Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 931975, 17 pages
doi:10.1155/2012/931975
Research Article
Exact Null Controllability for
Fractional Nonlocal Integrodifferential
Equations via Implicit Evolution System
Amar Debbouche1 and Dumitru Baleanu2, 3
1
Department of Mathematics, Faculty of Science, Guelma University, Guelma, Algeria
Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey
3
Institute of Space Sciences, P.O. Box, MG-23, 76900 Magurele-Bucharest, Romania
2
Correspondence should be addressed to Dumitru Baleanu, dumitru@cankaya.edu.tr
Received 1 May 2012; Revised 15 July 2012; Accepted 16 July 2012
Academic Editor: F. Marcellán
Copyright q 2012 A. Debbouche and D. Baleanu. 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.
We introduce a new concept called implicit evolution system to establish the existence results of
mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system,
then we prove the exact null controllability result of a class of fractional evolution nonlocal
integrodifferential control system in Banach space. As an application that illustrates the abstract
results, two examples are provided.
1. Introduction
In this paper, we study the fractional nonlocal integrodifferential system of the form
t
dα ut
At,
B
utut
ft,
B
ut
gt, s, B3 usds,
1
2
dtα
0
a
B4 u0 − u0 hutdt,
1.1
1.2
0
where 0 < α ≤ 1, t ∈ 0, a. Let −A be the infinitesimal generator of a C0 -semigroup in a
Banach space X, and {Bi t : i 1, 2, 3, 4} is a family of linear closed operators defined on
dense sets Si ⊃ DA, i 1, 2, 3, 4, respectively, in X into X. It is assumed that u0 ∈ X,
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f : I × X → X, g : Δ × X → X and h : CI : X → X are given abstract functions. Here,
I 0, a and Δ {s, t : 0 ≤ s ≤ t ≤ a}.
Basic researches in differential equations have showed that many phenomena in
nature are modeled more accurately using fractional derivatives and integrals; for more
detail, we can refer to 1–13 and the references therein. There are many applications where
the fractional calculus can be used, for example, viscoelasticity, electrochemistry, diffusion
processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals 14.
Controllability is a fundamental concept in mathematical control theory and plays an
important role in both finite and infinite dimensional spaces, that is, systems represented
by ordinary differential equations and partial differential equations, respectively. So it is
natural to extend this concept to dynamical systems represented by fractional differential
equations. Several fractional partial differential equations and integrodifferential equations
can be expressed abstractly in some Banach spaces, in many cases, the accurate analysis,
design and assessment of systems subjected to realistic environments must take into account
the potential of random loads and randomness in the system properties. Randomness is
intrinsic to the mathematical formulation of many phenomena such as fluctuations in the
stock market or noise in communication networks. Fu studied the controllability results of
some kinds of neutral functional differential systems, see 15, 16. In our previous work 17,
we established the controllability of fractional evolution nonlocal impulsive quasilinear delay
integrodifferential systems.
The existence results to evolution equations with nonlocal conditions in Banach space
were studied first by Byszewski 18, 19; subsequently, many authors have been studied the
same question, see for instance 20–23.
Deng 24 indicated that, using the nonlocal condition u0 hu u0 to describe for
instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give
better result than using the usual local Cauchy problem u0 u0 . Let us observe also that,
since Deng’s papers, the function h is considered
1.3
hu p
ck utk ,
k1
where ck , k 1, 2, . . . , p are given constants and 0 ≤ t1 < · · · < tp ≤ a.
In this paper, we introduce a new concept in the theory of Semigroup named “implicit
evolution system” to show the reader “what is the main difference between the solutions of
fractional 0 < α < 1 and classical first order homogeneous evolution equation?” which is
based on the work 17 and Pazy 25. A new form of nonlocal condition is also presented.
Our paper is organized as follows. Section 2 is devoted to a review of some essential
results which will be used in this work to obtain our main results. In Section 3, we use
the theory of semigroups 25 in order to introduce our new concept that is called implicit
evolution system. In Section 4, we establish the existence, uniqueness, and regularity of mild
solutions of a class of fractional evolution nonlinear integrodifferential systems with nonlocal
conditions in Banach space. In Section 5, we prove the exact null controllability of a class of
fractional evolution nonlocal integrodifferential control systems; the last section deals to give
examples that provide the abstract results.
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3
2. Preliminary Results
Definition 2.1. The fractional integral of order α with the lower limit zero for a function f ∈
C0, ∞ is defined as
1
I ft Γα
α
t
0
fs
t − s1−α
ds,
t > 0, 0 < α < 1,
2.1
provided the right side is pointwise defined on 0, ∞, where Γ is the gamma function.
Riemann-Liouville derivative of order α with the lower limit zero for a function f ∈
C0, ∞ can be written as
L
d
1
D ft Γ1 − α dt
α
t
0
fs
ds,
t − sα
t > 0, 0 < α < 1.
2.2
The Caputo derivative of order α for a function f ∈ C0, ∞ can be written as
C
Dα ft L Dα ft − f0 ,
t > 0, 0 < α < 1.
2.3
Remark 2.2. 1 If f ∈ C1 0, ∞, then
C
Dα ft 1
Γ1 − α
t
0
f s
ds I 1−α f t,
t − sα
t > 0, 0 < α < 1.
2.4
2 The Caputo derivative of a constant is equal to zero.
3 If f is an abstract function with values in X, then integrals which appear in
Definition 2.1 are taken in Bochner’s sense.
Definition 2.3. By a strong solution of the nonlocal Cauchy problem 1.1, 1.2, we mean a
function u with values in X such that
i u is a continuous function in t ∈ I and ut ∈ DA,
ii dα u/dtα exists and continuous on 0, a, 0 < α < 1, and u satisfies 1.1 on 0, a and
1.2.
It is suitable to rewrite 1.1, 1.2 in the form
t
α−1
1
ut u0 t−η
Γα 0
η
× −A η, B1 u η u η f η, B2 u η g η, s, B3 us ds dη,
2.5
0
see also 26, 27.
Let X and Y be two Banach spaces such that Y is densely and continuously embedded
in X. We denote by Z every Banach space ZI, X endowed with the usual norm, which is
given by uZ supt∈I ut, for u ∈ Z. The space of all bounded linear operators from X to
Y is denoted by BX, Y . We recall some definitions and known facts from Pazy 25.
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Definition 2.4. Let S be a linear operator in X, and let Y be a subspace of X. The operator S
{x ∈ DS ∩ Y : Sx ∈ Y } and Sx
Sx for x ∈ DS
is called the part of S
defined by DS
in Y .
Definition 2.5. Let Ω be a subset of X and for every t ∈ I and B1 u ∈ Ω, and let −At, B1 u be
the infinitesimal generator of a C0 -semigroup St,B1 u s, s ≥ 0, on X. The family of operators
{At, B1 u, t, B1 u ∈ I × Ω} is stable if there are constants M ≥ 1 and ω such that
ρAt, B1 u ⊃ ω, ∞
k
≤ Mλ − ω−k
R λ : A t j , B1 u
2.6
j1
for λ > ω every finite sequences 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ a, 1 ≤ j ≤ k.
The stability of {At, B1 u, t, B1 u ∈ I × Ω} implies that
k
⎛
⎞
k
Stj ,B1 u sj ≤ M exp⎝ω sj ⎠,
j1
sj ≥ 0,
2.7
j1
and any finite sequences 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ a, 1 ≤ j ≤ k, k 1, 2, . . ..
Definition 2.6. Let St,B1 u s, s ≥ 0 be the C0 -semigroup generated by At, B1 u, t, B1 u ∈ I × Ω.
A subspace Y of X is called At, B1 u-admissible if Y is invariant subspace of St,B1 u s, and
the restriction of St,B1 u s to Y is a C0 -semigroup in Y .
Let Ω ⊂ X be a subset of X such that for every t, B1 u ∈ I × Ω, At, B1 u is the
infinitesimal generator of a C0 -semigroup St,B1 u s, s ≥ 0 on X. We make the following
assumptions.
H1 The family {At, B1 u, t, B1 u ∈ I × Ω} is stable.
B1 u, t, B1 u ∈
H2 Y is At, B1 u-admissible for t, B1 u ∈ I × Ω, and the family {At,
B1 u of At, B1 u in Y is stable in Y .
I × Ω} of parts At,
H3 For t, B1 u ∈ I ×Ω, DAt, B1 u ⊃ Y , At, B1 u is a bounded linear operator from
Y to X and t → At, B1 u is continuous in the BY, X norm · .
H4 There is a constant L > 0 such that
At, B1 u − At, B1 vY → X ≤ Lu − vX
2.8
holds for every B1 u, B1 v ∈ Ω, and t ∈ I.
In the next section, we will introduce a new concept in the theory of semigroups.
3. Implicit Evolution System
Let Ω be a subset of X and {At, B1 u, t, B1 u ∈ I × Ω} a family of operators satisfying the
conditions H1 –H4 . If u ∈ CI : X has values in Ω, then there is a unique evolution system
Uα t, s; B1 u, 0 < α ≤ 1, 0 ≤ s ≤ t ≤ a, in X satisfying
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5
i Uα t, s; B1 u ≤ Meωt−s for 0 ≤ s ≤ t ≤ a, where M and ω are stability constants,
ii ∂α /∂tα Uα t, s; B1 uy As, B1 usUα t, s; B1 uy for y ∈ Y and 0 ≤ s ≤ t ≤ a,
iii ∂α /∂sα Uα t, s; B1 uy −Uα t, s; B1 uAs, B1 usy for y ∈ Y and 0 ≤ s ≤ t ≤ a.
Remark 3.1. 1 If B1 is the identity and α 1, then Ut, s; u is the explicit evolution system
given in Pazy 25 and in Zaidman 28.
2 Since, in our case, Ut, s; u is dependent of α and B1 , so we call it an implicit
evolution system generated by −At, B1 u.
3 For nonautonomous differential equations in a Banach space, the implicit evolution
system is similar to our concept α, u-resolvent family.
4 We can deduce that 1.1-1.2 is well posed if and only if −At, B1 u is the generator
of the implicit evolution system Ut, s; u.
Further, we assume the following.
H5 For every u ∈ CI : X satisfying ut ∈ Ω for 0 ≤ t ≤ a, we have
Ut, s; uY ⊂ Y,
0≤s≤t≤a
3.1
and Ut, s; u is strongly continuous in Y for 0 ≤ s ≤ t ≤ a.
H6 Y is reflexive.
H7 For every t, B2 u ∈ I × Ω, ft, B2 u ∈ Y .
H8 The operator B4 t λα I−1 exists in BX for any λ with Re λ ≤ 0 and
B4 t λα I−1 ≤
Cα
,
|λ| 1
t ∈ I,
3.2
where Cα is a positive constant independent of both t and λ.
H9 h : CI : Ω → Y is Lipschitz continuous in X and bounded in Y , that is, there
exist constants k1 > 0 and k2 > 0 such that
huY ≤ k1 ,
3.3
hu − hvY ≤ k2 max u − vX .
t∈I
For the conditions H9 and H10 , let Z be taken as both X and Y .
H10 g : Δ × Z → Z is continuous, and there exist constants k3 > 0 and k4 > 0 such
that
t
gt, s, B3 u − gt, s, B3 v
0
k4 max
Z
ds ≤ k3 u − vZ ,
t
gt, s, 0
0
u, v ∈ X,
Z
ds : t, s ∈ Δ .
3.4
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H11 f : I × Z → Z is continuous, and there exist constants k5 > 0 and k6 > 0 such
that
ft, B2 u − ft, B2 v
Z
≤ k5 u − vZ ,
k6 max ft, 0
Z
t∈I
u, v ∈ X,
.
3.5
Let us take M0 max Ut, s; uΩZ , 0 ≤ s ≤ t ≤ a, u ∈ Ω.
H12 There exist positive constants r > 0 and 0 < λ < 1 such that
M0 {u0 aCα k1 ark3 k5 k4 k6 } ≤ r,
λ Kau0 Y a2 Cα k1 K aM0 Cα k2 a{Kk3 k5 r k4 k6 a M0 k3 k5 }.
3.6
By a mild solution of 1.1, 1.2, we mean a function u ∈ CI : X with values in Ω and
u0 ∈ X satisfying the integral equation
ut Ut, 0; uu0 Ut, 0; uB4−1
t
0
a
hutdt
0
s
Ut, s; u fs, B2 us g s, η, B3 u η dη ds.
3.7
0
H13 Further, there exists a constant K > 0 such that for every u, v ∈ CI : X with
values in Ω and every ω ∈ Y we have
Ut, s; uω − Ut, s; vω ≤ KωY
t
uτ − vτdτ.
3.8
s
4. Existence Results
Theorem 4.1. Let u0 ∈ Y and Ω {u ∈ X : uY ≤ r}, r > 0. If −At, B1 u is the generator of
an implicit evolution system Ut, s; u and the assumptions (H5 )∼(H13 ) are satisfied, then 1.1, 1.2
has a unique mild solution on I.
Proof. Let S be a nonempty closed subset of CI : X defined by
S {u : u ∈ CI : X, uY ≤ r},
t ∈ I.
4.1
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Consider a mapping P on S defined by
P ut Ut, 0; uu0 Ut, 0; uB4−1
t
a
hutdt
0
4.2
s
Ut, s; u fs, B2 us g s, η, B3 u η dη ds.
0
0
For u ∈ S, we have
P utY ≤ Ut, 0; uu0 Ut, 0; uB4−1
t
a
hutdt
0
Ut, s; u fs, B2 us − fs, 0 fs, 0
0
s
g s, η, B3 u η −g s, η, 0 dη 0
s
g s, η, 0 dη
ds.
0
≤ M0 u0 aM0 Cα k1
t
M0 {k5 us k6 k3 us k4 }ds
0
≤ M0 u0 aM0 Cα k1 aM0 {k5 us k6 k3 us k4 }
≤ M0 {u0 aCα k1 ark3 k5 k4 k6 }
≤ r.
4.3
Thus, P maps S into itself. Now, we will show that P is a strict contraction on S which will
ensure the existence of a unique continuous function satisfying 3.7 on I.
If u, v ∈ S, then
P ut − P vt
≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; uB4−1
t
0
0
hutdt − Ut, 0; vB4−1
s
Ut, s; u fs, B2 us g s, η, B3 u η dη
a
hvtdt
0
0
s
−Ut, s; v fs, B2 vs g s, η, B3 v η dη ds
0
≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; uB4−1
a
Ut, 0; vB4−1
a
0
hutdt −
Ut, 0; vB4−1
a
0
hutdt − Ut, 0; vB4−1
a
hvtdt
0
a
hutdt
0
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t
s
Ut, s; u fs, B2 us g s, η, B3 u η dη
0
0
−Ut, s; v fs, B2 us s
g s, η, B3 u η dη
0
Ut, s; v fs, B2 us s
g s, η, B3 u η dη
0
−Ut, s; v fs, B2 vs s
g s, η, B3 v η dη
ds
0
≤ Kau0 Y max uτ − vτ a2 Cα k1 K max uτ − vτ
τ∈I
τ∈I
aM0 Cα k2 max uτ − vτ
τ∈I
s
t
K fs, B2 us g s, η, B3 u η dη
0
0
M0
τ∈I
fs, B2 us − fs, B2 vs
a max uτ − vτ
Y
s
dη
g s, η, B3 u η − g s, η, B3 v η
ds
0
≤ Kau0 Y a2 Cα k1 K aM0 Cα k2 max uτ − vτ
τ∈I
a{Kk3 k5 r k4 k6 a M0 k3 k5 } max uτ − vτ.
τ∈I
4.4
Thus,
P ut − P vt ≤ λ max uτ − vτ,
τ∈I
4.5
which means that P is a strict contraction map from S into S, and therefore by the Banach
contraction principle there exists a unique fixed point u ∈ S such that P u u. Hence, u is a
unique mild solution of 1.1, 1.2 on I.
Theorem 4.2. Assume the following.
i Conditions (H1 )∼(H13 ) hold.
ii The functions B2 tω and B3 tω are uniformly Hölder continuous in t ∈ I for every
element ω in S2 ∩ S3 .
iii There are numbers L1 , L2 > 0 and p, q ∈ 0, 1 such that
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9
ft1 , B2 u − ft2 , B2 v ≤ L1 |t1 − t2 |p B2 u − B2 v ,
g s1 , η, B3 u − g s2 , η, B3 v ≤ L2 |s1 − s2 |q
4.6
for all t1 , t2 ∈ I and all s1 , η, s2 , η ∈ Δ.
Then, the problem 1.1, 1.2 has a unique strong solution on I.
Proof. Applying Theorem 4.1, the problem 1.1, 1.2 has a mild solution u ∈ S. Now, we will
show that u is a unique strong solution of the considered problem on I.
According to ii, B2 u−B2 v is uniformly Hölder continuous in t ∈ I for every element
t
u in S2 ∩S3 , also iii implies that t → ft, B2 u and t → 0 gt, s, B3 uds are uniformly Hölder
continuous on I 20, 26.
Set
V t ft, B2 u t
gt, s, B3 uds.
4.7
0
Clearly V t is uniformly Hölder continuous in t ∈ I.
Consider the following nonlocal Cauchy problem:
dα vt
At, B1 utut V t,
dtα
a
hutdt.
B4 u0 − u0 4.8
0
From Pazy, 4.8 has a unique solution v on I given by
vt Ut, 0; uu0 Ut, 0; uB4−1
a
hutdt 0
t
Ut, s; uV sds.
4.9
0
Noting that 21, each term on the right hand side of 4.9 belongs to DA, thus vt ∈ DA,
using the uniqueness of V t, we have that ut vt. Hence, u is the unique strong solution
of 1.1, 1.2 on I.
In next section, some results are obtained from Sakthivel et al. 29, 30.
5. Exactly Null Controllability Results
Consider fractional nonlocal evolution integrodifferential control system of the form
t
dα ut
At, B1 utut Φμ t Ψ t, ft, B2 ut, gt, s, B3 usds ,
dtα
0
a
hutdt,
B4 u0 − u0 0
5.1
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where the unknown u· takes values in the Banach space X, the control function μ belongs to
the spaces L2 I, H, a Banach space of admissible control functions with H, a Banach space.
Further, Φ is a bounded linear operator from H into X, the function Ψ : I × X × X → X is
given, and the others terms are defined as above.
For all u0 ∈ X and admissible control μ ∈ L2 I, H, the problem 5.1 admits a mild
solution given by
uμ t Ut, 0; uu0 t
Ut, 0; uB4−1
a
hutdt
0
s
Ut, s; u Φμs Ψ s, fs, B2 us, gs, τ, B3 uτdτ ds.
0
5.2
0
Definition 5.1. We will say that system 5.1 is exactly null controllable on the interval I if
for a11 u0 ∈ X, there exists a control μ ∈ L2 I, H, such that the mild solution ut of 5.1
a
corresponding to μ verifies u0 − B4−1 0 hutdt u0 and ua 0.
In order to prove the controllability result, in addition, we consider the following
conditions.
H14 Ψ : I × X × X → X is continuous, and there exist constants N1 and N2 such that
for all xi , yi ∈ X, i 1, 2, we have
Ψ t, x1 , y1 − Ψ t, x2 , y2 ≤ N1 x1 − x2 y1 − y2 ,
N2 max Ψt, 0, 0.
5.3
t∈I
H15 Let
ρ aM0 M1 M2 ρ σ σ ≤ r,
5.4
where σ aM0 {N1 k3 k5 r k4 k6 N2 } and ρ M0 u0 aM0 Cα k1 , and let
λ Kau0 a2 Cα k1 K aM0 Cα k2 2a M0 M1 M2 ρ ap a Kqa p ,
5.5
where p M0 N1 k3 k5 , q N1 k3 k5 r k4 k6 N2 , and 0 ≤ λ < 1.
H16 The bounded linear operator W : L2 I, H → X defined by
Wμ a
Ua, s; uΦμds
5.6
0
has an induced inverse operator W −1 which takes values in L2 I, H/ker W and there exist
positive constants M1 , M2 , such that Φ ≤ M1 and W −1 ≤ M2 .
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Theorem 5.2. If hypotheses (H1 )∼(H16 ) are satisfied, then the control nonlocal fractional integrodifferential system 5.1 is exactly null controllable on I.
a
Proof. Let Sr {u : u ∈ CI : X, u0 − B4−1 0 hutdt u0 , u ≤ r, t ∈ I}.
We define an operator Q : Sr → Sr by
a
hutdt
Quμ t Ut, 0; uu0 Ut, 0; uB4−1
t
0
U t, η; u ΦW
−1
−Ua, 0; uu0 −
0
−
a
a
hutdt
0
Ua, s; uΨ s, fs, B2 us,
0
t
Ua, 0; uB4−1
s
Ut, s; uΨ s, fs, B2 us,
0
gs, τ, B3 uτdτ ds η dη
0
s
5.7
gs, τ, B3 uτdτ ds.
0
Using the hypothesis H14 , for an arbitrary function u·, we define the control
a
−1
−1
μt W −Ua, 0; uu0 − Ua, 0; uB4
hutdt
−
a
0
s
Ua, s; uΨ s, fs, B2 us,
0
gs, τ, B3 uτdτ ds t.
5.8
0
Using this controller, we will show that the operator Q has a fixed point. This fixed point is
then a solution of 5.2.
Clearly, Qμ ua 0, which means that the control μ steers system 5.1 from the initial
state u0 to origin in time a, provided we can obtain a fixed point of the nonlinear operator Q.
Now, we show that Q maps Sr into itself.
We have
Quμ t
≤ Ut, 0; uu0 t
Ut, 0; uB4−1
U t, η; u
ΦW −1
0
a
hutdt
0
Ua, 0; uu0 Ua, 0; uB4−1
a
0
Ua, s; u
a
hutdt
0
s
Ψ s, fs, B2 us, gs, τ, B3 uτdτ
0
− Ψs, 0, 0 Ψs, 0, 0 ds dη
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t
Ut, s; u
0
×
Ψ s, fs, B2 us,
s
gs, τ, B3 uτdτ
− Ψs, 0, 0 Ψs, 0, 0 ds.
0
≤ M0 u0 aM0 Cα k1
aM0 M1 M2 M0 u0 aM0 Cα k1 aM0 {N1 k5 r k6 k3 r k4 N2 }
aM0 {N1 k5 r k6 k3 r k4 N2 }
≤ r.
5.9
Thus, Q maps Sr into itself. Now, for u, v ∈ Sr , we have
Quμ t − Qvμ t
Ut, 0; uB4−1
≤ Ut, 0; uu0 − Ut, 0; vu0 t
0
a
hutdt −
0
a
U t, η; u ΦW −1 −Ua, 0; uu0 − Ua, 0; uB4−1
hutdt
−
a
0
s
Ua, s; uΨ s, fs, B2 us,
0
− U t, η; v ΦW −1 −Ua, 0; vu0 − Ua, 0; vB4−1
−
a
hvtdt
0
gs, τ, B3 uτdτ ds
0
hvtdt
0
Ua, s; vΨ s, fs, B2 vs,
s
dη
gs, τ, B3 vτdτ ds
0
0
−Ut, s; vΨ s, fs, B2 vs,
s
≤ Ut, 0; uu0 − Ut, 0; vu0 Ut, 0; vB4−1
t
a
0
Ut, 0; uB4−1
hutdt − Ut, 0; vB4−1
U t, η; u ΦW
−1
ds.
gs, τ, B3 vτdτ
0
0
a
s
Ut, s; uΨ s, fs, B2 us, gs, τ, B3 uτdτ
0
a
0
t
Ut, 0; vB4−1
−Ua, 0; uu0 −
a
hutdt −
0
Ut, 0; vB4−1
a
hvtdt
0
Ua, 0; uB4−1
a
hutdt
0
a
hutdt
0
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13
−
a
s
Ua, s; uΨ s, fs, B2 us, gs, τ, B3 uτdτ ds
0
− U t, η; v ΦW −1 −Ua, 0; vu0 − Ua, 0; vB4−1
−
a
s
Ut, s; uΨ s, fs, B2 us,
dη
gs, τ, B3 vτdτ ds
gs, τ, B3 uτdτ
0
hvtdt
0
0
s
0
0
Ua, s; vΨ s, fs, B2 vs,
0
t
a
−Ut, s; vΨ s, fs, B2 us,
s
gs, τ, B3 uτdτ
0
Ut, s; vΨ s, fs, B2 us,
s
gs, τ, B3 uτdτ
0
s
ds.
−Ut, s; vΨ s, fs, B2 vs, gs, τ, B3 vτdτ
0
≤ Kau0 a2 Cα k1 K aM0 Cα k2 max uτ − vτ
τ∈I
2a{M0 M1 M2 M0 u0 aM0 Cα k1 aM0 N1 k5 k3 } max uτ − vτ
τ∈I
a{aKN1 k5 r k6 k3 r k4 N2 M0 N1 k5 k3 } max uτ − vτ
τ∈I
≤ λ max uτ − vτ.
τ∈I
5.10
Therefore, Q is a contraction mapping, and hence there exists a unique fixed point u ∈ X,
such that Qut ut. Any fixed point of Q is a mild solution of 5.1 on I which satisfies
ua 0. Thus, system 5.1 is exactly null controllable on I.
6. Examples
To illustrate the abstract results, we give the following examples.
Example 6.1. Consider the nonlinear integropartial differential equation of fractional order
t
q
∂α ux, t
aq x, t; bq ux, t Dx ux, t Fx, t, w1 Gx, t, s, w2 sds,
∂tα
0
|q|≤2m
6.1
14
Journal of Applied Mathematics
with nonlocal condition
ux, 0 p
ck ux, tk gx,
6.2
k1
q
q
q
where 0 < α ≤ 1, 0 ≤ t1 < · · · < tp ≤ a, x ∈ Rn , Dx Dx11 · · · Dxnn , Dxi ∂/∂xi , q q1 , . . . , qn is
an n-dimensional multi-index, |q| q1 · · · qn , and wi , i 1, 2, is given by
wi x, t q
bqi x, tDx ux, t
|q|≤2m−1
q cqi x, tDy u y, t dy.
Ω q ≤2m−1
||
6.3
Let L2 Rn be the set of all square integrable functions on Rn . We denote by Cm Rn the set of
all continuous real-valued functions defined on Rn which have continuous partial derivatives
of order less than or equal to m. By C0m Rn , we denote the set of all functions f ∈ Cm Rn with compact supports. Let H m Rn be the completion of C0m Rn with respect to the norm
f
2
m
|q|≤m
Rn
!2
!
!
! q
!Dx fx! dx.
6.4
It is supposed that the following hold.
"
q
i The operator A − |q|≤2m aq x, t; bq ux, tDx is uniformly elliptic on Rn . In other
words, all the coefficients aq , |q| 2m, are continuous and bounded on Rn , and there is a
positive number c such that
−1m1
eq x, t; bq ξq ≥ c|ξ|2m ,
|q|2m
q
6.5
q
for all x ∈ Rn and all ξ /
0, ξ ∈ Rn , ξq ξ1 1 · · · ξnn , and |ξ|2 ξ12 · · · ξn2 .
ii All the coefficients aq , |q| 2m, satisfy a uniform Hölder condition on Rn . Under
these conditions, the operator A with domain of definition DA H 2m Rn generates an
evolution operator defined on L2 Rn , and it is well known that H 2m Rn is dense in X L2 Rn and the initial function gx is an element in Hilbert space H 2m Rn , see 26, page
438. Applying Theorem 4.1, this achieves the proof of the existence of mild solutions of the
problem 6.1, 6.2. In addition,
iii If the coefficients bq , cq , |q| ≤ 2m − 1 satisfy a uniform Hölder condition on Rn and
the operators F and G satisfy.
There are numbers L1 , L2 ≥ 0 and λ1 , λ2 ∈ 0, 1 such that
|q|≤2m−1
Rn
! !2
!
!2 q
q
!
!
!F x, t, Dx w1 − F x, s, Dx w1∗ ! dx ≤ L1 |t − s|λ1 !w1 − w1∗ ! dx ,
|q|≤2m−1
Rn
! !2
q
q
!
!
!G x, t, η, Dx w2 − G x, s, η, Dx w2 ! dx ≤ L2 |t − s|λ2 ,
6.6
Journal of Applied Mathematics
15
for all t, s ∈ I, t, η, s, η ∈ Δ, and all x ∈ Rn . Applying Theorem 4.2, we deduce that 6.1,
6.2 has a unique strong solution.
The second example is concerned with the controllability result.
Example 6.2. Consider the fractional nonlocal evolution integropartial differential control
system of the form
t
∂2 ux, t
∂α ux, t
ax, t, bux, t
ζx, t Υ t, μ1 t, ux, t, μ2 t, s, ux, tds ,
∂tα
∂x2
0
ux, 0 p
ck ux, tk gx,
x ∈ 0, π,
k1
u0, t uπ, t 0,
t ∈ I,
6.7
where 0 < α ≤ 1, 0 ≤ t1 < · · · < tp ≤ a, and the functions ax, t, ·, bx, t are continuous.
Let us take
X L2 0, π,
Sr y ∈ L2 0, π : y ≤ r .
6.8
Put Φμxt ζx, t, x ∈ 0, π where μt ζ·, t and ζ : 0, π×I → 0, π is continuous.
We define At, · : X → X by At, ·wx ax, t, ·w with domain DA {w ∈ X :
w, w are absolutely continuous, w ∈ X, w0 wπ 0}. Assume that −At, · generates
an evolution system Ut, s, · such that for every positive numbers n1 and n2 , Ut, s, · ≤ n1
and Ut, s, ·As, · ≤ n2 .
Also, define B1 t : DB1 S1 → X by B1 tzx bx, tz, for all z ∈ X and
x ∈ 0, π.
Assume that the linear operator W that is given by
Wμx a
Ua, s; uζx, sds,
x ∈ 0, π,
6.9
0
has a bounded invertible operator W −1 in L2 I, H/kerW.
Let us assume that the nonlinear functions Υ, μ1 , and μ2 satisfy the following Lipschitz
conditions
Υ t, z1 , y1 − Υ t, z2 , y2 ≤ c1 z1 − z2 y1 − y2 ,
μ1 t, x1 − μ1 t, x2 ≤ c2 x1 − x2 ,
t
μ2 t, s, v1 − μ2 t, s, v2 ds ≤ c3 v1 − v2 ,
0
where cj > 0, j 1, 2, 3, xi , yi , zi , vi ∈ X, i 1, 2, and s, t ∈ I.
6.10
16
Journal of Applied Mathematics
All the conditions stated in Theorem 5.2 are satisfied. Hence, system 6.7 is exactly
null controllable on I.
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