Approximate controllability of some nonlinear systems in Banach spaces

Boundary Value Problems20132013:50

DOI: 10.1186/1687-2770-2013-50

Received: 8 January 2013

Accepted: 25 February 2013

Published: 13 March 2013

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 [16] and the references therein. The approximate controllability of nonlinear systems when the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif, 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 L ( I , L ( X ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq3_HTML.gif. 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 [15, 7, 947]). 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 [2326] 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
y = y 0 + L B v + L F ( y , v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ1_HTML.gif
(1)
and define the following sets:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equa_HTML.gif
Here Y, X are separable reflexive Banach spaces and V is a Hilbert space, B L ( V , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq4_HTML.gif, L L ( Y , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq5_HTML.gif, Q L ( Y , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq6_HTML.gif, F : Y × V Y × V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq7_HTML.gif is a nonlinear operator, y 0 Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq8_HTML.gif, v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq9_HTML.gif. Q R ( L , F ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq10_HTML.gif is the set of points Qy, where y is a solution of (1), attainable from the point y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq11_HTML.gif. The set Q R ( L , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq12_HTML.gif is the set of points Qz, where z is a solution of
z = y 0 + L B v , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ2_HTML.gif
(2)
reachable from y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq11_HTML.gif. One can see that for each h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif the control
v ε = ( Q L B ) J ( ( ε I + Γ J ) 1 ( h Q y 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ3_HTML.gif
(3)
transfers equation (2) from y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq11_HTML.gif to
Q z ε = Q y 0 + Q L B v ε = Q y 0 + Γ J ( ( ε I + Γ J ) 1 ( h Q y 0 ) ) = h ε ( ε I + Γ J ) 1 ( h Q y 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equb_HTML.gif
where z ε = y 0 + L B v ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq15_HTML.gif. It is known that Q R ( L , 0 ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq16_HTML.gif if and only if
ε ( ε I + Γ J ) 1 ( h ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equc_HTML.gif

in the strong operator topology as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif, see [30]. Thus, the control (3) transfers system (2) from y 0 Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq8_HTML.gif to a small neighborhood of an arbitrary point h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif if and only if Q R ( L , 0 ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq16_HTML.gif.

The same idea is now used to investigate the controllability of semilinear system (1). To do so, for each ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif and h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, consider a nonlinear operator T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif from Y × V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq19_HTML.gif to Y × V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq19_HTML.gif defined by
T ε ( y , v ) = ( z , w ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ4_HTML.gif
(4)
where
{ z = y 0 + L B w + L F ( y , v ) , w = ( Q L B ) J ( ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y , v ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equd_HTML.gif
One can see that if the operator T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif has a fixed point ( y ε , v ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq20_HTML.gif, then the control v ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq21_HTML.gif steers control system (1) from y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq11_HTML.gif to
Q y ε = h J ( ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y ε , v ε ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Eque_HTML.gif
if ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif. We prove that Q y ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq22_HTML.gif is close to h provided that ε ( ε I + Q L B ( Q L B ) ) 1 ( h ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq23_HTML.gif converges strongly to zero as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif. Therefore, to prove the approximate controllability of (1), for each ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif and h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, we have to seek for a solution of the following equation:
{ y ε = y 0 + L B v ε + L F ( y ε , v ε ) , v ε = ( Q L B ) J ( ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y ε , v ε ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ5_HTML.gif
(5)

It is clear that the fixed points of the nonlinear operator T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif 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 Im Q L B X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq24_HTML.gif, 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 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq25_HTML.gif stand for its dual space with respect to the continuous pairing , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq26_HTML.gif. We may assume, without loss of generality, that X and X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq25_HTML.gif 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 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq25_HTML.gif given by the following relations:
J ( z ) = z , J ( z ) , z = z 2 for all  z X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equf_HTML.gif

is bijective, demicontinuous, i.e., continuous from X with a strong topology into X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq25_HTML.gif with weak topology and strictly monotonic. Moreover, J 1 : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq27_HTML.gif is also a duality mapping.

An operator Γ : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq28_HTML.gif is symmetric if
z 1 , Γ z 2 = z 2 , Γ z 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equg_HTML.gif

for all z 1 , z 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq29_HTML.gif. It is easy to see that Γ is linear and continuous. Γ is nonnegative if z , Γ z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq30_HTML.gif for all z X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq31_HTML.gif.

Lemma 1 [31]

For every h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif and ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif, the equation
ε z ε + Γ J ( z ε ) = ε h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ6_HTML.gif
(6)
has a unique solution z ε = z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq32_HTML.gif and
z ε ( h ) = J ( z ε ( h ) ) h . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ7_HTML.gif
(7)

Theorem 2 [31]

Let Γ be a symmetric operator. Then the following three conditions are equivalent:
  1. (i)

    Γ is positive, that is, z , Γ z > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq33_HTML.gif for all nonzero z X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq31_HTML.gif.

     
  2. (ii)

    For all h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, J ( z ε ( h ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq34_HTML.gif converges to zero as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif in the weak topology, where z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq35_HTML.gif is a solution of equation (6).

     
  3. (iii)

    For all h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq36_HTML.gif strongly converges to zero as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif.

     

We impose the following assumptions:

(A1) F : Y × V Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq37_HTML.gif is continuous and there exists C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq38_HTML.gif such that F ( y , v ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq39_HTML.gif for all ( y , v ) Y × V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq40_HTML.gif.

(A2) L : Y Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq41_HTML.gif is compact.

(A3) For all h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq42_HTML.gif strongly converges to zero as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif.

Note that the condition (A3) holds if and only if Im ( Q L B ) ¯ = Q R ( L , 0 ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq43_HTML.gif, i.e., system (2) is approximately controllable.

Definition 3 System (1) is approximately controllable if
Q R ( L , F ) ¯ = X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equh_HTML.gif

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

Proof Step 1. Show that the operator T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif has a fixed point in Y × V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq19_HTML.gif for all ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif. For our convenience, let us introduce the following notation:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equi_HTML.gif
Assume that r ( ε ) d ( ε ) + C c ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq44_HTML.gif. Then by (7) we have
w Q L B ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y , v ) ) 1 ε Q L B ( h + Q y 0 + C Q L ) = d 1 ( ε ) 4 + γ ( ε ) 4 a 1 C d 1 ( ε ) 4 a + γ ( ε ) 4 a C 1 4 a ( d ( ε ) + C c ( ε ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equj_HTML.gif
and
z y 0 + L B w + L F ( y , v ) a 2 + L B 1 4 a ( d ( ε ) + C c ( ε ) ) + L C d ( ε ) 4 + 1 4 ( d ( ε ) + c ( ε ) C ) + c ( ε ) 4 C 1 2 ( d ( ε ) + c ( ε ) C ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equk_HTML.gif

Thus we proved that T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif maps B ε = { ( z , w ) Y × V : ( z , w ) r ( ε ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq45_HTML.gif into itself. On the other hand, the operator T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif is continuous and T ε ( B ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq46_HTML.gif is relatively compact. By Schauder’s fixed-point theorem, for all ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif, T ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq18_HTML.gif has a fixed point in the ball B ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq47_HTML.gif.

Step 2. Assume Q R ( L , 0 ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq16_HTML.gif. By Step 1, the operator (4) has a fixed point ( y ε , v ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq48_HTML.gif. So, ( y ε , v ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq49_HTML.gif satisfies (5) and, moreover, it follows that for all h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif
Q y ε h = ε ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y ε , v ε ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ8_HTML.gif
(8)
So, z ε : = Q y ε h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq50_HTML.gif is a solution of the equation
ε z ε + Γ J ( z ε ) = ε ( Q L F ( y ε , v ε ) + Q y 0 h ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ9_HTML.gif
(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 { F ( y ε , v ε ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq51_HTML.gif, which weakly converges to say z Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq52_HTML.gif and L F ( y ε , v ε ) L z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq53_HTML.gif strongly in Y as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif. From (7) and strong convergence of the sequence { h ( z ε ) = h Q y 0 Q L F ( y ε , v ε ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq54_HTML.gif, it is easy to see that there exists C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq55_HTML.gif such that for all ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq14_HTML.gif
z ε = J ( z ε ) Q L F ( y ε , v ε ) + Q y 0 h C 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equl_HTML.gif
Then we can extract a subsequence, still denoted by z ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq56_HTML.gif, such that
J ( z ε ) J ( z ¯ 0 ) as  ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equm_HTML.gif
for some z ¯ 0 Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq57_HTML.gif. Applying J ( z ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq58_HTML.gif to equation (9) and taking the limit, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equn_HTML.gif
since Γ is positive. So, J ( z ε ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq59_HTML.gif as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq17_HTML.gif. Now, applying J ( z ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq60_HTML.gif to equation (9), dividing through by ε and taking the limit, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equo_HTML.gif

Thus lim ε 0 + Q y ε h = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq61_HTML.gif, consequently Q R ( L , F ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq62_HTML.gif. 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
{ D t α c x ( t ) = A x ( t ) + B u ( t ) + f ( t , x ( t ) , 0 t g ( t , s , x ( s ) ) d s ) , t [ 0 , b ] , x ( 0 ) = x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ10_HTML.gif
(10)

where the state variable x takes values in a separable reflexive Banach space X; D α c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq63_HTML.gif is the Caputo fractional derivative of order 1 2 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq64_HTML.gif; A is the infinitesimal generator of a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq65_HTML.gif semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq66_HTML.gif of bounded operators on X; the control function u is given in L 2 ( [ 0 , b ] , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq67_HTML.gif, U is a Hilbert space; B is a bounded linear operator from U into X, Δ = { ( t , s ) : 0 s t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq68_HTML.gif and g : Δ × X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq69_HTML.gif, f : I × X × X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq70_HTML.gif are continuous bounded functions and x 0 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq71_HTML.gif.

Definition 5 The fractional integral of order α 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%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equp_HTML.gif

provided the right-hand side is pointwise defined on [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq72_HTML.gif, where γ is the gamma function.

Definition 6 Riemann-Liouville derivative of order α with the lower limit 0 for a function f : [ 0 , ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq73_HTML.gif can be written as
D α L f ( t ) = 1 γ ( n α ) d n d t n 0 t f ( s ) ( t s ) α + 1 d s , t > 0 , n 1 < α < n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equq_HTML.gif
Definition 7 The Caputo derivative of order α for a function f : [ 0 , ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq74_HTML.gif can be written as
D α c f ( t ) = L D α ( f ( t ) k = 0 n 1 t k k ! f ( k ) ( 0 ) ) , t > 0 , n 1 < α < n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equr_HTML.gif
Remark 8
  1. (1)
    If f ( t ) C n [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq75_HTML.gif, then
    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 , n 1 < α < n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equs_HTML.gif
     
  2. (2)

    The Caputo derivative of a constant is equal to zero.

     
  3. (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
x ( t ) = S ˆ α ( t ) x 0 + 0 t ( t s ) q 1 S α ( t s ) f ( s , x ( s ) , 0 s g ( s , r , x ( r ) ) d r ) d s + 0 t ( t s ) q 1 S α ( t s ) B u ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ11_HTML.gif
(11)
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equt_HTML.gif

and η α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq76_HTML.gif is a probability density function defined on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq77_HTML.gif, that is, η α ( θ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq78_HTML.gif, θ ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq79_HTML.gif and 0 η α ( θ ) d θ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq80_HTML.gif.

Lemma 9 [34]

For any fixed t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq81_HTML.gif, the operators S ˆ α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq82_HTML.gif and S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq83_HTML.gif are linear compact and bounded operators, i.e., for any x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq84_HTML.gif, S ˆ α ( t ) x M x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq85_HTML.gif and S α ( t ) x M Γ ( α ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq86_HTML.gif.

Definition 10 A solution x ( ; x 0 , u ) C ( [ 0 , b ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq87_HTML.gif is said to be a mild solution of (10) if for any u L 2 ( [ 0 , b ] , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq88_HTML.gif and the integral equation (11) is satisfied.

Let x b ( x 0 ; u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq89_HTML.gif be the state value of (10) at terminal time b corresponding to the control u and the initial value x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq90_HTML.gif. Introduce the set ( b , x 0 ) = { x b ( x 0 ; u ) ( 0 ) : u L 2 ( [ 0 , b ] , U ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq91_HTML.gif, which is called the reachable set of system (10) at terminal time b, its closure in X is denoted by ( b , x 0 ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq92_HTML.gif.

Definition 11 System (10) is said to be approximately controllable on J if ( b , x 0 ) ¯ = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq92_HTML.gif, that is, given an arbitrary ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq93_HTML.gif, it is possible to steer from the point x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq90_HTML.gif to within a distance ϵ from all points in the state space X at time b.

Consider the following linear fractional differential system:
D t α x ( t ) = A x ( t ) + B u ( t ) , t [ 0 , b ] , x ( 0 ) = x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ12_HTML.gif
(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
Γ 0 b = 0 b ( b s ) 2 ( α 1 ) S α ( b s ) B B S α ( b s ) d s : X X , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equu_HTML.gif

where B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq94_HTML.gif denotes the adjoint of B and S α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq95_HTML.gif is the adjoint of S α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq96_HTML.gif. It is straightforward that the operator Γ 0 b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq97_HTML.gif is a linear bounded operator. By Theorem 2, linear fractional control system (12) is approximately controllable on [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq98_HTML.gif if and only if for any h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, z ε ( h ) = ε ( ε I + Γ 0 b J ) 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq99_HTML.gif converges strongly to zero as ε 0 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq100_HTML.gif.

Proposition 12 If S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif, are compact operators and 0 < 1 p < α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq101_HTML.gif, then the operator
L α f ( t ) = 0 t ( t s ) α 1 S α ( t s ) f ( s ) d s , f L p ( [ 0 , b ] , X ) , t [ 0 , b ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equv_HTML.gif

is compact from L p ( [ 0 , b ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq102_HTML.gif into C ( [ 0 , b ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq103_HTML.gif.

Proof According to the infinite dimensional version of the Ascoli-Arzela theorem, we need to show that
  1. (i)

    for arbitrary t [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq104_HTML.gif, the set { L α f ( t ) : f L p 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq105_HTML.gif is relatively compact in C ( [ 0 , b ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq106_HTML.gif;

     
  2. (ii)
    for arbitrary η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq107_HTML.gif, there exists δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq108_HTML.gif such that
    L α f ( t ) L α f ( s ) < η if  f L p 1 , | t s | δ , t , s [ 0 , b ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equw_HTML.gif
     
To prove (i), fix 0 < t < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq109_HTML.gif and define for 0 < η < t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq110_HTML.gif and δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq108_HTML.gif operators L α η , δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq111_HTML.gif from L p ( [ 0 , b ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq112_HTML.gif into X
( L α η , δ f ) ( t ) = α 0 t λ δ θ ( t s ) α 1 η α ( θ ) S ( ( t s ) α θ ) f ( s ) d s = α S ( λ α δ ) 0 t λ δ θ ( t s ) α 1 η α ( θ ) S ( ( t s ) α θ λ α δ ) f ( s ) d s , f L p ( [ 0 , b ] , X ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equx_HTML.gif
Since S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq113_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif, is a compact operator, the operators L α η , δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq111_HTML.gif are compact. Moreover, we have
( L α f ) ( t ) ( L α η , δ f ) ( t ) α 0 t 0 δ θ ( t s ) α 1 η α ( θ ) S ( ( t s ) α θ ) f ( s ) d θ d s + α t λ t δ θ ( t s ) α 1 η α ( θ ) S ( ( t s ) α θ ) f ( s ) d θ d s = : J 1 + J 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equy_HTML.gif
One can estimate J 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq114_HTML.gif and J 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq115_HTML.gif as follows:
J 1 α M 0 t ( t s ) α 1 f ( s ) d s ( 0 δ θ η α ( θ ) d θ ) α M ( 0 t ( t s ) ( α 1 ) q d s ) 1 / q f L p ( 0 δ θ η α ( θ ) d θ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equz_HTML.gif
and
J 2 α M t λ t ( t s ) α 1 f ( s ) d s ( δ θ η α ( θ ) d θ ) α M γ ( 1 + α ) ( 0 t ( t s ) ( α 1 ) q d s ) 1 / q ( 0 t f ( s ) p d s ) 1 / p = α M γ ( 1 + α ) ( η ( α 1 ) q + 1 ( α 1 ) q + 1 ) 1 / q f L p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equaa_HTML.gif
where we have used the equality
0 θ β η α ( θ ) d θ = γ ( 1 + β ) γ ( 1 + α β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equab_HTML.gif

Consequently, L α η , δ L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq116_HTML.gif in the operator norm so that L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq117_HTML.gif is compact and (i) follows immediately.

To prove (ii), note that, for 0 t t + h b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq118_HTML.gif and f L p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq119_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equac_HTML.gif
Applying the Hölder inequality, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equad_HTML.gif

It is clear that I 1 , I 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq120_HTML.gif as h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq121_HTML.gif. On the other hand, the compactness of S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq113_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif (and consequently S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq122_HTML.gif), implies the continuity of S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq123_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif, in the uniform operator topology. Then, by the Lebesque dominated convergence theorem, I 3 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq124_HTML.gif as h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq121_HTML.gif. Thus the proof of (ii), and therefore the proof of the proposition, is complete. □

Theorem 13 Suppose S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq113_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif, is compact and 1 2 < α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq125_HTML.gif. Then system (10) is approximately controllable on [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq126_HTML.gif if the corresponding linear system is approximately controllable on [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq127_HTML.gif.

Proof Let Y = L 2 ( [ 0 , b ] , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq128_HTML.gif, V = L 2 ( [ 0 , b ] , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq129_HTML.gif, and y 0 = S α ( ) x 0 Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq130_HTML.gif. Define the linear operators Q, L, L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq131_HTML.gif and the nonlinear operator F by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equae_HTML.gif

for y Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq132_HTML.gif, v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq9_HTML.gif. 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
{ D t α x ( t , θ ) = x θ θ ( t , θ ) + b ( θ ) u ( t ) + f ( t , x ( t , θ ) , 0 t g ( t , s , x ( s , θ ) ) d s ) , x ( t , 0 ) = x ( t , π ) = 0 , t > 0 , x ( 0 ) = x 0 , 0 < θ < π , 0 t b , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equ13_HTML.gif
(13)
where u L 2 [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq133_HTML.gif, X = L 2 [ 0 , π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq134_HTML.gif, h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq13_HTML.gif, 1 2 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq64_HTML.gif, and f : R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq135_HTML.gif, g : R × R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq136_HTML.gif are continuous and uniformly bounded. Let B L ( R , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq137_HTML.gif be defined as
( B u ) ( θ ) = b ( θ ) u , B h = 0 π h ( θ ) b ( θ ) d θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equaf_HTML.gif
where 0 θ π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq138_HTML.gif, u R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq139_HTML.gif, b ( θ ) L 2 [ 0 , π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq140_HTML.gif, and let A : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq141_HTML.gif be an operator defined by A z = z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq142_HTML.gif with the domain
D ( A ) = { z X z , z  are absolutely continuous,  z X , z ( 0 ) = z ( π ) = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equag_HTML.gif
Then
A z = n = 1 ( n 2 ) ( z , e n ) e n , z D ( A ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equah_HTML.gif
where e n ( θ ) = 2 / π sin n θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq143_HTML.gif, 0 x π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq144_HTML.gif, n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq145_HTML.gif . It is known that A generates a compact semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq1_HTML.gif, t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq2_HTML.gif, in X and is given by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equai_HTML.gif
Then B S α ( t ) z = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq146_HTML.gif for 0 t < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq147_HTML.gif implies
( z , e n ) ( b , e n ) = 0 for all  n = 1 , 2 , . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_Equaj_HTML.gif

Now if ( b , e n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq148_HTML.gif for all n, then ( z , e n ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq149_HTML.gif for all n and z = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq150_HTML.gif. Therefore, the associated linear system is approximately controllable provided that 0 π b ( θ ) e n ( θ ) d θ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq151_HTML.gif for n = 1 , 2 , 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq152_HTML.gif . Because of the compactness of the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq113_HTML.gif (and consequently S ˆ α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq153_HTML.gif, S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq154_HTML.gif) 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 [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-50/MediaObjects/13661_2013_Article_306_IEq126_HTML.gif.

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.

Declarations

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.

Authors’ Affiliations

(1)
Eastern Mediterranean University

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© Mahmudov; licensee Springer. 2013

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