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Open Journal of Applied Sciences, 2013, 3, 37-43
doi:10.4236/ojapps.2013.31B1008 Published Online April 2013 (http://www.scirp.org/journal/ojapps)
Trajectory Controllability of Semilinear Differential
Evolution Equations with Impulses and Delay
Maojun Bin, Yiliang Liu
College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, P. R. China
Email: [email protected], [email protected]
Received 2013
ABSTRACT
This paper researches trajectory controllability of semilinear differential evolution equations with impulses and delay.
The main techniques in our paper rely on the fixed point theorem and monotone operator theory. In the end of the paper,
an example is given to explain our main result.
Keywords: Trajectory Controllability; Monotone Operator Theory; Mild Solution; Fixed Point Theorem; Lipschitz
Continuity
1. Introduction
The impulsive differential systems originate from the
real world problems to describe the dynamics of proc-
esses in which sudden, discontinuous jumps occurs. Im-
pulsive differential equations have become more impor-
tant in many mathematical models of real processes and
phenomena studied in control, physics, chemistry, popu-
lation dynamics, aeronautics and engineering. Because of
their significance, many scholars have been researched
the solvability of impulsive differential equations in re-
cent years, especially in the area of impulsive differential
equations with fixed moments, see the monographs of
Bainov and Simeonov [3], Lakshmikantham et al. [14]
and Samoilenko and Perestyuk [17] and the papers of
[1,2,4,9-13,22. Another hand, differential equations with
delay was initiated about existence and stability by
Travis and Webb [19] and Webb [21]. Due to such equa-
tions are often more realistic to describe natural phe-
nomena than those without delay, they have been inves-
tigated in different aspects by many authors [2,15].
The concept of controllability (introduced by Kalman
1960) plays an important role in many areas of applied
mathematics. In recent years, significant progress has
been made in the controllability of linear and nonlinear
deterministic systems [8,16,18,20]. But it does not give
any idea about the control path along which trajectory
moves.
In[7], D. N. Chalishajar, R. K. George, A. K. Nanda-
kumaran, and F. D. Acharya studied trajectory controlla-
bility of the following fractional nonlinear integro-dif-
ferential systems
0
0
()()( ())(()
(()) [0]
(0)
t
wtAwt BtutFtwt
Gtsws dstJT
ww
 
 

where H and U are Hilbert spaces, the state ()wt H
and the control ()ut U
for each The operator tJ
()
A
DA HHBJ
 H
is a linear operator not necessarily
bounded. The maps
and
U HGH
F
JHH
{( )ts JH
0J s are nonlinear operators, where
}t T

Motivated by the above work, in this paper, we con-
sider the following equation:
()()(())()[0]
( )(( ))12()()0
tk
kkk
x
tAxtBtut ftxtJTtt
xtI xtkmxttrt
 
 
(1.1)
where []
rT

()
. Let X be a real Banach space, the
state
x
tX
and the control is a Banach
space of admissible control function with U a Banach
space,
() ()ut LJU
()
A
DA X
()TtX
0t
is the infinitesimal generator of
a C0-semigroup
 The maps and BJUX
f
JX X are nonlinear operators. {[Dr0] X

)

is continuous everywhere except for a finite number of
points s at which ()(
s
s

exist and () ()}
s
s
(0 )Dr
 for D
the norm of
is defined by
sup{( )tr0}t
D
 01
tt  0mm
k
1
tt
T
I
XX kkk
()() ()
x
txtxt

()
k
x
t and ()
k
x
t
denote the right and the left limits of ()
x
t at k
tt
12km
 
12
{}
m
For any continuous function x defined on
J
\tt t
 and tJ
we denote by t
x
the
Copyright © 2013 SciRes. OJAppS
M. J. BIN, Y. L. LIU
38
element of D defined by ()= ()
t
x
sxts
here
0rs 
()
t
x
represents the history of the state from tr
the present time t. up to
)
The rest of this paper is organized as follows: In sec-
tion 2, we present some preliminaries to prove our main
results. In section 3, by applying some standard fixed
point principles, we prove the existence of the mild solu-
tions for fractional nonlinear integro-differential equa-
tions. In section 4, the trajectory controllability of the
system (1.1) is proved by applying the tools of monotone
operator theory and set-valued analysis. In section 5, we
give an example to illustrate our main results.
2. Preliminaries
In this section, we introduce definitions and preliminar-
ies which are used throughout this paper, and then we
give the mild solutions of systems (1.1).
Let be a real Banach space. We denote by
the space of
X-valued continuous function on J,
with the norm
(X
()
X


CJ
sup{ }
x
xT J

 and by 1()LJX
the space of X-valued Bochner integral functions on J
with the norm 10
T
L()
f
ft dt

 Let
()Jx{PCxJX x  is continuous at k
tt
and
the left continuous at k the right limit tt()
k
x
t
exist,
It is easy to verify that
12k }m ()J XPC
is a
Banach space with the norm
max{sup(
tJ 0)sup(
tJ 0)}()
PC
x
xtxt

 
sup{( )FF
Bx
1}xx

denotes the Banach space of bounded linear operator
from X to X with norm ()BX



()
Definition 2.1. A function
x
X
)[(())(
))
k
is a solution
(mild solution) of the system (1.1) if it satisfies
0
() (0)(
()((
k
t
kk
tt
0
()
)]
s
x
tTt Tt
Ttt Ix



sBsusf
t

sxds
(2.1)
Definition 2.2. The system (1.1) is said to be com-
pletely controllability on J if for any 01
x
xR and
fixed T, there exists a control such that the
corresponding solution
2()uLJ
()
x
of (1.1) satisfied 1
()
x
Tx
Let be the set of all functions defined on
[0 ]
()z
J
T
2()J
such that 0 and z is
differentiable almost everywhere.
1
tJ
()
(0)( )zx
zTx 

Definition 2.3. The system (1.1) is said to be T-con-
trollability if for any there exists a control
such that the corresponding solution
z
uL
x
of
(1.1) satisfied ()()
x
tzt a.e. tJ
[][0]
if i
tt Tt
Definition 2.4. The system (1.1) is totally controllable
on J if for all subinterval J
 
the sys-
tem (2.1) is completely controllable.
Clearly, T-controllability Total controllability
Complete controllability.
Now, we give the following properties which would
be used to our main result in the next.
Lemma 2.1.([5]) Let X be a Banach space, and
PQ XX
two operators satisfying:
(i) P is a contraction, and
(ii) Q is completely continuous,
then either
(a) the operator equation ()()
x
Px Qx has a solu-
tion, or
(b) the set {()()(0
x
xX PQx x

1)}
 is
unbounded.
Lemma 2.2. ([6])(Main Theorem on Monotone Op-
erators) Let be a real, reflexive Banach space, and
let
X
X
X
 be a monotone, hemicontinuous, bounded,
and coercive operator, and Then there exists a
solution of the equation
bX
ub

3. Existence of Mild Solutions
In this section we prove the existence and uniqueness of
the mild solution of problem (1.1). Before stating and
proving the result, we assume the following conditions
hold:
(1)H There exist a constant such that 0M
()
sup{ }
BX
M
TtJ
 
(2)H B satisfies Caratheadory condition, i.e.,
()Bt UX

is continuous for tJ
and is meas-
urable for
()ByJ X
y
U
(3)H
()
f satisfies Caratheadory conditions like B, i.e.,
f
tXX
 is continuous for tJ
and
()
f
xJ X
 is measurable for
x
X
()at
(4)H There exist two functions 0 0
1
()bt L()JX
and two constants 11 0ab
 such that B and f satisfy
following growth conditions:
01
01
() ()
() ()
XU
XX
Btub tbuuUtJ
f
txataxtJxX
 
 
 
 
(5)( )
H
ftx is Lipschitz continuous with respect to
x
, i.e. there exist constants 0
such that
112 2112
()( )
f
txftxxx
 
for all 12
x
xXtJ

)
(6H
m
there exist constants 012
k
dk m
with
1
1
k
k
Md
such that
() ()
kk k
I
xIydxy xyX

Now, let us begin prove the existence and uniqueness
of the mild solution of (1.1).
Theorem 3.1. If the conditions hold,
then the problem (1.1) has at least one mild solution on
X.
(1) (6)HH
Copyright © 2013 SciRes. OJAppS
M. J. BIN, Y. L. LIU 39
Proof. Transform the problem (1.1) into a fixed problem.
Consider the two operators
defined by
() ()PQPCJXPCJX
 

0
0[0]
() ()(())
k
kk k
tt
tr
Px tTtt IxttJ



and
0
()[ 0]
()()(0)()[(())()]
t
s
ttr
QxtT tTtsBsusfsxds
tJ



Then the problem of finding the solution of problem
(1.1) is reduced to finding the solution of the operator
equation We shall show
that the operators P and Q satisfy all the conditions of
Lemma 2.1. For better readability, we break the proof
into a sequence of steps.
()() ()PxtQxtxttJ

Step 1. Q is continuous.
Let {}
n
x
be a sequence such that n
x
x in ()PC Jx
then for we have
tJ
0
0
()()()[()()]
()()
s
s
t
nn
t
ns
QxtQxtT tsfs xfs xds
Mfsxfsxds


s
Since ()
f
s is continuous for a.e.
s
J
by the
Lebesgue dominated convergence theorem, we have
()()0, as .
n
QxtQxtn 
Thus Q is continuous.
Step 2. Q maps bounded sets into bounded sets in
()PCJX
It is enough to show that for any 0
}
there exists a
positive constant l such that for each
{()
x
BxPCJXxl
  we have Qx l

So we choose
x
B

then for each we have
tJ
1
1
0
01 0
0
1
00 1
0
1
( )( )(0)()[(())()]
(0)[ ()()()
]
(0)[ ()()][
]
t
s
t
s
t
L
L
Qx tTtTtsBs usfsxds
MMbsbusas
axds
M
MbsasdsMax
bu

 
 
 



As then we have
1
00
() ()()atbt LJX
11
00
0
11
[()()]
t
LL
QxMMbsas ds
ax bu l
 

 
 
Step 3. Q maps bounded sets into equicontinuous sets
of ()PC JX
We consider B
as in step 2 and let
1212m
J
tt t

 
12
Thus if 0
and 12

 we have
21
21
1
21
21
21
00
21
00
21
2101
0
()()
()(0) ()(0)
()(())()(())
()()()()
()(0) ()(0)|
()()[()()]
ss
Qx Qx
TT
TsBsusdsTsBsus ds
TsfsxdsTsfsxds
TT
TsTsbsbusds




 


 


 


 
 


1
1
2
1
1
1
1
2
1
2101
201
2101
0
2101
201
()()[()()]
()[()()]
()()[()]
()()[()]
()[()]
s
s
s
TsTsbsbusds
Tsbsbusds
TsTsasaxds
TsTsasaxds
Tsasaxds
Q






  
 
 

 
1234567
QQQQQQ
We easily get,
1
1
1
1
1
1
12 112
22101
0
21
[0 ]
01
0
12
3210
()(0)()(0)0 as
()()[()()]
sup()()
[()]0
as 0
()()[()
s
L
QT T
QTsTsbsbus
TsTs
bsdsb u
QTsTsbsb




ds
 






 



 


11
1
1
1
2
1
2
1
1
1
21
[]
01
12
4201
01
12
521
0
()]
sup()()
[()]0
as 0
()[() ()]
([()()]0
as
()()[
s
L
us ds
TsTs
bsdsb u
QTsbsbusds
Mbsbusds
QTsTs












 





1
1
1
1
1
[]
11
1
1
01
[0 ]21
01
0
62101
21
01
() ]
sup()()
[()]0
as 0
()()[()]
sup()()
[()
s
s
s
L
s
asa xds
TsTs
asdsa x
QTsTsasaxd
TsTs
asdsa x














 






1
12
]0
as 0
L



s
Copyright © 2013 SciRes. OJAppS
M. J. BIN, Y. L. LIU
40
2
1
2
1
7201
01
()[() ]
([()]0 as 0
s
s
QTsasaxds
Masaxds
 
 
Then, we get 21
()() 0Qx Qx
 as 12

0

since is a strongly continuous operator
and the compactness of for implies the
continuity in the uniform operator topology. This proves
the equicontinuity for the case where k
It remains to examine the equicontinuity at
()
()Tt0t
tt
tt
Tt
12 m

t
0
k
tk
First, we prove equicontinuity at Fixed
i1
such that For
11
{}[
kii
tki tt

 ] 1
0h
we have
01
0
01
01
0
01
()()
()(0) ()(0)
()()[()()]
[()()]
()()[()]
[()]
i
i
i
i
i
i
ii
ii
th
ii
t
th
th
ii s
t
s
th
Qx tQx th
TtTth
TtsTthsb sb usds
Mbsbusds
TtsTthsa saxds
Masaxds


 


 

which tends to zero as
0h
D
efine
01
()()[0 ]QxtQxttt
and
1
() (]
() ()
ii
iii
Qx tttt
QxtQx ttt
 

Next, we prove equicontinuity at Fixed
i
tt 20
h
such that For
22
{}[
kii
tki tt

 ] 2
0
we have
01
0
01
01
0
01
()()
()(0)()(0)
()()[()()
[()()]
()()[()]
[()]
i
i
i
i
i
i
ii
ii
t
ii
th
t
t
ii s
th
s
t
Qx thQx t
Tt hTt
Tt hsTtsbsbusds
Mbsbusds
Tt hsTts as axds
Masaxds

 
 
 

 

]
which tends to zero as The equicontnuity for
the cases 12
0h
0
 and 12
0
follows from the
uniform continuity of φ on the interval
[0]r 
As consequence of Steps 1 to 3 together with Ar-
zela-Ascoli theorem it suffices to show that B maps B
into a precompact set in X.
Let be fixed and let
0tT 0t
 be a real
number. For
x
B
we define
0
()()(0)()()[(())
()]
t
s
QxtTtTTt sBsus
fsx ds

 

Since is a compact operator, the set
()Tt ()
X
t
{(Qxt)x B}
 is precompact in X for every
0t
 Moreover, for every
x
B

we have
01
01
()()()[()
()()]
t
s
t
Qx tQx tTtsasax
bsbus ds
 


Therefore, there are precompact sets arbitrarily close
to the set () {()}
X
tQxtxB
 
 
() }
Hence the set
() {
X
tQxtxB


() ()QPCJ XPCJ X

 
is precompact in X. Hence the
operator is completely con-
tinuous.
Step 4. P is a contraction.
Let ()
x
yPCJX
 then for t we have J
00
0
00
()()
()(()) ()(())
( ())(())
() ()
kk
k
kk
kk kkkk
tt tt
kk kk
tt
kk kk
tt tt
Px tPy t
TttIxt TttIyt
MIxtIyt
MdxtytMdxy
 

 

 
  
 



i.e.,
0
() ()
k
k
tt
Px tPytMdxy



since
0
1
k
k
tt
Md

then P is a contraction.
Step 5. A priori bounds.
It remains to show that the set
{()
x
x
PC JXxQxP

 for some 01}

is bounded.
Let
x
then
x
x
Qx P

 for some 01

Thus for each tJ
0
0
()()(0)()[(())()]
()(())
k
t
s
kk k
tt
x
tTt TtsBsusfsxd
x
Ttt It
 


 
s
Implying and for each we have
(4)H(6)HtJ
11
11
00
0
11 1
00
0
11
11
()(0)[( )( )]
(())
[()()]
(())(0)(0)
t
mx
kk
k
LL
t
LL
mm
x
kk kk
kk
xtMMbsas ds
aMxbM uMIt
MMbsasds
aMxbM u
MItI MI
 





 

 



 

 
1
1
11
1
001
0
11
00
10
11 1
((0))
[()()]
()
((0))[()()]
[()]
m
k
k
t
L
mx
kk
k
Lt
m
k
k
m
kk
k
LL
MI
Mbsasds aMx
bM uMdt
M
IMbsasd
Maxb uMdxt






 





 
s
Copyright © 2013 SciRes. OJAppS
M. J. BIN, Y. L. LIU 41
Now we consider the function
defined by
()sup{()}0txsrsttT

Then ()
s
x
t
 for all tJ
and there is a point
such that If by the
previous inequality we have for note
[t
 ]rt()
 
(J
( )txt
ttJ

tt
11
00
10
11 1
()((0))[ ()()]
[()]
t
m
k
k
m
k
k
LL
tMIMbsasds
Maxb uMdt

 



i.e.,
11
1
1
00
0
11
(1)()((0) )
[()()]
[]
mm
kk
k
k
t
LL
Md tMI
M
bs asds
Maxbu






 
Hence there exists a constant K such that,
1
11
1
1
1
00
0
11
() ((0))
[()()]
[]
mk
k
m
k
k
Md
T
LL
t{M I
Mbsasds
Maxb u} K







By the definition of
we have
sup()()
tJ
xxtTKx

 
This shows that the set
is bounded. As a conse-
quence of Theorem 3.1 we deduce that has a
fixed point which is a mild solution of problem (1.1).
PQ
The proof is completed.
4. T-Controllability Results
In this section, we are concerned with the trajectory con-
trollability of semilinear differential evolution equations
with impulses and delay.
We make the following additional assumption on B:
H(7): B satisfies monotonicity and coercivity condi-
tions, i.e.,
() ()0BtuBtv uvuvUtJ
and
()
lim
u
Btu u
u



 
Now let begin proving the T-controllability results for
the problem (1.1).
Theorem 4.1. Under the conditions the
problem (1.1) is T-controllable.
(1) (7)HH
Proof. Let be the prescribed trajectory with zT
(0)(0)
z
 we want to find a control u satisfying
0
0
()()(0)()[(())()]
()(())
k
t
s
kk k
tt
ztTtTtsBsusf szds
Ttt Izt


 
(4.1)
The equation (4.1) can be written as
0
0
0
()()(0)()()
()(())
()(())
k
t
s
kk k
tt
t
ztTtTtsfszds
TttIzt
TtsBsus ds




(4.2)
Differentiating with respect to t, we get
0
0
0
()()(0)()()()
()(())
()(())(( ))
k
t
s
t
kk k
tt
t
ztATtATt sfszdsftz
ATttIzt
ATtsBs usdsB tut




－
(4.3)
Equation (4.3) can be written as
0
0
()()()()
t
ytgtsysdsy t

(4.4)
where ()(()) ()()ytBtutgtsATt s
  and
0
0
0
()()() (0)
()()()
()(())
k
t
s
t
kk k
tt
ytzt ATt
A
Ttsfszdsftz
ATttIz t


 
Define an operator by
22
() (LJX LJX 
t
)
0
()()( )yt gtsysds
(4.5)
We easily know that 0 is continuous in ()yt ()
J
gts
is continuous in
J
J
then for any 2()
12
y
yLJX 
we have
0
12
12
00
12
0
12
()()()()
max()( )( )
tT
tt
T
yy
g
tsy sdsgtsysds
g
tsy sysds
Ly y









＝
n
is a contraction for sufficiently large n. Hence by
generalized Banach contraction principle, there exists a
unique solution y for (4.4) for given 2
0()
y
LJX
Therefore, T-controllability follows if we can extract
from the relation
()ut
()(())yt Btut

)
(4.6)
To see this, define an operator by
22
() (NLJXLJX
( ())t ut ()NutB (4.7)
By the conditions H(1) and H(4), N is well-defined,
continuous and bounded operator. Also, (())Btut
is
monotone and coercive, then we can easily get N is
monotone and coercive. A hemi-continuous monotone
mapping is of type (M). The nonlinear map N is onto. By
Lemma 2.2, there exists a control u satisfying (4.6). The
measurability of follows as u is in ()ut2()LJX
This
proves T- controllability of the problem (1.1).
Copyright © 2013 SciRes. OJAppS
M. J. BIN, Y. L. LIU
42
The proof is completed.
5. Application
Example. We consider the following semilinear impul-
sive equation:
2
1
()() [cos()cos()]
3
[0]
(0)(0)1 2
() ()
()() (())
k
kkkk
wtxw utxxtwt
ttJ Ttt
wx xinkm
wtxtx inJ
wtxwtxI wtx

 

 

 
(5.1)
where is abounded domain in with smooth
boundary
(1
n
Rn
( )L 
)
2
(0 )()xtx

 
2()XL
We take and define the operator
()
A
DAX X by
21
0
()()() ()DA HHAwAxDw
It is easily turned out that the operator A generates
equicontinuous -semigroup on X. re the control term
is linear,
0
C
)(()) (Btut ut
2
1
() [cos()cos()
3
t]
f
txxt wt
is Lipchitz conditions. It satisfies the conditions of
Theorem 4.1, then the problem (5.1) is T-controllable.
6. Acknowledgements
This work was financially supported by NNSF of China
Grant No.11271087, No.61263006, Guangxi Scientific
Experimental (China- ASEAN Research) Centre No. 20
120116, open fund of Guangxi Key laboratory of hybrid
computation and IC design analysis No.2012HC IC07,
and the Innovation Project of Guangxi Graduate Educa-
tion No. YCSZ2012062.
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