OBSERVABILITY OF N -DIMENSIONAL INTEGRO-DIFFERENTIAL SYSTEMS

. The aim of the paper is to show a reachability result for the solution of a multidimensional coupled Petrovsky and wave system when a non local term, expressed as a convolution integral, is active. Motivations to the study are in linear acoustic theory in three dimensions. To achieve that, we prove observability estimates by means of Ingham type inequalities applied to the Fourier series expansion of the solution.

Exponential kernels are used to model viscoelastic system, as well in the analysis of Maxwell fluids or Poynting-Thomson solids, see [14]. The proposed system models the vibration of viscoelastic membranes with exponential kernels coupled with plates. In the present paper we introduce a coupling term of a different nature.
In the second equation of (1.1) we assume the linear dependence on the displacement of the membrane. This coupling term is already used in the literature (see [5] and reference therein for string-beam coupling and [12] for viscoelastic string-beam 746 PAOLA LORETI AND DANIELA SFORZA coupling). The main novelty consists in considering a coupling term of higher order in the N -dimensional case, because in the first equation of (1.1) the coupling term is a u 2 instead of the term au 2 . To justify this new term, we recall that the operator measures the deficit (or concentration if we consider − ) of the function with respect to its integral average, the so-called local anomaly (see e.g. [13]). The coupling proposed requires an accurate spectral analysis, the inequalities leading to the reachability result are obtained with new proofs, adapted from previous results (see [12]).
Moreover, we study the model described by (1.1) in the N -dimensional case when the domain is an open ball of radius R. For this coupled system we will show a reachability result for time T > 2R, that is the same condition on T as in the uncoupled case.
More precisely, we prove the following A remark about the assumption η > 3β/2 is now in order. The proof of Theorem 1.1 (see Section 5) requires the Ingham type inequalities proved in [12] and recalled in this paper as Theorem 3.1. To apply Theorem 3.1 to our system we have to verify, in particular, an assumption involving the eigenvalues of the integro-differential operator, that is r n ≤ − ω n for large n, and that assumption is satisfied if the condition η > 3β/2 holds true.
We also note that the shape of domain Ω is crucial to solve the corresponding eigenvalue problem in the N -dimensional case by means of the Bessel functions.
Our approach to the controllability of systems modeled by linear partial differential equations goes back to the survey paper by D. Russell [16], to the books by J.-L. Lions [8,9], and to the book related to applications by J. Lagnese and J.-L. Lions [6]. We refer to [5] for the approach to controllability based on Fourier analysis and Ingham type inequalities (for the seminal paper by Ingham see [3]).
As is well known, to solve a reachability problem like ours is equivalent to establish observability estimates for the solution of the adjoint problem (see [16]).
We refer to [10] and [11] to get observability of a single viscoelastic equation in the 1-dimensional case and multidimensional case respectively. In [12] we explore, in 1-d, the problem with a weak coupling in a viscoelastic medium and we study the ensuing system, answering in a positive way to the question of exact controllability in time greater than 2π/γ, γ being the gap between eigenvalues. We refer to [4] for the exact controllability of membrane-plate coupled systems in the N -dimensional case without viscosity. For an overview concerning viscoelastic models and integrodifferential equations see [15,14]. Other applications are related to acoustic theory [2].
The plan of our paper is the following. In Section 2 we introduce some notations, a preliminary result, recall the structure of the eigenfunctions of the Laplace operator in a ball and briefly describe the Hilbert Uniqueness Method. In Section 3 we recall Ingham type inequalities. In Section 4 we give a Fourier series expansion for the solution of the adjoint system. Finally, in Section 5 we prove the reachability result for coupled systems with memory terms like (1.1).
N -DIMENSIONAL INTEGRO-DIFFERENTIAL SYSTEMS 747 2. Preliminaries. Throughout the paper, we will adopt the convention to write f g if there exist two positive constants c 1 and c 2 such that For any T > 0, we denote by L 1 (0, T ) resp. L 2 (0, T ) the usual space of measurable functions ϕ : We recall well-known results concerning integral equations, see for example [1, Theorem 2.3.5].
(i) For any ψ ∈ L 1 (0, T ) there exists a unique solution ϕ ∈ L 1 (0, T ) of the integral equation given by (2.1) Let Ω be an open ball Ω of radius R in R N , N ≥ 2. In the following we consider L 2 (Ω) and H 1 0 (Ω) endowed with the standard norms and H −1 (Ω) is endowed with the dual norm of · H 1 0 (Ω) . For the sake of completeness, we bring to mind some well-known arguments regarding the eigenfunctions of the Laplace operator in a ball, of which we will take advantage in the next sections.
First, we recall some basic facts regarding Bessel type functions (see e.g. [5]), which will be fundamental to introduce the eigenfunctions of the Laplace operator in a ball. Let us introduce the Bessel functions of the first kind of any real order p by the formula where Γ denotes the gamma function.
Lemma 2.2. Let p be a nonnegative real number. The following equality holds for every positive real number c: As for the location of the zeros of the Bessel functions, we have the next result. (a) For any given real number p, the positive zeros of J p (x) are simple and they form an infinite strictly increasing sequence {λ n } tending to infinity.
We may assume without loss of generality that Ω is the unit ball of R N : the general case then follows easily by a linear change of variables. We shall consider the case N ≥ 2. Let us also recall that the spherical harmonics of order m ∈ N are the restrictions to the unit sphere ∂Ω of the homogeneous polynomials of order m.
Lemma 2.4. The spherical harmonics of order m ∈ N form a finite-dimensional subspace S m in L 2 (∂Ω). These subspaces are mutually orthogonal and their linear hull is dense in L 2 (∂Ω).
where m ∈ N ∪ {0}, k ∈ N, H m ∈ S m and for each m we denote by {λ mk } k∈N the strictly increasing sequence of positive zeros of the Bessel function J m−1+ N 2 (x). The corresponding eigenvalue of the eigenfunction E mk (ρ, θ) is λ 2 mk . For reader's convenience, we will describe the Hilbert Uniqueness Method for coupled systems with memory, when the integral kernel is a general function k ∈ L 1 (0, T ). For another approach see e.g. [7,17].
Let Ω be an open ball of radius R in R N , N ≥ 2. We consider the following coupled system with null initial conditions For a reachability problem we mean the following. One can solve such reachability problems by the Hilbert Uniqueness Method. To see that, we proceed as follows.
Given (z 10 , z 11 , z 20 , z 21 ) ∈ (C ∞ c (Ω)) 4 , we introduce the adjoint system of (2.3), that is with final data The above problem is well-posed, see e.g. [14]. Thanks to the regularity of the final data, the solution (z 1 , z 2 ) of (2.6) -(2.7) is regular enough to consider the nonhomogeneous problem As in the non integral case, it can be proved that problem (2.8) admits a unique solution φ. Therefore, we can introduce the following linear operator: for any (z 10 , z 11 , z 20 , z 21 ) ∈ C ∞ c (Ω) 4 we define Ψ(z 10 , z 11 , z 20 , z 21 ) = (−φ 1t (T, ·), φ 1 (T, ·), −φ 2t (T, ·), φ 2 (T, ·)) . (2.9) The following identity holds Ψ(z 10 , z 11 , z 20 , z 21 ), (z 10 , z 11 , z 20 , z 21 ) L 2 (Ω) As a consequence, we can introduce a semi-norm on the space C ∞ c (Ω If Theorem 2.7 holds true, then we can define the Hilbert space F as the completion of C ∞ c (Ω) 4 for the norm (2.10). So, the operator Ψ can be extended uniquely to a continuous operator, denoted again by Ψ, from F to the dual space F in such a way that Ψ : F → F is an isomorphism. Moreover, if we prove observability estimates of the following type

Ingham type theorems.
To apply the Hilbert Uniqueness Method we need suitable observability estimates for the solution of the adjoint system, see (2.11). In the proof of our reachability result (see Theorem 5.1), to obtain such estimates we will take advantage of Ingham type inverse and direct inequalities that we proved in [12, Theorems 5.19 and 5.21]. For reader's convenience, in this section we recall those results, presented them in a slight modified version.
In the next theorem we consider functions of the following type R n e rnt + C n e iωnt + C n e −iωnt + D n e ipnt + D n e −ipnt , with r n , R n ∈ R and ω n , C n , p n , D n ∈ C. |R n | ≤ µ n ν |C n | ∀ n ≥ n , |R n | ≤ µ|C n | ∀ n ≤ n , m 1 |p n | 2 ≤ |d n | ≤ m 2 |p n | 2 ∀n ∈ N .
Then, for any T > 2π/γ we have
Theorem 4.1. For any t ≥ 0, 0 ≤ ρ ≤ 1 and θ ∈ ∂Ω we have where the numbers r mk ∈ R, ω mk , p mk , d mk ∈ C are defined by (4.5) and the spherical harmonics R mk (θ) ∈ R, C mk (θ) , D mk (θ) ∈ C are written in terms of the spherical harmonics of the initial data, see (4.2), in the following way Moreover, for any m ∈ N ∪ {0}, k ∈ N and θ ∈ ∂Ω one has (4.6) 5. The reachability result. In this section we will prove a reachability result for the coupled system by taking advantage of the Hilbert Uniqueness Method, recalled at the end of Section 2, and of the Ingham type inequalities given by Theorem 3.1. We may assume without loss of generality that Ω is the unit ball of R N .
So, our proof is complete.