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Boundary Value Problems

, 2013:50 | Cite as

Approximate controllability of some nonlinear systems in Banach spaces

  • Nazim I Mahmudov
Open Access
Research
Part of the following topical collections:
  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

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.

Keywords

Approximate Controllability Compactness Assumption Fractional Dynamical System Associate Linear System Separable Reflexive Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

The problems of controllability of infinite dimensional nonlinear (fractional) systems were studied widely by many authors; see [1, 2, 3, 4, 5, 6] and the references therein. The approximate controllability of nonlinear systems when the semigroup S ( t ) Open image in new window, t > 0 Open image in new window, 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 ) ) Open image in new window. Bian [9] investigated the approximate controllability for a class of semilinear systems. For abstract nonlinear systems, Carmichael and Quinn [10] used the Banach fixed-point theorem to obtain a local exact controllability in the case of nonlinearities with small Lipschitz constants. Zhang [11] studied the local exact controllability of semilinear evolution systems. Naito [7] and Seidman [12] used Schauder’s fixed-point theorem to prove invariance of the reachable set under nonlinear perturbations. Other related abstract results were given by Lasiecka and Triggiani [13].

In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]). A pioneering work has been reported by Bashirov and Mahmudov [17], Dauer and Mahmudov [28] and Mahmudov [31]. Sakthivel et al. [40] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. Klamka [23, 24, 25, 26] derived a set of sufficient conditions for constrained local controllability near the origin for semilinear dynamical control systems. Wang and Zhou [3] investigated the complete controllability of fractional evolution systems without assuming the compactness of characteristic solution operators. Sukavanam and Kumar [47] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and Schauder’s fixed-point theorem.

Consider an abstract semilinear equation
y = y 0 + L B v + L F ( y , v ) , Open image in new window
(1)
and define the following sets:
Here Y, X are separable reflexive Banach spaces and V is a Hilbert space, B L ( V , Y ) Open image in new window, L L ( Y , Y ) Open image in new window, Q L ( Y , X ) Open image in new window, F : Y × V Y × V Open image in new window is a nonlinear operator, y 0 Y Open image in new window, v V Open image in new window. Q R ( L , F ) Open image in new window is the set of points Qy, where y is a solution of (1), attainable from the point y 0 Open image in new window. The set Q R ( L , 0 ) Open image in new window is the set of points Qz, where z is a solution of
z = y 0 + L B v , Open image in new window
(2)
reachable from y 0 Open image in new window. One can see that for each h X Open image in new window, ε > 0 Open image in new window the control
v ε = ( Q L B ) J ( ( ε I + Γ J ) 1 ( h Q y 0 ) ) Open image in new window
(3)
transfers equation (2) from y 0 Open image in new window 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 ) , Open image in new window
where z ε = y 0 + L B v ε Open image in new window. It is known that Q R ( L , 0 ) ¯ = X Open image in new window if and only if
ε ( ε I + Γ J ) 1 ( h ) 0 Open image in new window

in the strong operator topology as ε 0 + Open image in new window, see [30]. Thus, the control (3) transfers system (2) from y 0 Y Open image in new window to a small neighborhood of an arbitrary point h X Open image in new window if and only if Q R ( L , 0 ) ¯ = X Open image in new window.

The same idea is now used to investigate the controllability of semilinear system (1). To do so, for each ε > 0 Open image in new window and h X Open image in new window, consider a nonlinear operator T ε Open image in new window from Y × V Open image in new window to Y × V Open image in new window defined by
T ε ( y , v ) = ( z , w ) , Open image in new window
(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 ) ) ) . Open image in new window
One can see that if the operator T ε Open image in new window has a fixed point ( y ε , v ε ) Open image in new window, then the control v ε Open image in new window steers control system (1) from y 0 Open image in new window to
Q y ε = h J ( ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y ε , v ε ) ) ) Open image in new window
if ε > 0 Open image in new window. We prove that Q y ε Open image in new window is close to h provided that ε ( ε I + Q L B ( Q L B ) ) 1 ( h ) 0 Open image in new window converges strongly to zero as ε 0 + Open image in new window. Therefore, to prove the approximate controllability of (1), for each ε > 0 Open image in new window and h X Open image in new window, 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 ) ) ) . Open image in new window
(5)

It is clear that the fixed points of the nonlinear operator T ε Open image in new window 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 Open image in new window, 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 Open image in new window stand for its dual space with respect to the continuous pairing , Open image in new window. We may assume, without loss of generality, that X and X Open image in new window 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 Open image in new window given by the following relations:
J ( z ) = z , J ( z ) , z = z 2 for all  z X Open image in new window

is bijective, demicontinuous, i.e., continuous from X with a strong topology into X Open image in new window with weak topology and strictly monotonic. Moreover, J 1 : X X Open image in new window is also a duality mapping.

An operator Γ : X X Open image in new window is symmetric if
z 1 , Γ z 2 = z 2 , Γ z 1 Open image in new window

for all z 1 , z 2 X Open image in new window. It is easy to see that Γ is linear and continuous. Γ is nonnegative if z , Γ z 0 Open image in new window for all z X Open image in new window.

Lemma 1 [31]

For every h X Open image in new window and ε > 0 Open image in new window, the equation
ε z ε + Γ J ( z ε ) = ε h Open image in new window
(6)
has a unique solution z ε = z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) Open image in new window and
z ε ( h ) = J ( z ε ( h ) ) h . Open image in new window
(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 Open image in new window for all nonzero z X Open image in new window.

     
  2. (ii)

    For all h X Open image in new window, J ( z ε ( h ) ) Open image in new window converges to zero as ε 0 + Open image in new window in the weak topology, where z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) Open image in new window is a solution of equation (6).

     
  3. (iii)

    For all h X Open image in new window, z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) Open image in new window strongly converges to zero as ε 0 + Open image in new window.

     

We impose the following assumptions:

(A1) F : Y × V Y Open image in new window is continuous and there exists C > 0 Open image in new window such that F ( y , v ) C Open image in new window for all ( y , v ) Y × V Open image in new window.

(A2) L : Y Y Open image in new window is compact.

(A3) For all h X Open image in new window, z ε ( h ) = ε ( ε I + Γ J ) 1 ( h ) Open image in new window strongly converges to zero as ε 0 + Open image in new window.

Note that the condition (A3) holds if and only if Im ( Q L B ) ¯ = Q R ( L , 0 ) ¯ = X Open image in new window, i.e., system (2) is approximately controllable.

Definition 3 System (1) is approximately controllable if
Q R ( L , F ) ¯ = X . Open image in new window

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

Proof Step 1. Show that the operator T ε Open image in new window has a fixed point in Y × V Open image in new window for all ε > 0 Open image in new window. For our convenience, let us introduce the following notation:
Assume that r ( ε ) d ( ε ) + C c ( ε ) Open image in new window. 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 ) ) , Open image in new window
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 ) . Open image in new window

Thus we proved that T ε Open image in new window maps B ε = { ( z , w ) Y × V : ( z , w ) r ( ε ) } Open image in new window into itself. On the other hand, the operator T ε Open image in new window is continuous and T ε ( B ε ) Open image in new window is relatively compact. By Schauder’s fixed-point theorem, for all ε > 0 Open image in new window, T ε Open image in new window has a fixed point in the ball B ε Open image in new window.

Step 2. Assume Q R ( L , 0 ) ¯ = X Open image in new window. By Step 1, the operator (4) has a fixed point ( y ε , v ε ) Open image in new window. So, ( y ε , v ε ) Open image in new window satisfies (5) and, moreover, it follows that for all h X Open image in new window
Q y ε h = ε ( ε I + Γ J ) 1 ( h Q y 0 Q L F ( y ε , v ε ) ) . Open image in new window
(8)
So, z ε : = Q y ε h Open image in new window 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 ) . Open image in new window
(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 ) } Open image in new window, which weakly converges to say z Y Open image in new window and L F ( y ε , v ε ) L z Open image in new window strongly in Y as ε 0 + Open image in new window. From (7) and strong convergence of the sequence { h ( z ε ) = h Q y 0 Q L F ( y ε , v ε ) } Open image in new window, it is easy to see that there exists C 1 > 0 Open image in new window such that for all ε > 0 Open image in new window
z ε = J ( z ε ) Q L F ( y ε , v ε ) + Q y 0 h C 1 . Open image in new window
Then we can extract a subsequence, still denoted by z ε Open image in new window, such that
J ( z ε ) J ( z ¯ 0 ) as  ε 0 + Open image in new window
for some z ¯ 0 Z Open image in new window. Applying J ( z ¯ 0 ) Open image in new window to equation (9) and taking the limit, we obtain
since Γ is positive. So, J ( z ε ) 0 Open image in new window as ε 0 + Open image in new window. Now, applying J ( z ε ) Open image in new window to equation (9), dividing through by ε and taking the limit, we obtain

Thus lim ε 0 + Q y ε h = 0 Open image in new window, consequently Q R ( L , F ) ¯ = X Open image in new window. 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 , Open image in new window
(10)

where the state variable x takes values in a separable reflexive Banach space X; D α c Open image in new window is the Caputo fractional derivative of order 1 2 < α < 1 Open image in new window; A is the infinitesimal generator of a C 0 Open image in new window semigroup S ( t ) Open image in new window of bounded operators on X; the control function u is given in L 2 ( [ 0 , b ] , U ) Open image in new window, U is a Hilbert space; B is a bounded linear operator from U into X, Δ = { ( t , s ) : 0 s t T } Open image in new window and g : Δ × X X Open image in new window, f : I × X × X X Open image in new window are continuous bounded functions and x 0 X Open image in new window.

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 , Open image in new window

provided the right-hand side is pointwise defined on [ 0 , ) Open image in new window, 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 Open image in new window 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 . Open image in new window
Definition 7 The Caputo derivative of order α for a function f : [ 0 , ) R Open image in new window 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 . Open image in new window
Remark 8
  1. (1)
    If f ( t ) C n [ 0 , ) Open image in new window, 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 . Open image in new window
     
  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 , Open image in new window
(11)

and η α Open image in new window is a probability density function defined on ( 0 , ) Open image in new window, that is, η α ( θ ) 0 Open image in new window, θ ( 0 , ) Open image in new window and 0 η α ( θ ) d θ = 1 Open image in new window.

Lemma 9 [34]

For any fixed t 0 Open image in new window, the operators S ˆ α ( t ) Open image in new window and S α ( t ) Open image in new window are linear compact and bounded operators, i.e., for any x X Open image in new window, S ˆ α ( t ) x M x Open image in new window and S α ( t ) x M Γ ( α ) x Open image in new window.

Definition 10 A solution x ( ; x 0 , u ) C ( [ 0 , b ] , X ) Open image in new window is said to be a mild solution of (10) if for any u L 2 ( [ 0 , b ] , U ) Open image in new window and the integral equation (11) is satisfied.

Let x b ( x 0 ; u ) Open image in new window be the state value of (10) at terminal time b corresponding to the control u and the initial value x 0 Open image in new window. Introduce the set ( b , x 0 ) = { x b ( x 0 ; u ) ( 0 ) : u L 2 ( [ 0 , b ] , U ) } Open image in new window, which is called the reachable set of system (10) at terminal time b, its closure in X is denoted by ( b , x 0 ) ¯ = X Open image in new window.

Definition 11 System (10) is said to be approximately controllable on J if ( b , x 0 ) ¯ = X Open image in new window, that is, given an arbitrary ϵ > 0 Open image in new window, it is possible to steer from the point x 0 Open image in new window 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 . Open image in new window
(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 , Open image in new window

where B Open image in new window denotes the adjoint of B and S α Open image in new window is the adjoint of S α Open image in new window. It is straightforward that the operator Γ 0 b Open image in new window is a linear bounded operator. By Theorem 2, linear fractional control system (12) is approximately controllable on [ 0 , b ] Open image in new window if and only if for any h X Open image in new window, z ε ( h ) = ε ( ε I + Γ 0 b J ) 1 ( h ) Open image in new window converges strongly to zero as ε 0 + Open image in new window.

Proposition 12 If S ( t ) Open image in new window, t > 0 Open image in new window, are compact operators and 0 < 1 p < α 1 Open image in new window, 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 ] , Open image in new window

is compact from L p ( [ 0 , b ] , X ) Open image in new window into C ( [ 0 , b ] , X ) Open image in new window.

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

    for arbitrary t [ 0 , b ] Open image in new window, the set { L α f ( t ) : f L p 1 } Open image in new window is relatively compact in C ( [ 0 , b ] , X ) Open image in new window;

     
  2. (ii)
    for arbitrary η > 0 Open image in new window, there exists δ > 0 Open image in new window such that
    L α f ( t ) L α f ( s ) < η if  f L p 1 , | t s | δ , t , s [ 0 , b ] . Open image in new window
     
To prove (i), fix 0 < t < b Open image in new window and define for 0 < η < t Open image in new window and δ > 0 Open image in new window operators L α η , δ Open image in new window from L p ( [ 0 , b ] , X ) Open image in new window 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 ) . Open image in new window
Since S ( t ) Open image in new window, t > 0 Open image in new window, is a compact operator, the operators L α η , δ Open image in new window 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 . Open image in new window
One can estimate J 1 Open image in new window and J 2 Open image in new window 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 ) , Open image in new window
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 , Open image in new window
where we have used the equality
0 θ β η α ( θ ) d θ = γ ( 1 + β ) γ ( 1 + α β ) . Open image in new window

Consequently, L α η , δ L α Open image in new window in the operator norm so that L α Open image in new window is compact and (i) follows immediately.

To prove (ii), note that, for 0 t t + h b Open image in new window and f L p 1 Open image in new window, we have
Applying the Hölder inequality, we obtain

It is clear that I 1 , I 2 0 Open image in new window as h 0 Open image in new window. On the other hand, the compactness of S ( t ) Open image in new window, t > 0 Open image in new window (and consequently S α ( t ) Open image in new window), implies the continuity of S α ( t ) Open image in new window, t > 0 Open image in new window, in the uniform operator topology. Then, by the Lebesque dominated convergence theorem, I 3 0 Open image in new window as h 0 Open image in new window. Thus the proof of (ii), and therefore the proof of the proposition, is complete. □

Theorem 13 Suppose S ( t ) Open image in new window, t > 0 Open image in new window, is compact and 1 2 < α 1 Open image in new window. Then system (10) is approximately controllable on [ 0 , b ] Open image in new window if the corresponding linear system is approximately controllable on [ 0 , b ] Open image in new window.

Proof Let Y = L 2 ( [ 0 , b ] , X ) Open image in new window, V = L 2 ( [ 0 , b ] , U ) Open image in new window, and y 0 = S α ( ) x 0 Y Open image in new window. Define the linear operators Q, L, L 1 Open image in new window and the nonlinear operator F by

for y Y Open image in new window, v V Open image in new window. 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 , Open image in new window
(13)
where u L 2 [ 0 , b ] Open image in new window, X = L 2 [ 0 , π ] Open image in new window, h X Open image in new window, 1 2 < α < 1 Open image in new window, and f : R × R R Open image in new window, g : R × R × R R Open image in new window are continuous and uniformly bounded. Let B L ( R , X ) Open image in new window be defined as
( B u ) ( θ ) = b ( θ ) u , B h = 0 π h ( θ ) b ( θ ) d θ , Open image in new window
where 0 θ π Open image in new window, u R Open image in new window, b ( θ ) L 2 [ 0 , π ] Open image in new window, and let A : X X Open image in new window be an operator defined by A z = z Open image in new window with the domain
D ( A ) = { z X z , z  are absolutely continuous,  z X , z ( 0 ) = z ( π ) = 0 } . Open image in new window
Then
A z = n = 1 ( n 2 ) ( z , e n ) e n , z D ( A ) , Open image in new window
where e n ( θ ) = 2 / π sin n θ Open image in new window, 0 x π Open image in new window, n = 1 , 2 , Open image in new window . It is known that A generates a compact semigroup S ( t ) Open image in new window, t > 0 Open image in new window, in X and is given by
Then B S α ( t ) z = 0 Open image in new window for 0 t < b Open image in new window implies
( z , e n ) ( b , e n ) = 0 for all  n = 1 , 2 , . Open image in new window

Now if ( b , e n ) 0 Open image in new window for all n, then ( z , e n ) = 0 Open image in new window for all n and z = 0 Open image in new window. Therefore, the associated linear system is approximately controllable provided that 0 π b ( θ ) e n ( θ ) d θ 0 Open image in new window for n = 1 , 2 , 3 , Open image in new window . Because of the compactness of the semigroup S ( t ) Open image in new window (and consequently S ˆ α ( t ) Open image in new window, S α ( t ) Open image in new window) 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 ] Open image in new window.

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.

Notes

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.

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© Mahmudov; licensee Springer. 2013

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

Authors and Affiliations

  1. 1.Eastern Mediterranean UniversityFamagusta, T.R. North CyprusTurkey

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