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

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.

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 fC(J×X,Y), HC(J×J×X,X) and gC(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 ds,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 ds= I n α f ( n ) (t),t>0,0n1<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)XY and E:D(E)XY 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 :YD(E) is continuous.

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

( H 4 ) For each t[0,a] and for some λρ(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 λρ(A), then S(t) is a compact semigroup.

From the above fact, A E 1 generates a compact semigroup {T(t),t0} 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

Wu= 0 a E 1 ( a s ) α 1 T α (as)Bu(s)ds

has an inverse bounded operator W 1 which takes values in L 2 (J,U)/kerW, where the kernel space of W is defined by kerW={x L 2 (J,U):Wx=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 tJ, the function f(t,):XY is continuous, and for each xX, the function f(,x):JY is strongly measurable.

  2. (ii)

    For each positive number kN, 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,):XX is continuous, and for each xX, the function H(,,x):J×JX 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×XY is continuous, and for each xX, HX, the function g(,x,y):J×JY is strongly measurable.

  2. (ii)

    For each positive number kN, 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)dθ=1.

Remark 0 θ ξ α (θ)dθ= 1 Γ ( 1 + α ) .

Definition 2.4 By a mild solution of the problem (1.1), we mean that the function xC(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 xX, S α (t)xMx, T α (t)x α M Γ ( α + 1 ) x;

  2. (II)

    { S α (t),t0} and { T α (t),t0} 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 xC=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 ={xC:x(0)= x 0 ,x(t)k,tJ}. 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)={(Qx)(t):x B k } is a precompact in X; for every tJ: This is trivial for t=0, since V k (0)={ x 0 }. Let t, 0<ta; 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 ={Qx:x B k } is an equicontinuous family of functions.

Let x B k and t,τJ 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 tτ. 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 (Qx)(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)XY and E:D(E)XY by Az= z x x and Ez=z z x x , where each domain D(A) and D(E) is given by {zX: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 sinnx, n=1,2, , is the orthonormal set of eigenvectors of A and (z, z n ) is the L 2 inner product. Moreover, for zX, we get

We assume that

( A 1 ): The operator B:UY, with UJ, is a bounded linear operator.

( A 2 ): The linear operator W:UX defined by

Wu= 0 a E 1 ( a s ) α 1 T α (as)Bu(s)ds

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

( A 3 ): The nonlinear operator μ 1 :J×XY satisfies the following three conditions:

  1. (i)

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

  2. (ii)

    For each zX, 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×XX satisfies the following two conditions:

  1. (i)

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

  2. (ii)

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

( A 5 ): The nonlinear operator μ 3 :J×J×X×XY 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 zX, 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×XY 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.

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Acknowledgements

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

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Correspondence to Hamdy M Ahmed.

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Ahmed, H.M. Controllability for Sobolev type fractional integro-differential systems in a Banach space. Adv Differ Equ 2012, 167 (2012). https://doi.org/10.1186/1687-1847-2012-167

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Keywords

  • fractional calculus
  • Sobolev type fractional integro-differential systems
  • controllability
  • compact semigroup
  • mild solution
  • Schauder fixed-point theorem