In this section we recall some non-harmonic Fourier analysis results and notions, bor-rowed from a paper of Kahane [40].
Let n ∈ N∗ be a natural number and I ⊂ Z a set of indices. We say that Λ = (λm)m∈I ⊂Rn is a regular sequence if there exists a constant γ >0 such that
m,linf∈I m6=l
|λm−λl|=γ. (1.30)
We introduce the notion of domain associated to a regular sequence, which allows to express some observability inequalities in terms of Fourier series.
Definition 1.12. We say that the open subset D ⊂ Rn is a domain associated to the regular sequence Λ = (λm)m∈I ⊂Rn if there exist constantsδ1(D), δ2(D)>0 such that,
for every sequence of complex numbers (am)m∈I with a finite number of non-vanishing
The following proposition (see Proposition III.1.2 from [40]) is an-dimensional equiv-alent of the classical Ingham inequality (Ingham [36]).
Proposition 1.13. Let Λ = (λm)m∈I ⊂ Rn be a regular sequence satisfying (1.30).
Then there exists a positive constantα (depending only on the space dimensionn) such that every ball in Rn of radius αγ is a domain associated to Λ.
Note that in the one dimensional case, the constant α, provided by the theorem of Ingham, is explicit and is equal to 2π. A consequence of this proposition is that we can prove easily the exact observability for systems with skew-adjoint generator and scalar output. More precisely, Tucsnak and Weiss proved the following result [80, Proposition 8.1.3].
Proposition 1.14. LetA:D(A)→Xbe a skew-adjoint operator generating the unitary group T. Assume that A is diagonalisable with an orthonormal basis (φm)m∈I in X formed by its eigenvectors and denote by (iλm)m ⊂ iR the eigenvalues corresponding to (φm). Assume that the eigenvalues of A are simple and there exists a bounded set J ⊂iR such that
λ,µ∈infσ(A)\J λ6=µ
|λ−µ|=γ >0.
Moreover, let C ∈ L(X1,C) be an observation operator for the semigroup T generated by A such that
minf∈I|Cφm|>0 and sup
m∈I|Cφm|<∞.
Then C is an admissible observation operator for T and the pair (A, C) is exactly ob-servable in any time τ > 2πγ .
Idea of the proof of Proposition 1.14. Using the facts that A is diagonalisable, that the eigenvalues of A are simple and that sup|Cφm| < ∞ we can easily obtain, applying Ingham inequality, the admissibility of C. For the observability of the pair (A, C), we firstly denote
V = span {φk|λk ∈J}⊥.
Applying Proposition 1.13 we have that (AV, CV) is exactly observable in any time τ > αγ. Then, from Proposition 1.9 and using that CΦm 6= 0, we obtain that (A, C) is exactly observable in any time τ > 2πγ.
The following results will be necessary for the proof of some exact observability and controllability results in Chapters 2 and 3.
Non-harmonic Fourier series and observability
Theorem 1.15 (Theorem III.3.1, Kahane [40]). Let Λ1, Λ2 be two regular sequences in Rn, and D1, D2 ⊂ Rn two domains associated to Λ1, respectively Λ2. If Λ = Λ1 ∪Λ2
is a regular sequence, then every set which contains the closure of D1+D2 is a domain associated to Λ.
The following theorem, proven in the same paper of Kahane [40], gives a condition on the density of a regular sequence such that any ball in Rn is a domain associated to this sequence.
Theorem 1.16. LetΛ be a regular sequence in Rn. Ford >0denote by ω(d)the upper limit when |b| → ∞ of the number of terms of Λ contained in the ball of center b and radius d. If ω(d) = o(d) when d→ ∞ then every ball in Rn of strictly positive radius is a domain associated to Λ.
In Chapter3we consider the linearized Berger equation (3.4)-(3.6) as a pertubation of the Euler-Bernoulli plates equation (3.1)-(3.3), and the eigenvalues of the perturbed operator have a “similar” asymptotic behavior as the eigenvalues of the unperturbed operator. For quantify this “similarity” we introduce the definition of asymptotically close sequences.
Definition 1.17. Let Λ = (λm)m∈I and ˜Λ = (˜λm)m∈I be two regular sequences in Rn. We say that the sequences Λ and ˜Λ are asymptotically close if for every α > 0 there exists an open ball B ⊂Rn large enough such that
|λm−λ˜m|< α (m∈ I such that λm,λ˜m ∈Rn\B).
The following theorem says that sequence wich are asymptotically close have the same associated domains.
Theorem 1.18 (Theorem III.2.2, Kahane [40]). Let Λ and Λ˜ be two regular sequences asymptotically close. Then an open set D ⊂ Rn is an associated domain to Λ if and only if is an associated domain to Λ.˜
2. Internal exact observability of a perturbed plate equation
In this chapter we study the exact internal observability of a linear perturbed plate equation. More precisely, we prove that the exact observability of the unperturbed Euler-Bernoulli plate equation implies the exact observability of the perturbed equation. If the plate is rectangular, the results from the unperturbed case are conserved, in particular, we have the exact observability in an arbitrarily small time.
Parts of this chapter were published in Cˆındea and Tucsnak [18].
2.1 Introduction and main results
Let Ω⊂Rn (n∈N∗) be an open and nonempty set with a C2 boundary or a rectangle.
We consider the following initial and boundary value problem :
¨
w(x, t)+∆2w(x, t)−a∆w(x, t)+b(x)·∇w(x, t)+c(x)w(x, t) = 0,(x, t)∈Ω×(0,∞) (2.1) w(x, t) = ∆w(x, t) = 0, (x, t)∈∂Ω×(0,∞) (2.2) w(x,0) =w0(x), w(x,˙ 0) =w1(x), x∈Ω, (2.3) where a > 0, b ∈ (L∞(Ω))n, c ∈ L∞(Ω), w0 ∈ H2(Ω)∩H01(Ω) and w1 ∈ L2(Ω). We consider the following output
y(t) = ˙w(·, t)|O, (2.4) where O is an open and nonempty subset of Ω and a dot denotes differentiation with respect to the time t:
˙
w= ∂w
∂t , w¨= ∂2w
∂t2 .
For n = 2 the equations (2.1)-(2.3) model the vibration of a perturbed Euler-Bernoulli plate with a hinged boundary.
The aim of this chapter is to study the exact observability of the system (2.1)-(2.4) in rapport with the exact observability of the unperturbed Euler-Bernoulli plate equation, i.e. (2.1)-(2.4) with a = 0, b = 0 and c = 0. More precisely, the main result of this chapter is the following theorem :
Theorem 2.1. Let O ∈ Ω be an open and nonempty subset of Ω such that (2.1)-(2.4) is exactly observable for a = 0, b = 0, c = 0. Then (2.1)-(2.4) is exactly observable for every a >0, b∈(L∞(Ω))n, c∈L∞(Ω).
A necessary and sufficient condition for the exact observability of the wave equation is that the observation regionO satisfies the geometric optic condition of Bardos, Lebeau and Rauch [9]. For the Euler-Bernoulli plate equation, it is known that the geometric optic condition is only a sufficient condition for the exact observability (see, for instance, Lebeau [46]). For example, if Ω is a rectangle, the Euler-Bernoulli plate equation is exactly observable in an arbitrarily small time for any open and nonempty observation region O (see, for instance, Jaffard [38] or Komornik [42]). This result is preserved for the perturbed plate equation (2.1)-(2.4), as it is formalized in the following theorem:
Theorem 2.2. Assume thatn = 2, Ωis a rectangle and let O be an open and nonempty subset of Ω. Then (2.1)−(2.4) is exactly observable for every a > 0, b ∈ (L∞(Ω))n, c∈L∞(Ω) in any time τ > 0.
For proving the above two theorems, we consider an abstract formulation of our exact observability problem. More precisely, in Section 2.2 we prove two observability results for two linear abstract perturbed systems.
In the third section of this chapter, we prove the theorems 2.1 and 2.2, applying the abstract results from Section2.2. A unique continuation result for the bi-Laplacian is proved in Section 2.3.1. For the proof of Theorem 2.2 we use some results of non-harmonic Fourier analysis introduced in Section 1.3.