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The controllability for the semi-discrete wave equation with a finite element method
Advances in Difference Equations volume 2013, Article number: 160 (2013)
In this paper, we study the controllability problem of the semi-discrete internally controlled one-dimensional wave equation with the finite element method. We derive the observability inequality and prove the exact controllability for the semi-discrete internally controlled wave equation, with the controls taken from a finite dimensional space.
In this paper, we discuss the topic related to the controllability for the semi-discrete internally controlled one-dimensional wave equations. First of all, we introduce certain notations. Let ω be an open and nonempty subset of . Define an operator by
for any . Let . The controlled wave equation, which we study in this paper, is as follows:
where the initial value belongs to and is a control function taken from the space .
System (1.1) is said to be exactly controllable from the initial value in time T if there exists a control function such that the solution of (1.1) matches that . We have already known that the controllability property for the above continuous one-dimensional wave equation holds for any given (see ).
In this work, we mainly focus on the issue of the controllability property of (1.1) under numerical approximation schemes with the finite element method. To this end, now we introduce the basis functions of the finite element space. Let h be a small enough positive number. Corresponding to each given h, we take nodal points , with , over the interval such that
Then we can define the basis function by
Corresponding to the state space , we build the finite element space as
Obviously, it is a subspace of . Let be the -projection from to , namely
Corresponding to the control space , we define the finite element space by
Throughout this paper, will be treated as an operator from to , by setting for all . Clearly, is a subspace of .
Write for the restriction of the operator E over , and project equation (1.1) into the following controlled ordinary differential equations:
Here, the initial value belongs to , the control is taken from the space , and the operator is defined by
In this paper, we mainly deal with the controllability for semi-discrete system (1.4). The main result of the paper is presented as follows.
Theorem 1.1 For each , controlled system (1.4) is exactly controllable in time T. Namely, there exists a control function such that the solution of (1.4) satisfies .
Remark 1.2 In this paper, our main purpose is to discuss whether or not the semi-discrete internally controlled systems have the exactly controllable property which the original controlled systems have. This is a very valuable problem in control theory. In , the authors established such a controllability result for the semi-discrete one-dimensional boundary controlled wave equation by the numerical approximation method. The authors also got that the semi-discrete systems are not uniformly controllable as the discretization parameter h goes to zero. The main differences between  and our paper are as follows. In , the authors focused on a one-dimensional boundary controlled wave equation and they obtained the controllability for the semi-discrete wave system, with the controls taken from an infinite dimensional space. In our paper, we discuss an internally controlled one-dimensional wave equation. In this case, we obtain the exact controllability of the semi-discrete wave equation, with the controls taken from a finite dimensional space. Regarding other works related to this problem, we would like to mention [3, 4], and .
The rest of the paper is structured as follows. In Section 2, we give a sufficient condition for controllability. By making use of this sufficient condition presented in Section 2, we provide the proof of Theorem 1.1 in Section 3.
2 The controllability and observability property
We first introduce the following auxiliary system.
where the initial data . Clearly, it is a controlled system governed by ordinary differential equations. However, the control functions for this system are taken from the infinite dimensional space .
In this section, we discuss some controllability result for system (2.1). More concisely, a sufficient condition for the exact controllability property of (2.1) will be presented. The proofs of the following Lemmas 2.1, 2.2 and 2.4 can be found in .
For any , consider the following homogeneous equation:
Lemma 2.1 The control drives the initial data of controlled system (2.1) to zero in time T if and only if
for all , where is the corresponding solution of equation (2.2).
Next, we define a functional from to ℝ by setting
where is the solution of (2.2) with initial data .
We have the following result.
Lemma 2.2 Suppose that is a minimizer of . If is the corresponding solution of equation (2.2) with initial , then
is a control which drives the initial data of controlled system (2.1) to zero in time T.
To get the sufficient condition that ensures the existence of a minimizer for , we need to give the following definition.
Definition 2.3 Equation (2.2) is observable in time T if there exists a positive constant L such that the following inequality holds:
for any , where is the solution of (2.2) with initial data .
Inequality (2.6) is called observability inequality. The following conclusion shows that observability inequality (2.6) is the sufficient condition for the exact controllability of system (2.1).
Lemma 2.4 Suppose equation (2.2) is observable in time T. Then the functional defined by (2.4) has a unique minimizer .
3 The proof of Theorem 1.1
Before giving the proof of the main result, we first present some preliminary lemmas. Assume that all distinct eigenvalues of the operator are , . For any given eigenvalue , , let be its multiplicity and be its eigenspace, with an orthogonal basis . It is easy to see that the family
forms an orthogonal basis of . The following two results are quoted from . They will be used later.
Lemma 3.1 For any non-zero vector in the space , we have , where , , is a normalized eigenfunction in the eigenspace , and are real numbers satisfying , where denotes the usual norm of .
Theorem 3.2 Suppose that is an eigenfunction to the operator , and ω is an open and nonempty subset of . Then .
Now, we will prove the controllability for system (2.1).
Theorem 3.3 For each , the solution of semi-discrete system (2.2) satisfies the following inequality:
Remark 3.4 Since is a finite dimensional space, thus all norms of this space are equivalent, and then we can get that inequality (3.1) implies observability of semi-discrete system (2.2).
Proof For any given , consider the following function defined by
where is the solution of (2.2) with initial data . Clearly, F is continuous. To prove inequality (3.1), it suffices to show that we can find a positive constant , where only depends on h and T, such that
To this end, we will give a proof by contradiction. Suppose that there is a unit vector in such that =0. Since , at least one of and is not 0. Without loss of generality, we can assume that . According to Lemma 3.1, can be presented by
where , with , is a normalized eigenfunction in the eigenspace , and are real numbers satisfying
Now, noting that is an dimensional space, we can choose which are normalized eigenfunctions of such that constitute a complete standard orthogonal basis of . Let be corresponding eigenvalues to eigenfunctions .
Now, for , we can write
and we can rewrite
where . By the classical theory of ODEs, equation (2.2) has a unique solution
This, together with the definition of the function F and the assumption that , indicates that
Hence, we necessarily have that
In the following, we are going to prove that
which leads to a contradiction to the assumption that , and then we can complete the proof of (3.2).
By taking in (3.4) and noting , we can get
where we use the fact that . Calculating the derivations twice to (3.4) and taking , we can get
Thus, by induction, we can get that
Noting that , , are the eigenfunctions of , we can see from Theorem 3.2 that for . Therefore, we can assume, without loss of generality, that with are linear independent in , and
With regard to the number α, there are only two possibilities: it either is equal to q or is less than q. If , (3.5) follows immediately from (3.6). If , we have the following presentation:
where , , , are real numbers. Since for all , we derive, from (3.7), the following fact:
On the other hand, combining (3.6) with (3.7) leads to
Since are linear independent in , it follows from the above identity that
Because are distinct positive numbers, we can deduce immediately from (3.9) that
In the above second identity, corresponding to each , we can take s as the number given in (3.8). Then it follows that for all . Hence, we prove (3.5) and finish the proof for this theorem. □
Proof of Theorem 1.1 According to Lemma 2.2, Lemma 2.4, and Theorem 3.3, we have that system (2.1) is controllable in time T. Suppose that is a minimizer of . If is the corresponding solution of equation (2.2) with initial data , then
is a control which drives the initial data of controlled system (2.1) to zero in time T. It is easy to see that . This completes the proof of this theorem. □
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The authors would like to thank professor Gengsheng Wang for his valuable suggestions on this paper. This work was partially supported by the National Natural Science Foundation of China (U1204105, 61203293), the Natural Science Foundation of Zhejiang (Y6110751), the Natural Science Foundation of Ningbo (2010A610096), the Key Foundation of Henan Educational Committee (13A120524, 12B120006), and the Fundamental and Frontier Technology Research Projects of Henan Province (132300410285).
The authors declare that they have no competing interests.
GZ provided the questions. GZ and XY gave the proof for the main result together. All authors read and approved the final manuscript.
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Zheng, G., Yu, X. The controllability for the semi-discrete wave equation with a finite element method. Adv Differ Equ 2013, 160 (2013). https://doi.org/10.1186/1687-1847-2013-160
- wave equation
- finite element approximation
- observability inequality