Jérémi Dardé

ITERATED QUASI-REVERSIBILITY METHOD APPLIED TO ELLIPTIC AND PARABOLIC DATA COMPLETION PROBLEMS

Keywords: 2010 Mathematics Subject Classification. Primary: 35A15, 35R25, 35R30; Secondary: 35N25 Elliptic inverse problems, Parabolic inverse problems, Quasireversibility method. 1

We study the iterated quasi-reversibility method to regularize illposed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data.

1. Introduction. We consider data completion problems for elliptic and parabolic operators. We start with elliptic operators: we consider a bounded domain Ω ⊂ R d , d ≥ 2, with Lipschitz boundary (see [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF]). Let ν ∈ L ∞ (∂Ω, R d ) be the exterior unit normal of ∂Ω, and Γ, Γ c ⊂ ∂Ω, such that ∂Ω = Γ ∪ Γ c and meas(Γ), meas(Γ c ) > 0. Let σ : Ω → R d×d be a real matrix valued function such that σ ∈ W 1,∞ (Ω) d×d and σ = σ T , c |ξ| 2 ≤ σξ • ξ, ∀ξ ∈ R d , a.e. in Ω.

The data completion problem is:

Problem. For f , g D and g N in L 2 (Ω) × L 2 (Γ) × L 2 (Γ), find u ∈ H 1 (Ω) such that    -∇ • σ∇u = f in Ω u = g D on Γ σ∇u • ν = g N on Γ.
This problem is well-known to be ill-posed (see [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF][START_REF] Belgacem | Why is the Cauchy problem severely ill-posed?[END_REF] and the references therein): it does not necessarily admit a solution for any data (f, g D , g N ), and if a solution exists, it does not depend continuously on the data. On the other hand, if the problem admits a solution u s , this solution is necessarily unique (see e.g. [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF][START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF]).

Such a problem is encountered in many practical applications, among others in plasma physic [START_REF] Ben Abda | Minimization of an energy error functional to solve a Cauchy problem arising in plasma physics: the reconstruction of the magnetic flux in the vacuum surrounding the plasma in a Tokamak[END_REF][START_REF] Blum | Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time[END_REF], or corrosion detection problems [8, [START_REF] Sincich | Lipschitz stability for the inverse Robin problem[END_REF]7,[START_REF] Fasino | An inverse Robin problem for Laplace's equation: theoretical results and numerical methods[END_REF][START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF]]. We will be particularly interested in the corrosion detection problem: in this problem, u is the electrical potential inside a conductive object Ω, σ is the conductivity of the object, g N represent a current imposed on Γ, accessible part of the boundary of Ω, and g D is the corresponding potential measured on Γ. The aim is to determine if some portion of the inaccessible part of the boundary Γ c is corroded.

J ÉR ÉMI DARD É

Mathematically, it exists a non-negative function µ define on Γ c such that σ∇u • ν + µ u = 0 on Γ c and the objective is to reconstruct µ: µ = 0 on the healthy part of Γ c , and µ > 0 on the corroded part. In section 6.1, we test our method on this problem.

The data completion problem is known to be severely, even exponentially illposed [START_REF] Belgacem | Why is the Cauchy problem severely ill-posed?[END_REF]. Therefore ones needs to use regularization methods to try to reconstruct u. Several methods have been proposed to stabilize the problem: see, e.g., [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Cimetière | Solution of the Cauchy problem using iterated Tikhonov regularization[END_REF][START_REF] Ben Belgacem | Extended-domain-Lavrentiev's regularization for the Cauchy problem[END_REF][START_REF] Azaïez | On Cauchy's problem: II. Completion, regularization and approximation[END_REF][START_REF] Boukari | A Convergent Data Completion Algorithm Using Surface Integral Equations[END_REF][START_REF] Burman | A stabilized nonconforming finite element method for the elliptic Cauchy problem[END_REF] and the references therein.

We are also interesting in the data completion problem for the heat equation, which is quite similar to the elliptic one, except that this time u solves a parabolic equation. Such inverse problem appears naturally in thermal imaging [START_REF] Bryan | An inverse problem in thermal imaging[END_REF] and inverse obstacle problems [START_REF] Harbrecht | On the numerical solution of a shape optimization problem for the heat equation[END_REF][START_REF] Ikehata | The enclosure method for the heat equation[END_REF]. For T > 0, we define Q := (0, T ) × Ω. Let f be in L 2 (Q), g D and g N in L 2 (0, T ; L 2 (Γ)). The data completion problem is then

Problem. find u ∈ H 1,1 (Q) := L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) such that    ∂ t u -∆u = f in Q u = g D on (0, T ) × Γ ∇u • ν = g N on (0, T ) × Γ
This parabolic data completion problem is also severely ill-posed (see e.g. [START_REF] Puzyrev | On an Ill-Posed Problem for the Heat Equation[END_REF]). Note that it is not mandatory to impose an initial condition u(0, .) on Ω to obtain the uniqueness of the solution (if such a solution exists). Again, regularization methods are needed to obtain a stable reconstruction of u from the data f, g D and g N .

The quasi-reversibility method is such a regularization method, introduced in the pioneering work of Lattès and Lions [START_REF] Lattès | The Method of Quasi-reversibility: Applications to Partial Differential Equations[END_REF] to regularize elliptic, parabolic (and even hyperbolic) data completion problems. The mean idea of the method is to approach the ill-posed data completion problem by a family of well-posed variational problems of higher order (typically fourth order problems) depending on a (small) parameter ε. The solution of the regularized problem converges to the solution of the data completion problem, when the parameter ε goes to zero. The quasi-reversibility method presents interesting features: first of all the variational problems appearing in the method are naturally discretized using finite element methods, thus the method can be used in complicated geometries, an interesting property when the method is used in an iterative algorithm with changing domain. Furthermore, the method is independent of the dimension. Since its introduction, the quasi-reversibility method has been successfully used to reconstruct the solution of elliptic [START_REF] Klibanov | A Computational Quasi-Reversibility Method for Cauchy Problems for Laplaces Equation[END_REF][START_REF] Clason | The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium[END_REF][START_REF] Bourgeois | Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation[END_REF][START_REF] Cao | A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation[END_REF][START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF] and parabolic [START_REF] Ames | Continuous dependence on modeling for some well-posed perturbations of the backward heat equation[END_REF][START_REF] Clark | Quasireversibility methods for non-well-posed problems[END_REF] ill-posed problems, and as a keystone in the resolution of inverse obstacle problems in the exterior approach [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF][START_REF] Dardé | The 'exterior approach': a new framework to solve inverse obstacle problems[END_REF][START_REF] Bourgeois | The "exterior approach" to solve the inverse obstacle problem for the Stokes system[END_REF].

In the present paper, we are interested in a natural extension of the quasireversibility method, the iterated quasi-reversibility method : it consists in solving iteratively quasi-reversibility problems, the solution of each one depending on the solution of the previous one. We therefore obtain a sequence of quasi-reversibility solutions, which converges to the exact solution of the data completion problem if exact data are provided, for any choice of the regularization parameter ε. This has interesting consequences from a numerical point of view: first of all, one can now choose a large value for the parameter of regularization ε, leading to an improvement in the conditioning of the finite-element problems, without lowering the quality of the reconstruction. This is not the case for the standard quasi-reversibility method, for which it is mandatory to use small ε to obtain a good reconstruction. Furthermore, in presence of noisy data, we present a method to choose when to stop the iterations according to the amplitude of noise on the data, based on the Morozov discrepancy principle, which ensure both stability and convergence of the method. The main drawback of this extension of the quasi-reversibility method, comparatively to the standard quasi-reversibility, is that several problems have to be solved to obtain a good reconstruction. However, as it is the same variational problem that appears in each iteration of the method, one can precompute a factorization of the finite-element matrix. Hence, the cost of the method is not significantly higher.

The paper is organized as follows. In section 2, we introduce an abstract setting to treat both data completion problems we are interested in at once. In section 3, we present the standard quasi-reversibility regularization in this abstract setting, and prove some results we need to study the iterated quasi-reversibility method. In section 4, we focus on the iterated quasi-reversibility method, both in the case of exact data and noisy data. In section 5, we show that the abstract setting apply to both elliptic and parabolic data completion problems. In section 6, numerical results are presented, demonstrating the feasibility and efficiency of the method for both problems.

2. An abstract setting for data completion problems. In this section, we set up an abstract setting corresponding to both data completion problems we are interested in.

Let X , Y be two Hilbert spaces endowed with respective scalar products (., .) X and (., .) Y , and corresponding norms denoted . X and . Y .

Let y ∈ Y. Both of our data completion problems can be written in the following way: find x ∈ X such that Ax = y, with A : X → Y a continuous linear operator with following properties:

• A is one-to-one

• A is not onto • Im(A) Y = Y.
In this setting, y plays the role of the data, and x the solution of our data completion problem. The problem is obviously ill-posed: indeed, as A is not onto, there exist y in Y for which the problem admits no solution. We define Y adm := Im(A) the set of admissible data, and

Y nadm = Y \ Y adm the set of non-admissible ones. By definition, Y adm is dense in Y. Actually, this is also true for Y nadm Proposition 1. The set Y nadm is dense in Y.
Proof. This is quite simple: suppose it exists ȳ ∈ Y adm and δ > 0 such that

y -ȳ Y ≤ δ ⇒ y ∈ Y adm . It exists x ∈ X s.t. Ax = ȳ.
Let y be any element of Y, y = ȳ. We define ỹ =

y -ȳ y -ȳ Y δ 2 + ȳ. Obviously, ỹ -ȳ Y ≤ δ.
Therefore, ỹ ∈ Y adm , and it exists x ∈ X such that Ax = ỹ. A simple computation shows then that

A 2 y -ȳ Y δ (x -x) + x = y.
Hence Im(A) = Y, contradicting the assumptions on A. Therefore, for any y ∈ Y adm , for any δ > 0, there exists y δ ∈ Y nadm such that y -y δ Y ≤ δ, which ends the proof, as Y = Y adm ∪ Y nadm .

In other word, for any admissible data y exists a non-admissible one ỹ arbitrary close to y. In particular, this leads to the high instability of the problem with respect to noise: Proposition 2. For any y ∈ Y, the exists a sequence x n ∈ X such that

x n X n→∞ ----→ +∞ and Ax n n→∞ ----→ Y y. Proof. We start with y ∈ Y nadm . As Im(A) is dense in Y, it exists a sequence x n ∈ X in such that Ax n n→∞ ----→ Y y.
This sequence cannot have any bounded subsequence: indeed, if such a subsequence would exist, there would be another subsequence, denoted x m here, such that x m weakly converges to an element x in X . The operator A being linear and strongly continuous, it is weakly continuous [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], hence Ax m weakly converges to Ax. But by definition Ax m strongly converges to y. By uniqueness of the limit, we have Ax = y, and y ∈ Y adm , in contradiction with the initial assumption. Therefore, we have It is then not difficult to verify that the sequence xm verifies the researched properties.

Remark 1. Actually, if y is not an admissible data, it is shown in the proof that any sequence

(x n ) n∈N ∈ X N such that Ax n n→∞ ----→ Y y verifies lim n→∞ x n X = +∞.
This proposition has for important consequences the fact that for any admissible data y, with corresponding solution x, one can find an admissible data ỹ, with corresponding solution x, such that ỹ is arbitrarily close to y and x is arbitrarily far from x.

We retrieve here the well-known fact that the problem of noisy data is crucial in data completion problems. Clearly, it is not sufficient to build a method that (approximately) reconstruct the solution of the data completion problem for any admissible data, it is also mandatory to propose a strategy for noisy data, as in practice data are always corrupted by some noise due to inaccurate measurements.

3. Standard quasi-reversibility method. We define b a symmetric bilinear nonnegative form on X , and denote by . b the induced seminorm on X . We suppose that it exists two strictly positive constants c, C such that

c 2 x 2 X ≤ Ax 2 Y + x 2 b ≤ C 2 x 2 X .
Therefore, the symmetric bilinear form (., .) A,b , define by

∀(x, x) ∈ X , (x, x) A,b = (Ax, Ax) Y + b(x, x),
is a scalar product on X , and X endowed with this scalar product is a Hilbert space. We denote . A,b the corresponding norm, which is equivalent to the . X norm.

Obviously, there exists such a form b: it suffices to take the whole scalar product in X , b(., .) = (., .) X .

Adapting the initial idea of Jacques-Louis Lions and Robert Lattès [START_REF] Lattès | The Method of Quasi-reversibility: Applications to Partial Differential Equations[END_REF], the quasi-reversibility method applied to the abstract data completion problem defined above relies on the resolution of the following regularized problem Problem. for y ∈ Y and ε > 0, find x ε ∈ X such that

(Ax ε , Ax) Y + ε b(x ε , x) = (y, Ax) Y , ∀x ∈ X .
The quasi-reversibility equation is the Euler-Lagrange equation corresponding to the minimization over X of the energy Ax -

f 2 Y + ε x 2 b .
In other words, it is a Tykhonov regularization of the data completion problem, ε > 0 being the parameter of regularization and . b the penalization (semi)norm. Since its introduction in 1963 by A.N. Tykhonov [START_REF] Tykhonov | Solution of incorrectly formulated problems and the regularization method[END_REF], this regularization has been widely studied and used to solve inverse problems (for a complete study on the topic, see [START_REF] Engl | Regularization of inverse problems[END_REF] and the references therein). There are various methods to study such regularization method: e.g. singular value decomposition if A is compact (which is not the case in our data completion problems, see section 5) or spectral theory. Here we propose another approach to study the method, based on the variational formulation of the quasireversibility method, and on the differentiability of the approximated solution with respect to the parameter of regularization, the later being useful in the study of the iterated quasi-reversibility method.

First of all, let us verify that the quasi-reversibility problem is well-posed.

Proposition 3. For any y ∈ Y and ε > 0, the quasi-reversibility problems admits a unique solution x ε , with the following estimates:

Ax ε Y ≤ f Y , Ax ε -y Y ≤ y Y , x ε A,b ≤ 1 min(1, √ ε) y Y .
Proof. let us define the bilinear form

a ε (x, x) := (Ax, Ax) Y + ε b(x, x), ∀x, x ∈ X .
It is obviously continuous. Furthermore, for all x ∈ X, we have

a ε (x, x) ≥ min(1, ε) x 2 A,b , and therefore it is coercive. Finally, as |(y, Ax) Y | ≤ A y Y x X ≤ A y Y x A,b
, we obtain the existence and uniqueness of x ε by Lax-Milgram theorem. By definition, we have

Ax ε 2 Y ≤ Ax ε 2 Y + ε x ε 2 b = (Ax ε , y) Y ≤ Ax ε Y y Y ⇒ Ax ε Y ≤ y Y . Furthermore, (Ax ε -y, Ax ε ) Y = -ε x ε 2 b ≤ 0 ⇒ Ax ε -y 2 Y ≤ -(y, Ax ε -y) Y ≤ y Y Ax ε -y Y , implying Ax ε -y Y ≤ y Y . Finally, we have Ax ε 2 Y + ε x ε 2 b = (Ax ε , y) Y ≤ Ax ε Y y Y ≤ Ax ε 2 Y + ε x ε 2 b y Y leading to min(1, √ ε) x ε A,b ≤ Ax ε 2 Y + ε x ε 2 b ≤ y Y .
Remark 2. In particular, we always have

x ε X ---→ ε→∞ 0.
Suppose there exists x ∈ X such that Ax = y (i.e. y ∈ Y adm ). It is easily seen that x is never the solution of the quasi-reversibility problem, except in the special case y = 0 (which is always in Y adm ) for which x = 0 = x ε . In other words, there is no ε > 0 such that the quasi-reversibility method reconstructs exactly the exact solution of the data completion problem. As seen in the following corollary, the solution of the quasi-reversibility problem is also never 0, except again in the special case y = 0. Corollary 1. The three following properties are equivalent:

(i) y = 0 (ii) ∃ ε > 0 s.t. x ε = 0 (iii) ∀ε > 0, x ε = 0.
Proof. obviously, (iii) implies (ii). Furthermore, as min(1,

√ ε) x ε A,b ≤ y Y , (ii) implies (i).
Suppose it exists ε > 0 such that x ε = 0. For that particular ε and for any x ∈ X, we would have (y,

Ax) Y = (Ax ε , Ax) Y + ε(x ε , x) b = 0. As Im(A) Y = Y,
this directly implies y = 0, so (i) implies (iii). Proposition 4. Let y ∈ Y, and x ε the solution of the corresponding quasi-reversibility problem. Then Ax ε strongly converges to y (even if y is not an admissible data).

Proof. As min(1, √ ε) x ε A,b ≤ y Y , we have, for any x ∈ X , ε| b(x ε , x)| ≤ ε min(1, √ ε) y Y x b ε→0 ---→ 0.
Let (ε m ) m∈N be a decreasing sequence of strictly positive real numbers such that lim m→∞ ε m = 0, and note x m := x εm . As Ax m Y ≤ y Y , it exists a subsequence (still denoted x m ) such that Ax m weakly converges to ỹ ∈ Y. But, for all x ∈ X , we have

(y -ỹ, Ax) Y m→∞ ← ----(y -Ax m , Ax) Y = ε m b(x m , x) m→∞ ----→ 0, that is y = ỹ as Im(A) Y = Y,
and Ax m weakly converges to y. As Ax m Y ≤ y Y (proposition 3), Ax m strongly converges to y. It is then not difficult to see that Ax ε strongly converges to y as ε goes to zero.

We can now state the main theorem regarding the standard quasi-reversibility method: Theorem 3.1. Suppose y ∈ Y adm , and let x s be the (necessarily unique) solution of the abstract data completion problem. Then x ε converges to x s as ε goes to zero, and we have the estimates

Ax ε -y Y ≤ √ ε x s b , x ε b ≤ x s b and x ε -x s b ≤ x s b . Suppose y ∈ Y nadm . Then lim ε→0 x ε b = +∞.
The theorem remains valid when the . b seminorm is replaced with the . A,b norm.

Proof. Suppose first that y ∈ Y nadm . Then, as x ε is a sequence in X such that Ax ε converges to y (proposition 4), proposition 2 and remark 1 imply lim ε→0 x ε X = +∞. As

x ε 2 A,b = Ax ε 2 Y + x ε 2 b ≥ c 2 x ε X , we have lim ε→0 x ε b = +∞.
Now, suppose it exists x s such that Ax s = y. Then, choosing x = x ε -x s as test function in the quasi-reversibility problem, we obtain

Ax ε -y 2 Y + εb(x ε , x ε -x s ) = 0, (1)
which in turn implies b(x ε , x ε -x s ) ≤ 0 ⇒ x ε b ≤ x s b ⇒ x ε A,b ≤ x s A,b .
Therefore, x ε is a bounded sequence in X , and up to a subsequence it weakly converges to x. As A is a linear continuous operator, and hence is weakly continuous, proposition 4 implies Ax = y, which implies x = x s as A is one-to-one. The uniqueness of the limit implies that the whole sequence weakly converges to x s . Finally as x ε A,p ≤ x s A,p , the sequence strongly converges to x s . Subtracting εb(x s , x ε -x s ) to equation 1, we obtain

Ax ε -y 2 Y + ε x s -x ε 2 b = -εb(x s , x ε -x s ) ⇒ x s -x ε 2 b ≤ |b(x s , x ε -x s )| and by Cauchy-Schwarz inequality, x ε -x s b ≤ x s b .
Finally, equation 1 implies

Ax ε -y 2 Y ≤ ε x ε b x ε -x s b ≤ ε x s 2 b
which ends the proof.

Next, we focus on the differentiability of the solution of the quasi-reversibility method with respect to ε, a result that will be useful in the study of the iterated quasi-reversibility method.

3.1. Differentiability of the quasi-reversibility solution with respect to ε. It turns out that x ε , solution of the quasi-reversibility problem, depends smoothly on the parameter of regularization ε. Indeed, let us define the map F : ε > 0 → x ε .

Proposition 5. The map F is continuous.

Proof. We choose ε > 0 and h such that ε -|h| > 0. For any x ∈ X , we have

(Ax ε+h , Ax) Y + (ε + h) b(x ε+h , x) = (y, Ax) Y (Ax ε , Ax) Y + ε b(x ε , x) = (y, Ax) Y .

Subtracting the two equations, and choosing

x = xε,h := x ε+h -x ε , lead to Ax ε,h 2 Y + ε xε,h 2 b = -h b(x ε+h , xε,h ). In conclusion, we have min(1, ε) xε,h 2 A,b ≤ h x ε+h b xε,h b ≤ h min(1, (ε + h) -1/2 ) y Y xε,h A,b
which ends the proof.

Remark 3. If the data completion problem admits a solution x s , then F extends continuously to R + by defining

F (0) = x s . Proposition 6. We have F ∈ C 1 (R + * ; X ). For all ε > 0, F (ε) = x (1) ε , unique element of X verifying (Ax (1) ε , Ax) Y + εb(x (1) ε , x) = -b(x ε , x), ∀x ∈ X . (2)
Furthermore, x

A,b ≤ min(1, ε -3 2 ) y Y . (1) ε
Proof. By Lax-Milgram theorem, there exists a unique x

ε ∈ X verifying 2, and it clearly verifies

min(1, ε) x (1) ε 2 A,b ≤ x ε b x (1) ε A,b ≤ min(1, ε -1 2 ) y Y x (1) ε A,b .
It is a continuous function of ε: indeed, for ε > 0 and h ∈ R s.t. ε -|h| > 0, we have, for all x ∈ X , (Ax

(1) ε+h , Ax) Y + (ε + h) b(x (1) ε+h , x) = -b(x ε+h , x) (Ax (1) ε , Ax) Y + ε b(x (1) ε , x) = -b(x ε , x). Choosing x = x(1) ε,h := x (1) ε+h -x (1)
ε ∈ X and subtracting the two equations lead to

Ax (1) ε,h 2 Y + ε x(1) ε,h 2 b = -hb(x (1) ε+h , x(1) ε,h ) -b(x ε,h , x(1) ε,h ). Therefore, x(1) ε,h A,b ≤ h min(1, (ε + h) -3/2 ) + min(1, (ε + h) -1/2 ) min(1, ε -1 ) y Y implying the continuity of the map R + * ε → x (1)
ε ∈ X . Remains to be proved that

F (ε) = x (1) ε . For ε > 0 and h ∈ R such that ε -|h| > 0, we have (Ax ε+h , Ax) Y + (ε + h) b(x ε+h , x) = (y, Ax) Y -(Ax ε , Ax) Y -ε b(x ε , x) = -(y, Ax) Y -h (Ax (1) ε , Ax) Y -h ε b(x (1) ε , x) = h b(x ε , x). Choosing x = xε,h := x ε+h -x ε -hx (1)
ε and adding the three above relations lead to

Ax ε,h 2 Y +ε xε,h 2 b = -hb(x ε,h , xε,h ) ⇒ xε,h A,b ≤ h 2 C(h, ε) y Y , with C(h, ε) > c > 0.
The result follows.

A simple induction leads then to the following theorem:

Theorem 3.2. F ∈ C ∞ (R + * ; X ). For ε > 0, for all m ∈ N, d m F dε m (ε) := x (m) ε with x (m) ε defined recursively by      x (0) ε := x ε , ∀m ∈ N, x (m+1) ε is the only element of X verifying (Ax (m+1) ε , Ax) Y + ε b(x (m+1) ε , x) = -(m + 1) b(x (m) ε , x), ∀x ∈ X .
In particular, x (m) ε verifies the following estimate:

x (m) ε A,b ≤ m ! min(1, ε m+1/2 ) y Y .
If the data completion problem admits a solution x s , it is not difficult to prove that

x (m) ε A,b ≤ min(1, ε m ) -1 m! x s A,b
. Finally, we have the following generalization of corollary 1: Corollary 2. the three following properties are equivalent:

(i) y = 0 (ii) ∃ε > 0, ∃m ∈ N s.t. x (m) ε = 0 (iii) ∀ε > 0, ∀m ∈ N, x (m) ε = 0. Proof. Clearly (iii) implies (ii). Furthermore, as min(1, ε m+ 1 2 ) x (m) ε A,b ≤ m ! y Y , (ii) implies (i).
Suppose it exists ε > 0 and m ∈ N such that x

(m) ε = 0. If m = 0, then corollary 1 implies y = 0. If m > 0, as (Ax (m) ε , Ax (m-1) ε ) Y + ε b(x (m) ε , x (m-1) ε ) = -m x (m-1) ε 2
b , we obtain x (m-1) ε = 0, and by induction x ε = 0, implying again y = 0. Therefore (i) implies (iii).

3.2. Monotonic convergence of the quasi-reversibility method. In this section, y = 0. Using the results on the derivatives of x ε with respect to ε, it is easy to prove that if the data completion problem admits a solution x s , then x ε converges monotonically to x s when ε goes to zero. This is of course not the only method to obtain such results (see for example [START_REF] Engl | Regularization of inverse problems[END_REF], where spectral theory is used), but it has the advantage to be quite simple.

The main result of this section is the following Theorem 3.3. Suppose the data completion problem admits a unique solution x s . Then x ε -x s A,b is strictly increasing with respect to ε.

We need to prove first the following two results, which are true whether or not the data completion problem admits a solution:

Lemma 3.4. For all m ∈ N, for all n ∈ N, (-1) m+n b(x (m) ε , x (n) ε ) > 0.
Proof. For m ∈ N, let us define the axiom of induction:

P (m) : ∀n ∈ {0, • • • , m} , (-1) m+n b(x (m) ε , x (n) ε ) > 0. Obviously, P (0) is true, as y = 0 ⇒ x ε = 0. Suppose P (M ) is true for some M ∈ N. Let k be in {0, • • • , M + 1}. • if k = M + 1, then (-1) 2M +2 b(x (M +1) ε , x (M +1) ε ) = x (M +1) ε 2 b > 0 (as y = 0) • if k = M , then, by definition of x (M +1) ε , (-1) 2M +1 b(x (M +1) ε , x (M ) ε ) = -b(x (M +1) ε , x (M ) ε ) = Ax (M +1) ε 2 Y + ε x (M +1) ε 2 b M + 1 > 0 • if k < M , then, using successively the definition of x (k+1) ε and x (M +1) ε , we obtain b(x (M +1) ε , x (k) ε ) = -1 k + 1 (Ax (M +1) ε , Ax (k+1) ε ) Y + εb(x (M +1) ε , x (k+1) ε ) = M + 1 k + 1 (x (M ) ε , x (k+1) ε ) b . As k + 1 ∈ {0, . . . , M }, P (M ) implies (-1) M +k+1 b(x (M ) ε , x (k+1) ε ) > 0 ⇒ (-1) M +k+1 b(x (M +1) ε , x (k) ε ) > 0.
Proposition 7. The quantity Ax ε -y Y is a strictly increasing function of ε.

Proof. Defining g : ε ∈ R + * → 1 2 Ax ε -y 2 Y , we have g (ε) = (Ax ε -y, x (1) ε ) Y = -εb(x ε , x (1) ε ) > 0. J ÉR ÉMI DARD É Proof of theorem 3.3. define g := ε ∈ R + * → 1 2 x ε -x s 2 b . We have g (ε) = b(x ε - x s , x (1) ε ) and g (ε) = x (1) ε 2 b + b(x ε -x s , x (2)
ε ). Therefore

εg (ε) = ε x (1) ε 2 b + εb(x ε -x s , x (2) ε ) = ε x (1) ε 2 b -(A(x ε -x s ), Ax (2) ε ) Y -2b(x ε -x s , x (1) ε ) ( definition of x (2) ε ) = ε x (1) ε 2 b + εb(x ε , x (2) ε ) -2b(x ε -x s , x (1) ε ) ( definition of x ε and Ax s = y) = ε x (1) ε 2 b + εb(x ε , x (2) ε ) -2g (ε).
So g verifies the following ODE:

εg (ε) + 2g (ε) = ε x (1) ε 2 b + εb(x ε , x (2)
ε ), that is d dε (ε 2 g (ε)) = ε 2 x (1) ε 2 b + ε 2 b(x ε , x (2) ε ) > 0.
Therefore, ε 2 g (ε) is a strictly increasing function. As

x ε -x s b ≤ x s b and x (1) ε b ≤ 1 min(1, ε)
x s b , we have

|ε 2 g (ε)| = |ε 2 b(x ε -x s , x (1) ε )| ≤ ε 2 x ε -x s b x 1 ε b ≤ ε x s b ---→ ε→0 0,
which leads to ε 2 g (ε) > 0 ⇒ g (ε) > 0, which ends the demonstration, as x ε -

x s A,b = Ax ε -y 2 Y + x ε -x s 2 b .
4. Iterated quasi-reversibility. As seen in the previous section, the quasi-reversibility method can be viewed as a Tykhonov regularization of our abstract data completion problem. Therefore, it seems natural to study a well-known extension of such regularization, namely the iterated Tykhonov regularization method, to our problem: we then obtain the iterated quasi-reversibility method.

The iterated quasi-reversibility method consists in solving iteratively quasi-reversibility problems, each one depending on the solution of the previous one. More precisely, we define a sequence of quasi-reversibility solutions by induction :

X -1 ε = 0 X and for all M ∈ N, X M ε is the unique element of X verifying (AX M ε , Ax) Y + εb(X M ε , x) = (y, Ax) Y + εb(X M -1 ε , x), ∀x ∈ X .
It is not difficult to verify that the sequence is well-defined. In particular, it is clear that X 0 ε = x ε , solution of the quasi-reversibility problem. Our study of the iterated quasi-reversibility method is based on the following result, which highlighted the link between the solutions of the iterated quasireversibility method (X M ε ) M ∈{-1}∪N and the derivatives of x ε with respect to the parameter of regularization ε: Theorem 4.1. For all ε > 0, for all M ∈ {-1} ∪ N, we have

X M ε = M m=0 (-1) m ε m m ! x (m) ε . Proof. Denote XM ε := M m=0 (-1) m ε m m ! x (m)
ε . For M = -1, the sum is empty, therefore we have X-

1 ε = 0 X = X -1 ε . For M = 0, we also have X0 ε = x ε = X 0 ε . Finally, for M ≥ 1 and 1 ≤ m ≤ M + 1, in virtue of the definition of x (m) ε , we have for all x ∈ X A (-1) m ε m m! x (m) ε , Ax Y + ε b (-1) m ε m m! x (m) ε , x = ε b (-1) m-1 ε m-1 (m -1)! x (m-1) ε , x .
Summing for m = 1 to M + 1, and adding the equation verified by

x (0) ε = x ε , we obtain (A XM+1 ε , Ax) Y + εb( XM+1 ε , x) = (f, Ax) Y + εb( XM ε , x).
A straightforward induction ends the proof.

From now on, we suppose y = 0: if not, we have X M ε = 0 for all ε and M .

4.1. Some estimates on X M ε and AX M ε . We start with estimates on the M -th iterated quasi-reversibility solution, valid for any data y, admissible or not. In other words, these estimates are valid whether or not the data completion problem has a solution.

Proposition 8. For all ε > 0, for all M ∈ N, we have

(a) X M -1 ε b < X M ε b (b) AX M ε Y < y Y (c) AX M ε -y Y < y Y (d) AX M ε -y Y < AX M -1 ε -y Y .
Proof. We start with estimate (a): as y = 0, we have 0 = X -1

ε b < x ε b = X 1 ε b . Furthermore, for M ∈ N, X M ε 2 b = M k=0 M m=0 (-1) k+m ε k+m k! m! b(x (k) ε , x (m) ε )
. Therefore, from lemma 3.4 we obtain

X M +1 ε 2 b -X M ε 2 b = 2 M +1 m=0 (-1) M +1+m ε M +1+m (M + 1)! m! b(x (M +1) ε , x m ε ) > 0.
Regarding estimates (b) and (c), we note that they hold for M = 0. Furthermore,

AX M +1 ε 2 Y = (y, AX M +1 ε ) Y + εb(X M ε , X M +1 ε ) -ε X M +1 ε 2 b . Estimate (a) implies b(X M +1 ε , X M ε ) < X M +1 ε 2
b , and

AX M +1 ε 2 Y < (y, AX M +1 ε ) Y .
Cauchy-Schwarz inequality implies then the estimate (b). Furthermore,

AX M +1 ε -y 2 Y = (AX M +1 ε -y, AX M +1 ε ) Y -(AX M +1 ε -y, y) Y < -(AX M +1 ε -y, y) Y
which leads to estimate (c). Finally, the case M = 0 of estimate (d) correspond to estimate (c) with same M . For M ∈ N, we note that

AX M +1 ε -y 2 Y = M +1 m=0 (-1) m ε m m! Ax (m) ε -y 2 Y = AX M ε -y 2 Y + (-1) M +1 2 ε M +1 (M + 1)! (Ax (M +1) ε , AX M ε -y) Y .
Therefore, it is sufficient to determine the sign of (-1) M +1 (Ax

(M +1) ε , AX M ε -y) Y . By definition, we have (Ax (M +1) ε , Ax ε -y) Y = -ε b(x (M +1) ε , x ε )
and for all m ∈ {1, . . . , M },

Ax (M +1) ε , A (-1) m ε m m! x (m) ε Y = (-1) m+1 ε m+1 m! b(x (M +1) ε , x (m) ε ) + (-1) m+1 ε m (m -1)! b(x (M +1) ε , x (m-1) ε ).
Summing these equations for m = 0 to M , we obtain

(Ax (M +1) ε , AX M ε -y) Y = (-1) M +1 ε M +1 M ! b(x (M +1) ε , x (M ) ε ).
In conclusion, (-1) M +1 (Ax

(M +1) ε , AX M ε -y) Y = ε M +1 M ! b(x (M +1) ε , x (M ) ε
) < 0 by lemma 3.4. The result follows.

Proposition 9. For all ε > 0, for all M ∈ {-1} ∪ N, X M ε b ≤ 2(M +1) ε y Y .
Proof. Proposition 9 is obviously true for M = -1. Let M ∈ N. We consider first the inductive sequence:

x 1 > 0 ∀M ∈ N, x M +1 > 0 and x 2 M +1 -x M +1 x M -x 2 1 = 0.
Note that the sequence is well defined as p M (x) := x 2 -x M x-x 2 1 verifies p M (0) < 0 and therefore has a unique strictly positive root.

We prove by induction that x M < √ 2 M x 1 . It obviously holds for M = 1. Suppose that x M < √ 2 M x 1 for some M ∈ N. Then

p M 2(M + 1)x 1 = 2(M + 1)x 2 1 -2(M + 1)x 1 x M -x 2 1 > (2M + 1)x 2 1 -2 √ M + 1 √ M x 2 1 = √ M + 1 - √ M 2 x 2 1 > 0 and therefore x M +1 < 2(M + 1)x 1 .
Now, we specify the sequence, defining

x 1 := 1 √ ε y Y , and prove by induction that X M ε b ≤ x M +1
. Suppose it holds for some M ∈ N. Note that by definition of X M +1 ε , we have

AX M +1 ε 2 Y + ε X M +1 ε 2 b = (y, AX M +1 ε ) Y + εb(X M ε , X M +1 ε ) ≤ y Y AX M +1 ε Y + ε X M ε b X M +1 ε b ≤ y 2 Y + ε x M +1 X M +1
ε b , (we used estimate (b) of proposition 8 here) which in particular implies

X M +1 ε 2 b -x M +1 X M +1 ε b -x 2 1 < 0. The definition of the sequence (x M ) M ∈N implies directly X M +1 ε b ≤ x M +2 .
As the result is true for M = 0, the proposition follows.

Remark 4. Actually, for M ∈ N, the inequality in proposition 9 is strict. 4.2. The case of exact data. From now on, we suppose that y ∈ Y adm , and denote x s the solution of the abstract data completion problem. We define R M ε := X M ε -x s , the discrepancy between the M -th iterated QR solution, and the exact solution. Note that by definition, R -1 ε = -x s and for all M ∈ N,

(AR M ε , Ax) Y + ε b(R M ε , x) = ε b(R M -1 ε , x), ∀x ∈ X . (3)
We aim to prove the following theorem:

Theorem 4.2. For all ε > 0, R M ε M →∞ ----→ X 0.
In other words, for any ε > 0, X M ε converges to x s as M goes to infinity.

As X M ε = M m=0 (-1) m ε m m ! x (m)
ε , it means that the sum converges as M goes to infinity. In other words, it means that if it exists x s ∈ X solution of Ax s = y, then

x s = ∞ m=0 (-1) m ε m m ! x (m) ε ,
hence the solution of the data completion problem can be seen as a series of derivatives of the quasi-reversibility solution w.r.t. the parameter ε.

Let ε > 0 be fixed. We start with the following estimates Proposition 10. For all ε > 0, for all

M ∈ {-1} ∪ N, -R M +1 ε b < R M ε b -X M ε b ≤ x s b . As a consequence, we have R M ε b ≤ x s b for all M ∈ {-1} ∪ N and all ε > 0. Proof. Choosing x = R M ε in (3), we obtain AR M ε 2 Y +ε R M ε 2 b = εb(R M -1 ε , R M ε ) ⇒ R M ε 2 b < b(R M ε , R M -1 ε ) ≤ R M ε b R M -1 ε b ,
hence the first estimate is valid. Note that in particular, as

R M ε b ≤ R 0 ε b = x ε -x s b , we have R M ε b ε→0 ---→ 0 for any M ∈ N.
Let us now focus on the second estimate, which is directly true for M = -1.

For M ∈ N, let us define g := ε ∈ R + * → 1 2 X M ε 2 b . As by definition, X M ε = M m=0 (-1) m ε m m! x (m) ε , we have d dε X M ε = M m=0 (-1) m ε m m! x (m+1) ε + M m=1 (-1) m ε m-1 (m -1)! x (m) ε = (-1) M ε M M ! x (M +1) ε .
Therefore, we have

g (ε) = b( d dε X M ε , X M ε ) = M m=0 (-1) M +m ε M +m M ! m! b(x (M +1) ε , x (m) ε ) < 0, implying in particular that X M ε b ≤ lim η→0 X M η b = x s b , as R M ε b ε→0 ---→ 0.
Proposition 11. The series

M AR M ε 2 Y converges, therefore lim M →∞ AR M ε Y = 0.
Proof. For any M ∈ N, we have

AR M +1 ε 2 Y + ε R M +1 ε 2 b = εb(R M ε , R M +1 ε )
which leads to

AR M +1 ε 2 Y < εb(R M ε , R M +1 ε ) ≤ ε R M ε b R M +1 ε b < ε R M ε 2 b = εb(R M -1 ε , R M ε )-AR M ε 2
Y . Therefore, we obtain

AR M +1 ε 2 Y + AR M ε 2 Y < εb(R M -1 ε , R M ε ) and by an immediate induction M m=1 AR m ε 2 Y ≤ ε R 0 ε 2 b ≤ ε x s 2 b .
Therefore , the series AR m ε 2 Y converges. The property follows.

Theorem 4.2 follows from the previous proposition: indeed, let ϕ : N → N be a strictly increasing map, and define RM

ε := R ϕ(M ) ε . As RM ε 2 A,b = A RM ε 2 Y + RM ε 2 b ≤ y 2 Y + x s 2 b
we have that ( RM ε ) M ∈N is a bounded sequence in X . Consequently, there exists ϑ : N → N, a strictly increasing map such that RM ε := Rϑ(M) ε weakly converges to R ∞ in X . As A is linear and continuous, we directly obtain from proposition 11 that AR ∞ = 0 Y , which implies R ∞ = 0 X as A is one-to-one.

We hence have obtained that RM strongly converges to x s , and it is then not difficult to show that the whole sequence X M ε strongly converges to x s as M goes to infinity.

4.3.

The case of noisy data. In this section, we suppose that our exact data, denoted y ex ∈ Y, for which the data completion problem admits a unique solution x s ∈ X , is corrupted by some noise. The obtained perturbed data, denoted y δ ∈ Y, is supposed to verify y δ -y ex Y ≤ δ: in other words, we know the amplitude of noise on the data. On the other hand, there might or might not be x ∈ X such that Ax = y δ : we don't know if y δ is an admissible solution or not. From now on, for any y ∈ Y, we will denote X M ε (y) the M-th iterated quasireversibility solution with y as data. Our main objective in this section is to propose an admissible strategy to choose M as a function of δ, the amplitude of noise, to ensure that, when δ goes to zero, X M (δ) ε tends to the exact solution x s . As pointed out in proposition 2 and remark 1, this is a crucial point in the study of data completion problems.

A first important remark is the following: AX M ε (y) always converges to y, regardless of the admissibility of y as data for the data completion problem. Proof. As Y adm is dense in Y, for any η > 0, it exists y η ∈ Y adm such that y η -

y Y ≤ η 3 . Proposition 8 (b) implies AX M ε (y) -AX M ε (y η ) Y ≤ y -y η Y ≤ η 3 .
Finally, as y η ∈ Y adm , there exists M η > 0 such that for any

M ≥ M η , AX M ε (y η ) -y η Y ≤ η 3 .
The result follows.

Next proposition defines the admissible choices of M to ensure the desired convergence: Proposition 13. For any ε > 0, for any choice of

M := M (δ) such that M (δ) ---→ δ→0 +∞ and δ M (δ) ---→ δ→0 0, we have X M (δ) ε (y δ ) X ---→ δ→0 x s .
Proof. obviously, we have, for any ε > 0 and M ∈ N

X M ε (y δ ) -x s A,b ≤ X M ε (y δ ) -X M ε (y) A,b + X M ε (y) -x s A,b . If M := M (δ) verifies lim δ→0 M (δ) = +∞, theorem 4.2 implies directly that X M (δ) ε (y) -x s A,b ---→ δ→0 0.
Furthermore, propositions 8 and 10 imply

X M ε (y δ ) -X M ε (y) 2 A,b ≤ 1 + 2(M + 1) ε y δ -y 2 Y = 1 + 2(M + 1) ε δ 2 .
The result follows.

Proposition 13 defines the admissible strategies to choose M depending on δ. An admissible choice could be M (δ) := 1 √ δ for example. But such a choice, if it guarantees the convergence of the method, does not correspond to any precise objective. We therefore focus on another method to choose M (δ).

Let r > 1. For a fixed ε > 0, we define M δ := M ∈ N, AX M ε (y δ ) -y δ Y ≤ rδ . Proposition 12 implies that M δ is non-empty, and we define M (δ) as the minimum element of M δ : M (δ) := min {M ∈ M(δ)}.

M (δ) is chosen accordingly to the Morozov discrepancy principle: it is the first M ∈ N such that the distance between AX M ε and y δ is (approximately) equal to the distance between Ax s = y and y δ :

AX M ε -y δ Y ≈ Ax s -y δ Y .
This method to choose M depending on δ has two interesting characteristics:

1. with this choice, one does the minimum number of iterations of the iterated quasi-reversibility method required to obtain an error in the residual AX M εy δ Y of same order of the error on the data. 2. such choice is admissible, in the sense of proposition 13.

We now prove that M (δ) is an admissible choice.

Proposition 14. M (δ) δ→0 ---→ +∞.
Proof. Suppose it is not the case. Then there exists a sequence of strictly positive real numbers δ n and a positive constant

C such that δ n n→∞ ----→ 0 and M (δ n ) ≤ C. It implies the existence of a subsequence (still denoted δ n ) and M ∞ ∈ N such that δ n n→∞ ----→ 0 and M (δ n ) n→∞ ----→ M ∞ . In particular, it exists N ∈ N such that for all n ≥ N , M (δ n ) = M ∞ .
For n ≥ N , the definition of M (δ) implies

AX M∞ ε (y δn ) -y ex Y ≤ AX M∞ ε (y δ n ) -y δn Y + y δn -y ex Y ≤ (r + 1)δ n n→∞ ----→ 0.
Consequently, using proposition 8 we have

AX M∞ ε (y ex ) -y ex Y ≤ AX M∞ ε (y ex ) -AX M∞ ε (y δn ) Y + AX M∞ ε (y δn ) -y ex Y ≤ y ex -y δn Y + (r + 1)δ n = (r + 2)δ n ----→ n→∞ 0, that is AX M∞ ε (y ex ) = y ex . If M ∞ = -1,
we directly obtain y ex = 0, in contradiction with the hypothesis. If M ∞ = 0, we obtain x ε = x s , which again (corollary 1) implies y s = 0 Finally, if M ∞ > 0, as for all x ∈ X ,

(AX M∞ ε (y ex ), Ax) Y + εb(X M∞ ε (y ex ), x) = εb(X M∞-1 ε (y ex ), x) + (y ex , Ax) Y , we obtain X M∞ ε (y ex ) = X M∞-1 ε (y ex ), or equivalently x (M∞) ε
(y ex ) = 0, which again implies y ex = 0 by corollary 2. We obtain once again a contradiction, which ends the proof.

Proposition 15. lim δ→0 δ M (δ) = 0.
Proof. by definition, we have AX

M (δ)-1 ε (y δ ) -y δ Y > rδ. Therefore AX M (δ)-1 ε (y ex ) -y ex Y = AX M (δ)-1 ε (y ex ) -AX M (δ)-1 ε (y δ ) + AX M (δ)-1 ε (y δ ) -y δ + y δ -y ex Y ≥ AX M (δ)-1 ε (y δ ) -y δ Y -AX M (δ)-1 ε (y ex ) -AX M (δ)-1 ε (y δ ) -(y ex -y δ ) Y > (r -1)δ
as proposition 8, estimate (c) implies

AX M (δ)-1 ε (y ex ) -AX M (δ)-1 ε (y δ ) -(y ex -y δ ) Y ≤ y ex -y δ Y ≤ δ. We hence have obtained: (M (δ)-1)δ 2 (r-1) 2 ≤ (M (δ)-1) AX M (δ)-1 ε (y ex )-y ex 2 Y = (M (δ)-1) AR M (δ)-1 ε (y ex ) 2 Y . As AR m+1 ε (y ex ) Y < AR m ε (y ex ) Y and m AR m ε (y ex ) 2 Y converges, m AR m ε (y ex ) 2 Y
tends to zero as m goes to infinity. Therefore, (M (δ) -1) AR

M (δ)-1 ε (y ex ) 2
Y goes to zero as δ tends to zero, implying that lim δ→0 (M (δ) -1)δ 2 (r -1) 2 = 0. The result follows.

5.

Quasi-reversibility methods for data completion problems for the Poisson's equation and the heat equation. We will now go back to the data completion problems described in the introduction, and verify that they correspond to the abstract setting introduced in section 2.

Poisson's equation. As mentioned in the introduction, the data completion problem for Poisson's equation is

: for (f, g D , g N ) ∈ L 2 (Ω) × L 2 (Γ) × L 2 (Γ), find u ∈ H 1 (Ω) s.t.    -∇ • σ∇u = f in Ω u = g D on Γ σ∇u • ν = g N on Γ.
We could directly use this formulation of the problem to obtain a quasi-reversibility regularization. However, if we do so, we obtain a fourth-order variational problem, which is rather difficult to discretize as we would need C 1 or non-conforming finite elements which are seldom available in numerical solvers. Therefore, we first modify the problem by introducing the flux p := σ •∇u as an additional unknown, following the idea introduced in [START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF]

]. It verifies -∇ • p = -∇ • σ∇u = f ∈ L 2 (Ω) and p • ν = σ∇u • ν = g N ∈ L 2 (Γ), hence p ∈ Hdiv (Ω) := q ∈ L 2 (Ω) d , ∇ • q ∈ L 2 (Ω), q • ν ∈ L 2 (Γ) .
Hdiv (Ω), endowed with the scalar product (p, q) Hdiv :=

Ω p • q + (∇ • p) (∇ • q) dx + Γ (p • ν) (q • ν) dS
is an Hilbert space [START_REF] Fernandes | Magnetostatic and Electrostatic Problems in Inhomogeneous Anisotropic Media with Irregular Boundary and Mixed Boundary Conditions[END_REF]. We modify the data completion problem the following way:

Problem. For f , g D and g N in respectively L 2 (Ω), L 2 (Γ) and L 2 (Γ), find (u, p) ∈ H 1 (Ω) × Hdiv (Ω) such that        σ∇u = p in Ω -∇ • p = f in Ω u = g D on Γ p • ν = g N on Γ.
Obviously, this is exactly the same problem as previously. However, this small modification will lead to a second-order variational quasi-reversibility regularization in the product space H 1 × Hdiv , easily discretized using standard finite-elements.

To fit in our abstract setting, we introduce the operator

A :(u, p) ∈ X = H 1 (Ω) × Hdiv (Ω) → (σ∇u -p, -∇ • p, u |Γ , p • ν |Γ ) ∈ Y = L 2 (Ω) d × L 2 (Ω) × L 2 (Γ) × L 2 (Γ).
The spaces X and Y, endowed respectively with the scalar products (

X := (u, v) H 1 + (p, q) Hdiv and (F, f, g, h), ( F, f , g, h) Y := Ω F • F + f f dx + Γ g g + h h ds u, p), (v, q)
are obviously Hilbert spaces, and the data completion problems can be rewritten: find (u, p) ∈ X such that A(u, p) = (0, f, g D , g N ) ∈ Y.

Proposition 16. A is linear, continuous, one-to-one. It is not onto but has dense range. Additionally, A is not a compact operator.

Proof. Clearly, A is linear continuous. As the data completion problem for Poisson's equation is known to admits at most a solution, but may have no solution, A is oneto-one but not onto. Let us prove that Im(A)

Y = Y. Let (F, f, g, h) ∈ Y such that A(u, p), (F, f, g, h) Y = 0, ∀(u, p) ∈ Y, that is Ω (σ∇u-p)•F-(∇•p) f dx+ Γ u g+(p•ν) h ds = 0, ∀(u, p) ∈ H 1 (Ω)× Hdiv (Ω).
Choosing u = ϕ ∈ C ∞ c (Ω) and p = 0, we obtain -∇ • σ T F = 0, and in particular σ T F ∈ H div (Ω). Choosing u = 0 and p = Ψ ∈ C ∞ c (Ω) d , we obtain ∇f = F. Therefore, f ∈ H 1 (Ω), and verifies -∇ • σ T ∇f = 0. Hence, taking u ∈ H 1 (Ω) and p = 0, and using the Green formula, we obtain

σ T ∇f • ν, u = Ω σ T ∇f • ∇u dx = Ω σ∇u • ∇f dx = - Γ u g ds, ∀u ∈ H 1 (Ω),
implying that σ T ∇f • ν = -g on Γ and 0 on Γ c . Taking u = 0 and p ∈ Hdiv (Ω), and using the divergence theorem, we obtain

p • ν, f = Ω p • ∇f + ∇ • p f dx = Γ (p • ν) h ds,
and therefore f = h on Γ and 0 on Γ c . We have obtain that f ∈ H 1 (Ω) verifies -∇ • σ T ∇f = 0 in Ω and f = σ T ∇f • ν = 0 on Γ c : by uniqueness of the solution of the elliptic data completion problem, necessarily f = 0, which implies directly F = 0 and g = h = 0.

Finally, let us prove that A is not a compact operator. Consider e n an Hilbert basis of L 2 (Ω), and

u n in H 1 (Ω) verifying Ω u n dx = 0, -∇ • σ∇u n = e n in
Ω and σ∇u n • ν = 0 on ∂Ω. It is not difficult to show that u n exists and is unique. Furthermore, u n H 1 (Ω) ≤ C(Ω, σ), and in particular u n L 2 (Γ) ≤ C(Ω, σ). Defining p n := σ∇u n ∈ Hdiv (Ω), we obtain p n Hdiv ≤ C(Ω, σ), hence (u n , p n ) is a bounded sequence in X . But A(u n , p n ) = (0, e n , u n |Γ , 0) does not admit any convergent subsequence.

To define our quasi-reversibility approach to this data completion problem, we choose b(., .) such that the corresponding norm . A,b is equivalent to the norm . X . Of course, we could choose the whole X -scalar product. But it might be interesting to use another form, to soften the regularization: here we define b (u, p), (v, q) := Ω (∇u • ∇v + p • q) dx which is a symmetric bilinear non-negative form in X (but obviously not a scalar product). Using Poincaré inequality, it is easy to obtain the existence of c, C > 0 such that c v,

q X ≤ v, q A,b ≤ C v, q X .
As for any (v, q) ∈ X , v, q b ≤ v, q X , for a fixed ε the regularization term in the quasi-reversibility method is smaller. Applying the abstract setting to this problem, we obtain the following quasireversibility regularization: for ε > 0, find (u ε , p ε ) ∈ H 1 (Ω) × Hdiv(Ω) such that for all (v, q) ∈ H 1 (Ω) × Hdiv (Ω),

Ω (σ∇u ε -p ε ) • (σ∇v -q) dx + Ω (∇ • p ε ) (∇ • q) dx + Γ u ε v + (p ε • ν) (q • ν) ds +ε Ω (∇u ε • ∇v + p ε • q) dx = - Ω f (∇ • q)dx + Γ g D v + g N (q • ν) ds.
This problem always admits an unique solution (u ε , p ε ). We know from our study that if the data completion problem for the Poisson's equation admits a solution (u s , p s ), then (u ε , p ε ) converges monotonically to (u s , p s ) as ε goes to zero, with the estimate

σ∇u ε -p ε 2 L 2 (Ω) d + ∇ • p ε -f 2 L 2 (Ω) + u ε -g D 2 L 2 (Γ) + p ε • ν -g N 2 L 2 (Γ) ≤ √ ε u s , p s b .
If not, we know that (u ε , p ε ) b ---→ ε→0 +∞.

The quasi-reversibility method we obtain in this study is close to the one proposed in [START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF] to stabilize the data completion problem, which was: find (u ε , p ε ) ∈ H 1 (Ω)× Hdiv(Ω) , u ε = g D and p ε • ν = g N on Γ, such that for all (v, q) ∈ H 1 (Ω) × Hdiv (Ω), v = q • ν = 0 on Γ,

Ω (σ∇u ε -p ε )•(σ∇v -q) dx + Ω (∇ • p ε ) (∇ • q) dx + ε (u ε , p ε ), (v, q) X = - Ω f (∇ • q)dx.
The two differences are first the use of b(., .) instead of (., .) X in the regularization term, and secondly in the way the boundary condition are included in the problem. In the formulation proposed in [START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF], they are strongly imposed, which presents two main issues: one is theoretical, as the regularized problem might not have solution if g D is in L 2 (Γ), and not in H 1/2 (Γ), as in that case there is no v ∈ H 1 (Ω) such that v = g D on Γ, and therefore u ε cannot exist. The second one is practical: it is not a good idea to strongly impose data which might extremely noisy, as in that case the noise is somehow imposed to the solution. In the quasi-reversibility regularization obtain in the present paper, the boundary conditions are weakly imposed, which solves both of the problems: the regularized problem always admits a solution, even in the case where g D is not the trace on Γ of a H 1 function, and the noise is regularized directly by the formulation, leading to a stabler formulation.

Finally, the abstract iterated quasi-reversibility method applied to the elliptic data completion problem is: for ε > 0, (u -1 ε , p -1 ε ) = (0, 0) and for all M ∈ N,

(u M ε , pM ε ) ∈ H 1 (Ω) × Hdiv (Ω) verifies Ω (σ∇u M ε -p M ε ) • (σ∇v -q) dx + Ω (∇ • p M ε ) (∇ • q) dx + Γ u M ε v + (p M ε • ν) (q • ν) ds + ε b (u M ε , p M ε ), (v, q) = - Ω f (∇ • q)dx + Γ g D v + g N (q • ν) ds + ε b (U M -1 ε , P M -1 ε ), (v, q) .
and we directly know that u M ε and p M ε converge to u and σ∇u when M goes to infinity. In the case where noisy data f δ , g δ D and g δ N are available, such that

g D -g δ D 2 L 2 (Γ) + g N -g δ N 2 L 2 (Γ) + f -f δ 2 L 2 (Ω) ≤ δ 2 ,
in accordance with the result of section 4.3, we stop the iterations the first time that

A(u M ε , p M ε ) -(0, f δ , g δ D , g δ N )
Y ≤ rδ, with r > 1 close to 1. Remark 5. Actually, in the following numerical results, we use r = 1.

Heat equation.

As for the Poisson problem, we modify the data completion problem defined in the introduction, introducing the flux p := ∇u as an additional unknown:

Problem. find (u, p) in L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) × L 2 (0, T ; Hdiv (Ω)) such that        ∂ t u -∇ • p = f in Q ∇u = p in Q u = g D on (0, T ) × Γ p • ν = g N on (0, T ) × Γ Again, we define A :(u, p) ∈ X := L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) × L 2 (0, T ; Hdiv (Ω)) → (∇u -p, ∂ t u -∇ • p, u |Γ , p • ν |Γ ) ∈ Y = L 2 (0, T ; L 2 (Ω) d ) × L 2 (0, T ; L 2 (Ω)) × L 2 (0, T ; L 2 (Γ)) × L 2 (0, T ; L 2 (Γ)).
Here, the spaces X and Y are endowed with their natural scalar products, respectively (u, p), (v, q)

X := T 0 Ω ∂ t u ∂ t v + ∇u • ∇v + u v+(∇ • p) (∇ • q) + p • q dx dt + T 0 Γ (p • ν) (q • ν) dS dt and (F 1 , f 1 , g 1 , h 1 ),(F 2 , f 2 , g 2 , h 2 ) Y := T 0 Ω F 1 • F 2 + f 1 f 2 dx dt + T 0 Γ g 1 g 2 + h 1 h 2 dS dt.
It is then not difficult to verify the Proposition 17. A is a linear continuous. It is one-to-one but not onto, and has dense image. Furthermore, it is not a compact operator.

Proof. We will just prove prove that A has dense range, as it is not difficult to be convinced that A is non compact, and the rest of the proposition follows directly from the definition of A, X and Y, and the ill-posedness of the corresponding data completion problem. Let F ∈ L 2 (0, T ; L 2 (Ω) d ), f ∈ L 2 (0, T ; L 2 (Ω)), g ∈ L 2 (0, T ; L 2 (Γ)) and h ∈ L 2 (0, T ; L 2 (Γ)) be such that for all v ∈ L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) and all q ∈ L 2 (0, T ; Hdiv (Ω)), A(v, q), (F, f, g, h)

Y = 0, that is T 0 Ω (∇v -q) • F + (∂ t v -∇ • q) f dx + T 0 Γ v g + (q • ν)h ds = 0.
First of all, choosing q = Υ ∈ C ∞ c (Ω) d , we obtain F = ∇f , and therefore f ∈ L 2 (0, T ; H 1 (Ω)). So we have

T 0 Ω (∇v -q) • ∇f + (∂ t v -∇ • q) f dx + T 0 Γ v g + (q • ν)h ds = 0. ( 4
)
For all q ∈ L 2 (0, T ; Hdiv (Ω)), for almost all t ∈ (0, T ), we have

Ω q • ∇f + (∇ • q)f dx = q • ν, f H -1/2 (∂Ω),H 1/2 (∂Ω)
which leads by integration in time to 4) leads to ∂ t f + ∆f = 0 in (0, T ) × Ω. We see that V := (f, ∇f ) ∈ H div (Q), and we can apply the divergence theorem: for all

T 0 q • ν, f H -1/2 (∂Ω),H 1/2 (∂Ω) dt = - T 0 Γ (q • ν)hds dt that is f = -h on (0, T ) × Γ and f = 0 on (0, T ) × Γ c . Now, taking v = ϕ ∈ C ∞ c (Ω) in (
v ∈ H 1,1 (Q) = L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), we have Q (∇ t,x • V)v + V • ∇ t,x v dx dt = V • ν, v H -1/2 (∂Q),H 1/2 (∂Q) . that is, taking any v ∈ H 1,1 (Q) such that v(0, x) = v(T, x) = 0, T 0 Ω (∂ t f + ∆f ) =0 v+f ∂ t v+∇f •∇v dxdt = ∇f •ν, v H -1/2 (∂Ω),H 1/2 (∂Ω) H -1/2 (0,T ),H 1/2 (0,T ) leading to ∇f • ν = -g on (0, T ) × Γ and ∇f • ν = 0 on (0, T ) × Γ c . Therefore, f verifies ∂ t f + ∆f = 0 in (0, T ) × Ω, f = ∇f • ν = 0 on (0, T ) × Γ c , hence f ≡ 0 in (0, T ) × Γ, leading to F = 0 and g = h = 0.
Similarly as for the previous regularization, we introduce the symmetric bilinear non-negative form b (u, p), (v, q) :=

T 0 Ω ∂ t u ∂ t v + ∇u • ∇v + p • q dx dt.
It is easy to check that the bilinear form (A(u, p), A(v, q)) Y + b((u, p), (v, q)) is a scalar product on X , and that it exists two constants c, C > 0 such that

c u, p X ≤ u, p A,b ≤ C u, p X .
The quasi-reversibility regularization we consider is therefore: for ε > 0, find (u ε , p ε ) ∈ X such that for all (v, q) ∈ X , we have

T 0 Ω (∂ t u ε -∇ • p ε )(∂ t v -∇ • q) + (∇u ε -p ε ) • (∇v -q) dx dt + T 0 Γ u ε v + (p ε • ν)(q • ν) dS dt + ε T 0 Ω ∂ t u ε ∂ t v + ∇u ε • ∇v + p ε • q dx dt = T 0 Ω f (∂ t v -∇ • q) dx dt + T 0 Γ g D v + g N (q • ν) dS dt.
According to our present study, this problem always admits a unique solution (u ε , p ε ) that converges to (u, ∇u) when ε goes to zero. The corresponding iterated quasi-reversibility is:

(u -1 ε , p -1 ε ) = (0, 0) and for all M ∈ N, (u M ε , p M ε ) is such that for all (v, q), T 0 Ω (∂ t u M ε -∇ • p M ε )(∂ t v -∇ • q) + (∇u M ε -p M ε ) • (∇v -q) dx dt + T 0 Γ u M ε v+(p M ε •ν)(q•ν) dS dt+ε T 0 Ω ∂ t u M ε ∂ t v+∇u M ε •∇v+p M ε •q dx dt = T 0 Ω f (∂ t v -∇ • q) dx dt + T 0 Γ g D v + g N (q • ν) dS dt +ε T 0 Ω ∂ t u M -1 ε ∂ t v + ∇u M -1 ε • ∇v + p M -1 ε • q dx dt.
6. Numerical results.

∈ C 2 (Γ c ) such that        -∆u = 0 in Ω u = g δ D on Γ ∂ ν u = g δ N on Γ ∂ ν u + η u = 0 on Γ c
The Cauchy data (g δ D , g δ N ) ∈ L 2 (Γ) × L 2 (Γ) is supposed to correspond to an exact data (g D , g N ) corrupted by some noise of amplitude δ :

g δ D -g D 2 L 2 (Γ) + g δ N -g N 2 L 2 (Γ) ≤ δ 2 .
Our strategy to reconstruct η is therefore to compute u Then we corrupt the Dirichlet data g D pointwise with a normal noise having zero mean and variance one, to obtained the corrupted Dirichlet data g δ D . The noise is scaled so that g δ D -g D ∞ = α g D ∞ that is the relative amplitude of noise in L ∞ -norm is α. In the experiments, we have chosen α = 1%, 2% and 5%. The exact Neumann data is used (i.e. g δ N = g N ), as in practical situations it is the imposed data (the net current), whereas the g D is the measured data (the corresponding voltages). Therefore g N is known quite precisely compared to g D . We then compute the corresponding amplitude of noise for the L 2 norm δ = δ(α, g D ∞ ), which defined our stopping criterion for the iteration of the method. The iterated quasi-reversibility problem is then solved using a conforming finiteelement method using P 2 Lagrange finite elements for u M ε and RT 1 Raviart-Thomas finite elements for p M ε [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF]. The study of convergence of the finite-element approximation of the quasi-reversibility approximation toward the continuous solution is just a slight adaptation of section 4.4 in [START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF], as the formulations are quite similar, and therefore is omitted in the present study. To avoid an inverse crime, the direct and inverse problems are solved on different meshes. According to our study, the choice of ε is completely arbitrary in the iterated quasi-reversibility method. Therefore, we have chosen ε = 1 in the experiments, as it leads to a good conditioning of the finite-element matrices.

First of all, we present in figure 3 the evolution of the residual

∇ • p M ε 2 L 2 (Ω) + ∇u M ε -p M ε 2 L 2 (Ω) + u ε -g δ D 2 L 2 (Γ) + p ε • ν -g δ N L 2 (Γ)
until the stopping criterion is reached. As expected theoretically, the greater is the noise, the smaller is M (δ). Now we present the reconstruction results: in figure 4, the exact solution is compared to the reconstructed one in the whole domain of study. In figure 5, we focus on the boundary Γ: we compare the exact data g D , the noisy one g δ D used in the iterated quasi-reversibility method, and finally the trace of the reconstructed solution u M (δ) ε . Note that the iterated quasi-reversibility method gives good result even with severely corrupted data.

Finally, on figure 6, we present the reconstructed Robin coefficient on Γ c , which was our main objective. Again, the reconstruction is still acceptable for high level of noise on the data. 6.2. One-dimensional heat equation. We now focus on the data-completion problem for a one-dimensional heat equation. The problem reads: find u ∈ H 1,1 ((0, T )× Note that, as ∂ xx u = ∂ t u ∈ L 2 (0, T ; L 2 (a, b)), we have p := ∂ x u ∈ L 2 (0, T ; H 1 (a, b)), and hence g N (t) = p(a, t) ∈ L 2 (0, T ) without additional assumption, which is not the case for the multi-dimensional case. Hence the equivalent data-completion problem with additional unknown p reads: for (g δ D , g δ N ) ∈ L 2 (0, T ) × L 2 (0, T ), find u ∈ H 1,1 ((0, T ) × (a, b)) and p ∈ L 2 (0, T ; H 1 (a; b)) such that

       ∂ t u = ∂ x p in (0, T ) × (a, b) ∂ x u
= p in (0, T ) × (a, b) u(t, a) = g δ D , t ∈ (0, T ) p(t, a) = g δ N , t ∈ (0, T ). According to our study, the quasi-reversibility regularization of this problem is: for ε > 0, find u ε ∈ H 1,1 ((0, T ) × (a, b)) and p ε ∈ L 2 (0, T ; H 1 (a, b)) such that for all v ∈ H 1,1 ((0, T ) × (a, b)), for all q ∈ L 2 (0, T ; H 1 (a, b))

T 0 b a (∂ t u ε -∂ x p ε ) (∂ t v -∂ x q) + (∂ x u ε -p ε ) (∂ x v -q) dx dt
(∂ t u M ε -∂ x p M ε ) (∂ t v -∂ x q) + (∂ x u M ε -p M ε ) (∂ x v -q) dx dt
∂ t u M -1 ε ∂ t v+∂ x u M -1 ε ∂ x v+p M -1 ε q dx dt.
We discretize the space H 1,1 (Q) and L 2 (0, T ; H 1 (a, b)) using a tensorial product of Lagrange finite elements, namely P 1 ⊗ P 1 finite elements for H 1,1 and P 0 × P 1 for L 2 (H 1 ).

In our simulations, we choose T = 1, a = 1 and b = 2. We consider two exact solution of the heat equation u 1 (t, x) := 1 8

x 3 3 + x (1 + 2 t) and u 2 (t, x) := e -t/4 sin (t/2). The corresponding exact data (g D , g N ) are corrupted pointwise by a normal noise with zero means and variance one, which is scaled so that the noisy data (g δ D , g δ N ) In our experiments, we test our method with α = 2% and α = 5%. As in the elliptic case, we choose ε = 1, and stop the iterations of the method once the stopping criterion is reached.

In figures 8 and 9, we present the relative error over Q, defined as the ratio u M ε (δ) -u u ∞ for both solutions u 1 and u 2 . We see that the iterated quasi-reversibility method gives also good reconstruction for this parabolic problem, even for high level of noise on both Dirichlet and Neumann data.

Finally, in figures 10, we present the evolution of the residual quantity during the iterations of the method, until the stopping criterion is reached.

∂tu M ε -∂xp M ε 2 L 2 (Q) + ∂xu M ε -p M ε 2 L 2 (Q) + u M ε (., a) -g δ

  x n X n→∞ ----→ +∞. Now, consider y ∈ Y adm . The previous proposition implies the existence of a sequence y m ∈ Y nadm such that y m m→∞ ----→ Y y. For a fixed m, we now know the existence of a sequence x m,n ∈ X such that Ax m,n n→∞ ----→ Y y m and x m,n X n→∞ ----→ +∞. In particular, for any m ∈ N, there exists n(m) ∈ N such that xm := x m,n(m) verifies at the same time xm X ≥ m and Ax m -y m Y ≤ 1 m .

  ≤ x s A,b , implying that XM ε

Proposition 12 .

 12 For any y ∈ Y, for any ε > 0, AX M ε

  tions of u and ∇u with the prescribed noisy Cauchy data on Γ and no data at all on Γ c , using the iterated quasi-reversibility method for the Poisson problem. Then, we obtain an approximation η ε of η on Γ c by simply taking the ratio η ε = -p experiments, η = 0.5 + 0.3 sin(2 (θ -5π/4)), θ being the polar angle of a point x on Γ c , and g N = 1. The corresponding Dirichlet data is obtained by solving the direct problem -∆u = 0 in Ω, ∂ ν u = 0.2 on Γ and ∂ ν u + ηu = 0 on Γ c using a finite-element method, and defining g D := u |Γ .

Figure 1 .

 1 Figure 1. Exact solution.

Figure 2 .

 2 Figure 2. The direct and inverse meshes used in the numerical simulation.

Figure 3 .

 3 Figure 3. Residual vs. number of iterations, until stopping criterion is reached, for α = 1%, 2% and 5%.

+ T 0 u

 0 ε (s, a) v(s, a)+p ε (a, s) q(a, s) dt+ε T 0 b a ∂ t u ε ∂ t v+∂ x u ε ∂ x v+p ε q dx dt

Figure 4 .

 4 Figure 4. |u M (δ) ε -u|, for α = 1%, 2% and 5%.

∂

  a) v(s, a)+p M ε (a, s) q(a, s) dt+ε t u M ε ∂ t v+∂ x u M ε ∂ x v+p M ε q dx dt =

Figure 5 .

 5 Figure 5. Exact Dirichlet data, noisy Dirichlet data, and trace of u M (δ) ε on Γ, for α = 1%, 2% and 5%.

D 2 L 2 2 L 2

 2222 (0,T ) + p M ε (., a) -g δ N (0,T )

Figure 6 .

 6 Figure 6. Exact (η) and reconstructed (η ε ) Robin coefficient on Γ c , for α = 1%, 2% and 5%.

Figure 7 .

 7 Figure 7. Exact solutions u 1 and u 2 in Q.

  6.1. Elliptic equation. We consider a domain Ω ⊂ R 2 , with exterior boundary Γ and interior boundary Γ c defined by We consider the problem of reconstructing a Robin coefficient η on ∂Γ c from the knowledge of a noisy Cauchy data (g δ D , g δ N ) on ∂Γ. Mathematically, we want to reconstruct a function u ∈ H 1 (Ω) and a function η

	∂Γ := r(t)	cos(t) sin(t)	t ∈ [0, 2 π]	∂Γ c := r c (t)	cos(t) sin(t)	t ∈ [0, 2 π] ,
	with
	r(t) = 1 + 0.1 cos(2 t) -0.05 sin(3 t), r c (t) = 0.5 -0.02 cos(t) + 0.1 sin(t).


Mahmoud Abdelrahim

Romain Postoyan
email: romain.postoyan@univ-lorraine.fr.

Jamal Daafouz
email: jamal.daafouz@univ-lorraine.fr.

Dragan Nešić

Input-to-state stabilization of nonlinear systems using event-triggered output feedback controllers

We synthesize robust output-based event-triggered controllers for a class of nonlinear systems subject to external disturbances and noises in both the plant measurement and the control input. We follow an emulation approach to this purpose: we first assume that we know a robust feedback law in continuous time, we then take into account the sampling and we explain how to construct the triggering condition to preserve stability. The triggering strategy enforces a minimum time between two consecutive transmissions by combining ideas from event-triggered control and periodic sampled-data control. The closed-loop system is shown to satisfy an input-to-state stability (ISS) property with respect to the external disturbances, the noises as well as their time-derivatives. The analysis reveals a tradeoff between the enforced minimum inter-transmission time and the magnitude of the ISS gains. The results are also new in the particular case where the triggering condition is only time-dependent as in traditional sampled-data control.

I. INTRODUCTION

Event-triggered control is a sampling paradigm in which the sequence of transmission instants is determined based on the violation of a state-dependent criterion. In that way, a substantial reduction of the amount of transmissions may be achieved compared to conventional time-triggered setups as the loop is closed only when it is necessary in view of the control objective, see e.g., [1]-[5] and the references therein. Event-triggered implementations are particularly interesting for networked control systems and embedded systems due to their limited computation and communication resources.

, for instance, that the popular technique of [3] is non-robust to external disturbances as it may generate Zeno phenomenon.

uncertainties are present. Several works have therefore proposed event-triggered control solutions which are robust to exogenous inputs, see e.g., [START_REF] Borgers | Event-separation properties of eventtriggered control systems[END_REF], [START_REF] Sijs | On integration of event-based estimation and robust MPC in a feedback loop[END_REF]- [START_REF] Shi | Event-triggered maximum likelihood state estimation[END_REF]. Most of these references assume that the full state of the plant can be measured, see [START_REF] Persis | On a small-gain approach to distributed event-triggered control[END_REF], [START_REF] Lehmann | Event-triggered pi control: Saturating actuators and anti-windup compensation[END_REF], [START_REF] Lehmann | Event-triggered model predictive control of discrete-time linear systems subject to disturbances[END_REF], [START_REF] Shi | Event-triggered maximum likelihood state estimation[END_REF]. Our objective is to go beyond this assumption and to construct event-triggered output-based controllers to stabilize a class of nonlinear systems subject to external disturbances and corrupted measurements of the plant output and the control input. It is more challenging in this case to guarantee the existence of a minimum time between two consecutive transmissions when only an output of the plant is available, even for the disturbance free case as shown in [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralised eventtriggering[END_REF]. Related results of the literature mainly focus on linear time-invariant (LTI) systems, see [START_REF] Borgers | Event-separation properties of eventtriggered control systems[END_REF], [START_REF] Sijs | On integration of event-based estimation and robust MPC in a feedback loop[END_REF], [START_REF] Lehmann | Event-based output-feedback control[END_REF], [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralised eventtriggering[END_REF]. Compared to the latter results, we address nonlinear systems, we investigate a different setup and we propose a different type of triggering conditions. Furthermore, we also take into account the possible corruption of the control input by an external signal which is not the case in the aforementioned references. Considering this type of exogenous signals is useful in practice as the control input may be subject to computation error or quantization error.

The solution we propose is inspired by our recent work in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF] which did not consider systems with exogenous inputs. We follow an emulation approach, as we start from a robust feedback law which is designed in continuous time, we then take sampling into account and we design the triggering condition to preserve stability. The triggering condition consists in waiting a fixed amount of time T after each transmission instant and then to evaluate a criterion which only involves the noisy output of the plant (and potentially the noisy control input). We show that the system satisfies an ISS property with respect to the external disturbances, as well as the different noises and their time-derivative. The fact that the guaranteed stability property involves the time-derivative of the noises is common in the sampleddata control literature, see [START_REF] Nešić | A note on input-to-state stabilization for nonlinear sampled-data systems[END_REF] for instance. Furthermore, the analysis reveals a tradeoff between the ISS gains and the guaranteed minimum time T between transmissions. On the other hand, the triggering condition we propose can easily be made time-dependent only. In that way, we obtain a periodic sampled-data controller which generalizes the results in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] to systems with exogenous inputs.

While the same type of transmission rule is proposed in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF], the fact that the overall system is affected by exogenous inputs leads to non-trivial challenges. Indeed, we first need to carefully select the coordinates of the hybrid dynamical model, in particular the sampling-induced errors. This point is essential for the construction of the eventtriggering condition and the stability analysis. Second, the thorough analysis in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF] has to be revisited because global asymptotic stability of nonlinear systems does not imply ISS with respect to bounded inputs, even for continuoustime systems, see [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], [START_REF] Freeman | Global internal stabilizability does not imply global external stabilizability for small sensor disturbances[END_REF] for counter examples. Third, the presence of exogenous inputs may lead to the Zeno phenomenon as shown in [START_REF] Borgers | Event-separation properties of eventtriggered control systems[END_REF]. Finally, we have to adapt the lower bound T on the inter-transmission times in order to cope with the exogenous inputs (we cannot take the same value as in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF]).

The remainder of the paper is organised as follows. Preliminaries are given in Section II. The problem is formally stated in Section III. In Section IV, we give the main results and an illustrative example is provided in Section V. Conclusions are provided in Section VI. The proofs are given in the Appendix. and, for each s ∈ R ≥0 , γ(s, .) is decreasing to zero. We denote the minimum and maximum eigenvalues of the square, real, symmetric matrix A as λ min (A) and λ max (A), respectively. We write (x, y) to represent the vector [x T , y T ] T for x ∈ R n and y ∈ R m . We will consider locally Lipschitz Lyapunov functions (that are not necessarily differentiable everywhere), therefore we will use the generalized directional derivative of Clarke which is defined as follows. For a locally Lipschitz function V : R n → R ≥0 and a vector υ ∈ R n , V • (x; υ) := lim sup h→0 + , y→x (V (y +hυ)-V (y))/h. For a continuously differentiable function V , V • (x; υ) reduces to the standard directional derivative ∇V (x), υ , where ∇V (x) is the (classical) gradient. We will invoke the following result, see Lemma II.1 in [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF].

II. PRELIMINARIES Let

R := (-∞, ∞), R ≥0 := [0, ∞) and Z ≥0 := {0, 1, 2, ..}. The Euclidean norm is denoted |.|. A continuous function γ : R ≥0 → R ≥0 is of class K if it is zero at zero, strictly increasing, and it is of class K ∞ if in addition γ(s) → ∞ as s → ∞. A continuous function γ : R ≥0 × R ≥0 → R ≥0 is of class KL if for each t ∈ R ≥0 , γ(., t) is of class K,
Lemma 1 (Lemma II.1 [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF]). Consider two functions U 1 : R n → R and U 2 : R n → R that have well-defined Clarke derivatives for all x ∈ R n and υ ∈ R n . Introduce three sets

A := {x : U 1 (x) > U 2 (x)}, B := {x : U 1 (x) < U 2 (x)}, Γ := {x : U 1 (x) = U 2 (x)}. Then, for any υ ∈ R n , the function U (x) := max{U 1 (x), U 2 (x)} satisfies U • (x; υ) = U • 1 (x; υ) for all x ∈ A, U • (x; υ) = U • 2 (x; υ) for all x ∈ B and U • (x; υ) ≤ max{U • 1 (x; υ), U • 2 (x; υ)} for all x ∈ Γ.
In this paper, we consider hybrid systems of the following form using the formalism of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF]

ẋ = F (x, u) x ∈ C, x + = G(x, u) x ∈ D, (1)
where x ∈ R nx is the state, u ∈ R nu is the input, C is the flow set, F is the flow map, D is the jump set and G is the jump map. The vector fields F and G are assumed to be continuous and the sets C and D are closed. The solutions to system (1) are defined on so-called hybrid time domains.

A set E ⊂ R ≥0 × Z ≥0 is called a compact hybrid time domain if E = J-1 j=0 ([t j , t j+1 ], j) for some finite sequence of times 0 = t 0 ≤ t 1 ≤ ... ≤ t J and it is a hybrid time domain if for all (T, J) ∈ E, E ∩ ([0, T ] × {0, 1, ..., J}) is a compact hybrid time domain.
A hybrid signal is a function defined on a hybrid time domain. A hybrid signal u : dom u → R nu is called a hybrid input (or disturbance) if u(., j) is measurable and locally essentially bounded for each j. A hybrid signal x : dom x → R nx is called a hybrid arc if x(., j) is locally absolutely continuous for each j. A hybrid arc x : dom x → R nx is a solution to system (1) with hybrid input u : dom u → R nu if dom x = dom u, x(0, 0) ∈ C ∪ D, and: (i) for all j ∈ Z ≥0 , and almost all t such that (t, j) ∈ dom x, x(t, j) ∈ C and ẋ(t, j) = F (x(t, j), u(t, j)); (ii) for all (t, j) ∈ dom x such that (t, j + 1) ∈ dom x, x(t, j) ∈ D and x(t, j + 1) = G(x(t, j), u(t, j)). The following definition of L ∞ norm is considered in [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF], [START_REF] Sanfelice | Results on input-to-output and input-output-to-state stability for hybrid systems and their interconnections[END_REF] for hybrid signals.

Definition 1. Given a hybrid signal r, its L ∞ norm is given by ||r|| (t,j) := max ess sup

(t ′ ,j ′ )∈dom r\Γ(r), t ′ +j ′ ≤t+j |r(t ′ , j ′ )|, sup (t ′ ,j ′ )∈Γ(r), t ′ +j ′ ≤t+j |r(t ′ , j ′ )| , (2)
where Γ(r) := {(t, j) ∈ dom r : (t, j + 1) ∈ dom r}. Furthermore, a hybrid input u is said to belong to L ∞ when ||u|| (t,j) ≤ c for any (t, j) ∈ dom u, for some c ≥ 0.

III. PROBLEM STATEMENT

Consider the nonlinear plant model

ẋp = f p (x p , u, w), y = g p (x p , d y ), (3)
where x p ∈ R np is the plant state, u ∈ R nu is the control input, w ∈ R nw is a vector of unknown exogenous disturbances, y ∈ R ny is the available output of the plant which is affected by the vector of noises d y ∈ R n dy . We consider the following dynamic controller

ẋc = f c (x c , y), u = g c (x c , y, d u ), (4)
where x c ∈ R nc is the controller state and d u ∈ R n du is a measurement noise affecting the control input. The presence of d u may be due to quantization error or computational glitch, to give a few examples. We emphasize that the x c -system is not necessarily an observer and that (4) captures static feedbacks laws as a particular case by setting u = g c (y, d u ). We assume that the signals corresponding to w, d y , d u are unknown, bounded (according to the L ∞norm), differentiable for d y and d u with time-derivatives in L ∞ . Note that we do not need to know any bound on the norm of these signals to apply the presented results. The functions f p , f c are assumed to be continuous and the functions g p , g c are assumed to be continuously differentiable. We consider the scenario where controller (4) communicates with the plant via a digital channel. Hence, the plant output y and the control input u are sent only at transmission instants t i , i ∈ Z ≥0 . We are interested in an event-triggered implementation in the sense that the sequence of transmission instants is determined by a criterion based on the output measurement y and the control input u, see Figure 1. Note that the event-triggering mechanism only depends on the noisy variables y and u. The setup we envision allows the triggering condition to depend on both y and u. While this may be hard to achieve in practice, we focus on this general scenario because it allows us to cover the particular cases where the triggering condition only depends on y or u as particular cases. This will be the case for instance when the controller is static or when only y or u is sampled, to give a few examples.

At each transmission instant, y is sent to the controller which computes a new control input that is instantaneously transmitted to the plant. In that way, we obtain

ẋp = f p (x p , û, w) t ∈ [t i , t i+1 ] ẋc = f c (x c , ŷ) t ∈ [t i , t i+1 ] u = g c (x c , ŷ, d u ) y = g p (x p , d y ) ẏ = 0 t ∈ [t i , t i+1 ] u = 0 t ∈ [t i , t i+1 ] ŷ(t + i ) = y(t i ) û(t + i ) = u(t i ), (5)
where ŷ and û respectively denote the last transmitted values of y and u. We assume that zero-order-hold devices are used to generate the sampled values ŷ and û, which leads to ẏ = 0 and u = 0.

Because we only have access to noisy signals, we need to define the sampling-induced error using noisy variables. Hence, we consider e := (e y , e u ) ∈ R ne , where

e y := ŷ -y e u := û -u (6)
which is reset to 0 at each transmission instant. Note that e is available to the event-triggering mechanism. We model the event-triggered controlled system using the hybrid formalism of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] like in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF], [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralised eventtriggering[END_REF], [START_REF] Forni | Event-triggered transmission for linear control over communication channels[END_REF], [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], for which a jump corresponds to a transmission. In that way, we

obtain    ẋ ė τ    =    f (x, e, w, d) g(x, e, w, d, υ) 1    (x, e, τ ) ∈ C    x + e + τ +    =    x 0 0    (x, e, τ ) ∈ D, (7)
where τ is an additional clock variable introduced to describe the time elapsed since the last transmission, x := (x p , x c ) ∈ R nx , d := (d y , d u ) ∈ R n d , υ := (υ y , υ u ) ∈ R n d and υ y , υ u respectively denote the time-derivative of the noises corresponding to d y and d u . The functions f, g in ( 7) are given by (their arguments are omitted below)

f = f p (x p , g c (x c , y + e y , d u ) + e u , w) f c (x c , y + e y ) g =       -∂ ∂xp g p (x p , d y )f p (x p , g c (x c , y + e y , d u ) + e u , w) -∂ ∂dy g p (x p , d y )υ y -∂ ∂xc g c (x c , y + e y , d u )f c (x c , y + e y ) -∂ ∂du g c (x c , y + e y , d u )υ u       . ( 8
)
The flow and jump sets of ( 7) are defined according to the triggering condition we will synthesize. As long as the triggering condition is not violated, the system flows on C and a jump occurs when the state enters in D. When (x, e, τ ) ∈ C∩D, the solution may flow only if flowing keeps (x, e, τ ) in C, otherwise the system experiences a jump. The sets C and D will be closed.

The main objective of this paper is to design the flow and the jump sets of system [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF], i.e. the triggering condition, to ensure an ISS stability property for system [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] as well as to ensure the existence of a strictly positive lower bound on the inter-transmission times.

IV. MAIN RESULTS

We first present the conditions we assume on system (7), then we present the event-triggering law and we state the main result. Finally, we explain how to apply our results in the periodic sampled-data control.

A. Assumptions

We make the following assumption on system [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF], which is a generalization of Assumption 1 in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Assumption 1. There exist a locally Lipschitz positive definite function V : R nx → R ≥0 , a locally Lipschitz positive semi-definite function W : R ne → R ≥0 , a continuous function H : R nx → R ≥0 , real numbers L ≥ 0, γ > 0, α, α, α, µ, σ ∈ K ∞ and a continuous, nonnegative function δ : R ny → R ≥0 such that the following holds.

(i) For all x ∈ R nx α(|x|) ≤ V (x) ≤ α(|x|). (9)
The function V in Assumption 1 corresponds to a Lyapunov function for the closed-loop system in the absence of sampling. Property [START_REF] Persis | On a small-gain approach to distributed event-triggered control[END_REF] implies that the system ẋ = f (x, e, w, d) is L 2 -gain stable from (W, √ µ) to (H, √ δ). Items (i) and (ii) of Assumption 1 also imply that the system ẋ = f (x, e, w, d) satisfies a derivative ISS property, in the sense of [START_REF] Angeli | Input-to-state stability with respect to inputs and their derivatives[END_REF], with respect to w, d (since µ ∈ K ∞ and W is positive semi-definite and continuous). It has to be noted that the case where item (ii) of Assumption 1 holds with µ(|(w, d)|) instead of µ(|(w, d, υ)|), is a particular case of [START_REF] Persis | On a small-gain approach to distributed event-triggered control[END_REF]. We decided to consider the case where µ also depends on υ for the sake of generality. The last item of Assumption 1 is an exponential growth condition on the e-system, similar conditions are imposed in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Note that the dependence of σ on υ comes from the sampling of the noisy measurements, see [START_REF] Borgers | Event-separation properties of eventtriggered control systems[END_REF], which is typical when investigating sampled-data systems with external inputs, see for instance [START_REF] Nešić | A note on input-to-state stabilization for nonlinear sampled-data systems[END_REF].

B. Triggering condition

Inspired by [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF], we trigger transmissions whenever the condition below is violated

γ 2 W 2 (e) ≤ δ(y) or τ ∈ [0, T ], (12)
where γ, W and δ come from Assumption 1 and T > 0 is selected such that T < T (γ, η, L), where

T (γ, η, L) :=    1 Lr arctan(r) (1 + η)γ > L 1 L (1 + η)γ = L 1 Lr arctanh(r) (1 + η)γ < L (13)
with r := ( (1+η)γ

L

) 2 -1 . The constant η > 0 is a tuning parameter which can be arbitrarily chosen by the user. Note that T (γ, η, L) is a decreasing function in η and when η → 0 we recover the upper bound on the maximum allowable transmission interval (MATI) given by [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and used in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF]. Hence, a small value of η > 0 leads to a larger T (γ, η, L), however this will lead to a big ISS gain as we will see below.

The constant T (γ, η, L) is therefore different from the one in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF], which means that we have to adapt the enforced minimum time between two successive transmissions to the presence of exogenous inputs.

In view of ( 12), the flow and jump sets of system ( 7) are

C = (x, e, τ ) : γ 2 W 2 (e) ≤ δ(y) or τ ∈ [0, T ] D = (x, e, τ ) : γ 2 W 2 (e) ≥ δ(y) and τ ≥ T . (14)

C. Stability result

We are ready to state the main result. Its proof is provided in the Appendix.

Theorem 1. Suppose that Assumption 1 holds and consider system [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] with the flow and jump sets defined in [START_REF] Shi | Event-triggered maximum likelihood state estimation[END_REF], where the constant T is such that T ∈ (0, T (γ, η, L)). There exist β ∈ KL and ξ ∈ K ∞ such that any solution φ = (φ x , φ e , φ τ ) with w, d, υ ∈ L ∞ and dom (w, d, υ) = dom φ satisfies, for all (t, j) ∈ dom φ,

|φ x (t, j)| ≤ max β(|(φ x (0, 0), φ e (0, 0))|, t + j), ξ(||(w, d, υ)|| (t,j) ) , ( 15
) where ξ(s) := α -1 ρ -1 1 ε (µ(s) + 1 η σ 2 (s)) , ρ(s) := min{ρ 1 ( s 2 ), ηγθ s 2 } for s ≥ 0, ρ 1 ∈ K ∞ , η > 0 and ε, θ ∈ (0, 1)
. Furthermore, the inter-transmission times are lower bounded by the constant T .

Theorem 1 shows that system (7), ( 14) satisfies a derivative ISS property, see [START_REF] Angeli | Input-to-state stability with respect to inputs and their derivatives[END_REF], with respect to the external disturbance w and the measurement noise d. It is interesting to note that even when µ in ( 10) is independent of υ (the timederivative of the noises), the ISS gain ξ in (15) does depend on υ because of [START_REF] Lehmann | Event-triggered pi control: Saturating actuators and anti-windup compensation[END_REF]. This type of result is common in the sampled-data control literature, see [START_REF] Nešić | A note on input-to-state stabilization for nonlinear sampled-data systems[END_REF]. The constants θ and ε can be arbitrarily chosen in (0,1) as shown in the proof. We notice, in view of ( 13), [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF], that η provides a tradeoff between the guaranteed minimum time between transmissions and the magnitude of the ISS gain ξ. The value of T (γ, η, L) can be increased by taking η small, however the ISS (nonlinear) gain ξ in (15) will increase, and vice versa. When η → ∞ and ε → 1 (which implies that T → 0), we recover the ISS property ensured in the absence of sampling.

D. Time-triggered implementation

The results of this section covers periodic sampled-data control as a particular case by removing the output dependent condition γ 2 W 2 (e) ≤ δ(y) in [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved and decentralised eventtriggering[END_REF]. In that way, the flow and the jump sets of system [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] are

C = {(x, e, τ ) : τ ∈ [0, T ]} D = {(x, e, τ ) : τ = T }, ( 16
)
and T is selected strictly smaller than T (γ, η, L) in [START_REF] Lehmann | Event-triggered model predictive control of discrete-time linear systems subject to disturbances[END_REF] as before. We can then state the following result which generalizes [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] to systems affected by exogenous inputs. Its proof follows the same lines as the proof of Theorem 1, it is therefore omitted.

Corollary 1. Suppose that Assumption 1 holds and consider system [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF], [START_REF] Nešić | A note on input-to-state stabilization for nonlinear sampled-data systems[END_REF]. Then, the conclusions of Theorem 1 hold.

V. ILLUSTRATIVE EXAMPLE

We consider the controlled Lorenz model of fluid convection in [START_REF] Wan | Nonlinear feedback control with global stabilization[END_REF] affected by external disturbances and measurement noises ẋ1 = -ax

1 + ax 2 + w 1 ẋ2 = bx 1 -x 2 -x 1 x 3 + u + w 2 ẋ3 = x 1 x 2 -cx 3 + w 3 y = x 1 + d y , (17)
where x 1 is proportional to the intensity of the convective motion, x 2 is proportional to the temperature difference between the ascending and descending currents, x 3 is proportional to the distortion of the vertical temperature profile from being linear, u corresponds to the tilt angle of a closedloop of natural convection from the vertical, w 1 , w 2 , w 3 are external disturbances and d y is the measurement noise. The parameters a, b, and c are related to some physical constants and are positive, see [START_REF] Vincent | Control of a chaotic system[END_REF] for more detail. We consider the following static output feedback controller u = -( p1 p2 a + b)y, where p 1 , p 2 > 0. We assume that u is affected by some additive noise d u , hence the input to ( 17)

is u = - p 1 p 2 a + b y + d u . (18)
We consider the scenario where a network is used to transmit data between the plant and the controller. We therefore take into account the sampling-induced error e = ŷ -y.

Note that it is not necessary to consider the error in u as the controller is static, as explained in Section III. Denote x := (x 1 , x 2 , x 3 ), w := (w 1 , w 2 , w 3 ) and d := (d y , d u ). We can model the system as [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] with (the arguments of f, g are omitted below)

f =     -ax 1 + ax 2 + w 1 bx 1 -x 2 -x 1 x 3 -( p1 p2 a + b)(y s + e) + d u + w 2 x 1 x 2 -cx 3 + w 3     g = ax 1 -ax 2 -w 1 -υ y , (19)
recall that υ y denotes the time-derivative of the signal corresponding to d y .

Let W (e) = |e| for any e. For all x ∈ R 3 and almost all e

∈ R ne , υ y ∈ R n dy ∇W (e), g ≤ a|x 1 | + a|x 2 | + |w 1 | + |υ y |. (20)
Hence, condition [START_REF] Lehmann | Event-triggered pi control: Saturating actuators and anti-windup compensation[END_REF] holds with L = 0, H

(x) = a|x 1 | + a|x 2 |, σ(|(w, d, υ)|) = 2|(w, d, υ)|. Let V (x) = p 1 x 2 1 + p 2 x 2 2 + p 2 x 2 3
, where p 1 , p 2 > 0. Item (i) in Assumption 1 holds with α(s) = min{p 1 , p 2 }s 2 and α(s) = max{p 1 , p 2 }s 2 for any s ≥ 0. By taking p

1 > 2, p 2 > 2a, c > 1 2 , item (ii) in Assumption 1 is verified with α(s) = min{ 1 2 a(p 1 -2), (p 2 -2a), p 2 (2c -1)}s 2 , δ(y) = 1 4 a(p 1 -2)y 2 , γ = 4p 2 ( p1 p2 a + b) 2 and µ(|(w, d)|) = max{p 1 , 4p 2 }|(w, d)| 2 + max{4p 2 , 4p 2 ( p1 p2 a + b) 2 + 1 2 a(p 1 - 2)}|(w, d)| 2 .
Note that µ is independent on υ in this example.

We set the parameter values a = 10, b = 28, c = 8/3, as in [START_REF] Wan | Nonlinear feedback control with global stabilization[END_REF], and we take p 1 = 3, p 2 = 3a, η = 0.05. We obtain T (γ, η, L) = 0.0047 using [START_REF] Lehmann | Event-triggered model predictive control of discrete-time linear systems subject to disturbances[END_REF]. Hence, we select T = 0.0045 and (12) becomes 1.0092 × 10 5 |e| 2 ≤ 2.5y 2 or τ ∈ [0, 0.0045].

We run simulations with random disturbances w satisfying |w i (t)| ≤ 0.1 for any t ≥ 0, i ∈ {1, 2, 3} and the following noises signals d y (t) = d u (t) = 0.1 sin(50t) for any t ≥ 0. Figure 2 shows that the state trajectory of the plant converges to a neighbourhood to the origin as expected, where the initial condition in this case is (x(0, 0), e(0, 0), τ (0, 0)) = (-20, -20, 30, 0, 0). The generated inter-transmission times are provided in Figure 3, where we can see the interaction between the time-dependent and the output-dependent conditions of the triggering rule. It can be also observed from Figure 3 that the minimum inter-transmission time corresponds to the guaranteed lower bound T , represented by the red line, which justifies our event-triggering mechanism. This point is also supported by Figure 4 where Zeno phenomenon occurs when we set T = 0, i.e. without enforcing a minimum time between transmissions. Table I provides the minimum and the average inter-transmission times, respectively denoted as τ min and τ avg , for 100 randomly distributed initial conditions such that |x(0, 0)|, |e(0, 0)| ≤ 100 and τ (0, 0) = 0 for a simulation time of 10 seconds.

VI. CONCLUSION

Input-to-state stabilizing event-triggered output feedback controllers have been synthesized for a class of nonlinear systems subject to external disturbances and corrupted output measurements and control inputs. The developed technique generalizes the results in [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback laws[END_REF] and enforces a strictly positive lower bound on the inter-transmission times. Future work will investigate the L p stability of the closed-loop system and the application of the results to LTI systems in the context of co-design, like in [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for linear systems[END_REF].

Fig. 1 .

 1 Fig. 1. Event-triggered control schematic.

(

  ii) For almost all x ∈ R nx and all (e, w, d, υ) ∈ R ne+nw+2n d ∇V (x), f (x, e, w, d) ≤ -α(|x|) -H 2 (x) -δ(y)+γ 2 W 2 (e) + µ(|(w, d, υ)|). (10) (iii) For almost all e ∈ R ne and all (x, w, d, υ) ∈ R nx+nw+2n d ∇W (e), g(x, e, w, d, υ) ≤ LW (e) + H(x) +σ(|(w, d, υ)|).

Fig. 2 .

 2 Fig. 2. State trajectory of the plant.

Fig. 3 .

 3 Fig.3. Inter-transmission times for the first three units of time.

Fig. 4 .

 4 Fig. 4. Inter-transmission times with T = 0.

J. Daafouz is also with the Institut Universitaire de France (IUF). This work was supported by the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 257462: HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems" and by the ANR under the grant COMPACS (ANR-13-BS03-0004-02). D. Nešić is with the Department of Electrical and His work is supported by the Australian Research Council under the Discovery Projects.

APPENDIX

Proof of Theorem 1. We first introduce ζ : R ≥0 → R the solution to the following differential system

where θ ∈ (0, 1) and η > 0 is fixed. We denote T (θ, η, γ, L) the time it takes for ζ to decrease from θ -1 to θ. This time T (θ, η, γ, L) is a continuous function of θ, η which is decreasing in θ, η. On the other hand, we note that T (θ, γ, η, L) → T (γ, η, L) (where T (γ, η, L) is defined in ( 13)) as θ tends to 0. As a consequence, since T < T (γ, η, L), there exists θ such that T < T (θ, η, γ, L). We fix the value of θ.

Let q := (x, e, τ ). We define for all q ∈ C ∪ D R(q) := V (x) + max{0, γζ(τ )W 2 (e)}.

Let q ∈ D and G(q) := (x, 0, 0). We obtain, in view of [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] and the fact that W is positive semi-definite,

Let q ∈ C and (w, d, υ) ∈ R nw+2n d and suppose that ζ(τ ) < 0. As a consequence, R(q) = V (x) and it holds that τ > T . Indeed, ζ(τ ) is strictly decreasing in τ , in view of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], and [START_REF] Shi | Event-triggered maximum likelihood state estimation[END_REF] since q ∈ C. Consequently, in view of page 100 in [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF], Lemma 1, Assumption 1 and ( 22)

where F (q) := (f (x, e, w, d), g(x, e, w, d, υ), 1). Hence, in view of ( 9) and since α ∈ K ∞ , there exists

When ζ(τ ) > 0, we have

As above, in view of page 100 in [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF], Lemma 1, item (ii) of Assumption 1 and (21)

Using several times the fact that 2ab ≤ 1 κ a 2 + κb 2 for any a, b ≥ 0 and κ > 0 and since δ(y) ≥ 0, we deduce that

where χ := µ + 1 η σ 2 . By using the same argument as in [START_REF] Angeli | Input-to-state stability with respect to inputs and their derivatives[END_REF] and since ζ(τ ) ≤ θ -1 for all τ ≥ 0 in view of (21), we derive that R

where ρ 2 : s → ηγθs ∈ K ∞ . We deduce that there exists

where ρ(s) := min{ρ 1 ( s 2 ), ρ 2 ( s 2 )} for s ≥ 0. When ζ(τ ) = 0, we obtain, in view of ( 25), [START_REF] Khalil | Nonlinear Systems[END_REF] and Lemma 1

(31) Hence, (31) is satisfied in all cases.

Let φ := (φ x , φ e , φ τ ) be a solution to [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF], [START_REF] Shi | Event-triggered maximum likelihood state estimation[END_REF] with input (w, d, υ) ∈ L ∞ and dom (w, d, υ) = dom φ. In view of (31) and page 99 in [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF], it holds that, for some ε ∈ (0, 1),

) for all j and for almost all t ∈ I j where I j = {t : (t, j) ∈ dom φ}. As a consequence, using ( 23) and (32) and by following similar lines as in the end of the proof of Theorem 1 in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], we deduce that for any (t, j) ∈ dom φ R(φ(t, j)) ≤ max β(R(φ(0, 0)), 0.5t + 0.5T j),

On the other hand, in view of Assumption 1 and since W is continuous (since it is locally Lipschitz) and positive semidefinite, there exists

for all e ∈ R ne according to Lemma 4.3 in [START_REF] Khalil | Nonlinear Systems[END_REF]. As a result, in view of Assumption 1, ( 21) and ( 22), it holds that, for all q ∈ C ∪ D,

where α R : s → α(s) + γ θ α W (s) ∈ K ∞ . Hence, in view of (33) and (34), we deduce that

where β : (s 1 , s 2 ) → α -1 ( β(α R (s 1 ), s 2 )) ∈ KL, ξ(s) := α -1 ρ -1 ( 1 ε χ(s)) for s ≥ 0. Thus, (15) holds.