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 some nonlinear systems in Banach spaces

Nazim I Mahmudov

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

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Boundary Value Problems 2013, 2013:50  doi:10.1186/1687-2770-2013-50


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/50


Received:8 January 2013
Accepted:25 February 2013
Published:13 March 2013

© 2013 Mahmudov; 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, abstract results concerning the approximate controllability of semilinear evolution systems in a separable reflexive Banach space are obtained. An approximate controllability result for semilinear systems is obtained by means of Schauder’s fixed-point theorem under the compactness assumption of the linear operator involved. It is also proven that the controllability of the linear system implies the controllability of the associated semilinear system. Then the obtained results are applied to derive sufficient conditions for the approximate controllability of the semilinear fractional integrodifferential equations in Banach spaces and heat equations.

1 Introduction

The problems of controllability of infinite dimensional nonlinear (fractional) systems were studied widely by many authors; see [1-6] and the references therein. The approximate controllability of nonlinear systems when the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a>, generated by A is compact has been studied by many authors. The results of Zhou [6] and Naito [7] give sufficient conditions on B with finite dimensional range or necessary and sufficient conditions based on more strict assumptions on B. Li and Yong in [8] studied the same problem assuming the approximate controllability of the associated linear system under arbitrary perturbation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M3">View MathML</a>. Bian [9] investigated the approximate controllability for a class of semilinear systems. For abstract nonlinear systems, Carmichael and Quinn [10] used the Banach fixed-point theorem to obtain a local exact controllability in the case of nonlinearities with small Lipschitz constants. Zhang [11] studied the local exact controllability of semilinear evolution systems. Naito [7] and Seidman [12] used Schauder’s fixed-point theorem to prove invariance of the reachable set under nonlinear perturbations. Other related abstract results were given by Lasiecka and Triggiani [13].

In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [1-5,7,9-47]). A pioneering work has been reported by Bashirov and Mahmudov [17], Dauer and Mahmudov [28] and Mahmudov [31]. Sakthivel et al.[40] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. Klamka [23-26] derived a set of sufficient conditions for constrained local controllability near the origin for semilinear dynamical control systems. Wang and Zhou [3] investigated the complete controllability of fractional evolution systems without assuming the compactness of characteristic solution operators. Sukavanam and Kumar [47] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and Schauder’s fixed-point theorem.

Consider an abstract semilinear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M4">View MathML</a>

(1)

and define the following sets:

Here Y, X are separable reflexive Banach spaces and V is a Hilbert space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M9">View MathML</a> is a nonlinear operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M11">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M12">View MathML</a> is the set of points Qy, where y is a solution of (1), attainable from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13">View MathML</a>. The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M14">View MathML</a> is the set of points Qz, where z is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M15">View MathML</a>

(2)

reachable from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13">View MathML</a>. One can see that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a> the control

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M19">View MathML</a>

(3)

transfers equation (2) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13">View MathML</a> to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M21">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M22">View MathML</a>. It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M23">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M24">View MathML</a>

in the strong operator topology as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>, see [30]. Thus, the control (3) transfers system (2) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M10">View MathML</a> to a small neighborhood of an arbitrary point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M23">View MathML</a>.

The same idea is now used to investigate the controllability of semilinear system (1). To do so, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, consider a nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M32">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M32">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M34">View MathML</a>

(4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M35">View MathML</a>

One can see that if the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M37">View MathML</a>, then the control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M38">View MathML</a> steers control system (1) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M13">View MathML</a> to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M40">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a>. We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M42">View MathML</a> is close to h provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M43">View MathML</a> converges strongly to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>. Therefore, to prove the approximate controllability of (1), for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, we have to seek for a solution of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M47">View MathML</a>

(5)

It is clear that the fixed points of the nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> are the solutions of nonlinear control system (5) and vice versa.

To the best of our knowledge, the approximate controllability problem for semilinear abstract systems in Banach spaces has not been investigated yet. Motivated by this consideration, in this paper we study the approximate controllability of semilinear abstract systems in Banach spaces. The approximate controllability of (1) is derived under the compactness assumption of the linear operator involved. We prove that the approximate controllability of linear system (2) implies the approximate controllability of semilinear system (1) under some assumptions. On the other hand, it is known that if the operator L is compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M49">View MathML</a>, that is, linear system (2) is not exactly controllable. Thus the analogue of this result is not true for exact controllability, that is why we investigate just the approximate controllability. Notice that a similar result for semilinear equations in Hilbert spaces was obtained by Dauer and Mahmudov [27].

In Section 2 an abstract result concerning the approximate controllability of semilinear system (1) is obtained. It is proven that the controllability of (2) implies the controllability of (1). Finally, these abstract results are applied to the approximate controllability of semilinear fractional integrodifferential equations. These equations serve as an abstract formulation of a fractional partial integrodifferential equation arising in various applications such as viscoelasticity, heat equations and many other physical phenomena.

2 Approximate controllability of semilinear systems

Let X be a separable reflexive Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50">View MathML</a> stand for its dual space with respect to the continuous pairing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M51">View MathML</a>. We may assume, without loss of generality, that X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50">View MathML</a> are smooth and strictly convex by virtue of the renorming theorem (see, for example, [8,48]). In particular, this implies that the duality mapping J of X into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50">View MathML</a> given by the following relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M54">View MathML</a>

is bijective, demicontinuous, i.e., continuous from X with a strong topology into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M50">View MathML</a> with weak topology and strictly monotonic. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M56">View MathML</a> is also a duality mapping.

An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M57">View MathML</a> is symmetric if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M58">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M59">View MathML</a>. It is easy to see that Γ is linear and continuous. Γ is nonnegative if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M60">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M61">View MathML</a>.

Lemma 1[31]

For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a>, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M64">View MathML</a>

(6)

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M65">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M66">View MathML</a>

(7)

Theorem 2[31]

Let Γ be a symmetric operator. Then the following three conditions are equivalent:

(i) Γ is positive, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M67">View MathML</a>for all nonzero<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M61">View MathML</a>.

(ii) For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M70">View MathML</a>converges to zero as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>in the weak topology, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M72">View MathML</a>is a solution of equation (6).

(iii) For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M74">View MathML</a>strongly converges to zero as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>.

We impose the following assumptions:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M76">View MathML</a> is continuous and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M77">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M79">View MathML</a>.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M80">View MathML</a> is compact.

(A3) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M82">View MathML</a> strongly converges to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>.

Note that the condition (A3) holds if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M84">View MathML</a>, i.e., system (2) is approximately controllable.

Definition 3 System (1) is approximately controllable if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M85">View MathML</a>

Theorem 4Assume (A1)-(A3) hold. Then semilinear system (1) is approximate controllability.

Proof Step 1. Show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M32">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a>. For our convenience, let us introduce the following notation:

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M90">View MathML</a>. Then by (7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M91">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M92">View MathML</a>

Thus we proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M94">View MathML</a> into itself. On the other hand, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M96">View MathML</a> is relatively compact. By Schauder’s fixed-point theorem, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M31">View MathML</a> has a fixed point in the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M99">View MathML</a>.

Step 2. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M23">View MathML</a>. By Step 1, the operator (4) has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M101">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M102">View MathML</a> satisfies (5) and, moreover, it follows that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M104">View MathML</a>

(8)

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M105">View MathML</a> is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M106">View MathML</a>

(9)

By the assumptions (A1) and (A2), the operator F is continuous bounded and L is compact. So, there exists a subsequence, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M107">View MathML</a>, which weakly converges to say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M109">View MathML</a> strongly in Y as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>. From (7) and strong convergence of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M111">View MathML</a>, it is easy to see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M112">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M18">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M114">View MathML</a>

Then we can extract a subsequence, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M115">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M116">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M117">View MathML</a>. Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M118">View MathML</a> to equation (9) and taking the limit, we obtain

since Γ is positive. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M120">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M25">View MathML</a>. Now, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M122">View MathML</a> to equation (9), dividing through by ε and taking the limit, we obtain

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M124">View MathML</a>, consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M125">View MathML</a>. The theorem is proved. □

3 Fractional integrodifferential equations

The purpose of this section is to establish sufficient conditions for the approximate controllability of certain classes of abstract fractional integrodifferential equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M126">View MathML</a>

(10)

where the state variable x takes values in a separable reflexive Banach space X; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M127">View MathML</a> is the Caputo fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M128">View MathML</a>; A is the infinitesimal generator of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M129">View MathML</a> semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M130">View MathML</a> of bounded operators on X; the control function u is given in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M131">View MathML</a>, U is a Hilbert space; B is a bounded linear operator from U into X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M134">View MathML</a> are continuous bounded functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M135">View MathML</a>.

Definition 5 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/2013/1/50/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M136">View MathML</a>

provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M137">View MathML</a>, where γ is the gamma function.

Definition 6 Riemann-Liouville derivative of order α with the lower limit 0 for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M138">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M139">View MathML</a>

Definition 7 The Caputo derivative of order α for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M140">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M141">View MathML</a>

Remark 8

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M142">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M143">View MathML</a>

(2) The Caputo derivative of a constant is equal to zero.

(3) If f is an abstract function with values in X, then the integrals which appear in the above definitions are taken in Bochner’s sense.

For basic facts about fractional integrals and fractional derivatives, one can refer to [49].

In order to define the concept of a mild solution for problem (10), we associate problem (10) to the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M144">View MathML</a>

(11)

where

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M146">View MathML</a> is a probability density function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M147">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M150">View MathML</a>.

Lemma 9[34]

For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M151">View MathML</a>, the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M152">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M153">View MathML</a>are linear compact and bounded operators, i.e., for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M155">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M156">View MathML</a>.

Definition 10 A solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M157">View MathML</a> is said to be a mild solution of (10) if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M158">View MathML</a> and the integral equation (11) is satisfied.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M159">View MathML</a> be the state value of (10) at terminal time b corresponding to the control u and the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M160">View MathML</a>. Introduce the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M161">View MathML</a>, which is called the reachable set of system (10) at terminal time b, its closure in X is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M162">View MathML</a>.

Definition 11 System (10) is said to be approximately controllable on J if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M162">View MathML</a>, that is, given an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M164">View MathML</a>, it is possible to steer from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M160">View MathML</a> to within a distance ϵ from all points in the state space X at time b.

Consider the following linear fractional differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M166">View MathML</a>

(12)

The approximate controllability for linear fractional system (12) is a natural generalization of the approximate controllability of a linear first-order control system. It is convenient at this point to introduce the controllability operator associated with (12) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M167">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M168">View MathML</a> denotes the adjoint of B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M169">View MathML</a> is the adjoint of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M170">View MathML</a>. It is straightforward that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M171">View MathML</a> is a linear bounded operator. By Theorem 2, linear fractional control system (12) is approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M172">View MathML</a> if and only if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M174">View MathML</a> converges strongly to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M175">View MathML</a>.

Proposition 12If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a>, are compact operators and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M178">View MathML</a>, then the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M179">View MathML</a>

is compact from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M180">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M181">View MathML</a>.

Proof According to the infinite dimensional version of the Ascoli-Arzela theorem, we need to show that

(i) for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M182">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M183">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M184">View MathML</a>;

(ii) for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M185">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M186">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M187">View MathML</a>

To prove (i), fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M188">View MathML</a> and define for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M186">View MathML</a> operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M191">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M192">View MathML</a> into X

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M193">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a>, is a compact operator, the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M191">View MathML</a> are compact. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M197">View MathML</a>

One can estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M199">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M200">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M201">View MathML</a>

where we have used the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M202">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M203">View MathML</a> in the operator norm so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M204">View MathML</a> is compact and (i) follows immediately.

To prove (ii), note that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M206">View MathML</a>, we have

Applying the Hölder inequality, we obtain

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M209">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M210">View MathML</a>. On the other hand, the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a> (and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M213">View MathML</a>), implies the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a>, in the uniform operator topology. Then, by the Lebesque dominated convergence theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M216">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M210">View MathML</a>. Thus the proof of (ii), and therefore the proof of the proposition, is complete. □

Theorem 13Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a>, is compact and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M220">View MathML</a>. Then system (10) is approximately controllable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M221">View MathML</a>if the corresponding linear system is approximately controllable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M222">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M224">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M225">View MathML</a>. Define the linear operators Q, L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M226">View MathML</a> and the nonlinear operator F by

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M11">View MathML</a>. It is easy to see that by Proposition 12 all the conditions of Theorem 4 are satisfied and (10) is approximately controllable. This completes the proof. □

4 Application

Consider the partial differential system of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M230">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M128">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M236">View MathML</a> are continuous and uniformly bounded. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M237">View MathML</a> be defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M238">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M241">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M242">View MathML</a> be an operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M243">View MathML</a> with the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M244">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M245">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M248">View MathML</a> . It is known that A generates a compact semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M2">View MathML</a>, in X and is given by

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M252">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M253">View MathML</a> implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M254">View MathML</a>

Now if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M255">View MathML</a> for all n, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M256">View MathML</a> for all n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M257">View MathML</a>. Therefore, the associated linear system is approximately controllable provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M258">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M259">View MathML</a> . Because of the compactness of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M194">View MathML</a> (and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M262">View MathML</a>) generated by A, the associated linear system of (13) is not completely controllable but it is approximately controllable. Hence, according to Theorem 13, system (13) will be approximately controllable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/50/mathml/M221">View MathML</a>.

5 Conclusion

In this paper, abstract results concerning the approximate controllability of semilinear evolution systems in a separable reflexive Banach space are obtained. An approximate controllability result for semilinear systems is obtained by means of Schauder’s fixed-point theorem under the compactness assumption. It is also proven that the controllability of the linear system implies the controllability of the associated semilinear system. Then the obtained results are applied to derive sufficient conditions for the approximate controllability of the semilinear fractional integrodifferential equations in Banach spaces. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a wide class of fractional deterministic and stochastic differential equations.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

References

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