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Exact controllability for a wave equation with fixed boundary control

Lizhi Cui12* and Libo Song3

Author Affiliations

1 College of Applied Mathematics, Jilin University of Finance and Economics, Changchun, 130117, China

2 School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

3 Educational Administration, Jilin University of Finance and Economics, Changchun, 130117, China

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Boundary Value Problems 2014, 2014:47  doi:10.1186/1687-2770-2014-47

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/47


Received:24 August 2013
Accepted:17 February 2014
Published:24 February 2014

© 2014 Cui and Song; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper addresses the study of the 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 one moving. When the speed of the moving endpoint is less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M1">View MathML</a>, by the Hilbert Uniqueness Method, the exact controllability of this equation is established. Also, the explicit dependence of the controllability time on the speed of the moving endpoint is given.

Keywords:
exact controllability; non-cylindrical domain; wave equation

1 Introduction

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M2">View MathML</a>. Let us consider the non-cylindrical domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M3">View MathML</a>, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M4">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M5">View MathML</a>

(1.1)

Consider the following controlled wave equation in the non-cylindrical domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M6">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M7">View MathML</a>

(1.2)

where u is the state variable, v is the control variable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M8">View MathML</a> is any given initial value. Equation (1.2) may describe the motion of a string with a fixed endpoint and a moving one. The constant k is called the speed of the moving endpoint. By [1], for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M9">View MathML</a>, any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M11">View MathML</a>, (1.2) admits a unique solution in the sense of a transposition.

The main purpose of this paper is to study the exact controllability of (1.2). As we all know, there exists much literature on the controllability problems of wave equations in a cylindrical domain. However, there are only a few works on the exact controllability for wave equations defined in non-cylindrical domains. We refer to [1-3], and [4] for some known results in this respect. In [2], the exact controllability of a multi-dimensional wave equation with constant coefficients in a non-cylindrical domain was established, while the control entered the system through the whole non-cylindrical domain. In [1] and [3], some controllability results for the wave equations with Dirichlet boundary conditions in suitable non-cylindrical domains were investigated, respectively. But some additional conditions on the moving boundary were required, which entail the method used in [1] and [3] not to be applicable to the controllability problems of (1.2). In [1] and [3] in the one-dimensional case, the following condition seems necessary:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M12">View MathML</a>

It is easy to check that this condition is not satisfied for the moving boundary in (1.2). The control system of this paper is similar to that of [4]. But the control is put on a different boundary. We mainly use the multiplier method to overcome these difficulties and drop the additional conditions for the moving boundary. But the simple multiplier in [4] is not applicable to the controllability problem of (1.2). We choose the complicated multiplier which satisfies the first-order linear differential equation. But the result in this paper is not satisfactory. We hope that the controllability result is obtained when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M13">View MathML</a>. We hope that we obtain a modified multiplier in the forthcoming papers.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and the main results. In Section 3, we prove that the Hilbert Uniqueness Method (HUM) works very well for (1.2). Section 4 contains the proofs of the important inequalities used in Section 3.

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 Equation (1.2) is called exactly controllable at the time T, if for any initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M10">View MathML</a> and any target <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M15">View MathML</a>, one can always find a control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M11">View MathML</a> such that the corresponding solution u of (1.2) in the sense of a transposition satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M17">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M18">View MathML</a> for a controllability time. The main result of this paper is stated as follows.

Theorem 2.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M19">View MathML</a>. For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20">View MathML</a>, (1.2) is exactly controllable at timeTin the sense of Definition 2.1.

Remark 2.1 It seems natural to expect that the exact controllability of (1.2) holds when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M13">View MathML</a>. However, we did not have success in extending the approach developed in Theorem 2.1 to this case.

In order to prove Theorem 2.1, we first transform (1.2) into a wave equation with variable coefficients in a cylindrical domain. To this aim, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M22">View MathML</a>

Then, it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M23">View MathML</a> varies in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M24">View MathML</a>. Also, (1.2) is transformed into the following equivalent wave equation in the cylindrical domain Q:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M25">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M26">View MathML</a>

(2.2)

Equation (2.1) admits a unique solution in the sense of a transposition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M27">View MathML</a>

(see [5]).

Therefore, the exact controllability of (1.2) (Theorem 2.1) is reduced to the following main controllability result for the wave equation (2.1).

Theorem 2.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M19">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20">View MathML</a>. Then, for any initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M30">View MathML</a>and target<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M31">View MathML</a>, there exists a control<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M32">View MathML</a>such that the corresponding solutionwof (2.1) in the sense of transposition satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M33">View MathML</a>

The key proof of Theorem 2.2 is to prove two important inequalities for the following homogeneous wave equation in cylindrical domains. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M34">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M36">View MathML</a> is any given initial value, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M38">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M39">View MathML</a> are the functions given in (2.1). Similar to Theorem 3.2 in [5], we see that (2.3) has a unique weak solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M40">View MathML</a>

Define the following weighted energy for (2.3):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M41">View MathML</a>

where z is the solution of (2.3). It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M42">View MathML</a>

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 obtain the following two lemmas whose proof are found in [4].

Lemma 2.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M45">View MathML</a>

(2.4)

Lemma 2.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M46">View MathML</a>is any given function. Then any solutionzof (2.3) satisfies the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M47">View MathML</a>

(2.5)

In order to prove Theorem 2.2, we need the following two important inequalities. The proofs of two important inequalities are given in Section 4.

Theorem 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M2">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M36">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M50">View MathML</a>such that the corresponding solutionzof (2.3) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M51">View MathML</a>

(2.6)

Theorem 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M36">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M50">View MathML</a>such that the corresponding solutionzof (2.3) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M55">View MathML</a>

(2.7)

3 Application of HUM

In this section, we prove the exact controllability for the wave equation (2.1) in the cylindrical domain Q (Theorem 2.2) by HUM.

Proof of Theorem 2.2 We divide the proof of Theorem 2.2 into three parts. We use certain inequalities proved later in Section 4.

Step 1. First, we define a linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M56">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M43">View MathML</a>, denote by z the corresponding solution of (2.3). Consider the following homogeneous wave equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M58">View MathML</a>

(3.1)

Then it is well known that (3.1) admits a unique solution in the sense of a transposition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M59">View MathML</a>

Moreover, by Theorem 2.3 in Section 3, there exists a constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M60">View MathML</a>

(3.2)

Define a linear operator Λ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M61">View MathML</a>

where we use z to denote the solution of (2.3) associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M63">View MathML</a>, and η denotes the solution of and (3.1) associated to z.

Step 2. That Λ is an isomorphism is equivalent to the exact controllability of (2.1).

In fact, for any target <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M64">View MathML</a>, the following wave equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M65">View MathML</a>

(3.3)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M66">View MathML</a>.

Suppose that Λ is an isomorphism, for any initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M67">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M43">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M69">View MathML</a>

(3.4)

Note that η is the solution of (3.1) and that z is the solution of (2.3) associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M70">View MathML</a>. Then, by the definition of Λ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M71">View MathML</a>

(3.5)

By (3.4) and (3.5), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M72">View MathML</a>. If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M73">View MathML</a>, by the uniqueness of (3.3), then w is the solution of (2.1) associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M74">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M76">View MathML</a>. Therefore, we get the exact controllability of (2.1).

Step 3. Now we prove that Λ is an isomorphism, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20">View MathML</a>.

Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M78">View MathML</a> and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M79">View MathML</a> its conjugate space. Also, define a bilinear form A on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M80">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M81">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M82">View MathML</a>, where η denotes the solution of (3.1).

Multiplying the first equation of (3.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M83">View MathML</a> and integrating on Q, by (2.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M84">View MathML</a>

Combining the above equality with the definition of Λ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M85">View MathML</a>

(3.6)

By Theorem 2.3 and Theorem 2.4, it suffices to prove that Λ is surjective. Notice that Theorem 2.4 and (3.6) imply A is a coercive bilinear form. Moreover, by (3.2), it is easy to check that A is bounded. Therefore, by the Lax-Milgram Theorem, Λ is a surjection. It follows that Λ is an isomorphism. □

Remark 3.1 By the equivalent transformation in Section 2, Theorem 2.2 implies the exact controllability for in the non-cylindrical domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M3">View MathML</a> at the time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20">View MathML</a> (Theorem 2.1).

4 The proofs of important inequalities

In this section, we give proofs of Theorem 2.3 and Theorem 2.4.

Proof of Theorem 2.3 First, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M88">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M89">View MathML</a> in (2.5). Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M92">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M93">View MathML</a>

(4.1)

Next, we estimate every terms on the right side of (4.1). Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M95">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M96">View MathML</a>. By (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M97">View MathML</a>

(4.2)

On the other hand, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M99">View MathML</a>

(4.3)

Therefore, by (4.1)-(4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M100">View MathML</a>

 □

Remark 4.1 Theorem 2.3 implies that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M101">View MathML</a>, the corresponding solution z of (2.3) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M102">View MathML</a>.

In the following, we give a proof of Theorem 2.4.

Proof of Theorem 2.4 First, let q be the solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M103">View MathML</a>

(4.4)

e.g.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M104">View MathML</a>

It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M105">View MathML</a>

(4.5)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M106">View MathML</a>

then it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M107">View MathML</a>

By (2.5) and (4.5), after calculating, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M108">View MathML</a>

(4.6)

From (4.6), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M109">View MathML</a>

(4.7)

Next, we estimate the terms on the right side of (4.7). For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M111">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M112">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M113">View MathML</a>, then it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M114">View MathML</a>

This implies that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M44">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M116">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M117">View MathML</a>

(4.8)

On the other hand, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M118">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M119">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M120">View MathML</a>, then it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M121">View MathML</a>

By the value of M, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M122">View MathML</a>

From this, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M123">View MathML</a>

Then when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M124">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M125">View MathML</a>

(4.9)

Hence, by (4.7)-(4.9), we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M126">View MathML</a>

(4.10)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M127">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M20">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M129">View MathML</a>. Also,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M130">View MathML</a>

From this we get (2.7). This completes the proof of Theorem 2.4. □

Remark 4.2 It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M131">View MathML</a>

It is well known that the wave equation (1.2) in the cylindrical domain is null controllable at any time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M132">View MathML</a>. However, we do not know whether the controllability time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/47/mathml/M18">View MathML</a> is sharp.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All results belong to CL and SL. All authors read and approved the final manuscript.

Acknowledgements

This work is supported by the National Science Foundation of China 11171060, 11371084 and Department of Education Program of Jilin Province under grants 2012187 and 2013287. Moreover the authors are grateful to anonymous referees for their constructive comments and suggestions, which led to improvement of the original manuscript. The authors are grateful to Christopher D. Rualizo for patient and meticulous work.

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