DOI: 10.1515/auom-2016-0017
An. CSt. Univ. Ovidius ConstanCta
Vol. 24(1),2016, 29–55
Exact controllability of fractional neutral
integro-differential systems with
state-dependent delay in Banach spaces
S. Kailasavalli, D. Baleanu, S. Suganya and M. Mallika Arjunan
Abstract
In this manuscript, we have a tendency to execute Banach
contraction fixed point theorem combined with resolvent operator to
analyze the exact controllability results for fractional neutral integrodifferential systems (FNIDS) with state-dependent delay (SDD) in
Banach spaces. An illustration is additionally offered to exhibit the
achieved hypotheses.
1
Introduction
The rise of fractional calculus emerge new inquiries in basic physics,
which gives awesome complicated enthusiasm to the mathematicians and
physicists in the principle of fractional calculus. The fractional differential
equations (FDE) have been viewed as to be the beneficial tool, which can
summarize dynamical behavior of real life phenomena more precisely. Case
in point, the nonlinear oscillation of earthquake can be effectively displayed
with fractional derivatives. We can locate the various uses of FDE in control
theory, nonlinear oscillation of earthquake, the fluid-dynamic traffic model,
aerodynamics and in nearly every discipline of science and engineering. For
Key Words: Controllability, fractional order differential equations, state-dependent
delay, Banach fixed point theorem, resolvent operators, semigroup theory.
2010 Mathematics Subject Classification: Primary 93B05, 34K30, 26A33; Secondary
35R10, 47D06.
Received: 1.07.2015.
Revised: 2.08.2015.
Accepted: 3.10.2015.
29
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
30
fundamental certainties about fractional systems, one can make reference to
the books [1, 2], and the papers [3, 4, 5, 6, 7, 8, 9], and the references
cited therein. FDE with delay features happen in several areas such as
medical and physical with SDD or non-constant delay. These days, existence
and controllability results of mild solutions for such problems became very
attractive and several researchers working on it. As of late, few number of
papers have been released on the fractional order problems with SDD, see for
instance [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].
On the flip side, the idea of controllability has assumed a focal part all
through the historical backdrop of cutting edge control theory. This is the
qualitative property of control frameworks and is of specific significance in
control theory. Numerous dynamical frameworks are such that the control
does not influence the complete state of the dynamical framework yet just a
piece of it. Then again, frequently in real industrial procedures it is conceivable
to notice just a specific piece of the complete state of the dynamical framework.
In this way, it is essential to figure out if or not control of the complete state
of the dynamical framework is conceivable. In this way, here the idea of
complete controllability and approximate controllability emerges. Generally
discussing, controllability usually indicates that it is conceivable to steer
dynamical framework from a arbitrary beginning state to the coveted last
state utilizing the set of acceptable controls.
These days, the controllability for different sorts of fractional differential
and integro-differential frameworks in theoretical spaces have created
substantial passion among scientists, for instance, see [20, 22, 23, 24, 25, 26,
27, 28, 21]. As of late, Santos et al. [16, 22, 23] researched the existence of
solutions for FNIDE with unbounded or SDD delay in Banach spaces. Shu et
al. [24] looked into the existence results for FDE with nonlocal conditions
of order α ∈ (1, 2). In [25, 26], the writers present sufficient conditions
for the existence and approximate controllability of fractional order neutral
differential and stochastic differential system with infinite delay. Kexue et al.
[27] assessed the controllability of nonlocal FDE of order α ∈ (1, 2]. Sakthivel
et al. [28] identified the approximate controllability of fractional dynamical
system by selecting appropriate fixed point theorem whereas Rajivgandhi et al.
[29] established the approximate controllability of fractional stochastic integrodifferential equations with infinite delay of order 1 < α < 2 by utilizing the
Banach contraction principle. Lately, in [17, 18, 19], the authors discussed
the approximate controllability results for FNIDS with SDD by utilizing the
suitable fixed point theorem. Having said that, exact controllability results
for FNIDS with SDD in Bh phase space adages have not yet been entirely
inspected.
Roused by the exertion of the previously stated papers [14, 17, 19], the
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
31
essential motivation driving this manuscript is to research the controllability
of mild solutions for an FNIDS with SDD of the model
"
#
Z t
α
Dt x(t) + G t, x%(t,xt ) ,
e1 (t, s, x%(s,xs ) )ds
= A x(t)
0
Z
+
t
B(t − s)x(s)ds + F
Z
t, x%(t,xt ) ,
0
+ Cu(t),
x0 = ς(t) ∈ Bh ,
t
e2 (t, s, x%(s,xs ) )ds
0
t ∈ I = [0, T ],
x0 (0) = 0,
(1.1)
t ∈ (−∞, 0],
(1.2)
where the unknown x(·) needs values in the Banach space X having norm
k · k, Dtα is the Caputo fractional derivative of order α ∈ (1, 2), A , (B(t))t≥0
are closed linear operators described on a regular domain which is dense in
(X, k · k), the control function u(·) ∈ L2 (I , U ), a Banach space of admissible
control functions. Moreover, C is a bounded linear operator from U to X; and
Dtα σ(t) symbolize the Caputo derivative of α > 0 characterized by
Z t
dn
α
Dt σ(t) :=
µ
en−α (t − s) n σ(s)ds,
ds
0
β−1
where n ≥ α and µ
eβ (t) := tΓ(β) , t > 0, β ≥ 0. Further, G , F : I × Bh × X →
X, ei : D × Bh → X, i = 1, 2; D = {(t, s) ∈ I × I : 0 ≤ s ≤ t ≤ T }, % :
I ×Bh → (−∞, T ] are apposite functions, and Bh is a phase space recognized
in Preliminaries.
For almost any continuous function x characterized on (−∞, T ] and any
t ≥ 0, we designate by xt the part of Bh characterized by xt (θ) = x(t + θ)
for θ ≤ 0. Now xt (·) speaks to the historical backdrop of the state from every
θ ∈ (−∞, 0] likely the current time t.
We proceed ahead as comes after. Section 2 is fully committed to analysis
of some vital aspects that will be employed in this work to attain our principal
outcomes. In Section 3, we express and demonstrate the controllability
outcomes by implies of Banach fixed point theorem. In Section 4, as a final
point, a proper case is equipped to reveal the effectiveness of the abstract
techniques.
To the best of our understanding, there is no work gave a record of
the controllability results for FNIDS with SDD, which is conveyed in the
structure (1.1)–(1.2). To pack this crevice, in this manuscript, we mull over
this entrancing model.
2
Preliminaries
In this part, we present some primary components which are required to
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
32
confirm the principal outcomes.
Let L (X) symbolizes the Banach space of all bounded linear operators
from X into X endowed with the uniform operator topology, having its norm
recognized as k · kL (X) .
Let C(I , X) symbolize the space of all continuous functions from I into
X, having its norm recognized as k · kC(I ,X) . Moreover, Br (x, X) symbolizes
the closed ball in X with the middle at x and the distance r.
It needs to be outlined that, once the delay is infinite, then we should talk
about the theoretical phase space Bh in a beneficial way. In this manuscript,
we deliberate phase spaces Bh which are same as described in [30]. So, we
bypass the details.
We expect that the phase space (Bh , k · kBh ) is a semi-normed linear space
of functions mapping (−∞, 0] into X, and fulfilling the subsequent elementary
adages as a result of Hale and Kato ( see case in point in [31, 32]).
If x : (−∞, T ] → X, T > 0, is continuous on I and x0 ∈ Bh , then for
every t ∈ I the accompanying conditions hold:
(P1 ) xt is in Bh ;
(P2 ) kx(t)kX ≤ Hkxt kBh ;
(P3 ) kxt kBh ≤ D1 (t) sup{kx(s)kX : 0 ≤ s ≤ t} + D2 (t)kx0 kBh , where H > 0
is a constant and D1 (·) : [0, +∞) → [0, +∞) is continuous, D2 (·) :
[0, +∞) → [0, +∞) is locally bounded, and D1 , D2 are independent of
x(·).
(P4 ) The function t → ςt is well described and continuous from the set
R(%− ) = {%(s, ς) : (s, ς) ∈ [0, T ] × Bh },
into Bh and there is a continuous and bounded function J ς : R(%− ) →
(0, ∞) to ensure that kςt kBh ≤ J ς (t)kςkBh for every t ∈ R(%− ).
Recognize the space
BT = {x : (−∞, T ] → X : x|I is continuous and x0 ∈ Bh } ,
where x|I is the constraint of x to the real compact interval on I .
The function k · kBT to be a seminorm in BT , it is described by
kxkBT = kςkBh + sup{kx(s)kX : s ∈ [0, T ]}, x ∈ BT .
Lemma 1. [33, Lemma 2.1] Let x : (−∞, T ] → X be a function in a way that
x0 = ς, and if (P4 ) hold, then
kxs kBh ≤ (D2∗ + J ς )kςkBh + D1∗ sup{kx(θ)kX : θ ∈ [0, max{0, s}]},
s ∈ R(%− ) ∪ I ,
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
where J ς =
sup J ς (t),
t∈R(%− )
D1∗ = sup D1 (s),
D2∗ = sup D2 (s).
s∈[0,T ]
s∈[0,T ]
33
To be able to acquire our outcomes, we believe that the subsequent FIDS
Z t
Dtα x(t) = A x(t) +
B(t − s)x(s)ds,
(2.1)
0
x(0) = ς ∈ X,
x0 (0) = 0,
(2.2)
has an associated α-resolvent operator of bounded linear operators (Rα (t))t≥0
on X.
Definition 2.1. [22, Definition 2.1] A one parameter family of bounded linear
operators (Rα (t))t≥0 on X is called a α-resolvent operator of (2.1)-(2.2) if the
subsequent conditions are fulfilled.
(a) The function Rα (·) : [0, ∞) → L (X) is strongly continuous and
Rα (0)x = x for all x ∈ X and α ∈ (1, 2).
T
(b) For x ∈ D(A ), Rα (·)x ∈ C([0, ∞), [D(A )]) C 1 ((0, ∞), X), and for
every t ≥ 0, we receive
Z t
Dtα Rα (t)x = A Rα (t)x +
B(t − s)Rα (s)xds,
(2.3)
0
Z t
Dtα Rα (t)x = Rα (t)A x +
Rα (t − s)B(s)xds.
(2.4)
0
The existence of a α-resolvent operator for the model (2.1)-(2.2) was
analyzed in [20]. To be able to research our model, we need to consider
the conditions (P 1) − (P 3) which are same as stated in [22], consequently
we preclude it.
In view of the conditions (P 1) − (P 3), we determine the operator family
(Rα (t))t≥0 by
Z

 1
eλt Gα (λ)dλ, t > 0,
2πi
(2.5)
Rα (t) =
Γr,θ

I,
t = 0,
where
−1
b
ρα (Gα ) = {λ ∈ C : Gα (λ) := λα−1 (λα I − A − B(λ))
∈ L (X)}.
Hereafter, we expect that the conditions (P 1) − (P 3) are fulfilled. Further,
we need to talk about the mild solution for the model (1.1)-(1.2). For this
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
34
intent, it is necessary to discuss the subsequent non-homogeneous model
Z t
Dtα x(t) = A x(t) +
B(t − s)x(s)ds + F (t), t ∈ I ,
(2.6)
0
x(0) = x0 ,
x0 (0) = 0,
(2.7)
where α ∈ (1, 2) and F ∈ L1 (I , X).
Definition 2.2 ([22, Definition 2.6]). Let α ∈ (1, 2), we describe the family
(Sα (t))t≥0 by
Z t
Sα (t)x :=
µ
eα−1 (t − s)Rα (s)xds,
0
for each t ≥ 0.
The properties and auxiliary results of (2.1)-(2.2) and (2.6)-(2.7) are
broadly examined in [20, 22]. To maintain a strategic distance from the
redundancy, here we discard it.
In the subsequent result, we signify by (−A )ϑ the fractional power of the
operator −A , (see [34] for details).
Lemma 2 ([22, Lemma 3.1]). Suppose that the conditions (P1)–(P3) are
satisfied. Let α ∈ (1, 2) and ϑ ∈ (0, 1) such that αϑ ∈ (0, 1), then there exists
positive number C such that
k(−A )ϑ Rα (t)k ≤ Cert t−αϑ ,
(2.8)
k(−A )ϑ Sα (t)k ≤ Cert tα(1−ϑ)−1 ,
(2.9)
for all t > 0. If x ∈ [D((−A )ϑ )], then
(−A )ϑ Rα (t)x = Rα (t)(−A )ϑ x,
(−A )ϑ Sα (t)x = Sα (t)(−A )ϑ x.
(2.10)
Definition 2.3. Let xT (ς; u) be the state value of the model (1.1)-(1.2) at
terminal time T corresponding to the control u and the initial value ς ∈ Bh .
Present the set R(T, ς) = {xT (ς; u)(0) : u(·) ∈ L2 (I , U )}, which is known as
the reachable set of model (1.1)-(1.2) at terminal time T .
Definition 2.4. The model (1.1)-(1.2) is said to be exactly controllable on I
if R(T, ς) = X.
Assume that the fractional differential control model
Dα x(t) = A x(t) + (Cu)(t),
x0 = ς ∈ Bh ,
t ∈ I,
(2.11)
(2.12)
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
35
is exactly controllable. It is practical at this position to present the
controllability operator linked with (2.11)-(2.12) as
Z T
Sα (T − s)CC∗ S∗α (T − s)ds,
ΓT0 =
0
where C∗ and S∗α (t) denotes the adjoints of C and Sα (t), accordingly. It is
simple that the operator ΓT0 is a linear bounded operator [35, Theorem 3.2].
Lemma 3. If the linear fractional model (2.11)-(2.12) is exactly controllable
if and only then for some γ > 0 such that hΓT0 x, xi ≥ γkxk2 , for all x ∈ X and
as a result k(ΓT0 )−1 k ≤ γ1 .
Remark 2.1. Further, we assume that the linear fractional control system
(2.11)-(2.12) is exactly controllable.
3
Controllability Results
In this segment, we present and demonstrate the exact controllability
results for the structure (1.1)-(1.2) under Banach fixed point theorem.
Initially, we present the mild solution for the model (1.1)–(1.2).
Definition 3.1. [22, Definition 3.4] A function x : (−∞, T ] → X, is called
a mild solution of (1.1)-(1.2) on
[0, T ], if Zx0s = ς; x|[0,T ] ∈ C([0,
T ] : X);
the function s → A Sα (t − s)G s, x%(s,xs ) ,
e1 (s, τ, x%(τ,xτ ) )dτ and s →
0
Z s
Z τ
B(s − τ )Sα (t − s)G τ, x%(τ,xτ ) ,
e1 (τ, ξ, x%(ξ,xξ ) )dξ dτ is integrable on
0
0
[0, t) for all t ∈ (0, T ] and for t ∈ [0, T ],
Z t
x(t) = Rα (t) ς(0) + G (0, ς(0), 0) − G t, x%(t,xt ) ,
e1 (t, s, x%(s,xs ) )ds
0
Z t
Z s
−
A Sα (t − s)G s, x%(s,xs ) ,
e1 (s, τ, x%(τ,xτ ) )dτ ds
0
0
Z tZ s
Z τ
−
B(s − τ )Sα (t − s)G τ, x%(τ,xτ ) ,
e1 (τ, ξ, x%(ξ,xξ ) )dξ dτ ds
0
0
0
Z t
Z s
+
Sα (t − s)F s, x%(s,xs ) ,
e2 (s, τ, x%(τ,xτ ) )dτ ds
0
Z
+
0
t
Sα (t − s)Cu(s)ds.
0
Presently, we itemizing the subsequent suppositions:
(3.1)
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
36
(H1) The operator families Rα (t) and Sα (t) are compact for all t > 0,
and there exists a constant M in a way that kRα (t)kL (X) ≤ M and
kSα (t)kL (X) ≤ M for every t ∈ I and
k(−A )ϑ Sα (t)kX ≤ Mtα(1−ϑ)−1 ,
0 < t ≤ T.
(H2) The subsequent conditions are fulfilled.
(a) B(·)x ∈ C(I , X) for every x ∈ [D((−A )1−ϑ )].
(b) There is a function µ(·) ∈ L1 (I ; R+ ), to ensure that
kB(s)Sα (t)kL ([D((−A )ϑ )],X) ≤ Mµ(s)tαϑ−1 ,
0 ≤ s < t ≤ T.
(H3) The function F : I ×Bh ×X → X is continuous and we can find positive
e F > 0 and L∗ > 0 in ways that for all t ∈ I , x, y ∈ X,
constants LF , L
F
e F kx − ykX ,
kF (t, ψ1 , x) − F (t, ψ2 , y)kX ≤ LF kψ1 − ψ2 kBh + L
and
L∗F = max kF (t, 0, 0)kX .
t∈I
(H4) ei : D ×Bh → X is continuous and we can find constants Lei > 0, L∗ei > 0
to ensure that for all (t, s) ∈ D, (ς, ψ) ∈ Bh2 , i = 1, 2;
ei (t, s, ς) − ei (t, s, ψ) ≤ Lei kς − ψkB ,
h
X
and
L∗ei = max kei (t, s, 0)kX ,
t∈I
i = 1, 2.
(H5) The function G (·) is (−A )ϑ -valued, G : I × Bh × X → [D((−A )−ϑ )]
e G > 0 and L∗ > 0
is continuous and there exist positive constants LG , L
G
such that for all (t, ςj ) ∈ I × Bh , j = 1, 2; x, y ∈ X,
e G kx − ykX ,
k(−A )ϑ G (t, ς1 , x) − (−A )ϑ G (t, ς2 , y)kX ≤ LG kς1 − ς2 kBh + L
k(−A )ϑ G (t, ς, 0)kX ≤ LG kςkBh + L∗G ,
where
L∗G = max k(−A )ϑ G (t, 0, 0)kX .
t∈I
(H6) The following inequalities holds:
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
37
(i) Let
"
1 2 2
1 2 2
M MC T kxT k + 1 + M MC T
MM0 LG kςkBh + MM0 L∗G
γ
γ
!)
(
Z T
MT αϑ
∗
∗
e
1+
µ(τ )dτ
+ LG + LG T Le1
M0 +
αϑ
0
"
∗
∗
∗
e F T Le
eF T L
+ (D r + cn ) MT LF + L
+ MT L + L
F
(
MT αϑ
M0 +
αϑ
+
1
e2
Z
1+
2
!)
T
µ(τ )dτ
e G T Le
LG + L
1
##
≤ r,
0
for some r > 0.
(ii) Let
"
1 2 2
e F T Le
Λ = 1 + M MC T D1∗ MT LF + L
2
γ
!)
(
#
Z T
MT αϑ
e G T Le
1+
µ(τ )dτ
LG + L
+ M0 +
<1
1
αϑ
0
be such that 0 ≤ Λ < 1.
Theorem 3.1. Assume that the conditions (H1)-(H6) hold. Then the control
system (1.1)-(1.2) is exactly controllable on I .
Proof. Utilizing the hypothesis, for an arbitrary function x(·), choose the
feedback control function as follows:
"
ux (t) = C∗ S∗α (T − t)(ΓT0 )−1 xT − Rα (T ) ς(0) + G (0, ς(0), 0)
Z
+G
!
T
T, x%(T,xT ) ,
e1 (T, s, x%(s,xs ) )ds
0
Z
T
A Sα (T − s)G
+
0
Z
Z
s, x%(s,xs ) ,
s
e1 (s, τ, x%(τ,xτ ) )dτ
ds
0
T
Z
s
B(s − τ )Sα (T − s)
Z τ
(×)G τ, x%(τ,xτ ) ,
e1 (τ, ξ, x%(ξ,xξ ) )dξ dτ ds
+
0
0
0
(3.2)
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
Z
−
T
Sα (T − s)F
Z
s
e2 (s, τ, x%(τ,xτ ) )dτ
s, x%(s,xs ) ,
38
#
ds .
0
0
Presently, we determine the operator Υ : BT → BT by


Rα (t) ς(0) + G (0, ς(0), 0)


Z t




−G
t,
x
,
e
(t,
s,
x
)ds

1
%(t,xt )
%(s,xs )


0 
Z s
Z t





e1 (s, τ, x%(τ,xτ ) )dτ ds
A Sα (t − s)G s, x%(s,xs ) ,
−



0
Z0 t Z s



B(s − τ )Sα (t − s)
−
(Υx)(t) =
0 0
Z τ





(×)G τ, x%(τ,xτ ) ,
e1 (τ, ξ, x%(ξ,xξ ) )dξ dτ ds




0
Z t
Z s




+
Sα (t − s)F s, x%(s,xs ) ,
e2 (s, τ, x%(τ,xτ ) )dτ ds



0
0

Z

t




+
Sα (t − s)Cux (s)ds, t ∈ I .
0
Observe that the control (3.2) transfers the system (1.1)-(1.2) from the initial
state ς to the final state xT provided that the operator Υ has a fixed point.
To confirm the exact controllability outcome, it is adequate to demonstrate
that the operator Υ has a fixed point in BT .
We express the function y(·) : (−∞, T ] → X by
(
ς(t), t ≤ 0;
y(t) =
Rα (t)ς(0), t ∈ I ,
then y0 = ς. For every function z ∈ C(I , R) with z(0) = 0, we allocate as z
is characterized by
(
0,
t ≤ 0;
z(t) =
z(t), t ∈ I .
If x(·) fulfills (3.1), we are able to split it as x(t) = y(t) + z(t), t ∈ I , which
suggests xt = yt + zt , for each t ∈ I and also the function z(·) fulfills
z(t) = Rα (t)G (0, ς, 0) − G t, z%(t,zt +yt ) + y%(t,zt +yt ) ,
Z t
e1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds
0
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
Z
t
−
39
A Sα (t − s)G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z 0s
e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
Z tZ s
B(s − τ )Sα (t − s)G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
−
Z 0τ 0
e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ dτ ds
0
Z t
+
Sα (t − s)F s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z 0s
e2 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
Z t
+
Sα (t − s)Cuz+y (s)ds,
0
where
uz+y (t) =

































































"
C∗ S∗α (T
−
t)(ΓT0 )−1
xT − Rα (T )G (0, ς, 0)
+G T, z%(T,zT +yT ) + y%(T,zT +yT ) ,
Z T
e1 (T, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds
0Z
T
+
A Sα (T − s)G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z s0
e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0Z
T Z s
+
B(s − τ )Sα (T − s)G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
Z τ0 0
e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ dτ ds
0Z
T
−
Sα (T − s)F s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
#
Z s0
e2 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds .
0
Let BT0 = {z ∈ BT0 : z0 = 0 ∈ Bh }. Let k · kBT0 be the seminorm in BT0
described by
kzkBT0 = sup kz(t)kX + kz0 kBh = sup kz(t)kX ,
t∈I
t∈I
z ∈ BT0 ,
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
40
as a result (BT0 , k · kBT0 ) is a Banach space. We delimit the operator
Υ : BT0 → BT0 by


R
(t)G
(0,
ς,
0)
−
G
t, z%(t,zt +yt ) + y%(t,zt +yt ) ,

α


Z t





e1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds



0Z

t




−
A Sα (t − s)G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,



Z s0




e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds




0Z Z

s
t

B(s − τ )Sα (t − s)G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
−
(Υz)(t) =

Z τ0 0




e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ dτ ds



0Z


t




+
Sα (t − s)F s, z%(s,zs +ys ) + y%(s,zs +ys ) ,



Z s0




e
(s,
τ,
z
+
y
)dτ
ds

2
%(τ,zτ +yτ )
%(τ,zτ +yτ )


0Z


t



 +
Sα (t − s)Cuz+y (s)ds.
0
Remark 3.1. Let Br = {x ∈ X : kxk ≤ r} for some r > 0. From the above
discussion, we have the subsequent estimates:
(i)
kz%(s,zs +ys ) + y%(s,zs +ys ) kBh
≤ kz%(s,zs +ys ) kBh + ky%(s,zs +ys ) kBh
≤ D1∗ sup kz(τ )kX + (D2∗ + J ς )kz0 kBh + D1∗ |y(s)| + (D2∗ + J ς )ky0 kBh
0≤τ ≤s
≤
D1∗
sup kz(τ )kX + D1∗ kRα (t)kL (X) |ς(0)| + (D2∗ + J ς )kςkBh
0≤τ ≤s
≤ D1∗ sup kz(τ )kX + D1∗ MHkςkBh + (D2∗ + J ς )kςkBh
0≤τ ≤s
≤ D1∗ sup kz(τ )kX + (D1∗ MH + D2∗ + J ς )kςkBh .
0≤τ ≤s
In the event that kzkX < r, r > 0, then
kzρ(s,zs +ys ) + yρ(s,zs +ys ) kBh ≤ D1∗ r + cn ,
where cn = (D1∗ MH + D2∗ + J ς )kςkBh .
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
41
(ii) From suppositions (H1) and (H5), we sustain
kRα (t)kL (X) kG (0, ς, 0)kX ≤ MM0 LG kςkBh + L∗G ,
where k(−A )−ϑ k = M0 .
(iii)
Z t
G t, z%(t,z +y ) + y%(t,z +y ) ,
e1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds t
t
t
t
0
X
"
≤ k(−A )−ϑ k (−A )ϑ G t, z%(t,zt +yt ) + y%(t,zt +yt ) ,
Z
0
t
e1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds − (−A )ϑ G (t, 0, 0)
X
#
+ k(−A )ϑ G (t, 0, 0)kX
h
≤ M0 LG kzρ(t,zt +yt ) + yρ(t,zt +yt ) kBh
Z t
i
eG + L∗G
+L
e
(t,
s,
z
+
y
)ds
1
%(s,zs +ys )
%(s,zs +ys )
0
X
Z t "
eG
≤ M0 LG (D ∗ r + cn ) + M0 L
e1 (t, s, z%(s,z +y ) + y%(s,z
1
s
s
s +ys )
)
0
#
− e1 (t, s, 0) + ke1 (t, s, 0)kX ds + M0 L∗G
X
e G T Le kzρ(t,z +y ) + yρ(t,z +y ) kB
≤ M0 LG (D1∗ r + cn ) + M0 L∗G + M0 L
1
h
t
t
t
t
+ L∗e1
e G T Le (D1∗ r + cn ) + M0 L
e G T L∗e ,
≤ M0 LG (D1∗ r + cn ) + M0 L∗G + M0 L
1
1
and
Z t
e1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds
G t, z%(t,zt +yt ) + y%(t,zt +yt ) ,
0
Z t
− G t, z %(t,zt +yt ) + y%(t,zt +yt ) ,
e1 (t, s, z %(s,zs +ys ) + y%(s,zs +ys ) )ds 0
X
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
42
≤ k(−A ) k(−A )ϑ G t, z%(t,zt +yt ) + y%(t,zt +yt ) ,
Z t
e1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) )ds − (−A )ϑ G t, z %(t,zt +yt ) + y%(t,zt +yt ) ,
0
Z t
e1 (t, s, z %(s,zs +ys ) + y%(s,zs +ys ) )ds 0
X
h
i
e
≤ M0 LG kz%(t,zt +yt ) − z %(t,zt +yt ) kBh + LG T Le1 kz%(t,zt +yt ) − z %(t,zt +yt ) kBh
h
i
e G T Le kz − zkB0 ,
≤ M0 D1∗ LG + L
1
T
−ϑ
since
kz%(s,zs +ys ) − z %(s,zs +ys ) kBh
≤ D1∗ |z(s)| + (D2∗ + J ς )kz0 kBh − D1∗ |z(s)| − (D2∗ + J ς )kz 0 kBh
≤ D1∗ |z(s) − z(s)|X
≤ D1∗ kz − zkBT0 .
(iv)
Z
t
A Sα (t − s)G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
0
Z s
e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
X
"
Z t
≤
k(−A )1−ϑ Sα (t − s)kX k(−A )ϑ G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z
0
s
0
e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ − (−A )ϑ G (s, 0, 0)kX
#
+ k(−A )ϑ G (s, 0, 0)kX ds
Z
≤
"
t
M(t − s)
αϑ−1
LG kz%(s,zs +ys ) + y%(s,zs +ys ) kBh
0
Z
s
h
ke1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ) − e1 (s, τ, 0)kX
#
i
+ ke1 (s, τ, 0)kX dτ + L∗G ds
eG
+L
0
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
"
#
MT αϑ
∗
∗
∗
∗
e
e
≤
LG (D1 r + cn ) + LG T Le1 (D1 r + cn ) + LG T Le1 + LG
αϑ
n
o
αϑ
αϑ
MT αϑ ∗
e G T L∗e + MT L∗G ,
e G T Le + MT L
(D1 r + cn ) LG + L
≤
1
1
αϑ
αϑ
αϑ
and
Z t
A Sα (t − s)G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z0 s
e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
Z t
−
A Sα (t − s)G s, z %(s,zs +ys ) + y%(s,zs +ys ) ,
Z s0
e1 (s, τ, z %(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
X
0
"
Z t
≤
k(−A )1−ϑ Sα (t − s)kX (−A )ϑ G s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z
0
s
e1 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ
0
− (−A )ϑ G s, z %(s,zs +ys ) + y%(s,zs +ys ) ,
#
Z s
e1 (s, τ, z %(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
X
0
Z
≤
t
"
M(t − s)αϑ−1 LG kz%(s,zs +ys ) − z %(s,zs +ys ) kBh
0
#
e G T Le kz%(s,z +y ) − z %(s,z +y ) kB ds
+L
1
h
s
s
s
s
≤
i
MT αϑ ∗ h
e G T Le kz − zkB0 .
D1 LG + L
1
T
αϑ
(v)
Z Z
t s
B(s − τ )Sα (t − s)G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
0 0
Z τ
e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ dτ ds
0
X
43
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
Z tZ
"
s
αϑ−1
µ(s − τ )M(t − s)
≤
Z
0
τ
0
44
k(−A )ϑ G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
0
e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ − k(−A )ϑ G (τ, 0, 0)kX
#
+ k(−A )ϑ G (τ, 0, 0)kX dτ ds
!"
Z
MT αϑ T
≤
µ(τ )dτ
LG kz%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) kBh
αϑ
0
Z th
e
+ LG
ke1 (t, s, z%(s,zs +ys ) + y%(s,zs +ys ) ) − e1 (t, s, 0)kX
0
#
i
∗
+ ke1 (t, s, 0)kX ds + LG
MT αϑ
αϑ
≤
Z
!"
T
µ(τ )dτ
0
(D1∗ r
#
∗
∗
e
e
+ cn ){LG + LG T Le1 } + LG T Le1 + LG ,
and
Z Z
t s
B(s − τ )Sα (t − s)G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
0 0
Z τ
e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ dτ ds
0
Z tZ s
−
B(s − τ )Sα (t − s)G τ, z %(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
0
0
Z τ
e1 (τ, ξ, z %(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ dτ ds
0
X
"
Z tZ s
≤
µ(s − τ )M(t − s)αϑ−1 k(−A )ϑ G τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
Z
0
τ
0
e1 (τ, ξ, z%(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ
0
− (−A )ϑ G τ, z %(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ,
#
Z τ
e1 (τ, ξ, z %(ξ,zξ +yξ ) + y%(ξ,zξ +yξ ) )dξ kX dτ ds
0
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
MT αϑ
αϑ
≤
Z
!"
T
LG kz%(t,zt +yt ) − z %(t,zt +yt ) kBh
µ(τ )dτ
0
#
e G T Le kz%(t,z +y ) − z %(t,z +y ) kB
+L
1
h
t
t
t
t
MT αϑ
αϑ
≤
Z
!
T
µ(τ )dτ
0
h
i
e G T Le kz − zkB0 .
D1∗ LG + L
1
T
(vi)
Z
t
Sα (t − s)F s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
0
Z s
e2 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
X
"
Z t
≤
kSα (t − s)kL (X) kF s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
Z
0
s
0
e2 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ
#
− F (s, 0, 0)kX + kF (s, 0, 0)kX ds
Z t"
≤M
LF kz%(s,zs +ys ) + y%(s,zs +ys ) kBh
0
Z
s
h
ke2 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) ) − e2 (s, τ, 0)kX
#
i
+ ke2 (s, τ, 0)kX dτ + L∗F ds
eF
+L
0
"
≤ MT
(D1∗ r
#
n
o
∗
e
e
+ cn ) LF + LF T Le2 + LF T Le2 + LF ,
and
Z
t
Sα (t − s)F s, z%(s,zs +ys ) + y%(s,zs +ys ) ,
0
Z s
e2 (s, τ, z%(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
45
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
Z
46
t
Sα (t − s)F s, z %(s,zs +ys ) + y%(s,zs +ys ) ,
0
Z s
e2 (s, τ, z %(τ,zτ +yτ ) + y%(τ,zτ +yτ ) )dτ ds
0
X
"
Z
−
t
≤
0
kSα (t − s)kL (X) LF kz%(s,zs +ys ) − z %(s,zs +ys ) kBh
#
e F T Le kz%(s,z +y ) − z %(s,z +y ) kB ds
+L
2
h
s
s
s
s
n
o
e F T Le kz − zkB0 .
≤ MT D1∗ LF + L
2
T
(vii)
kCuz+y (s)k ≤
1
MM2C
γ
"
kxT k + MM0 LG kςkBh + MM0 L∗G
e G T L∗e
+ L∗G + L
1
+ MT
L∗F
(
MT αϑ
M0 +
αϑ
1+
!)
T
µ(τ )dτ
0
"
∗
∗
e
e F T Le
+ LF T Le2 + (D1 r + cn ) MT LF + L
2
(
+
Z
MT αϑ
M0 +
αϑ
Z
1+
!)
T
µ(τ )dτ
e G T Le
LG + L
1
##
0
and
kCuz+y (s) − Cuz+y (s)k
"
1
2
e F T Le
≤
MMC D1∗ MT LF + L
2
γ
!)
#
(
Z T
MT αϑ
e G T Le kz − zkB0 .
+ M0 +
1+
µ(τ )dτ
LG + L
1
T
αϑ
0
Therefore, we have
Z t
f f
Sα (t − s)Cuz+y (s)ds
≤ C1 + C2 ,
0
X
,
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
47
where
f1 =
C
f2 =
C
1 2 2
M MC T
γ
1 2 2
M MC T
γ
MT αϑ
+
αϑ
kxT k,
"
MM0 LG kςkBh +
Z
1+
!)
T
µ(τ )dτ
0
MM0 L∗G
+
L∗G
+
e G T L∗e
L
1
(
M0
e F T L∗e
+ MT L∗F + L
2
"
e F T Le
+ (D1∗ r + cn ) MT LF + L
2
(
+
MT αϑ
M0 +
αϑ
Z
1+
!)
T
µ(τ )dτ
e G T Le
LG + L
1
##
,
0
and
Z t
i h
Sα (t − s) Cuz+y (s) − Cuz+y (s) ds
0
X
"
1 2 2
e F T Le
≤
M MC T D1∗ MT LF + L
2
γ
!)
#
(
Z T
MT αϑ
e G T Le kz − zkB0 .
1+
µ(τ )dτ
LG + L
+ M0 +
1
T
αϑ
0
Now, we enter the main proof of this theorem. Initially, we demonstrate
that Υ maps Br (0, BT0 ) into Br (0, BT0 ). For any z(·) ∈ BT0 , by employing
Remark 3.1, we sustain
f1 + C
f3
k(Υz)(t)kX ≤ C
≤ r,
where
(
"
1
2
2
∗
∗
∗
eG T L
f3 = 1 + M M T
MM0 LG kςkBh + MM0 LG + LG + L
M0
C
C
e1
γ
!)
Z T
MT αϑ
e F T L∗e
+
1+
µ(τ )dτ
+ MT L∗F + L
2
αϑ
0
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
48
"
+
(D1∗ r
e F T Le
+ cn ) MT LF + L
2
(
+
MT αϑ
M0 +
αϑ
Z
1+
!)
T
µ(τ )dτ
e G T Le
LG + L
1
##
.
0
Therefore, Υ maps the ball Br (0, BT0 ) into itself. Finally, we show that Υ is
a contraction on Br (0, BT0 ). For this, let us consider z, z ∈ Br (0, BT0 ), then
from Remark 3.1, we sustain
k(Υz)(t) − (Υz)(t)kX
"
1 2 2
e F T Le
≤ 1 + M MC T D1∗ MT LF + L
2
γ
!)
#
(
Z T
MT αϑ
e G T Le
1+
µ(τ )dτ
LG + L
+ M0 +
1
αϑ
0
≤ Λkz − zkBT0 .
From the assumption (H6) and in the perspective of the contraction
mapping principle, we understand that Υ includes a unique fixed point z ∈ BT0 .
Thus, the model (1.1)-(1.2) is exactly controllable on I . The proof is now
completed.
4
Applications
To exemplify our theoretical outcomes, we treat the FNIDS with SDD of
the model
"
Z t
u(s − %1 (s)%2 (ku(s)k), x)
α
ds
Dt u(t, x) +
e2(s−t)
49
−∞
#
Z t
Z s
∂2
2(τ −s) u(τ − %1 (τ )%2 (ku(τ )k), x)
+
sin(t − s)
e
dτ ds =
u(t, x)
36
∂x2
0
−∞
Z t
∂2
+
(t − s)δ e−γ(t−s) 2 u(s, x)ds + ν(t, x)
∂x
0
Z t
u(s
−
%
1 (s)%2 (ku(s)k), x)
+
e2(s−t)
ds
9
−∞
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
t
Z
Z
s
u(τ − %1 (τ )%2 (ku(τ )k), x)
dτ ds,
25
e2(τ −s)
sin(t − s)
+
−∞
0
u(t, x) = ς(t, x),
(4.1)
t ∈ [0, T ],
u(t, 0) = 0 = u(t, π),
t ≤ 0,
49
(4.2)
x ∈ [0, π],
(4.3)
Dtα
where
is Caputo’s fractional derivative of order α ∈ (1, 2), δ and γ are
positive numbers and ς ∈ Bh . We consider X = L2 [0, π] having the norm
| · |L2 and determine the operator A : D(A ) ⊂ X → X by A w = w00 with the
domain
D(A ) = {w ∈ X : w, w0 are absolutely continuous, w00 ∈ X, w(0) = w(π) = 0}.
Then
Aw =
∞
X
n2 hw, wn iwn ,
w ∈ D(A ),
n=1
q
2
in which wn (s) =
π sin(ns), n = 1, 2, . . . , . is the orthogonal set of
eigenvectors of A . It is long familiar that A is the infinitesimal generator
of an analytic semigroup (T (t))t≥0 in X and is provided by
T (t)w =
∞
X
2
e−n t hw, wn iwn ,
for all
w ∈ X,
and every t > 0.
n=1
1
Hence (H1) is fulfilled. If we fix ϑ = 12 , then the operator (−A ) 2 is given by
1
(−A ) 2 w =
∞
X
1
nhw, wn iwn , w ∈ (D(−A ) 2 ),
n=1
(
1
2
in which (D(−A ) ) =
ω(·) ∈ X :
∞
X
)
nhω, wn iwn ∈ X
n=1
1
and k(−A )− 2 k = 1.
Therefore, A is sectorial of type and the properties (P1) hold. We also take
into account the operator B(t) : D(A ) ⊆ X → X, t ≥ 0, B(t)x = tδ e−γt A x
for x ∈ D(A ). In addition, it is simple to see that conditions (P2)-(P3)[for
more details, refer[22]] are fulfilled with b(t) = tδ e−γt and D = C0∞ ([0, π]),
where C0∞ ([0, π]) is the space of infinitely differentiable functions that vanish
at x = 0 and x = π. From the Lemma 2.4, it is simple to see that condition
(H2) is fulfills.
Z 0
1
For the phase space, we choose h = e2s , s < 0, then l =
h(s)ds = <
2
−∞
∞, for t ≤ 0 and determine
Z 0
kςkBh =
h(s) sup kς(θ)kL2 ds.
−∞
θ∈[s,0]
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
50
Hence, for (t, ς) ∈ [0, 1] × Bh , where ς(θ)(x) = ς(θ, x), (θ, x) ∈ (−∞, 0] ×
[0, π]. Set
u(t)(x) = u(t, x),
%(t, ς) = %1 (t)%2 (kς(0)k),
we have
G (t, ς, H ς)(x) =
Z
F (t, ς, H ς)(x) =
Z
0
e2(s)
−∞
0
ς
ds + (H ς)(x),
49
ς
e2(s) ds + (H ς)(x),
9
−∞
where
Z
t
Z
0
sin(t − s)
(H ς)(x) =
t
Z
sin(t − s)
(H ς)(x) =
0
ς
dτ ds,
36
e2(τ )
ς
dτ ds.
25
−∞
0
0
Z
e2(τ )
−∞
Further, we define the operator C : U → X by Cu(t, x) = ν(t, x), 0 < x < π, u ∈
U, where ν : [0, 1] × [0, π] → [0, π], then using these configurations, the system
(4.1)-(4.3) is usually written in the theoretical form of design (1.1)-(1.2).
To treat this system we assume that %i : [0, ∞) → [0, ∞), i = 1, 2 are
continuous. Now, we can see that for t ∈ [0, 1], ς, ς ∈ Bh , we have
1
k(−A ) 2 G (t, ς, H ς)kX
2 ! 12
Z π Z 0
Z t
Z 0
2(s) ς 2(τ ) ς ≤
e
k sin(t − s)k
e
ds +
dτ ds dx
49
36
0
−∞
0
−∞
2 ! 21
Z π Z 0
Z 0
1
1
≤
e2(s) sup kςkds +
e2(s) sup kςkds dx
49 −∞
36 −∞
0
√
√
π
π
kςkBh +
kςkBh
≤
49
36
e G kςkB ,
≤ LG kςkBh + L
h
eG =
where LG + L
√
85 π
1764 ,
1
and
1
k(−A ) 2 G (t, ς, H ς) − (−A ) 2 G (t, ς, H ς)kX
Z π Z 0
ς
ς ≤
e2(s) − ds
49
49
0
−∞
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
t
Z
0
Z
2(τ )
k sin(t − s)k
+
π
Z
≤
√
e
−∞
0
0
1
49
Z
0
e
2(s)
−∞
51
! 12
2
ς
− ς dτ ds dx
36 36 1
sup kς − ςkds +
36
√
π
π
kς − ςkBh +
kς − ςkBh
49
36
e G kς − ςkB .
≤ LG kς − ςkBh + L
h
Z
0
e
2(s)
2 ! 12
sup kς − ςkds dx
−∞
≤
Similarly, we conclude
kF (t, ς, H ς)kL2
2 ! 21
Z π Z 0
Z t
Z 0
ς ς ≤
e2(s) ds +
k sin(t − s)k
e2(τ ) dτ ds dx
9
25
0
−∞
0
−∞
2 ! 21
Z π Z 0
Z 0
1
1
≤
e2(s) sup kςkds +
e2(s) sup kςkds dx
9 −∞
25 −∞
0
√
√
π
π
≤
kςkBh +
kςkBh
9
25
e F kςkB ,
≤ LF kςkBh + L
h
eF =
where LF + L
√
34 π
225 ,
and
kF (t, ς, H ς) − F (t, ς, H ς)kL2
Z π Z 0
ς
ς
≤
e2(s) − ds
9 9
0
−∞
t
Z
Z
0
k sin(t − s)k
+
0
e
−∞
2(τ )
! 21
2
ς
ς
− dτ ds dx
25 25 Z 0
2 ! 21
Z 0
1
1
≤
e2(s) sup kς − ςkds +
e2(s) sup kς − ςkds dx
9 −∞
25 −∞
0
√
√
π
π
kς − ςkBh +
kς − ςkBh
≤
9
25
e F kς − ςkB .
≤ LF kς − ςkBh + L
h
Z
π
Therefore the conditions (H3) and (H5) are all fulfilled. Furthermore, we
assume that D1∗ = 12 , M0 = 1, M = 1, MC = 1, γ = 1, T = 1, α = 23 , Le1 =
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
Z
52
1
µ(τ )dτ = 1. Then
1, Le2 = 1 and
0
"
1 2 2
e F T Le
1 + M MC T D1∗ MT LF + L
2
γ
(
!)
#
Z T
MT αϑ
e G T Le
µ(τ )dτ
1+
LG + L
≈ 0.58098 < 1.
+ M0 +
1
αϑ
0
Thus the condition (H6) holds. Hence by Theorem 3.1, we realize that the
system (4.1)–(4.3) has a unique mild solution on [0, 1].
References
[1] D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional Dynamics and
Control, Springer, New York, USA, 2012.
[2] A. Kilbas, H. Srivastava and J. J. Trujillo, Theory and Applications of
Fractional Differential Equations, Elsevier, Amesterdam, 2006.
[3] G. Bonanno, R. Rodriguez-Lopez, S. Tersian, Existence of solutions to
boundary value problem for impulsive fractional differential equations,
Fractional Calculus and Applied Analysis, 17(3)(2014), 717-744.
[4] R. Rodriguez-Lopez and S. Tersian, Multiple solutions to boundary
value problem for impulsive fractional differential equations, Fractional
Calculus and Applied Analysis, 17(4)(2014), 1016-1038.
[5] R. P. Agarwal, V. Lupulescu, D. O’Regan and G. Rahman, Fractional
calculus and fractional differential equations in nonreflexive Banach
spaces, Communications in Nonlinear Science and Numerical Simulation,
20(1)(2015), 59-73.
[6] E. Keshavarz, Y. Ordokhani and M. Razzaghi, Bernoulli wavelet
operational matrix of fractional order integration and its applications in
solving the fractional order differential equations, Applied Mathematical
Modelling, 38(2014), 6038-6051.
[7] Z. Lv and B. Chen, Existence and uniqueness of positive solutions for a
fractional switched system, Abstract and Applied Analysis, Volume 2014
(2014), Article ID 828721, 7 pages.
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
53
[8] Y. Wang, L. Liu and Y. Wu, Positive solutions for a class of higher-order
singular semipositone fractional differential systems with coupled integral
boundary conditions and parameters, Advances in Difference Equations,
2014:268, 1-24.
[9] B. Ahmad, S. K. Ntouyas and A. Alsaed, Existence of solutions
for fractional q-integro-difference inclusions with fractional q-integral
boundary conditions, Advances in Difference Equations, 2014:257, 1–18.
[10] R. P. Agarwal, B. D. Andrade, On fractional integro-differential equations
with state-dependent delay, Comp. Math. App., 62(2011), 1143–1149.
[11] M. Benchohra, F. Berhoun, Impulsive fractional differential equations
with state-dependent delay, Commun. Appl. Anal., 14(2)(2010), 213–224.
[12] K. Aissani and M. Benchohra, Fractional integro-differential equations
with state-dependent delay, Advances in Dynamical Systems and
Applications, 9(1)(2014), 17-30.
[13] J. Dabas and G.R. Gautam, Impulsive neutral fractional integrodifferential equation with state-dependent delay and integral boundary
condition, Electronic Journal of Differential Equations, Vol. 2013 (2013),
No. 273, 1–13.
[14] S. Suganya, M. Mallika Arjunan, and J. J. Trujillo, Existence results for
an impulsive fractional integro-differential equation with state-dependent
delay, Applied Mathematics and Computation, 266(2015), 54-69.
[15] J. P. C. Dos Santos, C. Cuevas, B. de Andrade, Existence results for a
fractional equations with state-dependent delay, Advances in Difference
Equations, (2011), 1-15.
[16] J. P. C. Dos Santos, M. Mallika Arjunan, and C. Cuevas, Existence results
for fractional neutral integrodifferential equations with state-dependent
delay, Comput. Math. Appl., 62(2011), 1275–1283.
[17] Z. Yan, Approximate controllability of fractional neutral integrodifferential inclusions with state-dependent delay in Hilbert spaces, IMA
Journal of Mathematical Control and Information, 30(2013), 443–462.
[18] Z. Yan and X. Jia, Approximate controllability of partial fractional
neutral stochastic functional integro-differential inclusions with statedependent delay, Collect. Math., 66(2015), 93–124.
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
54
[19] V. Vijayakumar, C. Ravichandran and R. Murugesu, Approximate
controllability for a class of fractional neutral integro-differential
inclusions with state-dependent delay, Nonlinear Studies, 20(4)(2013),
513-532.
[20] R. P. Agarwal, J. P. C. Dos Santos, and C. Cuevas, Analytic resolvent
operator and existence results for fractional integro-differential equations,
J. Abstr. Differ. Equ. Appl., 2(2)(2012), 26-47.
[21] V. Vijayakumar, A. Selvakumar and R. Murugesu, Controllability for a
class of fractional neutral integro-differential equations with unbounded
delay, Applied Mathematics and Computation, 232(2014), 303-312.
[22] B. D. Andrade, J. P. C. Dos Santos, Existence of solutions for a fractional
neutral integro-differential equation with unbounded delay, Electronic
Journal of Differential Equations, Vol. 2012 (2012), No. 90, pp. 1–13.
[23] J. P. C. Dos Santos, V. Vijayakumar and R. Murugesu, Existence
of mild solutions for nonlocal Cauchy problem for fractional neutral
integro-differential equation with unbounded delay, Communications in
Mathematical Analysis, X(2011), 1–13.
[24] X. Shu and Q. Wang, The existence and uniqueness of mild solutions
for fractional differential equations with nonlocal conditions of order
1 < α < 2, Comput. Math. Appl., 64(2012), 2100–2110.
[25] Z. Yan and X. Jia, Impulsive problems for fractional partial neutral
functional integro-differential inclusions with infinite delay and analytic
resolvent operators, Mediterr. Math., 11(2014), 393–428.
[26] Z. Yan and F. Lu, On approximate controllability of fractional stochastic
neutral integro-differential inclusions with infinite delay, Applicable
Analysis, (2014), 1235-1258.
[27] L. Kexue, P. Jigen, G. Jinghuai, Controllability of nonlocal fractional
differential systems of order α ∈ (1, 2] in Banach spaces, Rep. Math.
Phys. , 71(2013), 33–43.
[28] R. Sakthivel, R. Ganesh, Y. Ren, S. Marshal Anthoni, Approximate
controllability of nonlinear fractional dynamical systems, Commun.
Nonlinear Sci. Numer. Simulat., 18(2013), 3498–3508.
[29] C. Rajivganthi, P. Muthukumar. and B. Ganesh Priya, Approximate
controllability of fractional stochastic integro-differential equations with
infinite delay of order 1 < α < 2, IMA Journal of Mathematical Control
and Information, (2015), 1-15, Available On line.
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
55
[30] J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for
an impulsive fractional integro-differential equation with infinite delay,
Mathematical and Computer Modelling, 57(3-4)(2013), 754-763.
[31] J. Hale and J. Kato, Phase space for retarded equations with infinite
delay, Funkcial. Ekvac., 21(1978), 11-41.
[32] Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations
with Unbounded Delay, Springer-Verlag, Berlin, 1991.
[33] X. Fu and R. Huang, Existence of solutions for neutral integro-differential
equations with state-dependent delay, Appl. Math. Comp., 224(2013),
743-759.
[34] A. Pazy, Semigroups of Linear Operators and Applications to Partial
Differential Equations, Springer-Verlag, New York, 1983.
[35] N. I. Mahmudov and A. Denker, On controllability of linear stochastic
systems, Int. J. Control, 73(2000), 144-151.
Subramanian Kailasavalli,
Department of Mathematics,
PSNA College of Engineering and Technology,
Dindigul-624622, Tamil Nadu, India.
Email: skvalli2k5@gmail.com
Dumitru Baleanu,
Department of Mathematics and Computer Sciences,
Faculty of Art and Sciences,
Cankaya University, 06530 Ankara,
Turkey and Institute of Space Sciences,
Magurele-Bucharest, Romania.
Email: dumitru@cankaya.edu.tr
Selvaraj Suganya,
Department of Mathematics,
C. B. M. College, Kovaipudur,
Coimbatore - 641 042, Tamil Nadu, India.
Email: selvarajsuganya2014@yahoo.in
Mani Mallika Arjunan,
Department of Mathematics,
C. B. M. College, Kovaipudur,
Coimbatore - 641 042, Tamil Nadu, India.
Email: arjunphd07@yahoo.co.in
Exact controllability of fractional neutral integro-differential systems
with state-dependent delay in Banach spaces
56