Florent Koudohode

Nicolas Espitia

Miroslav Krstic

Event-triggered boundary control of an unstable reaction diffusion PDE with input delay

Keywords: reaction-diffusion systems, first order hyperbolic equation, backstepping control design, event-triggered control, small-gain analysis

In chemical, biological, or population (epidemiological) processes the feedback action may be considerably delayed by time-consuming chemical measurements or biological tests. With such large delays on the control action in mind, and motivated by the fact that in some of these systems only piecewise-constant inputs can be applied between time instants at which measurements trigger changes in control, we consider the problem of event-triggered stabilization of 1-D reaction-diffusion PDE systems with input delay. The approach relies on reformulating the delay problem as an actuated transport PDE which cascades into the reaction-diffusion PDE, and on the emulation of backstepping control. The paper proposes a static (state-dependent) triggering condition which establishes the time instants at which the control value needs to be updated. It is shown that under the proposed event-triggered boundary control, there exists a minimal dwelltime (independent of the initial conditions) between two triggering times which allows to guarantee the well-posedness of the closed-loop system, and the exponential stability. The stability analysis is based on Input-to-State stability theory for PDEs and small-gain arguments. A simulation example is presented to validate the theoretical results.

Introduction

Literature on stabilization of reaction-diffusion PDEs with delay

Several physical phenomena arising in biology [START_REF] Perthame | Parabolic equations in biology[END_REF], chemistry [START_REF] Grzybowski | Chemistry in motion: reaction-diffusion systems for micro-and nanotechnology[END_REF], spatial ecology [START_REF] Cantrell | Spatial ecology via reaction-diffusion equations[END_REF], etc, are described by reactiondiffusion partial differential equations (PDEs). In these applications, the feedback action may be considerably delayed by e.g., time-consuming chemical measurements or biological tests. The presence of delay in the inputs may imply situations where there is a substantial time lag between an event occurring and its impact being felt in the system. This phenomenon is often observed in processes involving propagation, transport, communication delays, distribution of chemicals in biological tissues as highlighted in [START_REF] Malek-Zavarei | Time-delay systems: analysis, optimization and applications[END_REF][START_REF] Sen | Time-delay-induced instabilities in reaction-diffusion systems[END_REF][START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]. In real-world scenarios, particularly during a pandemic prior to the availability of commercial rapid tests, both individuals and public health authorities would typically wait a couple of days to receive test results. This waiting period can be categorized as a common sensor delay. Subsequently, it would

Email addresses: fkoudohode@laas.fr (Florent Koudohode), nicolas.espitia-hoyos@univ-lille.fr (Nicolas Espitia), krstic@ucsd.edu (Miroslav Krstic). take even more time for authorities to reach a consensus on a public health response. Similarly, certain medications or vaccine (control actions) require days to manifest their effects. All of these scenarios exemplify input delays. Henceforth, the delays of the inputs have to be taken into account in the control design, as they may induce instabilities in the closed-loop system [START_REF] Sen | Time-delay-induced instabilities in reaction-diffusion systems[END_REF].

The stabilization of reaction-diffusion PDEs with an arbitrary level of instability and under arbitrarily long input delay is a challenging problem, first formulated and solved in [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF] using the backstepping method for PDEs as the stabilization of a hyperbolic (transport) PDE which cascades into the reaction-diffusion PDE. Since then, control design for delay compensation (including known or unknown constant/timevarying delays) has evolved considerably and several results have been proposed for reaction-diffusion PDEs, see, e.g., [START_REF] Fridman | Exponential stability of linear distributed parameter systems with time-varying delays[END_REF][START_REF] Hashimoto | Stabilization of reaction diffusion equations with state delay using boundary control input[END_REF][START_REF] Prieur | Feedback stabilization of a 1-d linear reaction-diffusion equation with delay boundary control[END_REF][START_REF] Katz | Network-based boundary observer-controller design for 1D heat equation[END_REF][START_REF] Sano | Neumann boundary stabilization of one-dimensional linear parabolic systems with input delay[END_REF][START_REF] Djebour | Feedback stabilization of parabolic systems with input delay[END_REF][START_REF] Wang | Adaptive boundary control of reaction-diffusion pdes with unknown input delay[END_REF] and the references therein.

PDE backstepping [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] makes use of a Volterra transformation to map the PDE system into a suitable target PDE system on which one can perform Lyapunov stability analysis. An alternative method for stabilization of a parabolic PDE with input delay is modal decomposition [START_REF] Prieur | Feedback stabilization of a 1-d linear reaction-diffusion equation with delay boundary control[END_REF][START_REF] Katz | Network-based boundary observer-controller design for 1D heat equation[END_REF][START_REF] Djebour | Feedback stabilization of parabolic systems with input delay[END_REF] which relies on separating a finite-dimensional unstable part from a stable infinite-dimensional part of the PDE. Then, one applies the classical predictor-based techniques to the finitedimensional system and uses for example, spectral analysis, the pole-shifting theorem, and Lyapunov-based techniques. The application of a backstepping approach in boundary control design offers several key advantages. It enables the expression of designs applicable to an entire class of systems, regardless of the specific plant's unstable eigenvalues. Moreover, it allows for extension to parameter-adaptive use, with the real-time parameter estimation, where one does not know the number of unstable eigenvalues in the unknown plant. Additionally, this approach demonstrates the potential to achieve enhanced type of convergence, including the desirable outcomes of finite or prescribed-time convergence [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF][START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF]. Both of the aforementioned methods have been the object of further advances, which include, on the one hand, the Fredholm backstepping control for coupled parabolic PDEs with input/output delays [START_REF] Deutscher | Fredholm backstepping control of coupled linear parabolic pdes with input and output delays[END_REF], and on the other hand, finite-dimensional observer-based control design for parabolic PDEs with delays and sampled-data (using spectral reduction and LMIs-based stability conditions) [START_REF] Katz | Delayed finite-dimensional observer-based control of 1d parabolic PDEs via reduced-order LMIs[END_REF], among others. More specifically, in [START_REF] Katz | Delayed finite-dimensional observer-based control of 1d parabolic PDEs via reduced-order LMIs[END_REF], the authors design a finite-dimensional observer-based control, for the reaction-diffusion equation under fast-varying input (known or unknown) and output delayed measurements, thanks to the modal decomposition approach. Furthermore, this technique incorporates an LMI-based time-regularized dynamic eventtriggering mechanism, designed to reduce the workload of the network, addressing an open problem that persists with the new approach developed in this paper.

Event-triggered control of PDEs

In some systems modeled by reaction-diffusion PDEs, only piecewise-constant inputs can be applied between time instants, yet when the arising controller is continuous (e.g., boundary controllers designed by the backstepping method). Therefore, the issue of implementing the controller in a sampled-and-hold fashion has to be carefully studied. To that end, sampled-data and event-triggered control (ETC) can offer suitable approaches to be adopted towards digital realizations and realistic ways for the actuation.

Event-triggered control is a computer control strategy that only updates the control value when the system needs attention while preserving stability and performance. It has been particularly relevant in controlled networked systems because of the efficient usage of computational and communication resources such as power usage, bandwidth, etc (see, e.g., [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF][START_REF] Liu | Robust event-triggered control of nonlinear systems[END_REF] and the references therein in the framework of finite-dimensional systems).

ETC for PDEs has gained a lot of interest during the last few years, and several contributions have since then been proposed for wide classes of PDEs. For hyperbolic PDEs, [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF] proposes an output feedback event-triggered boundary controller for 1-D linear hyperbolic systems of conservation laws through Lyapunov techniques. Using the backstepping approach, [START_REF] Espitia | Event-based boundary control of a linear 2 × 2 hyperbolic system via backstepping approach[END_REF] and [START_REF] Espitia | Observer-based event-triggered boundary control of a linear 2 × 2 hyperbolic systems[END_REF] introduce dynamic triggering conditions to the event-triggered boundary controllers for the stabilization of coupled 2 × 2 linear hyperbolic systems by full-state feedback and output feedback, respectively. The methodology is further employed and advanced in [START_REF] Wang | Event-triggered output-feedback backstepping control of sandwiched hyperbolic PDE systems[END_REF] and [START_REF] Wang | Event-triggered adaptive control of coupled hyperbolic PDEs with piecewise-constant inputs and identification[END_REF], the latter proposing an event-triggered adaptive control for coupled hyperbolic PDEs. The results on event-triggered control using backstepping-based methods have been applied to load-moving cable systems [START_REF] Wang | Adaptive event-triggered PDE control for load-moving cable systems[END_REF] and traffic flow control on connected roads [START_REF] Espitia | Traffic flow control on cascaded roads by event-triggered output feedback[END_REF].

For parabolic PDEs, [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF] proposes a decentralized eventtriggered control to reduce the number of transmitted measurements, while [START_REF] Katz | Boundary delayed observercontroller design for reaction-diffusion systems[END_REF] builds on modal decomposition and comes up with sampled-data and observer-based event triggered boundary control for 1-D reaction-diffusion systems in the presence of time-varying input delays. Such a contribution includes a novel switching-based dynamic triggering condition depending on the finite modes of the estimated state and a suitable time regularization, allowing the avoidance of the Zeno phenomenon. On the other hand, using Input-to-State stability (ISS) properties for PDEs and small gain arguments, [START_REF] Espitia | Event-triggered boundary control of constant-parameter reaction-diffusion pdes: A small-gain approach[END_REF] proposes a backstepping-based full-state feedback ETC strategy for a 1-D reaction-diffusion system with constant parameters and Dirichlet boundary actuation. In addition, [START_REF] Rathnayake | Observerbased event-triggered boundary control of a class of reaction-diffusion pdes[END_REF] proposes an observer-based event-triggered backstepping boundary control in the case of Robin boundary actuation. The ETC strategy includes a dynamic triggering condition under which it is possible to obtain a dwelltime, thus avoiding the Zeno phenomenon. Moreover, [START_REF] Wang | Event-triggered adaptive control of a parabolic PDE-ODE cascade with piecewise-constant inputs and identification[END_REF] extendes the results of [START_REF] Karafyllis | Adaptive boundary control of constant-parameter reaction-diffusion PDEs using regulationtriggered finite-time identification[END_REF] and [START_REF] Rathnayake | Observerbased event-triggered boundary control of a class of reaction-diffusion pdes[END_REF] and proposes a novel adaptive event-triggered boundary control for a parabolic PDE-ODE system with uncertain parameters, whereas [START_REF] Rathnayake | Event-based boundary control of one-phase Stefan Problem: A static triggering approach[END_REF] goes further with applications to the Stefan problem. Eventtriggered control strategies for other classes of PDEs (including abstract infinite-dimensional systems [START_REF] Wakaiki | Event-triggered control of infinitedimensional systems[END_REF]) are reported in [START_REF] Kang | Event-triggered control of Korteweg-de-Vries equation under averaged measurements[END_REF] for nonlinear Korteweg-de Vries (KdV) under averaged measurements, [START_REF] Koudohode | Event-based control for the damped linear wave equation[END_REF] for the damped linear wave equation, and [START_REF] Koudohode | Event-based control of a damped linear Schrödinger equation[END_REF] for the damped linear Schrödinger equation, to mention a few.

Contributions

In this paper, we propose an event-triggered control scheme of the backstepping-based controller originally introduced in [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF]. Our contribution extends the results of [START_REF] Espitia | Event-triggered boundary control of constant-parameter reaction-diffusion pdes: A small-gain approach[END_REF] to the case of delayed input, reformulates the problem as a parabolictransport hyperbolic PDE-PDE, and provides a suitable statedependent event-triggering condition. The choice of the space norms (L 2 -norm for the reaction-diffusion PDE and supremum-norm for the hyperbolic PDE) is crucial in the design of the triggering policy and for the stability analysis, which is based on Input-to-State stability and small-gain arguments. We prove the existence of a uniform minimal dwell time (independent of the initial condition) between two consecutive triggering time instants, thus avoiding the Zeno Phenomenon. Consequently, we guarantee the existence and uniqueness of solutions to the closed-loop system and the global exponential stability of the closed-loop system.

Organization

The structure of the present work is as follows: Section 2 is devoted to the presentation of the class of reactiondiffusion parabolic systems, some preliminaries on stability and backstepping boundary control and the notion of existence and uniqueness of solutions. Section 3 provides the event-triggered boundary control and the main results. Section 4 provides a numerical example to illustrate the main results. Finally, conclusions and perspectives are given in Section 5.

Notation R + will denote the set of nonnegative real numbers. Let S ⊆ R n be an open set and let A ⊆ R n be an open set that satisfies S ⊆ A ⊆ S. By C 0 (A; Ω), we denote the class of continuous functions on A, which take values in Ω ⊆ R. By C k (A; Ω), where k ≥ 1 is an integer, we denote the class of functions on A, which takes values in Ω and has continuous derivatives of order k. L 2 (0, 1) denotes the equivalence class of Lebesgue measurable functions f :

[0, 1] → R such that f = 1 0 | f (x)| 2 dx 1/2 < ∞. Let u : R + × [0, 1] → R be given. u(t, •) denotes the profile of u at certain t ≥ 0, for all x ∈ [0, 1]. For an interval J ⊆ R + , the space C 0 (J; L 2 (0, 1)) is the space of continuous mappings J ∋ t → u(t, •) ∈ L 2 (0, 1). H 2 (0, 1) denotes the Sobolev space of functions f ∈ L 2 (0, 1)
with square integrable (weak) first and second-order derivatives f

′ (•), f ′′ (•) ∈ L 2 (0, 1). A function f : J → R is called right continuous on J if for every s ∈ J and ε > 0 there exists δ (ε, s) > 0 such that for all τ ∈ J with s ≤ τ < s + δ (ε, s) it holds that | f (τ) - f (s)| < ε. A right continuous function f : J → R is piecewise C 1 on J (and we denote it as C 1 rpw (J, R)) if there exists a finite set B ⊂ J such that f is C 1 on J\B and all meaningful limits lim h→0 + ( ḟ (s + h)), lim h→0 + ( ḟ (s -h)), lim h→0 + ( f (s + h)), lim h→0 + ( f (s -h)) exist for all s ∈ J\B and are finite. The sup-norm is defined by f ∞ = max x∈J (| f (x)|).
I m (•), J m (•) with m ∈ Z, denote the modified Bessel and (nonmodified) Bessel functions of the first kind.

Preliminaries and problem description

Let us consider the following scalar reaction-diffusion PDE with known constant input delay D > 0:

u t (t, x) = u xx (t, x) + λ u(t, x), (1) u(t, 0) = 0, (2) u(t, 1) = U(t -D), (3)
where λ ∈ R. As in [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF], we pose this delay problem as an actuated transport PDE (modeling the delay phenomenon) which cascades into the boundary of the reaction-diffusion PDE,

u t (t, x) = u xx (t, x) + λ u(t, x), (4) u(t, 0) = 0, (5) u(t, 1) = v(t, 0), (6) v t (t, x) = 1 D v x (t, x), (7) v(t, 1) = U(t). ( 8
) (t, x) ∈ R + × [0, 1]
, where u(t, •) and v(t, •) are respectively, the reaction-diffusion PDE and the transport PDE states at time t, with initial conditions u(0, x) = u 0 (x), and v(0, x) = v 0 (x), for all x ∈ [0, 1] (where u 0 , v 0 are given functions, belonging to appropriate spaces to be specified later from Subsection 2.2). U(t) ∈ R is the control input.

The solution of the input delay dynamics is given as

v(t, x) = v 0 ( 1 D t + x) for t ≤ D(1 -x) and v(t, x) = U(t + D(x -1)) for t ≥ D(1 -x).
When U(t) ≡ 0 and the reaction parameter λ > π 2 , the system ( 4)-( 8) is unstable; whereas for λ ≤ π 2 , ( 4)-( 8) is asymptotically stable. In [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF] the following continuous-time controller (nominal boundary feedback as we will call it in the sequel) was obtained by the Backstepping approach:

U(t) = 1 0 γ(1, y)u(t, y)dy + D 1 0 q(1, y)v(t, y)dy, (9) where γ(x, y) = 2 ∞ ∑ n=1 e D(λ -n 2 π 2 )x sin(nπy) 1 0 sin(nπζ )k(1, ζ )dζ , (10) with k(x, y) = -λ y I 1 λ (x 2 -y 2 ) λ (x 2 -y 2 ) , (11)
on T := {(x, y) : 0 ≤ y ≤ x ≤ 1}, where I n (•) denotes the modified Bessel function of first kind. In addition

q(x, y) = -γ y (x -y, 1). ( 12
)
Remark 1. In [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF], the continuous-time controller (9) obtained by the Backstepping approach is used to guarantee the global exponential stability of the closed-loop system (4) in H1 -norm. More precisely, we recall that if the initial conditions are such that (u 0 , v 0 ) ∈ L 2 (0, 1) × H 1 (1, 1 + D), then the system has a unique solution

(u(t, •), v(t, •)) ∈ C((0, ∞), L 2 (0, 1) × H 1 (1, 1 + D))
and there exists a positive continuous function M :

R 2 → R + such that V (t) ≤ M(λ , D)e cD V (0)e -min(2,c)t , ∀t ≥ 0
for any c > 0, where

V (t) = 1 0 u 2 (t, x)dx + 1+D 1 (v 2 (t, x) + v 2 x (t, x)dx.

Emulation of the boundary controller

Under the emulation approach 1 , the boundary controller is perfectly known (i.e., the nominal control given in ( 9)). We aim at stabilizing closed-loop system ( 4)-( 8) on events while updating the controller (9) at certain time {t j } j defined by an event-triggered mechanism. To that end, we consider the following event-triggered boundary control:

U d (t) = 1 0 γ(1, y)u(t j , y)dy + D 1 0
q(1, y)v(t j , y)dy, [START_REF] Grzybowski | Chemistry in motion: reaction-diffusion systems for micro-and nanotechnology[END_REF] for all t ∈ [t j ,t j+1 ). The updates times {t j } j form an increasing sequence and are such that the value of the control is held constant between two successive events and is updated when some triggering condition (state-dependent triggering condition) is verified. Thus, the boundary value of the state is modified as

v(t, 1) = U d (t), for all t ∈ [t j ,t j+1 ), j ≥ 0. Note that U d (t) = U(t) + d(t) with U(t)
given by ( 9) and d given by:

d(t) = 1 0 γ(1, y) (u(t j , y) -u(t, y))dy +D 1 0 q(1, y) (v(t j , y) -v(t, y))dy. ( 14
)
where d can be viewed as an actuation deviation (or input holding error).

Therefore, the control problem we aim at handling can be reformulated as follows:

u t (t, x) = u xx (t, x) + λ u(t, x), (15) u(t, 0) = 0, (16) u(t, 1) = v(t, 0), ( 17
) v t (t, x) = 1 D v x (t, x), ( 18
) v(t, 1) = U d (t), (19)
with U d (t) being defined in (13) for all t ∈ [t j ,t j+1 ), j ≥ 0.

Well-posedness issues

Similar to [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF] and [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF], in this paper we deal with a linear hyperbolic equation subject to a discontinuous boundary input (see [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF]). The discontinuous signal gets into the reactiondiffusion PDE through the boundary, i.e., [START_REF] Espitia | Traffic flow control on cascaded roads by event-triggered output feedback[END_REF]. Consequently, the well-posedness study requires to extend the case in [START_REF] Espitia | Event-triggered boundary control of constant-parameter reaction-diffusion pdes: A small-gain approach[END_REF] along with [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF][START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF] in order to be able to construct the solution for the closed-loop PDE-PDE system. This is done by means of the following proposition: R) and u 0 ∈ L 2 (0, 1), there exist unique solutions v, u to (15)- [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF] with the following properties: ]) where Ĩ = [0, lim j→∞ (t j )) \{t j : j = 0, 1, 2, ...} which also satisfies (15)-( 17) for t ∈ Ĩ.

Proposition 1. For every initial data v 0 ∈ C 1 rpw ([0, 1],
• v is the unique solution to (18)-(19) in the sense of characteristics on [0, lim j→∞ (t j )) × [0, 1]. Moreover, for all t ∈ [0, lim j→∞ (t j )), v(t, •) ∈ C 1 rpw ([0, 1], R) and for all x ∈ [0, 1], v(•, x) ∈ C 1 rpw ([0, lim j→∞ (t j )), R). • u ∈ C 0 [0, lim j→∞ (t j )); L 2 (0, 1) with u(t, •) ∈ C 2 ([0, 1]) for t ∈ (0, lim j→∞ (t j )) and u ∈ C 1 ( Ĩ × [0, 1
Proof. Let us focus first on the v-system of ( 18)- [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF]. Following similar arguments as in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF], let us define for k ∈ N, the interval ∆ k ; = [kD, (k + 1)D] ⊂ [0, lim j→∞ (t j )), where D is the time for the transport equation with velocity 1/D to cross the spatial domain [0, 1]. By the method of characteristics (see e.g., [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF]), the explicit solution of ( 18)-( 19), for a given initial data v(kD,

•) ∈ C 1 rpw ([0, 1], R) is as follows: v(t, x) = v kD, 1 D (t -kD) + x , kD ≤ t < kD + D(1 -x) U d (t + D(x -1)), kD + D(1 -x) ≤ t ≤ (k + 1)D. ( 20
) for all t ∈ ∆ k , k ∈ N. It follows then, from (20), that v is well-defined on ∆ k × [0, 1].
) it holds that v(t, •) ∈ C 1 rpw ([0, 1], R) for all t ∈ ∆ k and v(•, x) ∈ C 1 rpw (∆ k , R) for all x ∈ [0, 1]. This yields v(t, 0) ∈ C 1 rpw (∆ k , R)
which constitutes an allowable boundary input for the u-system. Indeed, since it is piecewise continuous with the required regularity properties, we can apply [20, Theorem 4.10] for system ( 15)-( 17) on the interval ∆ k . We obtain then, that for any initial data u(kD, •) ∈ L 2 (0, 1), there exists a unique function u

∈ C 0 ∆ k ; L 2 (0, 1) with u(t, •) ∈ C 2 ([0, 1]) for t ∈ ∆ k \{kD, (k + 1)D} and u ∈ C 1 ( Ĩk × [0, 1])
where Ĩk = ∆ k \{t j : j = 0, 1, 2, ...}, k ∈ N which also satisfies ( 15)-( 17) for t ∈ Ĩk . Therefore, by the step-by-step method, we can construct the solution for all [0, lim j→∞ (t j )), i,e., i) solutions v on [0, lim j→∞ (t j )) × [0, 1] such that for all t ∈ [0, lim j→∞ (t j )), v(t,

•) ∈ C 1 rpw ([0, 1], R) and for all x ∈ [0, 1], v(•, x) ∈ C 1 rpw ([0, lim j→∞ (t j )), R); ii) u ∈ C 0 [0, lim j→∞ (t j )); L 2 (0, 1) with u(t, •) ∈ C 2 ([0, 1]) for t ∈ (0, lim j→∞ (t j )) and u ∈ C 1 ( Ĩ × [0, 1]
) where Ĩ = [0, lim j→∞ (t j )) \{t j : j = 0, 1, 2, ...} which also satisfies ( 15)- [START_REF] Karafyllis | Event-triggered gain scheduling of reaction-diffusion pdes[END_REF] for t ∈ Ĩ. This concludes the proof.

Backstepping tranformation and target system

Since we need to assess the impact of the deviation d(t) to the closed-loop system under the event-triggered implementation, we use a backstepping transformations so that we can work on a target system with desired stability properties and that exhibits the deviation d(t) at the boundary. The backstepping transformation is defined as follows:

w(t, x) = u(t, x) - x 0 k(x, y)u(t, y)dy, ( 21
)
z(t, x) = v(t, x) -D x 0 q(x, y)v(t, y)dy - 1 0 γ(x, y)u(t, y)dy, ( 22
)
x ∈ [0, 1], with γ(x, y), k(x, y) and q(x, y) given, respectively, by [START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF], [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF] and [START_REF] Fridman | Exponential stability of linear distributed parameter systems with time-varying delays[END_REF]. Hence, the system ( 15)-( 19) is transformed into the following target system:

w t (t, x) = w xx (t, x), (23) w(t, 0) = 0, (24) w(t, 1) = z(t, 0), ( 25
) z t (t, x) = 1 D z x (t, x), ( 26
) z(t, 1) = d(t), ( 27
)
with initial conditions

w 0 (x) = u 0 (x) - x 0 k(x, y)u 0 (y)dy, ( 28
)
z 0 (x) = v 0 (x) - 1 0 γ(x, y)u 0 (y)dy -D x 0 q(x, y)v 0 (y)dy. (29)
Notice that when d(t) = 0, one has that the target system is globally exponential stable.

It is worth recalling that the backstepping transformation ( 21)-( 22) is invertible. The inverse transformation is given by

u(t, x) = w(t, x) + x 0 l(x, y)w(t, y)dy, ( 30
) v(t, x) = z(t, x) + 1 0 δ (x, y)w(t, y)dy + D x 0 p(x, y)z(t, y)dy, (31)
where

l(x, y) = -λ y J 1 λ (x 2 -y 2 ) λ (x 2 -y 2 ) , (32)
on T := {(x, y) : 0 ≤ y ≤ x ≤ 1} where J n (•) denotes the Bessel function of first kind,

δ (x, y) = 2 ∞ ∑ n=1 e -Dn 2 π 2 x sin(nπy) 1 0 sin(nπζ )l(1, ζ )dζ , (33)
and p(x, y) = -δ y (xy, 1).

Using the inverse transformation, we can rewrite ( 13) and ( 14) as a function of the states w and z, i.e.,

U d (t) = 1 0 δ (1, y)w(t j , y)dy + D 1 0
p(1, y)z(t j , y)dy, [START_REF] Sano | Neumann boundary stabilization of one-dimensional linear parabolic systems with input delay[END_REF] and

d(t) = 1 0 δ (1, y) (w(t j , y) -w(t, y)) dy + D 1 0
p(1, y) (z(t j , y)z(t, y)) dy [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF] for all t ∈ [t j ,t j+1 ), j ≥ 0.

Event-triggered boundary control and main results

In this section we introduce the event-triggered boundary control and the main results: avoidance of the Zeno phenomenon and the exponential stability of the event-triggered controlled system. The event-triggered boundary control considered in this paper involves a triggering condition (which determines the time instant at which the controller needs to be updated) and the backstepping boundary feedback which is applied as Zero-Order Hold. The proposed event-triggering condition is based on the evolution of the magnitude of the actuation deviation and the evolution of the norms (on suitable functional spaces) of both the reactiondiffusion and transport PDE states.

Definition of the event-triggered boundary control

Let β > 0 be a design parameter and define the following set:

E(t j ) := {t > t j : |d(t)| > β max t j ≤s≤t ( w(s, •) L 2 ) + β max t j ≤s≤t ( z(s, •) ∞ )} (37)
where w(t, •) and z(t, •) are the solution of ( 23)-( 27) for all t ≥ t j and d(t) is defined by [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF].

The event-triggered boundary control is defined by considering the following components: I) (The event-triggering condition) The times of the events t j ≥ 0 with t 0 = 0 form a finite or countable set of times which is determined by the following rules for some j ≥ 0: a) if E(t j ) = / 0 then the set of the times of the events is {t 0 , ...,t j }. b) if E(t j ) = / 0, then the next event time is given by:

t j+1 := inf E(t j ). (38)
II) (the control action) The boundary feedback law,

U d (t) = 1 0 δ (1, y)w(t j , y)dy + D 1 0
p(1, y)z(t j , y)dy, [START_REF] Wakaiki | Event-triggered control of infinitedimensional systems[END_REF] for all t ∈ [t j ,t j+1 ).

Remark 2. Notice that in definition of the event-triggering condition in ( 37)-( 38), we use the L ∞ -norm for the transport PDE subsystem ( 26)- [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF]. It is worth recalling that in [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF], a Lyapunov-based stability analysis is performed using the H 1 -norm for the transport PDE subsystem (instead of the L 2norm) mainly due to the unboundedness of the operator in the interconnection parabolic-transport PDEs. One may indeed have trace terms that cannot be estimated using the L 2norm. Therefore, H 1 -norm turned out to be suitable in that work (see Remark 1). Nevertheless, in this event-triggered framework, working with Lyapunov-based techniques using the H 1 -norm may not be appropriate as this needs to have regularity on the initial data and solutions to be at least absolutely continuous. This may not possible in the present setting as we deal with piecewise-constant input signals in the transport PDE and discontinuities propagating through the spatial domain. Therefore, a L ∞ -norm is suitable for the transport PDE subsystem, thanks to which it is possible: i) to rely on the required regularity for the analysis for the wellposedness of the overall closed-loop system (as established in Subsection 2.2), ii) to obtain a suitable ISS estimate allowing a subsequent small-gain-based stability analysis (see Section 3.3).

Avoidance of the Zeno phenomenon and wellposedness

Theorem 1. Consider the closed-loop system (15)-( 19) with the event-triggered boundary control (37)-( 39) with β > 0 be given. Then, there exists a minimal dwell-time between two triggering times, i.e. there exists a constant τ > 0 (independent of the initial conditions u 0 , v 0 ) such that t j+1t j ≥ τ, for all j ≥ 0.

Proof. Let us focus on the deviation of actuation given in [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF], expressed in terms of the dynamics of the target system ( 23)-( 27) and the kernels of the inverse transformation. Proposition 1 in conjunction with the backstepping transformations ( 21)-( 22) allow to assert that target system is well-posed as d(t) can be proved to belong to C 1 rpw ([0, lim j→∞ (t j )), R). Following similar arguments as in [19, Section 3], it can be further proved that the following differential equation holds, for t ∈ (t j ,t j+1 ), j ≥ 0:

ḋ(t) = - 1 0 δ (1, y)w t (t, y)dy -D 1 0 p(1, y)z t (t, y)dy = - 1 0 δ (1, y)w yy (t, y)dy - 1 0 p(1, y)z y (t, y)dy = -δ (1, 1)w y (t, 1) + δ (1, 0)w y (t, 0) + δ y (1, 1)w(t, 1)
-δ y (1, 0)w(t, 0) -p(1, 1)z(t, 1) + p(1, 0)z(t, 0) (41)

Moreover, the following inequality holds for t ∈ (t j ,t j+1 ):

| ḋ(t)| ≤ a 0 |d(t)| + 1 0 |δ yy (1, y)w(t, y)|dy + 1 0 |p y (1, y)z(t, y)|dy. ( 42
)
with

a 0 := p(1, 1) = λ 2 + λ 2 8 , (43)
which can be computed by using [START_REF] Prieur | Feedback stabilization of a 1-d linear reaction-diffusion equation with delay boundary control[END_REF] and [START_REF] Rathnayake | Observerbased event-triggered boundary control of a class of reaction-diffusion pdes[END_REF] together with the fact that d dρ (ρ -1 J 1 (ρ)) = -ρ -1 J 2 (ρ) and lim ρ→0 J n (ρ)

ρ n = 1 2 n n! .
Using

) 44
Using the Cauchy-Schwarz inequality and the fact d(t j ) = 0, we have from [START_REF] Wang | Adaptive boundary control of reaction-diffusion pdes with unknown input delay[END_REF] the following estimate:

|d(t)| ≤ 1 0 |p y (t, y)|dy t t j exp (a 0 (t -s)) z(s, •) ∞ ds + δ yy (1, •) L 2 t t j exp (a 0 (t -s)) w(s, •) L 2 ds. ( 45
)
Moreover, it holds for all t ∈ [t j ,t j+1 )

|d(t)| ≤ Γ a 0 (t -t j ) a 1 max t j ≤s≤t ( z(s, •) ∞ ) + a 2 max t j ≤s≤t ( w(s, •) L 2 ) , (46)
where

Γ a 0 (s) := 1 a 0 exp(a 0 (s)) -1 > 0, ( 47
)
a 1 := 1 0 |p y (1, y)|dy, ( 48
)
a 2 := δ yy (1, •) L 2 .
(49) Using ( 46) and assuming that an event is triggered at t = t j+1 , we have

|d(t j+1 )| ≤ Γ a 0 (t j+1 -t j ) a 1 max t j ≤s≤t j+1 ( z(s, •) ∞ ) + a 2 max t j ≤s≤t j+1 ( w(s, •) L 2 ) , (50)
which, together with the definition of the event-triggering condition (37)- [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] , yield the following inequality:

β max t j ≤s≤t j+1 ( z(s, •) ∞ ) + max t j ≤s≤t j+1 ( w(s, •) L 2 ) ≤ Γ a 0 (t j+1 -t j ) a 1 max t j ≤s≤t j+1 ( z(s, •) ∞ ) + a 2 max t j ≤s≤t j+1 ( w(s, •) L 2 ) , (51)
Therefore, from (51), we obtain

0 < β max{a 1 , a 2 } ≤ Γ a 0 (t j+1 -t j ). (52)
Using the definition (47) and from (52), we can conclude, for all j ≥ 0

t j+1 -t j ≥ 1 a 0 ln 1 + a 0 β max{a 1 , a 2 } =: τ > 0, (53)
which is a minimal dwell-time (independent on the initial conditions). This concludes the proof.

Theorem 1 allows to conclude that lim j→+∞ (t j ) = +∞ and therefore we can apply Proposition 1 to finally get the following well-posedness result of the closed-loop system ( 15)- [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF].

Corollary 1. For every initial data v 0 ∈ C 1 rpw ([0, 1], R) and u 0 ∈ L 2 (0, 1), there exist unique solutions v, u to (15)- [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF] with the following properties: ]) where I = R + \{t j : j = 0, 1, 2, ...} which also satisfies (15)-( 17) for t ∈ I.

• v is the unique solution to (18)-(19) on R + × [0, 1]. More- over, for all t ∈ R + , v(t, •) ∈ C 1 rpw ([0, 1], R) and for all x ∈ [0, 1], v(•, x) ∈ C 1 rpw (R + , R). • u ∈ C 0 R + ; L 2 (0, 1) with u(t, •) ∈ C 2 ([0, 1]) for t ∈ R + and u ∈ C 1 (I × [0, 1
Proof. It is an immediate consequence of Proposition 1 and Theorem 1 (which guarantees that no Zeno solution can appear).

Stability result

In this section, we derive the exponential stability result for the closed-loop system ( 15)- [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-D Hyperbolic PDEs with non-local terms[END_REF]. To that end, we seek an Input-to-State stability property of the target system ( 23)-( 27) with respect to the deviation d(t), and we follow smallgain arguments.

Theorem 2. Let β > 0 be a design parameter (involved in the triggering condition [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]) that is selected in such a way that the following condition is fulfilled:

β < exp (-D) 1 + √ 3 3 . ( 54
)
Then, the closed-loop system (15)-( 19) with event-triggered boundary control (38)-( 39) is globally exponentially stable. More specifically, there exist constants M, ξ > 0 such that:

u(t, •) L 2 + v(t, •) ∞ ≤ M exp(-ξ t)( u 0 L 2 + v 0 ∞ ) , (55
) for all t ≥ 0.

Proof. By virtue of condition (54), there exist constants ε, µ > 0, such that

β 1 √ 3 (1 + ε) 3 exp(µD) + (1 + ε) 2 exp(µD)) exp(D) < 1. ( 56
)
The existence of ε, and µ > 0 is guaranteed since the function

h 1 (ε, µ) := β 1 √ 3 (1 + ε) 3 exp(µD) + (1 + ε) 2 exp(µD)) exp(D)
is continuous at (0, 0) and satisfies h 1 (0, 0) < 1. Condition (56), in turn, implies the following condition:

β (1 + ε) 2 exp(µD) exp(D) < 1. (57)
Applying the results in [20, Chapters 3.2 and 6.4], the following ISS estimates for the solutions of the target system ( 23)-( 27) can be obtained:

w(t, •) L 2 ≤ exp(-π 2 t) w 0 L 2 + 1 √ 3 max 0≤s≤t (|z(s, 0)|), (58)
and

z(t, •) ∞ ≤ exp (-µ(t -D)) exp (D) z 0 ∞ + exp(µD) exp (D) max 0≤s≤t (|d(s)|) , (59)
for all t ≥ 0 and µ > 0. Moreover, using [20, Lemma 7.1], we guarantee that there exists ξ > 0 such that the following fading memory estimates hold for all t ≥ 0:

w(t, •) L 2 exp (ξ t) ≤ w 0 L 2 + 1 √ 3 (1 + ε) max 0≤s≤t ( z(s, •) ∞ exp (ξ s)), ( 60
)
z(t, •) ∞ exp (ξ t) ≤ exp (D) z 0 ∞ + exp(µD) exp (D) (1 + ε) max 0≤s≤t (|d(s)| exp (ξ s)) . (61)
We define the following quantities for all t ≥ 0:

w [0,t] := max 0≤s≤t ( w(s, •) L 2 exp (ξ s)) , (62)
z [0,t] := max 0≤s≤t ( z(s, •) ∞ exp (ξ s)) . (63)
Using ( 60)-( 61) and the definitions (62)-( 63), we get for all t ≥ 0,

w [0,t] ≤ w 0 L 2 + 1 √ 3 (1 + ε) z [0,t] , (64)
z [0,t] ≤ exp(D) z 0 ∞ + exp(µD) exp (D) (1 + ε) max 0≤s≤t (|d(s)| exp (ξ s)) . (65)
From the definition of the event-triggering condition (37)- [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], events are triggered to guarantee, for all t j ≥ 0 and t ≥ t j .

|d(t)| ≤ β max t j ≤s≤t ( w(s, •) L 2 ) + β max t j ≤s≤t ( z(s, •) ∞ ). (66)
Notice that (66) can be read as e.g.,

|d(t)| ≤ exp(-π(t -t j ))|d(t j )| + β max t j ≤s≤t ( w(s, •) L 2 ) + β max t j ≤s≤t ( z(s, •) ∞ ),
|d(t)| ≤ exp(-ξ t)|d(0)| + β (1 + ε) max 0≤s≤t ( w(s, •) L 2 exp (-ξ (t -s))) + β (1 + ε) max 0≤s≤t ( z(s, •) ∞ exp (-ξ (t -s))). ( 67
)
with ξ , ε as in (60)-(61). Hence, we obtain the following estimate (since |d(0)| = 0):

|d(t)| exp (ξ t) ≤β (1 + ε) max 0≤s≤t ( w(s, •) L 2 exp (ξ s)) + β (1 + ε) max 0≤s≤t ( z(s, •) ∞ exp (ξ s)). ( 68
)
Using definitions (62)-(63), we get max 0≤s≤t

(|d(s)| exp (ξ s)) ≤β (1 + ε) w [0,t] + β (1 + ε) z [0,t] . (69)
Therefore, using (69) along with (64)-( 65), we get

w [0,t] ≤ w 0 L 2 + 1 √ 3 (1 + ε) z [0,t] , (70)
and

z [0,t] ≤ exp (D) z 0 ∞ + exp(µD) exp (D) (1 + ε) 2 β w [0,t] + exp(µD) exp (D) (1 + ε) 2 β z [0,t] . (71)
From (71) and since (57) holds, we have

z [0,t] ≤ 1 -β (1 + ε) 2 φ -1 exp(D) z 0 ∞ + β (1 + ε) 2 φ 1 -β (1 + ε) 2 φ -1 w [0,t] , (72)
where φ := exp ((1 + µ)D).

(73) Then,

w [0,t] ≤ w 0 L 2 + 1 √ 3 (1 + ε) 1 -β (1 + ε) 2 φ -1 exp (D) z 0 ∞ + 1 √ 3 β (1 + ε) 3 φ 1 -β (1 + ε) 2 φ -1 w [0,t] . (74)
From ( 56), one has

β 1 √ 3 (1 + ε) 3 exp(µD) exp(D) < 1 -β (1 + ε) 2 exp(µD) exp(D)
which can be rewritten, using (73), as follows:

β 1 √ 3 (1 + ε) 3 φ < 1 -β (1 + ε) 2 φ .
Moreover, from (57) it holds that 1 -β (1 + ε) 2 φ -1 > 0, thus one obtains

β 1 √ 3 (1 + ε) 3 φ 1 -β (1 + ε) 2 φ -1 < 1, 1 -β 1 √ 3 (1 + ε) 3 φ 1 -β (1 + ε) 2 φ -1 > 0, that is, 1 -β (1 + ε) 3 ψ > 0 where ψ := 1 √ 3 φ 1 -β (1 + ε) 2 φ -1 . (75)
Therefore, from (74), one has

w [0,t] ≤ (1 -β (1 + ε) 3 ψ) -1 w 0 L 2 + 1 √ 3 (1 + ε) 1 -β (1 + ε) 3 ψ -1 × 1 -β (1 + ε) 2 φ -1 exp (D) z 0 ∞ , (76)
On the other hand, from (70) and (72), we have

z [0,t] ≤ 1 -β (1 + ε) 2 φ -1 exp (D) z 0 ∞ + β (1 + ε) 2 φ 1 -β (1 + ε) 2 φ -1 w 0 L 2 + 1 √ 3 β (1 + ε) 3 φ 1 -β (1 + ε) 2 φ -1 z [0,t] , (77)
and since (56) holds, then

z [0,t] ≤(1 -β (1 + ε) 3 ψ) -1 1 -β (1 + ε) 2 φ -1 × exp(D) z 0 ∞ + β (1 + ε) 2 (1 -β (1 + ε) 3 ψ) -1 × φ 1 -β (1 + ε) 2 φ -1 w 0 L 2 . (78)
Combining ( 76) and (78), we get

w [0,t] + z [0,t] ≤ (1 -β (1 + ε) 3 ψ) -1 × 1 + β (1 + ε) 2 φ 1 -β (1 + ε) 2 φ -1 w 0 L 2 + (1 -β (1 + ε) 3 ψ) -1 1 -β (1 + ε) 2 φ -1 × (1 + 1 √ 3 (1 + ε)) exp(D) z 0 ∞ . (79)
Hence,

w(t, •) L 2 + z(t, •) ∞ ≤ M 0 exp(-ξ t) w 0 L 2 + M 0 (1 + 1 √ 3 (1 + ε)) exp(D) exp(-ξ t) z 0 ∞ , (80)
with

M 0 := (1 -β (1 + ε) 3 ψ) -1 1 -β (1 + ε) 2 φ -1 ; further- more w(t, •) L 2 + z(t, •) ∞ ≤ M 1 exp(-ξ t) ( w 0 L 2 + z 0 ∞ ) , (81)
with

M 1 := M 0 (1 + 1 √ 3 (1 + ε)) exp(D)
). Next, we use the estimates of the backstepping transformations, i.e., 83) and ( 84)-(85), we finally obtain, for all t ≥ 0

w(t, •) L 2 ≤ k u(t, •) L 2 , (82) u(t, •) L 2 ≤ l w(t, •) L 2 , ( 83
) z(t, •) ∞ ≤ γ u(t, •) L 2 + q v(t, •) ∞ , (84) v(t, •) ∞ ≤ δ w(t, •) L 2 + p z(t, •) ∞ , (85)
u(t, •) L 2 + v(t, •) ∞ ≤ M exp(-ξ t)( u 0 L 2 + v 0 ∞ ) , ( 86
) with M := M 1 M 2 M 3 ,
where M 1 is as in (81), M 2 := max{( l + δ ), p} and M 3 := max{( k + γ), q}. This concludes the proof.

Remark 3. The small-gain condition β 1+ 54) is a delay-dependent condition which involves also the parameter β of the triggering condition [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. Notice that the larger D, the smaller β should be chosen to preserve the theoretical guarantees. This implies sampling faster, thus the boundary control input is updated more often. It is worth mentioning, however, that larger values of β can be taken (eventually violating (54)) and may be used in practice since the obtained estimates are conservative.

√ 3 √ 3 exp (D) < 1 from (

Numerical simulations

We illustrate the results by considering the reaction-diffusion PDE ( 15)-( 19) with λ = 12, input delay D = 0.5, and initial condition u 0

(x) = ∑ 3 n=1 √ 2 n sin(nπx) + 3(x 2 -x 3 ), v 0 (x) = 0, x ∈ [0, 1]
. We run simulations on a time horizon T = 1. For the numerical simulations: i) we implement an implicit Euler scheme for the parabolic subsystems ( 15)-( 17) and ( 23)- [START_REF] Koudohode | Event-based control for the damped linear wave equation[END_REF]. The discretization with respect to space and time is done with steps ∆ x = 1 × 10 -3 and ∆ t = 1 × 10 -3 , respectively. ii) We compute the numerical solution of the hyperbolic subsystems ( 18)-( 19) and ( 26)-( 27) by means of their explicit solutions, i,e.,

v(t, x) = v 0, 1 D t + x , 0 ≤ t < D(1 -x) U d (t + D(x -1)), t ≥ D(1 -x), (87) z(t, x) = z 0, 1 D t + x , 0 ≤ t < D(1 -x) d(t + D(x -1)), t ≥ D(1 -x). ( 88
)
We stabilize the system on events under the event-triggered control ( 38)- [START_REF] Wakaiki | Event-triggered control of infinitedimensional systems[END_REF] where the parameter β = 0.05 is selected according to (54) in Theorem 2. Conditions (56)-(57) (used just in the stability analysis) are also verified with e.g., ε = 0.1 and µ = 0.1. Since the event-triggering condition is monitored in terms of the states of the target system ( 23)-( 27) and the kernel of the inverse transformation, their numerical solutions can also be found according to ( 21)-( 22), along with the explicit expressions (32)- [START_REF] Rathnayake | Observerbased event-triggered boundary control of a class of reaction-diffusion pdes[END_REF]. In addition, using (53), we compute the minimal dwell-time τ = 8.7 × 10 -3 .

Figures 1 and2 show the numerical solution of the closedloop system ( 15)-( 19) with continuous-time boundary control [START_REF] Espitia | Event-triggered boundary control of constant-parameter reaction-diffusion pdes: A small-gain approach[END_REF] and with event-triggered control ( 38)-( 39), respectively. Figure 3 shows the numerical solutions of the target system ( 23)- [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF]. We can observe on the right that the piecewise-continuous signal d(t) appears at the boundary and the discontinuities propagate along the spatial domain of the transport PDE. The time-evolution of control functions under the continuous and event-triggered case is shown in Figure 4. The control value is kept constant between event times and updated according to the triggering law. We obtained in total 27 updates within the considered time horizon.

Conclusion

In this paper, an event-triggered boundary control is proposed for the stabilization of a 1-D reaction diffusion equation with input delay. The delay is treated as a transport PDE, thus the problem is reformulated as a cascade PDE-PDE controlled system. We performed emulation on the backstepping control and proposed a state dependent event-triggering mechanism. The existence of a minimal dwell-time between two triggering times is proved in order to avoid the Zeno behavior. Henceforth, we ensured the well-posedness of the closedloop system and the global exponential stability. In future work, we may consider an observer-based event-based control for more complex coupled reaction-diffusion systems with varying coefficients and subject to input/output delays inspired by the results of [START_REF] Deutscher | Fredholm backstepping control of coupled linear parabolic pdes with input and output delays[END_REF]. Besides, we expect to extend the results of [START_REF] Karafyllis | Event-triggered gain scheduling of reaction-diffusion pdes[END_REF] to the delay compensated event-triggered gain scheduling for the reaction-diffusion system with timeand space varying reaction coefficients. n sin(nπx)+3(x 2 -x 3 ), v 0 (x) = 0, x ∈ [0, 1] and under the event-triggered control (38)- (39) with β = 0.05. The parabolic PDE subsystem is depicted on the left and the transport PDE subsystem is depicted on the right. The piecewise-constant signal appears at the boundary and the discontinuities propagate along the spatial domain.

- 1 0δ 1 0 1 0δ 1 0

 1111 yy (1, y)w(t, y)dy + p y (1, y)z(t, y)dy.[START_REF] Wang | Adaptive event-triggered PDE control for load-moving cable systems[END_REF] Knowing that w(t, 0) = 0, and from (33)-[START_REF] Rathnayake | Observerbased event-triggered boundary control of a class of reaction-diffusion pdes[END_REF], one has δ (1, 1) = δ (1, 0) = 0 and -δ y (1, 1) = p(1, 0). Hence, from[START_REF] Wang | Adaptive event-triggered PDE control for load-moving cable systems[END_REF], we get ḋ(t) = -p(1, 1)d(t)yy (1, y)w(t, y)dy + p y (1, y)z(t, y)dy.

  = γ(x, •) L 2 , q := 1 + D max 0≤x≤1 x 0 |q(x, y)|dy, δ := δ (x, •) L 2 and p := 1 + D max 0≤x≤1 x 0 |p(x, y)|dy. Hence, from (81), along with (82)-(

Figure 1 . 2 n

 12 Figure 1. Numerical solutions of the closed-loop system with λ = 12, delay D = 0.5, initial condition u 0 (x) = ∑ 3 n=1

Figure 2 .

 2 Figure 2. Numerical solutions of the closed-loop system with λ = 12, delay D = 0.5, initial condition u 0 (x) = ∑ 3 n=1

Figure 3 .Figure 4 .

 34 Figure 3. Numerical solutions of the target system (23)-(27). The parabolic PDE subsystem is depicted on the left and the transport PDE subsystem is depicted on the right. The piecewise-continuous signal d(t) appears at the boundary and the discontinuities propagate along the spatial domain.

  Moreover, by definition of U d (being a piecewise constant function and assuming U d (t) ∈ C 1 rpw (∆ k , R)) we have that U d (t + D(x -1)) belongs to C 1 rpw with respect to t and belongs to C 1 rpw with respect to x. In addition, v kD, 1 D (t -kD) + x belongs to C 1 rpw with respect to t and belongs to C 1 rpw with respect to x. Therefore, from (20

  the absolute continuity of d(t) on (t j ,t j+1 ), we get, for all t ∈ [t j ,t j+1 ) |d(t)| ≤ exp (a 0 (tt j )) |d(t j )| +

			t
			t j	exp (a 0 (t -s))
		1		1
	×	0	|δ yy (1, y)w(s, y)|dy +	0	|p y (1, y)z(s, y)|dy ds.
					(

We recall that in the context of event-triggered control frameworks, there are typically two approaches: emulation and codesign. The emulation approach involves having a predefined controller and then designing the event-triggering mechanism to work with it. On the other hand, the co-design approach entails the simultaneous design of both the controller and the event-triggering mechanism.

Acknowledgements

We would like to thank Lucie Baudouin, Sophie Tarbouriech, and Iasson Karafyllis for their valuable comments on the preliminary version of this paper. The first author would like to thank the GdR MACS CNRS for the mobility financial support.


Zoran Tiganj

Mamadou Mboup

Spike detection and sorting: combining algebraic differentiations with ICA

A new method for action potentials detection is proposed. The method is based on a numerical differentiation, as recently introduced from operational calculus. We show that it has good performance as compared to existing methods. We also combine the proposed method with ICA in order to obtain spike sorting.

Introduction

Decoding the neural information is one of the most important problems in neuroscience. As it is well known, this information is conveyed by the spike train of electrical discharges, called action potential. Action potentials are generated when the equilibrium of electrical charges across the axonal membrane of a neuron is broken [START_REF] Rieke | Spikes: Exploring the neural code[END_REF]. Decoding of the communication between neurons is an extremely challenging problem. One of the difficulties stems from the impossibility, in general, to record the activity of a single neuron but only a mixture activity of all neurons in a measured region. Imperative requirement for the neural information decoding is the ability of action potential detection and sorting (finding out which action potentials are fired by the same neuron) [START_REF] Lewicki | A review of methods for spike sorting: the detection and classification of neural action potentials[END_REF], [START_REF] Delescluse | Efficient spike-sorting of multi-state neurons using interspike intervals information[END_REF], [START_REF] Land | Spike (action potential) detection and sorting[END_REF], [START_REF] Nenadic | Spike detection using the continious wavelet transform[END_REF].

To make the detection and sorting easier, recording systems usually consist of several electrodes. Each electrode receives the action potentials from all the surrounding neurons. The contribution from a single neuron depends on its distance from the electrode and on the type of the tissue that the action potentials go through. It is shown that ICA [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF], [START_REF] Comon | Independent component analysis, A new concept?[END_REF] has a strong potential in neural signals detection and sorting [START_REF] Mamlouka | Unsupervised spike sorting with ICA and its evaluation using GENESIS simulations[END_REF], [START_REF] Brown | Independent component analysis at the neural cocktail party[END_REF], even though it is well known that neurons are not independent sources -they are communicating with each other and the breaking of the equilibrium of electrical charges is caused by that communication.

In this paper we show how a new algebraic technique for numerical differentiation [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] can lead to a very good performances in neural spike detection. Further on, we use a combination of the technique with ICA to perform spike sorting.

In a neural recording settings, the number of neurons significantly exceeds that of the intracranial electrodes. We assume a measurement configuration where the distances between the neurons and the electrodes are such that the contribution of only few closest neurons will have a strong influence on the electrodes. The spiking activity of the remaining neurons is considered as the background noise [START_REF] Gerstner | Spiking Neuron Models. Single neurons, Populations, Plasticity[END_REF].

The neural signal is first represented using a local piecewise polynomial model. The occurrence of an action potential (a spike) is represented by a discontinuity point in the model. Now, such singularities are fully characterized and easy to handle by the differential algebra and operational calculus approach to parameter estimation, introduced in [START_REF] Fliess | An algebraic framework for linear identification ESAIM Contr[END_REF] (see also [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] and [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF]). Following this approach, we give in section 2 an explicit characterization of the spiking instants, as solutions of a polynomial equation, whose coefficients are composed of a short time window iterated integrals of the signal observation. Using classical numerical integration, we implement these coefficients as outputs of FIR filters. The filters are, as a matter of fact, numerical differentiators and moreover, they inherit the very important low pass property of the iterated integrals. The proposed method is compared in section 3 with two among the most successful methods in the literature. The simulation is performed using synthetic signal. We also combine our detection approach with ICA (see section 4). Benefit of such approach is not only a spike detection, but also a spike sorting, done by ICA. Some concluding remarks are pointed out in section 5.

Characterization of the spike instants

In this section we give an explicit characterization of the spiking instants. More precisely we show that if an action potential occurs at time instant t , then this may be described as a solution of a polynomial equation.

Mathematical model

To begin, the noisy observation spike train y(t) is represented by the piecewise regular model

y(t) = K i=1 χ [ti-1,ti] (t)p i (t -t i-1 ) + n(t), (1)
where χ [ti-1,ti] (t) is the characteristic function of the interval [t i-1 , t i ], the t i are the spiking instants and n(t) is the noise corruption. In the sequel, we denote by x(t) = y(t) -n(t) the unobserved noise-free signal. Each p i (t) is assumed to be a polynomial. Set t 0 = 0. Let T be given such that there is at most one discontinuity point in each interval

I T τ = (τ -T, τ ), τ ≥ T . Let set x τ (t) = x(τ - t), t ∈ [0, T ]
for the restriction of the signal in I T τ and redefine the discontinuity point, say t τ , relatively to I T τ with:

-

t τ = 0 if x τ (t) is smooth -0 < t τ ≤ T otherwise
Now, the N th order derivative of x τ (t) (in the sense of distributions theory [START_REF] Schwartz | Theorie des distributions[END_REF]) satisfies

d N dt N x τ (t) = [x (N ) τ ](t) + N k=1 µ 0 N -k δ(t) (k-1) + µ N -k δ(t -t τ ) (k-1) (2)
where:

-µ 0 k = x (k) (0+) -x (k) (0-) = x (k) (0) -µ k is the jump of the k th order derivative at the point t τ µ k = x (k) (t τ + ) -x (k) (t τ -)
with:

•

µ 0 = µ 1 = ... = µ N -1 = 0 if there is no spike (action potential) • µ k = 0, 0 ≤ t τ ≤ T if there is a spike (action potential) where the notation f (k) means k th order derivative of f . -[x (N )
τ ] represents the regular part of the N th order derivative of the signal. The action potential detection problem is now casted into the estimation of the location of the signal discontinuities, viz spike locations, t τ .

We assume that the degree of each p i (t) in ( 1) does not exceed N -1. The regular part [x (N ) τ ] then vanishes. Notice that different choices for N lead to different descriptions of t τ . In the sequel we set N = 2. Equation ( 2) then reduces to

d 2 dt 2 x τ (t) = 2 k=1 µ 0 2-k δ(t) (k-1) + µ 2-k δ(t -t τ ) (k-1) (3)

Finding the firing instants

In order to solve the equation (3) for t τ , the problem is transferred into the operational domain (it is possible to solve the equation in the time domain too, but it is easier to handle expression like δ(t) using operational calculus) where equation (3) reads as:

s 2 xτ (s) -sx τ (0) -ẋτ (0) = µ 1 e -t τ s + sµ 0 e -t τ s (4)
Unknown parameters µ 0 , µ 1 and initial conditions x τ (0) and ẋτ (0) are irrelevant for our purpose. We consider them as undesired perturbations which are easy to eliminate by a N th order differentiation with respect to s. This results in:

t 2 τ (s 2 xτ ) (2) + 2t τ (s 2 xτ ) (3) + (s 2 xτ ) (4) = 0 (5)
Multiplication by s k , k > 0 in the operational domain implies k order derivative in the time domain. Now differentiation is known to be difficult and illconditioned. On the other hand division by s ν , ν > 0 corresponds in the time domain to the ν th order iterated integral

û s ν → f (t) = 1 (ν -1)! t 0 (t -λ) ν-1 u(λ)dλ.
Let us then divide equation ( 5) by s ν , ν > 2. The resulting equation becomes in the time domain:

t 2 τ t 0 (t -λ) ν-1 λ 2 x (2) τ (λ)dλ -2t τ t 0 (t -λ) ν-1 λ 3 x (2) τ (λ)dλ + t 0 (t -λ) ν-1 λ 4 x (2) τ (λ)dλ = 0. ( 6
)
In the sequel we fix t to t = T . Next, we integrate by parts and we replace the unobserved signal x τ (t) by its noisy observation counterpart y τ (t) = y(τ -t) = x(τ -t) + n(τ -t). This leads to the explicit characterization of tτ :

tτ 2 T 0 (T -λ) ν-1 λ 2 (2) y(τ -λ)dλ -2 tτ T 0 (T -λ) ν-1 λ 3 (2) y(τ -λ)dλ + T 0 (T -λ) ν-1 λ 4 (2) y(τ -λ)dλ = 0 (7)
Let us define:

h g (λ) =    (T -λ) ν-1 λ g+1 (2) 0 ≤ λ < T 0 otherwise (8)
where g = 1, 2, 3. After inserting h g (λ) back in equation ( 7) we obtain tτ

2 T 0 h 1 (λ)y(τ -λ)dλ-2 tτ T 0 h 2 (λ)y(τ -λ)dλ+ T 0 h 3 (λ)y(τ -λ)dλ = 0 (9)
The coefficients of this equation can be easily implemented as FIR filters.

Only discrete samples of the observation are available. We assume a regular sampling with the sampling period T s . Let τ = nT s and T = M T s . Using a numerical integration method with abscissas mT s and weight W m , m = 0, ..., M , we obtain the discrete approximation of equation [START_REF] Brown | Independent component analysis at the neural cocktail party[END_REF] given by tτ

2 M m=0 h 1,m y n-m -2 tτ M m=0 h 2,m y n-m + M m=0 h 3,m y n-m = 0 (10)
where: h g,m = W m h g (mT s ) and y m = y(mT s ). Trapezoidal rule for numerical integration defines the parameters W m :

W 0 = W M = T s 2 , W i = 1, i = 1, 2, ..., M -1 3

Spike detection algorithm

Recall that if the input observation x τ is smooth, then all the three coefficients of the second order equation ( 6) are equal to 0 and we have t τ = 0. Otherwise, the spiking instant satisfies 0 t τ T . In the discrete time domain each of these coefficients can be derived as the input observation x τ filtered with one of the three filters {h g,m }, g = 1, 2, 3 from equation [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]. In the presence of a noise, y τ = x τ + n τ , each filter's output will be different from 0, wether x τ is smooth or not. But still, the values of the filters outputs will be highly correlated with the smoothness of the noise-free observation x τ (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] where it is shown that the filters behave as a numerical differentiators). Therefore we use a product of the three filter's outputs as the final spike detection decision parameter. In this way, the occurrences of the spikes are highly emphasized with respect to the noise.

Comparison of the proposed detection method with two methods among the most accurate ones in the literature is hereafter presented. These are the method based on a combination of median filter, matched filter and nonlinear energy operator (NEO) [START_REF] Land | Spike (action potential) detection and sorting[END_REF] and the method based on wavelet transformation [START_REF] Nenadic | Spike detection using the continious wavelet transform[END_REF]. The spike train observation data are simulated1 as in [START_REF] Land | Spike (action potential) detection and sorting[END_REF]. The value of the parameter T , that defines the filter window length, is set to be 20% above the simulated spike length (T = 100 samples). Figure 1 method compares favorably with respect to the others.

Combination with ICA

In this section, we use a more elaborated neural model for the simulation. This is described first. The description is followed by the implementation of the algebraic differentiation filters with ICA. Finally the simulation results are presented. For each neuron, we simulate first the firing instants as Poisson distributed with a refractory period of 6.7ms. To each location, we then associate randomly one of the above spike templates. The neurons are spatially randomly distributed in the form of a cube, with minimal and maximal mutual distances. Four electrodes are placed on one side of the cube. We assume an isotropic medium (constant conductivity of the tissue), so that the contribution of each neuron is determined only by its distance from each electrode.

Spike detection and sorting

Superposition of the distant neurons activity occurs as a noise on the electrodes -original action potential shape waveforms are destroyed. Only the action potentials coming from the neurons that are close to the electrodes are recorded as spike shape waveforms. Next, signals from the four electrodes are applied to each of the three filters in order to emphasize action potential shape waveforms. The product of the filters outputs at each electrode may be used as a decision function. This was done in the previous section. Here we use the filters outputs as inputs for ICA in order to perform both, spike detection and sorting. We apply ICA on each set of the four outputs from each of the three filters. The goal is to reconstruct the activity of one neuron per electrode (the closest one). FastICA MATLAB toolbox3 is used for ICA implementation. ICA gives three results for each of the three filters. The final detection and sorting result is given as a product of these three ICA outputs.

Results

To gauge the effects of the filters, we also consider the same settings, but with the filters replaced by the identity: we apply ICA directly on the signals from the electrodes and then raise each ICA output to the power 3. Figure 3 compares the two normalized results obtained with and without the filters, on one sample realization of the reconstructed activity of one neuron. The figure shows that the amplitudes of the spikes that are not fired by the neuron closest to the observed electrode are attenuated more with the proposed method than when only ICA is used. This is confirmed in Table 1 that displays the ratio of the Reconstruction of the neuron activity by using the combination of algebraic differentiations and ICA Fig. 3. Results of spike detection and sorting on one electrode with 30000 samples when only ICA is applied and when the combinations of algebraic differentiations and ICA is applied. mean amplitude of the correctly detected action potentials (15 in total) over the mean amplitude of the falsely detected ones (80 in total). At a given electrode, a correct detection is meant when the detected action potential stems from the closest neuron. Otherwise, the detection is qualified as false. The first row of the table concerns the combination of the algebraic differentiations and ICA while the second row is for ICA alone. The different columns correspond to the four electrodes.

Conclusion

We have proposed a piecewise regular model for the spike train where the occurrence of a spike (transient) is represented by a discontinuity points. Using operational calculus and basic algebraic manipulations we give an explicit characterization of these discontinuity points. The characterization provides a spike detection method which compares favorably with the existing methods. By combining this approach with ICA we obtain not only good results for detection, but also for spike sorting.

Fig. 1 .

 1 Fig. 1. ROC curves obtained by using the proposed method (Algebraic), NEO, and wavelet based method. Figures for two different firing rates (FR) are shown.

4. 1 Fig. 2 .

 12 Fig. 2. Cell attached recording and action potential templates

  electrectrode Reconstruction of the neuron activity by using only ICA

Table 1 .

 1 Comparison of action potentials (AP) sorting when ICA is combined with the algebraic differentiation filters and when only ICA is used

		Averaged normalized amplitudes of correctly detected AP / Av-
		eraged normalized amplitudes of falsely detected AP
		Neuron 1	Neuron 2	Neuron 3	Neuron 4
	AD and ICA 29.45	48.97	29.76	86.31
	ICA	19.72	46.80	15.41	78.12

The signal is created using a code provided by Bruce Land at http://www.nbb.cornell.edu/neurobio/land/PROJECTS/spikeSort.

The toolbox is developed at Laboratory of Information and Computer Science in the Helsinki University of Technology and it is available on http://www.cis.hut.fi/projects/ica/fastica/


Jason Govilas

Anouk Chevallier

Violaine Guicheret

Fabien Amiot

Johnny Beaugrand

Cédric Clevy

Vincent Placet

Single plant fiber transverse compression -Studying influent parameters and identifying material properties

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Single Plant Fiber Transverse Compression: Investigation of Influential Parameters and Identification of Mechanical Properties

Jason Govilas 1 , Anouk Chevallier 1 , Violaine Guicheret-Retel 1 , Fabien Amiot 1 , Johnny Beaugrand 2 , Cédric Clévy 1 , Vincent Placet Plant fibers offer an alternative to synthetic fibers and monofilaments that satisfy current societal and policy maker demands for renewable and environmentally friendly materials. Just like their man-made counterparts they can be used in composite or textile materials for various applications. However, their complex morphology and micro-structure makes them a more challenging material to characterize. While their longitudinal properties have been investigated under tensile tests, their transverse properties remain largely unknown. Composite bearing strength and ballistic cloths are examples where knowledge of these properties is crucial. To identify them, single fiber transverse compression tests can be used. Yet, to the authors knowledge, no such tests have been performed on plant fibers and the influence of several parameters on the fiber's identified properties have not been investigated, even in the case of synthetic fibers.

This communication will showcase the transverse mechanical behavior of nettle, flax and hemp fibers with the use of specialized micro-mechatronic setup. Through a combination of actuators, custom sensors and protocols, sub mNewton and µmeter precision is achieved while maintaining platen parallelism under 0.1°. With the use of numerical simulation and experimental studies the influence of several key parameters on the identified transverse elastic properties is also investigated. The impact of compression platen parallelism as well as fiber ellipticity, lumen and viscoelasticity will be presented.

Transverse material properties are obtained through inverse identification with the help of analytical models. However, by considering fibers as transversely isotropic elastic cylinders, these models are shown to be sensitive to all studied parameters. While some deviations can be minimized through experimental protocols, an evolution of these models or inverse identification based on finite element models could improve transverse property identification.

Figure 1 .

 1 Figure 1. Single fiber transverse compression: experimental configuration and examples of numerical studies.