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Boundary Controllability of Nonlinear Fractional Integrodifferential Systems
Advances in Difference Equations volume 2010, Article number: 279493 (2010)
Abstract
Sufficient conditions for boundary controllability of nonlinear fractional integrodifferential systems in Banach space are established. The results are obtained by using fixed point theorems. We also give an application for integropartial differential equations of fractional order.
1. Introduction
Let and
be a pair of real Banach spaces with norms
and
, respectively. Let
be a linear closed and densely defined operator with
and let
be a linear operator with
and
, a Banach space. In this paper we study the boundary controllability of nonlinear fractional integrodifferential systems in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ1_HTML.gif)
where and
is a linear continuous operator, and the control function
is given in
a Banach space of admissible control functions. The nonlinear operators
and
are given and
Let be the linear operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ2_HTML.gif)
The controllability of integrodifferential systems has been studied by many authors (see [1–6]). This work may be regarded as a direct attempt to generalize the work in [7, 8].
2. Main Result
Definition 2.1.
System (1.1) is said to be controllable on the interval if for every
there exists a control
such that
of (1.1) satisfies
To establish the result, we need the following hypotheses.
-
(H1)
and the restriction of
to
is continuous relative to the graph norm of
.
-
(H2) The operator
is the infinitesimal generator of a compact semigroup
and there exists a constant
such that
-
(H3) There exists a linear continuous operator
such that
,
for all
Also
is continuously differentiable and
for all
where C is a constant.
-
(H4) For all
and
,
. Moreover, there exists a positive constant
such that
-
(H5) The nonlinear operators
and
, for
satisfy
(2.1)where
and
-
(H6) The linear operator
from
into
defined by
(2.2)
where is a probability density function defined on
(see [9, 10]) and induces an invertible operator
defined on
and there exists a positive constant
and
such that
and
. Let
be the solution of (1.1). Then we define a function
and from our assumption we see that
. Hence (1.1) can be written in terms of
and
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ5_HTML.gif)
If is continuously differentiable on
, then
can be defined as a mild solution to be the Cauchy problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ6_HTML.gif)
and the solution of (1.1) is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ7_HTML.gif)
Since the differentiability of the control represents an unrealistic and severe requirement, it is necessary of the solution for the general inputs
Integrating (2.5) by parts, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ8_HTML.gif)
Thus (2.6) is well defined and it is called a mild solution of system (1.1).
Theorem 2.2.
If hypotheses (H1)–(H6) are satisfied, then the boundary control fractional integrodifferential system (1.1) is controllable on .
Proof.
Using assumption (H6), for an arbitrary function define the control
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ9_HTML.gif)
We shall now show that, when using this control, the operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ10_HTML.gif)
has a fixed point. This fixed point is then a solution of (1.1). Clearly, which means that the control
steers the nonlinear fractional integrodifferential system from the initial state
to
in time
, provided we can obtain a fixed point of the nonlinear operator
.
Let and
where the positive constant
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ11_HTML.gif)
Then is clearly a bounded, closed, and convex subset of
. We define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ12_HTML.gif)
Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ13_HTML.gif)
Since and
are continuous and
it follows that
is also continuous and maps
into itself. Moreover,
maps
into precompact subset of
. To prove this, we first show that, for every fixed
, the set
is precompact in
. This is clear for
, since
. Let
be fixed and for
define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ14_HTML.gif)
Since is compact for every
, the set
is precompact in
for every
,
Furthermore, for
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ15_HTML.gif)
which implies that is totally bounded, that is, precompact in
. We want to show that
is an equicontinuous family of functions. For that, let
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ16_HTML.gif)
By using conditions (H2)–(H6), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ17_HTML.gif)
The compactness of implies that
is continuous in the uniform operator topology for
Thus, the right hand side of (2.15) tends to zero as
So,
is an equicontinuous family of functions. Also,
is bounded in
, and so by the Arzela- Ascoli theorem,
is precompact. Hence, from the Schauder fixed point in
any fixed point of
is a mild solution of (1.1) on
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ18_HTML.gif)
Thus, system (1.1) is controllable on .
3. Application
Let be bounded with smooth boundary
Consider the boundary control fractional integropartial differential system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ19_HTML.gif)
The above problem can be formulated as a boundary control problem of the form of (1.1) by suitably taking the spaces and the operators
,
and
as follows.
Let ,
,
,
, the identity operator and
,
The operator
is the trace operator such that
is well defined and belongs to
for each
and the operator
is given by
,
where
and
are usual Sobolev spaces on
We define the linear operator
by
where
is the unique solution to the Dirichlet boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ20_HTML.gif)
We also introduce the nonlinear operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F279493/MediaObjects/13662_2009_Article_1270_Equ21_HTML.gif)
Choose and other constants such that conditions (H1)–(H6) are satisfied. Consequently Theorem 2.2 can be applied for (3.1), so (3.1) is controllable on
.
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Ahmed, H.M. Boundary Controllability of Nonlinear Fractional Integrodifferential Systems. Adv Differ Equ 2010, 279493 (2010). https://doi.org/10.1155/2010/279493
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DOI: https://doi.org/10.1155/2010/279493