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Boundary Value Problems

, 2013:118 | Cite as

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

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  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

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.

Keywords

Fractional Calculus Mild Solution Fractional Differential Equation Approximate Controllability Caputo Fractional Derivative 
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

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 [1, 2, 3, 4, 5, 6, 7, 8, 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] 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 ) , Open image in new window
(1)

in X α Open image in new window, where D t β C Open image in new window, 0 < β < 1 Open image in new window, stands for the Caputo fractional derivative of order β, and f : [ 0 , T ] × X α X Open image in new window, g : [ 0 , T ] × X α X Open image in new window, K : [ 0 , T ] R + Open image in new window, h : C ( [ 0 , T ] ; X α ) X α Open image in new window are given functions to be specified later. Here, ( A , D ( A ) ) Open image in new window is the infinitesimal generator of a compact analytic semigroup of bounded linear operators S ( t ) Open image in new window, t 0 Open image in new window, 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 ) Open image in new window defined in (5). We show that the control system (1) is approximately controllable on [ 0 , T ] Open image in new window 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 : = , Open image in new window. Let C ( [ 0 , T ] , X ) Open image in new window be the Banach space of continuous functions from [ 0 , T ] Open image in new window into X with the norm x = sup t [ 0 , T ] x ( t ) Open image in new window, here x C ( [ 0 , T ] , X ) Open image in new window. In this paper, we also assume that A : D ( A ) X X Open image in new window is the infinitesimal generator of a compact analytic semigroup S ( t ) Open image in new window, t > 0 Open image in new window, of uniformly bounded linear operator in X, that is, there exists M > 1 Open image in new window such that S ( t ) M Open image in new window for all t 0 Open image in new window. Without loss of generality, let 0 ρ ( A ) Open image in new window, where ρ ( A ) Open image in new window is the resolvent set of A. Then for any α > 0 Open image in new window, we can define A α Open image in new window by
A α : = 1 Γ ( α ) 0 t α 1 S ( t ) d t . Open image in new window

It follows that each A α Open image in new window is an injective continuous endomorphism of X. Hence we can define A α : = ( A α ) 1 Open image in new window, which is a closed bijective linear operator in X. It can be shown that each A α Open image in new window has dense domain and that D ( A β ) D ( A α ) Open image in new window for 0 α β Open image in new window. Moreover, A α + β x = A α A β x = A β A α x Open image in new window for every α , β R Open image in new window and x D ( A μ ) Open image in new window with μ : = max ( α , β , α + β ) Open image in new window, where A 0 = I Open image in new window, I is the identity in X. (For proofs of these facts, we refer to the literature [15, 20, 22].)

We denote by X α Open image in new window the Hilbert space of D ( A α ) Open image in new window equipped with norm x α : = A α x = A α x , A α x Open image in new window for x D ( A α ) Open image in new window, which is equivalent to the graph norm of A α Open image in new window. Then we have X β X α Open image in new window, for 0 α β Open image in new window (with X 0 = X Open image in new window ) and the embedding is continuous. Moreover, A α Open image in new window has the following basic properties.

Lemma 1 [42]

A α Open image in new window and S ( t ) Open image in new window have the following properties.
  1. (i)

    S ( t ) : X X α Open image in new window for each t > 0 Open image in new window and α 0 Open image in new window.

     
  2. (ii)

    A α S ( t ) x = S ( t ) A α x Open image in new window for each x D ( A α ) Open image in new window and t 0 Open image in new window.

     
  3. (iii)
    For every t > 0 Open image in new window, A α S ( t ) Open image in new window is bounded in X and there exists M α > 0 Open image in new window such that
    A α S ( t ) M α t α . Open image in new window
     
  4. (iv)

    A α Open image in new window is a bounded linear operator for 0 α 1 Open image in new window.

     

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 Open image in new window 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 3 The Caputo derivative of order α > 0 Open image in new window 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 . Open image in new window

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 ] , Open image in new window
(2)
provided that the integral in (2) exists. Applying the Laplace transform
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 , Open image in new window

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

For x X Open image in new window, we define two families { S β ( t ) : t 0 } Open image in new window and { P β ( t ) : t 0 } Open image in new window of operators by

respectively.

The following lemma follows from the results given in [37, 38, 39].

Lemma 4 The operators S β Open image in new window and P β Open image in new window have the following properties.
  1. (i)
    For any fixed t 0 Open image in new window, and any x X α Open image in new window, we have the operators S β ( t ) Open image in new window and P β ( t ) Open image in new window are linear and bounded operators, i.e. for any x X Open image in new window,
    S β ( t ) x α M x α and P β ( t ) x α M Γ ( β ) x α . Open image in new window
     
  2. (ii)

    The operators S β ( t ) Open image in new window and P β ( t ) Open image in new window are strongly continuous for all t 0 Open image in new window.

     
  3. (iii)

    S β ( t ) Open image in new window and P β ( t ) Open image in new window are norm continuous in X for t > 0 Open image in new window.

     
  4. (iv)

    S β ( t ) Open image in new window and P β ( t ) Open image in new window are compact operators in X for t > 0 Open image in new window.

     
  5. (v)

    For every t > 0 Open image in new window, the restriction of S β ( t ) Open image in new window to X α Open image in new window and the restriction of P β ( t ) Open image in new window to X α Open image in new window are norm continuous.

     
  6. (vi)

    For every t > 0 Open image in new window, the restriction of S β ( t ) Open image in new window to X α Open image in new window and the restriction of P β ( t ) Open image in new window to X α Open image in new window are compact operators in X α Open image in new window.

     
  7. (vii)
    For all x X Open image in new window and t ( 0 , ) Open image in new window,
    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 ) ) . Open image in new window
     

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 ) Open image in new window is said to be a mild solution of (1) if for any u L 2 ( [ 0 , T ] , U ) Open image in new window 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 , Open image in new window
(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 ) Open image in new window is bounded if 1 2 < β 1 Open image in new window. In what follows, we assume that 1 2 < β 1 Open image in new window.

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 Open image in new window 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 ) Open image in new window be the state value of (1) at terminal time T corresponding to the control u and the initial value x 0 Open image in new window. Introduce the set ( T , x 0 ) = { x ( T ; x 0 , u ) : u L 2 ( [ 0 , T ] , U ) } Open image in new window, which is called the reachable set of system (1) at terminal time T, its closure in X α Open image in new window is denoted by ( T , x 0 ) ¯ Open image in new window.

Definition 6 The system (1) is said to be approximately controllable on [ 0 , T ] Open image in new window if ( T , x 0 ) ¯ = X α 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 α Open image in new window 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 . Open image in new window
(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

respectively, where B Open image in new window denotes the adjoint of B and P β ( t ) Open image in new window is the adjoint of P β ( t ) Open image in new window. It is straightforward that the operator Γ 0 T Open image in new window is a linear bounded operator.

Theorem 7 [10]

Let Z be a separable reflexive Banach space and let Z Open image in new window stands for its dual space. Assume that Γ : Z Z Open image in new window is symmetric. Then the following two conditions are equivalent:
  1. 1.

    Γ : Z Z Open image in new window is positive, that is, z , Γ z > 0 Open image in new window for all nonzero z Z Open image in new window.

     
  2. 2.

    For all h Z 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. Here, J is the duality mapping of Z into Z Open image in new window.

     

Lemma 8 The linear fractional control system (4) is approximately controllable on [ 0 , T ] Open image in new window if and only if ε R ( ε , Γ 0 T ) 0 Open image in new window as ε 0 + Open image in new window 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 ] Open image in new window if and only if Γ 0 T x , x > 0 Open image in new window for all nonzero x X Open image in new window, see [7]. By Theorem 7, ε ( ε I + Γ 0 T ) 1 ( h ) 0 Open image in new window as ε 0 + Open image in new window for all h X Open image in new window. □

Remark 9 Notice that positivity of Γ 0 T Open image in new window is equivalent to Γ 0 T x , x = 0 x = 0 Open image in new window. In other words, since Γ 0 T x , x = 0 T ( T s ) β 1 B P β ( T s ) x 2 d s Open image in new window, approximate controllability of the linear system (4) is equivalent to B P β ( T s ) x = 0 Open image in new window, 0 s < T x = 0 Open image in new window.

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

(H1) f , g : [ 0 , T ] × X α × X α X Open image in new window are continuous and for each r N Open image in new window, there exists a constant γ [ 0 , β ( 1 α ) ] Open image in new window and functions φ r L 1 / γ ( [ 0 , T ] ; R + ) Open image in new window, ψ r L ( [ 0 , T ] ; R + ) Open image in new window such that

(H2) h : C ( [ 0 , T ] ; X α ) X α Open image in new window is a Lipschitz function with Lipschitz constant L h Open image in new window.

(H c ) The linear system (4) is approximately controllable on [ 0 , T ] Open image in new window.

Using the hypothesis (H c ), for an arbitrary function x C ( [ 0 , T ] ; X α ) Open image in new window, 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 ] . Open image in new window
(5)
Let B r = { x C ( [ 0 , T ] ; X α ) : x α r } Open image in new window, where r is a positive constant. Then B r Open image in new window is clearly a bounded closed and convex subset in C ( [ 0 , T ] ; X α ) Open image in new window. We will show that when using the above control the operator Λ ε : B k B k Open image in new window defined by
( Λ ε x ) ( t ) : = ( Φ ε x ) ( t ) + ( Π ε x ) ( t ) , t [ 0 , T ] , Open image in new window

has a fixed point in C ( [ 0 , T ] ; X α ) Open image in new window.

Theorem 10 Let the assumptions (H1) and (H2) be satisfied. Then for x 0 X α Open image in new window, the fractional Cauchy problem (1) with u = u ε ( t , x ) Open image in new window 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 , Open image in new window
(6)

Proof It is easy to see that for any ε > 0 Open image in new window the operator Λ ε Open image in new window maps C ( [ 0 , T ] ; X α ) Open image in new window into itself.

Let x B r Open image in new window and 0 t T Open image in new window. Using assumption (H1) yield the following estimations,
From (6) and the assumption (H2), it follows that for any ε > 0 Open image in new window there exists r ( ε ) > 0 Open image in new window such that

Therefore, from (7) and (8), it follows that for any ε > 0 Open image in new window there exists r ( ε ) > 0 Open image in new window such that Φ ε y + Π ε x B r ( ε ) Open image in new window for every x , y B r ( ε ) Open image in new window. Therefore, for any ε > 0 Open image in new window the fractional Cauchy problem (1) with the control (5) has a mild solution if and only if the operator Φ ε + Π ε Open image in new window has a fixed point in B r ( ε ) Open image in new window.

In what follows, we will show that Φ ε Open image in new window and Π ε Open image in new window satisfy the conditions of Krasnoselskii’s fixed-point theorem. From (H2) and (6), we infer that Φ ε Open image in new window is a contraction. Next, we show that Π ε Open image in new window is completely continuous on B r ( ε ) Open image in new window.

Step 1: We first prove that Π ε Open image in new window is continuous on B r ( ε ) Open image in new window. Let { x n } n = 1 B r ( ε ) Open image in new window be a sequence such that x n x Open image in new window as n Open image in new window in C ( [ 0 , T ] ; X α ) Open image in new window. Therefore, it follows from the continuity of f, g and u ε Open image in new window that for each t [ 0 , T ] Open image in new window,
Also, by (H1), we see that
using the Lebesgue dominated convergence theorem that for all t [ 0 , T ] Open image in new window, we conclude
( Π ε x n ) ( t ) ( Π ε x ) ( t ) α 0 , as  n , Open image in new window

implying that Π ε x n Π ε x α 0 Open image in new window as n Open image in new window. This proves that Π ε Open image in new window is continuous on B r ( ε ) Open image in new window.

Step 2. Π ε Open image in new window is compact on B r ( ε ) Open image in new window.

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 ) . Open image in new window
Let t [ 0 , T ] Open image in new window be fixed and δ , η > 0 Open image in new window be small enough. For x B r ( ε ) Open image in new window, 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 . Open image in new window
Therefore, from Lemma 4, we see that for each t ( 0 , T ] Open image in new window, the set { ( Π ε δ η x ) ( t ) : x B r ( ε ) } Open image in new window is relatively compact in X α Open image in new window. Since

approaches to zero as η 0 + Open image in new window, using the total boundedness, we conclude that for each t [ 0 , T ] Open image in new window, the set { ( Π ε δ η x ) ( t ) : x B r ( ε ) } Open image in new window is relatively compact in X α Open image in new window.

On the other hand, for 0 < t 1 < t 2 T Open image in new window and δ > 0 Open image in new window small enough, we have
( Π ε x ) ( t 1 ) ( Π ε x ) ( t 2 ) α I 1 + I 2 + I 3 + I 4 , Open image in new window
Therefore, it follows from (H1) and Lemma 4 that
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 ] , Open image in new window
from which it is easy to see that all I i Open image in new window, i = 1 , 2 , 3 , 4 Open image in new window, tend to zero independent of x B k Open image in new window as t 2 t 1 0 Open image in new window and δ 0 Open image in new window. Thus, we can conclude that
( Π ε x ) ( t 1 ) ( Π ε x ) ( t 2 ) α 0 as  t 2 t 1 0 , Open image in new window

and the limit is independent of x B r ( ε ) Open image in new window. The case t 1 = 0 Open image in new window is trivial. Consequently, the set { ( Π ε x ) ( t ) : t [ 0 , T ] , x B r ( ε ) } Open image in new window is equicontinuous. Now applying the Arzela-Ascoli theorem, it results that Π ε Open image in new window is compact on B r ( ε ) Open image in new window.

Therefore, applying Krasnoselskii’s fixed-point theorem, we conclude that Λ ε Open image in new window 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 Open image in new window and h : C ( [ 0 , T ] ; X α ) X α Open image in new window are bounded and M L h < 1 Open image in new window. Then the semilinear fractional system (3) is approximately controllable on [ 0 , T ] Open image in new window.

Proof It is clear that all assumptions of Theorem 10 are satisfied with σ 1 = σ 2 = 0 Open image in new window. Let x ε Open image in new window be a fixed point of F ε Open image in new window in B r Open image in new window. Any fixed point of F ε Open image in new window 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 ) Open image in new window
and satisfies the equality
x ε ( T ) = h ε R ( ε , Γ 0 T ) p ( x ε ) , Open image in new window
(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 ) . Open image in new window
Moreover, by the boundedness of the functions f and g and Dunford-Pettis theorem, we have that the sequences { f ( s , x ε ( s ) ) } Open image in new window and { g ( s , x ε ( s ) ) } Open image in new window are weakly compact in L 2 ( [ 0 , T ] ; X ) Open image in new window, so there are subsequences still denoted by { f ( s , x ε ( s ) ) } Open image in new window and { g ( s , x ε ( s ) ) } Open image in new window, that weakly converge to, say, f and g in L 2 ( [ 0 , T ] ; X ) Open image in new window. On the other hand, there exists h ˜ X α Open image in new window such that h ( x ε ) Open image in new window converges to h ˜ Open image in new window weakly in X α Open image in new window. 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 . Open image in new window
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 Open image in new window
as ε 0 + Open image in new window 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 ) . Open image in new window
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 Open image in new window
(10)

as ε 0 + Open image in new window. 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 , Open image in new window
(11)

where f , g : [ 0 , T ] × [ 0 , π ] × R R Open image in new window, k : [ 0 , π ] × [ 0 , π ] R Open image in new window, 0 < t 1 < < t p < T Open image in new window.

Let us take X = U = L 2 [ 0 , π ] Open image in new window and define the operator A by A w = w Open image in new window 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 } Open image in new window. Then
A w = n = 1 n 2 w , e n e n , w D ( A ) , Open image in new window
where e n ( z ) = 2 π sin n z Open image in new window, 0 z π Open image in new window, n = 1 , 2 , Open image in new window . Clearly −A generates a compact analytic semigroup S ( t ) Open image in new window, t > 0 Open image in new window in X and it is given by
S ( t ) w = n = 1 e n 2 t w , e n e n , w X . Open image in new window

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 ] Open image in new window; see [12].

The operator A 1 2 Open image in new window is given by
A 1 2 w = n = 1 n w , e n e n , w D ( A 1 2 ) , Open image in new window

where D ( A 1 2 ) = { w X : n = 1 n w , e n e n X } Open image in new window and A 1 2 = 1 Open image in new window.

Let X 1 2 : = ( D ( A 1 2 ) , 1 / 2 ) Open image in new window, where x 1 / 2 : = A 1 2 x X Open image in new window for x D ( A 1 2 ) Open image in new window. Assume that F , G : [ 0 , T ] × [ 0 , π ] × R R Open image in new window satisfies the following conditions:
  1. 1.

    The functions F ( , , ) Open image in new window, G ( , , ) Open image in new window are continuous and uniformly bounded.

     
  2. 2.

    F ( 0 , , ) = F ( π , , ) = G ( 0 , , ) = G ( π , , ) = 0 Open image in new window.

     
  3. 3.
    k : [ 0 , π ] × [ 0 , π ] R Open image in new window is continuously differentiable, k ( 0 , ) = k ( π , ) = 0 Open image in new window and
    0 π 0 π | 2 ξ 2 k ( ξ , y ) | 2 d y d ξ < . Open image in new window
     
Denote by E β , ζ Open image in new window, the Mittag-Leffler special function defined by
E β , ζ = k = 0 t k Γ ( ζ k + β ) , ζ , β > 0 , t R . Open image in new window
Then, for each x , y C ( [ 0 , T ] , X 1 / 2 ) Open image in new window 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 Open image in new window
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 . Open image in new window

It follows that h : C ( [ 0 , T ] ; X 1 / 2 ) X 1 / 2 Open image in new window is bounded and Lipschitz continuous. On the other hand, it is not difficult to verify that f , g : [ 0 , T ] × X 1 / 2 X Open image in new window are continuous.

Next, we show that the linear system corresponding to (11) is approximately controllable on [ 0 , T ] Open image in new window. It is clear that P β ( t ) : X 1 2 X 1 2 Open image in new window is defined as follows:

By Remark 9, the linear system corresponding to (11) is approximately controllable on [ 0 , T ] Open image in new window if and only if B P β ( T t ) x = 0 Open image in new window, 0 t < T Open image in new window implies that x = 0 Open image in new window. This follows from the representation of B P β ( T t ) x Open image in new window.

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 ] Open image in new window, 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 . Open image in new window

Notes

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.

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Copyright information

© Mahmudov and Zorlu; 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 UniversityGazimagusaTurkey

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