Observability of wave equation with Ventcel dynamic condition

The main purpose of this work is to prove a new variant of Mehrenberger's inequality. Subsequently, we apply it to establish several observability estimates for the wave equation subject to Ventcel dynamic condition.


1.
Introduction. In the present paper, we address the exact controllability of the standard wave equation with mixed boundary condition: dynamic Ventcel condition on one part of the boundary and Dirichlet's on the other part. More precisely, let Ω be an open rectangular domain in R n , n ≥ 2 with boundary Γ Let Γ i , i = 1, . . . , n, Γ V and S denote the following subsets of Γ: (0, l j ), i = 1, . . . , n, Given T > 0, we consider the initial boundary value problem: where ∆ T designates the tangential laplacien on Γ V , ∇ T the tangential gradient on Γ V and ∂ ν the normal derivative with ν denoting the unit outward normal field to Γ. The second equation is called dynamic Ventcel boundary condition on Γ V . Differential problems with Ventcel conditions are quite interesting from the practical point of view, for they can model several physical processes. We mention as an example, the vibrations of an elastic body with a thin layer of high rigidity at the boundary [15], [16], [17] (see also [19], [7], and the references therein for more on the physical meanings of such problems).
The existence and regularity of solutions to such systems have been first investigated by Lemrabet [15], [16] with static Ventcel condition and later, by Lemrabet and Teniou [17] with dynamic Ventcel condition. Thus, we need not worry about an ill-posed problem. The main issue we are discussing here is the exact controllability of (1); though already studied in [19], [7], it seems that, compared to stabilization (see, e.g. [3], [4], [21]), it has received less attention.
In a recent paper, Gal and Tebou [7] have been able to derive, under some geometrical restrictions, a rather complex Carleman estimate for a problem similar to the one under consideration. Subsequently, using two controls, one on Ventcel portion of the boundary and the other on Dirichlet's, the authors have applied this estimate to establish the controllability of nonconservative waves.
Thanks to the particular structure of our domain, it's possible to take a much simpler approach; say the one based on Fourier expansions and Ingham type theorems. Since Ingham's first inequalities [9], many variants have been introduced and used in control theory [20], [12], [13], [14]. In particular, there is the outstanding generalization due to Mehrenberger [20] which extends Ingham's theorem to cover n-dimensional intervals.
Although working with these estimates, no one has retrieved the optimal time condition given by multiplier techniques; this approach remains significantly simpler to implement than the other methods.
To our knowledge, few are the papers investigating the exact controllability of systems such as ours and none are those with Ingham's theorems as their tools. The existing literature involves especially Dirichlet and Neumann conditions.
The system (1) is said to be exactly controllable if, for any given initial data v 0 , v 1 , v 2 , v 3 (in suitable Hilbert spaces), there exist control functions w i , i = 1, . . . , n, w S such that v(x, We shall work out this problem using the celebrated Hilbert Uniqueness Method (HUM) of J.-L. Lions [18] which permits us to prove the controllability of a system through the observability of the corresponding homogeneous problem. Spectral analysis of the latter has shown that the situation here is different from the case where Dirichlet condition is imposed on the whole boundary [20]. This brought out the need for the new version of Mehrenberger's theorem which we give in theorem 5.3. This adaptation is then used to establish the boundary observability cf. theorem 5.2.
After its first application to the observability of the wave equation with Dirichlet condition [20], Mehrenberger's inequality has been improved by Komornik and Miara ( [13], proposition 3.1, p. 139) and applied to establish additional results on the observability of rectangular membranes. Motivated by the joint work of Komornik with Loreti [12], we represent our variant of Mehrenberger's estimate in such a way that it can be employed to attain more observability estimates.
We emphasize that in our first result (cf. thm 5.1), while minding the geometric condition of Bardos-Lebeau-Rauch [1], Dirichlet action has sufficed to steer the system to rest within a finite amount of time. This fact, though the settings are not the same, refutes Gal and Tebou final remarks; namely, the two controls (Dirichlet and Ventcel's) are equally necessary to kill the vibrations. Interestingly, even if we choose the controls to be supported on both Ventcel and Dirichlet boundary, the system is proven not to be exactly controllable only approximately.
For computational simplicity, the results are given in dimension 2, that is when the domain Ω is a rectangle. The case of higher dimensions can be handled similarly.
This article is organized as follows: For the reader's convenience, in section 2, following HUM we prove the wellposedness of the associated homogeneous problem to (1) and we establish an a priori estimate (direct inequality). In section 3, after providing the spectral properties needed, we expand the solution of the homogeneous problem as a nonharmonic series. The existence of weak solution to problem (1) is given in section 4. Section 5 is devoted to the proof of our adaptation of Mehrenberger's theorem and its application to the boundary observability. Finally, in section 6 we obtain through this adaptation results on internal and combined internal-boundary observability.
In what follows, the notation A B means that c 1 B ≤ A ≤ c 2 B for some constants c 1 , c 2 > 0.
2. Statement of problem. Henceforth, we are restricting ourselves to the case of a rectangular membrane Ω = (0, l 1 ) × (0, l 2 ), l 1 , l 2 > 0. Let Γ 1 , Γ 2 , Γ V and Γ D denote the following boundary portions Let's introduce the homogeneous problem associated with (1) Throughout, ν, τ stand for the unit outward normal vector on Γ, and the unit tangent vector oriented outside of Γ V at its endpoints.
The well-posedness of problems with Ventcel boundary conditions has been verified by many authors either by variational techniques or by semi-groups theory (see for instance [17], [7], [19]). Nonetheless, for the reader's convenience, we're going over the proof of the existence of solutions as well as some regularity properties required for the application of HUM.
2.1. Well-posedness and regularity. Denoting the scalar product and norm, respectively on L 2 (Ω) and L 2 (Γ V ), we introduce Hilbert spaces Endowed with the norms We define on H the operator A 0 by whose domain is given by A 0 is sometimes called the Ventcel Laplacian.
Remark 1. We can readily verify that A 0 is self-adjoint and positive on H. Further, applying Lax-Milgram theorem shows that Thus, 0 ∈ ρ(A 0 ) and the inverse operator (A 0 ) −1 is bounded and compact due to Sobolev injections The energy at time t of the solution u to problem (3) is defined by The existence and uniqueness of solution to (3) is provided by Then, problem (3) has a unique solution u(x, t) such that ). In both cases, the energy is conserved: Proof. Putting U := (u, u| Γ V , u , (u| Γ V ) ), U 0 := (u 0 , u 1 , u 2 , u 3 ). System (3) can be written in the form of an abstract Cauchy problem in X where the operator A is defined on X by For problem (8) to have solutions, we need to show that A is the infinitesimal generator of a continuous semigroup. However, there is an even better alternative: Stone's theorem (see e.g. [22], thm. 1.10.8 or [23], thm. 3.8.6.) that characterizes generators of unitary groups to be skew-adjoint. Simple integrations by parts over Ω show that A is skew-symmetric on X i.e. [23], it results that A is skew-adjoint and 0 ∈ ρ(A). Moreover, by Stone's theorem A generates a group of isometrics on X. Thus, if U 0 ∈ X, problem (8) admits a unique solution U ∈ C(R; X) ( [11], thm. 2.1) which implies (6).

IMEN BENABBAS AND DJAMEL EDDINE TENIOU
On the other hand, if U 0 ∈ D(A), the solution satisfies According to proposition 1 we have H 2 -regularity in Ω and on Γ V . Thus, the solution is a strong one. The conservation of energy follows from the fact that the group generated by A is isometric.
2.2. Direct inequality. The next result is often referred to as hidden regularity. It plays a key role in the application of HUM. Actually, once proven, we have half of the uniqueness theorem we are seeking.
Proof. Since D(A) is dense in X, it suffices to show estimate (10) for regular solutions with initial conditions belonging to D(A).

OBSERVABILITY OF VENTCEL WAVE EQUATION 551
Combining these two equalities leads to the identity Now, taking q k = x k e k , k = 1, 2 where e k is kth vector in the canonical basis of R 2 , the left-hand side of (11) becomes On the other hand, we have Proceeding similarly with the remaining terms of the right-hand side of (11) and taking account of (7), we get the desired estimate.
3. Spectral analysis of the homogeneous problem (3). In this section, we shall express the solution u(x, t) as a Fourier series. To do this we are going to show, here below, the infinitesimal generator A (cf. eq. (9)) to be diagonalizable; namely, its resolvent is nonempty and there exists a Riesz base in X consisting of eigenvectors of A. So first we determine the eigenvalues and eigenvectors which, upon separation of variables, turn out to be solutions to some Sturm-liouville problems; then we apply theorem 3.1 [11] and obtain an explicit formula of u.

3.1.
Sturm-Liouville problems. We begin by computing the eigenvectors of the Ventcel Laplacian A 0 cf. eq. (4), (5). Afterward, we deduce those of the operator A. We know that α ∈ R + (A 0 is self-adjoint and positive) is

IMEN BENABBAS AND DJAMEL EDDINE TENIOU
Now, setting Thus, the eigenvalue problem is reduced to a pair of second-order differential equa- The first ODE is a regular Sturm-liouville equation whose solutions are the classical trigonometric functions associated with the eigenvalues It is well-known that {ϕ k1 , k 1 ≥ 1} forms an orthonormal basis of L 2 (0, l 1 ), that is to say a complete orthonormal system in L 2 (0, l 1 ). As for the second ODE, notice that its sole apparent difference from (12) is the presence of the eigenvalue parameter in the boundary condition. This type of Sturm-liouville problems arises when separation of variables is applied to differential equations with dynamic boundary conditions. Many papers have treated such problems, we mention for instance [2], [5], [6], [24]. Let Spectral properties of problem (13) are listed in the proposition below: denote, respectively, the eigenvalues and eigenfunctions of (13). We have the following assertions: 1. The eigenvalues µ 2 k2 ∞ k2=0 are positive roots of the transcendental equation µ k2 sin(µ k2 l 2 ) = cos(µ k2 l 2 ), k 2 ∈ N.
The corresponding eigenfunctions are given by

The normalized vectors
, form an orthonormal basis in H 1 .
3. The cosine functions having the same form as ψ k2 are orthogonal in 4. As k 2 → ∞, we have the asymptotic formulae Proof. By Green's formula, we see that which confirms that the eigenvalues of (13) are nonnegative. One can easily show that for µ = 0, the only possible solution is ψ = 0. Then µ = 0 isn't an eigenvalue. Fix µ = 0. We then have ψ(x 2 ) = a sin(µx 2 ) + b cos(µx 2 ).
Using the condition at x 2 = 0, it follows ψ(x 2 ) = a sin(µx 2 ). By the second condition, we infer that µ k2 , k 2 ≥ 0 are the positive roots of equation (16) and the corresponding functions are given by (17). Hence we have our first assertion.
Since the second and third properties are related, we work them out at the same time. Let k 2 , k 2 ∈ N, Hence for {Ψ k2 } to be orthogonal, it suffices to show l2 0 cos(µ k2 x 2 ) cos(µ k 2 x 2 )dx 2 = 0.
From the discussion above and remark 1, we draw the following consequence.
be the eigenvalues and corresponding eigenvectors of A 0 . Then, we have , k ∈ N * × N.

Moreover, the normalized vectors {ζ
, constitute an orthonomal basis of H.
One significant advantage of A 0 being diagonalizable is that it allows us to better grasp the structure of Hilbert spaces V, H, X and their duals. In fact, they can be characterized using the eigenvalues and eigenvectors of A 0 . Remark 2. Let Z be the linear hull of the basis vectors U k . For fixed s ∈ R, we define D s to be the completion of Z with respect to the Euclidian norm Then, identifying H with its dual, we have (see Komornik [10]) Remark 3. The dual space V could equally be viewed as the space of restrictions of elements of (H 1 Γ D (Ω)) × H −1 (Γ V ) to Hilbert space V (see [18], remark 5.1., p. 376).

Fourier series representation.
In order to write the solution of (3) as a Fourier series, We need to figure out a Riesz basis for Hilbert space X. Since A is defined in terms of Ventcel Laplacian A 0 , one can easily check that its eigenvalues are where ω k = k 1 π l 1 2 + µ 2 k2 , ω −k = −ω k , and the associated eigenvectors are given by .
On the other hand, we have D(A) ⊂ X densely with continuous compact embedding. Thus, A is skew-adjoint with compact resolvent and the set {E k } k∈Z * ×Z forms an orthonormal basis for X. Moreover, X is characterized using spaces D s as follows The next result provides an explicit formula of u(x, t) solution to (3):

Then the solution u(x, t) of (3) is given by Fourier series
with suitable coefficients a k , k ∈ Z * × Z depending on the initial data such that Proof. Writing the initial data U 0 = ((u 0 , u 0 | Γ V ), (u 2 , u 3 )) ∈ X in the base {E k }

IMEN BENABBAS AND DJAMEL EDDINE TENIOU
we obtain, according to theorem 3.1 [11], that the solution of (8) is If we integrate by parts, formally, in time and in space, we shall arrive at the relation Thus we are led to consider showing the existence of a weak solution v in the sense of the identity above. In fact, we have Theorem 4.1. For any initial and boundary data There exists a unique weak solution of (1) in the sense of identity (24) such that Proof. Setting we have expression (24) equivalent to Taking into account the direct inequality (10), it follows that, for each s ∈ [0, T ], the right-hand side of this formula defines a bounded linear form of U 0 ∈ X.
On the other hand, the operator A generates a unitary group T (s) on X. Hence, the right-hand side of (25) is also linear and bounded as a function of U (s) ∈ X. Denoting this form by V (s) and noting that it's continuous with respect to s, we conclude our proof. (24), we get to define the observability operator B by
Hence, estimate (10) means that B is an admissible observability operator.

Main result.
Our main result concerning the exact controllability of system (1) reads Then, there exist T 0 > 0 and control functions (w 1 , w 2 , w S ) of minimal norm in (1) reaches its equilibrium state (2) in time T.
Following HUM, the task of proving this theorem amounts to making sure that the observability inequality below holds for solutions of the homogeneous problem (3): Then, for T > T 0 there exists a constant c > 0 such that There are many approaches to take when dealing with observability estimates (Multiplier method, Carleman estimates, Microlocal analysis). Here we are adopting the one based on nonharmonic Fourier series. Precisely, we are recasting a generalization of Ingham theorem due to Mehrenberger [20] with the view of employing it to establish (26).

A variant of Mehrenberger's Ingham theorem.
Looking closely, one can observe that, in Mehrenberger's inequality all the variables have the same sequence of weights (p l ) l∈N * . This seems to fit well with its application to the observability of the wave equation with Dirichlet boundary condition. Here, though, the difference in form of the eigenvalues k 1 π l 1 k1∈N * and (µ k2 ) k2∈N leads us to allow each variable its own sequence of weights. The following adaptation shall be justified subsequently.
Theorem 5.3. Let {λ k , k = (k 1 , k 2 ) ∈ N * × N} be a sequence of real numbers. Given (p k1 ) k1∈N * , (q k2 ) k2N ⊂ C, we assume the following gap conditions: there exist γ 1 , γ 2 > 0 such that for k 1 , k 1 ∈ N * , k 2 , k 2 ∈ N, we have whenever |q k2 | ≤ max |p k1 | , |p k 1 | , and Then, for T > 2π 1 there is a constant c = c(γ 1 , γ 2 , T ) > 0 such that the following inequality holds for all square summable sequences (α k ) k∈N * ×N : For the proof, we shall need the following Lemma 5.4. Let us consider the function f : R → R defined by Then its Fourier transform f is given by Furthermore, we have for γ > 2π T , l ∈ N * and |x| ≥ lγ Proof. One may turn to [20] or [13] for the proof. Now, we are ready to take up proving estimate (29) based on the techniques already used by Mehrenberger [20].
Proof of theorem 11. To make use of the lemma above, we will effect the integrations over − T 2 , T 2 . A classical translation s = t + T 2 enables to recover (29). For brevity, we set k = (k 1 , k 2 ) and where .
Seeing as f (t) ≥ 0, we can discard the last term of (30). Now, taking account of the gap assumptions (27), we have for T > 2π On the other hand, using Young inequality, we find These estimates allow us to minorize the first three terms of (30) as follows

IMEN BENABBAS AND DJAMEL EDDINE TENIOU
Recalling that ∞ l=1 1 4l 2 − 1 = 1 2 , we add up these inequalities to obtain for T > 2π Hence, we get Denoting this time, k = (k 1 , k 2 ) and it is evident that by performing the same computations on A k1 , we get for T > 2π Summing over k 1 yields Now, noting that for fixed k 1 ∈ N * , k 2 ∈ N Whence, for the desired inequality to hold, we should take T > 2π 1
• We proceed now with the proof of the observability estimate (26): We know that the solution of (3) is given by Fourier series Using this expression and the orthogonality of systems cos k 1 π l 1 x 1 , sin k 1 π l 1 x 1 in L 2 (0, l 1 ) and that of (cos µ k2 x 2 ), sin µ k2 x 2 sin µ k2 l 2 in L 2 (0, l 2 ) and L 2 (0, l 2 ) × C (cf. proposition 2), the initial energy is found to satisfy ∇u 0 2 Likewise, we explicate the right side of (26) in terms of Fourier series (31) µ k2 a k e iω k t + a −k e −iω k t sin k 1 π l 1 x 1 2 dx 1 dt.
Once again, the orthogonality of the sine systems implies Taking λ k = ω k , p k1 = k 1 π l 1 , q k2 = µ k2 , (k 1 , k 2 ) ∈ N * × N in the statement of theorem 5.3, the inverse inequality (26) follows immediately from (29). Thus far, we have been able to drive system (1) to rest through Dirichlet action on two adjacent sides of the boundary Γ D . It seems interesting to consider including Γ V in the controlled area and seeing if it is possible to steer the system to equilibrium. This called for the following remarks.

Remark 5.
Let's consider one particular solution of the homogeneous problem (3) Then we have Note that this could not be minorized in terms of the initial energy because of the factor sin µ k2 l 2 that goes to zero as k 2 tends to infinity. Actually, it's not possible to exactly control system (1) on Ventcel portion of the boundary, because neither the tangential nor the time derivatives of the solution of (3) are observable on this portion (they both contain sin(µ k2 l 2 )).
The same could be said if we impose Ventcel condition on the top and right side of Ω and attempt to control the system through Ventcel action only. For this time the solution of the associated homogeneous problem is given by with δ k1 = cot(δ k1 l 1 ), µ k2 = cot(µ k2 l 2 ) and ω k = δ 2 k1 + µ 2 k2 .
6. Further results on the observability of system (3). In this section, we shall employ once more our variant of Mehrenberger's theorem to the next two results concerning internal and combined internal-boundary observability.
Then, for T > 2π γ we get for all square summable sequences (α l ) +∞ l=−∞ of complex numbers.
Proof. We get this inequality by repeating verbatim the computations carried out by Komornik [13], p. 140-141. However, we call attention to the fact that here, the decomposition of the sum is done according to the sequence (p l ) +∞ l=−∞ .
Before moving on to the demonstrations of the observability theorems, we take the chance to state the following Remark 6. Let's go back to the situation considered in remark 5 when we have two controls, one on the top side of Γ and the other on the left side. The representation given in the theorem above allows us to see that after all system (3) is approximately controllable. Indeed, if we assume that Then, according to inequality (34) we have k 1 π l 1 2 + |ω k sin µ k2 l 2 | 2 |a k | 2 + |a −k | 2 = 0, ∀k ∈ N * × N, which implies that the solution of (3) is approximately observable on Γ 1 and Γ V .
The gap assumptions satisfied by (ω k ) k∈N * ×N (cf. lemma 5.5) together with theorem 6.3 help us draw the following Corollary 2. If T > 2π γ 1 = 2 √ 2 + 1 l 1 , then for every k 2 ∈ N we have If T > 2π γ 2 = 4 √ 2 + 1 l 2 , then for every k 1 ∈ N * we have We get by means of classical trigonometric relations (see [13], p. 136) that 0 < m a,b ≤ Now, we have the necessary material to conduct the proofs of theorems 6.1 and 6.2.
6.1. Proof of theorem 6.1. We are going to assess the terms of the right-hand side of inequality (32), using the expansion of u(x, t) as a Fourier series as well as the estimates of corollary 2. We start up with the first term iω k a k e iω k t − a −k e −iω k t sin( k 1 π l 1 x 1 ) sin(µ k2 x 2 ) 2 dx 1 dt = l 1 2 T 0 k1 k2 ω k a k e iω k t − a −k e −iω k t sin(µ k2 x 2 ) 2 dt.