Open Access

Controllability for Sobolev type fractional integro-differential systems in a Banach space

Advances in Difference Equations20122012:167

https://doi.org/10.1186/1687-1847-2012-167

Received: 19 March 2012

Accepted: 10 September 2012

Published: 25 September 2012

Abstract

In this paper, by using compact semigroups and the Schauder fixed-point theorem, we study the sufficient conditions for controllability of Sobolev type fractional integro-differential systems in a Banach space. An example is provided to illustrate the obtained results.

MSC:26A33, 34G20, 93B05.

Keywords

fractional calculusSobolev type fractional integro-differential systemscontrollabilitycompact semigroupmild solutionSchauder fixed-point theorem

1 Introduction

A Sobolev-type equation appears in a variety of physical problems such as flow of fluids through fissured rocks, thermodynamics and propagation of long waves of small amplitude (see [13]). Recently, there has been an increasing interest in studying the problem of controllability of Sobolev type integro-differential systems. Balachandran and Dauer [4] studied the controllability of Sobolev type integro-differential systems in Banach spaces. Balachandran and Sakthivel [5] studied the controllability of Sobolev type semilinear integro-differential systems in Banach spaces. Balachandran, Anandhi and Dauer [6] studied the boundary controllability of Sobolev type abstract nonlinear integro-differential systems.

In this paper, we study the controllability of Sobolev type fractional integro-differential systems in Banach spaces in the following form:
(1.1)

where E and A are linear operators with domain contained in a Banach space X and ranges contained in a Banach space Y. The control function u ( ) is in L 2 ( J , U ) , a Banach space of admissible control functions, with U as a Banach space. B is a bounded linear operator from U into Y. The nonlinear operators f C ( J × X , Y ) , H C ( J × J × X , X ) and g C ( J × J × X × X , Y ) are all uniformly bounded continuous operators. The fractional derivative D α c , 0 < α < 1 is understood in the Caputo sense.

2 Preliminaries

In this section, we introduce preliminary facts which are used throughout this paper.

Definition 2.1 (see [79])

The fractional integral of order α > 0 with the lower limit zero for a function f can be defined as
I α f ( t ) = 1 Γ ( α ) 0 t f ( s ) ( t s ) 1 α d s , t > 0

provided the right-hand side is pointwise defined on [ 0 , ) , where Γ is the gamma function.

Definition 2.2 (see [79])

The Caputo derivative of order α with the lower limit zero 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 .

If f is an abstract function with values in X, then the integrals appearing in the above definitions are taken in Bochner’s sense.

The operators A : D ( A ) X Y and E : D ( E ) X Y satisfy the following hypotheses:

( H 1 ) A and E are closed linear operators,

( H 2 ) D ( E ) D ( A ) and E is bijective,

( H 3 ) E 1 : Y D ( E ) is continuous.

The hypotheses H 1 , H 2 and the closed graph theorem imply the boundedness of the linear operator A E 1 : Y Y .

( H 4 ) For each t [ 0 , a ] and for some λ ρ ( A E 1 ) , the resolvent set of A E 1 , the resolvent R ( λ , A E 1 ) is a compact operator.

Lemma 2.1 [10]

Let S ( t ) be a uniformly continuous semigroup. If the resolvent set R ( λ ; A ) of A is compact for every λ ρ ( A ) , then S ( t ) is a compact semigroup.

From the above fact, A E 1 generates a compact semigroup { T ( t ) , t 0 } in Y, which means that there exists M > 1 such that
max t J T ( t ) M .
(2.1)

Definition 2.3 The system (1.1) is said to be controllable on the interval J if for every x 0 , x 1 X , there exists a control u L 2 ( J , U ) such that the solution x ( ) of (1.1) satisfies x ( a ) = x 1 .

( H 5 ) The linear operator W from U into X defined by
W u = 0 a E 1 ( a s ) α 1 T α ( a s ) B u ( s ) d s

has an inverse bounded operator W 1 which takes values in L 2 ( J , U ) / ker W , where the kernel space of W is defined by ker W = { x L 2 ( J , U ) : W x = 0 } , B is a bounded linear operator and T α ( t ) is defined later.

( H 6 ) The function f satisfies the following two conditions:
  1. (i)

    For each t J , the function f ( t , ) : X Y is continuous, and for each x X , the function f ( , x ) : J Y is strongly measurable.

     
  2. (ii)
    For each positive number k N , there is a positive function g k ( ) : [ 0 , a ] R + such that
    sup | x | k | f ( t , x ) | g k ( t ) ,
     
the function s ( t s ) 1 α g k ( s ) L 1 ( [ 0 , t ] , R + ) , and there exists a β > 0 such that
lim k inf 0 t ( t s ) 1 α g k ( s ) d s k = β < , t [ 0 , a ] .

( H 7 ) For each ( t , s ) J × J , the function H ( t , s , ) : X X is continuous, and for each x X , the function H ( , , x ) : J × J X is strongly measurable.

( H 8 ) The function g satisfies the following two conditions:
  1. (i)

    For each ( t , s , x ) J × J × X , the function g ( t , s , , ) : X × X Y is continuous, and for each x X , H X , the function g ( , x , y ) : J × J Y is strongly measurable.

     
  2. (ii)
    For each positive number k N , there is a positive function h k ( ) : [ 0 , a ] R + such that
    sup | x | k | 0 t g ( t , s , x , 0 s H ( s , τ , x ) d τ ) d s | h k ( t ) ,
     
the function s ( t s ) 1 α h k ( s ) L 1 ( [ 0 , t ] , R + ) , and there exists a γ > 0 such that
lim k inf 0 t ( t s ) 1 α h k ( s ) d s k = γ < , t [ 0 , a ] .
According to [11, 12], a solution of equation (1.1) can be represented by
x ( t ) = E 1 S α ( t ) E x 0 + 0 t ( t s ) α 1 T α ( t s ) E 1 f ( s , x ( s ) ) d s + 0 t ( t s ) α 1 E 1 T α ( t s ) B u ( s ) d s + 0 t ( t s ) α 1 E 1 T α ( t s ) { 0 s g ( s , τ , x ( τ ) , R ( τ ) ) d τ } d s , t J ,
(2.2)
where

with ξ α being a probability density function defined on ( 0 , ) , that is, ξ α ( θ ) 0 , θ ( 0 , ) and 0 ξ α ( θ ) d θ = 1 .

Remark 0 θ ξ α ( θ ) d θ = 1 Γ ( 1 + α ) .

Definition 2.4 By a mild solution of the problem (1.1), we mean that the function x C ( J , X ) satisfies the integral equation (2.2).

Lemma 2.2 (see [11])

The operators S α ( t ) and T α ( t ) have the following properties:
  1. (I)

    For any fixed x X , S α ( t ) x M x , T α ( t ) x α M Γ ( α + 1 ) x ;

     
  2. (II)

    { S α ( t ) , t 0 } and { T α ( t ) , t 0 } are strongly continuous;

     
  3. (III)

    For every t > 0 , S α ( t ) and T α ( t ) are also compact operators if T ( t ) , t > 0 is compact.

     

3 Controllability result

In this section, we present and prove our main result.

Theorem 3.1 If the assumptions ( H 1 ) - ( H 8 ) are satisfied, then the system (1.1) is controllable on J provided that α M E 1 Γ ( α + 1 ) ( β + γ ) [ 1 + a α M E 1 Γ ( α + 1 ) B W 1 ] < 1 .

Proof Using the assumption ( H 5 ) , for an arbitrary function x ( ) , define the control
u ( t ) = W 1 [ x 1 E 1 S α ( t ) E x 0 0 a ( a s ) α 1 E 1 T α ( a s ) f ( s , x ( s ) ) d s 0 a ( a s ) α 1 E 1 T α ( a s ) { 0 s g ( s , τ , x ( τ ) , R ( τ ) ) d τ } d s ] ( t ) .
It shall now be shown that when using this control, the operator Q defined by
( Q x ) ( t ) = E 1 S α ( t ) E x 0 + 0 t ( t s ) α 1 E 1 T α ( t s ) f ( s , x ( s ) ) d s + 0 t ( t s ) α 1 E 1 T α ( t s ) B u ( s ) d s + 0 t ( t s ) α 1 E 1 T α ( t s ) { 0 s g ( s , τ , x ( τ ) , R ( τ ) ) d τ } d s
from C ( J , X ) into itself for each x C = C ( J , X ) has a fixed point. This fixed point is then a solution of equation (2.2).
( Q x ) ( a ) = E 1 S α ( a ) E x 0 + 0 a ( a s ) α 1 E 1 T α ( a s ) f ( s , x ( s ) ) d s + 0 a ( a s ) α 1 E 1 T α ( a s ) B W 1 × [ x 1 E 1 S α ( a ) E x 0 0 a ( a τ ) α 1 E 1 T α ( a τ ) f ( τ , x ( τ ) ) d τ 0 a ( a τ ) α 1 E 1 T α ( a τ ) { 0 τ g ( τ , η , x ( η ) , R ( η ) ) d η } d τ ] ( s ) d s + α 0 a ( a s ) α 1 E 1 T α ( a s ) { 0 s g ( s , τ , x ( τ ) , R ( τ ) ) d τ } d s = x 1 .

It can be easily verified that Q maps C into itself continuously.

For each positive number k > 0 , let B k = { x C : x ( 0 ) = x 0 , x ( t ) k , t J } . Obviously, B k is clearly a bounded, closed, convex subset in C. We claim that there exists a positive number k such that Q B k B k . If this is not true, then for each positive number k, there exists a function x k B k with Q x k B k , that is, Q x k k , then 1 1 k Q x k , and so
1 lim k k 1 Q x k .
However,
a contradiction. Hence, Q B k B k for some positive number k. In fact, the operator Q maps B k into a compact subset of B k . To prove this, we first show that the set V k ( t ) = { ( Q x ) ( t ) : x B k } is a precompact in X; for every t J : This is trivial for t = 0 , since V k ( 0 ) = { x 0 } . Let t, 0 < t a ; be fixed. For 0 < ϵ < t and arbitrary δ > 0 ; take
( Q ϵ , δ x ) ( t ) = δ ξ α ( θ ) E 1 T ( t α θ ) E x 0 d θ + α 0 t ϵ δ θ ( t s ) α 1 ξ α ( θ ) E 1 T ( ( t s ) α θ ) f ( s , x ( s ) ) d θ d s + α 0 t ϵ δ θ ( t s ) α 1 ξ α ( θ ) E 1 T ( ( t s ) α θ ) × B W 1 [ x 1 0 ξ α ( θ ) E 1 T ( a α θ ) E x 0 d θ α 0 a 0 θ ( a τ ) α 1 ξ α ( θ ) E 1 T ( ( a τ ) α θ ) f ( τ , x ( τ ) ) d θ d τ α 0 a 0 θ ( a τ ) α 1 ξ α ( θ ) E 1 T ( ( a τ ) α θ ) × { 0 τ g ( τ , η , x ( η ) , R ( η ) ) d η } d θ d τ ] ( s ) d θ d s + α 0 t ϵ δ θ ( t s ) α 1 ξ α ( θ ) E 1 T ( ( t s ) α θ ) × { 0 s g ( s , τ , x ( τ ) , R ( τ ) ) d τ } d θ d s = T ( ϵ α δ ) δ ξ α ( θ ) E 1 T ( t α θ ϵ α δ ) E x 0 d θ + T ( ϵ α δ ) α 0 t ϵ δ θ ( t s ) α 1 ξ α ( θ ) E 1 T ( ( t s ) α θ ϵ α δ ) f ( s , x ( s ) ) d θ d s + T ( ϵ α δ ) α 0 t ϵ δ θ ( t s ) α 1 ξ α ( θ ) E 1 T ( ( t s ) α θ ϵ α δ ) × B W 1 [ x 1 0 ξ α ( θ ) E 1 T ( a α θ ) E x 0 d θ α 0 a 0 θ ( a τ ) α 1 ξ α ( θ ) E 1 T ( ( a τ ) α θ ) f ( τ , x ( τ ) ) d θ d τ 0 a 0 θ ( a τ ) α 1 ξ α ( θ ) E 1 T ( ( a τ ) α θ ) × { 0 τ g ( τ , η , x ( η ) , R ( η ) ) d η } d θ d τ ] ( s ) d θ d s + T ( ϵ α δ ) α 0 t ϵ δ θ ( t s ) α 1 ξ α ( θ ) E 1 T ( ( t s ) α θ ϵ α δ ) × { 0 s g ( s , τ , x ( τ ) , R ( τ ) ) d τ } d θ d s .
Since u ( s ) is bounded and T ( ϵ α δ ) , ϵ α δ > 0 is a compact operator, then the set V k ϵ , δ ( t ) = { ( Q ϵ , δ x ) ( t ) : x B k } is a precompact set in X for every ϵ, 0 < ϵ < t , and for all δ > 0 . Also, for x B k , using the defined control u ( t ) yields

Therefore, as ϵ 0 + and δ 0 + , there are precompact sets arbitrary close to the set V k ( t ) and so V k ( t ) is precompact in X.

Next, we show that Q B k = { Q x : x B k } is an equicontinuous family of functions.

Let x B k and t , τ J such that 0 < t < τ , then

Now, T ( t ) is continuous in the uniform operator topology for t > 0 since T ( t ) is compact, and the right-hand side of the above inequality tends to zero as t τ . Thus, Q B k is both equicontinuous and bounded. By the Arzela-Ascoli theorem, Q B k is precompact in C ( J , X ) . Hence, Q is a completely continuous operator on C ( J , X ) .

From the Schauder fixed-point theorem, Q has a fixed point in B k . Any fixed point of Q is a mild solution of (1.1) on J satisfying ( Q x ) ( t ) = x ( t ) X . Thus, the system (1.1) is controllable on J. □

4 Example

In this section, we present an example to our abstract results.

We consider the fractional integro-partial differential equation in the form
(4.1)

where t α c is the Caputo fractional partial derivative of order 0 < α < 1 .

Take X = Y = L 2 [ 0 , π ] and define the operators A : D ( A ) X Y and E : D ( E ) X Y by A z = z x x and E z = z z x x , where each domain D ( A ) and D ( E ) is given by { z X : z , z x  are absolutely continuous , z x x X , z ( 0 ) = z ( π ) = 0 } .

Then A and E can be written respectively as [13]
where z n ( x ) = 2 / π sin n x , n = 1 , 2 ,  , is the orthonormal set of eigenvectors of A and ( z , z n ) is the L 2 inner product. Moreover, for z X , we get

We assume that

( A 1 ) : The operator B : U Y , with U J , is a bounded linear operator.

( A 2 ) : The linear operator W : U X defined by
W u = 0 a E 1 ( a s ) α 1 T α ( a s ) B u ( s ) d s

has an inverse bounded operator W 1 which takes values in L 2 ( J , U ) / ker W , where the kernel space of W is defined by ker W = { x L 2 ( J , U ) : W x = 0 } , B is a bounded linear operator.

( A 3 ) : The nonlinear operator μ 1 : J × X Y satisfies the following three conditions:
  1. (i)

    For each t J , μ 1 ( t , z ) is continuous.

     
  2. (ii)

    For each z X , μ 1 ( t , z ) is measurable.

     
  3. (iii)
    There is a constant ν ( 0 < ν < 1 ) and a function h ( ) : [ 0 , a ] R + such that for all ( t , z ) J × X ,
    μ 1 ( t , z ) h ( t ) | z | ν .
     
( A 4 ) : The nonlinear operator μ 2 : J × J × X X satisfies the following two conditions:
  1. (i)

    For each ( t , s ) J × J , μ 2 ( t , s , z ) is continuous.

     
  2. (ii)

    For each z X , μ 2 ( t , s , z ) is measurable.

     
( A 5 ) : The nonlinear operator μ 3 : J × J × X × X Y satisfies the following three conditions:
  1. (i)

    For each ( t , s , z ) J × J × X , μ 3 ( t , s , z ) is continuous.

     
  2. (ii)

    For each z X , μ 3 ( t , s , z ) is measurable.

     
  3. (iii)
    There is a constant ν ( 0 < ν < 1 ) and a function g ( ) : [ 0 , a ] R + such that for all ( t , s , z , y ) J × J × X × X ,
    0 t μ 3 ( t , s , z , 0 s μ 2 ( s , τ , z ) d τ ) d s g ( t ) | z | ν .
     
Define an operator f : J × X Y by
f ( t , z ) ( x ) = μ 1 ( t , z x x ( x ) )
and let
Then the problem (4.1) can be formulated abstractly as:

It is easy to see that A E 1 generates a uniformly continuous semigroup { S ( t ) } t 0 on Y which is compact, and (2.1) is satisfied. Also, the operator f satisfies condition ( H 6 ) and the operator H and g satisfy ( H 7 ) and ( H 8 ) . Also all the conditions of Theorem 3.1 are satisfied. Hence, the equation (4.1) is controllable on J.

Declarations

Acknowledgements

I would like to thank the referees and professor Ravi Agarwal for their valuable comments and suggestions.

Authors’ Affiliations

(1)
Higher Institute of Engineering, El-Shorouk Academy

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© Ahmed; licensee Springer 2012

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