Nicolás Espitia
email: nicolas.espitia-hoyos@inria.fr

Andrey Polyakov

Emilia Fridman

On local finite-time stabilization of the Viscous Burgers equation via boundary switched linear feedback

This paper considers the problem of local finitetime stabilization of the viscous Burgers equation. A boundary switched linear control with state dependent switching law is designed based on the Backstepping approach. The strategy builds on discontinuous kernels which render the control function a piecewise continuous one. It is proved that such a control stabilizes locally the viscous Burgers equation and that the settling time depends on initial conditions. A simulation result is provided to validate the theoretical results.

I. INTRODUCTION

For many dynamical systems, most of the results on stabilization and estimation are based on asymptotic or exponential guarantees. However, when the time of control is strongly restricted and transient process has to be finished in a finite-time, the need to address a finite-time or fixed time stabilization (or a very rapid stabilization, otherwise) becomes a central issue and necessity.

Finite-time concepts have been extensively considered in the framework of linear and nonlinear ordinary differential equations (ODES) ( [START_REF] Haimo | Finite time controllers[END_REF], [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF], [START_REF] Polyakov | Finite-time and Fixedtime Stabilization: Implicit Lyapunov Function Approach[END_REF], [START_REF] Lopez-Ramirez | Finitetime and fixed-time observer design: Implicit Lyapunov function approach[END_REF], [START_REF] Song | Timevarying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]). For partial differential equations (PDEs), finite-time concepts have been gaining a lot of attention as PDEs may indeed describe many complex systems (e.g. hydraulic networks, tubular chemical reactors, etc) for which the convergence while meeting time constraints or just realizing the well-known separation principle are central issues. Synthesis of controllers to achieve these goals would bring more challenges than exponential stabilization. For hyperbolic PDEs, for instance, one can refer to [START_REF] Perrollaz | Finite-time stabilization of 2×2 hyperbolic systems on tree-shaped networks[END_REF], [START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation[END_REF] for the stabilization in finite-time with boundary control. For linear parabolic PDEs, on the other hand, the scenario is even more demanding. Some works have addressed some relevant issues on null controllability and finite-time stabilization (e.g. [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] Polyakov | On boundary finite-time feedback control for heat equation[END_REF]) by making use of the backstepping approach to design time-varying feedbacks; although with discontinuous kernels. It is worth recalling that the backstepping method has been used as a standard method to design boundary controllers for stabilization of PDEs [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF].

This topic is moved forward for continuous time-varying feedbacks in [START_REF] Espitia | On continuous boundary time-varying feedbacks for fixed-time stabilization of coupled reaction-diffusion systems[END_REF] and [START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF] under backstepping approach for fixed-time stabilization. The time of convergence can even be prescribed in the design. Other results on finite-time for linear parabolic PDEs using in-domain control can be found in e.g. [START_REF] Pisano | Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques[END_REF], [START_REF] Polyakov | Finite-time and Fixedtime Stabilization: Implicit Lyapunov Function Approach[END_REF] where the former uses some sliding mode techniques and the latter uses homogeneity arguments.

Nonlinear parabolic PDEs have been also useful for the modeling of large scale networks and complex processes coming from fluid mechanics. In particular, the viscous Burgers equation has been studied as a simplified model of the Navier-Stokes equations. The viscous Burgers equation can be in fact a tractable control-oriented model for which boundary and in-domain control strategies have been proposed e.g. [START_REF] Iwamoto | Optimal control of the viscous Burgers equation by the Hopf-Cole transformation and its properties[END_REF], [START_REF] Hasan | Local exponential stabilization of coupled burgers' pde-ode system[END_REF]. This motivates, from the practical point of view, to use this model for active flow control strategies to reduce the drag in aerodynamic applications and transportation industry. On the other hand, from theoretical level, this equation posses interesting challenges as sock-like stationary solutions may be developed (see e.g. [START_REF] Krstic | Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE[END_REF]) and that have been useful for formation control problems in multi-agent systems; see e.g. [START_REF] Freudenthaler | Backstepping-based extended Luenberger observer design for a Burger-type PDE for multiagent deployment[END_REF]. Thus, stabilization and estimation of the viscous Burgers equation is of the great interest. However, to the best of our knowledge, finite-time stabilization and estimation for this kind of PDE have not been widely addressed in the literature. This motivate to study this topic under a boundary control problem formulation.

The main contribution of this paper relies on the design of a boundary switched linear control such that a suitable switching law is dependent on the state of the system. Inspired by [START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF] and [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF], we build on the backstepping approach from which kernels of the transformation are piecewise-continuous. Under the switching law the control gain is such that it increases more and more while the solution of the closed-loop systems goes to zero in a finitetime. We prove that for sufficiently small initial condition, it is possible to achieve finite-time stabilization and that the settling time depends on initial data of the closed-loop system. We recall here that in this framework, finite-time convergence differs from the fixed-time one whenever the settling-time depends on the initial condition of the system as studied in this paper. This paper is organized as follows. In Section II, we introduce the viscous Burgers and some preliminaries on the backstepping approach. Section III provides our approach towards fine-time stabilization. Section IV provides a numerical example to illustrate the main results. Finally, conclusions and perspectives are given in Section V.

Notations: R + will denote the set of nonnegative real numbers. The set of all functions g : [0, 1] → R such that 1 0 g(x) 2 dx < ∞ is denoted by L 2 ((0, 1), R) and is equipped with the norm • L 2 ((0,1),R ) . I m (•), J m (•) with m ∈ Z, denote the modified Bessel and (nonmodified) Bessel functions of the first kind, respectively.

II. PROBLEM STATEMENT, PRELIMINARIES AND BASIC

ASSUMPTIONS

Let us consider the following viscous Burgers equation with Dirichlet boundary conditions:

u t (t, x) = θu xx (t, x) -u x (t, x)u(t, x) (1) u(t, 0) = 0 (2) u(t, 1) = U (t) (3)
and initial condition:

u(0, x) = u 0 (x) (4)
where u : [0, T )×[0, 1] → R is the system state and θ > 0 is the viscous term. In addition, U (t) ∈ R is the control input having the functional form

U (t) = K σ(t) [u(t, •)](1) (5)
with

K σ(t) [u(t, •)](1) = 1 0 k(1, y, λ σ(t) )u(t, y)dy (6)
which is the linear switching feedback law. Here, k will be a time-varying kernel gain (piecewise-continuous function) and σ : [0, T ) → Z is going to be a state dependent switching signal (piecewise-constant function), both to be characterized later on .

In open loop (e.g. U (t) = 0), the system (1)-( 4) is globally exponentially stable. The control aim is to steer the state of the system (1)-( 4) to zero in a finite-time. As it is going to be stated later, the time of convergence depends on initial conditions of the system.

A. Backstepping transformation and switching kernel equations

In this work we aim at providing a boundary linear switched control which is going to be designed based on the backstepping approach (inspired by [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF], [START_REF] Hasan | Local exponential stabilization of coupled burgers' pde-ode system[END_REF], [START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF] and [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF]). The backstepping method has been typically applied to linear PDE systems. However, in this work the method may be applicable as soon as one considers the nonlinear term u x u in (1) as a small force (as typically done for the Kortewegde Vries equation, e.g. [START_REF] Xiang | Small-time local stabilization for a Korteweg-de Vries equation[END_REF]). A characterization for the backstepping transformation as well as target system by taking into account the nonlinearity after transformation, is going to be discussed in the sequel. To that end, let us first bring back the following integral Volterra transformation as follows:

w(t, x) = u(t, x) - x 0 k(x, y, λ σ(t) )u(t, y)dy = K σ(t) (t)[u(t, •)](x) (7)
whose inverse is given by

u(t, x) = w(t, x) + x 0 l(x, y, λ σ(t) )w(t, y)dy = L σ(t) (t)[w(t, •)](x) (8)
where k and l are the time-varying kernels of the direct and inverse backstepping Volterra transformations, respectively.

Similar to [START_REF] Hasan | Local exponential stabilization of coupled burgers' pde-ode system[END_REF], the aim is to transform the system (1)-( 4) into the following target system:

w t (t, x) = θw xx (t, x) -θλ σ(t) w(t, x) -F [w(t, •), w x (t, •)](x) (9)
w(t, 0) = 0 (10) w(t, 1) = 0 [START_REF] Hasan | Local exponential stabilization of coupled burgers' pde-ode system[END_REF] with initial condition:

w 0 (x) = u 0 (x) - x 0 k(x, y, λ σ(0) )u 0 (y)dy (12)
where w : [0, T )×[0, 1] → R is the state of the target system. In our approach, the reaction term will be changing thanks to λ σ(t) > 0 and according to some switching law that needs to be designed to achieve finite-time stability property. In addition, the functional F [w, w x ] is given as follows:

F [w, w x ](x) = K σ(t) L σ(t) [w]w x + L σ(t) [w]L 1 σ(t) [w] (x) (13
) where the functional L 1 σ(t) (t)[w(t, •)](x) is as follows:

L 1 σ(t) (t)[w(t, •)](x) = l(x, x, λ σ(t) )w(t, x) + x 0 l x (x, y, λ σ(t) )w(t, y)dy (14)
A suitable estimate of the functional F [w, w x ] given by ( 13) can be deduced from [11, Lemma 2].

Following the standard methodology to find kernel equations and since σ is a piecewise-constant function, it can be proved that the kernel of (7) satisfies the following PDE system:

k xx (x, y, λ σ(t) ) -k yy (x, y, λ σ(t) ) = λ σ(t) k(x, y, λ σ(t) ) (15) k(x, 0, λ σ(t) ) = 0 (16) k(x, x, λ σ(t) ) = -1 2 xλ σ(t) ( 17
)
where k is defined on the domain T = {(x, y) ∈ R 2 : 0 ≤ y ≤ x ≤ 1} × R + , whose solution is well-known to admit a closed form as follows [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]:

k(x, y, λ σ(t) ) = -yλ σ(t) I 1 λ σ(t) (x 2 -y 2 ) λ σ(t) (x 2 -y 2 ) (18)
and similarly, for the inverse transformation [START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF], the kernel satisfies:

l xx (x, y, λ σ(t) ) -l yy (x, y, λ σ(t) ) = -λ σ(t) l(x, y, λ σ(t) ) (19)
l(x, 0, λ σ(t) ) = 0 (20)

l(x, x, λ σ(t) ) = -1 2 xλ σ(t) (21)
where k is defined on the domain T = {(x, y) ∈ R 2 : 0 ≤ y ≤ x ≤ 1} × R + , whose closed-form solution is as follows:

l(x, y, λ σ(t) ) = -yλ σ(t) J 1 λ σ(t) (x 2 -y 2 ) λ σ(t) (x 2 -y 2 ) (22)
In addition, from ( 7) and ( 8), one has the following relations:

w(t, •) L 2 ((0,1),R) ≤ Ψ(λ σ(t) ) u(t, •) L 2 ((0,1),R) (23)
and

u(t, •) L 2 ((0,1),R) ≤ Φ(λ σ(t) ) w(t, •) L 2 ((0,1),R) (24)
where estimates Ψ(λ σ(t) ) and Φ(λ σ(t) ) are given by:

Ψ(λ σ(t) ) := 1 + 1 0 x 0 |k(x, y, λ σ(t) )| 2 dy dx 1/2
(25) with k given by ( 18), and

Φ(λ σ(t) ) := 1 + 1 0 x 0 |l(x, y, λ σ(t) )| 2 dy dx 1/2
(26) with k given by [START_REF] Song | Timevarying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF].

Finally, let us point out the following result [START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF] that is going to be used in the proof of local finite-time stability property.

Lemma 1: For the target system (9)-( 11), [START_REF] Krstic | Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE[END_REF], there exists c such that the following estimate holds true:

2 1 0 w(t, x)F [w(t, •), w x (t, •)](x)dx ≤ e c √ λ σ(t) ( w(t, •) 4 L 2 ((0,1),R) + w(t, •) 6 L 2 ((0,1),R) ) + w x (t, •) 2 L 2 ((0,1),R) (27) Proof: See [5, Section 4].
Remark 1: A precise characterization of c in (27) is not provided here; however, it can be done by analyzing the growth-in-time of the functional F and kernels equations involved there. The analysis may follow similar ideas of [START_REF] Hasan | Local exponential stabilization of coupled burgers' pde-ode system[END_REF] which in turn follows [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF].

III. STATE DEPENDENT SWITCHING LAW AND FINITE-TIME STABILIZATION

A. On the state dependent switching law

Let us choose λ σ(t) = 2 σ(t) where the switching function σ(t) is governed by the following state dependent rule1 :

σ(t) = G(σ(t -), u(t -, •)) (28) with G : Z × L 2 ((0, 1), R) → Z, t -= t + 0 -and σ(0) ∈ Z. G(σ(t), u(t, •)) =      i + 1 if σ(t) = i and u(t, •) L 2 ≤ r i+1 i if σ(t) = i and r i+1 < u(t, •) L 2 < r i-1 i -1 if σ(t) = i and u(t, •) L 2 ≥ r i-1 (29)
where r i is a suitable sequence defined iteratively:

r 0 = e -c 2 Ψ(2 0 ) , r i = e -qi r i-1 , i ∈ Z (30)
with q i defined as follows:

q i = ln Ψ(2 i ) + ln Φ(2 i ) + c 2 √ 2 i (31)
and Ψ and Φ are defined in ( 25) and (26), respectively.

Proposition 1: Let q i be defined by (31). Then for q i > 0, it holds that r i → 0 as i → ∞ and

lim i→∞ q i+1 q i = √ 2 (32)
Proof: It follows the same lines of [18, Proposition 2].

B. On finite-time stabilization

We perform Lyapunov-based analysis on the target system and we use the bounded invertibility of the backstepping transformation in order to prove local finite-time stabilization. An estimate of the settling time is also derived.

Lemma 2: Let us consider the target system (9)-( 12) with λ σ(t) = 2 σ(t) . Assume that σ(t) = i k for all t ∈ [t k , t k+1 ) where the switching time instants of σ are denoted by

(t k ) k∈N . If w(t k , •) L 2 ((0,1),R) < e -c 2 √ 2 σ(t)
and θ2 i k > 1, then there exists a unique solution

w ∈ C 0 [t k , t k+1 ); L 2 ((0, 1), R) ∩ L 2 loc [t k , t k+1 ); H 1 0 ((0, 1), R)) such that w(t, •) L 2 ((0,1),R) ≤ e -(θ2 σ(t) -1)(t-t k ) w(t k , •) L 2 ((0,1),R) (33
) for all t ∈ [t k , t k+1 ).

Proof:

We consider the target system ( 9)-( 12), associated with each constant mode i k on [t k , t k+1 ) (i.e. σ(t) = i k ) where w(t, x) = w i k (t, x) for all t ∈ [t k , t k+1 ). That is,

w i k t (t, x) = θw i k xx (t, x) -θ2 i k w i k (t, x) -F [w i k (t, •), w i k x (t, •)](x)(34) w i k (t, 0) = 0 (35) w i k (t, 1) = 0 (36)
The local existence and uniqueness of the solution

w ∈ C 0 [t k , t k+1 ); L 2 ((0, 1), R) ∩ L 2 loc [t k , t k+1
); H 1 0 ((0, 1), R)) of target system (34)-(36) follow by considering the nonlinear term F [w, w x ](x) as a force term and using Banach fixed point theorem as proved in [START_REF] Coron | Small-time global stabilization of the viscous Burgers equation with three scalar controls[END_REF].

Let us prove now that the solution decreases exponentially on that interval. Consider the following Lyapunov function

V : L 2 ((0, 1), R) → R, V (w) = 1 0 w(x) 2 dx.
Computing the time derivative along the solutions of ( 34)-( 36), performing integration by parts and using the boundary conditions, yield, for all t ∈ [t k , t k+1 ),

V (t) ≤ -2θ2 i k V (t) -2θ 1 0 w x (x)dx + 2 1 0 w i k (t, x)F [w i k (t, •), w i k x (t, •)](x)dx ( 37
)
with F is as in [START_REF] Krstic | Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE[END_REF].

By Lemma 1, it holds:

2 1 0 w i k (t, x)F [w i k (t, •), w i k x (t, •)](x)dx ≤ e c √ 2 i k (V 2 (t) + V 3 (t)) + w x (t, •) 2 L 2 ((0,1),R) (38)
for all t ∈ [t k , t k+1 ). Thus,

V (t) ≤ -2θ2 i k V (t) + e c √ 2 i k (V 2 (t) + V 3 (t)) (39)
Now, if

w i k (t k , •) L 2 ((0,1),R) < e -c 2 √ 2 i k , we have on the one hand, V (t) < e -c √ 2 i k . Then, V 2 (t) < e -c √ 2 i k V (t) (40)
On the other hand, since

V (t) < 1, then V 3 (t) < V 2 (t).
Hence, from (39) and using (40), we get,

V (t) ≤ -2(θ2 i k -1)V (t) (41)
with θ2 i k > 1. Therefore, by the Comparison principle, it holds for all t ∈ [t k , t k+1 ), that:

w(t, •) L 2 ((0,1),R) ≤ e -(θ2 i k -1)(t-t k ) w i k (t k , •) L 2 ((0,1),R) (42) It concludes the proof.
Let us state the main result of the paper.

Theorem 1: For any initial condition u 0 ∈ L 2 ((0, 1), R) such that

σ(0) = i 0 ∈ Z with u 0 L 2 ((0,1),R) ∈ (r i0+1 , r i0 ] (43)
and θ2 σ(0) > 1, then the closed-loop system (1)-( 4) with linear switched controller

U (t) = 1 0 k(1, y, 2 σ(t) )u(t, y)dy (44)
with k(1, y, 2 σ(t) ) given in [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF], and σ(t) given in (28) according to (29)-(30); has a unique solution u ∈ C 0 [0, T ); L 2 ((0, 1), R) ∩ L 2 loc [0, T ); H 1 0 ((0, 1), R)) and is locally finite-time stable, i.e

u(t, •) L 2 ((0,1),R) → 0, as t → T ( 45
)
with a bounded settling time depending on initial data given by

T (u 0 ) ≤ +∞ i=i0 q i + q i+1 θ2 i + ∞ i=i0 -c 2 √ 2 i θ2 i < +∞ (46)
Proof: I. On the well-posedeness of the closed-loop solution. If u(t i , •) L 2 ((0,1),R) ∈ (r i+1 , r i ] and σ(t) = i then there always exists an instance of time t * separated from t i such that u(t * , •) L 2 ((0,1),R) = r i+1 or u(t * , •) L 2 ((0,1),R) = r i-1 . It means that σ is switched either to i + 1 or i -1 according to (29) and the switching instant t * will be always isolated (this prevents the Zeno phenomena). Then, due to local existence and uniqueness of the solution of the target system and due to the bounded invertibility of the backstepping transformation, it can be proved that for any initial condition u 0 ∈ L 2 ((0, 1), R) with (43), the closedloop system (1)-( 4), (44) has a unique solution such that u ∈ C 0 [0, T ); L 2 ((0, 1), R) ∩L 2 loc [0, T ); H 1 0 ((0, 1), R)) , T = sup t k , where t k > 0 are the sequence of switching instants.

II On the local finite-time stability. The proof for local finite-time stability is done recursively with the following induction properties, for k ≥ 0:

• u(t k , •) L 2 ((0,1),R) ∈ (r i k +1 , r i k ]; • u(t, •) L 2 ((0,1),R) < r i k -1 , ∀t ∈ [t k , t k+1 ); • i k + 1 = i k+1 .
Let us verify for k = 0, t ∈ [t 0 , t 1 ). From [START_REF] Xiang | Small-time local stabilization for a Korteweg-de Vries equation[END_REF], it holds

w 0 L 2 ((0,1),R) ≤ Ψ(2 i0 ) u 0 L 2 ((0,1),R) (47)
By initialization hypothesis (43), we get

w 0 L 2 ((0,1),R) ≤ Ψ(2 i0 )r i0 (48)
and using the recurrence relation (30) along with (31), it holds

w 0 L 2 ((0,1),R) ≤ Ψ(2 i0 )e -qi 0 r i0-1 ≤ Ψ(2 i0 )e -ln(Ψ(2 i 0 )Φ(2 i 0 )) e -c 2 √ 2 i 0 r i0-1 ≤ e -ln Φ(2 i 0 ) e -c 2 √ 2 i 0 r i0-1 (49)
Note that e -ln Φ(2 i 0 ) r i0-1 ≤ 1 by virtue of the definition of the sequence r i in (30). Hence, we have that

w 0 L 2 ((0,1),R) ≤ e -c 2 i k (θ2 i k -1) (59)
which can further be written as follows:

T ≤ ∞ i=i0 q i+1 + q i (θ2 i -1) - ∞ i=i0 c 2 √ 2 i (θ2 i -1) (60)
It can be proved that the first sum in (60) can be rewritten as

∞ i=i0 q i+1 + q i (θ2 i -1) = ∞ i=i0 q i (θ2 i -1) 3θ2 i -4 (θ2 i -2) - q i0 (θ2 i0 -1) 2(θ2 i0 -1) (θ2 i0 -2)
Applying the ratio test for convergence to the above series, and due to Proposition 1, we have that

(θ2 i -1)(3θ2 i+1 -4)(θ2 i -2) (θ2 i+1 -1)(θ2 i+1 -2)(3θ2 i -4) q i+1 q i → √ 2 2 < 1 (61)
and qi

(θ2 i -1) 3θ2 i -4 (θ2 i -2) → 0 as i → ∞.
For the second sum, the ratio test guarantees its convergence since we have

(θ2 i -1) (θ2 i+1 -1) √ 2 i+1 √ 2 i → √ 2 2 < 1, as i → ∞ (62)
and

√ 2 i (θ2 i -1) → 0 as i → ∞. Therefore, T (u 0 ) < ∞ for u 0 L 2 ((0,1),R) ∈ (r i0+1 , r i0 ].
Hence, we finally conclude that u(t, •) L 2 ((0,1),R) → 0 as t → T .

IV. SIMULATIONS

We illustrate the results by considering the viscous Burgers equation ( 1)-( 4) with θ = 1.1. For numerical simulations, we implement a Crank-Nicholson scheme. We consider two different initial conditions, u 0 = 0.1x(x -1) and u 0 = x(1x) satisfying (43) and we select i 0 = 0 with threshold sequence (30) and c = 1. Figure 1 shows the evolution of the L 2 -norm of the closed-loop system with boundary switched linear feedback U (t) given in (44) with kernel gain [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF] as well as the evolution of the L 2 -norm of the closed-loop system with a linear control feedback (as in e.g. [START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-d partial integro-differential equations[END_REF]) for exponential stabilization. The plots are in logarithmic scale to illustrate the convergence to zero in finite-time. It can be observed that settling times are indeed dependent on initial data.

V. CONCLUSION

In this paper we have addressed the problem of local finitetime stabilization of the viscous Burgers equation under a boundary linear switched control provided the norm of initial condition is small enough. The design has been carried out based on the backstepping approach for which the switching law depends on the state of the system and kernels of the backstepping transformation turn out to be time-varying but discontinuous (i.e. piecewise-continuous). In addition, a rigorous characterization of c involved in (27) still has to be done. This issue is then left for a future work. Future work also includes local fixed-time stabilization of the viscous Burgers equation by means of continuous time-varying feedbacks. For that, we expect to apply our recent results [START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF], [START_REF] Espitia | Some characterizations of boundary time-varying feedbacks for fixed-time stabilization of reaction-diffusion systems[END_REF] and [START_REF] Espitia | On continuous boundary time-varying feedbacks for fixed-time stabilization of coupled reaction-diffusion systems[END_REF] on fixed-time stabilization of reaction-diffusion PDEs where closed-form time-varying kernels are obtained and a qualitative analysis of the fixedtime convergence is carried out. We expect also to address finite/fixed-time estimation for the viscous Burgers equation.

10 Fig. 1 .

 101 Fig. 1. Evolution of the L 2 -norms of the closed-loop system (logarithmic scale) with the linear switched controller (black lines) and with linear continuous controller (red dashed lines) for exponential stabilization.

For hyperbolic PDEs, switching law based on Lyapunov techniques can be found for instance in[START_REF] Prieur | Stability of switched linear hyperbolic systems by Lyapunov techniques[END_REF].

*This work has been partially supported by CPER ELSAT 2020, Project, Contratech and by Project 14.Z50.31.0031 of the Ministry of Education and Science of Russian Federation and Grant 08-08 of the Government of Russian Federation. N. Espitia is with Valse team, Inria Lille -Nord Europe, and CRIStAL (UMR-CNRS 9189),

√ 2 i 0 . Consequently, hypothesis of Lemma 2 is verified and therefore (33) holds for t ∈ [t 0 , t 1 ).

Then, combining (33) and (24) we have, for all t ∈ [t 0 , t 1 ),

From (31), we have that (50) is rewritten as follows:

Furthermore, by initialization hypothesis (43), we get

Hence, using the recurrence relation (30), we get

from which it can be deduced that u(t, •) L 2 ((0,1),R) < r i0-1 . It means that norm does not reach the upper level of the threshold (preventing overshoot). In addition, it is clear, due to (53), that the norm decreases exponentially. This implies that at some time t = t 1 the norm will reach the lower level of threshold, i.e. u(t 1 , •) L 2 ((0,1),R) = r i0+1 . Therefore, by (29), the switch is done so as σ(t 1 ) = i 0 + 1 with i 0 + 1 = i 0+1 = i 1 . Hence, we can repeat the reasoning for [t 1 , t 2 ) provided the initial data σ(t

By continuing the reasoning recursively, we assume that induction properties hold for k > 0 and we verify for k + 1. For that purpose it is sufficient to take as initial conditions σ(t k+1 ) = i k+1 with u(t k+1 , •) L 2 ((0,1),R) ∈ (r i k+1 +1 , r i k+1 ] and to apply the same arguments and steps as above.

It remains to prove that the settling time is bounded. By Lemma 2 we have,

(54) Hence, we derive,

Using ( 23) and (24), the previous inequality is rewritten as follows:

Moreover, by (31) we obtain that (57) can be bounded as follows:

Hence,


Volker Schröder

To quote the organizers: Fittingly, the conference will kick off with a panel on the book trade in Lyon, including a paper by Dominique Varry on "Lyon, capitale de la contrefaçon au XVIIe siècle."

Producing contrefaçons -illicit reprints of texts for which others had obtained a privilège du roi -became somewhat of a specialty of Lyon libraires in the 17th century, which is traditionally considered a period of decline for the trade after its "golden age" during the Renaissance. This development was due in part to royal policy, which increasingly favored Parisian printers and publishers over their provincial counterparts, more difficult to monitor and control.

But not all Lyon reprints were piracies… and some of them can even reward the curious reader with bonuses not present in the Paris originals, nor anywhere else. A case in point is the

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Rare texts and images from early modern France About Lyon, the second printing center of the Ancien Régime after Paris, is the perfect place for such an investigation. Today, Lyon is still a special place for book history in France, through a strong network of schools and cultural institutions: the Book History Institute (IHL), the National Superior School for Library and Information Sciences (Enssib), the Lyon Public Library (BmL), the Museum of Printing and Graphic Communication (MICG). All of them have accepted to be our partners in the organization of the conference.

Hilaire Baritel is known in the annals of contrefaçon as the author of several texts denouncing the corrupt business practices of his colleagues. In a grievance filed in October 1702, he writes that during a recent stay in Paris he acquired "with much difficulty" a share of the privilège for four titles, including the works of Mme Deshoulières, and that his investment will be lost -and the authority of Louis XIV undermined -if other Lyon publishers are allowed to flood the market with their cheap

  Abbesse de S. Pierre de Lyon," and the third to "Madame de Chevry" -high-born nuns who had probably been close to Deshoulières when they lived in Paris (in the 1660s?), before moving to Lyon to lead the Benedictine abbey of Saint-Pierre. (The building now houses the Museum of Fine Arts.) These poems are absent from Sophie Tonolo's recent critical edition of Deshoulières' Poésies, which does not take into account the 1703 publication. They do appear in the earlier edition by Catherine Hémon-Fabre and Pierre-Eugène Leroy, but with misleading information about their source and addressees. While the texts published by Baritel are not perfect, they seem authentic and could be among the earliest epistles composed by Deshoulières. The poet herself did not preserve them, but her Lyonnaise friends did, ensuring their eventual publication and survival. They deserve further study and raise a number of questions, the answers to which may well lie in Lyon.


Many American libraries along the eastern seaboard, from Maine to Washington D.C. and beyond, preserve outstanding autograph collections whose riches remain to be fully explored. While their strengths lie typically, and quite naturally, in

  The letter is addressed to "Monsieur Despreaux," i.e. the poet Nicolas Boileau-Despréaux, Racine's friend and colleague as royal historiographer. The second sentence, concerning Boileau's Epître de l'amour de Dieu and his reference to theologian Antoine Arnauld, makes it clear that the letter was written in 1698 (when January 20 was indeed a Monday). Racine also mentions his wife, his aunt (abbess of Port-Royal), his eldest daughter Marie-Catherine, and his eldest son Jean-Baptiste, who just had a meeting with Fénelon, archbishop of Cambrai. A lot of information in these nineteen lines! In 1747, Racine's youngest son Louis published an abridged version of the letter, omitting the last seven lines. In 1806, a partial facsimile of the autograph (which was at that time in the Bibliothèque Impériale) was published in the OEuvres de Louis XIV:


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In effigiem Caroli Patin

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 1 Portrait of Charles Patin by Claude Lefebvre (left) and Antoine Masson (right) Claude Lefebvre (1632-1675) was his generation's leading portrait painter, perhaps best known today for the portrait of Jean-Baptiste Colbert that he presented as his reception piece to the Académie. Lefebvre was also a printmaker, but his work in this medium consists of just four portraits: those of his mother (no impressions known), of himself, of the print dealer Alexandre Boudan, and of the physician and numismatist Charles Patin (1633-1693). The latter was produced for Patin's 1663 folio edition of Fulvio Orsini's Familiae Romanae in Antiquis Numismatibus; in the same year, Lefebvre reworked his plate for inclusion in Patin's treatise on peat and its use as fuel. Modern print scholars appreciate especially Lefebvre's portrait of Boudan, a freely and vividly etched image which shows the influence of Van Dyck. In comparison, the more formal portrait of Patin appears "rather conventionally executed" -a conventionality surely dictated by its function as a frontispiece. But Lefebvre's work also calls for another comparison, namely with the portrait of Patin engraved in 1667 by Antoine Masson, one of the most virtuosic and original printmakers of the seventeenth century. Information This entry was posted on February 9, 2017 by VS in Book Illustration, Portraits, Printmaking and tagged Antoine Masson, Charles Patin, Claude Lefebvre, Contraband, Medicine, Numismatics. Shortlink https://wp.me/p8rKtd-4 address to subscribe to this blog and receive notifications of new posts by email. Email Address Subscribe Anecdota Rare texts and images from early modern France About Lefebvre vs Antoine Masson (click for larger image) Charles' father Guy, a prominent physician and prolific letter-writer, praised the "excellence" of Masson's burin: Like Lefebvre's portrait, Masson's is accompanied by a Latin quatrain composed by the preacher and writer François Ogier, a friend of the Patin family. In 1671 it was used to adorn another numismatic folio authored by Charles Patin, Imperatorum Romanorum Numismata. In the meantime, however, Patin had already fled France, just before being condemned to the galleys for trafficking in counterfeit and forbidden books. After several years of travels throughout Europe, he settled in Padua where he was appointed to a chair of medicine and continued to write (and trade) books. In 1682, he published Lyceum Patavinum, containing his and his fellow professors' biographies along with their portraits by Martial Desbois. A fourth and, it seems, final portrait shows Charles Patin surrounded by his wife and two daughters, all of whom were published authors in their own right and members of the Accademia dei Ricovrati. Volker Schröder February 9, 2017 J'ai céans des portraits de mon Carolus en taille-douce pour vous envoyer. Ils sont très bien faits, on lui en demande de toutes parts, tant à cause de lui, qu'on aime, qu'à cause de celui qui l'a gravé, qui est excellent en son art, nommé Masson.

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