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The controllability for the internally controlled 1-D wave equation via a finite difference method
- Juntao Li^{1, 2},
- Yuhan Zheng^{1} and
- Guojie Zheng^{1}Email author
https://doi.org/10.1186/s13662-016-1028-7
© The Author(s) 2016
- Received: 22 March 2016
- Accepted: 15 November 2016
- Published: 24 November 2016
Abstract
In this paper, we study the controllability of the semi-discrete internally controlled 1-D wave equation by using the finite difference method. In the discrete setting of the finite difference method, we derive the observability inequality and get the exact controllability for the semi-discrete internally controlled wave equation in the one-dimensional case. Then we also analyze whether the uniform observability inequality holds for the adjoint system as \(h\rightarrow0\).
Keywords
- wave equation
- finite difference
- observability inequality
MSC
- 49K20
- 35J65
1 Introduction
Problem (1.1) is said to be exactly controllable from the initial value \((y_{0},y_{1})\in H^{1}_{0}(0,1)\times L^{2}(0,1)\) in time T if there exists a control function \(u(\cdot)\in L^{2}(0,T;L^{2}(0,1))\), such that the solution of (1.1) satisfies \((y(T),\partial_{t}y(T))=(0,0)\). The problem of the controllability of wave equations has also been the object of numerous studies. Extensive related references can be found in [1–3] and the rich work cited therein.
Remark 1.1
In this paper, we will study whether the inequality (1.5) for adjoint system (1.4) holds. We are also interested in whether the constant \(C(T,h)\) is bounded as \(h\rightarrow0\). The main results of the paper are presented as follows.
Remark 1.2
Theorem 1.1 shows that the semi-discrete system (1.2) or (1.3) is controllable for any time \(T>0\).
Theorem 1.2
To the best of our knowledge, Infante and Zuazua made the first study of this topic in [5]. They studied a controllability result for the semi-discrete 1-D wave equation with boundary control. However, the uniform controllability for the semi-discrete systems in [5] cannot be derived as the discretization parameter \(h\rightarrow0\). The main differences between [5] and our paper are as follows. In [5], the authors focused on a one-dimensional boundary controlled wave equation, and we mainly study the internally controlled 1-D wave equation. In this case, the controller is more complicated than the case with controller on boundary. Regarding other works on this subject, we mention [4, 7, 8] and [6].
The paper is organized as follows: Section 2 briefly describes some preliminary results on the finite difference scheme. The proofs of Theorem 1.1 and Theorem 1.2 are provided in Section 3.
2 The finite difference scheme
Lemma 2.1
- (i)For any eigenvector \(w=(w_{1},w_{2},\ldots,w_{N})^{T}\) with eigenvalue \(\lambda(h)\) of matrix \(A_{h}\), the following identity holds:where \(w_{0}=w_{N+1}=0\).$$\begin{aligned} \sum_{j=0}^{N}\biggl|\frac{w_{j}-w_{j+1}}{h}\biggr|^{2}= \lambda(h)\sum_{j=1}^{N}|w_{j}|^{2}, \end{aligned}$$(2.4)
- (ii)If \(w_{k}=(w_{k,1},w_{k,2},\ldots,w_{k,N})^{T}\) and \(w_{l}=(w_{l,1},w_{l,2},\ldots,w_{l,N})^{T}\) are eigenvectors associated to eigenvalue \(\lambda_{k}\), \(\lambda_{l}\), and \(\lambda_{k}\neq\lambda_{l}\), then we havewhere \(w_{k,0}=w_{k,N+1}=0\), and \(w_{l,0}=w_{l,N+1}=0\).$$\begin{aligned} \sum_{j=0}^{N}(w_{k,j}-w_{k,j+1}) (w_{l,j}-w_{l,j+1})=0, \end{aligned}$$(2.5)
This lemma is quoted from [5].
Lemma 2.2
Proof
Especially, we can get the following property for the eigenvectors for the matrix \(A_{h}\), which will play a key role in the proof of the main results.
Proposition 2.1
Proof
In summary, taking \(L=\min\{\frac{b-a}{4}, \frac{M}{2}, L_{1}\}\), we can complete the proof of this conclusion. □
Remark 2.1
This theorem gives a fundamental property for the unit eigenvectors \(\vec{w}_{i}^{h}\) (\(i=1,\ldots,N\)) of the discrete Laplacian operator. It shows that the energy for these unit eigenvectors have an uniform lower boundary, which is positive and not dependent on h, in a nonempty and open subset \(\omega\subset(0,1)\).
Lemma 2.3
This lemma can easily be deduced from Lemma 2.1.
Remark 2.2
- (i)
According to Lemma 2.3, it is easy to find that the spaces \(X_{0}^{h}\) and \(X_{1}^{h}\) are both Banach spaces. In fact, \(X_{0}^{h}\) and \(X_{1}^{h}\) can be regarded as the discrete version of the space \(L^{2}(0,1)\) and \(H_{0}^{1}(0,1)\), respectively. Thus, Lemma 2.3 can be regarded as the discrete version of Poincaré’s inequality.
- (ii)Since \(\mathbb{R}^{N}\times\mathbb{R}^{N}\) is a finite dimensional space, thus all norms of this space are equivalent. In particular, there exist positive numbers \(C_{1}\), \(C_{2}\), such thathold for any \((z_{1},z_{2})\in\mathbb{R}^{N}\times\mathbb{R}^{N}\).$$\begin{aligned} C_{1}\bigl\| (z_{1},z_{2}) \bigr\| _{X_{0}^{h}\times X_{1}^{h}}\leq\bigl\| (z_{1},z_{2})\bigr\| _{\mathbb{R}^{N}\times\mathbb{R}^{N}}\leq C_{2}\bigl\| (z_{1},z_{2})\bigr\| _{X_{0}^{h}\times X_{1}^{h}} \end{aligned}$$(2.13)
3 The proof of Theorem 1.1 and Theorem 1.2
3.1 The proof of Theorem 1.1
Proof
From (3.1) with (2.13), it is easy to obtain the observability inequality (1.5) of the semi-discrete system (1.4). This completes the proof of this theorem. □
3.2 The proof of Theorem 1.2
Proof
Declarations
Acknowledgements
The authors would like to thank the reviewers for their valuable comments and insightful suggestions. This work was partially supported by the National Natural Science Foundation of China (61203293, 61374079), Program for Science and Technology Innovation Talents in Universities of Henan Province (13HASTIT040), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (14IRTSTHN023), Henan Higher School Funding Scheme for Young Teachers (2012GGJS-063).
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Authors’ Affiliations
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