数学专业英语论文(含中文版)
-
Differential Calculus
Newton and Leibniz,quite independently
of one another,were largely responsible for
developing
the ideas of integral
calculus to the point where hitherto
insurmountable problems could be solved
by more or less routine successful
accomplishments of these men were primarily due
to the fact that they were able to fuse
together the integral calculus with the second
main branch of
calculus,differential
calculus.
In this article,
we give su
ffi
cient conditions
for controllability of some partial neutral
functional
di
ff
erential
equations with infinite delay. We suppose that the
linear part is not necessarily densely
defined but satisfies the resolvent
estimates of the Hille
-Yosida theorem.
The results are obtained
using the
integrated semigroups theory. An application is
given to illustrate our abstract result.
Key words
Controllability;
integrated semigroup; integral solution; infinity
delay
1 Introduction
In
this
article,
we
establish
a
result
about
controllability
to
the
following
class
of
partial
neutral functional
di
ff
erential equations with
infinite delay:
Dxt
A
Dxt
Cu
(
t
)
F
(
t
,
xt
)<
/p>
,
t
0
(1)
t
x
0
<
/p>
where the state
variable
x
(.)
takes
values in a Banach space
(
E
,
.
)
and
the control
u
(.)
is given in
L
2
(
0
,
T
,
U
),
T
0
< br>,the Banach space of admissible control functions with U a Banach space. C
is a
bounded linear operator from U into E, A : D(A)
⊆
E → E is a
linear operator on E, B is the
phase space of functions mapping (−∞,
0] into E, which will be specified later, D is a
bounded
linear operator from B into E
defined by
D
(
0
)
D
0
,
B
D
0
is
a bounded linear operator from B into E and for
each x : (−∞, T ] → E,
T > 0, and t
∈
[0,
T ], xt represents, as usual, the
mapping from (−∞, 0] into E defined by
<
/p>
xt
(
)
x
(
t
),
(
,
0
]
F is an
E-valued nonlinear continuous mapping on
.
The problem of
controllability of linear and nonlinear systems
repr
esented by ODE in finit
dimensional space was extensively
studied. Many authors extended the controllability
concept to
infinite dimensional systems
in Banach space with unbounded operators. Up to
now, there are a lot
of works on this
topic, see, for example, [4, 7, 10, 21]. There are
many systems that can be written
as
abstract neutral evolution equations with infinite
delay to study [23]. In recent years, the theory
of neutral functional
di
ff
erential equations with
infinite delay in infinite
dimension was deve
loped and
it is still a field of research (see, for instance,
[2, 9, 14, 15] and the
references
therein). Meanwhile, the controllability problem
of such systems was also discussed by
many
mathematicians,
see,
for
example,
[5,
8].
The
objective
of
this
article
is
to
discuss
the
controllability for Eq. (1), where the
linear part is supposed to be
non-
densely defined but satisfies
the resolvent estimates of the Hille-
Yosida theorem. We shall assume conditions that
assure global
existence and give the
su
ffi
cient conditions for
controllability of some partial neutral functional
di
ff
erential
equations with infinite delay. The results are
obtained using the integrated semigroups
theory and Banach fixed point theorem.
Besides, we make use of the notion of integral
solution
and we do not use the analytic
semigroups theory.
Treating
equations with infinite delay such as Eq. (1), we
need to introduce the phase space B.
To
avoid
repetitions
and
understand
the
interesting
properties
of
the
phase
space,
suppose
that
(
B
,
.
B
)
is
a
(semi)normed
abstract
linear
space
of
functions
mapping
(−∞,
0]
into
E,
and
satisfies
the following fundamental axioms that were first
introduced in [13] and widely discussed
in [16].
(A)
There
exist
a
positive
constant
H
and
functions
K(.),
M
(.):
,with
K
continuous and M
locally bounded, such that, for any
and
a
0
,if
x : (−∞,
σ + a] → E,
x
B
and
x
(.)
is continuous
on [σ, σ+a], then, for every t in [σ,
σ+a], the following conditions
hold:
(i)
xt
B
,
(ii)
x
(
t
)
H
x
t
(iii)
xt
< br>B
B
,which is equivalent to
<
/p>
s
t
(
0
)
H
B
or every
B
B
K
(
t
)
sup
x
(
s
)
M
(
< br>t
)
x
(A) For the
function
x
(.)
in
(A), t → xt is a
B
-
valued continuous function
for t in [σ, σ + a].
(B)
The space B
is complete.
Throughout
this
article,
we
also
assume
that
the
operator
A
satisfies
the
Hille
-Yosida
condition :
(H1) There exist
and
，
such that
(
,
)
(
A
)
and
n
n
:
n
N
,
M
（
2
）
sup
(
)
(
I
A
)
Let A0 be
the part of operator A in
D
(
A
)
defined by
D
(
A
0
< br>)
x
D
(
A
)
:
Ax
D
(<
/p>
A
)
A
x
Ax
,
for
,
x
D
(
A
)
0
0
< br>It is well known that
D
(<
/p>
A
0
)
D
(
A
)
and the operator
A
0
generates a strongly continuous
semigroup
(
T
0
(
t
)
t
0
)
on
D
(
A
)
.
Recall that [19] for al
l
x
D
(<
/p>
A
)
and
t
0
,on
e has
f
0
t
< br>T
0
(
s
)
xds
D
(
A
0
)
and
t
A
0
T<
/p>
0
(
s
)
sds
x
T
0
(
t
)
x
.
< br>
We
also
recall
that
(
T
0
(
t
))
t
0
coincides
on
D
(
A
)
with
the
derivative
of
the
locally
Lipschitz integrated
semigroup
(
S
(
t
))
t
0
generated by A on E,
which is, according to [3, 17, 18],
a
family of bounded linear operators on E,
that satisfies
(i)
S(0) = 0,
(ii)
for any y
∈
E, t → S(t)y is
strongly continuous with values in E,
(iii)
S
(
s
)
S
(
t
)
(
S
(
t
r
)
s
(
r
))
dr
for all t, s ≥ 0, and for any τ >
0 there exists a
0
s
constant l(τ) >
0, such that
S
(
t
)
S
(
s
)
l
(
< br>
)
t
s
or all t, s
∈
[0, τ]
.
The
C0-semigroup
(
S
(
t
))
t
0
is
exponentially
bounded,
that
is,
there
exist
two
constants
M
and
,such that
S
(
t
)
M
e
< br>t
for all t ≥
0.
Notice
that
the
controllability
of
a
class
of
non-de
nsely
defined
functional
di
ff
erential
equations was studied in [12] in the
finite delay case.
、
2 Main Results
We start with introducing the following
definition.
Definition 1
Let
T > 0 and
ϕ
∈
B. We consider
the following definition.
We say that a function x := x(.,
ϕ
) : (−∞, T ) → E, 0 < T ≤
+∞, is an integral solution of Eq.
(1)
if
(i)
(ii)
(iii)
(iv)
x is
continuous on [0, T ) ,
t
0
Dxsds
D
(
A
)
for t
∈
[0, T ) ,
t
t
0
0
Dx
t
D
< br>A
Dxsds
Cu
(
s
)
F
(
s
,
x
s
)
ds
for t
∈
[0, T ) ,
x
(
t
)
(
t
)
for all t
∈
(−∞,
0].
We deduce
from [1] and [22] that integral solutions of Eq.
(1) are given for
ϕ
∈
B, such that
D
D
(
A
)
by the following system
t
Dxt
S
(
t
)
D
lim
S
(
t
s
)
B
(
Cu
(
s
)
F
(
s
,
x
s
))
ds
,
t
[
0
,
t
),
、
0
x
(
t
)
(
t
),
t
< br>
(
,
0
],
(3)
Where
B
(
I
A
)
<
/p>
1
.
To obtain
global existence and uniqueness, we supposed as in
[1] that
(H2)
K
(<
/p>
0
)
D
0
1
.
(H3)
F
:
[
0
,
]
E
is
continuous and there exists
F
(
t
,
1
)
F
(
t
,
2
)
< br>
0
0
> 0, such that
1
2
for
ϕ
1,
ϕ
2
∈
B and t ≥ 0.
(4)
Using Theorem 7 in [1], we obtain the
following result.
Theorem 1
Assume that (H1), (H2), and
(H3) hold. Let
ϕ
∈
B such that
D
ϕ
∈
D(A).
Then, there exists a
unique integral solution x(.,
ϕ
) of Eq. (1), defined on
(−∞,+∞) .
Definition 2
Under the above
conditions, Eq. (1) is said to be controllable on
the interval J =
[0, δ], δ > 0, if for
every initial function
ϕ
∈
B with
D
ϕ
∈
D(A) and for any e1
∈
D(A),
there exists a control u
∈
L2(J,U), such
that the solution x(.) of Eq. (1) satisfies
x
(
)
e
1
.
Theorem 2
Suppose that(H1), (H2), and (H3) hold. Let x(.) be
the integral solution of Eq. (1)
on
(−∞, δ) , δ > 0, and assume that (see [20]) the
linear operator W from U into D(A) defined by
Wu
<
/p>
lim
~
2
<
/p>
0
S
(
s
)
B
Cu
(
s
)
ds
，
< br>
（
5
）
nduces an invertible operator
W
on
L
(
J
,
U
)
/
KerW
，
su
ch that there exist positive constants
~
N
1
and
N
2
satisfying
C
N
1
and
W
1
N
2
，
then,
Eq.
(1)
is
controllable
on
J
provided
that
Where
(
D
0
)
0
M
e
<
/p>
0
N
1
N
2
M
2
e
2
)
K
1
，
（
6
）
0
t
K
:
max
K
(
t
)
.
Proof
Following
[1], when the integral solution x(.)
of
Eq. (1) exists on (−∞, δ) , δ > 0, it
is
given for all t
∈
[0, δ] by
d
x
(
t
)
D
0
x
t
S
(
< br>t
)
D
dt
Or
d
S
(
t
s
)
F
(<
/p>
s
,
x
)
ds
s
0
dt
t
S
(
t
s
)
Cu
(
< br>s
)
ds
0
t
x
(
t
)
D
0<
/p>
x
t
S
(
t
)
D
lim
S
(
t
s
)
B
(
s
,
x
s
)
ds
0<
/p>
t
t
lim
S
(
t
s
)
B
Cu
(
s
)
ds
< br>
0
Then, an
arbitrary integral solution x(.) of Eq. (1) on
(−∞, δ) , δ > 0, satisfies x(δ) = e1 if and only
if
e
1
D
0
x
S
(
)
D
d
d
< br>
0
S
(
s
)
F
(
s
,
x
s
)
ds
lim
0
S
(
t
s
)
B
< br>Cu
(
s
)
ds
t
This implies that, by
use of (5), it su
ffi
ces to
take, for all t
∈
J,
0
1
t
~<
/p>
1
u
(
t
)
W
lim
S
(
t
s
)
B
Cu
< br>(
s
)
ds
(
t
)
~
W
1
e
D
x
< br>
S
(
)
D
lim
S
(
t
s
)
B
(
s
,
x
)
ds
(
t
)
t
0
0
s
in order to have x(δ) = e1. Hence, we
must take the control as above, and consequently,
the proof
is reduced to the existence
of the integral solution given for all t
∈
[0, δ]
by
(
Pz
)(
t
)
:
D
0
z
t
S
(
t
)
D
d
dt
d
dt
S
< br>(
t
s
)
F
(
s
,
z
0
t
s
)
ds
t
0
~
S
(
t
s
)
C
W
1
z
(
)
D
0<
/p>
z
S
(
)
D
lim
S
(
)
B
F
(
,
z<
/p>
)
d
}(
s
)
ds
0
Without loss of generality,
suppose that
hat, for every
≥ 0. Using
similar arguments as in [1], we can see
z
1
,
z
2
Z
(
)
and t
∈
[0, δ]
,
(
Pz
1
)(
t
)
<
/p>
(
Pz
2
)(<
/p>
t
)
(
D
0
0
M
e
)
K
z
1
z
2
As K is continuous and
D
0
K
(
0
)
1
,we can choose δ >
0 small enough, such that
D<
/p>
0
0
M
e
0
N
1
N
2
M
2
e
2
)
K
<
/p>
1
.
Then, P is a strict contraction in
Z
(
)
,and the fixed point of P gives
the unique integral olution
x(.,
ϕ
) on (−∞, δ] that verifies
x(δ) = e1.
Remark 1
Suppose that all
linear operators W from U into D(A) defined by
Wu
li
m
S
(<
/p>
b
s
)
B
Cu
(
s
)
ds
0
~
2
0 ≤ a < b ≤ T, T > 0,
induce invertible operators
W
on
L
([
a
,
b<
/p>
],
U
)
/
KerW
,such that there
T
~
1
W
N<
/p>
C
N
exis
t positive constants N1 and N2 satisfying
,N
2
,taking
1
and
N
large enough and following
[1]. A similar argument as the above proof can be
used inductively
in
[
n
,
(
n
1
)
],
1
n
N
1
,to see that Eq. (1) is
controllable on [0, T ] for all T > 0.
Acknowledgements
The authors
would like to thank Prof. Khalil Ezzinbi and Prof.
Pierre Magal
for the fruitful
discussions.
References
[1] Adimy M,
Bouzahir H, Ezzinbi K. Existence and stability for
some partial neutral functional
di
ff
erenti
al equations
with infinite delay. J Math Anal Appl, 2004, 294:
438–
461