Abstract
This paper addresses the study of the controllability for a onedimensional wave equation
in domains with moving boundary. This equation characterizes the motion of a string
with a fixed endpoint and the other one moving. When the speed of the moving endpoint
is less than
Keywords:
exact controllability; noncylindrical domain; wave equation1 Introduction
Given
where
Consider the following controlled wave equation in the noncylindrical domain
where u is the state variable, v is the control variable and
The main purpose of this paper is to study the exact controllability of (1.2). As we all know, there exists much literature on the controllability problems of wave equations in a cylindrical domain. However, there are only a few works on the exact controllability for wave equations defined in noncylindrical domains. We refer to [13], and [4] for some known results in this respect. In [2], the exact controllability of a multidimensional wave equation with constant coefficients in a noncylindrical domain was established, while the control entered the system through the whole noncylindrical domain. In [1] and [3], some controllability results for the wave equations with Dirichlet boundary conditions in suitable noncylindrical domains were investigated, respectively. But some additional conditions on the moving boundary were required, which entail the method used in [1] and [3] not to be applicable to the controllability problems of (1.2). In [1] and [3] in the onedimensional case, the following condition seems necessary:
It is easy to check that this condition is not satisfied for the moving boundary
in (1.2). The control system of this paper is similar to that of [4]. But the control is put on a different boundary. We mainly use the multiplier method
to overcome these difficulties and drop the additional conditions for the moving boundary.
But the simple multiplier in [4] is not applicable to the controllability problem of (1.2). We choose the complicated
multiplier which satisfies the firstorder linear differential equation. But the result
in this paper is not satisfactory. We hope that the controllability result is obtained
when
The rest of this paper is organized as follows. In Section 2, we give some preliminaries and the main results. In Section 3, we prove that the Hilbert Uniqueness Method (HUM) works very well for (1.2). Section 4 contains the proofs of the important inequalities used in Section 3.
2 Preliminaries and main results
The goal of this paper is to study the exact controllability of (1.2) in the following sense.
Definition 2.1 Equation (1.2) is called exactly controllable at the time T, if for any initial value
Denote
Theorem 2.1Suppose that
Remark 2.1 It seems natural to expect that the exact controllability of (1.2) holds when
In order to prove Theorem 2.1, we first transform (1.2) into a wave equation with variable coefficients in a cylindrical domain. To this aim, set
Then, it is easy to check that
where
Equation (2.1) admits a unique solution in the sense of a transposition,
(see [5]).
Therefore, the exact controllability of (1.2) (Theorem 2.1) is reduced to the following main controllability result for the wave equation (2.1).
Theorem 2.2Suppose that
The key proof of Theorem 2.2 is to prove two important inequalities for the following homogeneous wave equation in cylindrical domains. We have
where
Define the following weighted energy for (2.3):
where z is the solution of (2.3). It follows that
In the sequel, we denote by C a positive constant depending only on T and k, which may be different from one place to another.
We obtain the following two lemmas whose proof are found in [4].
Lemma 2.1For any
Lemma 2.2Suppose that
In order to prove Theorem 2.2, we need the following two important inequalities. The proofs of two important inequalities are given in Section 4.
Theorem 2.3Let
Theorem 2.4Let
3 Application of HUM
In this section, we prove the exact controllability for the wave equation (2.1) in the cylindrical domain Q (Theorem 2.2) by HUM.
Proof of Theorem 2.2 We divide the proof of Theorem 2.2 into three parts. We use certain inequalities proved later in Section 4.
Step 1. First, we define a linear operator
For any
Then it is well known that (3.1) admits a unique solution in the sense of a transposition,
Moreover, by Theorem 2.3 in Section 3, there exists a constant C such that
Define a linear operator Λ:
where we use z to denote the solution of (2.3) associated to
Step 2. That Λ is an isomorphism is equivalent to the exact controllability of (2.1).
In fact, for any target
has a unique solution
Suppose that Λ is an isomorphism, for any initial value
Note that η is the solution of (3.1) and that z is the solution of (2.3) associated to
By (3.4) and (3.5), it follows that
Step 3. Now we prove that Λ is an isomorphism, when
Write
for any
Multiplying the first equation of (3.1) by
Combining the above equality with the definition of Λ, we have
By Theorem 2.3 and Theorem 2.4, it suffices to prove that Λ is surjective. Notice that Theorem 2.4 and (3.6) imply A is a coercive bilinear form. Moreover, by (3.2), it is easy to check that A is bounded. Therefore, by the LaxMilgram Theorem, Λ is a surjection. It follows that Λ is an isomorphism. □
Remark 3.1 By the equivalent transformation in Section 2, Theorem 2.2 implies the exact controllability
for in the noncylindrical domain
4 The proofs of important inequalities
In this section, we give proofs of Theorem 2.3 and Theorem 2.4.
Proof of Theorem 2.3 First, we choose
Next, we estimate every terms on the right side of (4.1). Notice that
On the other hand, for each
Therefore, by (4.1)(4.3), we have
□
Remark 4.1 Theorem 2.3 implies that for any
In the following, we give a proof of Theorem 2.4.
Proof of Theorem 2.4 First, let q be the solution of the following problem:
e.g.,
It is easy to check that
Set
then it is easy to check that
By (2.5) and (4.5), after calculating, we have
From (4.6), it follows that
Next, we estimate the terms on the right side of (4.7). For each
Take
This implies that for any
It follows that
On the other hand, for any given
Take
By the value of M, we have
From this, it follows that
Then when
we obtain
Hence, by (4.7)(4.9), we derive
Let
If
From this we get (2.7). This completes the proof of Theorem 2.4. □
Remark 4.2 It is easy to check that
It is well known that the wave equation (1.2) in the cylindrical domain is null controllable
at any time
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All results belong to CL and SL. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the National Science Foundation of China 11171060, 11371084 and Department of Education Program of Jilin Province under grants 2012187 and 2013287. Moreover the authors are grateful to anonymous referees for their constructive comments and suggestions, which led to improvement of the original manuscript. The authors are grateful to Christopher D. Rualizo for patient and meticulous work.
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