Applied Mathematics, 2011, 2, 1486-1496
doi:10.4236/am.2011.212211 Published Online December 2011 (http://www.SciRP.org/journal/am)
Controllability of Neutral Impulsive Differential Inclusions
with Non-Local Conditions
Dimplekumar N. Chalishajar1, Falguni S. Acharya2*
1Department of Mathematics and Computer Science, Virginia Military Institute (VMI),
Lexington, USA
2Department of Applied Sciences Humanities, Institute of Technology and Management Universe,
Vadodara, India
Received July 14, 2011; revised September 27, 2011; accepted October 7, 2011
Abstract
In this short article, we have studied the controllability result for neutral impulsive differential inclusions
with nonlocal conditions by using the fixed point theorem for condensing multi-valued map due to Martelli
[1]. The system considered here follows the P.D.E involving spatial partial derivatives with α-norms.
Keywords: Controllability, Neutral Impulsive Differential Inclusions, Spatial Partial Derivative, Martelli
Fixed Point Theorem
1. Introduction
In this paper we have discussed the controllability of
nonlocal Cauchy problem for neutral impulsive differen-
tial inclusions of the form
   


12
=0
d() ,()()() ,();
d
:=0,;
|=(); =1,2,; (0)()
k
tt k k
k
x
tFtxht AxtButGtxht
t
tJ btt
x
Ixtkmxgxx X












(1)
where the linear operator ()
A

generates an analytic
semigroup ; is a multi-valued map and

0
()t Tt G
==() () ttk k
k
x
xt xt

, 0 ()=( )
limh
kk
x
txt



k
h and

0
()=
limh
k
x
t


()
xth represent the right and left
limits of
x
t=t at k respectively, 0 t,
x
X
)\ :(FJ XPX

 is a multi-valued map [ is
the family of all subsets of X] and
()PX
)(,
g
CJ
2(, )uLJU
U
X. Also
the control function , a Banach space of
admissible control functions with U as a Banach space. B
is a bounded linear operator from to
X
and
X
is
a separable Banach space with norm .
:()
k
I
XDA; and . =1,2, ,km12
As a model we consider the following system of heat
equations;
,hh C(,JJ
2
2
0
(,)(,(cos, )),(cos, )
(, )
=(,),(cos,),(cos
(,0)=(,π)=0;
()()= (()),
;=1, 2,,
(0,)((,))=(), 0π,[0,1]
kkkk
k
z
ztx Ftztxtx
tx
ztx z
utxG tztxtx
x x
zt zt
ztztI zt
ttkm
zxgztx zxxt
 






















,),
(2)
Since
F
and involve spatial partial derivative,
the results obtained by other authors cannot be applied to
our system even if . This is the main motivation
of this paper.
G
g(.) = 0
The existence and controllability of the following sy-
stem is studied by Benchohra and Ntouyas [2]
=
d()(, )()()(,);
d
:= [0,];
|=(()); =1,2,,;()=(), (,0).
tt
k
ttk k
k
xtgt xAxtButFt x
t
tJ btt
xIxtk mxttt 











(3)
Here authors have proved exact controllability by
using fixed point theorem for condensing multi-valued
maps due to Martelli. In this paper, we have discussed
controllability results with α-norms as in [3] with de-
)

D. N. CHALISHAJAR ET AL.1487
viating arguments in terms involving spatial partial de-
rivatives.
As indicated in [4], and reference therein, the nonlocal
Cauchy problem 0
(0)() =
x
gx x can be applied in di-
fferent fields with better effect than the classical initial
condition 0
(0)
x
x. For example in [5], the author
described the diffusion phenomenon of a small amount
of gas in a transparent tube by using the formula
=0
()= (),
p
ii
i
g
xcxt

where are given constants and
01 In this case the above equation
allows the additional measurement at i, .
In the past several years theorems about controllability of
differential, integro-differential, fractional differential
systems and inclusions with nonlocal conditions have
been studied by Chalishajar and Acharya [6-9], Ben-
chohra and Ntouyas [10,11], and Hernandez, Rabello and
Henriquez [12] and the references therein. In [13],
Chalishajar discussed exact controllability of third order
nonlinear integro-differential dispersion system without
compactness of semigroup.
,= 0,1,,
i
ci p
, ,,.
p
tttb0<
t=0,1, ,ip
Xianlong Fu and Yueju Cao [14], has discussed the
existence of mild solution for neutral partial differential
inclusions involving spatial partial derivative with -

norms in Banach space. However in their work authors
impose some severe assumptions on the operator family
generator by ()
A
, i.e. ():()
A
DA XX

()Tt
is an
infinitesimal generator of a compact analytic semigroup
of a uniformly bounded linear operator 0t, which
imply that underlying space
X
has finite dimension
and so the example considered in [14], and subsequently
in Section 4 is ordinary differential equation but not
partial differential equation which shows lack of exi-
stence (exact controllability) in abstract (control) system
(refer [15]). This fact and several other applications of
neutral equation (inclusions) are the main motivation of
this paper.
In Section 3 (followed by Preliminaries) of present
paper we discuss the controllability of neutral impulsive
differential inclusion with nonlocal condition with devi-
ating arguments with α-norm, which is the genera-
lization of [14], in a finite dimensional space. The ex-
ample is given in Section 4 to support the theory. In
Section 5 we study exact controllability of same system
in infinite dimension space by dropping the compactness
assumption of semigroup

0
()t Tt e generalized
the result proved in Section 3.
. Here w
2. Preliminaries
In this section, we shall introduce some basic definitions,
notations and lemmas which are used throughout this
paper.
Let


,X

be a Banach space. is the
Banach space of continuous functions from
(, )CJX
J
into
X
with the norm defined by


:= sup():.
J
x
xtt J
Let be the Banach space of bounded linear
operators from
()BX
X
into
X
with standard norm


()
:=sup():= 1.
BX
NNxx
A measurable function :
x
JX is Bochner inte-
grable if and only if
x
is Lebesgue integrable. (For
properties of the Bochner integral see [16]). Let
denotes the Banach space of Bochner inte-
grable functions
1(, )LJX
:
x
JX with norm
10
=()d
b
L
x
xt t
 for all 1(,)
x
LJX
X
.
We use the notations

()=2:PX YY

,


closed()=():
cl
PXYPXY,


ounded()=():
b
PXYPXYb,


onvex()=():
c
PXYPXYc, and


Ycompact
:2
X
GX
()=():
cp
PXY PX.
A multi-valued map is convex (respec-
tively closed) valued if is convex (respectively
closed) for all
()Gx
x
X

.
The map is bounded on bounded sets if G
()= ()
xB
GB Gx

 is bounded in
X
for any bounded
set of B
X
.





..sup:( )< supxB iexxGx
.
G is called upper semi-continuous (u.s.c.) on
X
if for
each 0
x
X

, the set 0 is a nonempty closed
subset of
()Gx
X
and if for each open set of B
X
containing 0, there exists an open neighborhood ()Gx
A
of 0
x
such that ()GA B.
The map is said to be completely continuous if
is relatively compact for every bounded subset
G
()GB
BX.
If the multi-valued map is completely continuous
with nonempty compact values, then is u.s.c. if and
only if has a closed graph, That is, if
00
,
G
G
G
,
nn
x
xyy where nn then 00 (yGx)()yGx

.
has a fixed point if there is
G
x
X such that
()
x
Gx

.
A multi-valued map is said to be
measurable, if for each
:BC()GJ CX
x
X, the distance function
defined by
:YJ R

 
()=,() =inf:()Yt dxGtxzzGt
is measurable.
An upper semi-continuous map is said
to be condensing, if for any bounded subset ,
with
:2
X
GX
BX
() 0B


, we have , where

< ()B  ()GB 

denotes the Kuratowski measure of non-compactness.

D. N. CHALISHAJAR ET AL. 1488
We remark that a completely continuous multi-valued
map is the easiest example of a condensing map. For
more details on multivalued maps see the books of
Deimling [17].
Throughout this paper, :()
A
DA XX will be
the infinitesimal generator of a compact analytic semi-
group of uniformly bounded linear operator Let
().Tt
0()
A



, then it is possible to define the fractional
power , as a closed linear operator on
its domain
, 01Afor  
()DA

. Furthermore, the subspace ()DA

is dense in X and the expression
=; ()
x
AxxDA




defines a norm on ()DA

. Hereafter we denote by
,
X

the Banach space ()DA

normed with
x

.
Then for each 0< 1

,
X

is a Banach space, and

XXf0<or< 1

  and the imbedding is
compact whenever the resolvent operator of A is
compact.
Semigroup satisfies the following properties:

()t Tt 
10 a) there is a
M
 such that
() forall 0Tt Mt

;
b) for any 0<1,

 there exists a positive constant
C

such that
(); 0<
C
ATt t
t




For more details about the above preliminaries, we re-
fer to ([18,19]).
In order to define the solution of the system (1) we
shall consider the space


0
=:[0,];(,);=0,1,,
andthereexist ()and (); =0,1,,
with ()=( ),(0)()=,
kk
kk
kk
x
bXxCJXk m
xtxt km
xtxtxg xx







which is a Banach space with the norm

=;=0,1,
kJk
,
x
max xkm

where k
x
is the restriction of
x
to

1,=0,1,, k
=,
kk
J
tt km
  and
=()upxs.
ksJk
Jk
k
xs


For the system (1) we assume that the following hypo-
theses are satisfied for some (0,1) :


(H1) Let 2 :(,)WLJU X

 be the linear operator
defined by
0
=( )()d
b
WuT bs Bu ss

The 2 :(,)WLJU kerWX

 induces a bounded
invertible operator and there exists positive con-
1
W

1
2
WM

. 
(H2) i) there exists a constant (0,1)

 such that
:[0, ]
F
bX X



 is a continution, and ous func
:[0, ]
A
FbX

X


satisfies the Lipschitz condition,
stant >0L such that

that is, there exists a con

112 21212
(,)( ,)AFt xAFtxtxx

 ,L t



for any .
121 2
0, ;,ttbxxX



reover, there exists a constant ii) Mo1>0L such that
the inequality

1
(, )1,AFtxL x




x
X


holds for any .
lti-valued map )
sa
(H3) The mu, :(
ccp
GJ XPX


tisfies the following conditions:
i) for each ,tJ

the function ,
(,.):( )
ccp
GtXPX


is u.s.c. and for each ,
x
X


the function
,
(.,) :()
ccp
GxJP Xurable. Also fo is measr each fixed
y

 the set


,2
,):()(,(())) ..
Gx
SvJXvtGtxhtforaet
1
= (LJ
is nonempty.
positive number , there exists a
po
ii) for each lN
on lsitive function ()wl dependent such that
(, )()Gtx wl sup
xl
and ()
liminf= <
l
wl
l



where


(,) =sup:(,)Gtxv vGtx, 0 =( sup s)
x
xs



 .
(H4) (,),=1,2.
i
hCJJig X:



and satisfies that
is continuous
positive constants and i) there exists 2 L2 L

such
that
22
() forall .gyL yLy
 

 
ii)
A
g

is completely continuous map.
(H5) (,),=1,2,,,
stant 1
M
and 2
M
such that and 1 BM and
k
I
CX Xkm


 are all bounded,
that is, there exist constants ,=1,2
k
dk , ,,m such that
() ,
kk
I
xd f
x
X

.

or each
Now we define the mild solusystem (1).
-
lo
tion for the
DEFINITION 2.1 The system (1) is said to be non
cally controllable on the interval
J
if for every
(0)( )()
x
gx DA

 and 01 ,
x
zX, there exists a con-




0, :=,uL bLLJU

0, 2π such that the
22 2
trol
of (1) satcorresponding solution ()
x

isfies
i) 1
() ()=
x
bgx z

w0 (0)( );ith
x
gx x
1, 2,,; ii) ));
=ttk k
k
|( (
x
Ixt k
m 
there exists a function 1
vL
iii) J(, )X such that

D. N. CHALISHAJAR ET AL.1489
1
2
( )(,(( )))tGt xht a.e. on J and
( )(0,((0)))
v
0
0< <
( )=( )
(
()
t
tt
k
01
1
00
,
( ,(( )))
)( ,(( )))d()( )d
()()d
()(()); ,.
tt
kk kGx
x
tTtxgFtxhtxFxh
A
Tt
Tt s
Tt






The following lem
ma
[2
sFsxhssTtsvss
Bu ss
t I xttJvS

 





(2.1)
mas are crucial in the proof o
0] Let
f our
in theorem.
LEMMA 2.2
X
be a Banach space. Let
,,
:()
bclc
J XPX satisfies that G
i) For each
x
X
wi t a
,

,,tx Gt

is measurable
x
each tJ, th respect to for (, )(,)tx Gtx
is u.s.c. with respect to x .
ii) For each fixed
nd

,
x
CJX

, the set
 


1
,2
=, (()),fo
xvLJX ht:,r.. vtGtx aet G
S
is nonempty.
a
1) L

:,OS C

():=xOSx
(, )(, )CJXCJX
LEMMA 2.3 [1
set
J
r continuous mapping from
d graph operator in
.
et be a bounded and convex

Let  be a line
(,JX to (, )CJX then the operator
 

,, c
J XCJX,
Gcp


,GGx
S is a clo
P
se
7] L
F in Banach space X. :2\


 be an upper
semi-continuous and condeed map. If for
every x, ()
nsing multi-valu
F
x is closed and convex set in

,
then
F
has a fixed point in

.
3. Controlla
e are now ab
.1
bility Result
le to state and prove our main contro-
Let
W
llability result.
THEOREM 30.
x
X

 If the hypotheses (H1)-
(H5) are satisfied, then the system (1) is controllable
provided

001
1
:1 1M MC







(3.1)
LL


01201
1
11
11
1
1
MLL MML
CL C


 



 

(3.2)
and
31 1022
1
01 1
1
=1
=(1)(
(1) (1)(1)
() ,
1
m
k
k
Mz MxMLlL
C
0=MA

.
)
M
ML lL lb
CwlbM d



 






  


(3.3)
where,
:=( ,())CCJDA Proof. Let denote the Banac
of continuous functions from
h space
to ()DA
J
normed by


=():
C
x
supx ttJ
Using (.)
x
hypothesis (H5) for an arbitrary function
and 1(zDAtrol
1
() =()()((0,((0)))utWzgx Tbgxh
) define the con
110
11
0
0
)
(,( ()))()(,( ()))d
x
b
kk k
xxF
0< <
() ()d ()(())
b
tt
k
F
bxh bATbsFsxh ss


 


Tb svs sTtt Ixt

  





Using the above control, define a multi-valued map
:2N

 by


01
11
0
,
0<<
()=:()=( )()(0,((0)))
(,(()))()(, (()))d
()(());, .
t
kk kGx
tt
k
NxyytTtxgxFxh
00
() ()d ()()()d
tt
F
txh tATtsFsxhss
tt IxttJvS

  


 

Tt svs sTtsBuss
T
 


,,
k
F
gI and the fact that 0
x
X


, By assumption on
it is obvious that ()Xyt


.
ints of Clearly the fiN are mild solu
We shall show that N satisfies the hypotheses of Lemma
2.3. The proof will be given in several steps. Step 1:
There exists a positive number such that
xed potions to (1).
lN
(),
ll
H H where N


),0.lt=:(
l
Hx xt


  
For each positive number l, l
H
is clearly a bounded
closed convex set in .

We claim that there exists a
positive integer l such that () ,
l
NH H where
()=().
l
NHNx If it is ue, then fo
er l, the
l
tr
the functi
lx

positive integ
Hnot r each
eons re xist(.)
ll
x
H

and ()
ll
yNx,

but (.) ,
ll
yH that is ()>
l
yt l

for some ()[0, ],tl b

where denotes t is depen-
dent on l. However one other hand we have,
()tl
th
01
1
00
<( )( )()(0,((0)))
( ,(( )))()(,(
( )d()()( )d
ll
t
ll
ll
lytTxgxFxh
1
0
,
0< <
( )))d
( )
()(()), where
l
tt
kklkl Gx
l
tt
k
t
F
txhtATtsFsx
v sstsBuss

 




h ss
Tt sT
TttIx tvS




 

Hence,

D. N. CHALISHAJAR ET AL. 1490
01
1
1
1
0
00
0<<
<( )()(0,((0)))
(,(()))
() (,(()))d
( )()d( )()()d
( )(())
ll
l
t
l
tt
ll
kklk
tt
k
lTtxgx AAFxh
AAFtxht
ATtsAFsxhs s
Ttsvs sTtsBus s
TttIxt


















 




02201 01
1
11
(1 )
00
123
0=1
(1) (1)
(1)d ()d
() ()
d
()
tt
m
t
k
k
lMxLlL MLlMLl
CC
Llswl s
ts ts
CMM MsMd
ts



















Dividing on both sides byand taking the lower
limit as we get
l
l
1
01 201 11
() 1
bb
MLLMML CL C






 
 1.
This is a contradiction with Formula (2). Hence for
some positive integer
Step 2: is c
Indeed if) then ther
()
ll
NH H.
onvex for each ()Nx
yy
x.
e exists
12
,(Nx12 , ,Gx vv S

such that , we have for each tJ
01
( )=( )()(0,((0)))
i
ytTt xgxFxh
11
0
(, ( ()))()(,( ()))d
()()d ()()()d
t
tt
ii
kk k
00
0< <
()
(()); =1,2.
tt
k
F
txh tATtsFsxhss
Ttsvs sTtsBus s
 
 


Tt t Ixti



 

Let 01.

 Then for each tJha we

ve



1
11
0
12
0
12
0
0<<
() =((0)))
(,( ()))()(,( ()))d
() ()(1)()d
() ()(1)()d
()(())
t
t
t
kk k
tt
k
t h
Ftxh tATtsFsxh ss
Ttsv svss
TtsBu suss
Ttt Ixt



 
 
 





12 0
(1)()()(0,yyTtxgxFx    
 
Sinc convex becausehas convex v
( )y x
e isalues,
,Gx
S
12
)
G
(1 yN

 
Nx
 .
Step 3: is closed for each
Let x such that in
()

yN
x.
yy
0()
nn n.

Then
y
1
g the fact that Gt valuesa
to a sury to ge
and hence
y
tJ
and there exists
,
,nG
vSx such that for ever
01
1
0
0
0< <
()=()()(0,((0)))(,(()))
()( ,(( )))d
()(())
n
t
tt
tt
k
ytTtxgxFxhFtxh t
ATts Fsx hss
T
Tt txt

  







0
() ()d ()()()d
n
tsvssTtsBu ss
I
 

kk k
Usin has compac, we my pass
bsequence if necessat that vn conver-ges to
1(,J)vL X,Gx
vS

. Then for each ,tJ

01
11
0
00
()()=()()(0,((0)))
(,( ()))()(,(()))d
()()d()()()d
()(());.
n
t
tt
kkk
ytytTtxgxFxh
0< <tt
k
F
txh tATtsFsxhss
Ttsvs sTtsBus s
Ttt IxttJ







 
 



Hence ()yNx

.
ext we showe operator N c Step 4: that this u.s.
and condensing.
For this purpose, we decompose N as N = N1 + N2, where
the operators N1, N2 are defined on Hl respectively by
1
N
11
1
0
()( )=( ,()( ))()(0,((0))
()( ,(( )))d
t
NxtFt xhtTt Fxh
ATtsFsx hss





00
() ( )d()()()d
()(()); .
tt
20
,
0< <
=:()=()()(0,((0)
(
kk k Gx
tt
k
NxyytTt xgxFxh
1
1
))
1
0
(,(( )))()(,()))d
t
F
txh tATtsFsxhss 

Tt svs sTtsBuss
Ttt IxtvS

 
 


  

We will verify that N1 is a contraction while 2 is a
completely continuous operator.
To prove that N1 is a contraction, we take
N
12
,l
x
xH

(H2), arbitrarily. Then for each and by condition
we have that
tJ

1112
11 21
11 21
21
11
()()( )()
(,(()))(,( ()))
()[(0,( (0)))(0,((0)))]
(0,( (0))
Nx tNxt
Ftxh tFtxh t
TtF xhF xh
Fx hAF




 




11
0()(,(( )))(,(( ))) d
tAT tsFs xhsFs xhss 

11 2 1
=( ,(()))(,(()))
( )
AAFtxh tAFtxht
TtA A
 







21
1
0
11 21
1
012
1
00
01 12
0
012
0
(0,((0)))
()
( ,(( )))( ,(()))d
(1)d. ()() sup
()
1
(1)()()
sup
=()()
sup
t
t
s
s
s
xh
ATts
AFsxhsAF sxhss
C
M
MLLsxsxs
ts
LMMCxsxs
Lxsxs


























 




 








D. N. CHALISHAJAR ET AL.1491
Thus, 112 2012 NxN xLxx



tion 0
0< <1L, 1 N is a contraction.
Next we show that
. Therefore by
assump
s.c. and condensing.
i)
ii) is equi-continuous.
Let
2
is clearly bounded.
N is u.
2() l
NH
()NH
2l
121 2
,,<J

 . Let l
x
H and 2()yNx

.
Then thsucr eac ere exists , vSGx h that foh tJ

, we
have
)
Then,
00
00<<
()=()()()( )d
()()()d( )(()
t
t
kk k
tt
k
ytTtxgxTtsvs s
Tts Bu ssTttIxt




 





10 1
()(0,((0)))zgx xFxh
21
210
12
21 2
01
11
21
0
0
() ()
[()()](( ))
[() ()]()d()()d
()()
)()(
(
b
b
yy
TTxgx
TsTsvss Tsvss
TsTsBW
Tbxg
Tb





1
0A1
(,(()))()(, (()))dFbxh bTbFxh
)
(v




 



 
 









   

 
  



01
2
1
()
tk
Ts
BW



1
2
10 1
11
0
12
)( ()) d
( )()( )(0,((0)))
(, (()))()(, (()))
()()d ()(())d
kk k
b
b
kk k
tk
t Ixts
zgxTbxgxFxh
FbxhbATbFxhd
vTtt Ixts

0Tb
)d (Tt


  





  



 


The right hand side tends to zero as 0
 






21
()

 ,
compactness
in the uniform ope-
uous on .
l
since is strongly continuous and the
of implies the continuity
rator . Thus is equi-contin
()Tt

0
()t Tt 
topol ogy 2(.)N
H
act for each ,tJ
iii) 2
()(
lt )is relatively compNH

where

22
()=():()
ll
yt yNH.
condition (4)(),(
()NH t
Obviously, by 2)()
l
H
iiN Ht is rela-
tively compact in
X

for =0t. Let 0<tb

be
fixed and 0< <t

. For l
x
H and 2()yN
ch that
x, there
exists a
)
Define,
)
Since is compact, the set
function ,Gx vS su
00
0
0<<
()=()[()]()( )d
() ()d()()()d
()()()d( )(()
t
tt
t
t
kk k
ttt
k
ytTtxgxTtsvss
Ttsvs sTtsBus s
Tts Bu ssTttIxt









 
 
 




00
00< <
()=()()() ( )d
()()()d( )(()
t
t
kkk
tt
k
ytTtxgxTt svss
Tts Bu ssTttIxt





 


 



00
0
0< <
()=()()()() ( )d
()()( )()d
()(())
t
t
kk k
tt
k
ytTtxgxTTtsvss
TTtsBuss
Ttt Ixt








 







()Tt


2
()=( )
l
YtyNH():y t

is relatively compact in
X

for,0 << tevery

 .
Moreorevery yN
2( ), l
H over,f

12 1
01
()( )(0,((0)))
( ,(( )))
t
Tb xgxF xh
Fbxhb

 

1
1
0
0< <
() ()
=()()d ()()()d
()d
()
)d
(
tt
tt
t
t
t
tt
k
yty t
TtsvssTts Buss
Mwls
MM Mzgx
T




 









0()
(
)(()
b
kk k
Tb v
tt Ixt


()( ,(()))d
bATbFx h
 

















12122
022 01
1
01 1
(1 )
0
0=1
123
123
)d
()
()
()(1)
(1) (1)d
()
()d d
()
()
()
t
t
b
m
b
k
k
s
Mw l
MMMzLlL
MxLlLMLl
C
MLlLl
b
CwlM ds
b
Mw lMMMM
Mwl MMM
























 
















Therefore, letting 0


act sets arb
, we see that there re
relatively compitrarily close to the set
a


2
():( )
l
ytyN H
is relatively comp
. Hence the set

2
():( )
l
ytyN H
act in
X

.
As a consequence of (i),(ii), (iii) and together with the
can conclude that
:
Arzela-Ascoli theorem we
22
H
l
l
NH
map and, th
is a comply continuous multi-valued
erefore, a condensing mu valued map.
lete
lti-

D. N. CHALISHAJAR ET AL. 1492
iv) has a closed graph.
Froe above steps we can see, for every
2
N
m th
We must prove that there exists such that
)
where
,Gx
vS


,
l
x
H

which can
pact set.
2()Nx
be testified as i
Let
00
00< <
()=()()() ()d
()()()d( )(()
t
t
kk k
tt
k
yxTtxgxTtsvss
TtsBussTttIx t






 



is relatively compact and closed set,
n Step 3. Hence is a com
2()Nx
,
n
x
x  )
2nln n
,(
x
HyNx and .yy
n


We m
that there
ust show that
exists
22
(N
for each
();yNxy


,nGx
n
vS such that ,
)
nn
x means
.tJ

00
00< <
()=()()()( )d
()()()d( )(()
nn n
t
nkk
tt
k
ytTt xgxTtsvss
Tts BussTttIxt

 





t
)
nk
where
1
nk
1
10
11
0
00<<
()=()()()(0,((0)))
(, (()))()(, (()))d
()()d ()(())
xnn
n
b
b
nkk
tt
k
ut Wzgx TbxgxF xh
F
bxh bATbsFsxhss
Tbsv ssTttIx t










 




1
10 1
11
0
00<<
()
=( )()()(0,((0)))
(,(()))()( ,(( )))d
()()d ()(())
x
b
b
kk k
tt
k
ut
WzgxTbxgxFxh
F
bxh bATbsFsxh ss
TbsvssTttIx t




 







 




,=1, 2,,
k
I
km and
g
Clearly, since are continu-
ous we have that
000<<
000<<
()()()()()()d()(())
()()()()()()d()(())0,as .
t
nnn kknk
tt
k
t
kk k
tt
k
ytTt xgxTtsBussTttIxt
yxTtxgxTtsBussTttIx tn





 






 
 








Consider the linear continuous operator
.
From Lemma (H3) it follows that is a closed
graph operator.
Moreover, we obtain that
Gx



Since
G
OS

1
0
:, (,),=()d
t
LJXCJXvvtTt svss 


00
(( )()()(
nn n
ytxgxTt sBu 
 



,
0< <
)())d()(()) ().
t
kknk
n
tt
k
TtssTttIxtS


 

,
n
x
x  it follows from (H3) that
has a i
u-val p
value, c. On the other handis a con-
traction. nce is u.s.c. and ng
By La fixed point
00
()()( )()()y tTtxgxTtsBu





that is, there must exist a , () Gx vt S

 such that
,
0< <
()d()(()) ()
t
kk kGx
tt
k
ssTttIxtS





0
()( )yTtxgx 

00<<
() ()()d( )(())
t
kk k
tt
k
t
Tt s BussTttIx t

 


0
=
Therefore, 2 Nclosed graph. Since s a
completely continuous mltiued map with cact
()
(())= ()()d.
t
vtTtsvs s





is u.s.
2.3,
2
N
om
nsi
)
2
N
He
emma
1 N
conde
(.
12
=NNN
there exists
.
x
for N
on .
l
H
Therefore, the nonlocal Cauchym
ffect (1) is controllable on
proble with
impulsive e.
J
Particularly, if is a single-valued map,
then the system (1) will become





12
=
0
d(),()=()(),() ;
d
:= 0,;
|=(()); =1,2,,; ;
(0)() =
ttk kk
k
x
tFtxht AxtButGtxht
t
tJ b
xIxtk mtt
xgxx












(4)
by using Sadovskii’s fixed-point theorem for condensing
map, we can analogously study the controllability of the
system (4).
(H3)’ The function satisfies the fo-
llowing conditions:
i) for each
:GJ XX


2
(, (()))Gtxh t ,tJ

the function is (,.):Gt XX



D. N. CHALISHAJAR ET AL.1493
continuous; and for each ,
x
X

 the ion funct
is stroble.
positive num there exists a
such that
(.,) :GxJ X
ii) for each
positive function
ngly measura
ber
dependent
,lN
on l()wl
()
(,)() and li
l
l m= <
wl
supG txwl  
where
0
=(
sup
s
xx).s


THEOREM 3.2 Let
 
0
x
X


. If the hypotheses
(1),(2),(3),(4)and( 5)
H
HHH

the system (4) is controllabl
H
e on
are satisfied, then
J
provided (1), (2)
and (1) hold.
Proof The mild solution of the system (4) is given by
1
01
2
00
0< <
()=()( )(0,((0)))
)))d
( )(,(())d()()()d
()(()); .
tt
kk k
tt
k
xtT txg xFxh
1
0
( ,(()))( )(,((
t
F
txh tATtsFsx 
h ss
TtsGsxh ssTts Bu ss
Ttt IxttJ

 


 


We define the operator :N

 by
Then we can decompose as where
1
an
ve
epoorem
can be applied to the operato
ws th (4)
is controllable on the interval
01
11
0
20
0<<
( ()
(,( ()))()(,( ()))d
()(,(()))d ()()()d
()(())
t
tt
kkk
tt
k
NT tx
FtxhtATtsFsxhss
TtsGsxh ssTtsBu ss
Ttt Ixt

 


 




0
)()=() (0,((0)))xtgxFx h
N12 =,NNN
11
1
0
()( )=(,()( ))()(0,((0))
()(,(()))d
t
NxtFt xhtTtFxh
ATtsFsx hss



d
20 2
0
()( )=( )()()(,(()))d
)d
t
t
Nx tTtxgxTtsGsxhss
Tt s






0
and rify that 1 N is a contraction while 2 N is a
compact operator, thus Sadovskii’s fixd-
() ()(s Bu s
int the
at syste
r N and hence N has
atleast a fixed point on , which sho
.
m
J
T
π
0
2
2
( ,)( ,,)(sin ,)(sin ,)d
(, )
=(,),(sin,),(sin,
01, 0π, , =1,2,,;
),
= 0; ()(
=1,2, ,
(0, )
k
kk
z
ztxbtyx ztytyy
ty
ztx z
utyhtztxtx
x
x
txttkm
ztztz
km
zx














(,0)=(, π)) =(()),
kk
zt ztI t

 




  




π
0
0
=0
(,)(,)d=(), 0π
p
ii
i
kyxzsyyzxx












(4.1)
where p is a positive integer,
and
is defi



01
0<< <<<1,
p
sss
2
0
;( )=([0,π]).zxXL A

12
0< <<<<1
m
tt t
ned by =A



 with the domain


2
() =([0,]) =(.):,
ous,
()=0
DA HX
0
are absolutelycontinu
,(0)=X







 
Then A generates a strongly continuous semigroup
which is compact, analytic and self-adjoint.
a’) Also A has a discrete spectrum reresentation
;
(.)T
p
2
=1
=()<, >,(),
nn
n
A
nD 

 AnN
where 2 ()=(); =1,2,
π
nxsinnxn
  is the orthogonal
set of eigenvector of A. The eigenvalues are
b’) The operator
2,.nn N
1
2
A
is given by
he desireof is
similar to Step 4 of Theorem 3.1.
4. Example
As an application of Theorem 3.2, we study the
following impulsive partial function differential system
with nonlocal condition
d pro
1
2=A
=0
<,>
nn
n
n

 


on the space

2
=1
()=( <,>
nn
n
DA n 

perator 2 :(,)BLJXX

1
.) :.X X

The control o is defined by
()()()=(,); (0,π)Butyu tyy

which satisfies condition
(H5). Here B is dentity operator and the contan i
function u(.) is given in
We assume that the fo
2([0, π], )LU.
llowing conditions hol
rol
d:
unction b is measurable and i) The f
ππ
2
01
00
sup(,,)dd<.
tbtyxyx  

ii) The function
2(,bteasurable,
(, ,0)= (, ,π)=0,btybt y and
2
, )yx
x  is m
1
22
2
ππ (,, )
=supdd<.
bt yx
Nyx




10 12
00
tx 








D. N. CHALISHAJAR ET AL. 1494
iii) For the function R the fo-
llowing three conditions are satisfied:
1) For each is continuous.
2) For each
:[0,1]hRR
[0,1],( ,.,.)tht
1
2
,(.,,zXhzz)

 is measurable.
3) There is a positive number 1 suchthatc
1
(, ,),
g
tzzc z

for all 1
2
( ,)[0,1].tz X
iv) 11
22
,,=1,
k
ICXX k





,=1,,,
k
dk m such that
,and there exist cons-
tants
11
22
() , .
kk
I
zdzX
Here we choose 1
==.
2
 According to paper [21],
we know that, if 1
2
,zX then z is absolutely
ous , In view of
this result, for
continu ,zX
 and (0) =(π)=0.zz
1
2
( ,)tz [0,1] X,

  ( is defined
as in Section 3), we can define respectively that

π
0
(, )()(,,)( )( )d.=
F
tzxbtyxzyz yy



(,)()=(, (()),Gtzxhtzxz x

and
1
(())= (),
p
ii
gtKs 

),
=0
,
i
 
where 11
22
:
i
K
XX is completely continuous [16]
such that π
0
()()=(, )()d
ii
K
zx kyxzyy
 and
1
2
:[0,1]GXX It is easy to see that
1
2
11 11
22 22
:[0,1], :[0,1]FXXAFXX 
for each [0,1],t we have
In fact,

π
0
(, ),
12 (,,)
=()()d,cos()
π
n
Ftz
bt yxzyz yynx
nx





also,
,

2
π
12 (,,)
=(
bt yx
22
0
(, ),
)()d,sin().
π
n
Ftz
zyzyynx
nx





This shows that F and
1
2
A
F both take values in 1
2
X
e the in terms of properties (a’) and (b’), and therefor
function g. Since, for any 12 1
2
,
x
xX,
22 22
212121
=, ,
nn
=0 =0
2
nn
x
xxxznxxz



1
21
2
.xx
This inequality alongwith condition (ii) says that (H2)
is satisfied. Also G satisfies and g satisfies (H4). (3)H
By (i), (, )
F
tz is a bounaded liner operator on
X
.
(),( 3),(1),(24),(5)
H
HH HH

Thus are satisfied and
5.lla nite
Dimensional Space
It has been observed that the example in ([2-11,22])
overed as special case of the abstract result.
up is compact then the assumption (H1) in
Se finite dimension
applications are restricted to
overcome to this problem in
Seother
whe system (4) in
infinite dimension space.
LEMMA 5.1 Let
the system (1) is controllable on [0,1].
Exact Controbility in Infi
cannot be rec
If the semigro
ction 2 is valid only inal space so the
ordinary differential control
system but not to partial differential equations (refer
[15]). We have tried to
ction 3 for the inclusion (1). Here we present an
ay of exact controllability result of t
 ([0, ]),)
X


be a space
formed by normalized piecewise continuous function
([0, ],).X


Let  h tha  suc


=:
kk
VV

; where
tively compact if an
e space
Assume that the function F and G
ve mptions

1
(); ,
()
(); =
kk
k
kk
Vtttt
Vt
Vtt t








The set   is relad only if
each set k

 is relatively compact in th
).C
1
([ ,];
kk
tt X
 
THEOREM
rify the assu
5.2
(1)
H
and (2)H respectively
an ndiare fulfilled: d suppose that the following co
a1) For every >0r and all
tions
>0

there are com-
pact sets su
,1, 2,
i
r
UX
(, )Fs
, =i

1
U
ch that
,r

()TA

 and 2
,
() r TG U(, )s for every

 
():(0, ).
r
FJ



b1) Conditions (3)
H
and (4)
H
e
are satisfied. Then
there exists a mof th
Proof Consider the system (3.4). As a main portion of
eorem, we prov completely continuous
operator.
ild solution system (3.4).
the the that N is

D. N. CHALISHAJAR ET AL.1495
The mild solution given in Theorem (3.2) can be
splitted up into following four parts:


(1)
1
() =()(0)()(0,((0)))
(, (()))
Nt TtxgxFxh
txh t
 1
F
(2) ()=(,(( )))d
t
NtATsxhss

(3)
0
( )=(,()()d
t
Ntz TsxhsBuss

(4)
0
() =
tt
k
Nt


()i
N
1
0()tsF
20
()( )))d(
t
tsGs s Tt

()(()),foreach .
kk k
TttIxt tJ



Obviously each is continuous. To prove that
is compact operatwe will show separately that
is relatively compact in for every
.
is relatively com
or
()
N
()i
N
x
((0,))
r
=(0
rr

1: Let

pact in
,)
(1)
NStep r.

oundLet is uniformly bed
on [0,us fonorm of the operator in
(0,b], we can observe that the sets
is relatively compact for every
act in
(1)
=(). Let
r
VN
b] and continuo
(.)T
r the
1
,
)) ;
kr
VT AU



 
 it follows that k
V

1
[,
kk
ttt


()
r is relatively comp
1
( )(0,((0)tFxh
],=0,1,2,,.k m 
Step 2: Let (2) N.

ompWe first shvely cact
for each For vial. Assume that
ow that 2(())
r
Nt is relati
.tJ
tb
=0t it is tri
0<2 <

 and let 1
,r
U

pothesis (1a
be the com
the hy. Since
pact set
introduced in) (.)
A
T

is
strongly cs on [,]b ontinuou

, it fo
11
llows that


is relatively com-
pact in

,
=():[,],
r
UATsxsbxU

 
 
.
X

usinNow g mean value theorem for Bochner inteal,
we can write
gr
2
(2)
N1
1
0
1
1
2
()=()()(,(())d
() (,(())d
(2 )()()(0,)
t
t
t
r
xtATtsTAFsxhs s
ATtsAFsxhs s
tcoUX












 



for each where (),
r
x()co U

denotes convex hull
of U

and


*
11
=21 .rCLx






Thus 2()()
r
Nt is relavepact in .

tily com
X

tinuous.Next we show that (2) i-c()
r
N
. Then
is equon
Let 0 0<ttb
(2) (1)
1
0
0
0
() ()
=()( ,(( )))d
()( ,(()))d
(
t
t
Nxt Nxt
01
0
(2)
00
1
=( ( )))
()(,(()))d
t
t
A
TtsFsxh ss
A
TtsFsxh ss
xtITtt N
A
TtsFsxh ss



Sinc for ;
r
x






(2)
0
();Nxt e the elements

 are
included in a compact set, it follows that th on
ght hand side convergey to zer
e first term
o as
he fun
ris uniforml0.tt
ctionSimilarly it follows from (1)b that t
1
()( ,(( ))),r tsFsx hsxAT

 are equi-int
rm on ri
egrable, which
nd sideimply that the second teght ha also
converges uniformly to zero as 0.tt
This show that (2)()
r
N is equi-continuous from the
right at 0.t Similarly it can be prove that 2()
r
N is
equi-continuous from the left at 0>0.t Thus (2) ()
r
N
is equi-continuous and hence (2) ()
r
N is relatively
compact in

.
Step 3: By using same argument as in Step 2 we can
prove that the set (3)().
r
N is relatively compact in .

Step 4: The relatively compactness of (4)().
r
N is
consequence of assumption (H4) and Lemnce
th
compactnesption o.2 (a1) and growth
condition (H2) (5). If the maps F and
ma 5.1 He
e proof.
Remark Throughout Section 5 we have used
s assumf Theorem 5
ii) and (H
,=1, 2,,
k
I
km
(i) and (H5) instead
satisfy some Lipschitz conditions (H2)
of compactness in (a1) then also we
can prove controllability result.
6. References
artellcompact
al
lusionite D
Point Theory and Aol. 2, No. 1, 2007,
pp. 11-51.
[3] C. C. Travis anebb, “Existence, Stab
α-Norm for Partial
s,” Tran of A
[4]
ear
em,” Journal of Mathematical Analysis and Appli-
utions of Semilinear
Paprabolic Equations with Nonlocal Initial Conditions,”
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179, 1993, pp. 630-637. doi:10.1006/jmaa.1993.1373
[1] M. Mi, “A Rothe’s Type Theorem for Non
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matica Italiana, Vol. 2, 1975, pp. 70-76.
[2] M. Benchohra and S. K. Ntouyas, “Existence Results for
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Fixed pplications, V
d G. F. W
th
ility and
Compactnes wiFunctional Differ-
ential Equationnsactiomerican Mathematical
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L. Byszewski, “Theorems about the Existence and
Uniqueness of a Semilin Evolution Nonlocal Cauchy
Probl
cations, Vol. 162, 1991, pp. 496-505.
[5] K. Deng, “Exponential Decay of Sol
s
[6] D. N. Chalishajar, “Controllability of Second Order Se-
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2009, pp. 1-15.
[7] D. N. Chalishajar and F. S. Acharya, “Controllability of
lsive
entific and.
r
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D. N. CHALISHAJAR ET AL. 1496
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