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 [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.

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 $u(\cdot )$ 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.

Definition 2.1 (see [7–9])

The fractional integral of order $\alpha >0$ with the lower limit zero for a function f can be defined as

$$I^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t\frac{f(s)}{{(t-s)}^{1-\alpha }}ds,t>0$$

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

Definition 2.2 (see [7–9])

The Caputo derivative of order α with the lower limit zero for a function f can be written as

$${}^{c}D^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^t\frac{f^{(n)}(s)}{{(t-s)}^{\alpha +1-n}}ds=I^{n-\alpha }{f}^{(n)}(t),t>0,0\le n-1<\alpha <n.$$

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 $A:D(A)\subset X\to Y$ 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

$$\underset{t\in J}{max}\parallel T(t)\parallel \le M.$$

(2.1)

Definition 2.3 The system (1.1) is said to be controllable on the interval J if for every $x_0,x_1\in X$, 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

$$Wu={\int }_0^a{E}^{-1}{(a-s)}^{\alpha -1}{T}_{\alpha }(a-s)Bu(s)ds$$

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 $kerW=\{x\in L^2(J,U):Wx=0\}$, 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 $s\to {(t-s)}^{1-\alpha }{g}_k(s)\in L^1([0,t],R^{+})$, and there exists a $\beta >0$ such that

$$\underset{k\to \mathrm{\infty }}{lim}inf\frac{{\int }_0^t{(t-s)}^{1-\alpha }{g}_k(s)ds}{k}=\beta <\mathrm{\infty },t\in [0,a].$$

$(H_7)$ 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

  $$\underset{|x|\le k}{sup}\left|{\int }_0^{t}g(t,s,x,{\int }_0^{s}H(s,\tau ,x)d\tau )ds\right|\le h_k(t),$$

the function $s\to {(t-s)}^{1-\alpha }{h}_k(s)\in L^1([0,t],R^{+})$, and there exists a $\gamma >0$ such that

$$\underset{k\to \mathrm{\infty }}{lim}inf\frac{{\int }_0^t{(t-s)}^{1-\alpha }{h}_k(s)ds}{k}=\gamma <\mathrm{\infty },t\in [0,a].$$

According to [11, 12], a solution of equation (1.1) can be represented by

$$\begin{array}{rcl}x(t)& =& E^{-1}{S}_{\alpha }(t)E{x}_0+{\int }_0^t{(t-s)}^{\alpha -1}{T}_{\alpha }(t-s)E^{-1}f(s,x(s))ds\\ +{\int }_0^t{(t-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(t-s)Bu(s)ds\\ +{\int }_0^t{(t-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(t-s)\left\{{\int }_0^{s}g(s,\tau ,x(\tau ),R(\tau ))d\tau \right\}ds,t\in J,\end{array}$$

(2.2)

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 ${\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )d\theta =1$.

Remark ${\int }_0^{\mathrm{\infty }}\theta {\xi }_{\alpha }(\theta )d\theta =\frac{1}{\mathrm{\Gamma }(1+\alpha )}$.

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

$$\begin{array}{rcl}u(t)& =& W^{-1}[x_1-E^{-1}{S}_{\alpha }(t)E{x}_0-{\int }_0^a{(a-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-s)f(s,x(s))ds\\ -{\int }_0^a{(a-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-s)\left\{{\int }_0^{s}g(s,\tau ,x(\tau ),R(\tau ))d\tau \right\}ds](t).\end{array}$$

It shall now be shown that when using this control, the operator Q defined by

$$\begin{array}{rcl}(Qx)(t)& =& E^{-1}{S}_{\alpha }(t)E{x}_0+{\int }_0^t{(t-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(t-s)f(s,x(s))ds\\ +{\int }_0^t{(t-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(t-s)Bu(s)ds\\ +{\int }_0^t{(t-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(t-s)\left\{{\int }_0^{s}g(s,\tau ,x(\tau ),R(\tau ))d\tau \right\}ds\end{array}$$

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).

$$\begin{array}{rcl}(Qx)(a)& =& E^{-1}{S}_{\alpha }(a)E{x}_0+{\int }_0^a{(a-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-s)f(s,x(s))ds\\ +{\int }_0^a{(a-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-s)B{W}^{-1}\\ \times [x_1-E^{-1}{S}_{\alpha }(a)E{x}_0-{\int }_0^a{(a-\tau )}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-\tau )f(\tau ,x(\tau ))d\tau \\ -{\int }_0^a{(a-\tau )}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-\tau )\left\{{\int }_0^{\tau }g(\tau ,\eta ,x(\eta ),R(\eta ))d\eta \right\}d\tau ](s)ds\\ +\alpha {\int }_0^a{(a-s)}^{\alpha -1}{E}^{-1}{T}_{\alpha }(a-s)\left\{{\int }_0^{s}g(s,\tau ,x(\tau ),R(\tau ))d\tau \right\}ds=x_1.\end{array}$$

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

$$1\le \underset{k\to \mathrm{\infty }}{lim}{k}^{-1}\parallel Q{x}_k\parallel .$$

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 $0<ϵ<t$ and arbitrary $\delta >0$; take

$$\begin{array}{rcl}\left(Q^{ϵ,\delta }x\right)(t)& =& {\int }_{\delta }^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )E^{-1}T\left(t^{\alpha }\theta \right)E{x}_{0}d\theta \\ +\alpha {\int }_0^{t-ϵ}{\int }_{\delta }^{\mathrm{\infty }}\theta {(t-s)}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(t-s)}^{\alpha }\theta \right)f(s,x(s))d\theta ds\\ +\alpha {\int }_0^{t-ϵ}{\int }_{\delta }^{\mathrm{\infty }}\theta {(t-s)}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(t-s)}^{\alpha }\theta \right)\\ \times B{W}^{-1}[x_1-{\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )E^{-1}T\left(a^{\alpha }\theta \right)E{x}_{0}d\theta \\ -\alpha {\int }_0^a{\int }_0^{\mathrm{\infty }}\theta {(a-\tau )}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(a-\tau )}^{\alpha }\theta \right)f(\tau ,x(\tau ))d\theta d\tau \\ -\alpha {\int }_0^a{\int }_0^{\mathrm{\infty }}\theta {(a-\tau )}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(a-\tau )}^{\alpha }\theta \right)\\ \times \left\{{\int }_0^{\tau }g(\tau ,\eta ,x(\eta ),R(\eta ))d\eta \right\}d\theta d\tau ](s)d\theta ds\\ +\alpha {\int }_0^{t-ϵ}{\int }_{\delta }^{\mathrm{\infty }}\theta {(t-s)}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(t-s)}^{\alpha }\theta \right)\\ \times \left\{{\int }_0^{s}g(s,\tau ,x(\tau ),R(\tau ))d\tau \right\}d\theta ds\\ =& T\left({ϵ}^{\alpha }\delta \right){\int }_{\delta }^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )E^{-1}T(t^{\alpha }\theta -{ϵ}^{\alpha }\delta )E{x}_{0}d\theta \\ +T\left({ϵ}^{\alpha }\delta \right)\alpha {\int }_0^{t-ϵ}{\int }_{\delta }^{\mathrm{\infty }}\theta {(t-s)}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T({(t-s)}^{\alpha }\theta -{ϵ}^{\alpha }\delta )f(s,x(s))d\theta ds\\ +T\left({ϵ}^{\alpha }\delta \right)\alpha {\int }_0^{t-ϵ}{\int }_{\delta }^{\mathrm{\infty }}\theta {(t-s)}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T({(t-s)}^{\alpha }\theta -{ϵ}^{\alpha }\delta )\\ \times B{W}^{-1}[x_1-{\int }_0^{\mathrm{\infty }}{\xi }_{\alpha }(\theta )E^{-1}T\left(a^{\alpha }\theta \right)E{x}_{0}d\theta \\ -\alpha {\int }_0^a{\int }_0^{\mathrm{\infty }}\theta {(a-\tau )}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(a-\tau )}^{\alpha }\theta \right)f(\tau ,x(\tau ))d\theta d\tau \\ -{\int }_0^a{\int }_0^{\mathrm{\infty }}\theta {(a-\tau )}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T\left({(a-\tau )}^{\alpha }\theta \right)\\ \times \left\{{\int }_0^{\tau }g(\tau ,\eta ,x(\eta ),R(\eta ))d\eta \right\}d\theta d\tau ](s)d\theta ds\\ +T\left({ϵ}^{\alpha }\delta \right)\alpha {\int }_0^{t-ϵ}{\int }_{\delta }^{\mathrm{\infty }}\theta {(t-s)}^{\alpha -1}{\xi }_{\alpha }(\theta )E^{-1}T({(t-s)}^{\alpha }\theta -{ϵ}^{\alpha }\delta )\\ \times \left\{{\int }_0^{s}g(s,\tau ,x(\tau ),R(\tau ))d\tau \right\}d\theta ds.\end{array}$$

Since $u(s)$ is bounded and $T({ϵ}^{\alpha }\delta )$, ${ϵ}^{\alpha }\delta >0$ is a compact operator, then the set $V_k^{ϵ,\delta }(t)=\{(Q^{ϵ,\delta }x)(t):x\in B_k\}$ is a precompact set in X for every ϵ, $0<ϵ<t$, and for all $\delta >0$. Also, for $x\in B_k$, using the defined control $u(t)$ yields

Therefore, as $ϵ\to 0^{+}$ 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 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.

We consider the fractional integro-partial differential equation in the form

(4.1)

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

$$Wu={\int }_0^a{E}^{-1}{(a-s)}^{\alpha -1}{T}_{\alpha }(a-s)Bu(s)ds$$

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 $kerW=\{x\in L^2(J,U):Wx=0\}$, 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 $(t,s,z,y)\in J\times J\times X\times X$,

  $$\parallel {\int }_0^t{\mu }_3(t,s,z,{\int }_0^s{\mu }_2(s,\tau ,z)d\tau )ds\parallel \le g(t){|z|}^{\nu }.$$

Define an operator $f:J\times X\to Y$ by

$$f(t,z)(x)={\mu }_1(t,z_{xx}(x))$$

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 ${\{S(t)\}}_{t\ge 0}$ 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.