1. No category

Qin et al. Advances in Difference Equations 2014, 2014:322
http://www.advancesindifferenceequations.com/content/2014/1/322
RESEARCH
Open Access
Approximate controllability and optimal
controls of fractional evolution systems in
abstract spaces
Haiyong Qin1 , Jianwei Liu1* , Xin Zuo1 and Lishan Liu2
*
Correspondence: [email protected]
Department of Automation, China
University of Petroleum (Beijing),
Changping, Beijing 102249, China
Full list of author information is
available at the end of the article
1
Abstract
In this paper, under the assumption that the corresponding linear system is
approximately controllable, we obtain the approximate controllability of semilinear
fractional evolution systems in Hilbert spaces. The approximate controllability results
are proved by means of the Hölder inequality, the Banach contraction mapping
principle, and the Schauder fixed point theorem. We also discuss the existence of
optimal controls for semilinear fractional controlled systems. Finally, an example is
also given to illustrate the applications of the main results.
MSC: 26A33; 49J15; 49K27; 93B05; 93C25
Keywords: fractional evolution systems; approximate controllability; optimal
controls; semigroup theory; fixed point theorem
1 Introduction
During the past few decades, fractional differential equations have proved to be valuable
tools in the modeling of many phenomena in viscoelasticity, electrochemistry, control,
porous media, and electromagnetism, etc. Due to its tremendous scopes and applications,
several monographs have been devoted to the study of fractional differential equations;
see the monographs [–]. Controllability is a mathematical problem. Since approximately
controllable systems are considered to be more prevalent and very often approximate controllability is completely adequate in applications, a considerable interest has been shown
in approximate controllability of control systems consisting of a linear and a nonlinear
part [–]. In addition, the problems associated with optimal controls for fractional systems in abstract spaces have been widely discussed [–]. Wang and Wei [] obtained
the existence and uniqueness of the PC-mild solution for one order nonlinear integrodifferential impulsive differential equations with nonlocal conditions. Bragdi [] established exact controllability results for a class of nonlocal quasilinear differential inclusions
of fractional order in a Banach space. Machado et al. [] considered the exact controllability for one order abstract impulsive mixed point-type functional integro-differential
equations with finite delay in a Banach space. Approximate controllability for one order
nonlinear evolution equations with monotone operators was attained in []. By the wellknown monotone iterative technique, Mu and Li [] obtained existence and uniqueness
results for fractional evolution equations without mixed type operators in nonlinearity.
©2014 Qin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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Wang and Zhou [] studied a class of fractional evolution equations of the following
type:
Dq x(t) = –Ax(t) + f (t, x(t)),
x() = x ,
t ∈ [, T], q ∈ (, ),
where Dq is the Caputo fractional derivative of order  < q < , –A is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators. A suitable
α-mild solution of the semilinear fractional evolution equations is given, and the existence
and uniqueness of α-mild solutions are also proved. Then by inducing a control term, the
existence of an optimal pair of systems governed by a class of fractional evolution equations is also presented.
Mahmudov and Zorlu [] considered the following semilinear fractional evolution system:
C
q
Dt x(t) = –Ax(t) + Bu(t) + f (t, x(t), (Gx)(t)),
x() = x ,
t ∈ [, T],
q
where C Dt is the Caputo fractional derivative of order  < q < , the state variable x takes
values in a Hilbert space X, A is the infinitesimal generator of a C -semigroup of bounded
operators on the Hilbert space X, the control function u is given in L ([, T], U), U is a
t
Hilbert space, B is a bounded linear operator from U into Xα . (Gx)(t) :=  K(t, s)x(s) ds is a
Volterra integral operator. They studied the approximate controllability of the above controlled system described by semilinear fractional integro-differential evolution equation
by the Schauder fixed point theorem. Very recently, Wang et al. [] researched nonlocal
problems for fractional integro-differential equations via fractional operators and optimal
controls, and they obtained the existence of mild solutions and the existence of optimal
pairs of systems governed by fractional integro-differential equations with nonlocal conditions. Subsequently, Ganesh et al. [] presented the approximate controllability results
for fractional integro-differential equations studied in [].
In this paper, we concern the following fractional semilinear integro-differential evolution equation with nonlocal initial conditions:
C
Dq x(t) = –Ax(t) + a(t)f (t, x(t), (Hx)(t)) + Bu(t),
x() = g(x) + x ∈ Xα ,
t ∈ I = [, b],
(.)
where C Dq denotes the Caputo derivative,  < q < , the state variable x takes values in a
√
Hilbert space X with the norm · = ·, ·, –A : D(A) → X is the infinitesimal generator
of a C -semigroup of uniformly bounded linear operators, that is, there exists M >  such
that T(t) ≤ M for all t ≥ , a ∈ Lp ([, b], R+ ), p > . We denote by Xα a Hilbert space of
√
D(Aα ) equipped with norm xα = Aα x = Aα x, Aα x for all x ∈ D(Aα ), which is equivalent to the graph norm of Aα ,  < α < . The control function u is given in L ([, b], U),
U is a Hilbert space, B is a bounded linear operator from U into Xα . The Volterra integral
t
operator H is defined by (Hx)(t) =  h(t, s, x(s)) ds. The nonlinear term f and the nonlocal
term g will be specified later.
Here, it should be emphasized that no one has investigated the approximate controllability and further the existence of optimal controls for the fractional evolution system (.)
Qin et al. Advances in Difference Equations 2014, 2014:322
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in a Hilbert space, and this is the main motivation of this paper. The main objective of this
paper is to derive sufficient conditions for approximate controllability and existence of optimal controls for the abstract fractional equation (.). The considered system (.) is of
a more general form, with a coefficient function in front of the nonlinear term. Finally, an
example is also given to illustrate the applications of the theory. The previously reported
results in [, , ] are only the special cases of our research.
The rest of this paper is organized as follows. In Section , we present some necessary
preliminaries and lemmas. In Section , we prove the approximate controllability for the
system (.). In Section , we study the existence of optimal controls for the Bolza problem. At last, an example is given to demonstrate the effectiveness of the main results in
Section .
2 Preliminaries and lemmas
Unless otherwise specified, · Lp [,b] represents the Lp (I, R+ ) norm,  ≤ p ≤ ∞, C(I, Xα )
is a Banach space equipped with supnorm given by x∞ = supt∈I xα for x ∈ C(I, Xα ).
Let  ∈ ρ(A), here ρ(A) is the resolvent set of A. Define
A–α =

(α)
∞
t α– T(t) dt.
(.)

It follows that each A–α is an injective continuous endomorphism of X. So we can define
Aα = (A–α )– , which is a closed bijective linear operator in X. It can be shown that Aα has
a dense domain and D(Aβ ) ⊂ D(Aα ) for  ≤ α ≤ β. Moreover, Aα+β x = Aα Aβ x = Aβ Aα x,
x ∈ D(Aμ ) with μ := max{α, β, α +β}, where A = I, I is the identity in X. We have Xβ → Xα
for  ≤ α ≤ β (with X = X), and the embedding is continuous. Moreover, Aα has the
following basic properties.
Lemma . (see []) Aα and T(t) have the following properties:
() T(t) : X → Xα , for each t >  and α ≥ .
() Aα T(t)x = T(t)Aα x, for each x ∈ D(Aα ) and t ≥ .
() For every t > , Aα T(t) is bounded in X, and there exists Mα >  such that
α
A T(t) ≤ Mα t –α .
(.)
() A–α is a bounded linear operator for  ≤ α ≤ , and there exists Cα >  such that
A–α ≤ Cα .
Definition . The fractional integral of order q with the lower limit zero for a function
f is defined as

I f (t) =
(q)
t
q

f (s)
ds,
(t – s)–q
t > , q > ,
(.)
provided that the right side is point-wise defined on [, +∞), where (·) is the gamma
function.
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Definition . The Riemann-Liouville derivative of the order q with the lower limit zero
for a function f : [, ∞] → R can be written as
L

dn
D f (t) =
(n – q) dt n
t
f (s)
ds,
(t – s)–n+q
q

t > , n –  < q < n.
(.)
Definition . The Caputo derivative of the order q for a function f : [, ∞] → R can be
written as
C
Dq f (t) = L Dq f (t) –
n– k
t
k!
k=
f (k) () ,
t > , n –  < q < n.
(.)
Remark .
() If f (t) ∈ C n [, ∞), then
C
Dq f (t) =

(n – q)
t

f (n) (s)
ds = I n–q f (n) (t),
(t – s)–n+q
t > , n –  < q < n.
(.)
() The Caputo derivative of a constant equals zero.
() If f is an abstract function with values in X, then the integrals which appear in
Definitions ., ., and . are taken in Bochner’s sense.
Definition . A solution x ∈ C(I, Xα ) is said to be a mild solution of the system (.), we
mean that for any u(·) ∈ L (I, U), the following integral equation holds:
x(t) = T (t) x + g(x) +
t
(t – s)q– S (t – s)a(s)f s, x(s), (Hx)(s) ds

t
+
(t – s)q– S (t – s)Bu(s) ds,
t ∈ I,
(.)

where
∞
T (t) =
ξq (θ)T t q θ dθ ,
∞
S (t) = q

θ ξq (θ)T t q θ dθ ,



ξq (θ) = θ –– q q θ – q ≥ ,
q
q (θ) =
(.)

(.)
∞
 (nq + )
sin(nπq),
(–)n– θ –qn–
π n=
n!
θ ∈ (, ∞),
(.)
ξq is a probability density function defined on (, ∞), that is,
ξq (θ) ≥ ,
∞
θ ∈ (, ∞) and
ξq (θ) dθ = .
(.)

Definition . The system (.) is said to be approximately controllable on [, b] if
R(b, x ) = Xα , that is, given an arbitrary ε > , it is possible to steer from the point x
to within a distance ε >  for all points in the state space Xα at time b. Here R(b, x ) :=
{x(b; x , u) : u ∈ L ([, b], Uad )}, R(b, x ) is called the reachable set of the system (.) at
terminal time b, x(b; x , u) is the state value at terminal time b corresponding to the control u and the initial value x , R(b, x ) represents its closure in Xα .
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Denote
b =
b
(b – s)q– S (b – s)BB∗ S ∗ (b – s) ds : Xα → Xα ,
(.)

–
R ε, b = εI + b : Xα → Xα ,
ε > ,
(.)
where B∗ denotes the adjoint of B and V ∗ (t) is the adjoint of V (t). Obviously, b is a linear
bounded operator. We define the following linear fractional control system:
t ∈ I = [, b],
C
Dq x(t) = Ax(t) + Bu(t),
x() = x ∈ Xα .
(.)
Lemma . (see []) The linear fractional control system (.) is approximately controllable on [, b] if and only if εR(ε, b ) →  as ε → + in the strong operator topology.
Lemma . (see []) The operators T and S have the following properties:
() For fixed t ≥ , T (t) and S (t) are linear and bounded operators, that is, for any x ∈ X,
S (t)x ≤ M x.
(q)
T (t)x ≤ Mx,
(.)
() (T (t))t≥ and (S (t))t≥ are strongly continuous.
() For every t > , T (t) and S (t) are also compact if T(t) is compact.
() For any x ∈ X, α ∈ (, ) and β ∈ (, ), we have
AS (t)x = A–β S (t)Aβ x,
t ∈ I,
α
A S (t) ≤ Mα q( – α) t –αq ,
( + q( – α))
(.)
 < t ≤ b.
() For fixed t ≥  and any x ∈ Xα , we have
T (t)x ≤ Mxα ,
α
S (t)x ≤ M xα .
α
(q)
(.)
() Tα (t) and Sα (t) are uniformly continuous, that is, for each fixed t >  and > , there
exists h >  such that
Tα (t + ) – Tα (t) < ε,
α
Sα (t + ) – Sα (t) < ε,
α
for t + ≥  and || < h,
for t + ≥  and || < h,
(.)
where
∞
Tα (t) =
q
ξq (θ)Tα t θ dθ ,

∞
Sα (t) = q
θ ξq (θ)Tα t q θ dθ .
(.)

Lemma . (see []) For σ ∈ (, ] and  < c ≤ c we have |cσ – cσ | ≤ (c – c )σ .
Lemma . (Schauder’s fixed point theorem) If B is a closed bounded and convex subset
of a Banach space X and Q : B → B is completely continuous, then Q has a fixed point in B.
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3 Approximate controllability
In this section, we impose the following assumptions:
(H ) f : I × Xα × Xα → X is continuous and there exist m , m >  such that
f (t, x , x ) – f (t, y , y ) ≤ m x – y α + m x – y α ,
(.)
for all xi , yi ∈ Xα , i = , , and t ∈ I.
(H ) h : × Xα → Xα , there exists a function m(t, s) ∈ C(, R+ ) and
∗
k = sup
t∈I
b
m(t, s) ds < ∞

such that
h(t, s, x) – h(t, s, y) ≤ m(t, s)x – yα ,
α
(.)
for each (t, s) ∈ and x, y ∈ Xα , where = {(t, s) ∈ R ;  ≤ s, t ≤ b}.
(H ) g : C(I, Xα ) → Xα is continuous and there exists a constant lg >  such that
g(x) – g(y) ≤ lg x – y∞ ,
α
(.)
for any x, y ∈ C(I, Xα ).
(H ) The function ε : I → R+ defined by
ε (t) = Mlg +
Mα q( – α)
Bα sup B∗ S∗ (b – t)Mblg
ε( + q( – α))
≤t≤b
pp –
 q–  –αq
p – 
Mα q( – α)(m + m k ∗ )aLp [,b]
+
t p
( + q( – α))
p + p (q – ) – 
t
× +
(.)
q – p – αq + 
√
–α
satisfies  < ε (t) <  for all t ∈ I, where max{ p , –+(–α)
} < q < .
Theorem . Assume that conditions (H )-(H ) are satisfied. In addition, the functions
f and g are bounded and the linear system (.) is approximately controllable on [, b].
Then the fractional system (.) is approximately controllable on [, b].
Proof For arbitrary x ∈ Xα , define a control function as follows:
b
uε,x (t) = B S (b – t)R ε,  x – T (b) x + g(x)
∗
∗
b
–
(b – s)

q–
S (b – s)a(s)f s, x(s), (Hx)(s) ds
(.)
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and define the operator Qε by
(Qε x)(t) = T (t) x + g(x) +
t
(t – s)q– S (t – s)a(s)f s, x(s), (Hx)(s) ds

t
+
(t – s)q– S (t – s)Buε,x (s) ds.
(.)

Obviously Qε is well defined on C(I, Xα ).
For x, y ∈ C(I, Xα ), we have
(Qε x)(t) – (Qε y)(t)
α
≤ T (t) g(x) – g(y) (t – s)q– a(s)S (t – s) f s, x(s), (Hx)(s) – f s, y(s), (Hy)(s) α ds
t
(t – s)q– S (t – s) Buε,x (s) – Buε,y (s) α ds
+

+

≤ Mg(x) – g(y)
α
t
(t – s)q– a(s)Aα S (t – s) f s, x(s), (Hx)(s) – f s, y(s), (Hy)(s) ds
t
(t – s)q– Aα S (t – s) Buε,x (s) – Buε,y (s) ds
+
α
t

+

≤ I + I + I.

(.)
By (H )-(H ), Lemma . and the Hölder inequality, we have I  ≤ Mlg x – y∞ and
I  ≤ t –αq
Mα q( – α)
m x(s) – y(s)α
( + q( – α))
t
(t – s)q– a(s) ds

t
Mα q( – α)
+ t –αq
m (Hx)(s) – (Hy)(s)α
(t – s)q– a(s) ds
( + q( – α))

b
Mα q( – α)
m x – y∞ + m
m(s, τ )x(τ ) – y(τ )α dτ
≤ t –αq
( + q( – α))

×
≤
t
(t – s)
p
(q–) p –
pp – 

t
a(s)
p
p
ds


∗
Mα q( – α)(m + m k )aLp [,b]
( + q( – α))
p – 
×
p + p (q – ) – 
I ≤

pp –

t
q– p –αq

x – y∞ ,
(.)
Mα q( – α)
Bα sup B∗ S∗ (b – t)Mbg(x) – g(y)∞
ε( + q( – α))
≤t≤b
Mα q( – α)(m + m k ∗ )aLp [,b]
+
( + q( – α))(q – p – αq + )
p – 
×
p + p (q – ) – 
pp –

t
q– p –αq+

x – y∞ .
(.)
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Then we can deduce that
Qε x – Qε y∞
Mα q( – α)
Bα sup B∗ S∗ (b – t)Mblg
≤ Mlg +
ε( + q( – α))
≤t≤b
Mα q( – α)(m + m k ∗ )aLp [,b]
( + q( – α))
pp –
 q–  –αq
p – 
t p
×
+
p + p (q – ) – 
q–
+

p
t
– αq + 
x – y∞
≤ ε (t)x – y∞ .
(.)
From (H ) and the contraction mapping principle, we conclude that the operator Qε has
a fixed point in C(I, Xα ). Since f and g are bounded, for definiteness and without loss of
generality, let xε be a fixed point of Qε in Br(ε) , where Br(ε) = {x ∈ C([, b], Xα ) | xα ≤ r(ε)}.
From the boundedness of xε , there is a subsequence denoted by {xε } which converges
weakly to x as ε → + , and xε α → xα as ε → + . Then limε→+ xε – xα = . Any
fixed point xε is a mild solution of (.) under the control
uε,x (t) = B∗ S ∗ (b – t)R ε, b x – T (b) x + g(xε )
b
–
(b – s)q– S (b – s)a(s)f s, xε (s), (Hxε )(s) ds .
(.)

Then
xε (t) = T (t) x + g(xε ) +
t
(t – s)q– S (t – s)a(s)f s, xε (s), (Hxε )(s) ds

t
+

(t – s)q– S (t – s)BB∗ S ∗ (b – t)R ε, b p(xε ) ds,
(.)
where
p(xε ) = x – T (b) x + g(xε )
b
(b – s)q– S (b – s)a(s)f s, xε (s), (Hxε )(s) ds.
–
(.)

Therefore we have
xε (b) = x – εR ε, b p(xε ).
(.)
Define
w = x – T (b) x + g(x) –
b

(b – s)q– S (b – s)a(s)f s, x(s), (Hx)(s) ds,
(.)
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it follows that
b
p(xε ) – w ≤ Mg(xε ) – g(x) + (b – s)q– S (b – s)a(s) f s, xε (s), (Hxε )(s)
α
α

– f s, x(s), (Hx)(s) ds
.
(.)
α
By assumptions (H )-(H ), it is easy to get p(xε ) – wα →  as ε → + . Then
xε (b) – x ≤ εR ε, b (w) + εR ε, b p(xε ) – w


α
α
α
b
≤ εR ε,  (w) α + p(xε ) – w α → .
(.)
This proves the approximate controllability of (.).
In order to obtain approximate controllability results by the Schauder fixed point theorem, we pose the following conditions:
(H ) (T(t))t> is a compact analytic semigroup in X.
(H ) There exist constants α ≤ β ≤  such that f : [, b] × Xα × Xα → Xβ and f satisfies:
() For each (x, y) ∈ Xα × Xα , the function f (·, x, y) is measurable.
() For each t ∈ [, b], the function f (t, ·, ·) : Xα × Xα → Xβ is continuous.
() For any r > , there exist functions ϕr ∈ L∞ ([, b], R+ ) such that
sup f (t, x, y)β : xα ≤ r, yα ≤ k ∗ br ≤ ϕr (t),
t ∈ [, b]
(.)
and there exists a constant γ >  such that
lim inf
r→+∞

r

t
a(s)ϕr (s)
ds ≤ γ < +∞,
(t – s)–q
(.)
where k ∗ has been specified in assumption (H ).
(H ) g : C(I, Xα ) → Xα is completely continuous. For any r > , there exist constants ψr
such that
g(x) : x∞ ≤ r ≤ ψr
α
(.)
and there exists a constant γ >  such that
lim
r→+∞
ψr
≤ γ < +∞.
r
(.)
(H ) The following inequality holds:
γ
 bq M
 bq M
MCβ–α
C B + γ M  +
CB < ,
+
(q)
ε q (q)
ε q (q)
where CB will be specified in the following theorem.
(.)
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Theorem . Assume that conditions (H ), (H )-(H ) are satisfied. In addition, the linear
system (.) is approximately controllable on [, b]. Then the fractional system (.) is
approximately controllable on [, b].
Proof For r(ε) > , we set Br(ε) = {x ∈ C([, b], Xα ) | xα ≤ r(ε)}. For arbitrary x ∈ Xα ,
define the control function as follows:
b
uε,x (t) = B S (b – t)R ε,  x – T (b) x + g(x)
∗
∗
b
–
(b – s)
q–
S (b – s)a(s)f s, x(s), (Hx)(s) ds
(.)

and define the operator Qε by
(Qε x)(t) = T (t) x + g(x) +
t
(t – s)q– S (t – s)a(s)f s, x(s), (Hx)(s) ds

t
+
(t – s)q– S (t – s)Buε,x (s) ds.
(.)

We divide the proof into five steps.
Step : Qε maps bounded sets into bounded sets, that is, for arbitrary ε > , there is a
positive constant r(ε) such that Qε (Br(ε) ) ⊂ Br(ε) .
Let x ∈ Br(ε) , from (.), (.), and (.), we have
Buε,x (t) ≤  Aα BB∗ S ∗ (b – t) Aα x + T (b)Aα x + T (b)Aα g(x)
α
ε
b
q– α–β
β
+
(b
–
s)
A
S
(b
–
s)A
a(s)f
s,
x(s),
(Hx)(s)
ds


≤ Aα B sup B∗ S ∗ (b – t)
ε
≤t≤b
MCβ–α b
q–
× x α + Mx α + Mψr +
(b – s) a(s)ϕr (s) ds
(q) 

≤ Cu ,
ε
(.)
where
MCβ–α
Cu = CB x α + Mx α + Mψr +
(q)
∗ ∗
CB = Bα sup B S (b – t).
b
(b – s)
q–
a(s)ϕr (s) ds ,
(.)

(.)
≤t≤b
Then we get

t
 tq
Cu .
(t – s)q– Aα Buε,x (s) ds ≤
ε q
(.)
If operator Qε is not bounded, for each r > , there would exist x ∈ Br(ε) and tr ∈ [, b]
such that
Qin et al. Advances in Difference Equations 2014, 2014:322
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r <(Qε x)(t)α
≤ Mx α + Mg(x)α + +
tr

Page 11 of 22
(tr – s)q– S (tr – s)a(s)f s, x(s), (Hx)(s) ds
α
q–
(tr – s) S (tr – s)Buε,x (s) ds

α
tr
≤ Mx α + Mg(x)α +
(tr – s)q– Aα–β S (tr – s)a(s)Aβ f s, x(s), (Hx)(s) ds
tr

tr
+
(tr – s)q– S (tr – s)Aα Buε,x (s) ds

MCβ–α tr
(tr – s)q– a(s)ϕr (s) ds
(q) 
 bq M
Bα sup B∗ S ∗ (b – t)
+
ε q (q)
≤t≤b
MCβ–α b
q–
× x α + Mx α + Mψr +
(b – s) a(s)ϕr (s) ds .
(q) 
≤ Mx α + Mψr +
(.)
Dividing both sides by r and taking the lower as r → ∞, we have
MCβ–α
 bq M
 bq M
+
C B + γ M  +
CB > ,
γ
(q)
ε q (q)
ε q (q)
(.)
which is a contradiction to (H ). Then Qε maps bounded sets into bounded sets.
Step . Qε is continuous.
Let {xn } ⊂ Br(ε) and xn → x ∈ Br(ε) as n → ∞. From assumptions (H )-(H ), for each
s ∈ [, b], we have
a(s)f s, xn , (Hxn )(s) – f s, x, (Hx)(s) β ≤ a(s)ϕr (s),
(.)
Buε,x (s) – Buε,x (s) ≤  Cu .
n
α
ε
(.)
By the Lebesgue dominated convergence theorem, for each s ∈ [, b], we get
(Qε xn )(t) – (Qε x)(t)
α
t
≤ Mlg xn – x∞ + (t – s)q– Aα–β S (t – s)Aβ a(s)

× f s, xn , (Hxn )(s) – f s, x, (Hx)(s) ds
t
+ (t – s)q– S (t – s)Aα B uε,xn (s) – uε,x (s) ds

MCβ–α t
(t – s)q– a(s)
(q) 
× f s, xn (s), (Hxn )(s) – f s, x(s), (Hx)(s) β ds
t
M
+
(t – s)q– B uε,xn (s) – uε,x (s) α ds → ,
(q) 
≤ Mlg xn – x∞ +
which implies that Qε : Br(ε) → Br(ε) is continuous.
(.)
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Step . For each ε > , the set V (t) = {(Qε x)(t) : x ∈ Br(ε) } is relatively compact in Xα .
The case t =  is trivial, V () = {(Qε x)() : x(·) ∈ Br(ε) } = {x + g(x)} is compact in Xα (see
(H )). So let t ∈ (, b] be a fixed real number, and let h be given a real number satisfied
 < h < t. For any δ > , define Vh (t) = {(Qh,δ
ε x)(t) : x ∈ Br(ε) },
Qh,δ
ε x
∞
ξq (θ)T t q θ dθ x + g(x)
(t) =
δ
t–h ∞
θ (t – s)q– ξq (θ)T (t – s)q θ a(s)f s, x(s), (Hx)(s) dθ ds
+q

δ
t–h ∞
θ (t – s)q– ξq (θ)T (t – s)q θ Buε,x (s) dθ ds
+q

= T hq δ
δ
∞
δ
ξq (θ)T t q θ – hq δ dθ x + g(x)
t–h ∞
+ T hq δ q

δ

δ
θ (t – s)q– ξq (θ)T (t – s)q θ – hq δ a(s)
× f s, x(s), (Hx)(s) dθ ds
t–h ∞
θ (t – s)q– ξq (θ)T (t – s)q θ – hq δ Buε,x (s) dθ ds
+ T hq δ q
q
(.)
= T h δ y(t, h).
Since T(hq δ) is compact in Xα and y(t, h) is bounded on Br(ε) , then the set Vh (t) is a relatively compact set in Xα . On the other hand,
(Qε x)(t) – Qh,δ x (t)
ε
α
δ
q ≤
ξq (θ)T t θ dθ x + g(x) 
α
t δ
q–
q
+ q
θ (t – s) ξq (θ)T (t – s) θ a(s)f s, x(s), (Hx)(s) + Buε,x (s) dθ ds
+
q


α
t
t–h
∞
θ (t – s)q– ξq (θ)T (t – s)q θ
δ
× a(s)f s, x(s), (Hx)(s) + Buε,x (s) dθ ds
≤ Mx α
×

t
δ
ξq (θ) dθ + ψr
(t – s)q– a(s)

α
δ


ξq (θ) dθ + qM Cβ–α ϕr L∞ [,b] + Cu
ε
δ
θ ξq (θ) dθ ds


+ qM Cβ–α ϕr L∞ [,b] + Cu
ε
t
(t – s)
t–h
q–
∞
θ ξq (θ) dθ ds.
a(s)
(.)

This implies that there are relatively compact sets arbitrarily close to the set V (t) for each
t ∈ (, b]. Then V (t), t ∈ (, b] is relatively compact in Xα . Since it is compact at t = , we
have the relatively compactness of V (t) in Xα for all t ∈ [, b].
Step . V := {Qε x ∈ C([, b], Xα ) | x ∈ Br(ε) } is an equicontinuous family of functions on
[, b].
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For  < t < t < b,
Qε x(t ) – Qε x(t )
α
≤ T (t )x – T (t )x α + T (t )g(x) – T (t )g(x)α
t
q–
+
(t – s) S (t – s)a(s)f s, x(s), (Hx)(s) t
α
t
q–
q–
+
S (t – s)a(s)f s, x(s), (Hx)(s) (t – s) – (t – s)

α
t
+
(t – s)q– S (t – s) – S (t – s) a(s)f s, x(s), (Hx)(s) 
α
t
q–
+
(t
–
s)
S
(t
–
s)Bu
(s)


ε,x
t
α
t
q–
q–
+
–
s)
–
(t
–
s)
S
(t
–
s)Bu
(s)
(t



ε,x

α
t
+
(t – s)q– S (t – s) – S (t – s) Buε,x (s)

α
=: I + I + I + I + I + I + I .
(.)
Form the Hölder inequality, Lemmas ., ., and assumption (H ), we obtain
I = t
t
≤
(t – s)q– S (t – s)a(s)f s, x(s), (Hx)(s) ds
α
(t – s)q– Aα–β S (t – s)Aβ a(s)f s, x(s), (Hx)(s) ds
t
t
MCβ–α ϕr L∞ [,b] q–
≤
(t
–
s)
a(s)
ds

(q)
t
t
MCβ–α ϕr L∞ [,b]
≤
(q)
≤
t
(t – s)
pp –
ds

t
MCβ–α ϕr L∞ [,b] aL/q [t ,t ]
(q)
p (q–)
p –
p – 
p q – 
× aLp [t ,t ]
pp –

(t – t )
q– p

.
(.)
From Lemma ., we have
I = (t – s)q– – (t – s)q– S (t – s)a(s)f s, x(s), (Hx)(s) t 
t
MCβ–α ϕr L∞ [,b] q–
q–
≤
–
s)
–
(t
–
s)
(t
a(s)
ds


(q)

t
MCβ–α ϕr L∞ [,b]
a(s)
(t – t )–q
ds.
≤
–q
(q)
(t – s)–q
 (t – s)
α
(.)
By (.), it is easy to see that
I = t
t
M(t – t )q 
Cu .
(t – s)q– S (t – s)Buε,x (s) ds
≤
q(q) ε
α
(.)
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Similar to (.), we obtain
I = t (t – s)
q–

≤
M
(t – t )–q
(q)
– (t – s)
q–
S (t – s)Buε,x (s) ds
α
t



ds Cu .
(t – s)–q (t – s)–q ε
(.)
For t = ,  < t ≤ b, it can easily be seen that I = I = . For t > , when η >  is small
enough, we have
t –η
I ≤

(t – s)q– a(s)S (t – s) – S(t – s) · f s, x(s), (Hx)(s) α ds
t
+
t –η
(t – s)q– a(s)S (t – s) – S(t – s) · f s, x(s), (Hx)(s) α ds
sup S (t – s) – S (t – s)
≤
s∈[,t –η]
p q–
p –  pp q–
p – η  – – t  aLp [t ,t ] Cβ–α ϕr L∞ [,b]
× –
p q – 
MCβ–α ϕr L∞ [,b] p –  pp q–
η  – aLp [t –η,t ]
(q)
p q – 
+
(.)
and
t –η
I ≤
(t – s)q– S (t – s) – S (t – s)Buε,x (s) ds

t
+
(t – s)q– S (t – s) – S (t – s)Buε,x (s) ds
t –η
≤

q
sup S (t – s) – S (t – s) Cu t – ηq
qε
s∈[,t –η]
+
MCα 
Cu ηq .
(q) qε
(.)
Since we have assumption (H ), S (t), t >  in t is continuous in the uniformly operator
topology, it can easily be seen that I and I tend to zero independently of x ∈ Br(ε) as
t → t , η → . It is clear that Ii → , i = , , . . . , , as t → t . Then V (t) is equicontinuous
and bounded. By the Ascoli-Arzela theorem, V (t) is relatively compact in C(I, Xα ). Hence
Qε is a completely continuous operator. From the Schauder fixed point theorem, Qε has a
fixed point, that is, the fractional control system (.) has a mild solution on [, b].
Step . Similar to the proof in Theorem ., it is easy to show that the semilinear fractional system (.) is approximately controllable on [, b].
Since the nonlinear term f is bounded, for any xε ∈ Br(ε) , there exists a constant N > 
such that

b
f s, xε (s), (Hxε )(s)  ds ≤ N.
α
(.)
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Consequently, the sequence {f (s, xε , (Hxε )(s))} is bounded in L ([, b], Xα ), then there is
a subsequence denoted by {f (s, xε , (Hxε )(s))}, which converges weakly to f (s, x, (Hx)(s)) in
L ([, b], Xα ).
It follows that
b
p(xε ) – w ≤ Mg(xε ) – g(x) + (b – s)q– S (b – s)a(s) f s, xε (s), (Hxε )(s)
α
α

– f s, x(s), (Hx)(s) ds
.
(.)
α
·
Now, by the compactness of the operator l(·) →  (· – s)q– S (· – s)l(s) ds : L (I, Xα ) →
C(I, Xα ) and (H ), it is easy to get p(xε ) – wα →  as ε → + . Then
xε (b) – x = εR ε, b (w) + εR ε, b p(xε ) – w


α
α
α
b
≤ εR ε,  (w)α + p(xε ) – wα → .
(.)
This proves the approximate controllability of (.). The proof is completed.
4 Optimal controls
In this section, we assume that Y is another separable reflexive Banach space from which
the controls u take the values. We define the admissible control set Uad = {u ∈ Lp (E) |
u(t) ∈ w(t) a.e.},  < p ≤ p < ∞, where the multifunction w : I → wf (Y) is measurable,
wf (Y) represents a class of nonempty closed and convex subsets of Y, and w(·) ⊂ E, E is a
bounded set of Y.
We consider the following controlled system:
C
Dq x(t) = –Ax(t) + a(t)f (t, x(t), (Hx)(t)) + C(t)u(t),
x() = g(x) + x ∈ Xα ,
t ∈ I, u ∈ Uad ,
(.)
where a ∈ Lp (I, R+ ), C ∈ L∞ (I, L(Y, Xα )), it is easy to see that Cu ∈ Lp (I, Xα ) for all u ∈
Uad .
Let xu denote a mild solution of the system (.) corresponding to u ∈ Uad . Here we
consider the Bolza problem (P), which means that we shall find an optimal pair (x , u ) ∈
C(I, Xα ) × Uad such that
J x , u ≤ J xu , u ,
for all u ∈ Uad ,
(.)
where
J xu , u = φ xu (b) +
b
l t, xu (t), u(t) dt.
(.)

We list here some suitable hypotheses on the operator C, φ, and l as follows:
(HL) () The functional l : I × Xα × Y → R ∪ {∞} is Borel measurable.
() l(t, ·, ·) is sequentially lower semicontinuous on Xα × Y for almost all t ∈ I.
() l(t, x, ·) is convex on Y for each x ∈ Xα and almost all t ∈ I.
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() There exist d ≥ , e > , and λ ∈ L (I, R) such that
p
l(t, x, u) ≥ λ(t) + dxα + euY .
() The functional φ : X → R is continuous and nonnegative.
() The following inequality holds:
γ
 bq M
MCβ–α
+
C∞ sup B∗ S ∗ (b – t)
(q)
ε q (q)
≤t≤b
q
b M
+ γ M  +
C∞ sup B∗ S ∗ (b – t) < .
ε q (q)
≤t≤b
(.)
Theorem . Assume that assumptions (H ), (H )-(H ), and (HL) are satisfied. Then the
Bolza problem (P) admits at least one optimal pair on C(I, Xα ) × Uad provided that
Mlg < .
(.)
Proof Firstly, we show that the system (.) has a mild solution corresponding to u given
by the following integral equation:
x (t) = T (t) x + g xu +
t
u
+
(t – s)q– S (t – s)a(s)f s, xu (s), Hxu (s) ds

t
(t – s)q– S (t – s)C(s)u(s) ds.
(.)

From Lemmas ., ., and (.), we have
t
(t – s)q– S (t – s)C(s)u(s) ds

α
 bq M
CL∞ B∗ S ∗ (b – t)
≤
ε q (q)
MCβ–α b
q–
× x α + Mx α + Mψr +
(b – s) a(s)ϕr (s) ds ,
(q) 
(.)
where · L∞ is the norm of Banach space L∞ (I, L(Y, X)). Meanwhile, assumptions (H )(H ) and (HL) are satisfied. Similar to the proof of Theorem ., we can verify that the
system (.) has a mild solution xu corresponding to u easily.
Secondly, we discuss the existence of optimal controls. If inf{J(xu , u) | u ∈ Uad } = +∞,
there is nothing to prove. We assume that inf{J(xu , u) | u ∈ Uad } = J < +∞. Using condition (HL), we know
J xu , u = J ≥ φ xu (b) +
≥ –J > –∞,
xu (t) dt + e
α
b
λ(t) dt + d

b

u(t)p dt
Y
b

(.)
here J ≥ –J , J >  is a constant.
By the definition of an infimum, there exists a minimizing sequence of the feasible pair
{xn , un } ⊂ Aad , where Aad := {(x, u) | x is a mild solution of the system (.) corresponding
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to u ∈ Uad }, such that J(xn , un ) → J as n → +∞. Since {un } ⊆ Uad , {un } is bounded in
Lp (I, Y), there exists a subsequence, relabeled as {un }, and u ∈ Lp (I, Y) satisfies
un → u
(weakly)
(.)
in Lp (I, Y). Since Uad is closed and convex, by the Marzur lemma, we have u ∈ Uad .
Let xn be a mild solution of the system (.) corresponding to un (n = , , , . . .), then xn
satisfies the following integral equation:
xn (t) = T (t) x + g xn +
t
(t – s)q– S (t – s)a(s)f s, xn (s), Hxn (s) ds

t
+
(t – s)q– S (t – s)C(s)un (s) ds.
(.)

Noting that f (·, xn (s), (Hxn )(s)) is a bounded continuous operator from I into Xβ , we have
f ·, xn (s), Hxn (s) ∈ Lp (I, Xβ ).
(.)
Furthermore, {f (·, xn (s), (Hxn )(s))} is bounded in Lp (I, Xβ ), so there exists a subsequence,
relabeled {f (s, xn (s), (Hxn )(s))} such that
f s, xn (s), Hxn (s) → f s, x(s), (Hx)(s) (weakly),
(.)
where f (·, x(s), (Hx)(s)) ∈ Lp (I, Xβ ).
We denote the operators Q and Q by
t
(Q x)(t) =
(t – s)q– S (t – s)a(s)f s, x, (Hx)(s) ds,
(.)
(t – s)q– S (t – s)C(s)u(s) ds.
(.)

(Q x)(t) =
t

Since {f (s, x(s), (Hx)(s))} ⊆ Lp (I, Xβ ) is bounded, similar to the proof of Theorem ., it
is easy to see that (Q x)(t)α is bounded. It is easy to verify that (Q x)(t) is compact and
equicontinuous in Xα . Due to the Ascoli-Arzela theorem, {(Q x)(t)} is relatively compact
in C(I, Xα ). Obviously, Q is linear and continuous, then Q is a strongly continuous operator, and we obtain
·
(· – s)q– S (· – s)a(s)f s, xn (s), Hxn (s) ds

→
·
(· – s)q– S (· – s)a(s)
f s, x(s), (Hx)(s) ds (strongly).
(.)

Similarly, we can verify Q is a strongly continuous operator and

·
·
(· – s)q– S (· – s)C(s)un (s) ds →

(· – s)q– S (· – s)C(s)u (s) ds
(strongly). (.)
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Now, we turn to considering the following controlled system:
Dq x(t) = –Ax(t) + a(t)
f (t, x(t), (Hx)(t)) + C(t)u (t),
x() = g(x) + x ∈ Xα .
C
t ∈ I, u ∈ Uad ,
(.)
By Theorem ., the above system has a mild solution x corresponding to u , and
x) +
x(t) = T (t) x + g(
t
(t – s)q– S (t – s)a(s)
f s, x(s), (Hx)(s) ds

t
+
(t – s)q– S (t – s)C(s)u (s) ds.
(.)

From Lemma . and (H ), we obtain
I := T (t) g xm – g(
x∞
x) α ≤ Mlg xm –
(.)
and
t
m m
q–
I := (t
–
s)
a(s)
T
(t
–
s)
f
s,
x
(s),
Hx
(s)
–
f
s,
x(s),
(Hx)(s)

t
≤
(t – s)
p
(q–) p –

α
pp –

ds

×

≤
p p
a(s)  S (t – s) f s, xm (s), Hxm (s) – f s, x(s), (Hx)(s) α ds
t
p – 
p + p (q – ) – 
pp –

b
p +p (q–)–
p
t
a(s)
p

p S (t – s) f s, xm (s), Hxm (s)

p
–
f s, x(s), (H)(s) α ds
p

.
(.)
Similarly, we have
I
t
m
q–

:= (t – s) a(s)T (t – s)C(s) u (s) – u (s) ds

p +p (q–)–
p – 
p
b
CL∞
p + p (q – ) – 
t
p
m
p
p 

×
S (t – s) u (s) – u (s) α ds
.
a(s)
≤
α
p –
p
(.)

By (.), (.), and the Lebesgue dominated convergence theorem, we can deduce that
I , I →  as m → .
For each t ∈ I, xn (·),
x(·) ∈ Xα , we have
n
x (t) –
x(t)α ≤ Mlg xn (t) –
x(t)α + I + I .
(.)
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Noting (.), we get
n
I + I
x –
x∞ ≤   .
 – Mlg
(.)
Then
x in C(I, Xα ) as n → ∞ (strongly).
xn →
(.)
Furthermore, we can infer that
f s, xn (s), Hxn (s) → f s,
x(s), (H
x)(s) in C(I, Xα ) as n → ∞ (strongly).
(.)
Using the uniqueness of the limit, we have
f s, x(s), (Hx)(s) = f s,
x(s), (H
x)(s) .
(.)
Therefore
x) +
x(t) = T (t) x + g(
t
(t – s)q– S (t – s)a(s)f s,
x(s), (H
x)(s) ds

t
+
(t – s)q– S (t – s)C(s)u (s) ds,
(.)

which is just a mild solution of the system (.) corresponding to u .
Since C(I, Xα ) → L (I, Xα ), using assumption (HL) and the Balder theorem, we have
J = lim φ xn (b) + lim
n→∞
x(b) +
≥φ n→∞ 
b
b
l t, xn (t), un (t) dt

x, u ≥ J .
l t,
x(t), u (t) dt = J (.)

This implies that J attains its minimum at (
x, u ) ∈ C(I, Xα ) × Uad .
5 Applications
Example . Consider optimal controls for the following fractional controlled system:
⎧ 
⎪
a(t)e–t
∂
∂

 t
s
⎪
⎪  x(t, y) = ∂y x(t, y) + et +e–t cos[  x(t, y) +   sin(  )x(s, y) ds] +
⎪

⎪
∂t
⎨

+  k (t, s)u(s, y) ds, y ∈ [, ], t ∈ [, ], u ∈ Uad ,
⎪
⎪
x(t, ) = x(t, ) = , t > ,
⎪
⎪
⎪
⎩x(, y) = σ  k (t, s)x(s , y) dy + σ  k (t, s) ∂ x(s , y) dy,
i
i
i=  
i=  
∂y
a(t)e–t
et +e–t
(.)
with the cost function

x(t, y) dy dt +

J(x, u) =





u(t, y) dy dt +

x(b, y) dy,

(.)

where σ ∈ N,  < s < s < · · · < sσ < , k ∈ C([, ] × [, ], R+ ), and k , k ∈ L ([, ] ×
[, ], R+ ).
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Let X = Y = (L ([, ]), ·  ). The operator A : D(A) → X is defined by D(A) = {x ∈ X |
x , x ∈ X, x() = x() = } with Ax = –x , then A generates a compact, analytic semigroup
T(·) of uniformly bounded linear operator. Clearly, assumption (H ) is satisfied. Moreover,
the eigenvalues of A are n π  and the corresponding normalized eigenvectors are en (u) =
√
 sin(nπu), n = , , . . . .
Define the control function u : Tx([, ]) → R such that u ∈ L (Tx([, ])). It means that
t → u(t, ·) going from [, ] into Y is measurable. Set U(t) = {u ∈ Y | uY ≤ ϑ} where
ϑ ∈ L (I, R+ ). We restrict the admissible controls Uad to be all the u ∈ L (Tx([, ])) such
that u(t, ·) ≤ ϑ(t).



Let X  = (D(A  ), ·  ), where ·  = A   and the operator A  is given by


A =

∞

z, en en ,
(.)
n=

– 
for each z ∈ D(A  ) = {f ∈ X | ∞
n= z, en en ∈ X} and A = .
Suppose that C(I, X  ) is a Banach space equipped with the supnorm · ∞ , x(t)(y) =

x(t, y), C(t)u(t)(y) = (  k (t, s)u(s) ds)(y). Define f : [, ] × X  × X  → X  by



f t, x(t), (Hx)(t) (y)
 t
e–t

s
e–t
x(t) +
x(s) ds (y) + t –t
sin
= t –t cos
e +e

 

e +e
(.)
and g : C(I, X  ) → X  by

g(x)(y) =

σ
(Kx)(ti ) (y)
for x ∈ C(I, X  ),
(.)

i=
where K : X  → X  is defined by


(Kx)(s) =


k (t, s)x(s) ds +

k (t, s)x (s) ds,
for all x ∈ X  .
(.)
k (t, s)(x – y )(s) ds,
for all x ∈ X  . (.)


Obviously, we have
K(x – y) (s) =


k (t, s)(x – y)(s) ds +



The system (.) can be transformed into
C
Dq x(t) = –Ax(t) + a(t)f (t, x(t), (Hx)(t)) + C(t)u(t),
x() = g(x) + x ∈ Xα ,
t ∈ I, u ∈ Uad
(.)
with the cost function
J(x, u) = x(b)  +


x(t) + u(t) dt,
Y
 
(.)
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and we can verify (HL)()-() are satisfied. It is also not difficult to see that
–t
f t, x(t), (Hx)(t)  ≤ e
= ϕ(t),

et + e–t
ϕ(t) ∈ L∞ I, R+ ,
(.)
then there exists ψr (t) ≡ ψ and γ =  such that (.) holds, and conditions (H ) is satisfied. Meanwhile, it comes from the example in [] that g is a completely continuous
operator from C(I, X  ) → X  and there exist constants c and c such that


g(x)  ≤ σ (c + c )x∞ ,

g(x) – g(z)  ≤ σ (c + c )x – z∞ .

(.)
Let ψr ≡ σ (c + c ) = lg , γ = , it is easy to verify that (H ) and (H ) hold. Since γ =
γ = , condition (H ) and condition (HL)() are satisfied automatically. By Theorem .,
we can conclude that the system (.) has at least one optimal pair while the condition
Mσ (c + c ) <  holds.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have contributed to this work on an equal basis. All authors read and approved the final manuscript.
Author details
1
Department of Automation, China University of Petroleum (Beijing), Changping, Beijing 102249, China. 2 School of
Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China.
Acknowledgements
The fourth author was supported financially by the National Natural Science Foundation of China (11371221), the
Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the
Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
Received: 6 May 2014 Accepted: 4 December 2014 Published: 18 Dec 2014
References
1. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
2. Miller, KS, Ross, B: A Introduction to Fractional Calculus and Differential Equations. Wiley, New York (1993)
3. Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems. Cambridge Scientific
Publishers, Cambridge (2009)
4. Trasov, VE: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media.
Springer, New York (2010)
5. Diethelm, K: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, New York
(2010)
6. Mahmudov, NI, Zorlu, S: On the approximate controllability of fractional evolution equations with compact analytic
semigroup. J. Comput. Appl. Math. 259, 194-204 (2014)
7. Ganesh, R, Sakthivel, R, Mahmudov, NI, Anthoni, SM: Approximate controllability of fractional integrodifferential
evolution equations. J. Appl. Math. 2013, Article ID 291816 (2013)
8. Mahmudov, NI: Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces. Abstr.
Appl. Anal. 2013, Article ID 502839 (2013)
9. Mahmudov, NI: Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal. 68,
536-546 (2008)
10. Sakthivel, R, Ren, Y, Mahmudov, NI: On the approximate controllability of semilinear fractional differential systems.
Comput. Math. Appl. 62, 1451-1459 (2011)
11. Wang, J, Xiang, X, Wei, W: A class of nonlinear integrodifferential impulsive periodic systems of mixed type and
optimal controls on Banach spaces. J. Appl. Math. Comput. 34, 465-484 (2010)
12. Peng, Y, Xiang, X: Necessary conditions of optimality for second order nonlinear impulsive integro-differential
equations on Banach spaces. Nonlinear Anal., Real World Appl. 11, 3121-3130 (2010)
13. Peng, Y, Xiang, X, Wei, W: Second order nonlinear impulsive integrodifferential equations of mixed type with
time-varying generating operators and optimal controls on Banach spaces. Comput. Math. Appl. 57, 42-53 (2009)
14. Liu, G, Xiang, X, Peng, Y: Nonlinear integrodifferential equations and optimal control problems on time scales.
Comput. Math. Appl. 61, 155-169 (2011)
15. Wang, J, Zhou, Y, Wei, W, Xu, H: Nonlocal problems for fractional integrodifferential equations via fractional operators
and optimal controls. Comput. Math. Appl. 62, 1427-1441 (2011)
Qin et al. Advances in Difference Equations 2014, 2014:322
http://www.advancesindifferenceequations.com/content/2014/1/322
16. Wang, J, Zhou, Y, Medved’, M: On the solvability and optimal controls of fractional integrodifferential evolution
systems with infinite delay. J. Optim. Theory Appl. 152, 31-50 (2012)
17. Zhang, N, Zhang, L: Optimal control problem of positive solutions to second order impulsive integrodifferential
equations. J. Comput. Inf. Syst. 8(20), 8311-8318 (2012)
18. Wang, J, Zhou, Y, Wei, W: Optimal feedback control for semilinear fractional evolution equations in Banach spaces.
Syst. Control Lett. 61, 472-476 (2012)
19. Liu, X, Liu, Z, Han, J: The solvability and optimal controls for some fractional impulsive equation. Abstr. Appl. Anal.
2013, Article ID 914592 (2013)
20. Wang, J, Zhou, Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12,
262-272 (2011)
21. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
22. Wang, J, Feˇckan, M, Zhou, Y: On the new concept solutions and existence results for impulsive fractional evolutions.
Dyn. Partial Differ. Equ. 8, 345-361 (2011)
23. Wang, J, Wei, W: A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. Results
Math. 58, 379-397 (2010)
24. Bragdi, M: Controllability of fractional nonlocal quasilinear evolution inclusions with resolvent families. Int. J. Differ.
Equ. 8(1), 15-25 (2013)
25. Machado, JA, Ravichandran, C, Rivero, M, Trujillo, JJ: Controllability results for impulsive mixed type functional
integro-differential evolution equations with nonlocal conditions. Fixed Point Theory Appl. 2013, Article ID 66 (2013)
26. Kang, YH, Jeong, JM, Rho, HH: Controllability for nonlinear evolution equations with monotone operators. J. Inequal.
Appl. 2013, Article ID 534 (2013)
27. Mu, J, Li, Y: Monotone iterative technique for impulsive fractional evolution equations. J. Inequal. Appl. 2011, Article
ID 125 (2011)
10.1186/1687-1847-2014-322
Cite this article as: Qin et al.: Approximate controllability and optimal controls of fractional evolution systems in
abstract spaces. Advances in Difference Equations 2014, 2014:322
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