Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 279493, 9 pages
doi:10.1155/2010/279493
Research Article
Boundary Controllability of Nonlinear Fractional
Integrodifferential Systems
Hamdy M. Ahmed
Higher Institute of Engineering, El-Shorouk Academy, P.O. 3 El-Shorouk City, Cairo, Egypt
Correspondence should be addressed to Hamdy M. Ahmed, hamdy 17eg@yahoo.com
Received 21 July 2009; Accepted 11 January 2010
Academic Editor: Toka Diagana
Copyright q 2010 Hamdy M. Ahmed. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 E and U be a pair of real Banach spaces with norms · and | · |, respectively. Let σ be a
linear closed and densely defined operator with Dσ ⊆ E and let τ ⊆ X be a linear operator
with Dσ and Rτ ⊆ X, a Banach space. In this paper we study the boundary controllability
of nonlinear fractional integrodifferential systems in the form
t
dα xt
σxt
ft,
xt
gt, s, xsds, t ∈ J 0, b,
dtα
0
1.1
τxt B1 ut,
x0 x0 ,
where 0 < α ≤ 1 and B1 : U → X is a linear continuous operator, and the control function u
is given in L2 J, U, a Banach space of admissible control functions. The nonlinear operators
f : J × E → E and g : Δ × E → E are given and Δ : t, s; 0 ≤ s ≤ t ≤ b.
Let A : E → E be the linear operator defined by
DA {x ∈ Dσ; τx 0},
Ax {σx, for x ∈ DA}.
1.2
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.
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2. Main Result
Definition 2.1. System 1.1 is said to be controllable on the interval J if for every x0 , x1 ∈ E
there exists a control u ∈ L2 J, U such that x· of 1.1 satisfies xb x1 .
To establish the result, we need the following hypotheses.
H1 Dσ ⊂ Dτ and the restriction of τ to Dσ is continuous relative to the graph
norm of Dσ.
H2 The operator A is the infinitesimal generator of a compact semigroup T t and there
exists a constant M1 > 0 such that T t ≤ M1 .
H3 There exists a linear continuous operator B : U → E such that σB ∈
LU, E, τBu B1 u, for all u ∈ U. Also But is continuously differentiable and
Bu ≤ CB1 u for all u ∈ U, where C is a constant.
H4 For all t ∈ 0, b and u ∈ U, T tBu ∈ DA. Moreover, there exists a positive
constant K1 > 0 such that AT t ≤ K1 .
H5 The nonlinear operators ft, xt and gt, s, xs, for t, s ∈ J, satisfy
ft, xt ≤ L1 ,
gt, s, xs ≤ L2 ,
2.1
where L1 ≥ 0 and L2 ≥ 0.
H6 The linear operator W from L2 J, U into E defined by
b ∞
θt − sα−1 ξα θ T b − sα θ σ − AT b − sα θ Busdθ ds,
Wu α
0
2.2
0
where ξα θ is a probability density function defined on 0, ∞ see 9, 10 and
induces an invertible operator W −1 defined on L2 J, U/kerW, and there exists a
positive constant M2 > 0 and M3 > 0 such that B ≤ M2 and W −1 ≤ M3 . Let
xt be the solution of 1.1. Then we define a function zt xt − But and from
our assumption we see that zt ∈ DA. Hence 1.1 can be written in terms of A
and B as
dα xt
Azt σBut ft, xt dtα
xt zt But,
t
gt, s, xsds,
t ∈ J,
2.3
0
x0 x0 .
If u is continuously differentiable on 0, b, then z can be defined as a mild solution to be the
Cauchy problem
dα zt
dα ut
Azt σBut − B
ft, xt α
dt
dtα
z0 x0 − Bu0,
t
gt, s, xsds,
0
t ∈ J,
2.4
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and the solution of 1.1 is given by
xt ∞
ξα θT tα θx0 − Bu0dθ But
0
t ∞
α
θt − sα−1 ξα θT t − sα θ fs, xsdθ ds
0
0
t ∞
dα us
α
θt − sα−1 ξα θT t − sα θ σBus − B
dθ ds
dsα
0 0
2.5
t ∞
s
α−1
α α
θt − s ξα θT t − s θ
gs, τ, xτdτ dθ ds
0
0
0
see 11–13.
Since the differentiability of the control u represents an unrealistic and severe
requirement, it is necessary of the solution for the general inputs u ∈ L2 J, U. Integrating
2.5 by parts, we get
xt ∞
t ∞
ξα θT t θx0 dθα
θt−sα−1 ξα θ T t−sα θ σ −AT t−sα θ Busdθ ds
α
0
0
0
t ∞
α
θt − sα−1 ξα θT t − sα θ fs, xsdθ ds
0
0
t ∞
s
α
θt − sα−1 ξα θT t − sα θ
gs, τ, xτdτ dθ ds.
0
0
0
2.6
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 J.
Proof. Using assumption H6, for an arbitrary function x· define the control
ut W
−1
x1 −
∞
b ∞
ξα θT bα θx0 dθ − α
θb − sα−1 ξα θT b − sα θ fs, xsdθ ds
0
0
0
b ∞
s
α−1
α −α
θb − s ξα θT b − s θ
gs, τ, xτdτ dθ ds t.
0
0
0
2.7
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We shall now show that, when using this control, the operator defined by
Φxt ∞
ξα θT tα θx0 dθ
0
t ∞
α
θt − sα−1 ξα θ T t − sα θ σ − AT t − sα θ Busdθ ds
0
0
2.8
t ∞
α
θt − sα−1 ξα θT t − sα θ fs, xsdθ ds
0
0
t ∞
s
θt − sα−1 ξα θT t − sα θ
gs, τ, xτdτ dθ ds
α
0
0
0
has a fixed point. This fixed point is then a solution of 1.1. Clearly, Φxb x1 , which
means that the control u steers the nonlinear fractional integrodifferential system from the
initial state x0 to x1 in time T , provided we can obtain a fixed point of the nonlinear operator
Φ.
Let Y CJ, X and Y0 {x ∈ Y : xt ≤ r, for t ∈ J}, where the positive constant r
is given by
r M1 x0 bα M1 σ K1 M2 M3 x1 M1 x0 M1 L1 bα M1 L2 bα1
2.9
M1 L1 bα M1 L2 bα1 .
Then Y0 is clearly a bounded, closed, and convex subset of Y . We define a mapping Φ : Y →
Y0 by
Φxt ∞
t ∞
ξα θT t θx0 dθ α
θt − sα−1 ξα θ T t − sα θ σ − AT t − sα θ BW −1
α
0
0
× x1 −
∞
0
b ∞
ξα θT b θx0 dθ − α
θb − sα−1 ξα θT b − sα θ fs, xsdθ ds
α
0
0
0
b ∞
s
α−1
α θb − s ξα θT b − s θ
gs, τ, xτdτ dθ ds sdθ ds
−α
0
0
0
t ∞
θt − sα−1 ξα θT t − sα θ fs, xsdθ ds
α
0
0
t ∞
s
α
θt − sα−1 ξα θT t − sα θ
gs, τ, xτdτ dθ ds.
0
0
0
2.10
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Consider
Φxt
t ∞
∞
α−1
α α α
≤ ξα θT t θx0 dθ α θt−s ξα θ T t − s θ σ −AT t−s θ dθ
B
0
0
0
b ∞
∞
−1 α−1
α α
× W x1 − −
α
ξ
θx
dθ
θb
−
s
ξ
θ
dθ
−
s
θT
b
θT
b
α
0
α
0
0
0
b ∞
α−1
α × fs, xsds − α θb
−
s
ξ
θ
dθ
−
s
θT
b
α
0
0
s
×
gs, τ, xτdτ ds sds
0
t ∞
α−1
α fs, xsds
θt
−
s
ξ
θ
dθ
α −
s
α θT t
0
0
t ∞
s
α−1
α α θt − s ξα θT t − s θ dθ
gs, τ, xτdτ ds
0
0
0
≤ M1 x0 bα M1 σ K1 M2 M3 x1 M1 x0 M1 L1 bα M1 L2 bα1
M1 L1 bα M1 L2 bα1 ≤ r.
2.11
Since f and g are continuous and Φxt ≤ r, it follows that Φ is also continuous and maps
Y0 into itself. Moreover, Φ maps Y0 into precompact subset of Y0 . To prove this, we first show
that, for every fixed t ∈ J, the set Y0 t {Φxt : x ∈ Y0 } is precompact in X. This is clear
for t 0, since Y0 0 {x0 }. Let t > 0 be fixed and for 0 < < t define
Φ xt ∞
t− ∞
ξα θT t θx0 dθα
θt−sα−1 ξα θ T t−sα θ σ −AT t−sα θ BW −1
α
0
0
× x1 −
∞
0
b ∞
ξα θT b θx0 dθ − α
θb − sα−1 ξα θT b − sα θ fs, xsdθ ds
α
0
0
0
b ∞
s
α−1
α θb − s ξα θT b − s θ
gs, τ, xτdτ dθ ds sdθ ds
−α
0
0
0
t− ∞
θt − sα−1 ξα θT t − sα θ fs, xsdθ ds
α
0
0
t− ∞
s
α
θt − sα−1 ξα θT t − sα θ
gs, τ, xτdτ dθ ds.
0
0
0
2.12
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Advances in Difference Equations
Since T t is compact for every t > 0, the set Y t {Φ xt : x ∈ Y0 } is precompact
in X for every , 0 < < t. Furthermore, for x ∈ Y0 , we have
Φxt − Φ xt
∞
t
≤ α
θt − sα−1 ξα θ T t − sα θ σ − AT t − sα θ BW −1
t− 0
× x1 −
∞
b ∞
ξα θT bα θx0 dθ − α
θb − sα−1 ξα θT b − sα θ fs, xsdθ ds
0
0
0
b ∞
s
α−1
α −α
θb − s ξα θT b − s θ
gs, τ, xτdτ dθ ds sdθ ds
0 0
0
∞
t
α−1
α α
θt − s ξα θT t − s θ fs, xsdθ ds
t− 0
s
∞
t
α−1
α α
θt − s ξα θT t − s θ
gs, τ, xτdτ dθ ds
t− 0
0
≤ α M1 σ K1 M2 M3 x1 M1 x0 M1 L1 bα M1 L2 bα1 M1 L1 α M1 L2 α b,
2.13
which implies that Y0 t is totally bounded, that is, precompact in X. We want to show that
ΦY0 {Φx : x ∈ Y0 } is an equicontinuous family of functions. For that, let t2 > t1 > 0. Then
we have
Φxt1 − Φxt2 t1 ∞
≤ α
θξα θ t1 − sα−1 T t1 − sα θ σ − AT t1 − sα θ
0 0
− t2 − sα−1 T t2 − sα θ σ − AT t2 − sα θ
× BW
−1
x1 −
∞
b ∞
ξα θT b θx0 dθ − α
θb − sα−1 ξα θT b − sα θ fs, xsdθ ds
α
0
0
0
b ∞
s
α−1
α −α
θb − s ξα θT b − s θ
gs, τ, xτdτ dθ ds sdθ ds
0 0
0
t2 ∞
α
θξα θt2 − sα−1 T t2 − sα θ σ − AT t2 − sα θ BW −1
t1 0
× x1 −
∞
0
b ∞
ξα θT b θx0 dθ − α
θb − sα−1 ξα θT b − sα θ fs, xsdθ ds
α
0
0
b ∞
s
α−1
α −α
θb−s ξα θT b−s θ
gs, τ, xτdτ dθ ds sdθ ds
0 0
0
Advances in Difference Equations
t1 ∞
α
θξα θ t1 − sα−1 T t1 − sα θ − t2 − sα−1 T t2 − sα θ fs, xsdθ ds
0 0
7
t ∞
s
α−1
α α−1 α α
θξ θ t1 −s
T t1 −s θ − t2 −s T t2 −s θ
gs, τ, xτdτ
0 0 α
0
× dθ ds
t2 ∞
α−1 α α
θξα θt2 − s T t2 − s θ fs, xsdθ ds
t1 0
s
t2 ∞
α−1
α α
θξα θt2 − s
T t2 − s θ
gs, τ, xτdτ dθ ds.
t1 0
0
2.14
By using conditions H2–H6, we get
Φxt1 − Φxt2 t1 ≤ α t1 − sα−1 T t1 − sα θ σ − AT t1 − sα θ
0
− t2 − sα−1 T t2 − sα θ σ − AT t2 − sα θ × M2 M3 x1 M1 x0 M1 L1 bα L2 bα1
ds
t2 − t1 α M1 M2 M3 σ K1 M2 M3 x1 M1 x0 M1 L1 bα L2 bα1
t1 α t1 − sα−1 T t1 − sα θ − t2 − sα−1 T t2 − sα θ 0
× L1 L2 bα ds t2 − t1 α M1 L1 L2 bα .
2.15
The compactness of T t, t > 0, implies that T t is continuous in the uniform operator
topology for t > 0. Thus, the right hand side of 2.15 tends to zero as t2 → t1 . So, ΦY0 is an equicontinuous family of functions. Also, ΦY0 is bounded in Y , and so by the ArzelaAscoli theorem, ΦY0 is precompact. Hence, from the Schauder fixed point in Y0 , any fixed
point of Φ is a mild solution of 1.1 on J satisfying
Φxt xt ∈ X.
Thus, system 1.1 is controllable on J.
2.16
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3. Application
Let Ω ⊂ Rn be bounded with smooth boundary Γ.
Consider the boundary control fractional integropartial differential system
∂α yt, x
− Δyt, x F t, yt, x ∂tα
yt, 0 ut, 0
t
G t, s, ys, x ds,
in Y 0, b × Ω,
0
on Σ 0, b × Γ,
y0, x y0 x,
t ∈ 0, b,
3.1
for x ∈ Ω.
The above problem can be formulated as a boundary control problem of the form of 1.1 by
suitably taking the spaces E, X, U and the operators B1 , σ, and τ as follows.
Let E L2 Ω, X H −1/2 Γ, U L2 Γ, B1 I, the identity operator and Dσ {y ∈
2
L Ω : Δy ∈ L2 Ω}, σ Δ. The operator τ is the trace operator such that τy y|Γ is well
defined and belongs to H −1/2 Γ for each y ∈ Dσ and the operator A is given by A Δ,
DA H01 Ω ∪ H 2 Ω where H k Ω, H β Ω, and H01 Ω are usual Sobolev spaces on Ω, Γ.
We define the linear operator B : L2 Γ → L2 Ω by Bu wu where wu is the unique solution
to the Dirichlet boundary value problem
Δwu 0 in Ω,
wu u
in Γ.
3.2
We also introduce the nonlinear operator defined by
ft, xt F t, yt, x ,
gt, s, xs G t, s, ys, x .
3.3
Choose b 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 0, b.
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