Controllability for a Wave Equation with Moving Boundary
We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than , by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint.
1. Introduction and Main Results
This paper concerns a finite vibrating string described by a wave equation. The left boundary endpoint of the string is fixed, while the right boundary endpoint is moving. Given that , write for the following noncylindrical domain: where
Consider the following wave equation in the noncylindrical domain : where is the control variable and is put on fixed endpoint. The constant is called to be the speed of the moving endpoint. By , for , any , and , (3) admits a unique transposition solution.
The exact controllability problem for (3) is formulated as follows: given large enough. For each and for each , is it possible to find a control such that the corresponding solution of (3) satisfies
The main purpose of this paper is to study the exact controllability of (3). As we all know, there exist pieces of literature on the controllability problems of wave equations in a cylindrical domain. However, as far as we know, there are only a few works on the exact controllability for wave equations defined in noncylindrical domains. We refer to [1–3] for some known results in this respect. In , the boundary controllability problem for a multidimensional wave equation with constant coefficients in a noncylindrical domain was discussed. However, in  in the one-dimensional case, the following condition seems necessary: It is easy to check that this condition is not satisfied for the moving boundary in (3). In , the exact controllability of a multidimensional wave equation with constant coefficients in a noncylindrical domain was established, while a control entered the system through the whole noncylindrical domain. In , the exact controllability of the following system is studied: But the control is put on moving endpoint. In order to overcome these difficulties and drop the additional conditions for the moving boundary, we use sidewise energy estimates method to obtain observability inequality.
The main result of this paper is stated as follows.
Theorem 1. Suppose that . For any given , (3) is exactly controllable at time .
Remark 2. will be defined during the course of the proof.
Remark 3. It seems natural to expect that the exact controllability of (3) holds when . However, we do not succeed in extending the approach developed in Theorem 1 to this case.
In order to prove Theorem 1, we first transform (3) into a wave equation with variable coefficients in a cylindrical domain. For this aim, set Then, it is easy to check that varies in . Also, (3) is transformed into the following equivalent wave equation in the cylindrical domain : where Equation (8) admits a unique solution in the sense of transposition (see ).
Therefore, the exact controllability of (3) (Theorem 1) is reduced to the following controllability result for the wave equation (8).
Theorem 4. Suppose that . Let . Then, for any initial value and target , there exists a control such that the corresponding solution of (8) in the sense of transposition satisfies
In order to obtain Theorem 4, we will use Hilbert uniqueness method. The main idea is to define a weighted energy function for the following wave equation with variable coefficients in cylindrical domains: where , is any given initial value, and , , and are the functions given in (8). Similar to Theorem 3.2 in , we have that (12) has a unique weak solution
In the sequel, we denote by a positive constant depending only on and , which may be different from one place to another.
The energy function of system (12) is defined as In particular Note that this weighted energy is different from the usual one, but they are equivalent. We will obtain explicit energy equality. Using this energy equality, we will first prove the following observability result.
Theorem 5. Let . For any , there exists a constant such that the corresponding solution of (12) satisfies
Then, applying Hilbert uniqueness method, we will deduce Theorem 4.
The rest of this paper is organized as follows. In Section 2, we derive Theorem 5. Section 3 is devoted to a proof of Theorem 4.
2. Observability: Proof of Theorem 5
In this section, first we give the following two lemmas (see the detailed proof in ).
Lemma 6. For any and , any solution of (12) satisfies the following estimate:
Lemma 7. Suppose that is any given function. Then any solution of (12) satisfies the following estimate:
Now, we give a proof of Theorem 5.
Proof. We first give the proof of the second inequality in (16).
Note that The derivative of the functional of is where By (12), it follows that from which and using integrating by parts, we have that We conclude, using (24), that
We will choose later which satisfies From (25) and (26), for , it concludes that Take ; then it is easy to check that Hence By Gronwall inequality, there exists such that Integrating (30) in , we have By (17), we deduce that Now choose that also satisfies Then from (19), (31), and (32), it follows that from which and from (20), we have that Let When , by (35), (16) follows.
In the following, when , we choose which satisfies (33) and (26). In (26), for , we have that Assume that We must take and satisfies From (39), we derive Let . Then it follows that that is, By (42), we deduce that , Integrating into , we have
Hence, we choose which satisfies (33) and (26). It follows that From the definition of (see (36)) and (46), we deduce that ,
In the following, we give the proof of the second inequality in (16).
We choose for in (18). Noting that , , and , it follows that
Next, we estimate every term in the right side of (48). Notice that and for any . By (17), we have
On the other hand, for each , it holds that Therefore, by (48)–(50), we have
Remark 8. Theorem 5 implies that, for any , the corresponding solution of (12) satisfies .
3. Controllability: Proof of Theorem 4
In this section, we prove the exact controllability for the wave equation (8) in the cylindrical domain (Theorem 4) by Hilbert uniqueness method. The main idea is to seek a control in the form , where is the solution of (12) for some suitable initial data. The other proof is similar to the proof of Theorem 2.1 in .
Remark 9. It is easy to check that It is well known that the wave equation (3) in the cylindrical domain is null controllable at any time . However, we do not know whether the controllability time is sharp.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by the NSF of China under Grants 11171060 and 11371084 and Department of Education Program of Jilin Province under Grants 2012187 and 2013287.
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