- Research
- Open Access
- Published:
The wave equation with internal control in non-cylindrical domains
Advances in Difference Equations volume 2017, Article number: 267 (2017)
Abstract
In this paper, we shall be concerned with interior controllability for a one-dimensional wave equation in a domain with moving boundary. When the speed of the moving endpoint is less than a certain constant which is less than the characteristic speed, we obtain exact controllability for this equation.
1 Introduction and main results
Given \(T>0\). For any \(0< k<1\), set
Also, define the following non-cylindrical domains:
and for any \(0< m< m^{\prime}< n^{\prime}< n<1\),
Consider the following controlled wave equation:
where \(v\in[H^{1}(\widehat{Q_{1}})]^{\prime}\) is control variable, u is state variable, \((u^{0}, u^{1} )\in L^{2}(0, 1)\times H^{-1}(0, 1)\) is any given initial value and \(\widehat{B}\in C_{0}^{\infty}(\widehat{Q}_{T}^{k})\),
By [1], it is easy to check that (1.2) has a unique weak solution u:
The main purpose of this paper is to study exact controllability of (1.2) in the following sense.
Definition 1.1
Equation (1.2) is called exactly controllable at the time T, if for any initial value \((u^{0}, u^{1} )\in L^{2}(0,1)\times H^{-1}(0,1)\) and any target \((u_{d}^{0}, u_{d}^{1} )\in L^{2}(0,\alpha_{k}(T))\times H^{-1}(0,\alpha_{k}(T))\), one can always find a control \(v\in[H^{1}(\widehat{Q_{1}})]^{\prime}\) such that the corresponding weak solution u of (1.2) satisfies
The main result of this paper is stated as follows.
Theorem 1.1
Suppose that \(0<\tilde{k}<1, 0<k<\tilde{k}\). For any given \(T>T^{*}_{k}\), (1.2) is exactly controllable at time T in the sense of Definition 1.1.
Remark 1.1
k̃ and \(T^{*}_{k}\) will be defined during proof of this theorem.
There is a variety of literature on interior and boundary controllability problems of wave equations in a cylindrical domain (see e.g. [2–6]). However, there is only little work of wave equations defined in non-cylindrical domains. We refer to [7–14] for some known results in this respect. In practical situations, many processes evolve in domains whose boundary has moving parts. A simple model, e.g., is the interface of ice water mixture when temperature increases. To study the controllability problem of wave equations with moving boundary or free boundary is very meaningful. In [7–13], boundary controllability for wave equations with a moving boundary has been obtained. In [7], some controllability results for wave equations with Dirichlet boundary conditions in suitable non-cylindrical domains were investigated. In [7], in the one-dimensional case, the following condition seems necessary:
In [8–13], the above condition is removed. In [14], exact controllability of a multi-dimensional wave equation with constant coefficients in a non-cylindrical domain was established, while a control entered the system through the whole non-cylindrical domain. Now we consider interior controllability for a one-dimensional wave equation with moving boundary when the moving endpoint moves along a line. Meanwhile, in our paper, we consider locally distributed control of a one-dimensional wave equation in a certain non-cylindrical domain. In order to overcome this difficulty, we transform (1.2) into an equivalent wave equation with variable coefficients in the cylindrical domain and establish exact interior controllability of this equation. In [6], a one-dimensional wave equation with variable coefficients with locally distributed control in cylindrical domains was proved. The variable coefficients are dependent of the space variable x, not dependent of the time variable t. Moreover, in the published work, variable coefficients are only dependent of space variable x in most cases. Meanwhile, in our paper, the variable coefficients are dependent of space variable x and time variable t. To solve this, motivated by [6], the key point is to construct a suitable multiplier different from that in [6].
The rest of this paper is organized as follows. In Section 2, we reduce the controllability problem of (1.2) to that of a wave equation with variable coefficients in a cylindrical domain. Section 3 is devoted to proving an observability inequality of a wave equation with variable coefficients in a cylindrical domain.
2 Reduction to controllability problems in a cylindrical domain
When \(0< k<1\), in order to prove Theorem 1.1, we first transform (1.2) into a wave equation with variable coefficients in a cylindrical domain in this section. Set
To this aim, set
Then it is easy to check that \((x, t)\) varies in Q. Also, (1.2) is transformed into the following equivalent wave equation in Q:
where
For any given initial value \((w^{0}, w^{1})\in L^{2}(0,1)\times H^{-1}(0,1)\) and any control \(\overline{v}\in[H^{1}(Q_{1})]^{\prime}\), (2.1) admits a unique weak solution
Therefore, exact controllability of (1.2) (Theorem 1.1) is reduced to the above interior controllability result for (2.1).
To prove this, we first solve the following system.
For any given \((w^{0}, w^{1})\in L^{2}(0,1)\times H^{-1}(0,1)\), this system has a unique weak solution
Then in order to obtain interior controllability of (2.1), we only prove interior controllability result for the following wave equation:
Theorem 2.1
Let \(T>T^{*}_{k}\). Then, for any target \((w_{d}^{0}, w_{d}^{1})\in L^{2}(0,1)\times H^{-1}(0,1)\), there exists a control \(\overline{v}\in[H^{1}(Q_{1})]^{\prime}\) such that corresponding weak solution η of (2.4) satisfies
Write \(H=[H^{1}(Q_{1})]^{\prime}\), \(F=L^{2}(0,1)\times H^{-1}(0,1)\) and \(F'=H_{0}^{1}(0, 1)\times L^{2}(0, 1)\). Define a linear operator A:
Then A is surjective is equivalent to interior controllability of the wave equation (2.4). And A is surjective is derived from an observability inequality of the form
for the dual operator \(A^{\prime}: F^{\prime}\rightarrow H^{\prime}\) for \(T>T^{*}_{k}\).
3 Observability inequality of wave equations with variable coefficients
First, we define \(A^{\prime}\). \(A^{\prime}\) is described by the following homogeneous wave equation:
where \(k\in(0, 1)\), \((z^{0}, z^{1})\in H_{0}^{1}(0, 1)\times L^{2}(0, 1)\) is any given initial value, and \(\alpha_{k}\), \(\beta_{k}\) and \(\gamma_{k}\) are given in (2.2). Equation (3.1) has a unique weak solution,
Set \(B^{\prime}\) the adjoint of the extension operator B in (1.2), and if \(\overline{v}\in[H^{1}(Q_{1})]^{\prime}\), then \(B^{\prime}: H^{1}(Q)\rightarrow H^{1}(Q_{1})\). Hence \(A^{\prime}\) is defined as follows:
where z is the solution of (3.1). Therefore, (2.5) is equivalent to the following inequality:
In the following, we shall give a proof of (3.2) by the multiplier method.
Define the following weighted energy for (3.1):
It follows that
We obtain the following lemma (see the detailed proof in [8]).
Lemma 3.1
For any \((z^{0}, z^{1})\in H^{1}_{0}(0, 1)\times L^{2}(0, 1)\) and \(t\in[0, T]\), any solution z of (3.1) satisfies the following estimate:
Equation (3.2) is derived with a special multiplier. Set
It is easy to check \(F_{x}(x,k)<0,(x,k)\in[0,1]\times(0,1)\). We have
We see \(F_{k}(1,k)= \frac{-4k}{(1-k^{2})^{2}}<0, k\in(0,1)\). Therefore, we obtain, for any \(\eta>0\),
Hence, we derive
Assume that \(\lambda\in(0,1)\) and a point \(x_{0}\in(m,n)\) to be unknown for now. We require the multiplier to satisfy the following lemma.
Lemma 3.2
Assume that \(p(x)\) be a solution of first-order linear differential equation
then there exist a unique \(\lambda\in(0,1)\) and a unique \(x_{0}\in(m,n)\) such that \(p(x)\) belongs to \(C[0,1]\) and satisfies
Proof
By (3.5), (3.6) and the constant variation method, it follows that
Note that \(p(x)\in C[0,1]\), we have
From this, we obtain
By (3.7) and (3.8), we have \(\lambda_{-}\) and \(\lambda_{+}\) are monotone decreasing and increasing with respect to \(x_{0}\) and satisfy
Hence there exists a unique \(x_{0}\in(m,n)\) of the equation \(\lambda_{-}=\lambda_{+}\), the corresponding value
 □
Remark 3.1
It is easy to verify that
In the following, we prove (3.2) by the above multiplier \(p(x)\). Multiplying the first equation of (3.1) by \(qz_{x}\) and integrating on Q, we have
In the following, we calculate the above three integrals \(D_{i}\ (i=1, 2, 3)\), respectively. It is easy to check that
and
Write
By Lemma 3.2, it follows that
and
Then we have
and
From this, we have
Similarly, we obtain
for any \(\eta>0\), \(k\in(0,1-\eta]\),
and
By (3.13)-(3.17), it follows that for any \(\eta>0\), \(k\in(0,1-\eta]\),
Therefore, we have, for any \(\eta>0\), \(k\in(0,1-\eta]\),
For each \(t\in[0, T]\) and \(\varepsilon>0\), we have
Take \(\varepsilon= \frac{1-k}{\sqrt{1+kt}M}\), then it is easy to check that
This implies that, for any \(t\in[0, T]\),
It follows that
For each \(\varepsilon\in (0, 1-\lambda )\), we have
Define
We see
By (3.20), we obtain, for each \(\varepsilon\in (0, 1-\lambda )\),
Take \(\varepsilon= \sqrt{M_{1}}<1-\lambda\), then it is easy to check that
i.e.,
From the above inequality, it follows that
From (3.21), we get
Write
By (3.3), (3.18), (3.20) and (3.22), we derive, for each \(k\in(0,\tilde{k})\),
Set
If \(T>T^{*}_{k}\), we have \(\frac{1-\lambda-\sqrt{M_{1}}}{k}\ln(1+kT)- \frac{2M}{1-k}>0\). Also,
Equation (3.2) is deduced by (3.24).
Remark 3.2
We can finally check that
It is well known that (1.2) in the cylindrical domain is interiorly controllable at any time \(T>T^{0}\). However, we do not know whether \(T^{*}_{k}\) is sharp.
4 Conclusions
In this paper, we consider interior controllability for a one-dimensional wave equation in a domain with moving boundary. When the speed of the moving endpoint is less than a certain constant which is less than the characteristic speed, we obtain exact controllability for this equation. In the future, we hope that we may consider controllability problem of wave equations with free boundary.
References
Lasiecka, I, Lions, JL, Triggiani, R: Non homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. 65, 149-192 (1986)
Lions, JL: Exact controllability, stabilizability and perturbation for distributed systems. SIAM Rev. 30, 1-68 (1988)
Zuazua, E: Exact controllability for semilinear wave equation in one space dimension. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, 109-129 (1993)
Yao, P: On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37, 1568-1599 (1999)
Fu, X, Yong, J, Zhang, X: Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46, 1578-1614 (2007)
Ho, LF: Exact controllability for the one-dimension wave equation with locally distributed control. SIAM J. Control Optim. 28, 733-748 (1990)
Miranda, MM: Exact controllability for the wave equation in domains with variable boundary. Rev. Mat. Univ. Complut. Madr. 9, 435-457 (1996)
Cui, L, Liu, X, Gao, H: Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. 402, 612-625 (2013)
Cui, L, Song, L: Exact controllability for a wave equation with fixed boundary control. Bound. Value Probl. 2014, 47 (2014)
Cui, L, Gao, H: Exact controllability for a wave equation with mixed boundary conditions in a non-cylindrical domain. Electron. J. Differ. Equ. 2014, 101 (2014)
Cui, L, Song, L: Controllability for a wave equation with moving boundary. J. Appl. Math. 2014, 827698 (2014)
Cui, L, Jiang, Y, Wang, Y: Exact controllability for a string equation in domains with moving boundary in one dimension. Bound. Value Probl. 2015, 208 (2015)
Sun, H, Li, H, Lu, L: Exact controllability for a one-dimensional wave equation with the fixed endpoint control. Electron. J. Differ. Equ. 2015, 98 (2015)
Bardos, C, Chen, G: Control and stabilization for the wave equation, part III: domain with moving boundary. SIAM J. Control Optim. 19, 123-138 (1981)
Debbouche, A, Baleanu, D: Exact null controllability for fractional nonlocal integrodifferential equations via implicit evolution system. J. Appl. Math. 2012, Article ID 931975 (2012)
Kerboua, M, Debbouche, A, Baleanu, D: Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. Abstr. Appl. Anal. 2013, Article ID 262191 (2013)
Kerboua, M, Debbouche, A, Baleanu, D: Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces. Electron. J. Qual. Theory Differ. Equ. 2014, 58 (2014)
Acknowledgements
The authors are deeply grateful to the anonymous referee and the editor for their careful reading, valuable comments and correcting some errors, which have greatly improved the quality of the paper. This work is partially supported by the NSF of China under grants 11171060, 11371084.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Cui, L. The wave equation with internal control in non-cylindrical domains. Adv Differ Equ 2017, 267 (2017). https://doi.org/10.1186/s13662-017-1284-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-017-1284-1
Keywords
- interior controllability
- wave equation
- non-cylindrical domains