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
Copyrightq2010 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
LetEandUbe a pair of real Banach spaces with norms · and| · |, respectively. Letσbe a linear closed and densely defined operator withDσ⊆Eand letτ ⊆Xbe a linear operator withDσandRτ⊆X, a Banach space. In this paper we study the boundary controllability of nonlinear fractional integrodifferential systems in the form
dαxt
dtα σxt ft, xt t
0
gt, s, xsds, t∈J 0, b ,
τxt B1ut,
x0 x0,
1.1
where 0< α≤1 andB1:U → Xis a linear continuous operator, and the control functionu is given inL1J, U,a Banach space of admissible control functions. The nonlinear operators
f:J×E → Eandg:Δ×E → Eare given andΔ:t, s; 0≤s≤t≤b.
LetA:E → Ebe the linear operator defined by
DA {x∈Dσ;τx0}, Ax{σx, forx∈DA}. 1.2
The controllability of integrodifferential systems has been studied by many authorssee1–
2. Main Result
Definition 2.1. System1.1is said to be controllable on the intervalJif for everyx0, x1 ∈ E there exists a controlu∈L2J, Usuch thatx·of1.1satisfiesxb x
1. To establish the result, we need the following hypotheses.
H1Dσ ⊂ Dτand the restriction ofτ to Dσis continuous relative to the graph norm ofDσ.
H2The operatorAis the infinitesimal generator of a compact semigroupTtand there exists a constantM1>0 such thatTt ≤M1.
H3There exists a linear continuous operator B : U → E such that σB ∈ LU, E,τBu B1u,for allu ∈ U.AlsoButis continuously differentiable and
Bu ≤CB1ufor allu∈U,where C is a constant.
H4For all t ∈ 0, b and u ∈ U,TtBu ∈ DA. Moreover, there exists a positive constantK1>0 such thatATt ≤K1.
H5The nonlinear operatorsft, xtandgt, s, xs, fort, s∈J,satisfy
ft, xt ≤L1, gt, s, xs ≤L2, 2.1
whereL1 ≥0 andL2≥0.
H6The linear operatorWfromL2J, UintoEdefined by
Wuα b
0
∞
0
θt−sα−1ξαθTb−sαθσ−ATb−sαθBusdθ ds, 2.2
where ξαθ is a probability density function defined on 0,∞ see9,10 and induces an invertible operatorW−1 defined onL2J, U/kerW,and there exists a positive constantM2 > 0 andM3 > 0 such thatB ≤ M2andW−1 ≤ M3. Let
xtbe the solution of1.1. Then we define a functionzt xt−Butand from our assumption we see thatzt∈DA. Hence1.1can be written in terms ofA
andBas
dαxt
dtα Azt σBut ft, xt t
0
gt, s, xsds, t∈J,
xt zt But, x0 x0.
2.3
Ifuis continuously differentiable on0, b , thenzcan be defined as a mild solution to be the Cauchy problem
dαzt
dtα Azt σBut−B dαut
dtα ft, xt t
0
gt, s, xsds, t∈J,
and the solution of1.1is given by
xt ∞
0
ξαθTtαθx0−Bu0 dθBut
α t
0
∞
0
θt−sα−1ξαθTt−sαθfs, xsdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθσBus−Bd αus
dsα dθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθ s
0
gs, τ, xτdτ dθ ds
2.5
see11–13 .
Since the differentiability of the control u represents an unrealistic and severe requirement, it is necessary of the solution for the general inputsu ∈ L1J, U.Integrating
2.5by parts, we get
xt ∞
0
ξαθTtαθx0dθα
t
0
∞
0
θt−sα−1ξαθTt−sαθσ−ATt−sαθBusdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθfs, xsdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθ s
0
gs, τ, xτdτ dθ ds.
2.6
Thus2.6is well defined and it is called a mild solution of system1.1.
Theorem 2.2. If hypotheses (H1)–(H6) are satisfied, then the boundary control fractional integrodif-ferential system1.1is controllable onJ.
Proof. Using assumptionH6, for an arbitrary functionx·define the control
ut W−1
x1−
∞
0
ξαθTbαθx0dθ−α
b
0
∞
0
θb−sα−1ξαθTb−sαθfs, xsdθ ds
−α b
0
∞
0
θb−sα−1ξαθTb−sαθ s
0
gs, τ, xτdτ dθ ds
t.
We shall now show that, when using this control, the operator defined by
Φxt ∞
0
ξαθTtαθx0dθ
α t
0
∞
0
θt−sα−1ξαθTt−sαθσ−ATt−sαθBusdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθfs, xsdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθ s
0
gs, τ, xτdτ dθ ds
2.8
has a fixed point. This fixed point is then a solution of1.1. Clearly,Φxb x1,which means that the controlusteers the nonlinear fractional integrodifferential system from the initial statex0tox1in timeT, provided we can obtain a fixed point of the nonlinear operator
Φ.
LetY CJ, XandY0{x∈Y :xt ≤r, fort∈J},where the positive constantr is given by
r M1x0bαM1σK1 M2M3
x1M1x0M1L1bαM1L2bα1
M1L1bαM1L2bα1.
2.9
ThenY0is clearly a bounded, closed, and convex subset ofY. We define a mappingΦ:Y →
Y0by
Φxt ∞
0
ξαθTtαθx0dθα
t
0
∞
0
θt−sα−1ξαθTt−sαθσ−ATt−sαθBW−1
×
x1−
∞
0
ξαθTbαθx0dθ−α
b
0
∞
0
θb−sα−1ξαθTb−sαθfs, xsdθ ds
−α b
0
∞
0
θb−sα−1ξαθTb−sαθ s
0
gs, τ, xτdτ dθ ds
sdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθfs, xsdθ ds
α t
0
∞
0
θt−sα−1ξαθTt−sαθ s
0
Consider
SinceTtis compact for everyt >0, the setYt {Φxt:x∈Y0}is precompact
which implies thatY0tis totally bounded, that is, precompact inX. We want to show that
α is an equicontinuous family of functions. Also,ΦY0is bounded inY, and so by the Arzela-Ascoli theorem,ΦY0is precompact. Hence, from the Schauder fixed point inY0,any fixed point ofΦis a mild solution of1.1onJsatisfying
Φxt xt∈X. 2.16
3. Application
LetΩ⊂Rnbe bounded with smooth boundaryΓ.
Consider the boundary control fractional integropartial differential system
∂αyt, x
∂tα −Δyt, x F
t, yt, x t
0
Gt, s, ys, xds, inY 0, b×Ω,
yt,0 ut,0 onΣ 0, b×Γ, t∈0, b ,
y0, x y0x, forx∈Ω.
3.1
The above problem can be formulated as a boundary control problem of the form of1.1by suitably taking the spacesE, X, Uand the operatorsB1,σ,andτ as follows.
LetEL2Ω,XH−1/2Γ,UL2Γ,B1I, the identity operator andDσ {y∈
L2Ω :Δy∈L2Ω},σ Δ.The operatorτ is the trace operator such thatτy y|
Γis well
defined and belongs toH−1/2Γfor eachy ∈ Dσand the operatorAis given byA Δ,
DA H01Ω∪H2ΩwhereHkΩ, HβΩ,andH01Ωare usual Sobolev spaces onΩ, Γ.
We define the linear operatorB:L2Γ → L2ΩbyBuw
uwherewuis the unique solution
to the Dirichlet boundary value problem
Δwu0 in Ω,
wuu inΓ.
3.2
We also introduce the nonlinear operator defined by
ft, xt Ft, yt, x, gt, s, xs Gt, s, ys, x. 3.3
Choose band other constants such that conditions H1–H6 are satisfied. Consequently
Theorem 2.2can be applied for3.1, so3.1is controllable on0, b .
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