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Controllability for Sobolev type fractional integrodifferential systems in a Banach space
Advances in Difference Equations volume 2012, Article number: 167 (2012)
Abstract
In this paper, by using compact semigroups and the Schauder fixedpoint theorem, we study the sufficient conditions for controllability of Sobolev type fractional integrodifferential systems in a Banach space. An example is provided to illustrate the obtained results.
MSC:26A33, 34G20, 93B05.
1 Introduction
A Sobolevtype 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 integrodifferential systems. Balachandran and Dauer [4] studied the controllability of Sobolev type integrodifferential systems in Banach spaces. Balachandran and Sakthivel [5] studied the controllability of Sobolev type semilinear integrodifferential systems in Banach spaces. Balachandran, Anandhi and Dauer [6] studied the boundary controllability of Sobolev type abstract nonlinear integrodifferential systems.
In this paper, we study the controllability of Sobolev type fractional integrodifferential systems in Banach spaces in the following form:
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.
The fractional integral of order $\alpha >0$ with the lower limit zero for a function f can be defined as
provided the righthand side is pointwise defined on $[0,\mathrm{\infty})$, where Γ is the gamma function.
The Caputo derivative of order 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 $x(a)={x}_{1}$.
$({H}_{5})$ The linear operator 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.
$({H}_{6})$ The function 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
$$\underset{x\le k}{sup}f(t,x)\le {g}_{k}(t),$$
the function $s\to {(ts)}^{1\alpha}{g}_{k}(s)\in {L}^{1}([0,t],{R}^{+})$, and there exists a $\beta >0$ such that
$({H}_{7})$ 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.
$({H}_{8})$ The function 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
$$\underset{x\le k}{sup}\left{\int}_{0}^{t}g(t,s,x,{\int}_{0}^{s}H(s,\tau ,x)\phantom{\rule{0.2em}{0ex}}d\tau )\phantom{\rule{0.2em}{0ex}}ds\right\le {h}_{k}(t),$$
the function $s\to {(ts)}^{1\alpha}{h}_{k}(s)\in {L}^{1}([0,t],{R}^{+})$, and there exists a $\gamma >0$ such that
According to [11, 12], a solution of equation (1.1) can be represented by
where
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
It shall now be shown that when using this control, the operator Q defined by
from $C(J,X)$ into itself for each $x\in C=C(J,X)$ has a fixed point. This fixed point is then a solution of equation (2.2).
It can be easily verified that Q maps C into itself continuously.
For each positive number $k>0$, let ${B}_{k}=\{x\in C:x(0)={x}_{0},\parallel x(t)\parallel \le k,t\in J\}$. Obviously, ${B}_{k}$ is clearly a bounded, closed, convex subset in C. We claim that there exists a positive number k such that $Q{B}_{k}\subset {B}_{k}$. 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
However,
a contradiction. Hence, $Q{B}_{k}\subset {B}_{k}$ 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 ${V}_{k}(t)=\{(Qx)(t):x\in {B}_{k}\}$ 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
Since $u(s)$ is bounded and $T({\u03f5}^{\alpha}\delta )$, ${\u03f5}^{\alpha}\delta >0$ is a compact operator, then the set ${V}_{k}^{\u03f5,\delta}(t)=\{({Q}^{\u03f5,\delta}x)(t):x\in {B}_{k}\}$ is a precompact set in 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.
Let $x\in {B}_{k}$ and $t,\tau \in J$ such that $0<t<\tau $, then
Now, $T(t)$ is continuous in the uniform operator topology for $t>0$ since $T(t)$ is compact, and the righthand side of the above inequality tends to zero as $t\to \tau $. Thus, $Q{B}_{k}$ is both equicontinuous and bounded. By the ArzelaAscoli theorem, $Q{B}_{k}$ is precompact in $C(J,X)$. Hence, Q is a completely continuous operator on $C(J,X)$.
From the Schauder fixedpoint 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.
We consider the fractional integropartial differential equation in the form
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 $\{z\in X:z,{z}_{x}\text{are absolutely continuous},{z}_{xx}\in X,z(0)=z(\pi )=0\}$.
Then A and E can be written respectively as [13]
where ${z}_{n}(x)=\sqrt{2/\pi}sinnx$, $n=1,2,\dots $ , is the orthonormal set of eigenvectors of 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.
$({A}_{2})$: The linear operator $W:U\to X$ 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.
$({A}_{3})$: The nonlinear operator ${\mu}_{1}:J\times X\to Y$ satisfies the following three conditions:

(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 $(t,z)\in J\times X$,
$$\parallel {\mu}_{1}(t,z)\parallel \le h(t){z}^{\nu}.$$
$({A}_{4})$: The nonlinear operator ${\mu}_{2}:J\times J\times X\to X$ satisfies the following two conditions:

(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.
$({A}_{5})$: The nonlinear operator ${\mu}_{3}:J\times J\times X\times X\to Y$ satisfies the following three conditions:

(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 $(t,s,z,y)\in J\times J\times X\times X$,
$$\parallel {\int}_{0}^{t}{\mu}_{3}(t,s,z,{\int}_{0}^{s}{\mu}_{2}(s,\tau ,z)\phantom{\rule{0.2em}{0ex}}d\tau )\phantom{\rule{0.2em}{0ex}}ds\parallel \le g(t){z}^{\nu}.$$
Define an operator $f:J\times X\to Y$ by
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 ${\{S(t)\}}_{t\ge 0}$ 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.
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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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Ahmed, H.M. Controllability for Sobolev type fractional integrodifferential systems in a Banach space. Adv Differ Equ 2012, 167 (2012). https://doi.org/10.1186/168718472012167
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Keywords
 fractional calculus
 Sobolev type fractional integrodifferential systems
 controllability
 compact semigroup
 mild solution
 Schauder fixedpoint theorem