Kaïs Ammari
email: kais.ammari@fsm.rnu.tn

Thomas Duyckaerts
email: duyckaer@math.univ-paris13.fr

Armen Shirikyan
email: armen.shirikyan@u-cergy.fr

Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation

Keywords: AMS subject classifications: 35L71, 93B52, 93B07 Non-linear wave equation, distributed control, feedback stabilisation, truncated observability inequality inequality . . . . . . . . . . . . . . . . . 23 3.3 Observability inequality for the wave equation . . . . . . . .

come

Local feedback stabilisation to a non-stationary solution

for a damped non-linear wave equation

∂ 2 t u + γ∂ t u -∆u + f (u) = h(t, x), x ∈ Ω, (0.1)
where Ω ⊂ R 3 is a bounded domain with a smooth boundary ∂Ω, γ > 0 is a parameter, h is a locally square-integrable function with range in L 2 (Ω), and f : R → R is a function satisfying some natural growth and regularity conditions ensuring the existence, uniqueness, and regularity of a solution. Equation (0.1) is supplemented with the Dirichlet boundary condition,

u ∂Ω = 0, (0.2)
and the initial conditions u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x), (0.3) where u 0 ∈ H 1 0 (Ω) and u 1 ∈ L 2 (Ω). It is well known that, even though problem (0.1), (0.2) is dissipative and possesses a global attractor (which is finitedimensional in the autonomous case), its flow is not locally stable, unless we impose very restrictive conditions on the nonlinear term f . That is, the difference between two solutions with close initial data, in general, grows in time. The purpose of this paper is to show that any sufficiently regular solution of (0.1), (0.2) can be stabilised with the help of a finite-dimensional feedback control localised in space. Namely, instead of (0.1), consider the controlled equation

∂ 2 t u + γ∂ t u -∆u + f (u) = h(t, x) + η(t, x), x ∈ Ω, (0.4)
where η is a control whose support in x contains a subset satisfying a geometric condition. For a time-dependent function v, we set

Φ v (t) = v(t), v(t) , E v (t) = Ω | v(t, x)| 2 + |∇ x v(t, x)| 2 dx,
where the dot over a function stands for its time derivative. The following theorem is an informal statement of the main result of this paper.

Main Theorem. Let û be the solution of (0.1)-(0.3) with sufficiently regular initial data and let ω ⊂ Ω be a neighbourhood of the boundary ∂Ω. Then there is a finite-dimensional subspace F ⊂ H 1 0 (ω) and a family of continuous operators K û(t) : H 1 0 × L 2 → F such that, for any [u 0 , u 1 ] ∈ H 1 0 × L 2 that is sufficiently close to Φ û(0), problem (0.2)-(0.4) with η(t) = K û(t)Φ u-û (t) has a unique solution u(t, x), which satisfies the inequality E u-û (t) ≤ Ce -βt E u-û (0), t ≥ 0, (0.5)

where C and β are positive numbers not depending on [u 0 , u 1 ].

We refer the reader to Section 4 (see Theorem 4.1) for the exact formulation of our result. Before outlining the main idea of the proof of this theorem, we discuss some earlier results concerning the semi-linear wave equation with localised control. This problem was studied in a number of works, and first local results were obtained by Fattorini [START_REF] Fattorini | Local controllability of a nonlinear wave equation[END_REF] for rectangular domains and Chewning [START_REF] Chewning | Controllability of the nonlinear wave equation in several space variables[END_REF] for a bounded interval. Zuazua [START_REF] Zuazua | Exact controllability for the semilinear wave equation[END_REF][START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] proved that the 1D wave equation with a nonlinear term f (u) growing at infinity no faster that u(log u) 2 possesses the property of global exact controllability, provided that the control time is sufficiently large. These results were extended later to the multidimensional case, as well as to the case of nonlinearities with a faster growth and "right" sign at infinity [START_REF] Zhang | Exact controllability of semilinear evolution systems and its application[END_REF][START_REF] Li | Exact controllability for semilinear wave equations[END_REF]. Furthermore, the question of stabilisation to the zero solution was studied in [START_REF]Exponential decay for the semilinear wave equation with locally distributed damping[END_REF].

The application of methods of microlocal analysis enables one to get sharper results. In the linear case, the exact controllability and stabilisation to a stationary solution are established under conditions that are close to being necessary; e.g., see [START_REF]Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]. Dehman, Lebeau, and Zuazua [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] proved the global exact controllability by a control supported in the neighbourhood of the boundary, provided that nonlinearity is subcritical and satisfies the inequalities f (u)u ≥ 0; see also [START_REF] Dehman | Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time[END_REF] for a refinement of this result. Fu, Yong, and Zhang [START_REF] Fu | Exact controllability for multidimensional semilinear hyperbolic equations[END_REF] established a similar result for the equations with variable coefficients and a nonlinearity f growing at infinity slower than u(log u) 1/2 . Coron and Trélat [START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF] proved exact controllability of 1D nonlinear wave equation in a connected component of steady states. Laurent [START_REF]On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold[END_REF] extended the result of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] to the case of the nonlinear Klein-Gordon equation with a critical exponent. Very recently, Joly and Laurent [START_REF] Joly | A note on the global controllability of the semilinear wave equation[END_REF] proved the global exact controllability of the NLW equation of a particular form, using a fine analysis of the dynamics on the attractor. In conclusion, let us mention that the problem of exact controllability and stabilisation for other type of semi-linear dispersive equations was investigated in a large number of works, and we refer the reader to the papers [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF][START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF] for an overview of the literature in this direction.

To the best of our knowledge, the problem of stabilisation of a nonstationary solution û by a finite-dimensional localised control was not studied earlier. Without going into detail, let us describe informally the main idea of our approach, which is based on the study of Eq. (0.1) linearised around û. We thus consider the equation

∂ 2 t u + γ∂ t u -∆u + b(t, x)u = η(t, x), x ∈ Ω, (0.6)
supplemented with the initial and boundary conditions (0.2) and (0.3). We wish to find a finite-dimensional localised control depending on the initial conditions [u 0 , u 1 ] such that the energy E u (t) goes to zero exponentially fast. Following a well-known idea coming from the theory of attractors (see [START_REF] Haraux | Two remarks on hyperbolic dissipative problems, Nonlinear partial differential equations and their applications[END_REF] and Section II.6 in [START_REF] Babin | Attractors of Evolution Equations[END_REF]), we represent a solution of (0.6), (0.3) in the form u = v +w, where v is the solution of (0.6) with b ≡ η = 0 issued from [u 0 , u 1 ]. Then w satisfies the zero initial conditions and Eq. (0.6) with η replaced by η -bv. Let us note that v goes to zero in the energy space exponentially fast and that w has better regularity properties than v. It follows, in particular, that the decay of a sufficiently large finite-dimensional projection of w will result in exponential stabilisation of u. Combining this with the general scheme used in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] and a new observability inequality established in Section 3, we construct a finitedimensional localised control η which squeezes to zero the energy norm of w and that of u. A standard technique enables one to prove the latter property can be achieved by a feedback control. We refer the reader to Section 2 for more details.

The paper is organised as follows. In Section 1, we recall some well-known results on the Cauchy problem for semi-linear wave equation and establish the existence and uniqueness of a solution for the linear problem with low-regularity data. Section 2 is devoted to the stabilisation to zero for the linearised equation by a finite-dimensional localised control. The key tool for proving this result is the truncated observability inequality established in Section 3. Finally, the main result on local stabilisation for the non-linear problem is presented in Section 4.

Notation

Let J ⊂ R be a closed interval, let Ω ⊂ R d be a bounded domain with C ∞ boundary ∂Ω, and let X and Y be Banach spaces. We shall use the following functional spaces.

C ∞ 0 (Ω) is the space of infinitely smooth functions f : Ω → R with compact support. L p = L p (Ω) is the usual Lebesgue space of measurable functions f : Ω → R such that f p L p := Ω |f (x)| p dx < ∞.
When p = 2, this norm is generated by the L 2 -scalar product (•, •) and is denoted by • .

H s = H s (Ω) is the Sobolev space of order s with the standard norm • s .

H s 0 = H s 0 (Ω) is the closure of C ∞ 0 (Ω) in H s . It is well known that H s 0 = H s for s < 1
2 and that H s with s ≤ 0 is the dual of H -s 0 with respect to (•, •).

C(J, X) is the space of continuous functions f : J → X with the topology of uniform convergence on bounded intervals. Similarly, C k (J, X) is the space of k time continuously differentiable functions f ∈ C(J, X).

L p (J, X) is the space of Bochner measurable functions f : J → X with a finite norm

f L p (J,X) = J f (t) p X dt 1/p .
In the case p = ∞, this norm should be replaced by ess sup t∈J f (t) X .

W k,p (J, X) is the space of functions f ∈ L p (J, X) such that ∂ j t f ∈ L p (J, X) for j = 0, . . . , k. L(X, Y ) denotes the space of continuous linear operators from X to Y with the usual operator norm • L(X,Y ) . When the choice of X and Y is clear, we simply write • L . Given a function of the time variable v(t), we write Φ v (t) = [v(t), v(t)], where the dot stands for the time derivative. We denote by C unessential numbers (which may vary from line to line) and by M i (a 1 , . . . , a n ) positive constants depending on the parameters a 1 , . . . , a n . We write J T = [0, T ], J T (s) = [s, s + T ], R + = [0, +∞), and R s = [s, +∞).

Preliminaries

In this section, we first recall a well-known result on the existence, uniqueness, and regularity of a solution for (0.1)-(0.3). We next turn to the linearised problem, for which we prove the well-posedness in a space of functions of low regularity. Finally, we derive some commutator estimates used in what follows.

Cauchy problem for a semi-linear wave equation

Let us consider the equation

∂ 2 t u + γ∂ t u -∆u + f (u) = g(t, x), x ∈ Ω, (1.1)
where Ω ⊂ R 3 is a bounded domain with the boundary ∂Ω ∈ C ∞ , γ > 0 is a parameter, g is a locally square-integrable function of time with range in L 2 (Ω), and f ∈ C 1 (R) is a function vanishing at u = 0 and satisfying the inequalities

|f ′ (u)| ≤ C(1 + |u| 2 ), F (u) := u 0 f (v) dv ≥ -C, u ∈ R. (1.2)
Equation (1.1) is supplemented with the initial and boundary conditions (0.2) and (0.3). A proof of the following result on the well-posedness of the initialboundary value problem for (1.1) and regularity of solutions can be found in Chapter 1 of [START_REF] Lions | Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires[END_REF] and Section 1.8 of [START_REF] Babin | Attractors of Evolution Equations[END_REF].

Proposition 1.1. Under the above-mentioned hypotheses, for any u 0 ∈ H 1 0 (Ω), u 1 ∈ L 2 (Ω), and g ∈ L 2 loc (R + , L 2 ), problem (1.1), (0.2), (0.3) has a unique solution

u ∈ C(R + , H 1 0 ) ∩ C 1 (R + , L 2 ) ∩ W 2,2 loc (R + , H -1 ). (1.3) Moreover, if in addition u 0 ∈ H 2 (Ω), u 1 ∈ H 1 0 (Ω), and ∂ t g ∈ L 2 loc (R + , L 2 ), then u ∈ C(R + , H 2 ) ∩ C 1 (R + , H 1 0 ) ∩ W 2,2 loc (R + , L 2 ).
(1.4)

We now formulate a result on the time boundedness of solution for (1.1) under some additional assumptions on f . Namely, let us assume that f ∈ C 2 (R) is such that

f (u)u ≥ c F (u) -C, f ′ (u) ≥ -C, |f ′′ (u)| ≤ C(1 + |u|), u ∈ R, (1.5)
where C and c are positive constants. Note that these conditions are satisfied for polynomials of degree 3 with positive leading coefficient and, more generally, for C 2 -smooth functions behaving at infinity as c|u| ρ-1 u with c > 0 and ρ ∈ [1, 3]. The following result is established by Zelik [START_REF] Zelik | Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent[END_REF].

Proposition 1.2. Let us assume that f ∈ C 2 (R) satisfies (1.2) and (1.5) and let g ∈ W 1,∞ (R + , L 2 ). Then, for any u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) and u 1 ∈ H 1 0 (Ω), the solution u(t, x) of (1.1), (0.2), (0.3) satisfies the inequality

u(t) 2 + u(t) 1 ≤ M 1 u 0 2 , u 1 1 , G , t ≥ 0, (1.6)
where we set

G := ess sup t≥0 g(t, •) + ∂ t g(t, •) < ∞.

Cauchy problem for the wave equation with low-regularity data

In this and the next subsections, we assume that the space dimension d ≥ 1 is arbitrary, even though the results obtained here will be used only for d = 3. We study the linearised problem

v + γ v -∆v + b(t, x)v = η(t, x), x ∈ Ω, (1.7) v ∂Ω = 0, (1.8) v(0, x) = v 0 (x), v(0, x) = v 1 (x), (1.9)
where b and η are functions of low regularity, and γ ∈ R. Namely, let {e j } be the complete set of L 2 -normalised eigenvectors for the Dirichlet Laplacian in Ω (denoted by -∆) and let {λ j } be the corresponding eigenvalues numbered in an increasing order. We define the scale of spaces associated with -∆ by the relation

H s D = H s D (Ω) = f ∈ L 2 (Ω) : j≥1 (f, e j ) 2 λ s j < ∞ for s ≥ 0 (1.10)
and denote by H -s D = H -s D (Ω) the dual of H s D with respect to the L 2 scalar product. It is well known that (see [START_REF] Fujiwara | Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order[END_REF])

H s D = H s 0 for 1 2 < s < 1 2 , H s D = H s for -3 2 < s < 1 2 .
(1.11)

The wave propagator for (1.7), (1.8) is well defined (with the help of the eigenfunction expansion) when b ≡ η ≡ 0. In this case, for any initial data

[v 0 , v 1 ] ∈ H s D := H s D × H s-1 D there is a unique solution v ∈ C(R, H s D ) ∩ C 1 (R, H s-1 D ),
which satisfies the inequality

Φ v (t) H s D ≤ e c|t| Φ v (0) H s D for t ∈ R, (1.12)
where c ≥ 0 depends only on γ. We denote by S(t) :

H s D → H s D the oper- ator taking [v 0 , v 1 ] to Φ v (t) and write S(t) = [S 0 (t), S 1 (t)], so that S 0 (t) is a continuous operator from H s D to H s D for any s ∈ R. Let us define H s = H s 0 × H s-1 0
and note that, in view of (1.11), we have

H s = H s D for s ∈ (-1 2 , 1 2
). The proof of the following result is rather standard and is based on the Duhamel representation, the Banach fixed point theorem, and an estimate for the Sobolev of the product of two functions.

Proposition 1.3. Let r and T be positive numbers, let

J T = [0, T ], and let b ∈ L ∞ (J T × Ω) ∩ L ∞ (J T , H r ). Then there is σ 0 (r) > 0 such that for any σ ∈ [0, σ 0 (r)], [v 0 , v 1 ] ∈ H -σ , and η ∈ L 1 (J T , H -σ-1 ) problem (1.7)-(1.9) has a unique solution v ∈ X σ T := C(J T , H -σ ) ∩ C 1 (J T , H -σ-1 ).
Moreover, there is

M 2 = M 2 T, b L ∞ (JT ,L ∞ ∩H r ) such that sup t∈JT Φ v (t) H -σ ≤ M 2 [v 0 , v 1 ] H -σ + η L 1 (JT ,H -σ-1 ) .
(1.13) Note that inequality (1.13) is true for σ = -1, provided that b ∈ L ∞ (J T ×Ω) for any T > 0; this is a simple consequence of the standard energy estimate for the wave equation and the Gronwall inequality.

Proof of Proposition 1.3. We need to construct a solution of the integral equation

v(t) = S 0 (t)[v 0 , v 1 ]- t 0 S 0 (t-s) 0, b(s)v(s) ds+ t 0 S 0 (t-s) 0, η(s) ds, (1.14)
where t ∈ J T . We shall prove the existence of a solution on small time interval J τ = [0, τ ] whose length does not depend on the size of the initial data and the right-hand side. The global existence will then follow by iteration.

Step 1. Bound on the Duhamel term. Let us set

Qg(t) = t 0 S 0 (t -s)[0, g(s)] ds. We first show that if g ∈ L 1 (J τ , H -1-σ ), then Qg ∈ X σ τ , Qg X σ τ ≤ C 1 g L 1 (Jτ ,H -1-σ ) , (1.15)
where C 1 does not depend on g.

Indeed, if 0 ≤ σ < 1 2 , then H -σ-1 = H -σ-1 D and thus the mapping s → {S 0 (t -s)[0, g(s)], t ∈ J τ } belongs to the space L 1 (J τ , Y σ τ ), where Y σ τ = C(J τ , H -σ
). Furthermore, by (1.12), we have

S 0 (t -s)[0, g(s)] -σ ≤ e cτ g(s) -σ-1 for all t, s ∈ J τ , It follows that Qg ∈ C(J τ , H -σ ) and Qg(t) -σ ≤ C g L 1 (Jτ ,H -σ-1 ) for t ∈ J τ .
Using the relation

∂ ∂t S 0 (t -s)[0, g(s)] = S 0 (t -s)[g(s), 0],
we see that ∂ t (Qg) ∈ C(J τ , H -σ-1 ) and

∂ t Qg(t) -σ-1 ≤ C g L 1 (Jτ ,H -σ-1 ) , t ∈ J τ .
This completes the proof of (1.15).

Step 2. Fixed point argument. We shall need the following lemma, whose proof is given at the end of this subsection.

Lemma 1.4. Let r ∈ [0, 1 + d/2], let Ω ⊂ R d be a bounded domain with smooth boundary, and let a ∈ L ∞ ∩ H r . Then, for any s ∈ [0, r], we have

af s ≤ C a L ∞ ∩H r f ps for f ∈ H ps , (1.16) af -sp ≤ C a L ∞ ∩H r f -s for f ∈ H -s , (1.17)
where p = d+2 2r , and C > 0 depends only on Ω. For v ∈ X σ τ , we define

P v(t) = t 0 S 0 (t -s) 0, b(s)v(s) ds, t ∈ J τ . (1.18)
Using (1.17), one can find a number σ 0 (r

) > 0 such that, if 0 < σ ≤ σ 0 (r), then bv ∈ L ∞ (J τ , H -σ-1 ) and bv L ∞ (Jτ ,H -σ-1 ) ≤ C b L ∞ (Jτ ,L ∞ ∩H r ) v X σ τ .
Hence, in view of (1.15), we have

P v X σ τ ≤ C τ b L ∞ (Jτ ,H r ∩L ∞ ) v X σ τ . (1.19)
Now note that (1.14) holds for all t ∈ J τ and a function v ∈ X σ τ if and only if

(Id + P )u = S(t)[v 0 , v 1 ] + Qη, u ∈ X σ τ . Choosing τ so small that Cτ b L ∞ (0,τ ;H r ∩L ∞ ) ≤ 1 2 , we see from (1.19) that Id + P is invertible in X σ τ and (Id + P ) -1 L(X σ τ ,X σ τ ) ≤ 2. (1.20)
We thus obtain the existence of a unique solution v ∈ X σ τ for (1.14), which can be represented in the form

v = (Id + P ) -1 (S(•)[v 0 , v 1 ] + Qη) .
Combining this with (1.12), (1.15), and (1.20), we obtain the required estimate (1.13). This completes the proof of the proposition.

Proof of Lemma 1.4. Let us consider the multiplication operator f → af . Using the continuity of the embedding H 1+d/2 ⊂ L ∞ , the fact that H s is a Banach algebra for s > d/2, and interpolation techniques, it is easy to prove that

af r ≤ C 1 a r f 1+d/2 , f ∈ H 1+d/2 .
On the other hand, it is obvious that

af ≤ a L ∞ f , f ∈ L 2 .
By interpolation, the above two inequalities imply that

af θr ≤ C θ 1 a θ r a 1-θ L ∞ f θ(1+d/2) , f ∈ H θ(1+d/2
) . Taking θ = s/r, we arrive at inequality (1.16).

To prove (1.17), note that p ≥ 1, whence it follows that the operator of multiplication by a sends H sp 0 to H s 0 . By duality, it is also continuous from H -s to H -ps , and inequality (1.17) is implied by (1.16).

Commutator estimates

Given a function ψ ∈ C ∞ 0 (R), we define ψ(-∆)f = ∞ j=1 ψ(λ j )(f, e j )e j , f ∈ L 2 .
The aim of this subsection is to derive some estimates for the commutator of ψ(-∆) with the multiplication operator. In what follows, given a function a ∈ L 2 (Ω), we denote by the same symbol the corresponding multiplication operator sending f to af . We begin with the case of a smooth function.

Lemma 1.5. Let a ∈ C ∞ (Ω), ψ ∈ C ∞ 0 (R), and α ∈ [0, 1/2).
Then there is M 3 = M 3 (a, ψ) such that, for h ∈ (0, 1], we have

ψ(-h 2 ∆), a L(H α+1 0 ,H α ) ≤ M 3 h 2 , (1.21) ψ(-h 2 ∆), a L(H -α ,L 2 ) ≤ M 3 h 1-α . (1.22)
Proof. We first prove (1.21). Using the Fourier inversion formula, we get

ψ(-h 2 ∆) = 1 2π R e -ish 2 ∆ ψ(s) ds,
where ψ is the Fourier transform of ψ. Thus

ψ(-h 2 ∆), a f = 1 2π R ψ(s)v(h 2 s) ds,
where v(s) = e -is∆ , a f is the solution of the problem

-i∂ s v + ∆v = [a, ∆](e -is∆ f ), v s=0 = 0. (1.23) Using the fact that H α+1 0 = H α+1 D , we derive [a, ∆] e -is∆ f α = (∆ a)e -is∆ f + 2∇a • ∇(e -is∆ f ) α ≤ C e -is∆ f α+1 = C f α+1 .
Combining this with (1.23), we obtain v(s) α ≤ C|s| f α+1 for all s ≥ 0, whence it follows that

ψ(-h 2 ∆), a f α ≤ C R v(h 2 s) α | ψ(s)| ds ≤ C h 2 f α+1 s ψ L 1 .
This completes the proof of (1.21).

To prove (1.22), we first note that inequality (1.21) with α = 0 implies by duality that a, ψ(-

h 2 ∆) L(L 2 ,H -1 ) ≤ Ch 2 . (1.24) Let ϕ ∈ C ∞ 0 (R) be such that ϕ = 1 on the support of ψ. Using the relation ψ(-h 2 ∆)ϕ(-h 2 ∆) = ψ(-h 2 ∆), we get a, ψ(-h 2 ∆) = a, ψ(-h 2 ∆) ϕ(-h 2 ∆) + ψ(-h 2 ∆) a, ϕ(-h 2 ∆) .
Combining with (1.21), (1.24), and the inequality

ϕ(-h 2 ∆) L(L 2 ,H α+1 0 ) + ψ(-h 2 ∆) L(H -1 ,H α ) ≤ Ch -1-α , we derive a, ψ(-h 2 ∆) L(L 2 ,H α ) ≤ Ch 1-α .
By duality, we obtain (1.22).

We now turn to the case of functions of low regularity, which will be important in the derivation of an observability inequality (see Section 3).

Lemma 1.6. For any r > 0 there is

σ > 0 such that, if ψ ∈ C ∞ 0 (R) and a ∈ H r (Ω) ∩ L ∞ (Ω), then ψ(-h 2 ∆), a L(H -σ ,H -1 ) ≤ M 4 h σ , (1.25)
where

M 4 = M 4 (ψ, a L ∞ ∩H r ).
Proof. There is no loss of generality in assuming that 0

< r ≤ 1. Let us set A h = a, ψ(-h 2 ∆) . Since ψ(-h 2 ∆) : L 2 → L 2 is bounded uniformly in h ∈ (0, 1], we have sup h∈(0,1] A h L(L 2 ,L 2 ) < ∞. (1.26) Furthermore, since ψ(-h 2 ∆) : H s D → H s D is also bounded uniformly in h ∈ (0, 1] for any s ∈ R, it follows from (1.16) that sup h∈(0,1] A h L(H ps D ,H s D ) < ∞ for s ∈ [0, r],
(1.27)

where p = d+2 2r . We next show sup

h∈(0,1] h -2 A h H 2 D →H -2 D < ∞. (1.28)
To this end, we write (cf. proof of Lemma 1.5)

A h f = 1 2π R ψ(τ )v(h 2 τ ) dτ, v(t) = ae -it∆ f -e -it∆ (af ).
(1.29) Now note that

∂ t v = i∆ e it∆ (af ) -ia∆ e it∆ f , v(0) = 0.
It follows that

∂ t v H -2 D ≤ e it∆ (af ) L 2 + C ∆e it∆ f L 2 ≤ C f 2 H 2 D , whence we conclude that v H -2 D ≤ C|t| f H 2 D .

Recalling (1.29), we get the inequality

A h f H -2 D ≤ Ch 2 τ ψ L 1 f H 2 D , which implies (1.28).
Interpolating (1.26) and (1.28), we derive sup

h∈(0,1] h -s A h L(H s D ,H -s D ) < ∞.
Interpolating with (1.27), we deduce sup

h∈(0,1] h -s/3 A h H (2p+1)s/3 D →H s/3 D < ∞.
Taking s = 3/(2p + 1), by duality we obtain inequality (1.25) with σ = s/3.

Stabilisation of the linearised equation

This section is devoted to the stabilisation of the linearised problem (1.7), (1.8), in which γ > 0, b is a given function, and η is a finite-dimensional control supported by a given subdomain of Ω. The main result of this section is the existence of a feedback control exponentially stabilising problem (1.7), (1.8). To this end, we first construct a finite-dimensional stabilising control and then use a standard technique to get a feedback law.

Main result and scheme of its proof

As before, we denote by Ω ⊂ R 3 a bounded domain with C ∞ boundary Γ. We shall always assume that the following two conditions are satisfied.

Condition 2.1. The smooth surface Γ has only finite-order contacts with its tangent straight lines.

In other words, let y ∈ Γ and τ y ⊂ R 3 be the tangent plane to Γ at the point y. In a small neighbourhood of y, the surface Γ can be represented as the graph of a smooth function ϕ y : τ y → R vanishing at y together with its first-order derivatives. Condition 2.1 requires that the restriction of ϕ y to the straight lines passing through y has no zero of infinite order at the point y.

To formulate the second condition, we first introduce some notation. Given x 0 ∈ R 3 , define Γ(x 0 ) as the set of points y ∈ Γ such that y -x 0 , n y > 0, where n y stands for the outward unit normal to Γ at the point y. Let ω be the support of the control function η entering (1.7).

Condition 2.2. There is x 0 ∈ R 3 \ Ω and δ > 0 such that Ω δ (x 0 ) := {x ∈ Ω : there is y ∈ Γ(x 0 ) such that |x -y| < δ} ⊂ ω.

Before formulating the main result of this section, let us make some comments about the above hypotheses. Condition 2.2 naturally arising in the context of the multiplier method (see [START_REF] Lions | Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués[END_REF]) ensures that the observability inequality holds for (1.7) in the energy norm. On the other hand, Condition 2.1 enables one to define a generalised bicharacteristic flow on Ω (see Section 24.3 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]). Together with Condition 2.2, this implies that if T is sufficiently large, then for any δ ′ ∈ (0, δ) the pair (Ω δ ′ (x 0 ), T ) geometrically controls Ω in the sense that every generalised bicharacteristic ray of length T meets the set Ω δ ′ (x 0 ). In view of [START_REF]Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], it follows that the observability inequality holds for Eq. (1.7) with b ≡ 0 in spaces of negative regularity. We shall combine these two results with some commutators estimates and a compactness argument to establish a truncated observability inequality for (1.7) (see Section 3.2), which is a key point of the proof of the theorem below.

Let us fix a function χ ∈ C ∞ 0 (R 3 ) such that supp χ ∩ Ω ⊂ ω and χ(x) = 1 for x ∈ Ω δ/2 (x 0 ). We denote by F m the vector span of the functions {χe 1 , . . . , χe m }, where {e j } is a complete set of L 2 normalised eigenfunctions for the Dirichlet Laplacian. The following theorem is the main result of this section.

Theorem 2.3. Let Condition 2.1 and 2.2 be satisfied, let R and r be positive numbers, and let b(t, x) be a function such that

|b| := ess sup t≥0 b(t, •) L ∞ ∩H r ≤ R.
(2.1)

Then there is an integer m ≥ 1, positive numbers C and β, and a family of continuous linear operators

K b (t) : H 1 0 × L 2 → F m , t
v 0 , v 1 ] ∈ H 1 0 × L 2 , problem (1.7), (1.8) with the right-hand side η(t) = K b (t)[v(t), v(t)] has a unique solution v ∈ C(R + , H 1 0 ) ∩ C 1 (R + , L 2 ) satisfying the initial conditions v(s, x) = v 0 (x), ∂ t v(s, x) = v 1 (x). (2.2)
Moreover, we have the inequality

E v (t) ≤ C e -β(t-s) E v (s), t ≥ s. (2.3)
Let us sketch the proof of this result. We first prove that, for any β > 0 and sufficiently large T > 0, there is a linear operator Θ s : H → L 2 (J T (s), F m ), where J T (s) = [s, s + T ] and H = H 1 0 × L 2 , such that the norm of Θ s is bounded uniformly in s ≥ 0, and for any [v 0 , v 1 ] ∈ H 1 0 × L 2 the solution of problem (1.7), (1.8), (2.2) with η = Θ s [v 0 , v 1 ] satisfies the inequality

E v (s + T ) ≤ e -βT E v (s).
(2.4)

For given initial data [v 0 , v 1 ] ∈ H 1 0 × L 2 , an exponentially stabilising control η can be constructed by the rule

η JT (0) = Θ 0 [v 0 , v 1 ], η JT (kT ) = Θ kT Φ v (kT ), k ≥ 1.
(2.5) Inequality (2.4) and the uniform boundedness of Θ s imply that (2.3) holds with s = 0. Once the existence of at least one exponentially stabilising control is proved, one can use a standard technique based on the dynamical programming principle to construct a feedback law possessing the required properties. The uniqueness of a solution is proved by a standard argument based on the Gronwall inequality. The rest of this section is organised as follows. In Subsection 2.2, we prove the existence of an operator Θ s with the above-mentioned properties. A key point of the proof is the truncated observability inequality established in Section 3. Subsection 2.3 deals with the construction of an exponentially stabilising feedback law. Its properties mentioned in the theorem are established in Subsection 2.4. In what follows, the domain Ω and its closed subset ω are assumed to be fixed, and we do not follow the dependence of other quantities on them.

Construction of a stabilising control

Proposition 2.4. Let the hypotheses of Theorem 2.3 hold and let T > 0 be sufficiently large. Then, for a sufficiently small σ > 0, there is a constant C and an integer m ≥ 1, depending only R and r, such that, for any s ≥ 0, one can construct a continuous linear operator Θ s : H → L 2 (J T (s), H σ ) satisfying the following properties.

Boundedness. The norm of Θ s is bounded by C for any s ≥ 0, and its image is contained in L 2 (J T (s), F m ).

Squeezing. Let [v 0 , v 1 ] ∈ H and η = Θ s [v 0 , v 1 ].
Then the solution of (1.7), (1.8), (2.2) satisfies inequality (2.4).

An immediate consequence of this proposition is the following result on the existence of a stabilising control. For β > 0 and a Banach space X, we denote by L 2 β (R + , X) the space of locally square-integrable functions f : R + → X such that

f L 2 β := sup t≥0 t+1 t e βs f (s) 2 X ds < ∞.
Corollary 2.5. Under the hypotheses of Theorem 2.3, there is β > 0 and a continuous linear operator Θ :

H → L 2 β (R + , F m )
, where F m is endowed with the norm of H σ with some σ > 0, such that the solution of problem (1.7), (1.8), (2.2) with η = Θ[v 0 , v 1 ] and s = 0 satisfies inequality (2.3) with s = 0.

Proof. Let us define a control η : R + → F m by relations (2.5). It follows from (2.4) that E v (kT ) ≤ e -βT k E v (0), k ≥ 0.

(2.6)

Since the norms of Θ s are bounded uniformly in s ≥ 0, it follows from (2.5) and (2.6) that

η| JT (kt) L 2 (JT (kT ),Fm) ≤ C Φ v (kT ) H ≤ Ce -βT k/2 [v 0 , v 1 ] H . (2.7) This inequality shows that η ∈ L 2 β (R + , F m ) and the operator [v 0 , v 1 ] → η is continuous from H to L 2 β (R + , F m ).
Furthermore, the continuity of the resolving operator for problem (1.7), (1.8) and inequalities (2.6) and (2.7) imply that sup

t∈JT (kT ) Φ v (t) H ≤ C Φ v (kT ) H + η| JT (kt) L 2 (JT (kT ),Fm) ≤ Ce -βT k/2 [v 0 , v 1 ] H .
This immediately implies the required estimate (2.3) with s = 0.

Proof of Proposition 2.4. We first describe the scheme of the proof. Define an energy-type functional for a trajectory v(t, x) by the relation

E v (t) = Ω | v| 2 + |∇v| 2 + αv v dx.
For small α > 0, this quantity is equivalent to E v (t):

C -1 E v (t) ≤ E v (t) ≤ CE v (t), t ∈ R + .
(2.8)

Let z be the solution of problem (1.7), (1.8), (2.2) with b ≡ η ≡ 0. Taking the scalar product in L 2 of the equation for z with 2 ż + αz, we can find δ > 0 such that

E z (t) ≤ CE z (t) ≤ Ce -δ(t-s) E z (s) ≤ C 2 e -δ(t-s) E z (s).
(2.9)

In particular, if T > 0 is sufficiently large, then

Φ z (s + T ) H ≤ 1 4 [v 0 , v 1 ] H .
(2.10)

We seek a solution in the form v = z + w. Then w must be a solution of the control problem

ẅ + γ ẇ -∆w + b(t, x)w = η(t, x) -b(t, x)z, x ∈ Ω, (2.11)
w ∂Ω = 0, (2.12) w(s, x) = 0, ẇ(s, x) = 0.

(2.13)

Given an integer N ≥ 1 and a constant ε > 0, we shall construct a control η such that the corresponding solution w satisfies the inequalities

Φ w (s + T ) H σ ≤ M 5 [v 0 , v 1 ] H , P N Φ w (s + T ) ≤ ε [v 0 , v 1 ] H , (2.14)
where P N stands for the orthogonal projection in L 2 (Ω) to the vector span of the first N eigenfunctions of the Dirichlet Laplacian, and σ > 0 and M 5 are constants not depending on N and ε. For an appropriate choice of N and ε, these two inequalities imply that

Φ w (s + T ) H ≤ 1 4 [v 0 , v 1 ] H . (2.15)
Combining this with (2.10), we see that

Φ v (s + T ) H ≤ 1 2 [v 0 , v 1 ] H .
This inequality is equivalent to (2.4) with β = T -1 log 2.

We now turn to the accurate proof. The derivation of inequality (2.10) is classical (e.g., see Section 6 in [BV92, Chapter 2]), and we shall confine ourselves to the construction of w. To simplify notation, we shall assume that s = 0; the case s > 0 can be treated by a literal repetition of the argument used for s = 0.

Step 1. We seek η in the form

η(t, x) = χ(x)P m ζ(t, •) , (2.16)
where ζ ∈ L 2 (J T × Ω) is an unknown function and m ≥ 1 is an integer that will be chosen later. Let us define the space

X T = C(J T , H 1 0 ) ∩ C 1 (J T , L 2 ) ∩ W 2,2 (J T , H -1
) and consider the following minimisation problem: Problem 2.6. Given initial data [v 0 , v 1 ] ∈ H and (small) positive numbers δ and σ, minimise the functional

J (w, ζ) = 1 2 T 0 ζ(t, •) 2 σ dt + 1 δ ∇P N w(T ) 2 + P N ẇ(T ) 2
in the class of functions (w, ζ) ∈ X T × L 2 (J T , H σ ) satisfying Eqs. (2.11), (2.13) with s = 0 and η given by (2.16).

This is a linear-quadratic optimisation problem, and it is straightforward to prove the existence and uniqueness of an optimal solution, which will be denoted by (w, ζ). The mapping z → (w, ζ) is linear, and therefore so is the mapping [v 0 , v 1 ] → η. Let us derive some estimates for the norms of w and ζ. To this end, we write the optimality conditions: q -γ q -∆q + b(t, x)q = 0, (2.17)

(-∆) σ ζ = P m (χq), (2.18) q(T ) = - 2 δ P N ẇ(T ), q(T ) = - 2 δ P N ∆w(T ) + γ ẇ(T ) , (2.19)
where q ∈ L 2 (J T , H 1 0 ) is a Lagrange multiplier. Note that, in view of (2.17) and the uniqueness of a solution for the linear wave equation, the function q must belong to X T , so that relations (2.19) make sense. Let us take the scalar product in L 2 (J T × Ω) of Eqs. (2.11) and (2.17) with the functions q and w, respectively, and take the difference of the resulting equations. After some simple transformations, for σ ∈ (0, 1 2 ) we obtain ( ẇ(T ) + γw(T ), q(T )) -(w(T ), q(T )) + T 0 (bz, q) dt -

T 0 ζ, P m (χq) dt = 0.
Using (2.18) and (2.19), we obtain

2 δ P N w(T ) 2 1 + P N ẇ(T ) 2 + T 0 P m (χq) 2 -σ dt = T 0 (bz, q) dt. (2.20)
There is no loss of generality in assuming that T is so large that inequality (3.2) holds and, hence, the truncated observability inequality (3.11) is true for small σ > 0. Combining this with (1.16), (1.13), and (2.9) we derive

|(bz, q)| ≤ bz σ q -σ ≤ C z 1 Φ q (0) H -σ ≤ C Φ z (0) H P m (χq) L 2 (JT ,H -σ ) .
The Cauchy-Schwarz inequality now implies that T 0 (bz, q) dt ≤ 1 2

T 0 P m (χq) 2 -σ dt + C v 0 2 1 + v 1 2 .
Substituting this into (2.20), we obtain

1 δ P N w(T ) 2 1 + P N ẇ(T ) 2 + T 0 P m (χq) 2 -σ dt ≤ C [v 0 , v 1 ] 2 H , (2.21)
where m = m(N ) ≥ 1 is an integer and C is a constant not depending on δ and N . Taking N ≫ 1 and δ ≪ 1, we obtain the second inequality in (2.14) with s = 0.

Step 2. Let un prove the boundedness of the operator Θ 0 : [v 0 , v 1 ] → η from H to L 2 (J T , H σ ) and inequality (2.15) with s = 0. This will complete the proof of Proposition 2.4.

It follows from (2.18) and (2.21) that

ζ L 2 (JT ,H σ ) ≤ C [v 0 , v 1 ] H .
Since the projection P m and multiplication by χ are bounded operators in H σ , the above inequality combined with relation (2.16) shows that Θ 0 is bounded.

To prove (2.15), we write

Φ w (T ) H ≤ P N Φ w (T ) H + (I -P N )Φ w (T ) H ≤ ε [v 0 , v 1 ] H + δ N (σ) Φ w (T ) H σ ,
where δ N (σ) → 0 as N → ∞. It follows that (2.15) will be established if we prove the first inequality in (2.14).

Duhamel representation for solutions of (2.11)-(2.13) and inequality (1.12) with s = σ imply that

Φ w (T ) H σ ≤ C η -bz -bw L 1 (JT ,H σ ) .
Combining this with (1.16) and using condition (2.1) and the boundedness of z and w in C(J T , H 1 0 ), we arrive at the required inequality.

Dynamic programming principle and feedback law

Once the existence of a stabilising control is established, an exponentially stabilising feedback law can be constructed using a standard approach based on the dynamic programming principle. Since the corresponding argument was carried out in detail for the more complicated case of the Navier-Stokes system (see Section 3 in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF]), we shall omit some of the proofs. Let us consider the following optimisation problem depending on the parameter s ≥ 0.

Problem 2.7. Given [v 0 , v 1 ] ∈ H and β > 0, minimise the functional

I s (v, ζ) = 1 2 ∞ s e βt ∇v(t) 2 + v(t) 2 + ζ(t) 2 dt in the class of functions (v, ζ) such that v ∈ C(R s , H 1 0 ) ∩ C 1 (R s , L 2 ) ∩ W 2,2 loc (R s , H -1 ), ζ ∈ L 2 loc (R s , L 2
), and Eqs. (1.7) and (2.2) hold with η given by (2.16). This is a linear-quadratic optimisation problem, and in view of Corollary 2.5, there is at least one admissible pair (v, ζ) for which I s (v, ζ) < ∞. It follows that there is a unique optimal solution (v s , ζ s ) for Problem 2.7, and the corresponding optimal cost can be written as

I s (v s , ζ s ) = 1 2 Q s [v 0 , v 1 ], [v 0 , v 1 ] H ,
where Q s : H → H is a bounded positive operator in the Hilbert space H. Moreover, repeating the argument used in the proof of Lemma 3.8 in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF], one can prove that Q s continuously depends on s in the weak operator topology, and its norm satisfies the inequality

Q s L(H) ≤ C e βs , s ≥ 0. (2.22)
We now consider the following problem depending on the parameter s > 0.

Problem 2.8. Given [v 0 , v 1 ] ∈ H and β > 0, minimise the functional

K s (v, ζ) = 1 2 s 0 e βt ∇v(t) 2 + v(t) 2 + ζ(t) 2 dt + 1 2 Q s [v 0 , v 1 ], [v 0 , v 1 ] H in the class of functions (v, ζ) such that v ∈ C(J s , H 1 0 ) ∩ C 1 (J s , L 2 ) ∩ W 2,2 (J s , H -1 ), ζ ∈ L 2 (J s , L 2 ),
and Eqs. (1.7) and (1.9) hold with η given by (2.16).

This is a linear-quadratic optimisation problem, which has a unique solution (ṽ s , ζs ). The following lemma establishes a link between Problems 2.7 and 2.8. Its proof repeats the argument used in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] (see Lemma 3.10) and is omitted. Lemma 2.9. Let (v, ζ) = (v 0 , ζ 0 ) be the unique solution of Problem 2.7 with s = 0. Then the restriction of (v, ζ) to the interval J s coincides with (ṽ s , ζs ) and the restriction of (v, ζ) to the half-line R s coincides with (v s , ζ s ) corresponding to the initial data [v(s), v(s)].

The optimality conditions for the restriction of (v, ζ) to J s imply, in particular, that qs -γ qs -∆q s + b(t, x)q s = e βt ∆v + β v + v , (2.23)

ζ(t) = e -βt P m (χq s (t)), (2.24)
q s (s) = -Q 1 s Φ v (s), (2.25)
where q s ∈ L 2 (J s , H 1 0 ) is a Lagrange multiplier and Q 1 s : H → L 2 is a continuous operator defined by the relation

Q s V = [Q 0 s V, Q 1 s V ] for V ∈ H.
Since the right-hand side of (2.23) belongs to L 2 (J s , H -1 ), it follows from Proposition 1.3 with σ = 0 that the function q s must belong to C(J s , L 2 ), so that relation (2.25) makes sense, and ζ is a continuous function of time with range in F m . Combining (2.24) and (2.25), we see that

ζ(s) = -e -βs P m χQ 1 s Φ v (s) .
Recalling that s > 0 is arbitrary and using (2.16), we conclude that the unique optimal solution (v, ζ) of Problem 2.7 with s = 0 satisfies Eq. (1.7) with η(t, x) = K b (t)Φ v (t), where the linear operator K b (t) : H → F m is given by

K b (t)V = -e -βt χ P m χQ 1 t V , t ≥ 0.
(2.26)

In the next section, we shall show that this operator satisfies all the properties mentioned in Theorem 2.3.

Conclusion of the proof of Theorem 2.3

The continuity of the function t → K b (t) in the weak operator topology follows from a similar property Q t , and the uniform boundedness of its norm is an immediate consequence of (2.22). To establish (2.3), we first consider the case s = 0. Let us define w(t, x) = e βt/2 v(t, x). Then there is C > 1 such that

C -1 E w (t) ≤ e βt E v (t) ≤ C E w (t), t ≥ 0. (2.27)
Furthermore, the function w must satisfy the equation

ẅ + γ ẇ -∆w = g(t, x), (2.28)
where we set

g(t, x) = e βt/2 K b (t)Φ v (t) + β 2 4 + γβ 2 -b v + β v .
Note that g(t) 2 ≤ C e βt E v (t), and since v is the optimal solution of Problem 2.7 with s = 0, we have

∞ 0 g(t) 2 dt ≤ C Q 0 [v 0 , v 1 ], [v 0 , v 1 ] H ≤ C E v (0). (2.29)
Taking the scalar product in L 2 of Eq. (2.28) with 2 ẇ + αw, carrying out some standard transformations, and using the Gronwall inequality, we derive

E w (t) ≤ Ce -δt E w (0) + C t 0 e -δ(t-θ) g(θ, •) 2 dθ.
Combining this with (2.27) and (2.29), we arrive at the required inequality (2.3) with s = 0.

To prove (2.3) with an arbitrary s = θ > 0, we repeat the above argument with the initial point moved to θ. Namely, considering an analogue of Problem 2.8 on the half-line R θ , one can prove by the same argument as above that

ζ θ (t) = -e -βt P m χQ 1 t Φ θ v (t) , t > θ. It follows that if v is the solution of (1.7), (2.2) with η(t) = K b (t)Φ v (t) and s = θ, then Q θ [v 0 , v 1 ], [v 0 , v 1 ] H = 1 2 ∞ θ e βt E v (t) + e -βt P m χQ 1 t Φ v (t) 2 dt ≤ C e βθ E v (θ).
We can now establish (2.3) by literal repetition of the argument used above for problem on the half-line R + . It remains to establish the uniqueness of solution. Let v(t, x) be a function that belongs to the space

C(R + , H 1 0 ) ∩ C 1 (R + , L 2 ) and satisfies Eqs. (1.7) and (2.2) with η(t) = K b (t)Φ v (t) and v 0 = v 1 ≡ 0. Since η ∈ L 1 loc (R + , L 2
), inequality (1.13) and the boundedness of the operator K b (t) imply that

Φ v (t) H ≤ C t 0 Φ v (s) H ds.
By the Gronwall inequality, we conclude that v ≡ 0. The proof of Theorem 2.3 is complete.

Observability inequalities

This section is devoted to the proof of a truncated observability inequality used in Section 2.2. Namely, let us consider the homogeneous equation

v -γ v -∆v + b(t, x)v = 0, x ∈ Ω, (3.1)
supplemented with the Dirichlet boundary condition (1.8). We first establish a "full" observability inequality for solutions of low regularity and then use a compactness argument to derive the required result.

Observability of low-regularity solutions

Theorem 3.1. Let the hypotheses of Theorem 2.3 be fulfilled, let χ ∈ C ∞ 0 (R 3 ) be such that supp χ ∩ Ω ⊂ ω and χ(x) = 1 for x ∈ Ω δ/2 (x 0 ), and let

T > 2 sup x∈Ω |x -x 0 |, (3.2)
where x 0 ∈ R 3 is the point entering Condition 2.2. Then there are positive constants σ 0 (r) and M 6 = M 6 (R, r, T, γ, χ) such that, for any initial data [v 0 , v 1 ] ∈ H -σ with 0 ≤ σ ≤ σ 0 (r) the solution v(t, x) of problem (3.1), (1.8), (1.9) satisfies the inequality

v 0 2 -σ + v 1 2 -σ-1 ≤ M 6 T 0 χv(t) 2 -σ dt. (3.3)
Proof. The proof is divided into three steps: we first establish a unique continuation property for low-regularity solutions; we then use the Bardos-Lebeau-Rauch observability inequality to establish a high-frequency observability; and, finally, these two results are combined to prove the required observability inequality. Without loss of generality, we shall assume that γ = 0; the general case can easily be treated by the change of variable v(t) = e γt/2 w(t).

Step 1. Unique continuation property. Let σ ∈ (0, 1) be so small that the initial-boundary value problem for (3.1) is well posed in H -σ and the conclusion of Lemma 1.6 is true. We claim that if a solution v(t, x) of problem (3.1), (1.8), (1.9) with initial data [v 0 , v 1 ] ∈ H -σ is such that χv = 0 on J T × Ω, then v ≡ 0. Indeed, let ϕ ∈ C ∞ 0 (R) be such that ϕ = 1 around 0. For h ∈ (0, 1], let v h be the solution of (3.1), (1.8) with the initial condition

v h t=0 = ϕ(-h 2 ∆)v 0 , ∂ t v h t=0 = ϕ(-h 2 ∆)v 1 .
We set

u h = v h -ϕ(-h 2 ∆)v. Then ∂ 2 t u h -∆u h + b(t, x)u h = [ϕ(-h 2 ∆), b]v, [u h , ∂ t u h ] t=0 = 0. (3.4)
By Proposition 1.3 (with σ = 0) and Lemma 1.6, for all t ∈ J T we have

Φ u h (t) H 0 ≤ M 2 T 0 ϕ(-h 2 ∆), b(s) v(s) -1 ds ≤ M 2 M 4 T 0 h σ v(s) -σ ds ≤ C h σ [v 0 , v 1 ] H -σ . (3.5)
Suppose we have shown that

Φ v h (t) H 0 ≤ C h σ + h 1-σ [v 0 , v 1 ] H -σ . (3.6)
Then these two inequalities imply that

lim h→0 ϕ(-h 2 ∆)v 0 + ϕ(-h 2 ∆)v 1 -1 = 0,
whence we conclude that v 0 = v 1 = 0 and, hence, v ≡ 0.

To prove (3.6), recall that, by [START_REF] Duyckaerts | On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials[END_REF], the observability inequality (3.3) is true with σ = 0. Combining this with (1.13), we derive

Φ v h (t) 2 H 0 ≤ M 2 2 Φ v h (0) 2 H 0 ≤ M 2 2 M 6 T 0 χv h (s) 2 ds ≤ C T 0 χu h (s) 2 ds + C T 0 χ ϕ(-h 2 ∆) v(s) 2 ds.
(3.7) By (3.5), the first term on the right-hand side does not exceed Ch 2σ [v 0 , v 1 ] 2 H -σ . To estimate the second term, we write

χ ϕ(-h 2 ∆) v = ϕ(-h 2 ∆)χv =0 + χ, ϕ(-h 2 ∆) v.
Using Lemma 1.5 and Proposition 1.3, we get

χϕ(-h 2 ∆)v(t) ≤ M 3 h 1-σ v(t) -σ ≤ M 3 M 2 h 1-σ [v 0 , v 1 ] H -σ .
Substituting these estimates into (3.7), we obtain the required inequality (3.6).

Step 2. High-frequency observability. We now prove the following weaker version of (3.3):

sup t∈JT Φ v (t) 2 H -σ ≤ C T 0 χv(t) 2 -σ dt + T 0 v(s) 2 -σ-1 ds . (3.8)
To this end, recall that inequality (3.3) is true1 for solutions of Eq. (3.1) with γ = b = 0 (see [START_REF]Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF]). Combining that inequality with (1.13), we derive

sup t∈JT S(t)[v 0 , v 1 ] 2 H -σ ≤ C T 0 χS 0 (t)[v 0 , v 1 ] 2 -σ dt,
where S(t) and S 0 (t) are defined in Section 1.2. Using now (1.14) with η ≡ 0, we see that the solution of (3.1), (1.8), (1.9) satisfies the inequality

sup t∈JT Φ v (t) 2 H -σ ≤ C T 0 χ v(t) - t 0 S 0 (t -s)[0, (bv)(s)] ds 2 -σ dt ≤ C T 0 χv(t) 2 -σ dt + C T 0 (bv)(t) 2 -σ-1 dt. (3.9)
If σ > 0 is sufficiently small, then inequalities (1.17) and (2.1) and compactness of the embedding H -σ ⊂ H -σ-δ for δ > 0 imply that

(bv)(t) -σ-1 ≤ C v(t) -σ-δ ≤ ε v(t) -σ + C ε v(t) -σ-1 ,
where ε > 0 can be chosen arbitrarily small. Substituting this into (3.9), we arrive at the required inequality (3.8).

Step 3. Conclusion of the proof. We now argue by contradiction. Suppose there is a sequence of solutions (v n ) for (3.1), (1.8), (1.9) such that

Φ v n (0) H -σ = 1, T 0 χv n (t) 2 -σ dt ≤ n -1 for n ≥ 1. (3.10)
Without loss of generality, we can assume that

v n → v weakly * in L ∞ (J T , H -σ ), ∂ t v n → ∂ t v weakly * in L ∞ (J T , H -σ-1 ),
where v ∈ X σ T is a solution of (3.1), (1.8). Combining this with (3.10) and inequality (3.8) applied to v n , we obtain

1 ≤ C T 0 v(s) 2
-σ-1 ds, so that v ≡ 0. On the other hand, it follows from the second relation in (3.10) that

T 0 χv n (t) 2 -σ-1 dt = lim n→∞ T 0 χv n (t) 2 -σ-1 dt = 0,
whence we see that χv ≡ 0. By the unique continuation property established in Step 1, we conclude that v ≡ 0. The contradiction obtained completes the proof of the theorem.

Truncated observability inequality

Let us recall that H N stands for the vector span of e 1 , . . . , e N and P N denotes the orthogonal projection in L 2 to H N The following result shows that if v (T ) belongs to H N × H N , then in (3.3) the function χv can be replaced by its projection to H m with a sufficiently large m.

Theorem 3.2. Under the hypotheses of Theorem 3.1, for any N ≥ 1 there is

an integer m ≥ 1 such that if v ∈ X σ T is a solution of (3.1), (1.8) satisfying the condition Φ v (T ) ∈ H N × H N , then Φ v (0) 2 H -σ ≤ 2M 6 T 0 P m (χv(t)) 2 -σ dt. (3.11)
Proof. We repeat the argument used in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] for the case of the linearised Navier-Stokes system. It suffices to prove that if v ∈ X σ T is a solution of (3.1), (1.8) satisfying the condition Φ

v (T ) ∈ H N × H N , then T 0 χv(t) 2 dt ≤ C T 0 χv(t) 2 -σ dt, (3.12)
where C > 0 depends only on N and R; see Section A.3 in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF]. We argue by contradiction. Suppose there are sequence (

v n ) ⊂ X σ T and (b n ) ⊂ L ∞ (J T , H r ∩ L ∞ ) such that Φ v n (T ) ∈ H N × H N , Φ v n (T ) H -σ = 1, |b n | ≤ R, (3.13) vn -γ vn -∆v n + b n (t, x)v n = 0, (3.14) T 0 χv n (t) 2 dt ≥ n T 0 χv n (t) 2 -σ dt. (3.15)
Passing to a subsequence, we can assume that there are function v and b such

that v ∈ L ∞ (J T , H -σ ), v ∈ L ∞ (J T , H -σ-2 ), b ∈ L ∞ (J T , H r ∩ L ∞ )
, and the following convergences hold:

Φ v n (T ) → Φ v (T ), v n → v weakly * in L ∞ (J T , H -σ ), vn v weakly * in L ∞ (J T , H -σ-1 ), vn → v weakly * in L ∞ (J T , H -σ-2 ), b n → b weakly * in L ∞ (J T × Ω) and L ∞ (J T , H r ).
It follows from the first two relations in (3.13) that 0 = Φ v (T ) ∈ H and Φ v n (T ) H ≤ C. By the uniqueness of solution for (1.7), the functions v n must belong to X 0 T , and their norms are uniformly bounded. Inequality (3.15) now implies that χv n L 2 (JT ,H -σ ) → 0, whence we conclude that χv ≡ 0. Suppose we have proved that that we can pass to the limit as n → ∞ in Eq. (3.14). Then the function v is a solution of the limiting equation (3.1) and, hence, the observability inequality (3.3) holds for it. It follows that v ≡ 0 and therefore Φ v (T ) = 0. The contradiction obtained proves the required inequality (3.12).

It remains to prove that one can pass to the limit in (3.14). The only nontrivial term is b n v n . We need to show that

b n v n , ϕ → bv, ϕ for any ϕ ∈ C ∞ 0 , (3.16)
when C ∞ 0 is considered on the open set (0, T ) × Ω, and •, • denotes the duality between the space of distributions and C ∞ 0 . It follows from Lemma 5.1 in [Lio69] the sequence {v n } is relatively compact in the space L p (J T , H -σ-ε ) for any p ∈ [1, ∞) and ε > 0. Therefore so is the sequence {v n ϕ}. Choosing σ and ε so small that σ + ε ≤ r, we now write

b n v n , ϕ = T 0 b n (t, •), (v n ϕ)(t, •) dt.
Since the weak * convergence in L ∞ (J T , H r ) is uniform on compact subsets of L 1 (J T , H -r ), the above-mentioned compactness of {v n ϕ} implies that

T 0 (b n , v n ϕ) dt = T 0 (b n -b, v n ϕ) dt + T 0 (bϕ, v n ) dt → T 0 (bϕ, v) dt,
where we used the fact that bϕ ∈ L ∞ (J T , H r ) ⊂ L 1 (J T , H σ ). This completes the proof of (3.16) and that of Theorem 3.2.

Observability inequality for the wave equation

In this section, we prove that if [v 0 , v 1 ] ∈ H -σ with σ ∈ (0, 1 2 ), then the solution v(t, x) of problem (3.1), (1.8), (1.9) with γ = 0 and b ≡ 0 satisfies (3.3). To this end, we first recall that H s = H s D for 1 2 < s < 3 2 , and therefore, by

Theorem 1.3 in [DL09], for any [u 0 , u 1 ] ∈ H σ+1 there is ζ ∈ L 2 (J T , H σ ) such that ζ L 2 (JT ,H σ ) ≤ C u 0 σ+1 + u 1 σ , (3.17)
and the solution u ∈ C(J T , H σ+1 0

) ∩ C 1 (J T , H σ ) of the problem ü -∆u = χ(x)ζ(t, x), u(0, x) = u 0 (x), u(0, x) = 1 (x), satisfies the relation u(T, x) = u(T, x) ≡ 0. (3.18) Now let v ∈ C(J T , H -σ ) ∩ C 1 (J T , H -σ-1
) be the solution of (3.1), (1.8), (1.9) with [v 0 , v 1 ] ∈ H -σ . Then, in view of (3.18), we have

(u 0 , v 1 ) -(u 1 , v 0 ) = T 0 d dt ( u, v) -(u, v) dt = T 0 (ü, v) -(u, v) dt = T 0 (∆u + χζ, v) -(u, ∆v) dt = T 0 (ζ, χv) dt ≤ ζ L 2 (JT ,H σ ) χv L 2 (JT ,H -σ ) . Taking u 0 = (-∆) -σ-1 v 1 ∈ H σ+1 0 and u 1 = -(-∆) -σ v 0 ∈ H σ
and using (3.17), we arrive at the required inequality (3.3).

Main result: stabilisation of the non-linear problem

Let us consider the nonlinear problem (0.1)-(0.3), where Ω ⊂ R 3 is a bounded domain with a C 2 -smooth boundary Γ. We assume that γ > 0, and the function f ∈ C 2 (R) satisfies conditions (1.2) and (1.5). Let us consider a solution û(t, x) with initial data [û 0 , û1 ] ∈ (H 1 0 ∩ H 2 ) × H 1 0 and a right-hand side h ∈ W 1,∞ (R + , L 2 ). By Proposition 1.2, there is C > 0 such that û(t, •) 2 + ∂ t û(t, •) 1 ≤ C for all t ≥ 0.

(4.1)

Let us define a function b by the relation b(t, x) = f ′ (û(t, x)). It follows from (4.1) and the conditions imposed on f that b satisfies (2.1) with r = 1. Therefore, if Conditions 2.1 and 2.2 are also satisfied, then Theorem 2.3 is applicable, and one can construct a feedback law K û(t) := K b (t) exponentially stabilising the linearised problem (1.7), (1.8). The following theorem, which is the main result of this paper, shows that the same law stabilises locally exponentially also the nonlinear problem.

Theorem 4.1. Under the above hypotheses, there are positive constants C and ε such that, for any initial data [u 0 , u 1 ] ∈ H 1 0 × L 2 satisfying the inequality

u 0 -û0 1 + u 1 -û1 ≤ ε, (4.2)
problem (0.4), (0.2), (0.3) with η(t, x) = K û(t)Φ u-û (t) has a unique solution u ∈ X := C(R + , H 1 0 ) ∩ C 1 (R + , L 2 ), for which inequality (0.5) holds.

Proof. The proof based on a fixed point argument is rather standard (cf. Section 4 in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF]), and therefore we shall only outline it.

We seek a solution of the form u = û + v. Then v must be a solution of the problem v + γ v -∆v + f (û + v) -f (û) = η(t, x), (4.3) v ∂Ω = 0, (4.4) v(0, x) = v 0 (x), ∂ t v(0, x) = v 1 (x), (4.5)

where v 0 = u 0 -û0 , v 1 = u 1 -û1 , and η(t) = K û(t)Φ v (t). Let us fix θ > 0 and define the metric space

Z θ := {v ∈ X : Φ v (0) = [v 0 , v 1 ], E v (t) ≤ θ e -βt E v (0) for t ≥ 0}.
If we construct a solution v ∈ Z θ , then the corresponding function u = û + v will be the required solution of the original problem. The fact that there are no other solutions can easily proved by a standard argument (e.g., see Chapter 1 of [START_REF] Lions | Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires[END_REF]).

Step 1. Let us endow Z θ with the metric generated by the norm

v Z = sup t≥0 e βt E v (t) 1/2 .
Define a mapping Ξ : Z θ → X that takes w ∈ Z θ to the solution of (1.7)-(1.9), in which b(t, x) = f ′ (û(t, x)), η(t, x) = K û(t)Φ v (t) -f (û + w) -f (û) -b(t, x)w .

The mapping Ξ is well defined. Indeed, the homogeneous problem (that is system (1.7), (2.2) with η = K û(t)Φ v (t)) is well posed in view of Theorem 2.3, while a solution of the inhomogeneous equation can be written in the form of the Duhamel integral. Suppose we have shown that, for an appropriate choice of θ, the mapping Ξ is a contraction in Z θ . Then the unique fixed point v ∈ Z θ for Ξ is a solution of (4.3)-(4.5), and it satisfies (0.5) with C = θ.

Step 2. Let us prove that Ξ maps the space Z θ into itself. Define H = H 1 0 ×L 2 and denote by U (t, s) : H → H the operator that takes [v 0 , v 1 ] to v(t), where v(t, x) is the solutions of (1.7), (1.8), (2.2) with η(t) = K û(t)Φ v (t). Then we can write (Ξw)(t) = U (t, 0)[v 0 , v 1 ] - 2 H . Combining this with (4.9) and assuming that θ is sufficiently large and θε 2 ≤ 1, we see that Ξ(Z θ ) ⊂ Z θ .

Step 3. It remains to prove that Ξ is a contraction. It follows from (4.6) that if w 1 , w 2 ∈ Z θ , then (Ξw 1 -Ξw 2 )(t) = t 0 U (t, s)[0, h(s)] ds, where h(s) = f (û + w 2 ) -f (û + w 1 ) -f ′ (û)(w 2 -w 1 ). Repeating the argument used in the derivation of (4.9), we obtain , where w = w 1 -w 2 . Substituting this into (4.10) and using (4.2), we derive Ξw 1 -Ξw 2 2 Z ≤ C 5 θε 2 w 2 Z . Hence, if ε > 0 is sufficiently small, then Ξ is a contraction. This completes the proof of Theorem 4.1.

2

  Cauchy problem for a semi-linear wave equation . . . . . . . . . 5 1.2 Cauchy problem for the wave equation with low-regularity data . 6 1.3 Commutator estimates . . . . . . . . . . . . . . . . . . . . . . . . 9 Stabilisation of the linearised equation 12 2.1 Main result and scheme of its proof . . . . . . . . . . . . . . . . . 12 2.2 Construction of a stabilising control . . . . . . . . . . . . . . . . 14 2.3 Dynamic programming principle and feedback law . . . . . . . . 17 2.4 Conclusion of the proof of Theorem 2.3 . . . . . . . . . . . . . . 19 0 Introduction Let us consider the damped non-linear wave equation (NLW)

Acknowledgements.

  This work was initiated when the third author was visiting The University of Monastir (Tunisia) in April of 2011. He thanks the institute for hospitality. The research of TD was supported by the ERC starting grant DISPEQ , the ERC advanced grant BLOWDISOL (No. 291214), and the ANR JCJC grant SchEq. The research of AS supported by the Royal Society-CNRS grant Long time behavior of solutions for stochastic Navier-Stokes equations (No. YFDRN93583) and the ANR grant STOSYMAP (No. ANR 2011 BS01 015 01).

  ≥ 0, such that the following properties hold. Time continuity and boundedness. The function t → K b (t) is continuous from R + to the space L(H 1 0 × L 2 , F m ) endowed with the weak operator topology, and its norm is bounded by C. Exponential decay. For any s ≥ 0 and [

  where we set g(t, x) = f (û + w) -f (û) -f ′ (û)w. By Theorem 2.3, the operator norm of U (t, s) satisfies the inequality It follows from the conditions imposed on f , the Taylor expansion, and inequality (4.1) that, for any w ∈ Z θ and [v 0 , v 1 ] ∈ H satisfying (4.2), we have

				U (t, s) 2 L ≤ C e -β(t-s) , t ≥ s ≥ 0.	(4.7)
	Combining this with (4.6), we see that
	(Ξw)(t) 2 H ≤ 2 U (t, 0) 2 L [v 0 , v 1 ]	H + 2 2	0	t	U (t, s) L g(s) ds	2
		≤ 2Ce -βt [v 0 , v 1 ]	2 H +	0	t	e βs/2 g(s) ds	2	.	(4.8)
	Now note that (see Section 4 in [BRS11])
	sup t≥0	0	t	e βs/2 g(s) ds	2	t≥0 ≤ C 1 sup	t	t+1	e 2βs g(s) 2 ds.
	Substituting this into (4.8), we derive
	Ξw 2 Z ≤ 2 [v 0 , v 1 ]	2 H + C 1 sup t≥0		t	t+1	e 2βs g(s) 2 ds.	(4.9)
								t
									U (t, s)[0, g(s)] ds,	(4.6)
								0

g(s) 2 ≤ C 2 1 + û(s) 2 1 + w(s) 2 1 w(s) 4 1 ≤ C 3 θε 2 (1 + θε 2 )e -2βs [v 0 , v 1 ]

The paper[START_REF]Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] deals with the boundary control and establishes the observability inequality (3.3) in the scale of Sobolev spaces. On the other hand, the paper[START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF] is devoted to the case of distributed control and proves (3.3) with σ = 0. Even though it is commonly accepted that the observability inequality is true for the scale of Sobolev spaces also for a distributed control, we were not able to find an accurate proof in the literature and outline it in Section 3.3 for the reader's convenience.


Nejib Zemzemi

Simon Labarthe

Rémi Dubois

Yves Coudière

From Body Surface Potential to Activation Maps on the Atria: a Machine Learning Technique

The treatment of atrial fibrillation has greatly changed in the past decade. Ablation therapy, in particular pulmonary vein ablation, has quickly evolved. However, the sites of the trigger remain very difficult to localize. In this study we propose a machine-learning method able to non-invasively estimate a single site trigger. The machine learning technique is based on a kernel ridge regression algorithm. In this study the method is tested on a simulated data. We use the monodomain model in order to simulate the electrical activation in the atria. The ECGs are computed on the body surface by solving the Laplace equation in the torso.

Introduction

The inverse problem in cardiac electrophysiology also known as electrocardiography imaging (ECGI) is a new and a powerful diagnosis technique. It allows the reconstruction of the electrical potential on the heart surface from electrical potentials measured on the body surface. This non-invasive technology and other similar techniques like the electroencephalography imaging interest more and more medical industries. The success of these technology would be considered as a breakthrough in the cardiac and brain diagnosis. However, in many cases the quality of reconstructed electrical potential is not sufficiently accurate. The difficulty comes from the fact that the inverse problem in cardiac electrophysiology is well known as a mathematically ill-posed problem. Different methods based on Thikhnov regularization [START_REF] Ghosh | Application of l1-norm regularization to epicardial potential solution of the inverse electrocardiography problem[END_REF] have been used in order to regularize the problem, but still the reconstructed electrical potential is not sufficiently satisfactory. In this study we present a machine learning method based on Reproducing Kernel Hilbert Space (RKHS) [START_REF] Scholkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF]. The idea is to learn on a data base of body surface potentials (BSPs) and their correspondent electrograms (EMGs) in order to construct a metamodel. This is called the "learning phase". Afterwards the evaluation of the metamodel on a new BSP provides a statical guess of the correspendant heart potential (EGM). This second phase is called the "reconstruction phase".

Methods

In the present work the data base used for the learning phase is made of a set of couples (BSPs,EGMs) provided by our ECG simulator. In paragraph 2.1, we present the mathematical models and methods used in order to simulate the electrical potential on the atria and its correspondent BSP. In paragraph 2.2, we present the machine learning method solving the inverse problem.

Forward problem

The bidomain model is considered as the state-of-the art model describing the electrical wave propagation inside the heart [START_REF] Sundnes | Computing the electrical activity in the heart[END_REF]. Less complex and easier to solve is the monodomain model. It has been shown that the modomain model is a good approximation of the bidomain model [START_REF] Sundnes | Computing the electrical activity in the heart[END_REF][START_REF] Boulakia | Mathematical modeling of electrocardiograms: a numerical study[END_REF]. The monodomain model is a reaction-diffusion equation coupled to an ordinary differential equations system,

               A m C m ∂ t v m + I ion (v m , w) -div(σ m ∇v m ) = I stim , in Ω H , σ m ∇v m .n H = 0, on Σ. ∂ t w + g(v m , w) = 0, in Ω H . v m (0, .) = v 0 , and w(0, .) = w 0 in Ω H .
(1) where v m is the transmembrane potential.Constants A m and C m represent the rate of membrane surface per unit of volume and the membrane capacitance, respectively. The myocardium conductivity tensor is represented by σ m . The function I stim and I ion are the stimulation and the transmembrane ionic currents. The field of variables w is a vector containing different chemical concentrations and various gate variables. Its time derivative is given by the vector of functions g. Constants v 0 and w 0 are the initial conditions of the monodomain problem. The precise definitions of g and I ion depend on the electrophysiological transmembrane ionic model. In the present work we make use of one of the human myocyte model Beeler Reuter [START_REF] Beeler | Reconstruction of the action potential of ventricular myocardial fibres[END_REF].Ω H stands for the heart domain which is in our case the atria. In order to simulate the body surface potential we first need to compute the extracellular potential in the heart. Supposing the the extracellular conductivity tensor σ e = λσ i , where σ i is the intracellular conductivity tensor and λ ∈ R. The extracellular potential satisfies

(λ + 1) div(σ i ∇u e ) = -div(σ i ∇v m ). (2)
One can easily check that

u e = 1 (1 + λ) (< v m > -v m ) (3)
is a solution of equation ( 2). Morover, it is the unique solution satisfying

ΩH u e = 0.
Here,

< v m >= 1 |Ω H | ΩH v m is the mean value of v m in space.
In order to compute the body surface potential we need to solve a Laplace equation on the torso with a Dirichlet boundary condition on the heart-torso interface [START_REF] Boulakia | Mathematical modeling of electrocardiograms: a numerical study[END_REF].

     div(σ T ∇u T ) = 0, in Ω T , σ T ∇u T .n = 0, on Γ ext , u T = u e , on Σ. (4)
The forward problem algorithm is: (a) compute the transmembrane potential by solving equation ( 1), (b) compute the extracellular potential using the formula (3), (c) compute the torso potential by solving (4). As it was explained in the introduction we build a synthetic data base of BSPs and their correspondant EGMs. Each sample of BSPs and EGMs corresponds to a stimulation location. We simulated n heart beats, each one corresponds to a given I stim . We use finite element method in order to solve equations ( 1) and ( 4), a space discretisation of the heart and torso domains is then needed. Since we are interested in targeting ectopic beats in the atria, we only consider the electrical activation in the atria. The finite element geometry of atria is given in Figure 2.1 (left). it was embedded in a torso geometry given in Figure 2.1 (right) [START_REF] Klepfer | The effects of inhomogeneities and anisotropies on electrocardiographic fields:a three-dimensional finite element study[END_REF].

Inverse problem: a regression method

As explained in the previous paragraph, we have n samples of BSP and EGMs.

The sequence

(BSP i , EGM i ) i=1...n ∈ R p×m × R q×m
is the data set of our metamodel. Here p (respectively, q) is the number of potential measurement locations on the body surface (respectively, on the heart surface), and m is the number of time steps. We denote by (x k ) k=1...p (respectively, (y k ) k=1...q ) the positions of the BSP measurements (respectively, positions of EGMs measurements) and (t l ) l=0...m-1 the times of the recordings. For the i th element of our data set we have, BSP i ∈ R p×m , where BSP i (l × p + k) = u T (x k , t l ), for k = 1, ..., p and l = 0, ..., m -1, and EGM i ∈ R q×m where, EGM i (l × q + k) = u e (y k , t l ), for k = 1, ..., q and l = 0, ..., m -1.

The main goal is to build a function f able to accurately map a BSP to an EGM. We use a kernel ridge regression method based on the gaussian kernel

K(x, y) = e - |x -y| 2 2σ 2 , ∀x, y ∈ R m×p .
We look for f in a Reproducing Kernel Hilbert Space (RKHS) (H, •, • H ) characterized by the following property,

∀f ∈ H, ∀x ∈ R m×p , f (x) = f (•), K(•, x) H .
(5) where •, • H is the inner product in H. The use of the Gaussian kernel can be motivated by the following property (see [START_REF] Saunders | Ridge regression learning algorithm in dual variables[END_REF]): given a compact subset K of R m×p , the set of the restriction to K of functions from H is dense in the set of continuous functions from K to R. In other word, H is rich enough to approximate any possible continuous mapping from the BSP to a given lead of the EGM.

Consider in H the following regularized least square problem: given a training set {(x i , y

i )} n i=1 ∈ (R m×p × R) n , solve min f ∈H 1 n n i=1 (f (x i ) -y i ) 2 + λ f 2 H , (6)
where λ > 0 is a given regularization coefficient and n is the number of the data set element. The representer theorem ( [START_REF] Klepfer | The effects of inhomogeneities and anisotropies on electrocardiographic fields:a three-dimensional finite element study[END_REF]) states that the solution to ( 6) can be written as:

n i=1 α i K(•, x i ),
where α = (α i ) i=1...n ∈ R n is the new unknown. Defining the Gram matrix K = (K ij ) i,j=1...n with K ij = K(x i , x j ), the training problem ( 6) is equivalent to solving:

min α∈R n 1 n (Kα -y) T (Kα -y) + λα T Kα ,
whose solution is given by

α = (K + λnI) -1 y. (7)
Here y ∈ R n , and the i th element of y represents an element of EGM i meaning the value of u e in a fixed position at at a given time. In order to consider all the positions on the atria surface and all the time sequence. The training phase consists in solving this system:

A = (K + λnI) -1 Y, (8)
with A, Y ∈ R n× (q×m) , where Y = (EGM T i ) i=1...n . Once the training phase is completed, the reconstruction of EGM from a given heart beat BSP ∈ R p×m is obtained, according to [START_REF] Beeler | Reconstruction of the action potential of ventricular myocardial fibres[END_REF], by

EGM = KA, (9)
with

K i = K(BSP, BSP i ), i = 1 . . . n.

Results

In this section we present numerical simulations of the forward and inverse problems. We first built n = 400 heart beats using our computer simulator as described in section (2.1). In this work we reconstruct the EGMs on all the whole atria surface (q = 1994), from 264 BSPs (p = 264). In figure 3 we present two example of simulation depending on the location of the stimulus, in each case we show two snapshots one at 10ms (left) and the other at 90 ms (right). In the top two figures we show an example of simulation where the atria is activated in the left superior pulmonary vein and in figure 3 (bottom) the right atria appendage is activated. After building the metamodel from equation ( 8), we propose to test the regression method on different example. As expected the reconstruction of EGMs from BSPs belonging to the training data set is perfectly accurate. On the contrary, if the BSP does not belong to the training data set, the accuracy of the EGMs reconstruction is not accurate especially in terms of amplitude where the l ∞ relative error could reach 50%, an example is given in figure 4 where the amplitude of the exact solution (continuous red line) is twice higher then the RKHS solution (dashed green line). Nevertheless, the

Discussion

In this work we showed an original approach based on a machine learning technique in order to solve the inverse problem in electrocardiography. This method allows solving the inverse problem in real time: since the metamodel could be build off-line, in practice we only need to evaluate the metamodel according to equation (9). This does not require solving any linear system. The reconstruction of EGMs is accurate if the training data base is sufficiently rich. The accuracy of the produced EGMs depends on how close is the BSP from the elements of the training set. This means that it is difficult to reproduce an information which is far from the training data base. It is for instance difficult to reproduce accurate EGM for a multiple sites activation based on a data set containing only simulation with single site stimulus. This limitation could be tackled by including multiple site stimulations in the training data set, at the same time this will increase the complexity of the training phase. But since this phase is off line, the computational time of the reconstruction would be slightly affected.

Conclusion

In this work we have considered a machine learning approach, based on the kernel ridge regression method, in order to construct activation maps on the atria from a set of BSP measurements. The procedure has been trained using synthetic data from numerical simulations, based on a 3D mathematical model of the ECG involving a mathematical description of the electrical activity of the heart and the torso. Several examples showed the method is able to reconstruct conveniently the EGM information included in the training data set, and is robust when reconstructing situations close to the training data set. In this study we showed that a single site stimulus is localized with 0.5 cm error. The accuracy could be improved if the training set is much rich. In the proposed approach, the solution depends on all training examples, which may be too expensive. This point could be improved in future works. In future works this method would be trained and tested on clinical data. The ECG simulator would be used in order to enrich the clinical data.

Figure 1 .

 1 Figure 1. Two-dimensional geometrical description: heart domain Ω H , torso domain Ω T (extramyocardial regions), heart-torso interface Σ and torso external boundary Γ ext .

Figure 2 .

 2 Figure 2. Finite element computational domains: Atria geometry with the different locations of stimulus used to construct the training data set (lfet). Torso geometry with different BSP measurements locations (right).

Figure 3 .

 3 Figure 3. Snapshots of the simulated extracellular potential. The atria is activated in the pulmonary vein (top) and in the right atria (bottom).

Figure 4 .

 4 Figure 4. Comparison between the exact and the reconstructed EGM at a given location on the atria: Exact solution (red continuous line) and the one reconstructed with RKHS (green dashed line).

Figure 5 .

 5 Figure 5. Comparison of the activation times: exact activation times (left) and activation times reconstructed with RKHS metamodel (right).

Acknowledgements

This work was partially supported by an ANR grant part of Investissements dAvenir program reference ANR-10-IAHU-04