View MathML</a>, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M161">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M162">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M163">View MathML</a> is a Banach space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M164">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M165">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166">View MathML</a>, for each r, is a bounded, closed subset of H. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167">View MathML</a>, by lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M168">View MathML</a>

For the sake of convenience, we divide the proof into several steps.

Step 1. We claim that there exists a positive number r such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M169">View MathML</a>. If this is not true, then, for each positive integer r, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M171">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M172">View MathML</a>, t may depending upon r. However, on the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M173">View MathML</a>

By using (H1)-(H4), Lemma 2.1 and Hölder’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M174">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M176">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M177">View MathML</a>

Dividing both sides by r and taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M178">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M179">View MathML</a>

which is a contradiction to our assumption. Thus, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M142">View MathML</a>, there exists some positive number r such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M169">View MathML</a>.

Next, we show that the operator Φ is condensing, for convenience, we decompose Φ as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M182">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M183">View MathML</a>

Step 2. We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M184">View MathML</a> is a contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M187">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M188">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M189">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M184">View MathML</a> is a contraction.

Step 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M191">View MathML</a> maps bounded sets into bounded sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M193">View MathML</a>

Therefore, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M195">View MathML</a>.

Step 4. The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M191">View MathML</a> is equicontinuous. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M198">View MathML</a>. Then, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M199">View MathML</a>

Noting the fact that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M200">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M201">View MathML</a> such that, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M202">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M205">View MathML</a>. Therefore, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M206">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M207">View MathML</a>

The right hand of the inequality above tends to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M209">View MathML</a>, hence the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M210">View MathML</a> is equicontinuous.

Step 5. The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M211">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M213">View MathML</a> be fixed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M214">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M217">View MathML</a>

Then from the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M218">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M219">View MathML</a> is relatively compact in H for every ϵ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M220">View MathML</a>. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M167">View MathML</a>, we can easily prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M222">View MathML</a> is convergent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M223">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M209">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M226">View MathML</a>, hence the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M211">View MathML</a> is also relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M166">View MathML</a>. Thus, by Arzela-Ascoli theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M191">View MathML</a> is completely continuous. Consequently, from Lemma 2.4, Φ has a fixed point, which is a mild solution of (1). □

Theorem 3.2Assume that (H1)-(H5) are satisfied, and the conditions of Theorem 3.1 hold. Further, if the functionsfandgare uniformly bounded, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M6">View MathML</a>is compact, then the system (1) is approximately controllable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M146">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M232">View MathML</a> be a solution of (1), then we can easily get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M233">View MathML</a>

In view of the assumptions that f and g are uniformly bounded on J, hence, there is a subsequence still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M235">View MathML</a>, which converges weakly to say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M236">View MathML</a> in H, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M237">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M30">View MathML</a>. On the other hand, by assumption (H5), the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M239">View MathML</a> strongly as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M112">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M241">View MathML</a>, and, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M242">View MathML</a>. Thus, the Lebesgue dominated convergence theorem and the compactness of yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M244">View MathML</a>

This gives the approximate controllability of (1), the proof is complete. □

4 An example

As an application, we consider an impulsive neutral stochastic partial differential equation with the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M245">View MathML</a>

(3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M248">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M249">View MathML</a>. To study the approximate controllability of (3), assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M250">View MathML</a> is measurable and continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M251">View MathML</a> and thus bounded by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M252">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M253">View MathML</a> is measurable and continuous with finite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M254">View MathML</a>.

We define the operator A by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M255">View MathML</a> with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M256">View MathML</a>. It is well known that A generates an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M257">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M94">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M261">View MathML</a> , is the orthogonal set of eigenvectors of A.

Define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M263">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M264">View MathML</a>

With the choice of A, h, f, g, (3) can be rewritten as the abstract form of system (1). Thus, under the appropriate conditions on the functions h, f, g and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/193/mathml/M14">View MathML</a> as those in (H1)-(H5), system (3) is approximately controllable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript.

Acknowledgements

We are very grateful to the anonymous referee and the associate editor for their careful reading and helpful comments. This work was substantially supported by the National Natural Sciences Foundation of China (No. 11071259), Research Fund for the Doctoral Program of Higher Education of China (No. 20110162110060).

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