Polynomial and Analytic Boundary Feedback Stabilization of Square Plate

abstract: We consider a boundary feedback stabilization problem of the plate equation in a square, in the case where the geometric condition of Ammari-Tucsnak [6] is not satisfied. We prove a polynomial decay for regular initial data. Moreover, we prove an exponential stability result for some subspace of the energy space. Finally, we give a precise estimate on the analyticity of reachable functions where we have an exponential stability.


S. Nouira
The controllability of the dynamical system modelling the vibrations of the plate with boundary control acting on the moment has been investigated in several works such as Ammari and Khenissi [4]- [10], Ammari and Vodev [9], Krabs, Leugering and Seidman [15], Leugering [20], Lebeau [17], [14] and in [23].In [16] the exact controllability of the same system has been established under the assumption that Ω is a square and under much weaker assumption on the controlled part of the boundary (Γ 1 is only supposed to contain non-empty vertical and horizontal subsects).The geometric optics condition introduced by Bardos, Lebeau and Rauch in [12] for the wave equation is thus not necessary in this case.In fact, recently, Ammari and Tucsnak (see [6]) have proved that the system is exponentially stable if and only if the controlled part of the boundary contains a vertical and horizontal part of non-zero length.
In this work, we study the polynomial stability for regular initial data and we study the exponential stability for some analytic initial data of a square Euler-Bernoulli plate with feedback.We use the methodology introduced in [5] (see also [13] for the bounded case), where the exponential stability for this problem is reduced to an observability inequality proved by [23] : where φ is the solution of the following undamped system associated to (1) : The paper is organized as follows.The statements of the main results are given in the following section.Section 3 is devoted to the observability inequality of high and low frequency.In Section 4, we give some background on a class of dynamical systems.Finally, Section 5 contains the proof of main results.

Main results
The system (1) is well-posed for initial condition satisfying (u 0 , u 1 ) ∈ E = H 1 0 (Ω) × H −1 (Ω), i.e there exists a unique solution (see [6]) The energy E(t) of system (1) is given by the following expression : The solution of (1) satisfies the following energy estimate: Polynomial and Analytic Boundary Feedback Stabilization of Square Plate 25 Let n 0 ∈ N * fixed, we denote by E n0 the following space : and by Where is the eigenvalues sequence of ∆ 2 .For all (u 0 , u 1 ) ∈ E, there exists (a n,k ) ∈ l 2 such that a n,k ϕ n,k .

S. Nouira
Let T > 2π √ 2 and u := (u 0 , u 1 ) ∈ E. For n ∈ N * , if we restrict to E n , there exists C(n) > 0 such that for φ solution of (3), it holds : We take for the C(n) the smallest constant for which the previous inequality is checked and we denote by According to H.U.M method, see [21], and to [3, chapter 1] we have : Thus for all We give, now the main results of this paper : For α > α S (T ), there exists a constant C α , γ α > 0 such that 1. We remark that all the elements of X α can be continued as an holomorphic function over the complex strip |ℑm(y)| < α• 2. The first assertion of the previous theorem implies that any analytic initial condition belongs to some S T for T large enough, i.e., any initial condition whose Fourier coefficients in y decrase like e −α n belongs to S T if T is larger than The system described by ( 1) is polynomial stable i.e., for all Γ 0 = ∅, there exists a constant C > 0 such as for all (u 0 , u 1 ) ∈ D(A) we have : where Polynomial and Analytic Boundary Feedback Stabilization of Square Plate 27

Inequality of observability
In this section we give the observability inequality at low and high frequency of the solution of (3) has been used for the proof of the main results.We specify the dependence of the constant which occurs in this estimation in function of the frequency of cut n.Proposition 3.1 (low frequencies estimate) For all ǫ > 0, δ > 0, there exist that for all n ≥ n 1 and for all u ∈ E n , the solution of problem (3) satisfies Proof.Since we do not have a uniform gap, we adapt the method proposed by Allibert and Micu in [4], which is a method inspired from the WKB technique.First we need the next technical lemma, for this proof we refer to [4], paragraph 4.3, pages 580-591.
The two above positive constants C, c depend only on q and ǫ.Moreover it is always possible to choose h k0,n ǫ,q even or odd, that we denote by h k0,n eǫ,q and h k0,n oǫ,q .
The analogue of Proposition 3.1 is proved in [4, Lemme 6].The proof is quite similar, but for the sake of completeness, let us give the main steps.
Proof of Proposition 3.1.Let n ∈ N * be such that n ≥ n 1 (q, ǫ), and let

S. Nouira
Then we have Hence for where K is the operator defined by If L ≥ k 0 , then by the point 3 of Lemma 3.1 we will have (a n,k0 + a n,−k0 ) sin ny h k0,n eǫ,q (t)e iλ n,k 0 t dt.
For point 4 of Lemma 3.1 we deduce that there exists a constant c > 0 such that Consequently, if L tends to infinity, we obtain In the same we obtain Then

These two estimates yield
Integrating this estimate in y ∈ (0, π) and using point 2 of Lemma 3.1, we obtain a constant c 1 > 0 such that As T 1 (q, ε) ≤ C q ε 1+q 1−q = C δ ε 1+δ and δ → 0 + for q → +∞, this shows Proposition 3.1.2 Lemma 3.2 (High frequencies) For all T 2 > 2π √ 2, there exists a constant C T2 > 0 such that for all integer n > 0 and initial data u in E 2 n the solution of problem (3) satisfies Proof.For u ∈ E 2 n , we have ∂ t φ(x, y, t) = k>n 2a n,k π λ n,k sin kx sin ny e itλ n,k .
Then if we use the Ingham inequality [11], we obtain Which implies

Some background on a class of dynamical systems
Let H a Hilbert space with the norm ||.|| H , and let A 1 : D(A 1 ) → H be a selfadjoint, positive and boundedly invertible operator.For α ≥ 0, we introduce the scale of Hilbert spaces H α = D(A α 1 ), with the norm z α = A α 1 z H .The space H −α is defined by duality with respect to the pivot space H as follows : H −α = H * α for α > 0. The operator A 1 can be extended (or restricted) to each H α , such that it becomes a bounded operator The second ingredient needed for our construction is a bounded linear operator , where U is another Hilbert space which will be identified with its dual.
The system we consider are described by The system ( 8)-( 9) is well-posed : For (w 0 , w 1 ) ∈ H 1 2 × H, the problem ( 8)-( 9) allows a unique solution : Moreover, w satisfies the energy estimate, for all t ≥ 0 : For (10) we remark that the mapping t → (w(t), ẇ(t)) 2 ×H is non-increasing.Consider the initial value problem: It is well known that ( 11)-( 12) is well posed in H 1 × H 1 2 and in H 1 2 × H. Now, we consider the unbounded linear operator where The result below, proved in [5], shows that, under a certain regularity assumption, the exponential stability of ( 8)-( 9) is equivalent to a strong observability inequality for ( 11)-( 12) and the polynomial stability of ( 8)-( 9) is a consequence of weak observability inequality.More precisely, we have : Polynomial and Analytic Boundary Feedback Stabilization of Square Plate 31 Theorem 4.1 (Ammari-Tucsnak [5]) Assume that for any γ > 0 we have Then 1. there exists C, δ > 0 such that for all t > 0 and for all (w 0 , w where ϕ(t) is the solution of system ( 11)-( 12).

Proof of the main results
5.1.Proof of the first assertion of the Theorem 2.1.For this proof, we need a result of the following lemma inspired by [19] (see also [7]).
Lemma 5.1 Let v ∈ E then v ∈ S T if and only if there exists a constant C v > 0 such that for any initial data u ∈ E, the solution u of problem (3) satisfies .
Then, let δ, ǫ > 0 and v ∈ X ǫ .We can put v(x, y) = n∈N * e −ǫn v n (x) sin ny, with As ∀ n ∈ N * , and ∀ u n ∈ E n , we have According to Proposition 3.1 and Lemma 3.2 This inequality according to Proposition 3.1 implies If we replace this estimate in (18), we obtain .
Which shows Theorem 2.2 thanks to Theorem 4.1 for α = 0.