Exact controllability for a one-dimensional wave equation with the fixed endpoint control
- Lizhi Cui^{1},
- Yang Jiang^{2}Email author and
- Yu Wang^{1}
Received: 13 July 2015
Accepted: 4 November 2015
Published: 14 November 2015
Abstract
This paper is devoted to the study of the exact 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 a moving one. The control is put on the fixed endpoint. When the speed of the moving endpoint is less than the characteristic speed, by the Hilbert uniqueness method (HUM), exact controllability of this equation is established.
Keywords
1 Introduction
The rest of this paper is organized as follows. In Section 2, we give some preliminaries and main results. In Section 3, using the multiplier method, we give a growth estimate of the energy function and obtain two important inequalities used in Section 4. In Section 4, we prove that HUM works very well for (1.2).
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
For \(k\in(0,1)\), denote \(T^{*}_{k}=\frac{ {e^{\frac {2k(1+k)}{(1-k)^{3}}}}-1}{k}\) for the controllability time. The main result of this paper is stated as follows.
Theorem 2.1
Suppose that \(0< k<1\). For any given \(T>T^{*}_{k}\), (1.2) is exactly controllable at time T in the sense of Definition 2.1.
Remark 2.1
We can obtain the same result as that of this paper for a more general function \(\alpha_{k}(t)\), as long as it meets the condition \(0<\alpha'_{k}(t)<1\).
Remark 2.2
In fact, if initial value \((u^{0}, u^{1} )\in H^{\frac {1}{2}}(0,1)\times H^{-\frac{1}{2}}(0,1)\), by extending \(\widehat{Q}_{T}^{k}\) to a cylindrical domain, we can obtain the same controllability result when \(\alpha_{k}(t)\in C[0.T]\). The control is the trace on the fixed endpoint of the solution defined in the cylindrical domain. The extension method also applies to the moving endpoint control in non-cylindrical domain. But by the extension method, the controllability time is greater.
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 have the following two important inequalities. The proofs of the two important inequalities are given in Section 3.
Theorem 2.2
3 Observation: proof of Theorem 2.2
In this section, in order to prove Theorem 2.2, we need the following lemmas.
Lemma 3.1
Proof
Lemma 3.2
Proof
By the above two lemmas, we obtain the following lemma concerning a growth estimate of the energy function.
Lemma 3.3
Proof
Remark 3.1
In the following, we give the proof of Theorem 2.2.
Proof of Theorem 2.2
Next, we estimate every term in the right side of (3.14).
Step 2. In the following, we give the proof of the first inequality in (2.2).
Step 3. In the following, we prove the second inequality in (2.2).
Remark 3.2
Theorem 2.2 implies that, for any \((z_{0}, z_{1})\in H^{1}_{0}(0, 1)\times L^{2}(0, 1)\), the corresponding solution z of (2.1) satisfies \(z_{x}(0, \cdot)\in L^{2}(0, T)\).
Remark 3.3
Remark 3.4
By a similar method, we obtain the following.
4 Controllability: proof of Theorem 2.1
In this section, we prove the exact controllability for the wave equation (1.2) (Theorem 2.1) by the Hilbert uniqueness method.
Proof of Theorem 2.1
We divide the proof of Theorem 2.1 into two parts.
Step 1. First, we define a linear operator \(\Lambda: H_{0}^{1}(0, 1)\times L^{2}(0, 1)\rightarrow H^{-1}(0, 1)\times L^{2}(0, 1)\).
Step 2. That Λ is an isomorphism is equivalent to the exact controllability of (1.2).
By the uniqueness of (4.1), u is the solution of (1.2) associated to \(v=z_{x}(0, \cdot)\). Furthermore, \((u(0), u_{t}(0))=(u^{0}, u^{1})\) and \((u(T), u_{t}(T))=(0, 0)\). Therefore, we get the exact controllability of (1.2). □
Declarations
Acknowledgements
This work is supported by the National Science Foundation of China 11171060, 11371084 and 11426157. Moreover, the authors are grateful to the anonymous referees for their constructive comments and suggestions, which led to an improvement of the original manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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