1. Introduction
Let
. Define
as a non-cylindrical domain on
:
where
We denote the conjugate space of with .
We study wave equation as follows:
where
is the control variable and
is the state variable.
is an any given initial value. The physical meaning of
is called the velocity of moving endpoint. By [
1], we know that
has a unique wake solution
in the transposed sense.
Applications of control problems can be found everywhere in life; for example, in engineering practice and in science and technology. In modern mathematics, the distributed parameter energy control theory is an important branch. Control can be divided into exact control, null control and approximate control. In wave equations, we know that exact controllability is equivalent to null controllability.
In cylindrical domains, there are many studies on controllability of wave equations. However, not much work was performed on the wave equations defined in non-cylindrical domains ([
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]). In [
4], exact controllability was studied where the control is put on moving endpoints. In [
5], exact controllability was discussed, and the system is as follows:
In [
6,
7], exact internal controllability was reviewed. We discuss one-dimensional wave equations with the Dirichlet–Neumann boundaries and the control is put on a fixed endpoint with the Neumann boundary condition. By performing the calculation directly in non-cylindrical domains, we obtain exact null controllability by using the Hilbert Uniqueness Method.
In
Section 2, the definition of exact null controllability and some main theorems is provided. In
Section 3, the dual system of system
by proving Theorem 2 can be obtained. In
Section 4, by the nature of Hilbert’s Uniqueness Method, we prove controllability of system
(Proof of Theorem 1).
2. Main Results and Preliminary Work
Definition 1. Equation is called null controllable at the time if for any given initial value one can always find a control such that solution of satisfies in the transposed sense.
Remark 1. If is a more general function that satisfies ; then, it leads to the same conclusion as in this paper.
We set controllability time as follows:
The next theorem, Theorem 1, is the main proof of this paper (controllability).
Theorem 1. In the sense of Definition 1, is called exactly controllable at time for any given .
In order to prove controllability, we prove observability of its dual system. The dual system of system
is as follows:
where
is any given initial values. System
has a unique weak solution (for details refer to [
1]):
Remark 2. is a positive constant. Its value may vary from position to position.
Next, we give two important inequalities (observability).
Theorem 2. When , for any , there exists a constant such that the solution of satisfies 3. Observability: Proof of Theorem 2
For
, we give the definition of the energy equation of
as follows:
Lemma 1.
When ,
for any ,
the solution of satisfies Proof. For any
multiplying
by
and integrating on
, we obtain
Since
it is easy to check
It follows from
that
Taking
, it holds that
Therefore, we can conclude that
Due to
and
we have
Therefore, with
,
,
and
, we obtain
□
Lemma 2. When
, for any
, the solution
of
satisfies Proof. For any
, multiplying
by
and integrating on
, we can deduce that
Considering
and
, it follows from
that
Combining
, we see
□
Lemma 3.
When , for any , the solution of satisfies Proof. For any
, multiplying
by
and integrating on
, we get
Considering
and
, it is follows that
With
and
, we have
□
Lemma 4. When
, for any
, the solution
of
satisfies Proof. According to Lemmas 2 and 3, we can conclude that
Combining Lemma 1, we have
This follows from Cauchy’s inequality:
From
and
, it follows from
that
and
Hence, we see that follows. □
Remark 3. We will give the proof of Theorem 2, which has three steps.
Step 1. Multiplying
by
and integrating on
, it follows that
Next, we calculate
Combining
, it follows that
With
, it is obvious that
Therefore, with
and
, we obtain
Considering
, it follows from
that
This inequality implies that
Step 2. From
,
,
and
, it follows from
that
If
, we have
This implies that one can find a positive constant
to satisfy
Step 3. From
,
,
and
, one concludes from
that
With and , we get the desired result in Theorem 2. □
Remark 4. In the non-cylindrical domain , for any time , it is well known that is controllable. However, is not sharp.
4. Controllability: Proof of Theorem 1
We use Hilbert’s Uniqueness Method to prove controllability. The specific proof is divided into three steps.
Step 1. Define linear operator
We consider
For any
,
is defined as:
Step 2. Multiplying
by
and integrating on
, we can derive
From
, we get
Part
can conclude that
With Theorem 2, is proved to be coercive and bounded. Further, combining with the definition of the Lax–Milgram Theorem, we are able to obtain that is an isomorphic mapping.
Step 3. For any given initial value
we can define
where
is the solution of
. There exists
satisfying
By combining the definitions of
we get
where
is the solution of
.
Therefore, the following equation holds:
Due to the uniqueness of
we can obtain
Therefore, we complete the proof of exact null controllability of .
Author Contributions
Conceptualization, L.C. and J.L.; methodology, L.C.; software, J.L.; validation, L.C. and J.L.; writing—original draft preparation, J.L.; writing—review and editing, L.C. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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