Exact Boundary Observability and Controllability of the Wave Equation in an Interval with two Moving Endpoints

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate $1/t$. We also establish observability results, at one or two endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.


Introduction
The wave equation is a simple mathematical model describing the transverse small vibrations of a homogeneous string under tension and constrained to move in a plane. Let T > 0. When the string endpoints are clamped and its length 0 > 0 is invariant with time, this model can be stated as    w tt − w xx = 0, for (x, t) ∈ (0, 0 ) × (0, T ) , w (0, t) = w ( , t) = 0, for t ∈ (0, T ) , w(x, 0) = w 0 , w t (x, 0) = w 1 , for x ∈ (0, 0 ) , see [21]. The function x → w (x, t) , x ∈ (0, 0 ) describes the shape of the string at time t. It is well known that for initial data satisfying w 0 ∈ H 1 0 (0, 0 ) , w 1 ∈ L 2 (0, 0 ) , the solution of this problem is unique and enjoys, in particular, the following properties : • The "energy" of the solution, given by is a conserved quantity, i.e. E 0 (t) = Constante, ∀t ≥ 0. • Due to the finite speed of propagation (here equal to 1), the observability and the controllability at one endpoint hold if and only if the length of the time interval satisfies T ≥ 2 0 . • The observability and the controllability at the two endpoints holds if and only if T ≥ 0 . If the length of the string varies in time, we wonder if the solution of the wave equation has some analogue proprieties. Such situation where the spacial domain is time-dependent appears in many different areas of physics, from optics, electromagnetism, fluid dynamics to quantum mechanics. See for instance [7,14,16,23] and the survey paper [8].
To be more precise, we consider the wave equation in an interval with two linearly moving endpoints. We suppose that the left endpoint moves to the left with a constant speed 1 and the other endpoint moves to the right at a constant speed 2 . We assume that 0 ≤ 1 , 2 < 1 and 1 + 2 > 0. (1.1) The Later condition ensures that the length of the string is increasing. Let us denote by L 0 its initial length. In order to simplify notation and also to make the computations easier, we take as an initial time and consider the following interval with moving ends In the xt−plan, we have a noncylindrical domain Q t 0 +T , and its lateral boundary Σ t 0 +T , defined as Q t 0 +T := (x, t) ∈ R 2 | − 1 t < x < 2 t, for t ∈ (t 0 , t 0 + T ) , The assumption 0 ≤ 1 , 2 < 1 in (1.1) ensures that Σ t 0 +T satisfies the so-called timelike condition. Let us now consider the wave equation, with homogeneous Dirichlet boundary conditions, in Q t 0 +T , φ (− 1 t, t) = φ ( 2 t, t) = 0, for t ∈ (t 0 , t 0 + T ) , φ(x, t 0 ) = φ 0 (x) , φ t (x, t 0 ) = φ 1 (x) , for x ∈ I t 0 .

(W P )
Under the assumption (1.1) and for every initial data there exists a unique solution to Problem (W P ) such that see [2,4,11]. We define the "energy" of the solution of Problem (W P ) as As we will see below, E (t) is decaying in time. This contrasts the conservation of the energy E 0 (t) defined above. Next, we consider the observability problem for (W P ) at an endpoint ξt for ξ ∈ {− 1 , 2 } . This problem can be stated as follows: to give sufficient conditions on the length of the time interval, denoted by T , such that there exists a constant C(T ) > 0 for which the observability inequality holds for all the solutions of (W P ). This inequality is also called the inverse inequality. It allows estimating the energy of solutions in terms of the energy localized at the boundary x = ξt. Observe that φ (ξt, t) = 0 yields (φ (ξt, t)) t = ξφ x (ξt, t) + φ t (ξt, t) = 0. Then denoting C ξ = C(T ) 1 − ξ 2 , the precedent inequality can be rewritten as The minimal value of T , for which Inequality (1.5) holds, is called the time of observability. Due to the finite speed of propagation, one expects that T > 0 depends on the initial length L 0 and also on the two speeds of expansion 1 and 2 . This dependence is simply denoted by subscript in the notation of T . On the other hand, we consider the following boundary controllability problem: given find a control function v ∈ L 2 (t 0 , t 0 + T ) , acting at one of the endpoints, say x = 2 t, such that the solution of the problem , for x ∈ I t 0 +T . Note that Problem (CW P ) admits a unique solution in the transposition sense, see [15].
We shall also pay attention to the problem of observability at both ends where inequality (1.5) is replaced by for some timeT and a constant C(T ). Then we consider the associated controllability problem at both ends. That is to say, for any for t ∈ (t 0 , t 0 + T ) , y(x, t 0 ) = y 0 (x) , y t (x, t 0 ) = y 1 (x) , for x ∈ I t 0 , (CW P 2) satisfies the final conditions Observability and controllability of the wave equation in noncylindrical domains were considered by several authors. Bardos and Chen [2] obtained the interior exact controllability of the wave equation, in noncylindrical domains, by a "controllability via stabilisation" argument. Miranda [15] used a change of variable to transform the noncylindrical problem to a cylindrical one, shows the exact boundary controllability by the Hilbert Uniqueness method (see [12]) then, going back to the noncylindrical problem, he obtain the desired results. In recent years, there is a renewed interest in the observability and controllability of such problems, see for instance [5,13,22].
The authors, in these cited works, relay on the multiplier method (see [9]) to establish the energy estimates and inequalities necessary to derive the observability and controllability results. In this work, we present a different approach. It was inspired by the work of Balazs [1] where he obtained the exact solution of the 1-d wave equation as the sum of a generalised Fourier series, i.e., a countable set of orthogonal functions in a weighted L 2 −spaces.
The key idea is that we analyse the series representation of solution and use it to derive the desired energy and observability estimates. Then using HUM, we establish the controllability of the wave equation.
Fourier series approach in control theory of problems in cylindrical domains is by now classic, see [10,18,24]. However, the use of this approach for problems in noncylindrical problems seems to be new. Recently we have successfully applied this approach [19,20] to the wave equation in an interval with one moving endpoint (0, 0 t) , 0 < 0 < 1. We showed that the boundary observability and controllability at one endpoint, whether it is the fixed or the moving one, holds in a sharp time T 0 = 2L 0 / (1 − 0 ). In the paper at hand, we consider a wave equation in an interval with two moving endpoints. We show observability results at one endpoint and also at the two endpoints. Once the observability is established, the controllability is derived by HUM.
The main contribution of this work can be summarized as follows: • The series formulas of the exact solution of (W P ) was given by Balazs [1]. He managed to calculate the coefficients of the series when only one endpoint is moving, e.g. 1 = 0. If the two endpoints are moving, i.e. 1 > 0 and 2 > 0, the question remained open as far as we know. Here we show explicitly how to calculate these coefficients in function of the initial data φ 0 and φ 1 , see Theorem 1. • The decay of the energy of the solution of (W P ) is not conserved and decay at the precise rate 1/t, see Theorem 2. • The observability of (W P ) and the controllability of (CW P ), at an endpoint, holds if and only if . (1.11) This value of T holds whether the endpoint is x = − 1 t or x = 2 t, see Theorems 3 and 4.
• Problem (W P ) is observable and Problem (CW P 2) is controllable, at both endpoints, if and only if These results are new to our knowledge. In particular, the observability and the controllability at both endpoints (CW P 2) seem to be not considered before. As mentioned in [19,20], when an endpoint is a fixed one, e.g. 1 = 0, the obtained time of observability is sharp and improves other results obtained by the multiplier method, see for instance [5,6,22].
The remainder of this paper is organized as follows. For the convenience of the reader, we recall in section 2 some definitions and establish some lemmas that are necessary for the sequel. In section 3, we show how to calculate the coefficients of the series formula that gives the solution of (W P ). In section 4, we derive a sharp estimate for the energy of the solution of (W P ). Then, the boundary observability and controllability at one and at both endpoints are considered in the fifth and sixth sections.

Preliminaries
First, let us fix the notation of some constants that will frequently appear in the sequel.
In these notation, T andT , defined in (1.11) and (1.12), can be expressed as Taking into account Assumption (1.1), we can also check that Let a, b ∈ R such that b > a, and consider a positive (weight) function ρ : (a, b) → R. In the sequel, we denote by L 2 (a, b, ρds) the weighted Hilbert space of measurable complex valued functions on R, endowed by the scalar product b a f (s) g (s)ρ (s) ds and its associated norm. As usual, we drop ρds in the L 2 space notation if ρ = 1.
If the set of functions {ϕ n } n∈Z is a complete orthonormal basis of L 2 (a, b, ρds) , then every function f ∈ L 2 (a, b, ρds) can be written as (2.1) In particular, the following Parseval equality holds see for instance [3,17]. Let us check the completeness of some sets of functions used in the next sections. is complete and orthonormal in the space L 2 (a, b, dz/z) .
Proof. The orthonormality follows easily. The completeness holds if and only if no nonzero function in L 2 (a, b, dz/z) is orthogonal to all the e iπn(κ /M ) log z , n ∈ Z. Indeed, let f ∈ L 2 (a, b, dz/z) and consider the change of variable for every n ∈ Z. A well-known result in analysis is that the set of functions e iπns n∈Z is complete and orthogonal in L 2 (0, 2) . Thus f (ae M s/κ ) = 0, for a.e. s ∈ (0, 2), i.e. f (z) = 0, for a.e. z ∈ (a, b) .
This ends the proof.
This means that f ∈ L 2 (a, b) if and only if f ∈ L 2 (a, b, dz/z) .

Lemma 2.
For every t ≥ t 0 , the set of functions is complete and orthonormal in the space Proof. For every t ≥ t 0 , we use the change of variable to obtain s ∈ (0, 2). The result follows for by arguing as in the proof of Lemma 1.
Remark 3. Note that (1.1) ensures that the weight functions 1/ (t ± x) are positive, hence

Exact solution
According to Balazs [1], the exact solution of Problem (W P ) is given by the series where c n are complex numbers, independent of t. Unfortunately, he showed how to calculate the coefficients c n , in function of the initial data, only when the interval has one moving endpoint. If the two endpoints are moving, i.e. 1 > 0 and 2 > 0, the determination of c n turns out to be a bit tricky as we will see below.
First, we note that the two orthogonal sets considered in (2.4) appear in the series formula (3.1). To use their orthogonality properties, we need to extend the function φ, defined only on (− 1 t, 2 t) , to the intervals (−L 1 t, 2 t) and (− 1 t, . This extension is realised as follows The obtained function is well defined since the first variable of φ remain in the interval (− 1 t, 2 t) . This extension ensures some odd-like symmetry in the variable x forφ with respect to the point x = − 1 t on the interval (−L 1 t, 2 t) and with respect to the point x = 2 t on the interval (− 1 t, L 2 t) .
Derivation in time does not effect the odd-like symmetry in x, hence φ t is extended as follows Taking the derivative of (3.2) in x, we obtaiñ x ∈ ( 2 t, L 2 t) . Remark 4. If 1 = 0, then L 1 = 2 andφ,φ t are odd functions andφ x is an even function on the interval (− 2 t, 2 t) , ∀t ≥ t 0 . Similarly, if 2 = 0, then L 2 = 1 andφ 0 ,φ 1 are odd function andφ x is an even function on the interval (− 1 t, 1 t) . This justifies the terminology "odd-like" and "even-like" used above.
The next theorem shows how the calculate the coefficient of the series (3.1).
whereφ 1 andφ 0 x are given by (3.3) and (3.4). Moreover, the sum n∈Z * |nc n | 2 is finite and is given by any of the two formula Proof. First, Let us denote or alternatively C n e inπκ log 1+ 1 1− 1 = c n since e inπκ log α β = 1. Due to (1.3) and Remark 3, we can derive term by term the series (3.1), it comes that, (3.11) Combining this, with (3.3) and (3.4), for t = t 0 , the extensions of the initial data are given bỹ On one hand, taking the sum of (3.12) and (3.13), we get In particular, Due to (3.9), we can write (3.14) Thanks to Lemma 3, we see that nc n is the n th coefficient of the function in the basis κ /2e inπκ log(t 0 +x) n∈Z , i.e.
On the other hand, taking the difference of (3.12) and (3.13), we get in particular Using (3.9), we can write (3.16) This means that nC n is the n th coefficient of the function in the basis κ /2e inπκ log(t 0 −x) n∈Z , i.e. (3.6) also holds.

Energy estimates
The following theorem gives the precise decay rate of the energy for the solution of (W P ).

Using (3.4), (3.3) and considering the change of variable
Thus, considering the sum of (4.3) and (4.4), we infer that Expanding (φ x ± φ t ) 2 and collecting similar terms, we get Recalling that E (t) is given by (1.4), then (4.1) holds as claimed. Finally, we use the algebraic inequality ±ab ≤ a 2 + b 2 /2 and |x| ≤ L to obtain Due to (4.1), it comes that This implies (4.2) and the theorem follows.
The next corollary compares E (t) to the initial energy E (t 0 ) .
Proof. Since (4.5) holds also for t = t 0 , then (4.6) follows by combining the two inequalities Remark 5. Since E (t) define a norm on the space H 1 0 (I t ) × L 2 (I t ) , for t ≥ 0, then (4.6) implies the uniqueness for the solution of (W P ).

Observability and controllability at one endpoint
In this section, we show the observability of (W P ) at one of the endpoints, say x = 2 t, then by applying HUM we deduce the exact boundary controllability for (CW P ).

5.1.
Observability at one endpoint. First, we can state the following lemma.
and it holds that

(5.2)
Proof. Thanks to (3.10), we can evaluate φ x at the endpoint x = 2 t. We obtain By Lemma 1 and Parseval's equality applied to the function tφ x ( 2 t, t) ∈ which is (5.1). The estimate (5.2) follows by using (4.5) for t = t 0 .
The next theorem gives the direct and inverse inequalities for the solution of Problem (W P ) at the endpoint x = 2 t. • For every T ≥ 0, the solution of (W P ) satisfies the direct inequality with a constant K (T ) depending only on 1 , 2 and T.
• If T ≥ T , Problem (W P ) is observable at x = 2 t and it holds that Conversely, if T < T , (5.5) does not hold.
Remark 6. We obtain the same time of observability at the left boundary x = − 1 t. The proof is parallel to the precedent one.
Remark 7. The time of observability T can be predicted by a simple argument, see Figure 1. An initial disturbance concentrated near x = 2 t 0 may propagate to the left, as t increases, and bounce back on the left boundary, then travel to reach the left boundary, when t is close to α β t 0 , see Figure 1 (left). Thus, the needed time to complete this journey is close to (α β − 1) t 0 , which is the sharp time of observability T . Figure 1 (right) shows that we need the same time T for an initial disturbance concentrated Propagation of a wave with a small support near an endpoint ( 2 < 1 ).

5.2.
Controllability at one endpoint. First, let us check that the controllability problem (CW P ) can be reduced to a null-controllability one, i.e., we can always assume that Indeed, consider the homogenous backward problem One can argue as in [15] to show that this problem has a solution in the transposition sense. Then assume that there exists a function v ∈ L 2 (t 0 , t 0 + T ) , acting at the endpoint x = 2 t, driving the solution of the forward problem to the rest, i.e. w(x, t 0 + T ) = w t (x, t 0 + T ) = 0. Then, v drives u = z + w, solution of (CW P ), to u(x, t 0 + T ) = u 0 T (x) and u t (x, t 0 + T ) = u 1 T (x). The null-controllability of (CW P ) at one of the endpoints is derived by mean of HUM. Since the proofs are similar, only the case of the endpoint x = 2 t is considered.
where K (T ) is a constant depending on 1 , 2 and T.
Conversely, if T < T , Problem (CW P ) is not controllable at x = 2 t.
Proof. • Let φ be the solution of problem (W P ). The idea of HUM is to seek a control v in the special form v = φ x ( 2 t, t) ∈ L 2 (t 0 , t 0 + T ), for a suitable choice of φ 0 and φ 1 . First, we consider the backward problem for x ∈ I t 0 +T . (5.14) We obtain the a linear map, that relates (φ 0 , φ 1 ) to the initial data (ψ t (t 0 ) , −ψ (t 0 )) , The space H 1 0 (I t 0 ) × L 2 (I t 0 ) is equipped with the energy norm. To check that it is possible to choose (φ 0 , φ 1 ) such that (ψ t (t 0 ) , −ψ (t 0 )) = (u 1 , −u 0 ), we argue as in [9]. Since the solution of (5.14) is taken in the transposition sense, it comes that where , X denotes the duality product between a Banach space X and its dual. Observing that the boundary condition φ ( 2 t, t) = 0 implies that Then, we can rewrite (5.15) as Thanks to Theorem 3, we deduce that This means that Λ 1 is an isomorphism for T ≥ T and therefore (φ 0 , φ 1 ) can be determined such that the control v = φ x ( t, t) drive the solution of (CW P ) from the initial data u 0 , u 1 to the rest, i.e. u (t 0 + T ) = u t (t 0 + T ) = 0.
• If T < T , then (W P ) is not observable by Theorem 3. This means that we can find non-zero initial data (φ 0 , φ 1 ) ∈ H 1 0 (I t 0 ) × L 2 (I t 0 ) such that φ x ( 2 t, t) = 0, ∀t ∈ (t 0 , t 0 + T ). (5.16) Then, the solution of (CW P ), in the transposition sense, satisfies Whatever the choice of the control function v ∈ L 2 (t 0 , t 0 + T ), the last integral term always vanishes due to (5.16). Hence and therefore we cannot have u t (t 0 + T ) = u(t 0 + T ) = 0 on I t 0 +T . This completes the proof.

Remark 8.
The proof of Theorem 4 shows that the controllability of (CW P ), at the endpoint x = 2 t, is equivalent to the observability of (CW P ) at the same endpoint.

Observability and controllability at both endpoints
If we can observe simultaneously the two endpoints of the interval, one expects a shorter time of observability. The proof is more challenging in this case.   1) and (1.2), the solution of (W P ) satisfies and it holds that Proof. First, we establish (6.1) for smooth initial data. Assume that φ 0 x and φ 1 are continuous functions. This ensures in particular that their generalized Fourier series are absolutely converging, see [3,17]. More precisely, the coefficients c n , given by (3.5), satisfy n∈Z * |nc n | < +∞.
If n = m, then

The last parentheses vanishes, hence
A nm + B nm = 0 if n = m, n, m ∈ Z * .
Thus, we can rewrite (6.7) as Proof. • The right-hand side of Inequality (6.2) yields and thus • The right-hand side of (6.2), for M = 1, yields and thus inequality (6.9) holds for T = max {α , β } t 0 −t 0 =T and therefore for every T ≥T as well.
Remark 9. The observability and controllability of the 1-d wave equation in noncylindrical domains, with other types of boundary conditions, are established by the same approach used in this paper. The results will be published elsewhere.