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

- Hamdy M Ahmed
^{1}Email author

**2012**:167

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

© Ahmed; licensee Springer 2012

**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

## 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 [1–3]). 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.

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 $u(\cdot )$ 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 $f\in C(J\times X,Y)$, $H\in C(J\times J\times X,X)$ and $g\in C(J\times J\times X\times X,Y)$ are all uniformly bounded continuous operators. The fractional derivative ${}^{c}D^{\alpha}$, $0<\alpha <1$ is understood in the Caputo sense.

## 2 Preliminaries

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

*f*can be defined as

provided the right-hand side is pointwise defined on $[0,\mathrm{\infty})$, where Γ is the gamma function.

*α*with the lower limit zero for a function

*f*can be written as

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 $A:D(A)\subset X\to Y$ and $E:D(E)\subset X\to Y$ satisfy the following hypotheses:

$({H}_{1})$ *A* and *E* are closed linear operators,

$({H}_{2})$ $D(E)\subset D(A)$ and *E* is bijective,

$({H}_{3})$ ${E}^{-1}:Y\to D(E)$ is continuous.

The hypotheses ${H}_{1}$, ${H}_{2}$ and the closed graph theorem imply the boundedness of the linear operator $A{E}^{-1}:Y\to Y$.

$({H}_{4})$ For each $t\in [0,a]$ and for some $\lambda \in \rho (-A{E}^{-1})$, the resolvent set of $-A{E}^{-1}$, the resolvent $R(\lambda ,-A{E}^{-1})$ is a compact operator.

**Lemma 2.1** [10]

*Let* $S(t)$ *be a uniformly continuous semigroup*. *If the resolvent set* $R(\lambda ;A)$ *of* *A* *is compact for every* $\lambda \in \rho (A)$, *then* $S(t)$ *is a compact semigroup*.

*From the above fact*, $-A{E}^{-1}$

*generates a compact semigroup*$\{T(t),t\ge 0\}$

*in*

*Y*,

*which means that there exists*$M>1$

*such that*

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

*W*from

*U*into

*X*defined by

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 $kerW=\{x\in {L}^{2}(J,U):Wx=0\}$, *B* is a bounded linear operator and ${T}_{\alpha}(t)$ is defined later.

*f*satisfies the following two conditions:

- (i)
For each $t\in J$, the function $f(t,\cdot ):X\to Y$ is continuous, and for each $x\in X$, the function $f(\cdot ,x):J\to Y$ is strongly measurable.

- (ii)For each positive number $k\in N$, there is a positive function ${g}_{k}(\cdot ):[0,a]\to {R}^{+}$ such that$\underset{|x|\le k}{sup}|f(t,x)|\le {g}_{k}(t),$

$({H}_{7})$ For each $(t,s)\in J\times J$, the function $H(t,s,\cdot ):X\to X$ is continuous, and for each $x\in X$, the function $H(\cdot ,\cdot ,x):J\times J\to X$ is strongly measurable.

*g*satisfies the following two conditions:

- (i)
For each $(t,s,x)\in J\times J\times X$, the function $g(t,s,\cdot ,\cdot ):X\times X\to Y$ is continuous, and for each $x\in X$, $H\in X$, the function $g(\cdot ,x,y):J\times J\to Y$ is strongly measurable.

- (ii)For each positive number $k\in N$, there is a positive function ${h}_{k}(\cdot ):[0,a]\to {R}^{+}$ such that$\underset{|x|\le k}{sup}\left|{\int}_{0}^{t}g(t,s,x,{\int}_{0}^{s}H(s,\tau ,x)\phantom{\rule{0.2em}{0ex}}d\tau )\phantom{\rule{0.2em}{0ex}}ds\right|\le {h}_{k}(t),$

with ${\xi}_{\alpha}$ being a probability density function defined on $(0,\mathrm{\infty})$, that is, ${\xi}_{\alpha}(\theta )\ge 0$, $\theta \in (0,\mathrm{\infty})$ and ${\int}_{0}^{\mathrm{\infty}}{\xi}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta =1$.

**Remark** ${\int}_{0}^{\mathrm{\infty}}\theta {\xi}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta =\frac{1}{\mathrm{\Gamma}(1+\alpha )}$.

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

**Lemma 2.2** (see [11])

*The operators*${S}_{\alpha}(t)$

*and*${T}_{\alpha}(t)$

*have the following properties*:

- (I)
*For any fixed*$x\in X$, $\parallel {S}_{\alpha}(t)x\parallel \le M\parallel x\parallel $, $\parallel {T}_{\alpha}(t)x\parallel \le \frac{\alpha M}{\mathrm{\Gamma}(\alpha +1)}\parallel x\parallel $; - (II)
$\{{S}_{\alpha}(t),t\ge 0\}$

*and*$\{{T}_{\alpha}(t),t\ge 0\}$*are strongly continuous*; - (III)
*For every*$t>0$, ${S}_{\alpha}(t)$*and*${T}_{\alpha}(t)$*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* $\frac{\alpha M\parallel {E}^{-1}\parallel}{\mathrm{\Gamma}(\alpha +1)}(\beta +\gamma )[1+\frac{{a}^{\alpha}M\parallel {E}^{-1}\parallel}{\mathrm{\Gamma}(\alpha +1)}\parallel B\parallel \parallel {W}^{-1}\parallel ]<1$.

*Proof*Using the assumption $({H}_{5})$, for an arbitrary function $x(\cdot )$, define the control

*Q*defined by

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

*C*. We claim that there exists a positive number

*k*such that $Q{B}_{k}\subset {B}_{k}$. If this is not true, then for each positive number

*k*, there exists a function ${x}_{k}\in {B}_{k}$ with $Q{x}_{k}\notin {B}_{k}$, that is, $\parallel Q{x}_{k}\parallel \ge k$, then $1\le \frac{1}{k}\parallel Q{x}_{k}\parallel $, and so

*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 ${V}_{k}(t)=\{(Qx)(t):x\in {B}_{k}\}$ is a precompact in

*X*; for every $t\in J$: This is trivial for $t=0$, since ${V}_{k}(0)=\{{x}_{0}\}$. Let

*t*, $0<t\le a$; be fixed. For $0<\u03f5<t$ and arbitrary $\delta >0$; take

*X*for every

*ϵ*, $0<\u03f5<t$, and for all $\delta >0$. Also, for $x\in {B}_{k}$, using the defined control $u(t)$ yields

Therefore, as $\u03f5\to {0}^{+}$ and $\delta \to {0}^{+}$, 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 $Q{B}_{k}=\{Qx:x\in {B}_{k}\}$ is an equicontinuous family of functions.

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\to \tau $. 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 $(Qx)(t)=x(t)\in X$. Thus, the system (1.1) is controllable on *J*. □

## 4 Example

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

where ${}^{c}\partial _{t}^{\alpha}$ is the Caputo fractional partial derivative of order $0<\alpha <1$.

Take $X=Y={L}^{2}[0,\pi ]$ and define the operators $A:D(A)\subset X\to Y$ and $E:D(E)\subset X\to Y$ by $Az=-{z}_{xx}$ and $Ez=z-{z}_{xx}$, where each domain $D(A)$ and $D(E)$ is given by $\{z\in X:z,{z}_{x}\text{are absolutely continuous},{z}_{xx}\in X,z(0)=z(\pi )=0\}$.

*A*and $(z,{z}_{n})$ is the ${L}^{2}$ inner product. Moreover, for $z\in X$, we get

We assume that

$({A}_{1})$: The operator $B:U\to Y$, with $U\subset J$, is a bounded linear operator.

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 $kerW=\{x\in {L}^{2}(J,U):Wx=0\}$, *B* is a bounded linear operator.

- (i)
For each $t\in J$, ${\mu}_{1}(t,z)$ is continuous.

- (ii)
For each $z\in X$, ${\mu}_{1}(t,z)$ is measurable.

- (iii)There is a constant
*ν*($0<\nu <1$) and a function $h(\cdot ):[0,a]\to {R}^{+}$ such that for all $(t,z)\in J\times X$,$\parallel {\mu}_{1}(t,z)\parallel \le h(t){|z|}^{\nu}.$

- (i)
For each $(t,s)\in J\times J$, ${\mu}_{2}(t,s,z)$ is continuous.

- (ii)
For each $z\in X$, ${\mu}_{2}(t,s,z)$ is measurable.

- (i)
For each $(t,s,z)\in J\times J\times X$, ${\mu}_{3}(t,s,z)$ is continuous.

- (ii)
For each $z\in X$, ${\mu}_{3}(t,s,z)$ is measurable.

- (iii)There is a constant
*ν*($0<\nu <1$) and a function $g(\cdot ):[0,a]\to {R}^{+}$ such that for all $(t,s,z,y)\in J\times J\times X\times X$,$\parallel {\int}_{0}^{t}{\mu}_{3}(t,s,z,{\int}_{0}^{s}{\mu}_{2}(s,\tau ,z)\phantom{\rule{0.2em}{0ex}}d\tau )\phantom{\rule{0.2em}{0ex}}ds\parallel \le g(t){|z|}^{\nu}.$

It is easy to see that $-A{E}^{-1}$ generates a uniformly continuous semigroup ${\{S(t)\}}_{t\ge 0}$ 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

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