Applied Mathematics, 2010, 1, 265-273
doi:10.4236/am.2010.14033 Published Online October 2010 (http://www.SciRP.org/journal/am)
A Characterization of Semilinear Surjective Operators
and Applications to Control Problems*
Edgar Iturriaga, Hugo Leiva
Grupo de Matemática Aplicada, Departamento de Matemáticas, Universidad de Los Andes,
Mérida, Venezuela
Received April 28, 2010; revised July 30, 2010; accepted August 3, 2010
Abstract
In this paper we characterize a broad class of semilinear surjective operators H GV Z given by the fol-
lowing formula ()
H
Gw Gw Hw ,wV where ,V Z are Hilbert spaces, ()GLVZ

 and HV Z
is a suitable nonlinear function. First, we give a necessary and sufficient condition for the linear operator G
to be surjective. Second, we prove the following statement: If ()
R
ang GZ

and H is a Lipschitz func-
tion with a Lipschitz constant h small enough, then ()
H
R
ang GZ

and for all zZ the equation
() GwH wz admits the following solution 111 ()( ())
z
wGGG IHGGGz
 
.We use these results
to prove the exact controllability of the following semilinear evolution equation () (())zAzButFtzut


,
0zZuUt, where
Z
, U are Hilbert spaces, ()ADA ZZ is the infinitesimal generator of
strongly continuous semigroup 0 {()}
t
Tt  in
Z

B

(),LU Z

the control function u belong to 2(0 )LU 


and [0]
F
ZU Z

 is a suitable function. As a particular case we consider the semilinear damped
wave equation, the model of vibrating plate equation, the integrodifferential wave equation with Delay, etc.
Keywords: Semilinear Surjective Operators, Evolution Equations, Controllability, Damped Wave Equation
1. Introduction
In this paper we characterize a broad class of semilinear
surjective operators
H
GV Z given by the following formula
()
H
GwGwHwwV  (1.1)
Where
Z
, V are Hilbert spaces, GV Z is a
bounded linear operator (continuous and linear) and
H
VZ is a suitable non linear function in general
nonlinear. First, we give a necessary and sufficient con-
dition for the linear operator G to be surj ective. Secon d,
we prove the following statement: If ()Rang GZ

and
H
is a Lipschitz function with a Lipschitz constant h
small enough, then ()
H
Rang GZ and for all zZ

the equation
()GwH wz admits the following solution
111
()(())
z
wGGG IHGGGz
 
 
We apply our results to prove the exact controllability
of the following semilinear evolution equation
() (())0zAzButFtzutzZuUt


 (1.2)
where Z and U are Hilbert spaces, ()
A
DA ZZ
is the infinitesimal generator of strongly continuous se-
migroup 0 {()}
t
Tt  in ()
Z
BLUZ

, the control func-
tion u belong to 2(0 )LU 

 and [0]
F
ZUZ


is a suitable function. We give a necessary and sufficien t
condition for the exact controllability of the linear sys-
tem
() 0zAzButzZuUt


 (1.3)
Under some conditions on F, we prove that the control-
lability of the linear system (1.3) is preserved by the semi-
linear system (1.2). In this case the control 2(0 )uL U  
steering an initial state 0 z to a final state 1 z at time
0

 (using the non linear system (1.2)) is given by the
following formula:
11
10
()()()(( ))utBTtIKzTz 
 

W
where
K
ZZ is non linear operator gi ven by:
0()(()()())
K
TsFszsSsds


 

 

and ()z


is the solution of (1.2) corresponding to the
*This work was supported by the CDCHTA-PROJECT: C-1667-09-
05-AA

E. ITURRIAGA ET AL. 266
control u define by:
1
()()()()[0 ]utS tBTtt

 

W
As an application we consider some control systems
governed by partial differential equations, integrodiffer-
ential equations and difference equations that can be
studied using these results. Particularly, we work in de-
tails the following controlled damped wave equation
() (())
01
(0)(1) 0
tt txxt
wcwdwutxftutxww
x
wt wt
tIR
 



 



(1.4)
where 0d,0c the distributed control 2
uL1 (0 t


2(0 1))L and the nonlinear term ()
f
twvu is a func-
tion 3
1
[0 ]
f
tIR
IR
 . A physical interpretation of
the nonlinear term ()
t
f
tuww  could be as an eternal
force like in the suspension bridge equation proposed by
Lazer and McKenna (see [1]).
The novelty in this work lies in the following facts:
First, the main results are obtained by standard and basic
functional analysis such as Cauchy-Schwarz inequality,
Hahn-Banach theorem, the open mapping theorem, etc.
Second, the results are so general that can be apply to
those control systems governed by evolutions equations
like the one studied in [1-3] and [4]. Third, we find a
formula for a control steering the system from the initial
state 0
z to a final state 1 z on time 0

, for both the
linear and the nonlinear systems, which is very important
from engineering point of view. Also, we present here a
variational approach to find solutions of the semilinear
equation ()GwH wz which is motivated by the one
used to prove the interior controllability for some control
system governed by PDE’s, see [5]. Finally, these results
can be used to motivate the study of semilinear range
dense operator in order to characterize the approximate
controllability of evolution equations.
2. Surjective Linear Operator
In this section we shall presents a characterization of
surjective bounded linear operator. To this end, we de-
note by ()LV Z the space of linear and bounded op-
erators mapping V to
Z
endow with the norm of the
uniform convergence, and we will use the following
lemma from [6] in Hilbert space:
Lemma 1. Let ()GLZV
 be the adjoint operator
of ()GLVZ. Then the following statements holds:
1) () 0Rang GZ

 such that
VZ
Gzz zZ 

2) ()(){0}Rang GZKer G
 .
In the same way as definition 4.1.3 from [7] we define
the following concept:
Definition 1. The generalize controllability gramian of
the operator G is define by:
GG ZZ


W (2.1)
Theorem 1. An operator ()GLVZ is surjective if,
and only if, the operator GG
W is invertible. Under
this condition, for all zZ

the equation
Gw z

 (2.2)
admits the following solution
11
()
z
wGGG zGz
 

W (2.3)
Moreover, this solution has minimum norm. i.e.,


inf
z
wwGwzwV

 (2.4)
and
zz
wwGwz ww

.
Proof Suppose G is surjective. Then, from the
foregoing Lemma there exists 0

 such that
222
VZ
Gzz z Z 

Therefore,
22
Z
GGzzzzzz Z 


  W (2.5)
This implies that W is one to one. Now, we shall
prove that W is surjective. That is to say
()Range()RZ

WW
For the purpose of contradiction, let us assume that
()RW is strictly contained in
Z
. Using Cauchy Sch-
warz’s inequality and (2.5) we get
2,
Z
zzzZ  W
which implies that ()RW is closed. Then, from Hahn
Banach’s Theorem there exists 0
zZ with 00z

such that
00, .zzz Z

W
In particular, putting 0
zz

we get from (2.5) that
22
00 0
0
Z
zz z 

 W
Then 00z

, which is a contradiction. Hence, W is
a bijection and from the open mapping Theorem 1

W is
a bounded linear operator.
Now, suppose W is invertible. Then, given zZ

we shall prove the existence of wV such that Gw z

.
This w can be taking as follows
1
z
wG z


W
In fact,
11
()
z
GwGGzGG GGzz


W
Now, we shall see that the solution

E. ITURRIAGA ET AL. 267
11
()
z
wGGG zGz
 
W of the Equation (2.2) has
minimum norm. In fact, let wV such that Gw z

and consider
222
2
()
2Re
zzz
z
zz
wwww w
www ww


On the other hand,
1
11
0
zz z
z
www G zww
zGw Gwzzz


 

W
WW
Hence, 22 2
0
zz
ww ww.
Therefore, z ww, a nd z ww

if z ww.
Corollary 1. If an operator ()GLVZ is surjective,
then the operator
SZ V defined by:
1
SG



W (2.6)
is a right inverse of G. i.e., GSI.
Definition 2. Under the condition of the above theo-
rem the operator
1
SGZ V

W
is called the generalize steering operator.
Lemma 2. An operator ()GLVZ

 satisfies
()Rang GZ if, and only if, ()Rang ZW.
Proof Suppose that ()Rang GZ. Then, from Lemma
1 part(2) we have that
00zzz ZzW (2.7)
For the purpose of contradiction, let us assume that
()Rang ZW
Then, from Hanh Banach’s Theorem there exists
00z such that
00 .zzz ZW
In particular, if we put 0 zz

, then 00 0zz

W,
which contradicts (2.7).
Now, suppose that ()Rang ZW. Then,
()Rang GGZ
, and consequently ()Rang GZ.
2.1. Variational Method to Obtain Solutions
The Theorem 1 gave a formula for one solution of the
system (2.2) which has minimum norma. But, it is not
the only way allowing to build solutions of this equation.
Next, we shall present a variational method to obtain
solutions of (2.2) as a minimum of the quadratic
functional
Z
IR,
2
1
() 2Gz Z


 (2.8)
Proposition 1. For a given zZ the Equation (2.2)
has a solution wV

if, and only if,
0wGzZ  


  (2.9)
It is easy to see that (2.9) is in fact an optimality con-
dition for the critical points of the quadratic functional
 define above.
Lemma 3. Suppose the quadratic functional  has a
minimizer z
Z


. Then,
z
z
wG



 (2.10)
is a solution of (2.2).
Proof. First, observe that  has the following form
1
() 2GG zZ 



Then, if
z

is a point where  achieves its minimum
value, we obtain that
{}( )0
zz
dGG z
d





So, z
GG z 


and
z
z
wG


 is a solution of (2.2).
Remark 1. Under the condition of Theorem 1, the
solution given by the formulas (2.10) and ( 2.3) coincide.
Theorem 2. The system (2.2) is solvable if, and only if,
the quadratic functional  defined by (2.8) has a mini-
mum for all zZ

.
Proof Suppose (2.2) is solvable. Then, the operator
G is surjective. Hence, from Lemma 1 there exists
0

 such that
222
GZ 

Then,
22
() 2zZ

 
Therefore,
lim( )


 
Consequently,  is coercive and the existence of a
minimum is ensured. The other way of the proof follows
as in proposition 1.
3. Surjective Semlinear Operators
In this section we shall look for conditions under which
the semilinear operator
H
GV Z given by:
(), ,
H
GwGwHww V

 (3.1)
is surjective. To this end, we shall use the following theo-
rem from non linear analysis.

E. ITURRIAGA ET AL. 268
Theorem 1. Let
Z
be a Banach space and
K
ZZ
a Lipschitz function with a Lipschitz con s tant 1k

and
consider ˆ()
GzzKz . Then ˆ
G is an homemorphisme
whose inverse is a Lipschitz function with a Lipschitz
constant 1
(1 )k .
Theorem 2. If ()Rang GZ and
H
is a Lipschitz
function with a Lipschitz constant h such that
1
()1hG GG

, then ()
H
Rang GZ and for all
zZ the equation
()
H
GwGw Hwz  (3.2)
admits the following solution
111
1
()( ())
()
z
wGGG IHGGGz
SIHSz
 

 


 (3.3)
where 1 ()SGGG

.
Proof Suppose that ()Rang GZ. Then, from Coro-
llary 1 we know that the operator S define by (2.6) is a
right inverse of G, so if we put HH
GGS, we get
the new operator
()
HH
GGSHS Z

 (3.4)
where 11
()SGGG G
 
W. Hence, if we define the
operator
K
ZZ by:
()
K
HS

 (3.5)
the operator
H
G can be written as follows
()
H
GKIKZ
 
  (3.6)
On the other hand, K is a Lipschitz function with a
Lipschitz constant 1
()1hG GG 


. Then, applying
Theorem 1 we get the result.
Theorem 3. If ()Rang GZ

and the operator K
given by (3. 5) i s linear an d 0K, then ()
H
Rang GZ

and for all zZ the equation
()
H
GwGw Hwz  (3.7)
admits the following solution
11
()
z
wG IKz
 
W
Proof Since H GSIK, then
2
()
H
GSzzz zZ
Then, in the same way as in the proof of Theorem 1
we get the result.
Corollary 1. Under the conditions of the above
Theorems, the operator
Z
V define by:
111
1
()( ())
()
zGGG IHGGGz
SIHSz
 

 


 (3.8)
is a right inverse of
H
G. i.e., H GId
Corollary 2. Under the conditions of the above
Theorems, the solution
1
()
z
wSIHSz

 of the Equation (3.2) depends
continuously on z. Moreover,
1
zyV Z
S
wwzy zyZ
hS



4. Controllability of Semilinear Evolution
Equations
In this section we shall characterize the exact controlla-
bility of the semilinear evolution equation
() (())
0
zAzButFtzut
zZuUt



 (4.1)
Where Z, U are Hilbert spaces, ()
A
DA ZZ is
the infinitesimal generator of strongly continuous semi-
group 0
{()}
t
Tt  in Z, ()BLUZ

, the control function
u belongs to 2(0 )LU 

 and [0 ]
F
ZUZ


is a suitable function.
4.1. Linear Systems
First, we shall study the controllability of the linear
system (1.3), and to this end, for all 0 zZ

and
2(0 )uL U


 the the initial value problem
0
()() 0
(0)
zAztButt
zz





 (4.2)
admits only one mild solution given by:
00
()()()() [0]
t
ztTtzTtsBusds t


 
 (4.3)
Definition 1. (Exact Controllability) The system (1.3)
is is said to be exactly controllable on [0]

, if for all
01
zzZ

 there exists a control 2(0 )uLU   such
that the solution ()zt of (4.3) corresponding to u, ve-
rifies: 1
()zz


.
Consider the following bounded linear operator:
2
0
(0)()( )GLUZGuT sBusds

  
 (4.4)
whose adjoint operator 2(0 )GZL U 
 is given
by:
()()() [0] GsBT ssZ 


 (4.5)
Then, the gramian GG ZZ

W takes the foll-
owing classical form
0() ()zTsBBTszds






W (4.6)
Then, the following Theorem from [7](pg. 47, Theorem
4.17) is a characterization of the exact controllability of
the linear system (1.3).

E. ITURRIAGA ET AL. 269
Theorem 1. For the system (1.3) we have the following
condition for exact controllability.
System (1.3) is exactly controllable on [0]

 if, and
only if, any one of the following condition hold for some
0

 and all zZ
1) 2 ((0))Range( )GLUG Z   ,
2) 2
Z
zz z  

W,
3) 22 2
0()()
Z
UZ ds z
Gz s Gz


  



4) 22
0() Z
Uz
BT sz



 



5) (){0}Ker G and ()Rang G is closed.
Remark 1. One can observe that the invertibility of
the operator W is not proved in the foregoing theorem
and, consequently, none formula for the control steering
the system (4.2) from initial state 0 z to a final state 1 z
on time 0

 is given.
Now, we are ready to formulate and prove a new result
on exact controllability of the linear system (1.3).
Theorem 2. The system (1.3) is exactly controllable
on [0 ]

 if, and only if, the operator W is invertible.
Moreover, the control 2(0 )uL U
  steering an initial
state 0
z to a final state 1 z at time 0

 is given by
the following formula:
110
()()(( ))utBTtz Tz 

 W (4.7)
Proof It follows directly from the above notation and
applying Theorem 1.
Corolla ry 1. If the system (1.3) is exactly controllable,
then the operator
2(0 )SZL U
  define by:
11
or ()()()SGSsBT s

 

WW (4.8)
is a right inverse of G. i.e., GSI
In this case the Equation (2.2) takes the following form
2(0 ),
Gu z
uL UzZ 


(4.9)
and the quadratic functional  given by (2.8) can be
written as follows
2
0
1
()( )
2BTs ds zZ



 
(4.10)
The following results follow from Proposition 1,
Lemma 3 and Theorem 1 respectively.
Proposition 1. For a given zZ the Equation (4.9)
has a solution 2(0 )uL U
  if, and only if,
0()() <u tBTt>ds<z>Z

 

 
(4.11)
It is easy to see that (4.11) is in fact an optimality con-
dition for the critical points of the quadratic functional
 define above.
Lemma 1. Suppose the quadratic functional

has a
minimizer z
Z


. Then,
()()[0]
z
ut BTtt




 (4.12)
is a solution of (4.9).
Theorem 3. The system (1.3) is exactly controllable if,
and only if, the quadratic functional  define by (1)
has a minimizer
z

for all zZ.
Moreover, under this condition we obtain that
1
() ()()[0]
z
ut BTtBTtzt

 
 

W (4.13)
and 1
zz


W.
4.2. Nonlinear System
We assume that
F
is good enough such that the Equa-
tion (4.1) with the initial condition 0 (0)zz and a con-
trol 2(0 )uL U


 admits only one mild solution given
by:
00
0
()()()()
()(()()),[0]
t
t
ztTtzTt sBusds
Tt sFszsusdst


 

 

 (4.14)
Definition 2. (Exact Controllability) The system (4.1)
is said to be exactly controllable on [0 ]

 if for all
01
zzZ

 there exists a control 2(0 )uL U   such
that the corresponding solution z of (4.14) satisfies
1
()zz



Define the following operator: 2(0 )
F
GL UZ  
by
00
0
()() ()(()())
()(()())
F
GuT sBusdsTsFszsusds
GuTs F s z su sds




 
 


(4.15)
where ()()ztztu

 is the solution of (4.14) correspon-
ding to th e contr ol u. Then, the following proposition is
trivial and characterizes the exact controllability of (4.1).
Proposition 2. The system (4.1) is exactly controllable
on [0 ]


if, and only if, ()
F
Rang GZ
So, in order to prove exact controllability of system
(4.1) we have to verify the condition of the foregoing
proposition. To this end, we need to assume that the
linear system (1.3) is exactly controllable. In this case we
know from corollary 1 that the steering operator S
defined by (4.8) is a right inverse of G, so if we put
FF
GGS, we get the following representation:
0()(() ()())
FF
GGSTsFszsSs ds


  


(4.16)
where ()z


is the solution of (4.14) corresponding to

E. ITURRIAGA ET AL. 270
the control u define by:
1
()()()() [0]utS tBTtt



W
Hence, if we define the operator
K
ZZ by:
0()(()()())
K
TsFszsSsds


  
 (4.17)
the operator
F
G can be written as follows
()
F
GKIK

    (4.18)
Now, we shall prove some abstract results making
assumptions on the operator
K
. After that, we will put
conditions on the nonlinear term
F
that imply condi-
tion on
K
.
Theorem 4. If the linear system (1.3) is exactly con-
trollable on [0]

 and the operator
K
is globally Lip-
schitz with a Lipschitz constant 1k then the non
linear system (4.1) is exactly controllable on [0 ]


and
the control steering the initial state 0 z to the final state
1
z is given by:
11
10
()()() (())utBTtI KzTz 
 
 W
Proof It follows directly from Equation (3.6) and Th e-
orem 1 or The or em 2.
Theorem 5. If the system (1.3) is exactly controllable
on [0 ]

 and the operator
K
is linear with 0K,
then the system (4.1) is exactly controllable on [0 ]


and the control ()ut steering the initial state 0
z to the
final state 1
z is given by:
11
01
()()() (())utBTtI KzTz 
 
 W
Proof It follows directly from Equation (3).
The proof of the following lemma follows as in lemma
5.1 from [1].
Lemma 2. If
F
satisfies the Lipschitz condition

22112 12 1
21 21
()()
[0]
F
tzuFtzuLzzuu
zzZuuUt 

 
then
21 2121
()
K
zKzKLzzzzZ
where 12
()
K
LLHH and
12
21
0
[]1
sup ()
ML
st
HMBLe H
M
BM Tts






 

W
Theorem 6. If
F
satisfies the foregoing Lipschitz
condition, the linear system (1.3) is exactly controllable
on [0 ]

 and
21
([ ]1)()1
ML
LMBLeM B



 W (4.19)
then the non linear system (4.1) is exactly controllable on
[0 ]

 and the control steering the initial state 0 z to the
final state 1 z is given by:
11
10
()()()(( ))utBTtI KzTz 
 

W
Proof From Lemma 2 we know that
K
is a Lipschitz
function with a Lipschitz constant k given by:
21
([ ]1)()
ML
kLMBLe MB




W
and from condition (4.19) we get that 1k. Hence, ap-
plying Theorem 4 we complete the proof.
5. Applications and Further Research
In this section we consider some control systems gov-
erned by partial differential equations, integrodifferential
equations and difference equations that can study using
these results. Particularly, we work in details the con-
trolled damped wave equation. Finally, we propose fu-
ture investigations an open problem.
5.1. The Controlled Semilinear Damped Wave
Equation
Consider the following control system governed by a
1D semilinear damped wave equation
() (())
01
(0)(1)0
tt txxt
wcwdw utxftutxww
x
wt wt
tIR









(5.1)
where 0d, 0c, the distributed control
22
1
(0(0 1))uLtL

 and the nonlinear term ()ftwvu


is a function 3
1
[0 ]
f
tIR
IR
 .
Abstract Formulation of the Problem.
Now we will choose the space in which problem (5.1)
will be set as an abstract first order ordinary differential
equation.
Let 2[0 1]XL

 and consider the linear unbounded
operator
()
A
DA XX defined by
x
x
A

 , where
() {areac,(0)(1)0}
xxx
DA XX
 

 
(5.2)
The operator
A
has the following very well known
properties:
1) The spectrum of
A
consists of only eigenvalues
12
0n




each one with multiplicity one.
2) There exists a complete orthonormal set {}
n

of
eigenvectors of
A
.
3) For all ()
x
DA

we have
11
nnn nn
nn
A
xx Ex  



 

(5.3)
where

  is the inner product in
X
and

E. ITURRIAGA ET AL. 271
22
and ()2sin()
nnnn
n
Ex xn
xnx



 

(5.4)
So, {}
n
E is a family of complete orthogonal projec-
tions in
X
and
1n n
x
Exx X




4)
A
generates an analytic semigroup {}
At
e given
by:
1
nt
At n
n
ex eEx





 (5.5)
and
5) The fractional powered spaces r
X
are given by:
22
1
()() 0
rr r
nn
n
XDA xXExr 



 



with the norm
12
22
1
rr r
rnn
n
x
AxExx X 

 





 

where
1
rr
nn
n
A
xEx 



 (5.6)
Also, for 0r we define r
r
Z
XX, which is a
Hilbert Space with norm give n by :
2
22
r
r
Z
wwv
v
 


Using the change of variables wv
, the system (5.1)
can be written as a first order systems of ordinary diffe-
rential equations in the Hilbert space
12 12
12 ()
Z
DAX XX


as:
12
(())0zAzBuFtzutzZt

 
(5.7)
where
00
X
XX
I w
zBA
I
dA cI v
 
 
 
 
 

 
 
 (5.8)
is an unbounded linear operator with domain
() ()DADA X and
0
()() Ft zu
f
tu wv

 




(59)

and the function F: [0,t112 12
]
Z
XZ


. Since 12
X

is continuously included in
X
we obtain for all
12 12
zz Z

 and 12 uu X that

12
22 11
211221 1
()()
[0]
Z
FtzuFtz u
Lzzu utt


  

(5.10)
Throughout this section, without lose of generality, we
will assume that
21
4cd



The following proposition follows from [8] and [1].
Proposition 1. The operator
A
given by (5.8), is the
infinitesimal generator of strongly continuous group


()t Tt
I
R in 12
Z

given by:
12
1
() n
At n
n
Ttze PzzZ




 
 (5.11)
where


0
nn
P is a family of complete orthogonal pro-
jections on the Hilbert space 1
2
Z given by:


1
nnn
PdiagEEn

 (5.12)
and
01 1
nnnn n
ABPB n
dc 


 



(5.13)
This group decays exponentially to zero. In fact, we
have the following estimate
2
()( )0
ct
Tt Mcde t


 (5.14)
where
2
22
4
()
sup 2(2)
22 44
nn
nnn
cdc
Mcd d
cd dc















(5.15)
The proof of the following theorem follows in the
same way as the one for Theorem 4.1 from [1].
Theorem 1. The system
12
0
0
(0)
zAzBuzZt
zz






 (5.16)
is exactly controllable on [0 ]

.
Theorem 2. If the following estimate holds

 
()2 1
()[1]1 ()1
McdL
LMcdLeMcd W




 (5.17)
then the system (5.7) is exactly controllable on [0 ]


.
Proof It follows from Theorem 6 one we observe that
in this case 1B

.
5.2. Future Research
These results can be applied to the following class of
second order diffusion system in Hilbert spaces
0()()
0
wAwutftwu
twWuU



 

E. ITURRIAGA ET AL. 272
Where W, U are Hilbert spaces, the control 2(0 )uL U
 
,
00
()
A
DAW W is an unbounded linear operator
in W with the spectral decomposition:
011 1
j
jkjkjjj
jk j
A
w<w> Ew

 


 


where 1
j
j
kjkj
k
Ew< w>






, {}
kj

 is a complete
orthonormal set of eigenvectors of 0
A
 correspondent
to the eigenvalues 12 n

  with multipli-
city n

and 0
A
 generates a strongly continuous semi-
group 0 {()}
t
Tt  given by:
1
(), , 0
jt
j
j
TtweEwwW t





and [0 ]
f
WUW

  is a suitable function.
Examples of this class are the following well known
systems of partial differential equations:
Example 1. The nD wave equation with Dirichlet
boundary conditions
2
2
0
0
() (())
0
()0
0
(0)( )
(0)( )
wwutx ftutxw
t
tx
wtx
tx
wx x
zxx
t
x


 

 




 









(5.18)
where  is a sufficiently smooth bounded domain in
N
I
R,

22
[0 ]( )uL rL
, 2
00 ()L
  and
f
is
a suitable function.
Example 2. The model of Vibrating Plate
22
2
0
0
() (())
0
0
0
(0)( )
(0)()
wwutxftutxw
t
tx
ww
tx
wx x
wxx
t
x


 

 



 
 









(5.19)
where  is a sufficiently smooth bounded domain in
2
I
R,


22
[0 ]( )uL rL

 , 2
00 ()L
 
and
f
is
a suitable function.
Others type of problems are the following control
problems:
Example 3. Interior Controllability of the 1D Wave
Equation
0
1()(())
in (0)( 01)
(0)(1)0 (0)
(0)() in [01]
tt xxt
yy utxftutxww
yt ytt
yx yx



 





 
 


(5.20)
where

is an open non empty subset of [0 1], 1

de-
notes the characteristic function of the set

, the
distributed control 22 (0[0 1])uL L 

 and the nonlinear
term ()
f
twvu

 is a function 3
1
[0 ]
f
tIR IR  .
For the interior controllability of the linear wave equa-
tion one can see [5].
Example 4. Exact Controllability of Integrodifferential
1D Wave Equation with Del ay.
0
0
1
()(()
in (0)( 01)
(0)(1) 0
(0 )
() ()
in [0] [01]
() ()
in [0] [01]
t
tt xx
t
yy utxpsysrxds
yt yt
t
ysxysx
r
ysx ysx
r


 




 




 

 


 

 


(5.21)
where the distributed control 22
(0[0 1])uL L  ,
01
[0][01]yy rIR

 are continuous functions
and the nonlinear term 1([][0 1])pLr 

 .
Example 5. Exact controllability of Semilinear Difference
Equations
0
( 1)()()()()
(() ())(0)
znAnzn Bnun
f
zn unnzz IN



(5.22)
where
Z
, U are Hilbert spaces, (())
A
lINLZ

,
(())BlINLUZ


, 2()ulINU

, ()LU Z denotes
the space of all bound ed lin ear oper ato rs fr om U to
Z
and ()()LZZLZ

. The nonlinear term
f
ZU Z is a continuous Lipschitzian function.
That is to say: For all 21
zz Z and 12 uu U we
have that

22112 12 1
()()
f
zufzuLzzuu

  (5.23)

E. ITURRIAGA ET AL. 273
5.3. Open Problem
The solution of the following problem is very important,
no only to study the approximate controllability of evo-
lution Equation (1.2), it also can be used to solve others
mathematical problems.
The problem can be formulate as follows: Let
Z
, W
be Hilbert spaces, ()GLWZ and
H
WZ is
a suitable nonlinear function. When the following state-
ment holds?
If ()Rang GZ and
H
is a Lipschitz function with
a Lipschitz constant h small enough, then
()Rang GHZ and for all zZ there is a sequence
0
{}wW
  such that equation

0
lim()GwHwz




6. References
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[2] E. Iturriaga and H. Leiva, “A Necessary and Sufficient
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Hilbert Spaces and Applications,” IMA Journal of
Mathematics, Control and Information, Vol. 25, No. 3,
2008, pp. 269-280.
[3] H. Leiva, “Exact Controllability of Semilinear Evolution
Equation and Applications,” International Journal of
Systems and communications, Vol. 1, No. 1, 2008, pp. 1-
12.
[4] H. Leiva and J. Uzcategui, “Exact Controllability of
Semilinear Difference Equation and Application,” Journal
of Difference Equations and Applications, Vol. 14, No. 7,
2008, pp. 671-679.
[5] J. L. Lions, “Optimal Control of Systems Governed by
Partial Differential Equations,” Die Grundlehren der
mathematischen Wissenschaften in Einzeldarstellungen
Band 170, 1971.
[6] R. F. Curtain and A. J. Pritchard, “Infinite Dimensional
Linear Systems,” Le cture Notes in Control and I n for m at io n
Sciences, Vol. 8, Springer Verlag, Berlin, 1978.
[7] R. F. Curtain and H. J. Zwart, “An Introduction to Infinite
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Mathematics, Vol. 21. Springer Verlag, New York, 1995.
[8] H. Leiva, “A Lemma on C0-Semigroups and Applications
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