Wen Kang
email: kangwen@amss.ac.cn

Lucie Baudouin
email: baudouin@laas.fr

Emilia Fridman
email: emilia@eng.tau.ac.il

Event-triggered control of Korteweg-de Vries equation under averaged measurements

Keywords: Korteweg-de Vries equation, event-trigger, LMIs

This work addresses distributed event-triggered control law of 1-D nonlinear Korteweg-de Vries (KdV) equation posed on a bounded domain. Such a system, in a continuous framework, is exponentially stabilizable by a linear state feedback as a source term. Here we consider the situation where the feedback is sampled in time and piecewise averaged in space, and an event-triggering mechanism is designed to maintain stability of this infinite dimensional system. Both well-posedness of the closed-loop system and avoiding the Zeno behaviour issues are addressed. Sufficient LMI-based conditions are constructed to guarantee the regional exponential stability. Numerical examples illustrate the efficiency of the method.

Introduction

In fluid mechanics, the Korteweg-de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces in a rectangular channel, equation in which the effects of dispersion, dissipation and nonlinearity are taken into account. When adding a diffusion term, the KdV equation becomes Korteweg-de Vries Burgers (KdVB) equation. The study of KdV/ KdVB systems has been an active research topic because of its potential applications, see e.g. [START_REF] Baudouin | Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback[END_REF][START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF][START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF][START_REF] Coron | Control and Nonlinearity[END_REF][START_REF] Kang | Distributed stabilization of Kortewegde Vries-Burgers equation in the presence of input delay[END_REF][START_REF] Marx | Output feedback stabilization of the Korteweg-de Vries equation[END_REF]. In the field of automatic control, a backstepping approach has been applied in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF][START_REF] Coron | Control and Nonlinearity[END_REF][START_REF] Marx | Output feedback stabilization of the Korteweg-de Vries equation[END_REF] for the feedback stabilization of KdV equation, and Lyapunov-based arguments have been employed to ensure the stability of the original system under the proposed control law. On the other hand, the survey paper [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF] gives a detailed overview of bound-ary controllability and internal stabilization approaches and results for the KdV equation. One can read in [START_REF] Baudouin | Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback[END_REF] two different approaches (from a Lyapunov functional or from an observability inequality) employed to exponentially stabilize the nonlinear KdV equation via delayed boundary damping terms.

In [START_REF] Kang | Distributed stabilization of Kortewegde Vries-Burgers equation in the presence of input delay[END_REF], distributed control of KdVB system has been suggested under point or averaged localised measurements in space but the proof rely strongly on the presence of a diffusion term that is missing in the KdV equation. Such distributed control was introduced for heat equation under point [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF] and under averaged [START_REF] Fridman | Sampled-Data Distributed H∞ Control of Transport Reaction Systems[END_REF] measurements. In the latter papers, sampled-data control via time-delay approach and Lyapunov-Krasovskii functionals were studied, and the results of [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF][START_REF] Fridman | Sampled-Data Distributed H∞ Control of Transport Reaction Systems[END_REF] were extended to event-triggered control in [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF]. However, since the Lyapunov-Krasovskii functionals for sampled-data control depend on the state-derivative (see Chapter 7 of [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]), this method cannot be applied to sampled-data control of KdVB equation. So [START_REF] Kang | Distributed stabilization of Kortewegde Vries-Burgers equation in the presence of input delay[END_REF] considered the constant input delay case.

To the best of our knowledge, no event-triggered control of KdV equation has been studied yet. The goal of event-triggering mechanism to a sampled control law is to update the control input only at meaningful instants.

Its drawback, well-known in hybrid systems problematics, could be the exhibition of a Zeno behaviour. This can be summed up as the law bringing an infinite number of updates in a finite amount of time. The present paper aims at contributing to the study of this topic via a Lyapunov approach, where sufficient LMI-based conditions for the closed-loop system with the avoidance of Zeno behaviour will be investigated.

In recent years, event-triggered control systems have been extensively studied (see e.g. [START_REF] Espitia | Stabilization of boundary controlled hyperbolic PDEs via Lyapunovbased event triggered sampling and quantization[END_REF][START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF][START_REF] Seuret | LQ-based event-triggered controller co-design for saturated linear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]), bringing an important alternative to periodic sampling of control laws. There are many important results on event-triggering mechanisms [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]. In order to reduce out the number of updates, three main event-triggering mechanisms are proposed as follows: continuous event-triggering mechanism (see e.g. [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]), periodic event-triggering mechanism (see e.g. [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF]), and event-triggering mechanism with a dwell time (see e.g. [START_REF] Selivanov | Event-triggered H∞ control: a switching approach[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]). It is worth pointing out that most works focus on event-triggered control of finite-dimensional systems. However, to the best of our knowledge, there are few papers studying this technique in the infinite-dimensional systems framework (see e.g. [START_REF] Espitia | Event-triggered boundary control of constant-parameter reaction-diffusion PDEs: a small-gain approach[END_REF][START_REF] Espitia | Stabilization of boundary controlled hyperbolic PDEs via Lyapunovbased event triggered sampling and quantization[END_REF][START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF]).

In this work, the main contribution lies in the construction of the event-triggering mechanism and the design of event-triggered control law for nonlinear KdV equation. It can also be stressed that the Lyapunov-Krasovskii approaches for sampled-data control design under point/averaged measurements cannot work for KdV equation. As a by-product, the distributed control via the spatial decomposition (or sampling) for PDEs introduced in [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF] and [START_REF] Fridman | Sampled-Data Distributed H∞ Control of Transport Reaction Systems[END_REF] for systems with diffusion terms, is, for the first time, extended to KdV equation that has no such a term. This is achieved thanks to using a V µ term in Lyapunov functional V defined by (4.5). Such a term is borrowed from [START_REF] Baudouin | Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback[END_REF].

This work addresses the event-triggered control design for KdV system under in domain measurements averaged in space, and for the record, [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF] gathers the results for distributed continuous-in-time controller to stabilize the KdV equation exponentially. Our concern here is then mainly to prove that distributed event-triggered control can still bring, under appropriate assumptions and choice of triggering mechanism, the expected exponential stability. Finally, different from our present work but somehow related to the same area of interest, the exact boundary controllability for the KdV equation was studied in [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF], and [START_REF] Rosier | Control and stabilization of the Korteweg-de Vries equation: recent progresses[END_REF] is devoted to the design of distributed control for KdV equation on a periodic domain and to the design of boundary control for KdV equation on a finite domain.

The remainder of this work is organized as follows. The problem setting is described in Section 2 while Section 3 details the main result of this paper and give some remarks. We suggest finite-dimensional feedback controllers which are distributed on the whole domain or on subdomains under averaged measurements. For both cases, we provide the event-triggering mechanism. Section 4 is devoted to the technical proofs, both of well-posedness of the closed loop system, avoidance of the Zeno behaviour that an event triggering mechanism could introduce, and of the main regional exponential stability theorem. Section 5 contains an extension to distributed on subdomains control and Section 6 presents numerical examples to illustrate the effectiveness of the proposed control strategy. Finally, Section 7 briefly concludes the article.

Notation. For any matrix P in R n×n , P 0 means that P is symmetric positive definite. For a partitioned matrix, the symbol * stands for symmetric blocks and I is the identity, 0 the zero matrix. Using L 2 (0, L) for the Hilbert space of square integrable scalar functions, one writes

u 2 L 2 (0,L) = u, u = L 0 |u(x)| 2 dx
, and we also define the Sobolev spaces

H 1 (0, L) = {u ∈ L 2 (0, L), u ∈ L 2 (0, L)} and its norm by u 2 H 1 (0,L) = u 2 L 2 (0,L) + u 2 L 2 (0,L) , H 1 0 (0, L) = {u ∈ H 1 (0, L), u(0) = u(L) = 0}
where all the derivatives are to be considered in the weak sense. Finally, L ∞ (0, L) denotes the space of essentially bounded function. For a function y of several variables, the partial derivative with respect to a variable ξ is denoted ∂ ξ y = ∂y ∂ξ .

2 Preliminaries and problem formulation

             ∂ t z + z∂ x z + ∂ x z + ν∂ xxx z -λz = f (x, t), ∀x ∈ (0, L), t ≥ 0, z(0, t) = z(L, t) = 0, ∂ x z(L, t) = 0, ∀ t ≥ 0, z(x, 0) = z 0 (x), ∀x ∈ (0, L), (2.1)
with the initial state z 0 ∈ L 2 (0, L) and the source input f ∈ L 1 (0, T ; L 2 (0, L)), where ν > 0, λ ≥ 0, and z = z(x, t) is the state of the nonlinear KdV equation. For λ > 0, the open-loop system may be unstable (see the example below). Note that destabilizing λ > 0 was considered in [START_REF] Tang | Stabilization of linearized Korteweg-de Vries Systems with anti-diffusion[END_REF]. Also, λ > 0 may stand for the desired decay rate achieved after stabilization of (2.1) with λ = 0 (see Remark 3 below). By selecting the control law

f (x, t) = -Kz(x, t), K > λ, (2.2)
one obtains a closed-loop system that is globally exponentially stable, as it will be shown later. In this article, we would like to address the question of the robustness of this stability with respect to the presence of both an event triggering in time and a localized averaging in space of the feedback control law. Noticing that we cannot really apply infinite dimensional feedback control law, we will consider here a finite dimensional approximation of (2.2) that still stabilizes the system (see Section 3 below).

More precisely, we will consider that the control law will be implemented in such a way that for all x ∈ (0, L),

for all t ∈ [t k , t k+1 ), f (x, t) = -K N j=1 zj (t k )1 Ωj (x), K > λ, (2.3)
where the sampling times t k are following an appropriate event trigger law to be given later, while {1 Ωj } j are the characteristic functions of the intervals {Ω j } j covering (0, L), and zj (t) = 1 |Ωj | Ωj z(x, t)dx. We will also consider the case that the event-triggered controller does not cover the whole domain [0, L], which is distributed on some parts of subdomains (see (5.1) in Section 5 below).

Well-posedness and exponential stabilization result under (2.2)

The proof of existence and regularity of solutions for the KdV equation has been investigated in many references, in particular in the field of controllability studies and even if several results rely on the smallness of the initial and source data (e.g. [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF][START_REF] Coron | Control and Nonlinearity[END_REF]), one can find in [START_REF] Chapouly | Global controllability of a nonlinear Korteweg-de Vries equation[END_REF] the proof of the following general result: Lemma 1. For any T > 0, L > 0, if z 0 ∈ L 2 (0, L) and f ∈ L 1 (0, T ; L 2 (0, L)), then the Cauchy problem (2.1) is well posed in the space C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)), meaning that there exists a unique solution z to the system (2.1) that satisfies, for a constant c = c(T, L) > 0,

z L ∞ (0,T ;L 2 (0,L)) + z L 2 (0,T ;H 1 (0,L)) ≤ c z 0 L 2 (0,L) + c f L 1 (0,T ;L 2 (0,L)) .
The proof of this lemma is detailed in [START_REF] Chapouly | Global controllability of a nonlinear Korteweg-de Vries equation[END_REF] and relies on a fixed point argument for the small time wellposedness (as also referenced and described in [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF]) of the problem, that allows to handle the non-linearity z∂ x z, and on clever a priori estimates of the local solution to extend arbitrarily the time frame and get a global existence and regularity result.

The proof of the well-posedness of the closed-loop system (2.1)-(2.2) stems from the same arguments and is not detailed here. Besides, it is easy to prove its exponential stability, stated here: Lemma 2. Let L > 0, T > 0, K > λ and z 0 ∈ L 2 (0, L). The closed-loop KdV system (2.1)-(2.2) is exponentially stable in the sense that

z(•, t) 2 L 2 (0,L) ≤ e -2(K-λ)t z 0 2 L 2 (0,L) , ∀t ≥ 0.
Indeed : define the energy (that will act as a Lyapunov functional) of the solution of a KdV equation by

E(t) = z(•, t) 2 L 2 (0,L) , ∀t ≥ 0. (2.4)
Taking the time derivative of

E(t) along (2.1)-(2.
2), we have, for any t ≥ 0,

Ė(t) ≤ -2(K -λ) L 0 |z(x, t)| 2 dx -ν|∂ x z(0, t)| 2 ≤ -2(K -λ)E(t) implying E(t) ≤ e -2(K-λ)t E(0), ∀t ≥ 0.
Furthermore, [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF] gathers several internal stabilization results for nonlinear KdV equations, and specifically, the stabilization through a localized distributed internal damping f (x, t) = -a(x)z(x, t) with a ∈ L ∞ (0, L) such that a(•) ≥ a 0 > λ in some subdomain ω of (0, L), is actually also true, see e.g. [START_REF] Pazoto | Unique continuation and decay for the Korteweg-de Vries equation with localized damping[END_REF][START_REF] Menzala | Stabilization of the Korteweg-de Vries equation with localized damping[END_REF]. However, in our study, we focus on the case described by (2.3).

As already mentioned before, in this paper we will use a Lyapunov approach to deal with an event-triggered control of the KdV equation under averaged measurements. The next section is devoted to the description of our technical setting.

Problem formulation and main result

We consider the following closed-loop KdV system:

                     ∂ t z + z∂ x z + ∂ x z + ν∂ xxx z -λz = -K N j=1 zj (t k )1 Ωj (x) in (0, L) × [t k , t k+1 ), k ∈ N, z(0, t) = z(L, t) = 0, ∂ x z(L, t) = 0, ∀ t ≥ 0, z(x, 0) = z 0 (x), ∀x ∈ (0, L). (3.1)
where the chosen control law for (2.1) is (2.3),

zj (t k ) = 1 |Ω j | Ωj z(x, t k )dx. (3.2)
This closed-loop system is defined under the following assumptions:

• Space averaging: As in [START_REF] Azouani | Feedback control of nonlinear dissipative systems by finite determining parameters -A reaction-diffusion Paradigm[END_REF][START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF][START_REF] Fridman | Sampled-Data Distributed H∞ Control of Transport Reaction Systems[END_REF][START_REF] Lunasin | Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study[END_REF], we assume that the points 0

= x 0 < x 1 < • • • < x N = L divide the inter- val [0, L] into N intervals Ω j = [x j-1 , x j ) covering it all.
The width of each sub-interval is supposed to be upper bounded by some constant: 0 < x j -x j-1 = |Ω j | ≤ ∆ and as expected, the characteristic functions 1 Ωj (x) are such that

1 Ωj (x) = 0, x / ∈ Ω j , 1 Ωj (x) = 1, otherwise , j = 1, • • • , N. (3.3)
• Time sampling: The update instants satisfy 0

= t 0 < t 1 < • • • < t k < t k+1 , lim k→∞ t k = ∞.
We define the event trigger mechanism by the law

t k+1 = inf t ≥ t k such that z(•, t) -z(•, t k ) 2 L 2 (0,L) ≥ γE(t) + γ 0 E(0)e -2θt (3.4)
where the energy E is defined by (2.4) as the L 2 (0, L)-norm of the state, and γ, γ 0 and θ are positive constants to be determined. It should be noticed that due to the term "γ 0 E(0)e -2θt ", here no dwell time is needed to be defined. • Though the feedback is of finite dimension, both t k and ∆ depend on the initial data. The larger initial data is, the smaller t k and ∆ need to be.

Our main objective is to design a regionally stabilizing event-trigger controller

u j (t) = -K zj (t k )1 [t k ,t k+1 ) (t)
that has a control gain K > λ to be determined later. In other words, we aim at deriving sufficient conditions for regional exponential stability of the closed-loop system (3.1) and to find a bound on the domain of attraction.

Theorem 1. Let L > 0, T > 0. Given a desired decay rate δ > 0, a control gain K > λ + δ, a length bound ∆ > 0, and positive tuning parameters λ 0 , R, θ > δ, γ 0 > 0, assume that there exist positive scalars µ, λ 1 , λ 2 , γ, and Γ that solve the following optimization problem:

min Γ subject to -3µν + λ 1 + λ 2 + 2 3 µRL √ L < 0, (3.5) Φ =      φ 11 K(1 + µL) K(1 + µL) * -λ 2 π 2 ∆ 2 0 * * -λ 0      ≺ 0, (3.6) (1 + µL) 1 + λ 0 γ 0 2(θ -δ) < R 2 Γ, (3.7)
where

φ 11 = -2K + 2λ + µ + λ 0 γ -λ 1 π 2 L 2 + 2δ. (3.8)
Then for any initial function z 0 ∈ L 2 (0, L) satisfying

z 0 L 2 (0,L) < 1 √ Γ
, the closed-loop system (3.1) under the event-triggering mechanism (3.4) is exponentially stable:

E(t) ≤ 1 + λ 0 γ 0 2(θ -δ) (1 + µL)E(0)e -2δt (3.9)
for all t ≥ 0. Moreover, if the above LMIs hold with δ = 0, then the closed-loop system is exponentially stable with a small enough decay rate. Remark 1. One could wish here that we do not make the assumption K ≥ λ+δ on the gain we need to apply to stabilize our system, but we shall recall that the decay rate of the exponential stability of the system with continuous feedback law -Kz is excatly δ = K -λ (Lemma 2) so that it's not reasonable to expect better when applying an approximated feedback law as we do.

               ∂ t z + z∂ x z + ∂ x z + ν∂ xxx z = -K N j=1 zj (t k )1 Ωj (x), in (0, L) × [t k , t k+1 ), k ∈ N, z(0, t) = z(L, t) = 0, ∂ x z(L, t) = 0, ∀ t ≥ 0, z(x, 0) = z 0 (x), ∀x ∈ (0, L), (3.10
) where zj (t k ) is given by (3.2). Let z = e λt z. It is easy to see that z is governed by

                     ∂ t z + e -λt z∂ x z + ∂ x z + ν∂ xxx z -λz = -K N j=1 ẑj (t k )1 Ωj (x), in (0, L) × [t k , t k+1 ), k ∈ N, z(0, t) = z(L, t) = 0, ∂ x z(L, t) = 0, ∀ t ≥ 0, z(x, 0) = z 0 (x), ∀x ∈ (0, L), (3.11)
where

ẑj (t k ) = 1 |Ω j | Ωj z(x, t k )dx.
From the proof of Theorem 1, it follows that LMIs of this Theorem guarantee stability of (3.11) since the nonlinear term "e -λt z zx " with the multiplier e -λt ≤ 1 will not change the proof of stability. Hence, if the LMI conditions of Theorem 1 hold with δ = 0, then the decay rate λ of original system (3.10) can be guaranteed since z = e -λt z.

4 Technical proofs 4.1 Well-posedness of the controlled system and avoidance of Zeno behaviour

From Lemma 1, the following well-posedness result can be obtained by an induction approach. Proposition 1. Let L > 0, T > 0 and assume that z 0 ∈ L 2 (0, L). Then system (3.1) under the event triggering law (3.4) has a unique solution z satisfying z ∈ C(0, T ; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)). Furthermore, the Zeno phenomenon is avoided.

Proof. In oder to prove the existence, uniqueness and regularity of the solution, we proceed by induction.

(i) Initialization. On the first time interval, (3.1) reads

               ∂ t z + z∂ x z + ∂ x z + ν∂ xxx z -λz = -K N j=1 z0 j 1 Ωj (x), ∀x ∈ (0, L), t ∈ [0, t 1 ), z(0, t) = z(L, t) = 0, ∂ x z(L, t) = 0, ∀ t ≥ 0, z(x, 0) = z 0 (x), ∀x ∈ (0, L),
where z0 j = zj (0) = 1 |Ωj | Ωj z 0 (x)dx, and K > λ. This is a nonlinear KdV equation with initial data z 0 ∈ L 2 (0, L) and source term f = -K N j=1 z0 j 1 Ωj ∈ L 1 (0, t 1 ; L 2 (0, L)). Lemma 1 allows to conclude that there exists a unique solution z ∈ C([0, t 1 ]; L 2 (0, L)) ∩ L 2 (0, t 1 ; H 1 0 (0, L)) to the latter system.

(ii) Heredity. Let us only highlight that the previously obtained solution satisfies z(t 1 ) ∈ L 2 (0, L) so that system (3.1) considered on the next time interval [t 1 , t 2 ) has an initial condition z(t 1 ) ∈ L 2 (0, L) and a source term

-K N j=1 z1 j 1 Ωj ∈ L 1 (t 1 , t 2 ; L 2 (0, L)
where z1 j = zj (t 1 ). Therefore, the same argument using Lemma 1 holds again and the heredity is proved similarly at any step k ∈ N.

(iii) Conclusion. By induction, for any k ∈ N, z ∈ C([t k , t k+1 ]; L 2 (0, L)) ∩ L 2 (t k , t k+1 ; H 1 (0, L)).
Therefore, from the extension by continuity at the instants t k , one can conclude that (3.1) has a unique solution z ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)).

(iv) Convergence. The solution will never blow up before T as a contrary of the Zeno behaviour (i.e. ∃t k > T ). Now we aim at showing that the event-triggering mechanism (3.4) rules out the Zeno behaviour, where an infinite number of updates may occur in a finite amount of time. It is actually sufficient to show that for a given T > 0, there exists τ * > 0 such that all the sampling instants t k ≤ T complying to (3.4) satisfy t k+1 -t k ≥ τ * . Let us denote by e k the deviation from the continuous time position: for any x ∈ [0, L] and t ∈ (0, T ), there exists k ∈ N such that t ∈ [t k , t k+1 ), and we set e k (x, t) z(x, t) -z(x, t k ).

(4.1)

Since the solution of closed-loop system (3.1) satisfies z ∈ C([0, T ]; L 2 (0, L)) and [0, T ] is a compact set, this error function e k is uniformly continuous in time with values in L 2 (0, L). This means that for any > 0 there exists τ * > 0 such that for all t, s ∈ [0, T ], if |t -s| < τ * then we have e k (•, t) -e k (•, s) L 2 (0,L) < . Thus, the following reasoning by contraposition holds:

∀ > 0, ∃τ * > 0, ∀t, s ∈ [0, T ], e k (•, t) -e k (•, s) L 2 (0,L) ≥ =⇒ |t -s| ≥ τ * . (4.2)
Since e k (t k ) = 0, we have

e k (•, t k+1 ) -e k (•, t k ) L 2 (0,L) = e k (•, t k+1 ) L 2 (0,L) .
Next the substitution t → t k+1 and s → t k into (4.2), together with the definition of t k+1 in (3.4), leads to

e k (•, t k+1 ) 2 L 2 (0,L) ≥ γE(t k+1 ) + γ 0 E(0)e -2θt k+1 ≥ γ 0 z 0 2 L 2 (0,L) e -2θT
implying that |t k+1 -t k | ≥ τ * . Indeed, given z 0 = 0, we choose = γ 0 z 0 2 L 2 (0,L) e -2θT so that there exists τ * > 0, depending on z 0 , θ, γ 0 , γ and T for which for any k such that t k , t k+1 ∈ [0, T ], one has t k+1 -t k ≥ τ * , so that the Zeno behaviour is avoided.

Regional stability analysis

Now we focus on the regional stability analysis of the closed-loop system and prove Theorem 1. Let us mention two things. On the one hand, the event-triggering mechanism (3.4) yields that the event-triggering error function is bounded on each time sub-interval as follows: ∀t ∈ [t k , t k+1 )

e k (•, t) 2 L 2 (0,L) ≤ γE(t) + γ 0 E(0)e -2θt . (4.3)
On the other hand, f defined by (2.3) can be rewritten as

f (x, t) = -K N j=1 1 Ωj (x) [z(x, t) -f j (x, t) -ρ j (t)] , ∀x ∈ [0, L], ∀t ∈ [t k , t k+1 ), ∀k ∈ N (4.4)
where

f j (x, t) = z(x, t) -zj (t) = z(x, t) - 1 |Ω j | Ωj z(x, t)dx, ρ j (t) = zj (t) -zj (t k ) = 1 |Ω j | Ωj e k (x, t)dx.
Proof of Theorem 1

Writing V µ (t) = µ L 0 x|z(x, t)| 2
dx with µ > 0, we define the following functional (see [START_REF] Baudouin | Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback[END_REF]):

V (t) E(t)+V µ (t) = L 0 |z(x, t)| 2 dx+µ L 0 x|z(x, t)| 2 dx.
(4.5) First, this Lyapunov functional candidate V (t) is equivalent to the energy of the system E(t) in the sense that

E(t) ≤ V (t) ≤ (1 + µL)E(t). (4.6)
Then, let us estimate its time derivative. For t ∈ [t k , t k+1 ), substituting (4.4) into (3.1) and differentiating V (t) along (3.1), one gets

V (t) = Ė(t) + Vµ (t) = 2 L 0 z(x, t)∂ t z(x, t)dx + 2µ L 0 xz(x, t)∂ t z(x, t)dx = 2 L 0 (1 + µx)z(x, t) -ν∂ xxx z(x, t) -z(x, t)∂ x z(x, t) -∂ x z(x, t) + λz(x, t) -Kz(x, t) dx + 2K N j=1 Ωj (1 + µx)z(x, t) f j (x, t) + ρ j (t) dx.
Hence,

V (t) = -ν|∂ x z(0, t)| 2 -3µν L 0 |∂ x z(x, t)| 2 dx + µ L 0 |z(x, t)| 2 dx + 2 3 µ L 0 z 3 (x, t)dx -2(K -λ) L 0 (1 + µx)|z(x, t)| 2 dx + 2K N j=1 Ωj (1 + µx)z(x, t)[f j (x, t) + ρ j (t)]dx.
Using (4.3), for any λ 0 > 0 we can deduce that

V (t) ≤ V (t) + λ 0 γE(t) + γ 0 E(0)e -2θt -e k (•, t) 2 L 2 (0,L) ≤ -3µν L 0 |∂ x z(x, t)| 2 dx-(2K -µ -λ 0 γ -2λ) L 0 |z(x, t)| 2 dx + 2 3 µ L 0 z 3 (x, t)dx -(2K -2λ)µ L 0 x|z(x, t)| 2 dx + 2K N j=1 Ωj (1 + µx)z(x, t)[f j (x, t) + ρ j (t)]dx + λ 0 γ 0 E(0)e -2θt -λ 0 e k (•, t) 2 L 2 (0,L) .
(4.7) Several estimates can now be obtained to deal with each of these terms and bring this into a quadratic form. First, Cauchy-Schwarz inequality and Sobolev's inequality (see Lemma A.3) leads to

L 0 z 3 (x, t)dx ≤ z(•, t) 2 L ∞ (0,L) L 0 |z(x, t)|dx ≤ L √ L ∂ x z(•, t) 2 L 2 (0,L) z(•, t) L 2 (0,L) (4.8)
Then from Lemma A.4, Wirtinger's inequality yields

λ 1 ∂ x z(•, t) 2 L 2 (0,L) - π 2 L 2 z(•, t) 2 L 2 (0,L) ≥ 0. (4.9)
for any λ 1 > 0. Moreover, since Ωj f j (x, t)dx = 0, from Lemma A.5, Poincaré's inequality rewrites

f j (•, t) 2 L 2 (Ωj ) ≤ ∆ 2 π 2 ∂ x z(•, t) 2 L 2 (Ωj ) ,
bringing for any λ 2 > 0

λ 2 N j=1 ∂ x z(•, t) 2 L 2 (Ωj ) - π 2 ∆ 2 f j (•, t) 2 L 2 (Ωj ) ≥ 0.
(4.10) Applying the Cauchy-Schwarz inequality, we obtain

N j=1 Ωj ρ 2 j (t) = N j=1 ρ 2 j (t)|Ω j | ≤ N j=1 1 |Ω j | Ωj e k (x, t)dx 2 ≤ N j=1 Ωj e 2 k (x, t)dx = L 0 e 2 k (x, t)dx. (4.11) Hence, λ 0 e k (•, t) 2 L 2 (0,L) - N j=1 Ωj ρ 2 j (t) ≥ 0. (4.12)
Set η(x, t) = col{z(x, t), f j (x, t), ρ j (t)}. Substituting (4.8) and (4.11) into (4.7), and adding (4.9), (4.10) and (4.12) to V (t), we obtain

V (t) + 2δV (t) ≤ N j=1 Ωj η(x, t) Φ(x)η(x, t) -3µν -λ 1 -λ 2 - 2µ 3 L √ L z(•, t) L 2 (0,L) ∂ x z(•, t) 2 L 2 (0,L) -2(K -λ -δ)µ L 0 x|z(x, t)| 2 dx + λ 0 γ 0 E(0)e -2θt , (4.13
) where (3.8). Applying Schur complement theorem [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF], one gets that Φ(x) ≺ 0 is equivalent to

Φ(x) =      φ 11 K(1 + µx) K(1 + µx) * -λ 2 π 2 ∆ 2 0 * * -λ 0      and φ 11 = -2K + 2λ + µ + λ 0 γ -λ 1 π 2 L 2 + 2δ as in
φ 11 + K 2 (1 + µx) 2 ∆ 2 λ 2 π 2 + λ -1 0 < 0,
that also writes

-2K + 2λ + µ + λ 0 γ -λ 1 π 2 L 2 + 2δ + K 2 (1 + µx) 2 ∆ 2 λ 2 π 2 +λ -1 0 < 0 (4.14)
Since we need that property for all x ∈ [0, L], and since we have 1

≤ (1 + µx) 2 ≤ (1 + µL) 2 , then it proves Φ(L) ≺ 0 =⇒ Φ(x) ≺ 0, ∀x ∈ [0, L].
Hence, denoting Φ = Φ(L) so that (3.6) holds, we have proved that

N j=1 Ωj η(x, t) Φ(x)η(x, t) ≤ 0. ( 4
V (t)+2δV (t) ≤ λ 0 γ 0 E(0)e -2θt ≤ λ 0 γ 0 V (0)e -2θt , ∀t ≥ 0. Now let θ > δ.
Then, for all t ≥ 0 we can write

V (t) ≤ e -2δt V (0) + λ 0 γ 0 e -2δt V (0) t 0 e -2(θ-δ)s ds ≤ e -2δt V (0) + λ 0 γ 0 V (0) 2(θ -δ) e -2δt -e -2θt ≤ 1 + λ 0 γ 0 2(θ -δ) e -2δt V (0) - λ 0 γ 0 2(θ -δ) V (0)e -2θt
From (4.6) it follows that

E(t) ≤ 1 + λ 0 γ 0 2(θ -δ) (1 + µL)e -2δt E(0) - λ 0 γ 0 2(θ -δ) E(0)e -2θt
, (4.17)

which implies (3.9). In order to end the proof of Theorem 1, we need to prove that (4.16) holds. On the one hand, for t = 0, inequality (4.16) holds by hypothesis in Theorem 1, so that E(0) < 1 Γ . On the other hand, let (4.16) be false for some t > 0 and let t * be the smallest instant such that

E(t * ) ≥ R 2 . Since E is continuous in time, we have E(t * ) = R 2 and E(t) < R 2 for t ∈ [0, t * ).
Therefore, the feasibility of inequality (3.5) and LMI (3.6) guarantee that (4.17) is true for all t ∈ [0, t * ). Hence, by continuity,

E(t) ≤ 1 + λ 0 γ 0 2θ -2δ (1 + µL)e -2δt E(0) ≤ 1 + λ 0 γ 0 2θ -2δ (1 + µL) 1 Γ , ∀t ∈ [0, t * ].
The above inequality, together with the assumption (3.7), implies

E(t) ≤ 1 + λ 0 γ 0 2θ -2δ (1 + µL) 1 Γ < R 2
for all t ∈ [0, t * ], which contradicts the definition of t * . Therefore, (4.16) holds. Note that the feasibility of the strict LMI (3.6) with δ = 0 implies its feasibility with a slightly larger δ 0 > 0. Therefore, if the strict LMI (3.6) holds for δ = 0, then the closed-loop system is exponentially stable with a small decay rate. Remark 4. It must be stressed that the present Lyapunov function cannot work for the case of a simple sampled-data control under averaged measurement and that the event-triggered law is critical in the proof of stability. Remark 5. Given K > λ + δ, the LMI conditions of Theorem 1 are always feasible for small enough γ, γ 0 , ∆ and large enough λ 0 such that λ 0 γ is small. By Schur complement, Φ ≺ 0 is equivalent to (4.14) with x = L. The latter holds for µ = λ 1 = γ = ∆ = 0, and large enough λ 0 . Thus, LMIs hold for small enough µ, λ 1 , γ, ∆, γ 0 , R with appropriate (large enough) λ 0 . Remark 6. Let us explain here what prevents us from obtaining such results under point measurements. For the case of averaged measurements, in the proof of Theorem 1, we need the Lyapunov functional to be continuous in L 2 -norm. For the case of point measurements, for a matter of continuity in the space variable, we need to guarantee that the Lyapunov functional is continuous in H 1 -norm. But this requires that the solution is in C([0, T ]; H 1 (0, L)), therefore requiring more regular initial and boundary data than it is the case here.

Extension to the controller distributed on subdomains

In this subsection, we are concerned with the case that the actuation does not cover the whole domain Ω and the averaged measurements are measured over the parts of the subdomains. As in [START_REF] Wang | Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems[END_REF], let

0 ≤ x1 < x2 ≤ x3 < x4 ≤ • • • ≤ x2N-1 < x2N ≤ L, [x 2j-1 , x2j ] ⊂ [x j-1 , x j ], j = 1, 2, • • • , N. Denote Ω j [x 2j-1 , x2j
]. Now we study the system (2.1) under the event-triggered controller

f (x, t) = -K N j=1 zj (t k )1 Ωj (x), K > λ, (5.1)
where

zj (t k ) = 1 | Ω j | Ωj z(x, t k )dx, | Ω j | = x2j -x2j-1 .
(5.2) By applying the first mean value theorem, since z ∈ C([0, T ], L 2 (0, L)) we obtain that there exists a point xj t ∈ Ω j such that 1

| Ω j | Ωj z(x, t)dx = z(x j t , t). (5.3)
Then the controller (5.1) can be rewritten as

f (x, t) = -K N j=1 [z(x j t , t) -ρj (t)]1 Ωj (x), (5.4)
where ρj (t) = 1

| Ω j | Ωj e k (x, t)dx.
This leads to the closed-loop system

                     ∂ t z + z∂ x z + ∂ x z + ν∂ xxx z -λz = -K N j=1 [z(x j t , t) -ρj (t)]1 Ωj (x), in (0, L) × [t k , t k+1 ), k ∈ N, z(0, t) = z(L, t) = 0, ∂ x z(L, t) = 0, ∀ t ≥ 0,
z(x, 0) = z 0 (x), ∀x ∈ (0, L).

(5.5) Denote l j max{x 2j -x j-1 , x j -x2j-1 }.

(5.6)

x j-1 x j x2j-1 xj t x2j l j
Given a desired decay rate δ > 0, a control gain K > λ + δ, length bounds l > 0, ∆ > 0, and positive tuning parameters λ 0 , R, θ > δ, γ 0 > 0, assume that there exist positive scalars µ, λ 1 , β i (i = 1, 2), γ, and Γ that solve the following optimization problem:

min Γ subject to

-3µν + λ 1 + β 2 + 2 3 µRL √ L < 0, (5.8)
Φ =         φ 11 β 2 π 2 4l 2 0 K(1 + µL) * -2K ∆ -β 2 π 2 4l 2 -KµL 0 * * -β 1 0 * * * -λ 0         ≺ 0, (5.9)
(1 + µL) 1 + λ 0 γ 0 2(θ -δ) < R 2 Γ, (5.10)
where

φ 11 = µ + λ 0 γ + (2λ + 2δ)(1 + µL) + β 1 -β 2 π 2 4l 2 -λ 1 π 2 L 2 . (5.11)
Then for any initial function z 0 ∈ L 2 (0, L) satisfying

z 0 L 2 (0,L) < 1 √ Γ
, the closed-loop system (5.5) under the event-triggering mechanism (3.4) is exponentially stable in the sense that (3.9) holds. Moreover, if the above LMIs hold with δ = 0, then the closed-loop system is exponentially stable with a small enough decay rate.

Proof. Consider V (t) given by (4.5). Differentiating V (t) along (5.5), for any λ 0 > 0 one gets

V (t) ≤ V (t) + λ 0 γE(t) + γ 0 E(0)e -2θt -e k (•, t) 2 L 2 (0,L) ≤ -ν|∂ x z(0, t)| 2 -3µν L 0 |∂ x z(x, t)| 2 dx +(µ + λ 0 γ) L 0 |z(x, t)| 2 dx + 2 3 µ L 0 z 3 (x, t)dx +2λ L 0 (1 + µL)|z(x, t)| 2 dx -2K N j=1 z 2 (x j t , t)| Ω j | -2Kµ N j=1 Ωj x1 Ωj (x)z(x, t)z(x j t , t)dx +2K N j=1 Ωj 1 Ωj (x)(1 + µx)z(x, t)ρ j (t)dx +λ 0 γ 0 E(0)e -2θt -λ 0 e k (•, t) 2 L 2 (0,L) .
(5.12) Cauchy-Schwarz's inequality yields

Ωj [1 Ωj (x)ρ j (t)] 2 dx = | Ω j |ρ 2 j (t) = 1 | Ω j | Ωj e k (x, t)dx 2 ≤ Ωj |e k (x, t)| 2 dx ≤ Ωj |e k (x, t)| 2 dx so that λ 0 N j=1 Ωj |e k (x, t)| 2 -[1 Ωj (x)ρ j (t)] 2 dx ≥ 0.
From Ω j ⊂ Ω j , one has β 1 > 0 such that

β 1 N j=1 Ωj |z(x, t)| 2 -[1 Ωj (x)z(x, t)]
β 2 N j=1 Ωj |∂ x z(x, t)| 2 - π 2 4l 2 [z(x, t) -z(x j t , t)] 2 dx ≥ 0
for some constant β 2 > 0. Set η(x, t) = {z(x, t), z(x j t , t), 1 Ωj (x)z(x, t), 1 Ωj (x)ρ j (t)}. Using (4.8), (4.9), (5.7), (5.12) and applying Sprocedure, we have

V (t) + 2δV (t) ≤ V (t)+2δV (t)+λ 0 γE(t)+γ 0 E(0)e -2θt -e k (•, t) 2 L 2 (0,L) +λ 1 ∂ x z(•, t) 2 L 2 (0,L) - π 2 L 2 z(•, t) 2 L 2 (0,L) +λ 0 N j=1 Ωj |e k (x, t)| 2 -[1 Ωj (x)ρ j (t)] 2 dx +β 1 N j=1 Ωj |z(x, t)| 2 -[1 Ωj (x)z(x, t)] 2 dx +β 2 N j=1 Ωj |∂ x z(x, t)| 2 - π 2 4l 2 [z(x, t) -z(x j t , t)] 2 dx ≤ N j=1 Ωj η(x, t) Φ(x)η(x, t) + λ 0 γ 0 E(0)e -2θt -3µν -λ 1 -β 2 - 2µ 3 L √ L z(•, t) L 2 (0,L) ∂ x z(•, t) 2 L 2 (0,L) , where Φ(x) =         φ 11 β 2 π 2 4l 2 0 K(1 + µx) * -2K ∆ -β 2 π 2 4l 2 -Kµx 0 * * -β 1 0 * * * -λ 0        
, with φ 11 as in (5.11). Thus, by Schur complement, the LMIs (5.8), (5.9) yield (3.9).

+ λ -1 0 K 2 (1 + µL) 2 -(β 2 π 2 4l 2 ) 2 [-2K ∆-β 2 π 2 4l 2 +β -1 1 K 2 µ 2 L 2 ] -1 < 0. Choose β 2 = 2K ∆ 4l 2 π 2 .
The latter holds for µ = λ 1 = γ = 0, small enough β 1 and large enough λ 0 . Thus, LMIs hold for small enough µ, λ 1 , β 1 , γ, γ 0 , R < 9ν 2L √ L with appropriate (large enough) λ 0 , (small enough) l such that λ 0 γ and 2K ∆ 4l 2 π 2 are small.

Numerical examples

Consider the KdV system:

               ∂ t z + z∂ x z + ∂ x z + ν∂ xxx z -λz = f (x, t), ∀0 < x < L, t ≥ 0 z(0, t) = z(L, t) = ∂ x z(L, t) = 0, z(x, 0) = z 0 (x) = 0.32 1 -cos( 2πx L ) , x ∈ [0, L],
where ν > 0 will be chosen below. We will give simulation for the following cases:

• Open-loop system without input (i.e. f (x, t) = 0)

• Closed-loop system under continuous-time controller f (x, t) = -Kz(x, t) • Closed-loop system under event-triggered controller distributed on the whole domain with averaging f (x, t) = -K N j=1 zj (x, t k )1 Ωj (x) • Closed-loop system under event-triggered controller distributed on subdomains with averaging f

(x, t) = -K N j=1 zj (x, t k )1 Ωj (x)
where K > λ is a controller gain.

Example 1: For the event-triggered control law (2.3) under averaged measurements, we verify LMI conditions of Theorem 1 with K = L = 1, λ = 0.5, ν = 0.3, δ = 0.4, ∆ = 0.1, R = 0.5. We find that the closed-loop system under event-triggering mechanism (3.4) with θ = 2, γ = 0.00029 and γ 0 = 0.02 is exponentially stable for µ = 0.5401 and for any initial values satisfying z 0 L 2 (0,1) < 1 √ 6.1997 ≈ 0.4. A finite difference method is used to illustrate the effect of the proposed event-triggered control law. The steps of space and time are chosen as 0.05 and 0.0001, respectively. Fig. 2 illustrates the evolution of the state of the for the open-loop system, the closed-loop system under continuous-time controller, and the closed-loop system under the event-triggered controller. The simulations show that the event-triggered controller improves the performance. Example 2: For the event-triggered control law (5.1) under averaged and localized measurements, we verify LMI conditions of Proposition 2 with K = L = 1, λ = 0.5, λ 0 = 1, ν = 0.3, δ = 0.4, R = 0.5, l = 0.2, ∆ = 1/3. We find that the closed-loop system under eventtriggering mechanism (3.4) with θ = 2, γ = 0.0013 and γ 0 = 0.02 is exponentially stable for µ = 0.0235 and for any initial values satisfying z 0 L 2 (0,1) < 1 √ 4.5603 ≈ 0.46. We proceed further with numerical simulations of the closed-loop KdV system under the event-triggering mechanism t k+1 = inf t ≥ t k e k 2 L 2 (0,L) ≥ 0.0013E(t) + 0.02E(0)e -4t .

Let x 0 = 0, x 1 = 0.3, x 2 = 0.6, x 3 = 0.9 and x 4 = 1. Set x1 = 0.1, x2 = 0.2, x3 = 0.4, x4 = 0.5, x5 = 0.7, x6 = 0.8, x7 = 0.9, x8 = 1. The simulations show that the state of closed-loop KdV system converges to zero (see Fig. 6).

Conclusion

The present work discusses event-triggered control of the nonlinear KdV equation. An event-triggering mechanism has been proposed to reduce the number of control update. By constructing an appropriate Lyapunov functional, sufficient LMI-based conditions have been investigated while ensuring that the closed-loop system is regionally exponentially stable. The avoidance of Zeno behaviour is guaranteed. The presented method gives efficient tools for various event-triggered controller and observer design problems for nonlinear PDEs.

Remark 2 .

 2 If γ and γ 0 are small enough, then the event-triggering mechanism (3.4) gets more sensitive to the output change and transmits the signals more often, what makes the control more similar to the stabilizing continuous-time controller. Remark 3. Consider (3.1) with λ = 0

Fig. 1 .

 1 Fig. 1. Subdomain Ωj = [x2j-1, x2j], point xj t and ljThen we have the following result: Proposition 2. Consider the closed-loop system (5.5). Let L > 0, T > 0. Denote

Remark 7 .

 7 Given ∆ < 1 and K > λ + δ ∆ , the LMI conditions of Proposition 2 are always feasible for small enough γ, γ 0 , R < 9ν 2L √ L , and large enough λ 0 . By Schur complement, Φ ≺ 0 ⇐⇒ φ 11

Fig. 2 .

 2 Fig. 2. State of the open-loop system with λ = 0.5 open-loop KdV system. It is seen that the open-loop system is unstable.

Fig. 3 2 L 2

 322 Fig. 3 illustrates the evolution of the state of the closed-loop KdV system under the event-triggering mechanism

Fig. 3 .Fig. 4 .

 34 Fig. 3. State of the closed-loop system with the event-triggered control law (2.3) distributed over all domain

Fig. 5

 5 Fig. 5 demonstrates the time evolution of ln(E(t))for the open-loop system, the closed-loop system under continuous-time controller, and the closed-loop system under the event-triggered controller. The simulations show that the event-triggered controller improves the performance.

Energy

  Fig. 5. ln(E(t)) of the open-loop system, closed-loop system under continuous-time/event-triggered controller

Fig. 6 .

 6 Fig.6. State of the closed-loop system with the event-triggered control law (5.1) distributed over subdomains

This work was supported by Israel Science Foundation (Grant No. 1128/14), National Natural Science Foundation of China (Grant No. 61803026), Fundamental Research Funds for the Central Universities (Grant No. FRF-TP-18-032A1), Joint Research Project HetCPS: Ministry of Science & Technology of Israel and CNRS.

Appendix A Lemma A.3 (Sobolev embedding and inequality). The embedding [START_REF] Fridman | Sampled-Data Distributed H∞ Control of Transport Reaction Systems[END_REF][START_REF] Hardy | Inequalities. Mathematical Library[END_REF]). Assume that g ∈ H 1 (0, L) with


Meriadeg Christophe Rodriguez

Ar Gouilh

Nicolas Weiss

Sébastian Stroer

Karima Mokhtari

Danielle Seilhean

Bertrand Mathon

Vanessa Demontant

Melissa N'debi

Guillaume Gricourt

Paul-Louis Woerther

Jean-Michel Pawlotsky

Karl Stefic

Julien Marlet

Pierre-François Dequin

Antoine Guillon

Valérie Pourcher

David Boutolleau

Astrid Vabret

Sonia Burrel

P.-L Woerther

J.-M Pawlotsky

M Ar Gouilh

S Weiss

K Stroer

D Mokhtari

B Seilhean

V Mathon

D Pourcher

S Boutolleau

Burrel

P.-F Dequin

J Stefic

Marlet

Fatal Measles Inclusion-Body Encephalitis in an Adult with Untreated AIDS, France

Keywords: measles inclusion-body encephalitis, MIBE, HIV/AIDS and other retroviruses, meningitis/encephalitis, viruses, brain biopsy, metagenomic, France

We report a fatal case of measles inclusion-body encephalitis occurring in a woman with AIDS. After an extensive but unsuccessful diagnostic evaluation, a pan-pathogen shotgun metagenomic approach revealed a measles virus infection. We identified no mutations previously associated with neurovirulence.

The Study

We report a fatal case of a 28-year-old woman of Romanian origin with untreated AIDS, initially admitted to the hospital on September 22, 2018 (day 0) for afebrile generalized motor seizure that began focally in the right lower limb. Magnetic resonance imaging (MRI) was initially normal and electroencephalogram (EEG) showed slight abnormalities related to slow frontal activity. The patient recovered fully and was discharged with antiepileptic therapy. Of note, the patient had stopped antiretroviral therapy (ART) 1 year earlier and declined to restart therapy following this hospital admission. In the next week, she had several relapses of focal seizures, requiring hospital readmission on day 7. Despite antiepileptic therapy adjustments, the myoclonic seizures persisted and became resistant to high doses of anticonvulsants and clonazepam add-on therapy (day 31). Consequently, the patient was hospitalized in intensive care unit. EEG showed a pattern of frontal-lobe epilepsy. MRI showed hyperintense cortical signals in frontal and left temporal cortex without hemorrhage lesions and without any signs of cerebral venous thrombosis.

Biologic investigation revealed HIV replication and 26/mm 3 CD4 T-cell count at day 35. We initiated antiretroviral medications on day 43. During her hospitalization, the patient showed a gradual impairment of consciousness (Glasgow coma score 6 on day 61) and was mechanically ventilated. MRI showed increase of the cortical hyperintensities and EEG showed diffuse encephalopathy pattern (Figure 1). We analyzed cerebrospinal fluid (CSF) samples taken on days 37, 62, and 64 for pathogens: viruses (herpes simplex virus, varicella zoster virus, enterovirus, cytomegalovirus, Epstein-Barr virus, human herpesvirus 6, HIV, and polyomavirus JC), bacterial and mycobacteria, fungi (Aspergillus spp., Cryptococcus neoformans), and parasites (Toxoplasma gondii); no pathogens were detected. All CSF were paucicellular with normal protein and glucose levels. Autoimmune antibodies were also negative. We performed a brain biopsy of the left frontal lobe on day 71 to determine the cause of encephalitis by the underlying neurologic symptoms, abnormal imaging features, and biologic findings. Neuropathology analysis revealed scarce inflammatory activation of glial cells (Figure 2). Because all the firstline microbiology testing assays remained negative on the biopsy, we considered using shotgun metagenomic (SMg) for pan-pathogen RNA/DNA detection to analyze the clinical samples with an unbiased approach. In brief, we performed an extraction combining bead beating and chemical and enzymatic lysis before library preparation. We performed sequencing on NextSeq500 with High Output Kit version 2.5 (300 cycles) (Illumina, https://www.illumina.com) [START_REF] Rodriguez | Multidisciplinary Necrotizing Fasciitis Study Group. Pathogen identification by shotgun metagenomics of patients with necrotizing soft-tissue infections[END_REF]. We analyzed sequencing data using MetaMIC software, which performed microorganism identification, genome reconstruction, and variant calling [START_REF] Rodriguez | Multidisciplinary Necrotizing Fasciitis Study Group. Pathogen identification by shotgun metagenomics of patients with necrotizing soft-tissue infections[END_REF]. A total of 5 samples were tested by SMg: CSF (day 61), bronchoalveolar lavage (days 63 and 76), brain biopsy (day 71), whole blood (day 68). Only the brain biopsy sample was found to be positive for measles virus (MeV); >4,800,000 of 2 ×150 paired reads were assigned to the virus. These results were confirmed by specific MeV real-time reverse transcription PCR. The detected MeV sequences in the brain biopsy allowed the reconstruction of a nearly-complete genome assembly (99.5% with a median depth coverage >25,000; GenBank accession no. MN893225). BLAST analysis showed this MeV exhibited 99.3% identity (15,762/15,876 nt) with the closest available fully sequenced B3 genotype (MVs/California.USA/05.14/[B3]; GenBank accession no. KY969477). These results excluded the infection by a MeV vaccine strain and oriented toward a European lineage origin. However, it was not possible to determine the precise origin of the virus because a high number of MeV harboring an identical C-terminal hypervariable domain (450 nt) of the nucleoprotein N gene N-450, the only genetic data available on the WHO database (2), were co-circulating in Europe at the time (data not shown) [START_REF] Katoh | MAFFT multiple sequence alignment software version 7: improvements in performance and usability[END_REF][START_REF] Drummond | BEAST: Bayesian evolutionary analysis by sampling trees[END_REF][START_REF] Drummond | Relaxed phylogenetics and dating with confidence[END_REF][START_REF] Darriba | jModelTest 2: more models, new heuristics and parallel computing[END_REF]. In addition, we identified no previously reported mutations suspected for neurovirulence [START_REF] Hashiguchi | Structures of the prefusion form of measles virus fusion protein in complex with inhibitors[END_REF][START_REF] Plattet | Measles virus fusion protein: structure, function and inhibition[END_REF]. We observed a mutational hotspot within the virus-encoded matrix protein (M). Of interest, the F1 5′ end of the fusion protein (F) contained the most variable sites found along the genome. This hydrophobic F1 part is associated with hyperfusogenicity and neurodegenerative disorders (data not shown) [START_REF] Hashiguchi | Structures of the prefusion form of measles virus fusion protein in complex with inhibitors[END_REF][START_REF] Plattet | Measles virus fusion protein: structure, function and inhibition[END_REF].

Several characteristics supported measles inclusion-body encephalitis (MIBE) diagnosis:

immunocompromised patient, undetectable MeV RNA with no intrathecal synthesis of anti-MeV antibodies in CSF (MeV-specific IgG were detected in serum), and scarce inflammatory infiltrates on brain biopsy, despite the absence of characteristic inclusions or multinucleated giant cells [START_REF] Fisher | Measles-induced encephalitis[END_REF]. Moreover, the retrospective clinical investigation revealed that the patient, who was not vaccinated against measles, had close contact with a sibling who was acutely ill with measles in April 2018 in Romania. Because of late MIBE diagnosis and despite supportive treatment, the patient's neurologic status continued to deteriorate rapidly, and she died at day 109 with severe brain damages exemplified by pejorative MRI evolution (Figure 1) showing bilateral, symmetric, and diffuse distribution of lesions between days 59-94.

Conclusions

Ongoing measles resurgence may lead to an increase of measles-induced encephalitis cases with life-threatening outcomes. We reported a fatal case in a woman with AIDS who had an encephalitic syndrome with no initial clear etiologic diagnosis, retrospectively tagged as MIBE.

The patient did not receive ribavirin therapy [START_REF] Baldolli | Measles inclusionbody encephalitis (MIBE) in a immunocompromised patient[END_REF] for MeV infection because the first-line extensive diagnostic testing was unsuccessful. As a last resort, a SMg approach detected MeV in a brain biopsy, despite the known result that CSF MeV detection in MIBE is often negative [START_REF] Fisher | Measles-induced encephalitis[END_REF][START_REF] Baldolli | Measles inclusionbody encephalitis (MIBE) in a immunocompromised patient[END_REF][START_REF] Bitnun | Measles inclusion-body encephalitis caused by the vaccine strain of measles virus[END_REF].

Of interest, the brain biopsy did not reveal histopathologic features consistent with MIBE; we observed no immunoreactive inclusions or multinucleated giant cells within glial cells or neurons [START_REF] Baldolli | Measles inclusionbody encephalitis (MIBE) in a immunocompromised patient[END_REF][START_REF] Bitnun | Measles inclusion-body encephalitis caused by the vaccine strain of measles virus[END_REF]. However, MIBE lesions are scanty and can be missed in a small biopsy sample.

Unfortunately, there was no material available for electron microscopy and an autopsy was not done. The encephalitic syndrome developed in this unvaccinated patient ≈6 months after a close contact with a documented measles case-patient; however, she did not report any rash or clinical symptoms of measles infection. It is noteworthy that MeV real-time reverse transcription PCR performed on the brain biopsy sample could have been sufficient to detect the virus. However, in this case, the advantage of SMg for the diagnosis of encephalitis is that, aside from pathogen identification, it was possible to generate full genome sequence for B3-genotype MeV. In conclusion, this case highlights the advantage to have a reliable pan-pathogen SMg tool to diagnose atypical encephalitis with no clear etiology on an early brain biopsy sampling.

Corresponding author: Dr. Sonia Burrel, Assistance Publique-Hopitaux de Paris, La Pitie Salpetrier University Hospital, Virology Department, 83 boulevard de l'hopital Paris, Île-de-France 75013, France; email: sonia.burrel@aphp.fr

Figure 1 .

 1 Figure 1. Fluid-attenuated inversion recovery (FLAIR) images of magnetic resonance examinations after 1 week (A, D, G, J), 2 weeks (B, E, H, K) and 5 weeks (C, F, I, L) at the same brain levels. The first examination shows focal cortical hyperintensities (yellow arrows) in the left and right frontal cortex. After 2weeks, these cortical hyperintensities have widened and are spreading to the cingulum and the insula (green arrows). At 5 weeks, cortical hyperintensities involve a larger part of the neocortex, but also spread to the basal ganglia, amygdala (red arrows), and hippocampus (the hippocampus changes may also be induced by status epilepticus) and to the posterior areas of the pons.

Figure 2 .

 2 Figure 2. Histology (A) and immunohistochemical staining (B) of the cerebral cortex. A) Moderate increased cellular density. Absence of nuclear inclusion bodies (hematoxylin and eosin stain, 100× magnification). B) Microglial activation with high CD163 immunoreactivity (immunohistochemistry, antihuman CD163 monoclonal antibody, 100× and 400× magnifications).

Figures 2A/ 2B

  Figures 2A/ 2B

Acknowledgments

We thank the WHO MeaNS database administrator, curators, and national reference labs that contribute to this global surveillance system, especially Kevin Brown, Luiza Ustea, Mihaela Lazar, Jeff Connel, and Fabio Magurano.

About the Author

Dr. Rodriguez is an assistant professor in the Department of Microbiology at Henri Mondor Hospital, University of Paris-Est-Créteil, Créteil, France, and the head of the institution's genomics platform. He has implemented a clinical metagenomics technique in diagnostic routine and is working on the exploration of complex infectious diseases using this tool.