Let
be a state value of the system (SCE) at time
corresponding to the function
, the nonlinear term
, and the control
. We define the reachable sets for the system (SCE) as follows:
Definition.
The system (NCE) is said to be approximately controllable in the time interval
if for every desired final state
and
, there exists a control function
such that the solution
of (NCE) satisfies
, that is, if
where
is the closure of
in
, then the system (NCE) is called approximately controllable at time
.
We need the following hypothesis:
for any
and
there exists a
such that
where
is a constant independent of
.
As seen in [18], we obtain the following results.
Proposition.
Under the assumptions (G1-G2), (A), and (B), the following system
is approximately controllable on
, that is,
.
Let
. Then it is well-known that
for almost all point of
.
Definition.
The point
which permits (4.3) to hold is called the Lebesgue point of
.
Let
be a solution of (SCE) such that
in
is a solution of (NCE). First we consider the approximate controllability of the system (SCE) in case where the controller
is the identity operator on
under the Lipschitz conditions (G1-G2) on the nonlinear operator
in Proposition 4.2. So,
obviously.
Proposition.
Let
be solution of (4.2) corresponding to a control
. Then there exists a
such that
Proof.
Let
be a Lebesgue point of
so that
For a given
, we define a mapping
by
It follows readily from definition of
and Lemma 2.4 that
By a well-known contraction mapping principle,
has a unique fixed point
in
if the condition (4.5) is satisfied. Let
Then from (G1-G2), Lemma 2.4, and Proposition 2.5, it follows that
Thus, from which, we have
And we obtain
If
is a Lebesgue point of
, then we can solve the equation in
with the initial value
and obtain an analogous estimate to (4.10) and (4.12). If not, we can choose
to be a Lebesgue point of
. Since the condition (4.5) is independent of initial values, the solution can be extended to the interval
, and so we have showed that there exists a
such that
.
Now, we consider the approximate controllability for the following semilinear controlsystem in case where
is the identity operator,
Let us define the reachable sets for the system (4.13) as follows:
Theorem.
Under the assumptions (G1-G2), (A), and (B), we have
Therefore, if the system (4.2) with
is approximately controllable, then so is the semilinear system (4.13).
Proof.
Let
and let
be a solution of (4.2) corresponding to a control
. Consider the following semilinear system:
The solution of (4.2) and (4.16), respectively, can be written as
Then from Proposition 2.5, it is easily seen that
, that is,
as
in
. Let
be given. For
, set
Then we have
So, for fixing
, we choose some constant
satisfying
and from (2.13), or (2.16) it follows that
Thus, we know that
as
in
for
. Noting that
from (2.13), or (2.16), it follows that
Since the condition (4.20) is independent of
, by the step by stem method, we get
as
in
, for all
. Therefore, noting that
,
, every solution of the linear system with control
is also a solution of the semilinear system with control
, that is, we have that
in case where
.
From now on, we consider the initial value problem for the semilinear parabolic equation (SCE). Let
be some Banach space and let the controller operator
be a bounded linear operator from
to
.
Theorem.
Let us assume that there exists a constant
such that
Assume that assumptions (G1-G2), (A), and (B) are satisfied. Then we have
that is, the system (SCE) is approximately controllable on
.
Proof.
Let
be a solution of the smoothing system (SCE) corresponding to (NCE). Set
where
is a solution of (4.2) corresponding to a control
. Then as seen in Theorem 4.5, we know that
. Consider the following semilinear system:
If we define
as in proof of Theorem 3.2, then we get
So, as similar to the proof of Theorem 3.2, we obtain that
.
From Theorems 3.2 and 4.6, we obtain the following results.
Theorem.
Under the assumptions (G1-G2), (A), (B), and (B1), the system (NCE) is approximately controllable on
.