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

Boundary Value Problems20132013:118

DOI: 10.1186/1687-2770-2013-118

Received: 12 December 2012

Accepted: 22 March 2013

Published: 8 May 2013

Abstract

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

1 Introduction

Controllability is one of the fundamental concepts in mathematical control theory, which plays an important role in control systems. The controllability of nonlinear systems represented by evolution equations or inclusions in abstract spaces and qualitative theory of fractional differential equations has been extensively studied by several authors. An extensive list of these publications can be found in [144] and the references therein. Recently, the approximate controllability for various kinds of (fractional) differential equations has generated considerable interest. A pioneering work on the approximate controllability of deterministic and stochastic systems has been reported by Bashirov and Mahmudov [5], Dauer and Mahmudov [8] and Mahmudov [10]. Sakthivel et al. [28] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. On the other hand, the fractional differential equation has gained more attention due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Yan [45] derived a set of sufficient conditions for the controllability of fractional-order partial neutral functional integro-differential inclusions with infinite delay in Banach spaces. Debbouche and Baleanu [1] established the controllability result for a class of fractional evolution nonlocal impulsive quasi-linear delay integro-differential systems in a Banach space using the theory of fractional calculus and fixed point technique. However, there exists only a limited number of papers on the approximate controllability of the fractional nonlinear evolution systems. Sakthivel et al. [28] studied the approximate controllability of deterministic semilinear fractional differential equations in Hilbert spaces. Wang [40] investigated the nonlocal controllability of fractional evolution systems. Surendra Kumar and Sukavanam [33] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order using the contraction principle and the Schauder fixed-point theorem. More recently, Sakthivel et al. [27] derived a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations.

In this paper, we discuss the approximate controllability of nonlinear fractional integro-differential system under the assumption that the corresponding linear system is approximately controllable. We consider the following fractional integro-differential control system involving nonlocal conditions,
D t β C x ( t ) = A x ( t ) + f ( t , x ( t ) ) + 0 t K ( t s ) g ( s , x ( s ) ) d s + B u ( t ) , x ( 0 ) = x 0 + h ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ1_HTML.gif
(1)

in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif, where D t β C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq2_HTML.gif, 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq3_HTML.gif, stands for the Caputo fractional derivative of order β, and f : [ 0 , T ] × X α X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq4_HTML.gif, g : [ 0 , T ] × X α X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq5_HTML.gif, K : [ 0 , T ] R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq6_HTML.gif, h : C ( [ 0 , T ] ; X α ) X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq7_HTML.gif are given functions to be specified later. Here, ( A , D ( A ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq8_HTML.gif is the infinitesimal generator of a compact analytic semigroup of bounded linear operators S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq9_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq10_HTML.gif, on a real Hilbert space X. B is a linear bounded operator from a real Hilbert space U to X.

The rest of this paper is organized as follows. In Section 2, we give some preliminary results on the fractional powers of the generator of an analytic compact semigroup and introduce the mild solution of system (1). In Section 3, we study the existence of mild solutions for system (1) under the feedback control u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq11_HTML.gif defined in (5). We show that the control system (1) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq12_HTML.gif provided that the corresponding linear system is approximately controllable. Finally, an example is given to demonstrate the applicability of our result.

2 Preliminaries

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

We assume that X is a Hilbert space with norm : = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq13_HTML.gif. Let C ( [ 0 , T ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq14_HTML.gif be the Banach space of continuous functions from [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq15_HTML.gif into X with the norm x = sup t [ 0 , T ] x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq16_HTML.gif, here x C ( [ 0 , T ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq17_HTML.gif. In this paper, we also assume that A : D ( A ) X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq18_HTML.gif is the infinitesimal generator of a compact analytic semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq9_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif, of uniformly bounded linear operator in X, that is, there exists M > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq20_HTML.gif such that S ( t ) M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq21_HTML.gif for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq10_HTML.gif. Without loss of generality, let 0 ρ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq22_HTML.gif, where ρ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq23_HTML.gif is the resolvent set of A. Then for any α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq24_HTML.gif, we can define A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq25_HTML.gif by
A α : = 1 Γ ( α ) 0 t α 1 S ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equa_HTML.gif

It follows that each A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq25_HTML.gif is an injective continuous endomorphism of X. Hence we can define A α : = ( A α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq26_HTML.gif, which is a closed bijective linear operator in X. It can be shown that each A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq27_HTML.gif has dense domain and that D ( A β ) D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq28_HTML.gif for 0 α β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq29_HTML.gif. Moreover, A α + β x = A α A β x = A β A α x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq30_HTML.gif for every α , β R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq31_HTML.gif and x D ( A μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq32_HTML.gif with μ : = max ( α , β , α + β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq33_HTML.gif, where A 0 = I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq34_HTML.gif, I is the identity in X. (For proofs of these facts, we refer to the literature [15, 20, 22].)

We denote by X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif the Hilbert space of D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq35_HTML.gif equipped with norm x α : = A α x = A α x , A α x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq36_HTML.gif for x D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq37_HTML.gif, which is equivalent to the graph norm of A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq27_HTML.gif. Then we have X β X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq38_HTML.gif, for 0 α β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq29_HTML.gif (with X 0 = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq39_HTML.gif ) and the embedding is continuous. Moreover, A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq27_HTML.gif has the following basic properties.

Lemma 1 [42]

A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq27_HTML.gif and S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq9_HTML.gif have the following properties.
  1. (i)

    S ( t ) : X X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq40_HTML.gif for each t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif and α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq41_HTML.gif.

     
  2. (ii)

    A α S ( t ) x = S ( t ) A α x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq42_HTML.gif for each x D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq43_HTML.gif and t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq10_HTML.gif.

     
  3. (iii)
    For every t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif, A α S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq44_HTML.gif is bounded in X and there exists M α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq45_HTML.gif such that
    A α S ( t ) M α t α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equb_HTML.gif
     
  4. (iv)

    A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq25_HTML.gif is a bounded linear operator for 0 α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq46_HTML.gif.

     

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

Definition 2 The fractional integral of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq24_HTML.gif with the lower limit 0 for a function f is defined as
I α f ( t ) = 1 Γ ( α ) 0 t f ( s ) ( t s ) 1 α d s , t > 0 , α > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equc_HTML.gif

provided the right-hand side is pointwise defined on [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq47_HTML.gif, where Γ is the gamma function.

Definition 3 The Caputo derivative of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq24_HTML.gif with the lower limit 0 for a function f can be written as
D α C f ( t ) = 1 Γ ( n α ) 0 t f ( n ) ( s ) ( t s ) α + 1 n d s = I n α f ( n ) ( t ) , t > 0 , 0 n 1 < α < n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equd_HTML.gif

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

According to Definitions 2 and 3, it is suitable to rewrite the problem (1) in the equivalent integral equation
x ( t ) = x 0 + 1 Γ ( q ) 0 t ( t s ) α 1 × [ A x ( s ) + B u ( s ) + f ( s , x ( s ) ) + 0 s K ( s r ) g ( r , x ( r ) ) d r ] d s , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ2_HTML.gif
(2)
provided that the integral in (2) exists. Applying the Laplace transform
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Eque_HTML.gif
to (2) and using the method similar to that used in [38] we get
x ( t ) = 0 Ψ β ( θ ) S ( t β θ ) x 0 d θ + β 0 t 0 θ ( t s ) α 1 Ψ β ( θ ) S ( ( t s ) β θ ) × [ B u ( s ) + ( f ( s , x ( s ) ) + 0 s K ( s r ) g ( r , x ( r ) ) d r ) ] d θ d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equf_HTML.gif
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equg_HTML.gif

Here, Ψ β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq48_HTML.gif is a probability density function defined on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq49_HTML.gif, that is Ψ β ( θ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq50_HTML.gif, θ ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq51_HTML.gif and 0 Ψ β ( θ ) d θ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq52_HTML.gif.

For x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq53_HTML.gif, we define two families { S β ( t ) : t 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq54_HTML.gif and { P β ( t ) : t 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq55_HTML.gif of operators by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equh_HTML.gif

respectively.

The following lemma follows from the results given in [3739].

Lemma 4 The operators S β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq56_HTML.gif and P β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq57_HTML.gif have the following properties.
  1. (i)
    For any fixed t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq10_HTML.gif, and any x X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq58_HTML.gif, we have the operators S β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq59_HTML.gif and P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq60_HTML.gif are linear and bounded operators, i.e. for any x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq53_HTML.gif,
    S β ( t ) x α M x α and P β ( t ) x α M Γ ( β ) x α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equi_HTML.gif
     
  2. (ii)

    The operators S β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq59_HTML.gif and P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq61_HTML.gif are strongly continuous for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq10_HTML.gif.

     
  3. (iii)

    S β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq59_HTML.gif and P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq62_HTML.gif are norm continuous in X for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif.

     
  4. (iv)

    S β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq59_HTML.gif and P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq62_HTML.gif are compact operators in X for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif.

     
  5. (v)

    For every t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif, the restriction of S β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq59_HTML.gif to X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif and the restriction of P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq62_HTML.gif to X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif are norm continuous.

     
  6. (vi)

    For every t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif, the restriction of S β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq59_HTML.gif to X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif and the restriction of P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq62_HTML.gif to X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif are compact operators in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif.

     
  7. (vii)
    For all x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq53_HTML.gif and t ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq63_HTML.gif,
    A α P β ( t ) x C α t α β x , where C α : = M α β Γ ( 2 α ) Γ ( 1 + β ( 1 α ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equj_HTML.gif
     

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

Definition 5 A function x ( ; x 0 , u ) C ( [ 0 , T ] , X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq64_HTML.gif is said to be a mild solution of (1) if for any u L 2 ( [ 0 , T ] , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq65_HTML.gif the integral equation
x ( t ) = S β ( t ) ( x 0 + h ( x ) ) + 0 t ( t s ) β 1 P β ( t s ) B u ( s ) d s + 0 t ( t s ) β 1 P β ( t s ) [ f ( s , x ( s ) ) + 0 s K ( s r ) g ( r , x ( r ) ) d r ] d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ3_HTML.gif
(3)

is satisfied.

It is clear that L 0 t : = 0 t ( t s ) β 1 P β ( t s ) B u ( s ) d s : L 2 ( [ 0 , T ] , U ) C ( [ 0 , T ] , X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq66_HTML.gif is bounded if 1 2 < β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq67_HTML.gif. In what follows, we assume that 1 2 < β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq68_HTML.gif.

3 Approximate controllability

In this section, we state and prove conditions for the approximate controllability of semilinear fractional control integro-differential systems. To do this, we first prove the existence of a fixed point of the operator Λ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq69_HTML.gif defined below using Krasnoselskii’s fixed-point theorem. Secondly, in Theorem 11, we show that under the uniform boundedness of f and g the approximate controllability of fractional systems (1) is implied by the approximate controllability of the corresponding linear system (4).

Let x ( T ; x 0 , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq70_HTML.gif be the state value of (1) at terminal time T corresponding to the control u and the initial value x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq71_HTML.gif. Introduce the set ( T , x 0 ) = { x ( T ; x 0 , u ) : u L 2 ( [ 0 , T ] , U ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq72_HTML.gif, which is called the reachable set of system (1) at terminal time T, its closure in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif is denoted by ( T , x 0 ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq73_HTML.gif.

Definition 6 The system (1) is said to be approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq74_HTML.gif if ( T , x 0 ) ¯ = X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq75_HTML.gif, that is, given an arbitrary ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq76_HTML.gif it is possible to steer from the point x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq71_HTML.gif to within a distance ε from all points in the state space X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq77_HTML.gif at time T.

Consider the following linear fractional differential system:
D t β x ( t ) = A x ( t ) + B u ( t ) , t [ 0 , T ] , x ( 0 ) = x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ4_HTML.gif
(4)
The approximate controllability for linear fractional system (4) is a natural generalization of approximate controllability of linear first order control system [9, 10, 12]. It is convenient at this point to introduce the controllability and resolvent operators associated with (4) as
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equk_HTML.gif

respectively, where B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq78_HTML.gif denotes the adjoint of B and P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq79_HTML.gif is the adjoint of P β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq60_HTML.gif. It is straightforward that the operator Γ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq80_HTML.gif is a linear bounded operator.

Theorem 7 [10]

Let Z be a separable reflexive Banach space and let Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq81_HTML.gif stands for its dual space. Assume that Γ : Z Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq82_HTML.gif is symmetric. Then the following two conditions are equivalent:
  1. 1.

    Γ : Z Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq83_HTML.gif is positive, that is, z , Γ z > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq84_HTML.gif for all nonzero z Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq85_HTML.gif.

     
  2. 2.

    For all h Z z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq86_HTML.gif strongly converges to zero as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq87_HTML.gif. Here, J is the duality mapping of Z into Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq81_HTML.gif.

     

Lemma 8 The linear fractional control system (4) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq15_HTML.gif if and only if ε R ( ε , Γ 0 T ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq88_HTML.gif as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq87_HTML.gif in the strong operator topology.

Proof The lemma is a straightforward consequence of Theorem 7. Indeed, the system (4) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq89_HTML.gif if and only if Γ 0 T x , x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq90_HTML.gif for all nonzero x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq53_HTML.gif, see [7]. By Theorem 7, ε ( ε I + Γ 0 T ) 1 ( h ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq91_HTML.gif as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq87_HTML.gif for all h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq92_HTML.gif. □

Remark 9 Notice that positivity of Γ 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq80_HTML.gif is equivalent to Γ 0 T x , x = 0 x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq93_HTML.gif. In other words, since Γ 0 T x , x = 0 T ( T s ) β 1 B P β ( T s ) x 2 d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq94_HTML.gif, approximate controllability of the linear system (4) is equivalent to B P β ( T s ) x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq95_HTML.gif, 0 s < T x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq96_HTML.gif.

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

(H1) f , g : [ 0 , T ] × X α × X α X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq97_HTML.gif are continuous and for each r N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq98_HTML.gif, there exists a constant γ [ 0 , β ( 1 α ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq99_HTML.gif and functions φ r L 1 / γ ( [ 0 , T ] ; R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq100_HTML.gif, ψ r L ( [ 0 , T ] ; R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq101_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equl_HTML.gif

(H2) h : C ( [ 0 , T ] ; X α ) X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq102_HTML.gif is a Lipschitz function with Lipschitz constant L h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq103_HTML.gif.

(H c ) The linear system (4) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq74_HTML.gif.

Using the hypothesis (H c ), for an arbitrary function x C ( [ 0 , T ] ; X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq104_HTML.gif, we choose the feedback control function as follows:
u ε ( t , x ) = B P β ( T t ) ( ε I + Γ 0 T ) 1 [ S β ( T ) ( x 0 + h ( x ) ) 0 T ( T s ) β 1 P β ( T s ) [ f ( s , x ( s ) ) + 0 s K ( s , r ) g ( r , x ( r ) ) d r ] d s ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ5_HTML.gif
(5)
Let B r = { x C ( [ 0 , T ] ; X α ) : x α r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq105_HTML.gif, where r is a positive constant. Then B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq106_HTML.gif is clearly a bounded closed and convex subset in C ( [ 0 , T ] ; X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq107_HTML.gif. We will show that when using the above control the operator Λ ε : B k B k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq108_HTML.gif defined by
( Λ ε x ) ( t ) : = ( Φ ε x ) ( t ) + ( Π ε x ) ( t ) , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equm_HTML.gif
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equn_HTML.gif

has a fixed point in C ( [ 0 , T ] ; X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq109_HTML.gif.

Theorem 10 Let the assumptions (H1) and (H2) be satisfied. Then for x 0 X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq110_HTML.gif, the fractional Cauchy problem (1) with u = u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq111_HTML.gif has at least one mild solution provided that
L C + C α T ( 1 α ) β ε ( 1 α ) β M Γ ( β ) L B 2 L C < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ6_HTML.gif
(6)
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equo_HTML.gif

Proof It is easy to see that for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq76_HTML.gif the operator Λ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq69_HTML.gif maps C ( [ 0 , T ] ; X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq112_HTML.gif into itself.

Let x B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq113_HTML.gif and 0 t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq114_HTML.gif. Using assumption (H1) yield the following estimations,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equp_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ7_HTML.gif
(7)
From (6) and the assumption (H2), it follows that for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq76_HTML.gif there exists r ( ε ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq115_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ8_HTML.gif
(8)

Therefore, from (7) and (8), it follows that for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq76_HTML.gif there exists r ( ε ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq115_HTML.gif such that Φ ε y + Π ε x B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq116_HTML.gif for every x , y B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq117_HTML.gif. Therefore, for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq76_HTML.gif the fractional Cauchy problem (1) with the control (5) has a mild solution if and only if the operator Φ ε + Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq118_HTML.gif has a fixed point in B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq119_HTML.gif.

In what follows, we will show that Φ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq120_HTML.gif and Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq121_HTML.gif satisfy the conditions of Krasnoselskii’s fixed-point theorem. From (H2) and (6), we infer that Φ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq120_HTML.gif is a contraction. Next, we show that Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq122_HTML.gif is completely continuous on B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq123_HTML.gif.

Step 1: We first prove that Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq122_HTML.gif is continuous on B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq119_HTML.gif. Let { x n } n = 1 B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq124_HTML.gif be a sequence such that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq125_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq126_HTML.gif in C ( [ 0 , T ] ; X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq107_HTML.gif. Therefore, it follows from the continuity of f, g and u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq127_HTML.gif that for each t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq128_HTML.gif,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equq_HTML.gif
Also, by (H1), we see that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equr_HTML.gif
Since
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equs_HTML.gif
using the Lebesgue dominated convergence theorem that for all t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq128_HTML.gif, we conclude
( Π ε x n ) ( t ) ( Π ε x ) ( t ) α 0 , as  n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equt_HTML.gif

implying that Π ε x n Π ε x α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq129_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq126_HTML.gif. This proves that Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq122_HTML.gif is continuous on B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq123_HTML.gif.

Step 2. Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq122_HTML.gif is compact on B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq130_HTML.gif.

For the sake of brevity, we write
N ( x ( s ) ) : = f ( s , x ( s ) ) + 0 s K ( s , r ) g ( r , x ( r ) ) d r + B u ε ( s , x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equu_HTML.gif
Let t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq128_HTML.gif be fixed and δ , η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq131_HTML.gif be small enough. For x B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq132_HTML.gif, we define the map
( Π ε δ η x ) ( t ) = 0 δ η β r ( t s ) β 1 Ψ β ( r ) S ( ( t s ) β r ) N ( x ( s ) ) d r d s = S ( δ β η ) 0 δ η β r ( t s ) β 1 Ψ β ( r ) S ( ( t s ) β r δ β η ) N ( x ( s ) ) d r d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equv_HTML.gif
Therefore, from Lemma 4, we see that for each t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq133_HTML.gif, the set { ( Π ε δ η x ) ( t ) : x B r ( ε ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq134_HTML.gif is relatively compact in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif. Since
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equw_HTML.gif

approaches to zero as η 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq135_HTML.gif, using the total boundedness, we conclude that for each t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq128_HTML.gif, the set { ( Π ε δ η x ) ( t ) : x B r ( ε ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq136_HTML.gif is relatively compact in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif.

On the other hand, for 0 < t 1 < t 2 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq137_HTML.gif and δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq138_HTML.gif small enough, we have
( Π ε x ) ( t 1 ) ( Π ε x ) ( t 2 ) α I 1 + I 2 + I 3 + I 4 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equx_HTML.gif
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equy_HTML.gif
Therefore, it follows from (H1) and Lemma 4 that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equz_HTML.gif
and
I 4 C α 0 t 1 | ( t 1 s ) β 1 α β ( t 2 s ) β 1 α β | × ( ( φ r ( ε ) ( s ) + K ψ r ( ε ) L ) + 1 ε M Γ ( β ) L B 2 L u ( r ( ε ) ) ) d s C α ( 1 γ ( 1 α ) β γ ) 1 γ φ r ( ε ) L 1 / γ [ t 1 ( 1 α ) β γ ( t 2 ( 1 α ) β γ 1 γ ( t 2 t 1 ) ( 1 α ) β γ 1 γ ) 1 γ ] + C α 2 K ( 1 α ) β ψ r ( ε ) L [ t 1 ( 1 α ) β t 2 ( 1 α ) β ( t 2 t 1 ) ( 1 α ) β ] + C α 2 L B 2 L u ( r ( ε ) ) ( 1 α ) β M Γ ( β ) ψ r ( ε ) L [ t 1 ( 1 α ) β t 2 ( 1 α ) β ( t 2 t 1 ) ( 1 α ) β ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equaa_HTML.gif
from which it is easy to see that all I i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq139_HTML.gif, i = 1 , 2 , 3 , 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq140_HTML.gif, tend to zero independent of x B k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq141_HTML.gif as t 2 t 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq142_HTML.gif and δ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq143_HTML.gif. Thus, we can conclude that
( Π ε x ) ( t 1 ) ( Π ε x ) ( t 2 ) α 0 as  t 2 t 1 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equab_HTML.gif

and the limit is independent of x B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq144_HTML.gif. The case t 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq145_HTML.gif is trivial. Consequently, the set { ( Π ε x ) ( t ) : t [ 0 , T ] , x B r ( ε ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq146_HTML.gif is equicontinuous. Now applying the Arzela-Ascoli theorem, it results that Π ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq122_HTML.gif is compact on B r ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq119_HTML.gif.

Therefore, applying Krasnoselskii’s fixed-point theorem, we conclude that Λ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq69_HTML.gif has a fixed point, which gives rise to a mild solution of Cauchy problem (1) with control given in (5). This completes the proof. □

Theorem 11 Let the assumptions (H1), (H2) and (H c ) be satisfied. Moreover, assume the functions f , g : [ 0 , T ] × X α × X α X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq147_HTML.gif and h : C ( [ 0 , T ] ; X α ) X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq102_HTML.gif are bounded and M L h < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq148_HTML.gif. Then the semilinear fractional system (3) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq149_HTML.gif.

Proof It is clear that all assumptions of Theorem 10 are satisfied with σ 1 = σ 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq150_HTML.gif. Let x ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq151_HTML.gif be a fixed point of F ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq152_HTML.gif in B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq106_HTML.gif. Any fixed point of F ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq152_HTML.gif is a mild solution of (3) under the control
u ε ( t , x ε ) = B P β ( T t ) R ( ε , Γ 0 T ) ( h S β ( T ) ( x 0 + h ( x ε ) ) 0 T ( T s ) β 1 P β ( T s ) [ f ( s , x ε ( s ) ) + 0 s K ( s τ ) g ( τ , x ε ( τ ) ) d τ ] d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equac_HTML.gif
and satisfies the equality
x ε ( T ) = h ε R ( ε , Γ 0 T ) p ( x ε ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ9_HTML.gif
(9)
where
p ( x ε ) = ( h S β ( T ) ( x 0 + h ( x ε ) ) 0 T ( T s ) β 1 P β ( T s ) [ f ( s , x ε ( s ) ) + 0 s K ( s τ ) g ( τ , x ε ( τ ) ) d τ ] d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equad_HTML.gif
Moreover, by the boundedness of the functions f and g and Dunford-Pettis theorem, we have that the sequences { f ( s , x ε ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq153_HTML.gif and { g ( s , x ε ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq154_HTML.gif are weakly compact in L 2 ( [ 0 , T ] ; X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq155_HTML.gif, so there are subsequences still denoted by { f ( s , x ε ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq156_HTML.gif and { g ( s , x ε ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq157_HTML.gif, that weakly converge to, say, f and g in L 2 ( [ 0 , T ] ; X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq155_HTML.gif. On the other hand, there exists h ˜ X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq158_HTML.gif such that h ( x ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq159_HTML.gif converges to h ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq160_HTML.gif weakly in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq1_HTML.gif. Denote
w = h S β ( x 0 + h ˜ ) ) 0 T ( T s ) β 1 P β ( T s ) [ f ( s ) + 0 s K ( s τ ) g ( τ ) d τ ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equae_HTML.gif
It follows that
p ( x ε ) w α S β ( T ) h ( x ε ) S β ( T ) h ˜ α + 0 T ( T s ) β 1 P β ( T s ) ( f ( s , x ε ( s ) ) f ( s ) ) d s α + 0 T ( T s ) β 1 P β ( T s ) 0 s K ( s τ ) ( g ( τ , x ε ( τ ) g ( τ ) ) d τ d s α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equaf_HTML.gif
as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq87_HTML.gif because of compactness of the operator
l ( ) 0 ( s ) β 1 P β ( s ) l ( s ) d s : L 2 ( [ 0 , T ] , X ) C ( [ 0 , T ] , X α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equag_HTML.gif
Then from (9), we obtain
x ε ( T ) h α ε R ( ε , Γ 0 T ) ( w ) α + ε R ( ε , Γ 0 T ) p ( x ε ) w α ε R ( ε , Γ 0 T ) ( w ) α + p ( x ε ) w α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ10_HTML.gif
(10)

as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq87_HTML.gif. This proves the approximate controllability of (1). □

4 Applications

Example 1 As an application to Theorem 11, we study the following simple example. Consider a control system governed by the fractional partial differential equation of the form
{ t 3 4 c x ( t , z ) = z 2 x ( t , z ) + u ( t , z ) + F ( t , z , x ( t , z ) ) + 0 t K ( t , s ) G ( s , z , x ( s , z ) ) d s , t [ 0 , T ] , z [ 0 , π ] , x ( t , 0 ) = x ( t , π ) = 0 , x ( 0 , z ) = x 0 ( z ) + k = 1 p 0 π k ( z , r ) cos ( x ( t k , r ) ) d r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equ11_HTML.gif
(11)

where f , g : [ 0 , T ] × [ 0 , π ] × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq161_HTML.gif, k : [ 0 , π ] × [ 0 , π ] R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq162_HTML.gif, 0 < t 1 < < t p < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq163_HTML.gif.

Let us take X = U = L 2 [ 0 , π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq164_HTML.gif and define the operator A by A w = w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq165_HTML.gif with the domain D ( A ) = { w ( ) L 2 [ 0 , π ] , w , w  are absolutely continuous,  w L 2 [ 0 , π ] , w ( 0 ) = w ( π ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq166_HTML.gif. Then
A w = n = 1 n 2 w , e n e n , w D ( A ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equah_HTML.gif
where e n ( z ) = 2 π sin n z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq167_HTML.gif, 0 z π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq168_HTML.gif, n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq169_HTML.gif . Clearly −A generates a compact analytic semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq170_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq19_HTML.gif in X and it is given by
S ( t ) w = n = 1 e n 2 t w , e n e n , w X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equai_HTML.gif

Clearly, the assumption (H1) is satisfied. On the other hand, it can be easily seen that the deterministic linear system corresponding to (11) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq15_HTML.gif; see [12].

The operator A 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq171_HTML.gif is given by
A 1 2 w = n = 1 n w , e n e n , w D ( A 1 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equaj_HTML.gif

where D ( A 1 2 ) = { w X : n = 1 n w , e n e n X } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq172_HTML.gif and A 1 2 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq173_HTML.gif.

Let X 1 2 : = ( D ( A 1 2 ) , 1 / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq174_HTML.gif, where x 1 / 2 : = A 1 2 x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq175_HTML.gif for x D ( A 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq176_HTML.gif. Assume that F , G : [ 0 , T ] × [ 0 , π ] × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq177_HTML.gif satisfies the following conditions:
  1. 1.

    The functions F ( , , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq178_HTML.gif, G ( , , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq179_HTML.gif are continuous and uniformly bounded.

     
  2. 2.

    F ( 0 , , ) = F ( π , , ) = G ( 0 , , ) = G ( π , , ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq180_HTML.gif.

     
  3. 3.
    k : [ 0 , π ] × [ 0 , π ] R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq162_HTML.gif is continuously differentiable, k ( 0 , ) = k ( π , ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq181_HTML.gif and
    0 π 0 π | 2 ξ 2 k ( ξ , y ) | 2 d y d ξ < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equak_HTML.gif
     
Denote by E β , ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq182_HTML.gif, the Mittag-Leffler special function defined by
E β , ζ = k = 0 t k Γ ( ζ k + β ) , ζ , β > 0 , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equal_HTML.gif
Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equam_HTML.gif
Define
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equan_HTML.gif
Then, for each x , y C ( [ 0 , T ] , X 1 / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq183_HTML.gif we have
h ( x ) 1 / 2 2 = A 1 / 2 h ( x ) ( ) L 2 [ 0 , π ] 2 = n = 1 n 2 e n L 2 [ 0 , π ] 2 | h ( x ) ( ) , e n | 2 = 2 π n = 1 n 2 | 0 π h ( x ) ( ξ ) sin ( n ξ ) d ξ | 2 = n = 1 1 n 2 | 0 π 2 ξ 2 h ( x ) ( ξ ) e n ( ξ ) d ξ | 2 π 2 6 2 ξ 2 h ( x ) ( ξ ) L 2 [ 0 , π ] 2 = π 2 6 2 ξ 2 k = 0 p 0 π k ( ξ , y ) cos ( x ( t k , y ) ) d y L 2 [ 0 , π ] 2 = π 2 6 0 π | k = 1 p 0 π 2 ξ 2 k ( ξ , y ) cos ( x ( t k , y ) ) d y | 2 d ξ p π 3 6 0 π 0 π | 2 ξ 2 k ( ξ , y ) | 2 d y d ξ = p π 3 6 2 ξ 2 k ( ξ , y ) L 2 [ 0 , π ] × [ 0 , π ] 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equao_HTML.gif
and
h ( x ) h ( y ) 1 / 2 2 = A 1 / 2 h ( x ) ( ) A 1 / 2 h ( y ) ( ) L 2 [ 0 , π ] 2 π 2 6 2 ξ 2 k = 0 p 0 π k ( ξ , r ) [ cos ( x ( t k , r ) ) cos ( y ( t k , r ) ) ] d r L 2 [ 0 , π ] 2 = π 2 6 0 π | k = 0 p 0 π 2 ξ 2 k ( ξ , r ) [ cos ( x ( t k , r ) ) cos ( y ( t k , r ) ) ] d r | 2 d ξ p π 2 6 0 π 0 π | 2 ξ 2 k ( ξ , r ) | 2 d r d ξ sup 0 t π 0 π | x ( t , r ) y ( t , r ) | 2 d r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equap_HTML.gif

It follows that h : C ( [ 0 , T ] ; X 1 / 2 ) X 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq184_HTML.gif is bounded and Lipschitz continuous. On the other hand, it is not difficult to verify that f , g : [ 0 , T ] × X 1 / 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq185_HTML.gif are continuous.

Next, we show that the linear system corresponding to (11) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq15_HTML.gif. It is clear that P β ( t ) : X 1 2 X 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq186_HTML.gif is defined as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equaq_HTML.gif

By Remark 9, the linear system corresponding to (11) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq15_HTML.gif if and only if B P β ( T t ) x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq187_HTML.gif, 0 t < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq188_HTML.gif implies that x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq189_HTML.gif. This follows from the representation of B P β ( T t ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq190_HTML.gif.

Now, we note that the problem (11) can be reformulated as the abstract problem. Thus, by Theorem 11, the system (11) is approximately controllable on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_IEq15_HTML.gif, provided that
M L h = p π 2 6 0 π 0 π | 2 ξ 2 k ( ξ , r ) | 2 d r d ξ < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-118/MediaObjects/13661_2012_Article_371_Equar_HTML.gif

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

Authors’ Affiliations

(1)
Eastern Mediterranean University

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