Exact controllability for a wave equation with fixed boundary control
© Cui and Song; licensee Springer. 2014
Received: 24 August 2013
Accepted: 17 February 2014
Published: 24 February 2014
This paper addresses the study of the controllability for a one-dimensional 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 , by the Hilbert Uniqueness Method, the exact controllability of this equation is established. Also, the explicit dependence of the controllability time on the speed of the moving endpoint is given.
where u is the state variable, v is the control variable and is any given initial value. Equation (1.2) may describe the motion of a string with a fixed endpoint and a moving one. The constant k is called the speed of the moving endpoint. By , for , any and , (1.2) admits a unique solution in the sense of a transposition.
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 . 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  is not applicable to the controllability problem of (1.2). We choose the complicated multiplier which satisfies the first-order linear differential equation. But the result in this paper is not satisfactory. We hope that the controllability result is obtained when . We hope that we obtain a modified multiplier in the forthcoming papers.
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.
Denote for a controllability time. The main result of this paper is stated as follows.
Theorem 2.1 Suppose that . For any given , (1.2) is exactly controllable at time T in the sense of Definition 2.1.
Remark 2.1 It seems natural to expect that the exact controllability of (1.2) holds when . However, we did not have success in extending the approach developed in Theorem 2.1 to this case.
Therefore, the exact controllability of (1.2) (Theorem 2.1) is reduced to the following main controllability result for the wave equation (2.1).
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 .
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.
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 .
where we use z to denote the solution of (2.3) associated to and , and η denotes the solution of and (3.1) associated to z.
Step 2. That Λ is an isomorphism is equivalent to the exact controllability of (2.1).
has a unique solution .
By (3.4) and (3.5), it follows that . If we set , by the uniqueness of (3.3), then w is the solution of (2.1) associated to . Furthermore, and . Therefore, we get the exact controllability of (2.1).
Step 3. Now we prove that Λ is an isomorphism, when .
for any , where η denotes the solution of (3.1).
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 Lax-Milgram 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 non-cylindrical domain at the time (Theorem 2.1).
4 The proofs of important inequalities
In this section, we give proofs of Theorem 2.3 and Theorem 2.4.
Remark 4.1 Theorem 2.3 implies that for any , the corresponding solution z of (2.3) satisfies .
In the following, we give a proof of Theorem 2.4.
From this we get (2.7). This completes the proof of Theorem 2.4. □
It is well known that the wave equation (1.2) in the cylindrical domain is null controllable at any time . However, we do not know whether the controllability time is sharp.
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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