Gaëtan Beneux

Pierre Riedinger

Jamal Daafouz

Louis Grimaud

Adaptive stabilization of switched affine systems with unknown equilibrium points: application to power converters

Keywords: Switched control, Switched affine systems, DC-DC power converter

come

Adaptive stabilization of switched affine systems with

unknown equilibrium points: Application to power converters

Introduction

A considerable interest has been devoted to switched systems by many researchers both for theoretical and practical reasons. These hybrid systems consists of continuous or discrete time dynamical subsystems and a switching rule that determines at each instant of time the active subsystem [START_REF] Liberzon | Switching in systems and control, Systems & Control: Foundations and Applications[END_REF]. They are encountered in many applications such as embedded systems, automotive, aerospace, and many other fields. From a theoretical point view, the analysis and design problems are very challenging and many contributions have been proposed during the last decades (see [START_REF] Lin | Stability and stabilizability of switched linear systems: A survey of recent results[END_REF][START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF]) and references therein).

Here, we focus on a specific class of switched systems called switched affine systems. This class captures essential features of many applications including power systems and power electronics [START_REF] Albea | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF][START_REF] Buisson | On the Stabilisation of Switching Electrical Power Converters[END_REF][START_REF] Naim | H ∞ control applied to boost power converters[END_REF][START_REF] Sira-Ramirez | Passivity-based controllers for the stabilization of DC-DC power converters[END_REF]. It is also characterized by the fact that many challenging problems have not been solved yet mainly because the affine nature of the linear subsystems introduces additional difficulties in the analysis and control design problems. Among these difficulties, the one related to equilibrium points is of particular interest. There are contributions dedicated to the design of sta-bilizing switching rules under the assumption that the equilibrium point is perfectly known in advance [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF][START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF][START_REF] Hetel | Robust sampled data control of switched affine systems[END_REF][START_REF] Theunisse | Robust global stabilization of the DC-DC boost converter via hybrid control[END_REF]. However, these approaches cannot be applied in the realistic case where uncertainty or parameter variations are considered as it is the case of power converters with a lack of measurement on the input voltage and/or the output load [START_REF] Shahin | Approximate novel loss formulae estimation for optimization of power controller of DC-DC converter[END_REF]).

The problem is open and one may find two types of contributions in the literature. The first type is switched systems oriented and there are very few papers dedicated to switched affine systems with uncertain or unknown equilibrium points. In a very recent paper, the problem of stabilization of switched affine systems is addressed in the case where the equilibrium is uncertain and characterized by a given convex combination [START_REF] Senger | Switching rule design for affine switched systems with guaranteed cost and uncertain equilibrium condition[END_REF]. Under these assumptions, globally stabilizing switching rules are determined by solving an LMI problem. The second type of contributions is power converters oriented. More contributions can be found in this case (see [START_REF] Sira-Ramirez | An algebraic parameter estimation approach to the adaptive observer-controller based regulation of the boost converter[END_REF]) and references therein) but the results are specific to the considered application. As an example, for boost converters with unknown, piecewise constant, load resistance values an interesting contribution is proposed in [START_REF] Sira-Ramirez | An algebraic parameter estimation approach to the adaptive observer-controller based regulation of the boost converter[END_REF]) based on a nonlinear algebraic parameter estimation associated to a reduced order observer and a static output feedback.

Here, we propose a theoretical contribution dedicated to switched affine systems with unknown equilibrium points without restricting the set of equilibrium points to some given structure and we apply this approach to power converters with load and input voltage variations. We formulate the problem both in the case where the set of affine subsystems is finite and also in the case where the set of affine subsystems is not finite and given by a convex polytope, i.e., the convex hull of finitely many affine subsystems. The main contribution is an adaptive control design methodology with a global asymptotic stability property. A preliminary version of this work appeared in (Beneux et al. 2017a,b). The result in (Beneux et al. 2017a) concerns only the case where the set of affine subsystems is finite while (Beneux et al. 2017b) discusses this preliminary result in the case where a DC-DC power converter may operate in the so-called discontinuous conduction mode.

The paper is organized as follows. In the next section, we first recall the classical state feedback switching control design problem for switched affine systems and its solution in the case where the set of affine subsystems is finite. We also formulate the problem in the case where the set of affine subsystems is not finite and we give the solution in the case where the equilibrium point is perfectly known. Section 3 is dedicated to the main contribution of this paper which is an adaptive and switching control design methodology for switched affine systems with unknown parameters. Using a parametrization of the admissible set of equilibrium points and an estimation of the unknown parameters, we propose an adaptive and switched control law with global asymptotic stability guarantees. In section 4, the results are applied to a DC-DC power converter subject to load and input variations. We show how to take into account the fact that the load and the input voltage are not known a priori and we formulate the corresponding power converters stabilisation problem as a control problem for switched affine systems with unknown equilibrium points. We also show that the proposed method preserves the global asymptotic stability property when the DC-DC converter operates in the discontinuous conduction mode (DCM). We end the paper by a conclusion.

Notations:

The set composed by the N first integers is denoted by

K = {1, • • • , N }. The (N -1)-dimension simplex is denoted Λ := {λ ∈ R N | ∀i ∈ K, λ i ≥ 0, N i=1 λ i = 1}. The convex combination of a set of ma- trices A = {A 1 , • • • , A N } is denoted A(λ) = N i=1 λ i A i , with λ ∈ Λ.
The transpose of a matrix M is denoted M T . The identity matrix (or the null matrix) is denoted by I (or 0) for any dimension. For a square symmetric matrix, M 0 (M ≺ 0) indicates that it is positive (negative) definite. co stands for the convex hull. For a given vector v, the notation v [k1,k2] refers to a vector of dimen-

sion k 2 -k 1 + 1 formed by the components of v within the range [k 1 , k 2 ]. Finally, D = diag(d i , i = 1, • • • , n) is a diagonal matrix of dimension n × n whose entries are the d i s, i = 1, • • • , n.

State feedback stabilization

We consider the class of continuous-time switched affine systems given by:

ẋ(t) = A σ x(t) + b σ (1)
where x : R → R n is the state and σ : R n → K refers to the state dependent switching law that selects at each time one of the N subsystems characterized by the pairs

(A i , b i ), i ∈ K.
Our main objective in this paper is the design of adaptive and global stabilizing control laws for this class of hybrid systems. By adaptive, we mean control laws that take into account some parameter variations to be defined in the next section.

We also consider the class of dynamical systems obtained by taking the convex combination of these N subsystems and which is given by:

ẋ(t) = A(λ)x(t) + b(λ). (2)
The control is now λ and it takes its values in the whole simplex Λ. An important feature of this class of dynamical systems is the fact that it allows to characterize the set of equilibrium points of the switched affine systems (1) in already noticed in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF][START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF][START_REF] Hauroigne | Switched affine systems using sampled-data controllers: robust and guaranteed stabilization[END_REF][START_REF] Hetel | Robust sampled data control of switched affine systems[END_REF]. We recall this characterization.

Definition 1 Let Λ H be the subset of Λ such that A(λ) is Hurwitz, that is:

Λ H := {λ ∈ Λ : ∃P 0, A T (λ)P + P A(λ) ≺ 0}.
The set Λ H is open in Λ by construction. It is assumed to be nonempty Definition 2 Let X e be the set of the equilibrium points related to system (2) and defined as:

X e := {x e ∈ R n , x e = -A(λ e ) -1 b(λ e ), λ e ∈ Λ H }
As it has been shown in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF][START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF][START_REF] Hauroigne | Switched affine systems using sampled-data controllers: robust and guaranteed stabilization[END_REF][START_REF] Hetel | Robust sampled data control of switched affine systems[END_REF], state x of system (1) can be stabilized on any equilibrium point x e ∈ X e using the switched state feedback law, given in our framework by:

σ(x) ∈ arg min i∈K (x -x e ) T P (A i x + b i ), (3)
where P 0 satisfies the Lyapunov inequality A(λ e ) T P + P A(λ e ) ≺ 0.

Another interest of the class of dynamical systems given by ( 2) is related to the property of density of the trajectories of system (1) into trajectories generated by (2) which allows to use (2) instead of (1) for stability analysis and/or control design [START_REF] Bengea | Optimal control of switching systems[END_REF][START_REF] Hauroigne | Switched affine systems using sampled-data controllers: robust and guaranteed stabilization[END_REF][START_REF] Ingalls | An infinitetime relaxation theorem for differential inclusions[END_REF]. To be more precise, this corresponds in [START_REF] Bengea | Optimal control of switching systems[END_REF], which is dedicated to optimal control design for switched systems, to the fact that the switched system is embedded into a larger family of systems and the optimization problem is formulated for the latter. The authors show that the set of trajectories of the switched system is dense in the set of trajectories of the embedded system and the relationship between the two sets of trajectories motivates the shift of focus from the original problem to the more general one. They also underly the engineering relevance of the study of the second problem. Indeed, this density property is useful to establish some bridges from control laws dedicated to (1) to the ones dedicated to (2) through the notion of averaging. Conversely, control methods designed for (2) can be applied to systems given by (1) using Pulse-Width Modulation techniques (PWM). Owing to these interests, we provide in the next proposition a way to design what we will call in the sequel the embedded control λ.

Proposition 1 Consider the dynamical system (2). The equilibrium point corresponding to the pair (x e , λ e ) is globally asymptotically stable under the state feedback law:

λ [1,N -1] (x, x e ) = λ e [1,N -1] -Ky(x, x e ), λ N (x, x e ) = λ e N + 1 T Ky(x, x e ),
with y(x, x e ) = B T (x)P (x -x e ), and B(x) whose columns are given by B

i (x) = (A i -A N )x + (b i -b N )
and where P 0 satisfies the Lyapunov inequality

A(λ e ) T P + P A(λ e ) + Q ≺ 0.
for some given Q 0.

The matrix K is such that K = diag(k i , i = 1, • • • , N -1) with positive real numbers k i > 0 and 1 = (1, • • • , 1) (N -1 times).
PROOF. For a given (x e , λ e ), system (2) can be rewriten:

ẋ = A(λ e )(x -x e ) + A(λ -λ e )x + b(λ -λ e ),
then, using relation :

λ N = 1 - N -1 i=1 λ i , we have: ẋ = A(λ e )(x -x e ) + N -1 i=1 (λ -λ e ) i B i (x), with B i (x) = (A i -A N )x + (b i -b N )
. By denoting B(x) the matrix whose columns are the B i (x)'s, i = 1 • • • , N -1, we finally obtain:

ẋ = A(λ e )(x -x e ) + B(x)(λ -λ e ) [1,N -1] .
Consider the Lyapunov function

V (x, x e ) = (x -x e ) T P (x -x e ),
where P 0 satisfies the Lyapunov inequality A(λ e ) T P + P A(λ e ) + Q ≺ 0, for a given Q 0. Denote by y(x, x e ) = B T (x)P (x -x e ) and let us define the weight matrix

K = diag(k i , i = 1, • • • , N -1)
for some chosen positive real numbers k i > 0. By a derivation along the trajectory of V (x, x e ) in the direction defined by the control λ [1,N -1] (x, x e ) = λ e [1,N -1] -Ky(x, x e ), one can get the result:

V (x, x e ) =2(x -x e ) T P (A(λ e )(x -x e ) -B(x)Ky(x, x e )) ≤ -(x -x e ) T Q(x -x e ) -2y(x, x e ) T Ky(x, x e ) <0 if x -x e = 0. (4)
As the control domain Λ is bounded, the embedded control of Proposition 1 must be limited by a saturation function as follows:

Definition 3 For a given control of the form λ = λ e +δλ with λ e ∈ Λ and i∈K δλ i = 0, its saturation denoted by sat(λ), is defined by its projection in the direction δλ on Λ:

sat(λ) = P roj(λ; δλ) = λ e + αδλ (5) where α = max{α ∈ [0 1] : λ e + αδλ ∈ Λ}.
Lemma 1 The values of α are determined by the following relations: α = min j∈K α j and

α j =        min 1, 1-λ e j δλj if δλ j > 0 min 1, -λ e j δλj if δλ j < 0 1 if δλ j = 0 PROOF.
As λ e ∈ Λ and as i∈K δλ i = 0, the sum λ = λ e + δλ is inside the set Λ if the components of δλ satisfy for all i ∈ K,

1 -λ e i ≥ δλ i ≥ -λ e i .
3 Following the sign of δλ i , we deduce the following conditions, for all i ∈ K:

1 ≤ -λ e i δλ i if δλ i < 0 and 1 ≤ 1 -λ e i δλ i if δλ i > 0.
If there exists at least one subscript i such that the above relations are not satisfied, then there exists a reduction ratio α, 0 < α < 1 such that these relations are satisfied when replacing δλ by αδλ. The greater reduction ratio is obviously given by α = min i α i where

α i =        min 1, 1-λ e i δλi if δλ i > 0 min 1, -λ e i δλi if δλ i < 0 1 if δλ i = 0
We are now in position to state the main result of this section.

Theorem 2 Consider the dynamical system (2). The equilibrium point corresponding to the pair (x e , λ e ) is globally asymptotically stable under the state feedback control:

sat(λ(x)) = λ e [1,N -1] -αKy(x, x e ), λ e N + 1 T αKy(x, x e ), (6)
with sat(•) and α given by ( 5) and the control λ provided by Proposition 1.

PROOF. The proof can be easily obtained from the expression of the derivative of V and noticing that the function sat does not modify the sign of the derivative by substituting in (4) the term -2y(x, x e ) T Ky(x, x e ) by -2αy(x, x e ) T Ky(x, x e ).

A key assumption behind the state feedback control laws (3) and ( 6) is the knowledge of the pair (x e , λ e ) which allows to determine the feedback law in real time. A question of practical interest which one may ask is whether these control laws can be used in the case where the affine system under interest is subject to parameter variations or uncertainties. We answer this question in the next section.

Adaptive stabilization

Let a parameter dependent switched affine system described by:

ẋ(t) = A σ x(t) + b σ + G σ p y(t) = Cx (7)
where the matrices A i , G i and the vectors b i , i ∈ K, are constant and given, x ∈ R n is the state, y ∈ R r is a measured output, σ : R n → K is the switching control and p ∈ R m is a constant ( ṗ(t) = 0) but unknown parameter. In this setting, we assume that the dependency of the system with respect to the unknown parameter p is linear. We show in the section dedicated to the application the interest of this modeling in the case of power converters. Now, we assume that only a part of the state has to be controlled to some prescribed values. Without loss of generality, we denote this part by x 2 ∈ R n2 . The remaining components of the state are denoted x 1 ∈ R n1 and are such that x = (x 1 , x 2 ). This assumption is necessary in many applications and is related to the degrees of freedom provided by the control λ to fix the equilibrium by the relation

x e = -A(λ) -1 [b(λ) + G(λ)p] for a given p.
Thus, the dimension of x 2 is at most the number of free control variables, here N -1.

Parametrization of the equilibrium point

A non trivial difficulty in the case of parameter dependent switched affine systems is related to the definition of the equilibrium point which is a priori unknown. To define the set of admissible equilibrium points in this context, we introduce the following definition.

Definition 4 For an arbitrary small ξ ≥ 0 fixed, we define the set:

Λ H (ξ) = {λ ∈ Λ : A(λ + ν) is Hurwitz, ∀ν ∈ B(0, ξ)},
where B(0, ξ) denoted a closed ball of radius ξ in R N centered on 0.

As Λ H is assumed to be non empty, there always exists ξ > 0 such that Λ H (ξ) = ∅. This set ensures, for any λ which belongs to the closure of Λ H (ξ) in Λ, that the matrix A(λ) is Hurwitz. In the sequel, it is useful in order to guarantee a bound on the decay rate of the Lyapunov function. One can notice that Λ H (ξ) ⊆ Λ H (0) ⊆ Λ.

The set of admissible equilibrium points is defined as follows:

Definition 5 For ξ ≥ 0 fixed, we define the set of the equilibrium points as:

X e (ξ) = {x e ∈ R n : x e = -A(λ) -1 [b(λ) + G(λ)p] , (p, λ) ∈ R m × Λ H (ξ)}. Definition 6 The restriction of X e (ξ) to x 1 (respec- tively x 2 ) denoted by X e (ξ)| x1 (respectively X e (ξ)| x2 ) is defined by the set X e (ξ)| x1 = {x e 1 : x e = (x e 1 , x e 2 ) ∈ X e (ξ)}. (respectively X e (ξ)| x2 = {x e 2 : x e = (x e 1 , x e 2 ) ∈ X e (ξ)}).
Consider the C ∞ -class function ψ defined by:

ψ : Λ H (ξ) × X e (ξ)| x1 × X e (ξ)| x2 × R m → R n (λ, x 1 , x 2 , p) → A(λ)(x 1 , x 2 ) + b(λ) + G(λ)p.
For any λ ∈ Λ H (ξ), any fixed p, as A(λ) is invertible, there exists a unique x such that:

ψ(λ, x 1 , x 2 , p) = 0.
For a given (x e 2 , p), we assume that there exists a parametrization of x e 1 and λ e as follows.

Assumption 1 For all x e 2 ∈ X e (ξ)| x2 , p, there exists a unique C 1 -class function φ = (φ 1 , φ 2 ) defined by:

φ 1 : X e (ξ)| x2 × R m → X e (ξ)| x1 (x e 2 , p) → x e 1 = φ 1 (x e 2 , p) (8) φ 2 : X e (ξ)| x2 × R m → Λ H (ξ) (x e 2 , p) → λ e = φ 2 (x e 2 , p) (9)
fulfilling the relation ψ(λ e , x e 1 , x e 2 , p) = 0.

Moreover, we assume that there exists a constant η(x e 2 , ξ) > 0 such that:

∂φ 1 (x e 2 , •) ∂p ∞ ≤ η(x e 2 , ξ). ( 10
)
To overcome the fact that the equilibrium point depends on the parameter p which is unknown a priori, the control method we propose, uses an observer to estimate this equilibrium point in real time. This is explained in the next subsection.

Parameter estimation and reference point update

Here we present a switched observer which allows to estimate the state and the unknown parameter z = (x, p). This switched observer will be used in the stabilizing control strategy proposed in this paper and it is given by: ż

= Ãσ ẑ + bσ + L σ (y -ŷ) ŷ = C ẑ , (11)
where Ãi =

A i G i 0 0 , bi = b i 0 , with L i = L i L i and C = C 0 .
Let the error between z and ẑ be e(t) = z(t) -ẑ(t). Its dynamics is given by:

ė = ( Ãσ -L σ C)e. ( 12
)
The gains L i are designed to ensure that lim t→∞ (e(t) = z(t) -ẑ(t)) = 0 for any switching law σ and this can be done using the following classical result [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Proposition 3 If there exist a matrix P obs 0, matrices R i and scalars γ i > 0 satisfying for all i ∈ K :

ÃT i P obs + P obs Ãi -CT R T i -R i C + γ i P obs < 0 i = 1, 2, . . . , N
then the observer gains L i = P -1 obs R i ensure that the error dynamics (12) is exponentially globally stable for any switching law σ and it is characterized by a decay rate at least equal to γ = min

i∈K γ i .
PROOF. The proof is obvious from [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]) (chapter 5, section 5.1) Remark 1 The observer design is explained in the case of switched affine systems of the form (1). The same can be done for systems of the form (2) and one obtains a polytopic observer which is a relaxed version of the switched one. One can now see the important role played by the parametrization (8)-( 9) in the approach we propose. Indeed, it allows to estimate the reference point based on an estimation p of p and a given reference x e 2 for x 2 as follows:

xe 1 = φ 1 (x e 2 , p) (13) λe = φ 2 (x e 2 , p). ( 14
)
From now, we will refer to this estimation by xe = (x e 1 , x e 2 ). The dynamics of xe 1 for x e 2 fixed is :

ẋe 1 = ∂φ 1 ∂p ṗ = ∂φ 1 ∂p ( ṗ -ṗ).
Taking into account ṗ = 0 and replacing ṗ by its expression (11), we have:

ẋe 1 = ∂φ 1 ∂p L σ Ce (15) ẋe 2 = 0. ( 16
)

Adaptive stabilizing control laws

We are now ready to present the main result of this paper. The following assumptions will be used.

Assumption 2 For all λ ∈ Λ such that A(λ) is Hurwitz, there exists at least one subscript i such that λ i = 0 and A i is Hurwitz.

Assumption 3 Assume there exists a matrix P 0 such that

A T i P + P A i + α i P ≤ 0 i = 1, . . . , N (17)
with α i = α i > 0 if A i is Hurwitz α i = 0 if not
Assumptions 2 and 3 allow to impose a performance constraint and in particular to specify a decay rate index. The switched and adaptive stabilizing control law is given in the following Theorem.

Theorem 4 Consider the switched affine system (7) with unknown parameters p and let x e = (x e 1 , x e 2 ) with x e 1 = φ 1 (x e 2 , p). Assume that Proposition 3 is satisfied. Under assumptions 1, 2 and 3, for all x e 2 ∈ X e (ξ)| x2 with ξ > 0, the feedback law defined by

σ * (x, xe , p) ∈ arg min i∈K (x -xe ) T P (A i x + b i + G i p) (18)
ensures global asymptotic stability of the equilibrium point x e = (φ 1 (x e 2 , p), x e 2 ).

PROOF. For a given x e 2 and an estimated value p of p, denote by w = (x -xe ) (see equation ( 13)). For λe = φ 2 (x e 2 , p) ∈ Λ H (ξ) (see equation ( 14)), we have by ( 17) :

2w T P A( λe )w + α( λe )w T P w ≤ 0 (19) with α( λe ) = N i=1 λe i α i .
By assumption 1, as A( λe )x e + b( λe ) + G( λe )p = 0, it follows :

2w T P (A( λe )x + b( λe ) + G( λe )p) ≤ -α( λe )w T P w N i=1 λe i 2w T P (A i x + b i + G i p) ≤ -α( λe )w T P w As α( λe ) = N i=1
λe i α i cannot be zero by Assumption 2, we can conclude that the control law (18) leads to:

2w T P (A σ * x + b σ * + G σ * p) ≤ -α( λe )w T P w (20) < 0, if w = 0. ( 21
)
Consider the Lyapunov function V (x, x e ) = (x -x e ) T P (x -x e ).

Using (x, xe ) instead of (x, x e ) and taking into account the dynamics of xe (see ( 15) and ( 16)), the directional derivative in a direction specified by λ is given by:

V (x, xe ; λ) = 2w T P (A(λ)x + b(λ) + G(λ)p +(L(λ) -ΦL(λ)) Ce) with Φ = ∂φ1 ∂p 0 .
Therefore, adding to both sides of the inequality (20) the term:

2w T P (L σ * -ΦL σ * ) Ce,
we obtain:

V (x, xe ; σ * ) ≤ -α( λe )w T P w + 2w T P (L σ * -ΦL σ * ) Ce
As e converges exponentially to 0 for any switching rule, and thus, particularly for the switching rules (18), with γ = min i γ i , there exists K > 0, e(t) ≤ Ke -γt/2 e 0 .

Hence:

V (x, xe , σ * ) ≤ -α( λe )w T P w + 2β max w

×( L σ * + ∂φ1(x e 2 ,•) ∂p ∞ L σ * )
× C Ke -γt/2 e 0 with β max the maximal eigenvalue of P . A sufficient condition to guarantee

V (x, xe ; σ * ) ≤ - α( λe ) 2 w T P w < 0 is w >4 β max α min β min max i ( L i + ∂φ 1 (x e 2 , •) ∂p ∞ L i ) × C e 0 Ke -γt/2 (22)
where β min is the minimum eigenvalue of P and

α min = min λ∈Λ H (ξ) α(λ).
Note that α min > 0 for ξ > 0 under Assumption 2 and by definition of Λ H (ξ).

As there exists η(x e 2 , ξ) > 0 such that

∂φ 1 (x e 2 , •) ∂p ∞ ≤ η(x e 2 , ξ)
we can conclude that there exists a constant M (x e 2 , ξ) > 0 such that if w > M (x e 2 , ξ)e -γt/2 e 0 then V (x, xe ; σ * ) ≤ -α( λe ) 2 w T P w < 0.

As we can claim that ∀e 0 , ∀ > 0, ∃t 1 , such that M e -γt/2 e 0 < , ∀t > t 1 , we necessarily have V (x, xe ; σ * ) ≤ -αmin 2 w T P w < 0, for all w such that w > and t > t 1 . This means that for any w(0), there exists t 2 such that w(t) ≤ for all t > t 2 . We can conclude that w converges asymptotically to 0. As e converges also to zero, we have

x → x → xe = xe 1 x e 2 with xe 1 = φ 1 (x e 2 , p) because p → p.
Remark 2 If Assumptions 2 and 3 are not satisfied (for example if the matrices

A i , i = 1 • • • , N
are not Hurwitz then it is possible to weaken these assumptions by an alternative design of P . Assume that there exists a polytopic approximation

Λ poly := {ζ ∈ R m | ∀i ∈ {1, • • • , m}, ζ i ≥ 0, m i=1 ζ i = 1} of the set Λ H (ξ) such that : a) Λ H (ξ) ⊆ Λ poly ⊂ Λ, b) A(υ i ), i = 1, • • • , m are Hurwitz,
where the υ i 's, i = {1, • • • , m} denote the vertices of Λ poly (as the vertices υ

i , i = 1, • • • , m of Λ poly belong to Λ, there exists λ(υ i ) ∈ Λ, A(υ i ) = j=1,...,N λ j A j , i = 1, • • • , m and the expression A(υ i ) is well defined).
Now, if there exists a matrix P 0 such that

A(υ i ) T P + P A(υ i ) + α υi P ≤ 0, i = 1, • • • , m
for given α υi > 0 then using this matrix P , the same steps of the proof of Theorem 4 can be applied to obtain the global stability property. The only difference concerns equation (19) where α( λe ) must be replaced by α min = min i α υi (since λe ∈ Λ poly ).

We give in the next Theorem the expression of the adaptive stabilizing control for the polytopic version of (7) defined by:

ẋ(t) = A(λ)x(t) + b(λ) + G(λ)p (23)
and where the control is now the adaptive and embedded control λ(t) ∈ Λ.

Theorem 5 Consider the relaxed affine system (23) with unknown parameters p and let x e = (x e 1 , x e 2 ) with x e 1 = φ 1 (x e 2 , p). Assume that Proposition 3 is satisfied. Under assumptions 1, 2 and 3, for all x e 2 ∈ X e (ξ)| x2 with ξ > 0, the feedback law sat(λ * ) (see (5) for definition of sat) given by:

sat(λ * (x, p)) = λe [1,N -1] -αKy(x, p), λe N + 1 T αKy(x, p),
with y(x, p) = B T (x, p)P (x -xe ), the columns of matrix B(x, p) given by B PROOF. We only provide a sketch of the proof since the steps are similar to the proof of Theorem 4. Applying the same type of calculus used in the proof of Proposition 1 allows to establish the dynamics of w = x -xe when a control λ is applied:

i (x, p) = (A i -A N )x+(b i -b N )+(G i - G N )p, for i = 1, • • • , N -
ẇ = A( λe )(x -xe ) + B(x, p)(λ -λe ) [1 N -1] + D(λ) Ce with columns of matrix B(x, p) given by B i (x, p) = (A i - A N )x + (b i -b N ) + (G i -G N )p for i = 1, • • • , N -1, D(λ) = N i=1 λ i (L i -ΦL i ) and Φ = ∂φ1 ∂p 0 .
Consider once again the Lyapunov function V (x, x e ) = (x -x e ) T P (x -x e ) using (x, xe ) instead of (x, x e ). The directional derivative of V in direction provided by the proposed feedback sat(λ * ) leads to: V (x, xe ; sat(λ * )) = 2w T P A( λe )w -2αy T (x, p) Ky(x, p) + 2w T P D(sat(λ * )) Ce with y(x, p) = B T (x, p)P (x -xe ) and where α is provided by the function sat(λ * ) (see ( 5)).

As D(sat(λ * )) ≤ max i ( L i + ∂φ1(x e 2 ,•) ∂p ∞ L i ), it
appears that equation ( 22) is easily achieved and the proof follows the same steps as in Theorem 4.

The conditions on φ in Assumption 1 can be relaxed as follows.

Assumption 4 We assume that the application φ = (φ 1 , φ 2 ) in Assumption 1 is no more valid for all p but only for all p in a closed convex domain, denoted P(x 2 , ξ) function of x 2 and ξ. We also assume that the nominal value of the parameter p belongs to P(x e 2 , ξ) (where P stands for the interior of P) and λ = φ 2 (x e 2 , p) ∈ Λ H (ξ).

Proposition 6 Under assumption 4, Theorem 4 and Theorem 5 hold true if the estimated values (x e 1 , λe ) are such that

xe 1 = φ 1 (x e 2 , Πp) λe = φ 2 (x e 2 , Πp)
where Πp is the orthogonal projection of p on P(x e 2 , ξ).

PROOF. As P(x e 2 , ξ) is closed and convex, the orthogonal projection Πp of p on P(x e 2 , ξ) is continuous. We consider the function φ and we use the projection Π such that p → φ i (x e 2 , Πp), i = 1, 2 for x e 2 fixed. As λe = φ 2 (x e 2 , Πp) ∈ Λ H (ξ), we see that

xe 1 = φ 1 (x e 2 , Πp) = -C x1 A( λe ) -1 [b( λe ) + G( λe )Πp]
where the matrix C x1 is such that x 1 = C x1 (x 1 , x 2 ). The last right term is well defined since A( λe ) is invertible. Thus, the projection Π allows the computation of an estimated point (x e 1 , λe ) for all t. Of course, the obtained estimated operating point xe = (x e 1 , x e 2 ) with a projection of p is not an equilibrium point of the system when p / ∈ P(x e 2 , ξ).

As p ∈ P(x e 2 , ξ) where P stands for the interior of P, the convergence to zero of the error e for any switching rule and the continuity of φ ensure that there exists a time t 1 (e 0 ) such that λe = φ 2 (x e 2 , p) ∈ Λ H (ξ) and p ∈ P(x e 2 , ξ) for all t > t 1 as soon as p is close enough to p. For all t > t 1 , Πp = p and one concludes the proof following similar steps as in the proof of Theoreom 4 and Theoreom 5.

Application to DC-DC power converters

The theoretical approach we propose is applied to a DC-DC power converter subject to load and input variations. The so-called Flyback converter is depicted in Fig. 1 and it is composed of passive components (a resistor R, an inductor L, a capacitor C), a transformer and two types of switches: a controlled switch (transistor S) and an uncontrolled switch (diode D). The behavior of the Flyback converter implies switching among three different modes summarized in Table 1 and the operating domain of the converter is the first orthant. The Mode S = 1 and D = 1 does not appear for physical reasons. Indeed, if D = 1, the current i 2 passing through the diode is positive and this means that the current i 1 is negative. As S = 1 means that the current i 1 is positive this means that it is impossible to have both S = 1 and D = 1. The switching rules between the three modes are summarized in Fig. We recall that only Mode 1 and Mode 2 can be directly controlled. It is possible to have Mode 3 activated for physical reasons but its activation cannot be controlled.

+ - V e L + - V L S i L i 1 i 2 + - D C + - v C i C R + - v R i R Fig. 1. A Flyback converter
The design of a stabilizing switching control law that takes into account these three modes is very challenging. We first explain how the approach we propose in this paper can be applied when the converter works in CCM. The DCM phase is discussed at the end of this section.

The Flyback converter in the CCM mode is a switched affine system given by:

ẋ = A σ x + B σ u,
where x(t) ∈ R n is the state, u(t) ∈ R m is the input considered as fixed (u(t) = V e ) and cannot be used as a control variable. Therefore, denoting b σ = B σ u allows to use the framework developed in this paper. The control is the switching rule σ : R n → K which indicates the active Mode at each time instant. The state vector x T = [i L , v C ] is composed by i L (the inductor current) and v C (the capacitor voltage). Based on Fig. 1 and using Kirchhoff's laws, the sets

A = {A 1 , • • • , A N } and B = {B 1 , • • • , B N } with N = 2
are given by:

A 1 = 0 0 0 -1 RC , A 2 = 0 -nt L nt C -1 RC , B 1 = 1 L 0 , B 2 = 0 0 .
The interest of considering unknown parameters appears when robustness issues are formulated. For example, a lack of measurement on the input voltage and the output load can be represented by two parameters that can be linked to the resistor R and the voltage u as shown in Fig. 3 and4. The voltage parameter p 2 allows to identify the input error directly as explained in (25), and the current parameter p 1 allows to identify the output load as explained in (24). To model the lack of measurement on the input voltage and the output load, the parameters given by ( 24) and ( 25) are virtually added to the Flyback converter as shown in Fig. 5. This modified converter is a parameter dependent switched affine system given by:

+ - v p 1 R i = v R = p 1 + v R R = 1 1 R + p 1 v (24)
ẋ = A σ x + B σ u + G σ p, + - u + - p 2 L - + V L ẋ1 = v L L = u-p2 L û = u -p 2 (25)
Fig. 4. Parameter p2 to estimate û

+ - u + - p 2 L + - V L i L S i 1 i 2 + - D C + - V C i C p 1 R + - V R i R
Fig. 5. Flyback converter with parameters estimation with p the vector of unknown parameters and

G 1 = 0 -1 L -1 C 0 , G 2 = 0 -1 L -1 C 0 .
The application of the approach proposed in this paper and in particular the use of the observer allows to estimate the unknown parameters and gives a key information related to the steady state point which is not known a priori. This is clearly of practical interest and relevant for DC-DC power converters when the load or the input voltage are subject to variations.

Simulation results in CCM

The Flyback converter described by Fig. 5 depends on five parameters that, for simulation purpose, have been considered with the following nominal values : V e = 28V , R = 75Ω, L = 200µH, C = 2.6µF and n t = 2. Let the target point be x e 2 = 15V . We first present the simulation results in the case where the Flyback converter works with its nominal values to make a comparison between the switched based control σ of Theorem 4 and the embedded based control λ of Theorem 5. With α 1 = 0, α 2 = 772 the solution of the LMI conditions ( 17) is:

P = 3.2170 -0.0032 -0.0032 0.0418 .
The additional parameter k i in the expression λ i = λ e i -k i y i can be used to meet some performance requirements. Fig. 6 shows the transient for three values of this parameter k 1 = 3.10 -7 , 1.10 -6 , 7.10 -6 and also the transient corresponding to the switched case. These simulations are obtained using a control λ and a Pulse-width modulation with a frequency f s = 10 M Hz.

The switched control is simulated with a sampling time T e = 1 2fs in order to have for both σ and λ a similar number of switchings.

Compared to the switched based control, a better transient can be obtained using the control λ and an appropriate choice of k 1 . The ability to tune the smoothness of the embedded control using the gain k i is clearly of practical interest to meet performance requirements and not only stability properties. One can notice that large values for k i lead to a saturation of the control and tends to recover the switched case. Conversely, very small values correspond to an open loop control λ(t) ≈ λ e for all t. Obviously, the later is not suitable. To ease the presentation and comparison with the adaptive strategy, we decided to keep in the sequel only the simulation results of the embedded control λ with k 1 = 1.10 -6 . Now in order to show the effectiveness of the proposed adaptive design, we assume that R and V e may change and we do not measure these parameters. We use as a scenario a piecewise constant function with variations between 50Ω and 100Ω for the load, and variations between 20V and 40V for the input. First, let us show that all the required assumptions are satisfied. Assumption 2 is obvious. To show that Assumptions 4 is also satisfied, notice that for a given x e 2 and p, the functions φ i , i = 1, 2 are provided by the relations:

x e 1 = φ 1 (x e 2 , p) = (x e 2 + Rp 1 )(V e + nx e 2 ) nR(p 2 -V e )
and

λ e = φ 2 (x e 2 , p) =    λ e 1 = p2+nx e 2 Ve+nx e 2 λ e 2 = 1 -λ e 1 ( 26
)
Note that the above relations for φ i , i = 1, 2 are deduced from equation 0 = A(λ e )x e + b(λ e ) + G(λ e )p assuming the pair (x e 1 , λ e ) as the unknowns and (p, x e 2 ) are fixed. For a given ξ > 0, the set Λ H (ξ) is determined by

λ 1 ∈ [0 1 -ξ] λ 2 = 1 -λ 1
since only A 1 is not Hurwitz. Then from (26), it is simple to check that if P(x e 2 , ξ) is defined by:

P(x e 2 , ξ) = {p ∈ R 2 : |p 1 | ≤ p 1max , -nx e 2 ≤ p 2 ≤ (1 -ξ)V e -ξnx e 2 }
where p 1max is chosen arbitrarily large, then λ e ∈ Λ H (ξ) and x e 1 is well defined for this set P(x e 2 , ξ). Moreover, as ∂φ1 ∂p is given by:

∂φ 1 ∂p =   Ve+nx e 2 n(p2-Ve) - (x e 2 +Rp1)(V +nx e 2 ) nR(p2-Ve) 2   ,
it is well defined on the set P(x e 2 , ξ) and ( 10) is clearly satisfied by compacity of P(x e 2 , ξ). Hence, Assumption 4 is satisfied for our application.

We solve the LMI conditions of Proposition 3 and we obtain the following gains L i with γ 1 = γ 2 = 3.3 × 10 4 :

Li = 10 3        362 20 20 1589 243 -3152 -3152 -243        , i ∈ K.
Applying the adaptive control of Theorem 5, using Assumption 4 instead of Assumption 1 and thanks to Proposition 6, we obtain the results depicted in Fig. 7.

We start the scenario of the simulation with the nominal values (ie V e = u = 28V and R = 75Ω) to see the behavior of the closed loop system in the case where all the parameters are perfectly known. The variations of R and V e are introduced after 0.5 ms. From 0 to 0.5ms, the reference x e 2 is reached for both the adaptive control (Theorem 5) and the non adaptive one (the embedded control λ with k 1 = 1.10 -6 ). After 0.5 ms, stability is preserved by these control laws in the presence of load and input voltage variations. However, a huge steady state error appears when using the non adaptive law. Using the control proposed in Theorem 5, the parameter p converges to a new value each time R or V e changes and these changes have limited effect on the controlled output x 2 . The adaptive embedded control λ rejects nicely the perturbations induced by the parameter variations and allows the controlled output x 2 to follow its reference while the current x 1 adapts its value to support these variations.

DCM mode case

The proof of global asymptotic stability of the proposed adaptive control laws (see Theorem 4 or Theorem 5) assumes that the DC-DC converter does not operate in the DCM mode (The uncontrolled Mode 3 is never activated). This means that the control law is established with K restricted to the set {1, 2} instead of {1, 2, 3}. To prove stability when the converter enters in DCM mode, one has to take into account the fact that the uncontrolled mode (Mode 3) may appear when x 1 = 0, eventually many times, in the switching sequence. This mode is characterized by:

A 3 = 0 0 0 -1 RC , B 3 = 0 0 , G 3 = 0 0 -1 C 0
Denote the derivative of V (x) in the direction d by: V (x; d) = 2(x -x e ) T P d.

and by

f i (x) = A i x + B i V e + G i p, i = 1, 2, 3.
For the case of switched based control as the one given by (3), we refer to two studies in the literature taking into account the DCM mode. The first one is discussed in (Beneux et al. 2017b) where we prove that Mode 3 may only occur along the line ∆ := {x : x 1 = 0} when V (x; f 3 (x)) < 0 which preserves the stability property. The second study in presented in [START_REF] Theunisse | Robust global stabilization of the DC-DC boost converter via hybrid control[END_REF] where it is shown that a sufficient condition for the stability in DCM can be obtained by defining a Krasovskii regularization F 1 (x) = {f 1 (x)} and F 2 (x) = co{f 2 (x), f 3 (x)} and fulfilling the following relation: min

i=1,2 ( max ξ∈Fi(x) V (x; ξ)) < 0 (27)
along the line ∆.

Here, we focus on the embedded based control of Theorem 5 and we show that stability property is also preserved when the DCM mode occurs. To this end, we first recall the following result.

Proposition 7 When x ∈ ∆, the vector field f 3 (x) satisfies:

f 3 (x) = γf 1 (x) + (1 -γ)f 2 (x)
with γ(x 2 ) = nx2+p2 nx2+Ve (28)

where f i (x) = A i x + B i u + G i p, i = 1, 2, 3. Moreover, the function γ is a monotonically increasing function of x 2 and upper bounded by the value 1.

PROOF. The relation is easy to check, see (Beneux et al. 2017b).

Let Λ + (x) = {λ ∈ Λ : [1 0]f (x, λ) ≥ 0 with x ∈ ∆}, where f (x, λ) = λ 1 f 1 (x) + λ 2 f 2 (x). This set defines the set of admissible controls along ∆ (any λ / ∈ Λ + (x) is such that ẋ1 < 0 and leads to x 1 < 0 which is not allowed by the diode). It is simple to show using Proposition 7 that the set Λ + (x) is characterized by:

λ 1 ∈ [max(0, γ(x 2 )) 1] λ 2 = 1 -λ 1
We are now in position to state the following result. V (x; ξ)) = min i=1,2 ( V (x; f i (x))) < 0. (30) Note that the above relations are always satisfied: Eq. ( 29) as proved in (Beneux et al. 2017b) since if V (x; f 3 (x)) ≥ 0 then V (x; f 1 (x)) < 0 and Eq. ( 30) by definition of V (it involves only modes 1 and 2).

Denote by η the embedded control (η = sat(λ) obtained from ( 6)). Following the value of γ(x 2 ), we have two cases.

• If γ(x 2 ) < 0, the control domain is not modified since Λ + (x) = Λ(x) and the diode cannot be blocking when applying the embedded control λ. Thus, Mode 3 will not occur and by Theorem 5, V (x; f η (x)) < 0 where f η (x) = i=1,2 η i f i (x). • If γ(x 2 ) ≥ 0, Mode 3 occurs only if η / ∈ Λ + (x), or equivalently, only if the first component of η satisfies 0 ≤ η 1 < γ(x 2 ) (< 1) Therefore, if Mode 3 occurs, two cases must be distinguished from the relation (29): a) If S = 1 where S = arg min i=1,3 ( V (x; f i (x))), we have clearly V (x; f 1 (x)) ≤ V (x; f 3 (x)) ≤ V (x; f η (x)) < 0.

The inequalities are obtained by linearity of the derivative since 1 ≥ γ(x 2 ) ≥ η 1 and using (28). The strict inequality is obtained from the global stability property of the control η. b) If S = 3 then the relation (29) implies V (x; f 3 (x)) < 0.

As a conclusion, in both cases V (x; f 3 (x)) < 0 and the embedded control strategy of Theorem 5 preserves stability when the DCM occurs.

Finally, we illustrate this discussion in Figure 8 with a step variation leading to a transient in DCM mode. As expected, Mode 3 appears when the current x 1 vanishes, and stability property is preserved.

Conclusion

In this paper, it has been shown that an adaptive and switched-embedded based control can be designed to stabilize switched affine systems with unknown parameters.

The results have been applied to a DC/DC Flyback converter where some of the parameters are not known (in practice, the load and/or the input voltage). The proposed simulations confirm the interest of our approach. Moreover, it has also been shown that stability of the adaptive and switched control is preserved when the discontinuous conduction mode occurs.

  1 and P satisfying (17), ensures global asymptotic stability of the equilibrium point x e .

Fig. 2 .

 2 Fig. 2. Flyback modes and switching rules during the cycle, the converter works in CCM (Continuous Conduction Mode). If Mode 3 is active, the system works in DCM (Discontinuous Conduction Mode) and this means that all the energy stored during Mode 1 in the inductor is transferred to the capacitor during Mode 2.

Fig. 3 .

 3 Fig. 3. Parameter p1 to estimate R

Fig. 6 .

 6 Fig. 6. Start-up transient: Plot 1 shows the current x1, Plot 2 shows the voltage x2. Plot 3 shows the control (in average for the switched based control case). Switched based control (Blue); Embedded based control (Purple, k1 = 3.10 -7 )(Red, k1 = 1.10 -6 )(Yellow, k1 = 7.10 -6 ). Time is given in (ms).

Fig. 7 .

 7 Fig.7. Plot 1 shows respectively the current reference (Blue), the current x1(Yellow) when the adaptive embedded control λ is used, its estimation x1 (Red) and the current x1(Purple) when only the embedded based control λ is used without load and input estimation. Time is given in (ms). Plot 2 shows the voltage x2 using the same convention. Plot 3 shows the variation of the estimated value p1 (respectively p2) of parameter p1 (p2). Finally, Plot 4 shows the load R and input Ve variations.

Proposition 8

 8 The embedded control preserves global asymptotic stability when Mode 3 is activated.PROOF. If γ(x 2 ) ≥ 0 then, by linearity of the derivative and using (28), the relation (27) reduces to: min

Fig. 8 .

 8 Fig. 8. DCM transient. Plot 1 shows the current x1 (yellow), Plot 2 shows the voltage x2 (yellow) when a step variation of the reference (blue) leads to DCM mode.

Table 1

 1

	Possible mode of functioning
	Mode S D
	1	1	0
	2	0	1
	3	0	0


Artur Avila

Thomas Roblin
email: thomas.roblin@upmc.fr

UNIFORM EXPONENTIAL GROWTH FOR SOME SL(2, R) MATRIX PRODUCTS

Given a hyperbolic matrix H ∈ SL(2, R), we prove that for almost every R ∈ SL(2, R), any product of length n of H and R grows exponentially fast with n provided the matrix R occurs less than o(

Introduction

For t, θ ∈ R, let H = H(t) be the hyperbolic matrix exp 1 2 t 0 0 exp -1 2 t

and let R = R(θ) be the rotation matrix cos θ sin θ sin θ cos θ . For a finite word w = w n . . . w 1 on the symbols H and R, we let |w| denote its length and we let m(w) denote the number of occurrences of R in w. For any such word, and for any choice of parameters t and θ, we let A w (t, θ) denote the corresponding matrix product in SL(2, R), and denote by A w (t, θ) its norm. By the Oseledets Theorem, for a typical large word w on H and R, the size of the matrix product is given up to subexponential error, by e L(t,θ)|w| , where L(t, θ) is the Lyapunov exponent of the Bernoulli product giving equal probabilities for H and R. By Furstenberg's Theorem (cf [START_REF] Furstenberg | Harry Noncommuting random products[END_REF]), L(t, θ) > 0 unless t = 0 or θ = π/2 mod π, thus hyperbolic behavior prevails under a very mild "transversality condition" on the pair (H, R).

Here we are interested in the following subtler question: Assuming some stronger transversality condition on the pair (H, R), can one ensure hyperbolic behavior just by limiting the frequency of rotation elements in the word? A basic question in this direction, raised by Bochi and Fayad in [START_REF] Bochi | Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, R) cocycles[END_REF], is whether for almost every t and θ, a condition of the type C(t, θ)m(w) ≤ |w| implies that A w (t, θ) grows exponentially. While this question is still open, in [START_REF] Fayad | Exponential growth of product of matrices in SL(2,R)[END_REF], Fayad and Krikorian showed that for almost every t and θ, one has exponential growth provided m(w) ≤ |w| α with 0 < α < 1/2. Our goal in this paper will be to show that the weaker condition C(t, θ)m(w) log m(w) log log m(w) ≤ |w| suffices.

Theorem 1. For every t > 0, 0 < γ < t 2 and almost every θ ∈ R, there exists ǫ > 0 such that for any word w on H and R, if m(w) ≤ ǫ|w|(log |w| log log |w|) -1 , then the spectral radius of A w (t, θ) is at least e |w|γ .

In fact, our proof allows us to take for R a general matrix of SL(2, R), presented in its Cartan decomposition form, as follows.

This research was partially conducted during the period A.A. served as a Clay Research Fellow.

Theorem 2. For every t > 0, s > 0, α ∈ R, 0 < γ < t 2 and almost every θ ∈ R, there exists ǫ > 0 such that for any word w on H = H(t) and R = R(θ)H(s)R(α), if m(w) ≤ ǫ|w|(log |w| log log |w|) -1 , then the spectral radius of A w is at least e |w|γ .

Corollary. For every t > 0, 0 < γ < t 2 and almost every R ∈ SL(2, R) with respect to the Haar measure, there exists ǫ > 0 such that for any word w on H = H(t) and R, if m(w) ≤ ǫ|w|(log |w| log log |w|) -1 , then the spectral radius of A w is at least e |w|γ .

Proof of the theorems

We now give a detailed proof of theorem 1. Then we shall indicate how theorem 2 is obtained following the same lines.

From now on we fix t > 0, and drop the dependence on t from the notation. For a given word w we shall use the notations

w [i j] = w j . . . w i for 1 ≤ i ≤ j ≤ |w|. We also let a w , b w , c w , d w : R → R be defined so that A w (θ) = a w (θ) b w (θ) c w (θ) d w (θ) .
Let us say that a function ψ :

Z + → R + is good if (1) ∀k, l ≥ 1, ψ(k) + ψ(l) ≤ ψ(k + l) -log 2.
We will mostly work with multiples (by reals greater than 1) of the functions

ψ 1 (m) = m(1 + log 2 m) and ψ 2 (m) = m(1 + log m)(1 +
log log max{e, m}) (with 0 log 0 = 0). Both ψ 1 and ψ 2 are easily seen to be good. Given a good function ψ and 0 < γ ≤ t 2 , for any word w of length n, we let

F w (ψ, γ) = F w be the set of all θ ∈ [0, π) such that log |a w [1 k] | ≥ kγ -ψ(m(w [1 k] )) and log |a w [k+1 n] | ≥ (n -k)γ -ψ(m(w [k+1 n] )) for all 0 < k < n, but log |a w | < nγ -ψ(m(w)). (2)
Notice that if F w is not empty, necessarily w 1 = w n = R. In view of (1), it follows that on the set F w , (3)

|a w | ≤ 1 2 |a w [1 k] a w [k+1 n] |, ∀ 0 < k < n.
Lemma 1. For every w we have, writing |w| = n and m(w) = m:

(4) |F w | ≤ 8n 2 e ψ([ m 2 ])+ψ(m-[ m 2 ])-ψ(m) .
Proof. Since a ω is in general a polynomial of degree m(ω) in cos θ, as is easily checked, the set F w is the union of at most 4nm intervals. Now, in order to bound the size of such an interval, we show that the derivative of a w with respect to θ at any (fixed) point of F w is not too small. Since the derivative of R(θ) is R( π 2 )R(θ), using the product rule, it is easy to derive the following formula for the derivative of a w :

(5)

a ′ w = k, w k =R c w [1 k] a w [k+1 n] -a w [1 k] b w [k+1 n] .
On the one hand, we have, for all 0 < k < n,

(6) a w = a w [1 k] a w [k+1 n] + c w [1 k] b w [k+1 n] .
In view of (3), this shows that

(7) 1 2 ≤ - c w [1 k] b w [k+1 n] a w [1 k] a w [k+1 n] ≤ 3 2 .
In particular, for each 0

< k < n, c w [1 k] a w [k+1 n] and -a w [1 k] b w [k+1 n]
have the same sign.

On the other hand, one easily sees that ∀1 < k < n, the upper left entry of the matrix A w

[k+1 n] R( π 2 )A w [1 k] is c w [1 k] a w [k+1 n] -a w [1 k] b w [k+1 n] = c w [1 k-1] a w [k n] - a w [1 k-1] b w [k n] if w k = R and c w [1 k] a w [k+1 n] -a w [1 k] b w [k+1 n] = e -t c w [1 k-1] a w [k n] - e t a w [1 k-1] b w [k n] if w k = H (indeed, R( π 2 )H(t) = H(-t)R( π 2 ) = H(t)H(-2t)R( π 2
)). After finite iteration, we deduce from these observations that the quantities c w

[1 k] a w [k+1 n] and -a w [1 k] b w [k+1 n]
for k varying from 1 to n -1 have all the same sign; among them, the summands in (5). Therefore, taking k with

w k = R so that m(w [1 k] ) = [ m 2 ] where m = m(w), we have |a ′ w | ≥ |c w [1 k] a w [k+1 n] | + |a w [1 k] b w [k+1 n] | ≥ 2|a w [1 k] a w [k+1 n] c w [1 k] b w [k+1 n] | 1 2 .
From ( 7) and ( 2), we get (at any point θ ∈ F w ):

|a ′ w | ≥ |a w [1 k] a w [k+1 n] | ≥ e n t
From the above minoration, we deduce that any interval in F w as defined by ( 2)

is of length less than 2e ψ([ m 2 ])+ψ(m-[ m 2 ])-ψ(m)
. Since F w is the union of at most 4nm such intervals, the result follows.

Lemma 2. If F w = ∅ then n ≤ m(1 + 1 t ψ(m)),
where n = |w| and m = m(w). We have

(9) a w = e r t 2 a w [1 k] a w [k+r+1 n] + e -r t 2 c w [1 k] b w [k+r+1 n] .
Observe that in general max(a

2 ω + c 2 ω , b 2 ω + d 2 ω ) ≤ e |ω|t , so that here |c w [1 k] b w [k+r+1 n] | ≤ e (n-r) t 2 .
From ( 1),( 2) and (9), we get

2e n t 2 -ψ(m) ≤ e n t 2 -ψ(m(w [1 k] ))-ψ(m(w [k+r+1 n] )) ≤ e r t 2 |a w [1 k] a w [k+r+1 n] | < e n t 2 -ψ(m) + e (n-2r) t 2 .
Hence rt < ψ(m), which combined with (8) gives the result.

From now on, let E(ψ, γ) denote the set of all θ ∈ [0, π) such that (10) log |a w (θ)| < |w|γψ(m(w)) for some word w.

Lemma 3. There exists some constant c > 0 such that |E(λψ

1 , t 2 )| = O λ≥1 (e -cλ ). Proof. Let E n = E n (λψ 1 , t 2 ) ⊂ E = E(λψ 1 , t
2 ) be the set of θ such that n is the minimal length of a word w such that (10) holds. Clearly E is the disjoint union of the E n 's and each E n is covered by the F w 's with |w| = n.

We then apply lemmas 1 and 2 to estimate |E n | for n ≥ 2 as follows:

(11)

|E n | ≤ |w|=n |F w | ≤ 8n 2 m n m e λ(ψ1(m-[ m 2 ])+ψ1([ m 2 ])-ψ1(m)) ,
where the sum runs over the 2 ≤ m ≤ n such that n ≤ m(1 + 1 t λψ 1 (m)), which implies n ≤ C 0 λm 2 log 2 m. Here and in the sequel, C 0 , C 1 , . . . stand for positive constants independent of m, n or λ.

For n = 1, notice that

E 1 = { θ | | cos θ| < e t 2 -λ }. It is readily seen that ∀m ≥ 2, ψ 1 (m -[ m 2 ]) + ψ 1 ([ m 2 ]) -ψ 1 (m) ≤ -C 1 m log m.
On the other hand, by the use of Stirling's formula, we find that (12) n m ≤ e m log n-m log m+C2m .

So, summing over n in (11) and then reversing the order of summation yields

|E| ≤ |E 1 | + m≥2 e (C3-C1λ)m log m n≤C0λm 2 log 2 m n (m+2)
≤ C 4 e -λ + m≥2 e (C5-C1λ)m log m+(m+3) log λ .

For large λ, this sum is finite and less than e -cλ .

Lemma 4. Let 0 < γ < t 2 . There exists some constant c > 0 such that |E(λψ 2 , γ)\ E(λψ 1 , t

2 )| = O λ≥1 (e -cλ ). Proof. We first notice that if F w (λψ 2 , γ) \ E(λψ 1 , t

2 ) = ∅, then λψ 1 (m(w)) ≥ ( t 2γ)|w|. Thus, proceeding as in the previous lemma, we get (even for n = 1)

|E n (λψ 2 , γ) \ E(λψ 1 , t 2 )| ≤ 8n 2 λψ1(m)≥( t 2 -γ)n m≥2 n m e λ(ψ2(m-[ m 2 ])+ψ2([ m 2 ])-ψ2(m)) .
Here

∀m ≥ 2, ψ 2 (m -[ m 2 ]) + ψ 2 ([ m 2
])ψ 2 (m) ≤ -C 6 m(1 + log log max{e, m}). Using again (12), we obtain

|E(λψ 2 , γ) \ E(λψ 1 , t 2 )| ≤ m≥2 e (C7-C6λ)m(1+log log max{e,m})-m log m n≤C8λm log 2 m n (m+2)
≤ m≥2 e (C9-C6λ)m(1+log log max{e,m})+(m+3) log λ .

We conclude as before.

The lemmata 3 and 4 show that for 0 < γ < t 2 , the sum λ∈N * |E(λψ 2 , γ)| converges. By the Borel-Cantelli lemma, we conclude that for almost every θ, there exists λ ≥ 1 such that for all word w, log |a w (θ)| ≥ |w|γλψ 2 (m(w)).

It follows that for almost every θ, if |w| is large and m(w) is much smaller than |w|(log |w| log log |w|) -1 , then 1 |w| log A w (θ) is close to t 2 , as well as 1 |w| 2 log A ww (θ) .

But

A ww (θ) -A w trA w + id = A w (θ) 2 -A w trA w + id = 0, since A w ∈ SL(2, R), which shows that 1 |w| log |trA w | is close to t 2 , yielding the estimate on the spectral radius in theorem 1.

In order to prove theorem 2 by the same method, we consider, instead of the words on H and R, words w = w n . . . w 1 on H(t), R(θ), H(s) and R(α) such that the last three ones always appear consecutively, except maybe at the ends of the word, and m(w) is now the number of these occuring in w. Then the proof goes the same way, notably the considerations of sign in lemma 1.

Proof.

  Let us fix some θ ∈ F w , and write w = w [k+r+1 n] H r w [1 k] with r maximal. Since w 1 = w n = R, as we have already observed, one has 0 < k < nr, m(w [1 k] ), m(w [k+r+1 n] ) ≥ 1

-ψ([ m 2 ])-ψ(m-[ m 2 ])


David Krejcirik

Julien Royer

David Krej

Č I Ří

SPECTRUM OF THE WAVE EQUATION WITH DIRAC DAMPING ON A NON-COMPACT STAR GRAPH

come

Spectrum of the wave equation with Dirac damping on a non-compact star graph

Introduction

The primary motivation of this paper is the one-dimensional wave equation

ψ tt ´αδψ t ´ψxx " 0 , (1.1)
where t ą 0 and x P R are the time and space variables, respectively, δ is the Dirac delta function and α is a complex number. The distribution ´αδ models a highly localised damping: dissipation or supply of energy if α is negative or positive, respectively, while purely imaginary α admits a conservative, quantum-mechanical interpretation.

In the case of α real and the space variable x restricted to a bounded interval, say p´π 2 , π 2 q, the model (1.1) was introduced in [START_REF] Bamberger | A model for harmonics on stringed instruments[END_REF] in order to explain the playings of harmonics on stringed instruments. By a detailed spectral analysis, the authors argue that the optimal damping is α " ´2. The problem has been more recently analysed in [START_REF] Ammari | Optimal location of the actuator for the pointwise stabilization of a string[END_REF][START_REF] Ammari | Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string[END_REF][START_REF] Cox | Eliciting harmonics on strings[END_REF] (see also [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF]Sec. 4.1.1]). Qualitatively, spectral properties of (1.1) in the bounded case are the expected ones: the spectrum is composed of isolated eigenvalues of finite algebraic multiplicities. However, there is an abrupt change in basis properties depending on whether |α| " 2 or |α| " 2. While the root vectors form a Riesz basis in the former case, they are not even complete for the special values α " ˘2.

The objective of the present paper is to point out that the peculiar spectral transition is even more drastic in the unbounded situation considered here. Moreover, we allow for α being an arbitrary complex number). If α P Czt˘2u, the spectrum coincides with the imaginary axis iR, as in the damped-free case α " 0. However, the spectrum abruptly fills in the whole complex half-plane tz P C : ˘Repzq ě 0u as long as α " ˘2. This phenomenon was previously announced in [START_REF] Krejčiřík | From Lieb-Thirring inequalities to spectral enclosures for the damped wave equation[END_REF]Rem. 1], but no rigorous analysis was provided. Wild spectral properties for the damped wave equation with different unbounded dampings have been recently observed in [START_REF] Freitas | The damped wave equation with unbounded damping[END_REF][START_REF] Freitas | The damped wave equation with singular damping[END_REF].

What is more, we give an insight into the appearance of the "magical" values ˘2 by considering (1.1) in the more general situation of non-compact star graphs with N ě 1 edges. The real line R can be considered as the star graph with two edges, while the case N " 1 corresponds to the wave equation on the half-line with damping at the boundary. It turns out that the abrupt change in spectral properties happens precisely for α " ˘N .

The Laplacian on metric graphs with non-self-adjoint coupling conditions at the vertices has been recently analysed in [START_REF] Hussein | Non-self-adjoint graphs[END_REF][START_REF] Rivière | Spectrum of a non-selfadjoint quantum star graph[END_REF][START_REF] Hussein | Hidden symmetries in non-self-adjoint graphs[END_REF]. The damped wave equation on metric graphs is considered in [START_REF] Abdallah | Exponential stability of the wave equation on a star shaped network with indefinite sign damping[END_REF][START_REF] Freitas | Eigenvalue asymptotics for the damped wave equation on metric graphs[END_REF], however, just self-adjoint coupling conditions at the vertices (and finite edges) are considered. See also [START_REF] Assel | Optimal decay rate for the local energy of a unbounded network[END_REF] for an unbounded network with Dirichlet and Kirchhoff conditions. The present paper therefore opens a new direction of mathematically interesting and physically relevant research.

The organisation of this paper is as follows. In Section 2 we introduce our general model and formulate the main results. The preliminary Section 3 collects basic properties of the wave operator. In Section 4 we show that the Dirac damping can be realised as a limit of properly scaled regular dampings, in a norm-resolvent sense. The proofs of more involved spectral and evolution results are given in Sections 5 and 6, respectively. In the concluding Section 7, we provide a relationship between the damped wave equation and the Dirac equation in relativistic quantum mechanics, in order to motivate the present setting of unbounded geometries and non-real dampings.

Our model and main results

2.1. The damped wave equation and the wave operator. Let N P N ˚" t1, 2, . . . u. We set N " t1, . . . , N u. We consider a metric graph Γ given by N edges of infinite length linked together at a central vertex. Concretely, Γ consists of N copies of the open halfline R ˚"s0, `8r and the link between the edges will be encoded in the domain of our operator (see (2.5) below).

We set

L 2 pΓq " L 2 pR ˚qN .
It is endowed with its natural Hilbert structure. For k P N ˚we similarly define

9 H k pΓ ˚q " 9 H k pR ˚qN , H k pΓ ˚q " H k pR ˚qN ,
and for u " pu j q jPN P 9 H k pΓ ˚q we write u pkq for pu pkq j q jPN . In the sequel, when we consider a function u on Γ » pR ˚qN it will be implicitly understood that we denote by pu j q jPN its components.

We say that u P 9 H 1 pΓ ˚q is continuous at 0 if @j, k P N , u j p0q " u k p0q.

In this case we denote by up0q the common value of u j p0q, j P N . We set

9 H 1 pΓq " ! u P 9 H 1 pΓ ˚q : u is continuous at 0 ) .
To define the natural Hilbert structure, we have to take the quotient of 9 H 1 pΓq by constant functions. We denote by H1 pΓq this quotient. Then H1 pΓq is a Hilbert space for the inner product defined by

@u, v P H1 pΓq, xu, vy H1 pΓq " @ u 1 , v 1 D L 2 pΓq " N ÿ j"1 @ u 1 j , v 1 j D L 2 pR ˚q .
Throughout the paper we will identify a function in 9 H 1 pΓq and its equivalence class in H1 pΓq. This implicitly means that the results should be invariant by addition of a constant to the functions in 9 H 1 pΓq. We consider the damped wave equation on Γ with damping at the central vertex. Let I be an interval of R which contains 0. We introduce the problem B tt u j pt, xq ´Bxx u j pt, xq " 0, @j P N , @t P I, @x ě 0 (2.1)

with the continuity at the vertex u j pt, 0q " u k pt, 0q, @j, k P N , @t P I (2.2) (this also implies the continuity at the vertex for B t u) and the damping condition

N ÿ j"1
B x u j pt, 0q `αB t upt, 0q " 0, @t P I, (

where α P C. The Cauchy problem is completed by the initial conditions pu j , B t u j q| t"0 " pf j , g j q, @j P N ,

where f P 9 H 1 pΓq and g P L 2 pΓq. More precisely, a (strong) solution of (2.1)-(2.4) is a continuous function u : I Ñ H1 pΓq X 9

H 2 pΓ ˚q with continuous derivative B t u : I Ñ H 1 pΓq and continuous second derivative B tt u : I Ñ L 2 pΓq, such that (2.1) holds in L 2 pΓq for all t P I, with (2.2)-(2.3) for all t P I, and with (2.4) in H1 pΓq ˆL2 pΓq.

We set H " H1 pΓq ˆL2 pΓq and

DompW α q " # U " pu, vq P `H 1 pΓq X 9 H 2 pΓ ˚q˘ˆH 1 pΓq : N ÿ j"1 u 1 j p0q `αvp0q " 0 + .
(2.5) Then we define the unbounded operator W α in the Hilbert space H by @U " pu, vq

P DompW α q, W α ˆu v ˙" ˆv u 2 ˙. (2.6)
We endow DompW α q with the graph norm:

}U } 2 DompWαq " }W α U } 2 H `}U } 2 H , @U P DompW α q.
Then we see that u is a solution of (2.1)-(2.4) on I if and only if U " pu, B t uq belongs to C 1 pR `, H q X C 0 pR `, DompWqq and satisfies # U 1 ptq " WU ptq, @t P I, U p0q " pf, gq.

(2.7)

Our purpose in this paper is to describe the spectrum of the operator W α and the behaviour of the evolution problem (2.1)-(2.4) (or equivalently (2.7)). The first step is to check that W α is closed with non-empty resolvent set. In Section 3 we prove the following more precise result.

Theorem 2.1. The operator W α is maximal accretive (respectively maximal dissipative, respectively skew-adjoint) if Repαq ě 0 (respectively Repαq ď 0, respectively Repαq " 0).

An alternative approach, based on the dominant Schur complement, to introduce the wave operator with possibly highly irregular dampings (possibly distributions) has been recently developed by Gerhat [START_REF] Gerhat | Schur complement dominant operator matrices[END_REF].

2.2.

The Robin vertex condition. The definition of DompW α q contains continuity at 0 for u and v, and the additional condition will be referred to as the Robin condition.

Notice that for N " 1, the graph Γ reduces to the half-line s0, `8r and in this case we recover the usual Robin condition u 1 p0q `αvp0q " 0 at the boundary. When N " 2 we can identify the two edges with s0, `8r and s ´8, 0r (by considering the transformation x Þ Ñ u 2 p´xq on s´8, 0r), so Γ is identified with R and in this setting the Robin condition yields the usual jump condition u 1 p0 `q ´u1 p0 ´q " ´αvp0q.

Notice also that the two extreme situations α " 0 and α " 8 correspond, respectively, to Neumann (or Kirchhoff) and Dirichlet boundary conditions imposed at the central vertex.

The motivation for considering the vertex condition (2.3) for our wave equation is that it corresponds to a singular damping localised at the vertex. Formally, (2.1) and (2.3) mean B tt u ´Bxx u ´αδB t u " 0, where δ is a generalisation of the Dirac distribution on Γ. Similarly, for the corresponding wave operator we can formally write

W α ˆu v ˙" ˆv u 2 `αδ ˙.
This interpretation is supported by writing the quadratic form associated to the operator W α defined by (2.5)-(2.6). Indeed, for U " pu, vq P DompW α q we have

xW α U, U y H " xv 1 , u 1 y L 2 pΓq `xu 2 , vy L 2 pΓq " 2iImxv 1 , u 1 y L 2 pΓq `α |vp0q| 2 .
(2.8)

To go further, we compare our operator W α with an operator with a damping on the edges localised near the vertex.

For j P N we consider ρ j P L 1 pR ˚, R `q such that ş `8 0 ? xρ j pxq dx ă `8. We assume that ř N j"1 ş `8 0 ρ j pxq dx " 1. For n P N ˚and j P N we set ρ n j pxq " nρ j pxnq. Then we set ρ n " pρ n j q 1ďjďN . For n P N ˚we consider on H the operator

W α,n " ˆ0 1 B 2 αρ n ˙, defined on the domain DompW α,n q " # U " pu, vq P `H 1 pΓq X 9 H 2 pΓ ˚q˘ˆH 1 pΓq : N ÿ j"1 u 1 j p0q " 0 + .
A result similar to Theorem 2.1 holds for W α,n (see Proposition 3.7 below). In particular, setting C ˘" tz P C : ˘Repzq ą 0u , we deduce that if α P C ˘then for z P C ¯we have z P ρpW α q. In Section 4 we show that W α is the limit of W α,n in the sense of the norm of the resolvent.

Theorem 2.2. Let α P C ˘and z P C ¯. Then › › pW α ´zq ´1 ´pW α,n ´zq ´1› › LpH q ÝÝÝÝÑ nÑ`8 0.
2.3. Spectral properties of the wave operator. Having shown that W α is a well defined operator and a suitable model for the damped wave equation with damping at the central vertex, we can now turn to its spectral properties. We first notice that, as a consequence of Theorem 2.1, if α P C ˘then C ¯Ă ρpW α q and for z P

C ¯we have › › pW α ´zq ´1› › LpH q ď 1 ¯Repzq .
The main result of this paper is related to spectral properties on the other half-plane. There is no general theory for resolvent estimates inside the numerical range, but explicit computations provide a precise description of the spectrum and the resolvent for this particular problem.

Theorem 2.3. The spectrum of W α is $ ' & ' % iR if α P Czt˘N u, C `if α " N, C ´if α " ´N. Moreover,
(i) iR contains no eigenvalue nor residual spectrum of W α , (ii) if α " ˘N then any z P C ˘is an eigenvalue of W α of geometric multiplicity 1 and infinite algebraic multiplicity, (iii) there exist c 0 , C ą 0 such that for α P C ˘z t˘N u and z P C ˘we have

max ˆ1 |Repzq| , c 0 |z| |α ¯N | ˙ď › › pW α ´zq ´1› › LpH q ď C |Repzq| ˆ1 `1 |α ¯N | ˙.
2.4. The damped wave equation. Finally we go back to the time-dependant problem (2.1)-(2.4), or equivalently (2.7). If Repαq ď 0, the operator W α generates by Theorem 2.1 a contractions semigroup, so the problem (2.7) is well posed on R `. Moreover the energy Epu; tq

" }B t uptq} 2 L 2 pΓq `}B x uptq} 2 L 2 pΓq
of the solution u is non-increasing.

In this paragraph we address the question of well-posedness and growth of the energy for (2.7) (or equivalently for (2.1)-(2.4)) when Repαq ą 0. We have similar results for negative times when Repαq ă 0.

Theorem 2.4. Let α P C `and pf, gq P DompW α q.

(i) Assume that α ‰ N . The problem (2.1)-( 2.4) has a unique solution u on R `.

Moreover there exists C ą 0 independent of α and pf, gq such that for t ě 0 we have

Epu; tq ď C ˆ1 `1 |α ´N | 2 ˙Epu; 0q. (2.9)
(ii) Assume that α " N . Let

t 0 " sup # t ě 0 : N ÿ j"1
`f 1 j psq `gj psq ˘" 0, @s P r0, ts + P r0, `8s.

If t 0 ą 0 then (2.1)-(2.4) has an infinite number of solutions on r0, t 0 r. In particular there exists a solution u such that

Epu; tq Ý ÝÝ Ñ tÑt 0 `8.
If t 0 is finite then for any ε ą 0 the problem (2.1)-(2.4) has no solution on r0, t 0 `εr.

General properties of the wave operator

In this section we prove some basic properties for the wave operator W α . In particular, we give an expression for its resolvent with the spectral parameter lying in the suitable half-place (depending on the sign of Repαq) and deduce that W α is maximal accretive and/or maximal dissipative (Theorem 2.1). Notice that our proofs are quite robust and could be applied for the wave equation on more general graphs.

To prepare the proof of Theorem 2.2 in the next section, we proceed with the same analysis for W α,n , n P N ˚.

We first record the following direct consequence of formula (2.8).

Proposition 3.1. The operator W α is accretive (respectively dissipative, respectively skew-symmetric) if Repαq ě 0 (respectively Repαq ď 0, respectively Repαq " 0). In particular, if α P C ˘and z P C ¯then pW α ´zq is injective with closed range. For n P N ˚the operator W α,n has the same properties.

Next we mention the following symmetry result. Proposition 3.2. For α P C we have W α " ´W´ᾱ .

Proof. For U P DompW a q and Ũ P DompW ´ᾱ q we can check by direct computation that

@ W α U, Ũ D H " ´@U, W ´ᾱ Ũ D
H . This proves that DompW ´ᾱ q Ă DompW αq and that W α " ´W´ᾱ on DompW ´ᾱ q. Now let Ũ " pũ, ṽq P DompW αq and F " pf, gq " W α Ũ P H . For all U " pu, vq P DompW α q we have

@ W α U, Ũ D H " xU, F y H , which gives @ v 1 , ũ1 D L 2 pΓq `@u 2 , ṽD L 2 pΓq " @ u 1 , f 1 D L 2 pΓq `xv, gy L 2 pΓq . (3.1)
Let j P N . Applied with u " 0, v j P C 8 0 pR ˚q and v k " 0 for k ‰ j, this proves that ũ1 j P H 1 pR ˚q and ũ2 j " ´gj . Applied with v " 0, u j P C 8 0 pR ˚q and u k " 0 for k ‰ j, we deduce that there exists a constant β j such that ṽ1 j " ´f 1 j `βj in the sense of distributions. Since f 1 j and ṽj are in L 2 pR ˚q, we necessarily have β j " 0, so ṽ1 j " f 1 j belongs to L 2 pR ˚q.

We can rewrite (3.1) as

@ u 1 , f 1 D L 2 pΓq `xv, gy L 2 pΓq " ´@v, ũ2 D L 2 pΓq ´vp0q N ÿ j"1 ũ1 j p0q ´@u 1 , ṽ1 D L 2 pΓq ´N ÿ j"1 u 1 j p0qṽp0q.
This gives

´vp0q N ÿ j"1 ũ1 j p0q `αvp0qṽp0q " 0,
which implies that ř N j"1 ũ1 j p0q ´ᾱṽp0q " 0. Then DompW α q Ă DompW ´ᾱ q, and the proof is complete.

In Proposition 3.6 below, we will give an expression for the resolvent of W α valid in H 1 pΓq ˆL2 pΓq. This is a dense subset of H by the following classical lemma.

Lemma 3.3. H 1 pΓq is dense in 9 H 1 pΓq.
Proof. Let u " pu j q jPN P 9 H 1 pΓq. Let j P N . Let χ P C 8 pR `, r0, 1sq be equal to 1 on r0, 1s and equal to 0 on r2, `8r. For R ě 1 and x ě 0 we set χ R pxq " χ `x R ˘. We have › › pp1 ´χR qu j q 1 › ›

L 2 pR ˚q ď › › p1 ´χR qu 1 j › › L 2 pR ˚q `› › χ 1 R u j › › L 2 pR ˚q .
The first term goes to 0 as R Ñ 8 by the dominated convergence theorem. We estimate the second term. Let ε ą 0. There exists C j ą 0 such that for all x ě 1 we have |u j pxq| ď C j ?

x. Let x 0 ą 0 be so large that 2 ? 2C j }u 1 } L 2 px 0 ,8q ď ε, and let R ě 1 be so large that |upx 0 q| 2 ď εR 2 . Then for x ě maxpx 0 , Rq we have

|u j pxq| 2 L 2 pR ˚q ď |u j px 0 q| 2 `2 ż x x 0 |u j psq| ˇˇu 1 j psq ˇˇds ď |u j px 0 q| 2 `?2C j x › › u 1 j › › L 2 px 0 ,8q ď εx, so › › χ 1 R u j › › 2 ď }χ 1 } 2 8 R 2 ż 2R R |u j pxq| 2 dx ď 2ε › › χ 1 › › 2 8 .
The conclusion follows.

We denote by H ´1pΓq the space of continuous semilinear forms on H 1 pΓq (we have pϕ 1 `βϕ 2 q " pϕ 1 q `β pϕ 2 q for P H ´1pΓq, ϕ 1 , ϕ 2 P H 1 pΓq and β P C). In particular, δ : φ Þ Ñ φp0q belongs to H ´1pΓq. We refer to [11, pp. 3-4] for a discussion about this choice.

" cospθq › › w 1 › › 2 L 2 pΓq `cospηq |αz| xρ n w, wy L 2 pΓq `cospθq |z| 2 }w} 2 L 2 pΓq ě minp1, |z| 2 q cospθq }w} 2 H 1 pΓq .
By the Lax-Milgram Theorem, Q n,α pzq is invertible and

› › Q α,n pzq ´1› › LpH ´1pΓq,H 1
pΓqq is uniformly bounded in n P N ˚. We proceed similarly for Q α pzq.

We set R α pzq " Q α pzq ´1 and R α,n pzq " Q α,n pzq ´1. (i) Let w " R α pzqph`κδq P H 1 pΓq. Then for j P N we have w 2 j " z 2 w j ´hj P L 2 pR ˚q, and moreover

N ÿ j"1 w 1 j p0q `αwp0q " ´κ. (3.2)
(ii) Let w " R α,n pzqph `κρ n q P H 1 pΓq. Then ř jPN w 1 p0q " 0 and for j P N we have w 2 j " ´αzρ n w `z2 w j ´hj ´κρ n P L 2 pR ˚q.

Proof. Since h `κδ P H ´1pΓq, w is well defined as an element of H 1 pΓq by Proposition 3.4. For all φ P H 1 pΓq we have

@ w 1 , φ 1 D L 2 pΓq ´αzwp0qφp0q `z2 xw, φy L 2 pΓq " xh, φy L 2 pΓq `κφp0q. (3.3)
As in the proof of Proposition 3.2, by choosing φ supported away from the vertex we see that for all j P N we have in the sense of distributions w 2 j " z 2 w j ´hj . In particular w j P H 2 pR ˚q. Then, after integrations by parts in (3.3) we get ´N ÿ j"1 w 1 j p0qφp0q ´zαwp0qφp0q " κφp0q.

This gives (3.2). The second statement is similar.

We define B 2 P Lp 9 H 1 pΓq, H ´1pΓqq by @ B 2 ψ, ϕ D H ´1pΓq,H 1 pΓq " ´xψ 1 , ϕ 1 y L 2 pΓq for all ψ, ϕ P H 1 pΓq. In particular we have Q α pzq " ´B2 ´αzδ `z2 . Proposition 3.6. Let α P C ˘and z P C ¯. We have z P ρpW α q and

pW α ´zq ´1 " ˆ´z ´1`R α pzqB 2 `1˘´R α pzq ´Rα pzqB 2 ´zR α pzq ˙. (3.4)
Moreover, for F " H 1 pΓq ˆL2 pΓq we also have pW α ´zq ´1F " ˆRα pzqpαδ ´zq ´Rα pzq 1 `Rα pzqpzαδ ´z2 q ´zR α pzq ˙F.

(3.5)

For n P N ˚have the same results with W α , R α pzq and δ replaced by W α,n , R α,n pzq and ρ n .

Proof. For F P H 1 pΓq ˆL2 pΓq we denote by R α pzqF the right-hand side of (3.5). We set F " pf, gq and U " R α pzqF " pu, vq. We have u, v P H 1 pΓq. By Proposition 3.5, we have u 2 j P L 2 pR ˚q, u 2 j " z 2 u `zf `g and v j " f j `zu j for all j P N . On the other hand we have at the vertex:

N ÿ j"1 u 1 j p0q `αzup0q " ´αf p0q.
This gives the Robin condition. All this proves that U P DompW α q and pW α ´zqU " F .

In particular pW α ´zq has dense range. Since pW α ´zq is injective with closed range by Proposition 3.1, z belongs to ρpW α q and pW α ´zq ´1 " R α pzq on H 1 pΓq ˆL2 pΓq. Now we denote by Rα pzq the right-hand side of (3.4). From the properties of R α pzq we see that Rα pzq defines a bounded operator on H . Moreover, for F P H 1 pΓq ˆL2 pΓq we have Rα pzqF " R α pzqF " pW α ´zq ´1F . Since H 1 pΓq ˆL2 pΓq is dense in H , this proves that Rα pzq " pW α ´zq ´1.

The proof for W α,n is similar. We omit the details.

With Proposition 3.6 we can complete the statement of Proposition 3.1. This gives in particular Theorem 2.1.

Proposition 3.7. The operator W α is maximal accretive (respectively maximal dissipative, respectively skew-adjoint) if Repαq ě 0 (respectively Repαq ď 0, respectively Repαq " 0). For n P N ˚the operator W α,n has the same properties.

Damping at the vertex as a limit model for damping on the edges

In this section we establish Theorem 2.2. We first check that the sequence pρ n q nPN is an approximation of the Dirac distribution.

Spectrum of the wave operator

In this section we prove Theorem 2.3. By Proposition 3.2 it is enough to consider the case Repαq ě 0. In this case, we already know by Proposition 3.6 that C ´Ă ρpW α q. We use explicit computation to describe the spectrum on the right half-plane. Then Theorem 2.3 follows from Propositions 5.1, 5.2, 5.3 and 5.5 below.

Given a closed operator W in a Hilbert space H , we denote its point spectrum (i.e. the set of eigenvalues W) by σ p pWq. One says that λ P σpWq belongs to the continuous spectrum σ c pWq (respectively, residual spectrum σ r pWq) of W if λ R σ p pWq and the closure of the range of the shifted operator W ´λ equals H (respectively, the closure is a proper subset of H ).

We first consider the spectrum on the imaginary axis.

Proposition 5.1. Let α P C. Then iR Ă σ c pW α q.

Proof. Let θ P R. Let φ P C 8 0 pR ˚q be supported in [START_REF] Abdallah | Exponential stability of the wave equation on a star shaped network with indefinite sign damping[END_REF][START_REF] Ammari | Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string[END_REF] and such that }φ} L 2 pR ˚q " 1. For n P N ˚we define u n " pu n,j q jPN by u n,1 pxq " e iθx ? n φ ´x n ¯, u n,j pxq " 0, j P t2, . . . , N u, x ą 0.

Then we set U n " pu n , iθu n q P DompW α q. For n P N ˚we have }u

1 n } L 2 pΓq " |θ| `Opn ´1q, }iθu n } L 2 pΓq " |θ| and › › u 2 n `θ2 u n › › L 2 pΓq " Opn ´1q, so }U n } 2 H " 2θ 2 `Opn ´1q and }pW α ´iθqU n } 2 H " › › u 2 n `θ2 u n › › 2 L 2
pRq " Opn ´2q. This proves that iθ P σpW α q. Now let U " pu, vq P DompW α q such that W α U " iθU . That is, v " iθu and u 2 j " ´θ2 u j on R `for all j P N . Since u j P 9 H 1 pR `q, this implies that u j " 0. Then U " 0, so iθ cannot be an eigenvalue of W α . Finally, assume that iθ P σ r pW α q. Then ´iθ P σ p pW αq. By Proposition 3.2, iθ P σ p pW ´ᾱ q. However, the existence of eigenvalues on the entire imaginary axis has been already excluded.

Next we show that in the particular case α " N the right-half plane is filled with eigenvalues.

Proposition 5.2. Any z P C `is an eigenvalue of W N with geometric multiplicity 1 and infinite algebraic multiplicity.

Proof. Let z P C `. Assume that U " pu, vq P DompW N q is such that W N U " zU . Then v " zu and u 2 " zv " z 2 u. Let j P N . Since u j P L 2 pR ˚q, there exists A j P C such that u j pxq " A j e ´zx . By continuity at 0, the coefficients A j for j P N are all equal. Thus U is proportional to the vector U 1 " pu 1 , v 1 q defined by # u 1,j pxq " e ´zx , v 1,j pxq " ze ´zx .

Notice in particular that U 1 is radial (the expressions above do not depend on j P N ). Conversely, we can check that the vector U 1 defined in this way belongs to DompW N q and is an eigenvector corresponding to the eigenvalue z. This proves that z is a geometrically simple eigenvalue of W N .

For n ě 2 we define u n by u n,j pxq " p´1q n´1 x n´1 pn ´1q! e ´zx , j P N , x ą 0.

At the same time, we define v n by v n " zu n `un´1 .

It is straightforward to check that for all n ě 2 we have U n " pu n , v n q P DompW N q and W N U n " zU n `Un´1 . This proves that z has algebraic multiplicity `8 as an eigenvalue of W N .

The precedent proposition establishes part (ii) of Theorem 2.3. The other part that for α P C `ztN u there is no spectrum in the right-half plane will be proved in a moment. First, however, let us argue that even if there is no spectrum, the pseudospectra are highly non-trivial there and actually explode as α Ñ N . This is quantified by obtaining the following resolvent estimate. Proposition 5.3. There exists c 0 ą 0 such that for α P C `ztN u and z P C `X ρpW α q we have

› › pW α ´zq ´1› › LpH q ě max ˆ1 Repzq , c 0 |z| |N ´α| ˙.
Proof. For j P N and x ą 0 we set ηpxq " e ´αzx N . Then we define U " pu, vq by u j " η and v j " zη for all j P N . We have

U P DompW α q, }U } 2 H " N ˜|α| 2 N 2 `1¸| z| 2 }η} 2 L 2 pR ˚q and }pW α ´zqU } 2 H " N ˇˇˇα 2 N 2 ´1ˇˇˇˇ2 |z| 4 }η} 2 L 2 pR ˚q , so }pW α ´zqU } 2 H ď |z| 2 |α ´N | 2 |α `N | 2 N 2 p|α| 2 `N 2 q }U } 2 H .
This proves that

› › pW α ´zq ´1› › LpH q ě c 0 |z| |α ´N | , with c 0 " inf αPC `N b |α| 2 `N 2 |α `N | ą 0.
On the other hand we also have

› › pW α ´zq ´1› › LpH q ě 1 distpz, σpW α qq ě 1 Repzq ,
and the conclusion follows.

To show that for α P C `ztN u there is no spectrum in the right-half plane, we compute explicitely the resolvent of W α . This will also give the upper bound for the norm of the resolvent.

For z P C `and y P R we set

ρ z pyq " ze ´z|y| 2 . ( 5.1)
Notice that ρ z pyq " ´z2 G z pyq, where G z is the Green function for the Helmholtz equation in dimension 1. On a half-line (hence on a star graph), the solution of the Helmholtz equation is also given by convolution with G z or ρ z . Let h : r0, `8rÑ C and x ě 0. When this makes sense we set

pρ z ˚hqpxq " ż `8 0 ρ z px ´sqhpsq ds. (5.2)
We will use the following properties on this convolution product.

Lemma 5.4. Let z P C `.

(i) The convolution pρ z ˚hqpxq is well defined for h in L 2 pR ˚q or h P 9 H 1 pR ˚q and any x ě 0. (ii) For h P L 2 pR ˚q we have pρ z ˚hq P L 2 pR ˚q and

}ρ z ˚h} L 2 pR ˚q ď |z| Repzq }h} L 2 pR ˚q . (5.3)
(iii) For h in L 2 pR ˚q or 9 H 1 pR ˚q the convolution pρ z ˚hq belongs to 9 H 1 pR ˚q and for almost all x ą 0 we have pρ z ˚hq 1 pxq " zpρ z ˚hqpxq ´z2 ż x 0 e ´zpx´sq hpsq ds.

(5.4)

For h P 9 H 1 pR `q this also gives

pρ z ˚hqpxq " 1 z pρ z ˚h1 qpxq ´ż x 0
e ´zpx´sq h 1 psq ds `hpxq ´e´zx hp0q 2 and pρ z ˚hq 1 pxq " pρ ˚h1 qpxq `z 2 e ´zx hp0q.

(iv) For h P L 2 pR ˚q we have

|pρ z ˚hqp0q| ď |z| 2 a 2Repzq }h} L 2 pR ˚q .
Proof. Since ρ z decays exponentially, the integral (5.2) is well defined for h P L 2 pR ˚q or h P 9 H 1 pR `q (in this case hpsq grows at most like ? s), and for any x ě 0. If h P L 2 pR ˚q we extend h by 0 on s ´8, 0r to define a function h on R. Then pρ z ˚hq is the restriction to r0, `8r of the usual convolution pρ z ˚hq on R, and

}ρ z ˚h} L 2 pR ˚q ď }ρ z ˚h} L 2 pRq ď }ρ} L 1 pRq } h} L 2 pRq " |z| Repzq }h} L 2 pR ˚q .
The third statement is straightforward computation and the last property follows from the Cauchy-Schwarz inequality

|pρ z ˚hqp0q| " ˇˇˇż `8 0 ρ z p´sqhpsq ds ˇˇˇď }ρ z } L 2 pR ´q }h} L 2 pR ˚q " |z| 2 a 2Repzq }h} L 2 pR ˚q .
Now we are in a position to complete the proof of Theorem 2.3.

Proposition 5.5. Assume that α P C `ztN u. Then C `Ă ρpW α q and there exists C ą 0 independent of α such that for all z P C `we have

› › pW α ´zq ´1› › LpH q ď C Repzq ˆ1 `1 |α ´N | ˙.
Proof. Let z P C `. Let F " pf, gq P H . We prove that the equation

pW α ´zqU " F (5.5)
has a unique solution U " pu, vq P DompW α q. This will prove that z P ρpW α q, and the explicit expression for U will provide the estimates on pW α ´zq ´1. We identify f P H1 pΓq with any representative in 9 H 1 pΓq. We can check all along the proof that if we add a constant to f , then it only changes u by a constant and hence does not change its equivalence class in H1 pΓq. We could directly fix that f p0q " 0, but then the independence with respect to an additive constant would be less explicit. For clarity of the exposition, we divide the proof into several steps distinguished by the bullet mark. ' Assume that U " pu, vq P DompW α q satisfies (5.5). Let j P N . Then u j P H 2 loc pR ˚q and in the sense of distributions on R ˚we have u 2 j ´z2 u j " g j `zf j .

(

We identify u j with its continuous representative. Then there exist A j , B j P C such that, for all x ą 0, u j pxq " A j e ´zx `Bj e zx ´1 z 2 pρ z ˚gj qpxq ´1 z pρ z ˚fj qpxq, (

with ρ z defined by (5.1). Since u j , pρ z ˚gj q and pρ z ˚fj q belong to 9 H 1 pR ˚q we necessarily have B j " 0. If we set

Ãj " A j `f p0q 2z , (5.8)
then we have by Lemma 5.4 u 1 j pxq " ´z Ãj e ´zx ´1 z `ρz ˚pf 1 j `gj q ˘pxq `ż x 0 e ´zpx´sq g j psq ds (5.9) and v j pxq " zu j pxq `fj pxq " z Ãj e ´zx ´1 z `ρz ˚pf 1 j `gj q ˘pxq `ż x 0 e ´zpx´sq f 1 j psq ds.

(5.10)

By continuity of u at 0 we have for all j P N Ãj " up0q `f p0q z `1 z 2 `ρz ˚pf 1 j `gj q ˘p0q, (5.11) so the Robin condition in (2.5) reads N ÿ j"1 ˆ´zup0q ´f p0q ´2 z `ρz ˚pf 1 j `gj q ˘p0q ˙`αzup0q `αf p0q " 0.

This gives

up0q `f p0q z " 2 z 2 pα ´N q N ÿ "1
`ρz ˚pf 1 `g q ˘p0q.

(5.12)

Thus (5.11) gives an explicit expression for Ãj and, by (5.9)-(5.10), U is uniquely determined by F . This proves the injectivity of pW α ´zq : DompW a ´zq Ñ H . ' Conversely, let U " pu, vq be defined by (5.7) and (5.10), with B j " 0 and A j given by (5.8), (5.11) and (5.12). Let j P N . Then u j is a solution of (5.6). By Lemma 5.4 we have v j P H 1 pR ˚q and u 1 j P L 2 pR ˚q. Moreover, by (5.6), u 2 j " zv j `gj P L 2 pR ˚q. By construction, u is continuous at 0, then so is v, and the Robin condition in (2.5) holds. This proves that U P DompW α q and then that z P ρpW α q. ' It remains to prove that

}U } H À }F } H Repzq ˆ1 `1 |α ´N | ˙,
where the symbol À means that we have inequality up to a multiplicative constant which does not depend on F , α or z. For this we apply Lemma 5.4. By (5.12) we have

ˇˇˇu p0q `f p0q z ˇˇˇÀ }F } H |z| |α ´N | a Repzq .
Then (5.11) gives for all j P N

| Ãj | À }F } H |z| a Repzq ˆ1 `1 |α ´N | ˙.
Finally, with (5.9) and (5.10),

}U } H À N ÿ j"1 `}u 1 j } L 2 pR ˚q `}v j } L 2 pR ˚q˘À }F } H Repzq ˆ1 `1 |α ´N | ˙,
and the proof is complete.

Damped wave equation

In this section we prove Theorem 2.4 about the time-dependant problem (2.1)-(2.4).

Proof of Theorem 2.4. ' Assume that u is a solution of (2.1)-(2.4) on r0, τ r for some τ ą 0. For t P r0, τ r and j P N we identify u j ptq with its representative of class C 1 on R `. For j P N there exist ϕ j P C 1 p´τ, `8q X 9 H 1 p´τ, `8q X 9 H 2 p´τ, `8q and ψ j P C 1 pR `q X 9 H 1 pR `q X 9 H 2 pR `q such that ϕp0q " ψp0q " 0 and for all t P r0, τ r and x ą 0 we have u j pt, xq " up0, 0q `ϕj px ´tq `ψj px `tq. (6.1)

The initial condition gives for x ą 0 # ϕ 1 j pxq `ψ1 j pxq " B x up0, xq " f 1 j pxq, ´ϕ1 j pxq `ψ1 j pxq " B t up0, xq " g j pxq, so @x ą 0, ψ 1 j pxq "

f 1 j pxq `gj pxq 2 , ϕ 1 j pxq " f 1 j pxq ´gj pxq 2 . (6.2)
On the other hand, the continuity at the central vertex gives for j, k P N and t ą 0, after differentiation, @j, k P N , @t P r0, τ r, ϕ 1 j p´tq ´ϕ1 k p´tq " ψ 1 j ptq ´ψ1 k ptq. (6.3)

Finally, the damping condition yields

@t P r0, τ r, ´α N ´1¯N ÿ j"1 ϕ 1 j p´tq " ´α N `1¯N ÿ j"1 ψ 1 j ptq. (6.4)
In particular, we see that if α " N then we necessarily have ř N j"1 ψ 1 j ptq " 0 for all t P r0, τ r, so τ ď t 0 . On the other hand, if α ‰ N then u is uniquely determined. ' Conversely, we assume that α ‰ N and prove that (2.1)-(2.4) indeed has a solution on R `. For j P N and x ě 0 we set ϕ j pxq " 1 2 ż x 0 `f 1 j psq ´gj psq ˘ds, ψ j pxq "

1 2 ż x 0 `f 1 j psq `gj psq ˘ds. (6.5)
and then ϕpxq " pϕ j pxqq jPN and ψpxq " pψ j pxqq jPN , seen as column vectors. Then for s P R ˚we set ϕp´sq " ´M ´1 ´M`ψ psq, where

M ˘" ¨1 ´1 . . . . . . 1 ´1 α N ˘1 . . . α N ˘1 α N ˘1‹ ‹ ‹ ' . ( 6
α q ϕ 1 p´sq " M ´1 ´M`ψ 1 psq Ý ÝÝ Ñ sÑ0 M ´1 ´M`ψ 1 p0q " ϕ 1 p0 `q.
Thus ϕ P C 1 pRq X 9 H 1 pRq X 9 H 2 pRq. Then for t ě 0 and x ą 0 we define u by (6.1) (the choice of up0, 0q is not important since u is defined up to an additive constant). We have u P C 1 pR `, H 1 pΓqq X C 0 pR `, 9

H 2 pΓ ˚qq. On the other hand in the sense of distributions we have

B tt u " B xx u.
In particular, B tt u P C 0 pR `, L 2 pΓqq, so u P C 2 pR `, L 2 pΓqq. All this proves that u is a solution of (2.1)-(2.4) on R `. Moreover for t ě 0 we have

Epu; tq " }B t uptq} 2 L 2 pΓq `}B x uptq} 2 L 2 pΓq " 2 ż `8 0 `ˇϕ 1 px ´tq ˇˇ2 C N `ˇψ 1 px `tq ˇˇ2 C N ˘dx " 2 ż `8 ´t ˇˇϕ 1 psq ˇˇ2 C N ds `2 ż `8 t ˇˇψ 1 psq ˇˇ2 C N ds.
We have

ż t ´t ˇˇϕ 1 psq ˇˇ2 C N ds ď ż t 0 `ˇϕ 1 psq ˇˇ2 C N `› › M ´1 ´M`› › 2 ˇˇψ 1 psq ˇˇ2 C N ˘ds À ˆ1 `1 |α ´N | 2 ˙ż t 0 `ˇϕ 1 psq ˇˇ2 C N `ˇψ 1 psq ˇˇ2 C N ˘ds À ˆ1 `1 |α ´N | 2 ˙ż t 0 `ˇf 1 psq ˇˇ2 C N `|gpsq| 2 C N ˘ds. Since 2 ż `8 t `ˇϕ 1 psq ˇˇ2 C N `ˇψ 1 psq ˇˇ2 C N ˘ds " ż `8 t `ˇf 1 psq ˇˇ2 C N `ˇg 1 psq ˇˇ2 C N ˘ds,
the desired inequality (2.9) follows.

' Now assume that α " N and that t 0 ą 0. Let τ Ps0, t 0 s. On R `we define ϕ j and ψ j by (6.5). Let θ P C 8 pr0, τ r, Rq such that θp0q " 0 and θ 1 p0q " ϕ 1 1 p0q. For j P N and s Ps0, t 0 r we set ϕ j p´sq " θp´sq `ψj psq ´ψ1 psq.

In particular ϕ P C 1 ps ´τ, `8r; C N q and ϕ P 9 H 1 ps ´τ1 , `8r; C N q X 9 H 2 ps ´τ1 , `8r; C N q for all τ 1 Ps0, τ r. Finally we define u by (6.1). Then u is a solution of (2.1)-(2.4) on r0, τ r. Moreover for t P r0, τ r we have

Epu, tq ě ż `8 0 `ˇϕ 1 1 px ´tq ˇˇ2 `ˇψ 1 1 px `tq ˇˇ2 ˘dx ě ż 0 ´t ˇˇϕ 1 1 psq ˇˇ2 ds ě ż t 0 ˇˇθ 1 psq ˇˇds.
In particular we can choose θ in such a way that Epu; tq goes to `8 as t goes to τ .

Relationship with relativistic quantum mechanics

If the space variable x is restricted to a bounded interval and α is real, the classical interpretation of the wave equation (1.1) is that of a vibrating string, subject to a highly localised damping [START_REF] Bamberger | A model for harmonics on stringed instruments[END_REF] . Strictly speaking, the genuine damping (or friction) corresponds to negative α, while positive α models a supply of energy into the system. However, we use the word "damping" even in the more general situation whenever α has a non-zero real part, so that the physical system is not conservative.

The goal of this last section is to provide a physical motivation in the unconventional setting of unbounded geometries and/or non-real α. To this purpose, we recall a more or less well known relationship between the damped wave equation and the Dirac equation in relativistic quantum mechanics (see, e.g., [START_REF] Gesztesy | The damped string problem revisited[END_REF][START_REF] Gesztesy | Abstract wave equations and associated diractype operators[END_REF][START_REF] Cuenin | Eigenvalues of one-dimensional non-selfadjoint Dirac operators and applications[END_REF]). (Here the word relativistic stands for the original usage of the equation, recent years have brought motivations to consider the same mathematical problem for non-relativistic systems like graphene.)

For simplicity, let us restrict to the simplest graph N " 2, which can be identified with the real axis R. Instead of substituting pu, vq " pψ, B t ψq as above (2.7), let us write pξ, ηq " pB t ψ, B x ψ). Then the wave equation (1.1) is formally transferred to the first-order system B t ˆξ η ˙" iD α ˆξ η ˙with D α " ˆ´iαδ ´iB x ´iB x 0 ˙. (7.1)

Here D α is a Dirac-type operator considered in the Hilbert space L 2 pRq ˆL2 pRq. More specifically, D 0 with domain H 1 pRq ˆH1 pRq is the (self-adjoint) Dirac Hamiltonian modelling the propagation of relativistic (quasi-)particles in quantum mechanics. The perturbation `´iαδ 0 0 0 ˘(properly introduced via the jump condition ηp0 `q ´ηp0 ´q " ´αξp0q together with the continuity ξp0 `q " ξp0 ´q " ξp0q) represents neither a purely electric nor scalar potential, but it is self-adjoint (and therefore quantum-mechanically relevant) whenever α is purely imaginary. Moreover, the real part of α is potentially eligible in quasi-Hermitian quantum mechanics [START_REF] Krejčiřík | Pseudospectra in non-Hermitian quantum mechanics[END_REF].

It is interesting to compare the present model (7.1) with the δ-shell interaction Hamiltonian Dτ " ˆτ δ ´iB x ´iB x τ δ ˙(7.2)

intensively studied for real τ during the last decade (see [START_REF] Arrizibalaga | Shell interactions for Dirac operators[END_REF][START_REF] Ourmières-Bonafos | A strategy for self-adjointness of Dirac operators: Applications to the MIT bag model and delta-shell interactions[END_REF][START_REF] Holzmann | Dirac operators with Lorentz scalar shell interactions[END_REF][START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF] for counterparts of (7.2) in R 3 and its variants). Here Dτ is properly introduced via the transmission condition ˆτ {2 ´i ´i τ {2 ˙ˆξp0 `q ηp0 `q˙`ˆτ {2 i i τ {2 ˙ˆξp0 ´q ηp0

´q˙" ˆ0 0 ṫhat the functions from the operator domain of Dτ must satisfy. This definition makes sense even for complex τ . If τ is real, the operator Dτ is self-adjoint and its (purely continuous) spectrum coincides with the real axis R. In fact, the same spectral picture extends to be true for any τ P Czt˘2iu. However, if τ " 2i (respectively, τ " ´2i), then the whole upper (respectively, lower) complex plane belongs to the spectrum of Dτ . So we again see the presence of the not-any-more-magical number 2. Email address: david.krejcirik@fjfi.cvut.cz (J. Royer) Institut de mathématiques de Toulouse, Université Toulouse 3, 118 route de Narbonne, F-31062 Toulouse cedex 9, France Email address: julien.royer@math.univ-toulouse.fr

Proposition 3 . 5 .

 35 Let α P C ˘, z P C ¯and n P N ˚. Let h P L 2 pΩq and κ P C.

Lemma

(D. Krejčiřík )

 Krejčiřík Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00, Prague, Czech Republic

  For α P C, z P C and n P N ˚we define bounded operators Q α pzq and Q α,n pzq in LpH 1 pΓq, H ´1pΓqq by xQ α pzqψ, ϕy H ´1pΓq,H 1 pΓq " @ ψ 1 , ϕ 1 D L 2 pΓq ´αzψp0qϕp0q `z2 xψ, ϕy L 2 pΓq and xQ α,n pzqψ, ϕy H ´1pΓq,H 1 pΓq " @ ψ 1 , ϕ 1 D L 2 pΓq ´αz xρ n ψ, ϕy L 2 pΓq `z2 xψ, ϕy L 2 pΓq , for all ψ, ϕ P H 1 pΓq (the scalar products are linear on the left and semilinear on the right). Proposition 3.4. Let α P C ˘, z P C ¯and n P N ˚. Then Q α pzq and Q α,n pzq are invertible. Moreover, the norm › › Q α,n pzq LpH ´1pΓq,H 1 pΓqq is bounded uniformly in n P N We consider the case Repαq ě 0 and Repzq ă 0 (the case Repαq ď 0 and Repzq ą 0 is similar). Let θ " argpzq ´π P ‰ ´π 2 , π Re `e´iθ xQ α,n pzqw, wy H ´1pΓq,H 1 pΓq

	˚.
	Proof. 2	"	and η " argpαq P	"	´π 2 , π 2	‰ . For w P H 1 pΓq
	we have

´1›

›

  4.1. We have }ρ n ´δ} LpH 1 pΓq,H ´1pΓqq ÝÝÝÝÑ Let u, w P H 1 pΓq. For n P N ˚we have xpρ n ´δqu, wy H ´1pΓq,H 1 pΓq " Then there exists c ą 0 such that for all n P N ˚and u, w P H 1 pΓq we have ˇˇxpρ n ´δqu, wy H ´1pΓq,H 1 pΓq ˇˇď c ? n }u} H 1 pΓq }w} H 1 pΓq . Let α P C ˘and z P C ¯. We have }R α pzq ´Rα,n pzq} LpH ´1pΓq,H 1 pΓq ÝÝÝÝÑ The resolvent identity gives R α pzq ´Rα,n pzq " zR α pzq `αδ ´αρ n ˘Rα,n pzq. Since the size of R α,n pzq in LpH ´1pΓq, H 1 pΓqq does not depend on n P N ˚(recall Proposition 3.5), we conclude with help of Lemma 4.1.

	This concludes the proof of the lemma.
	Proposition 4.2. nÑ`8	0.
	Proof. Now we are in a position to establish Theorem 2.2.
	Proof of Theorem 2.2. For F P H 1 pΓq ˆL2 pΓq we have
	› › `pW α ´zq ´1 ´pW α,n ´zq ´1˘F › ›
					nÑ`8	0.
	Proof. N ÿ	ż `8	ρ n j pxq `uj pxqw j pxq ´up0qwp0q ˘dx.
		j"0	0
	For j P N we have
	ż `8	ρ n j pxq |pu j w j qpxq ´puwqp0q| dx ď	ż `8	ρ n j pxq ˇˇˇż	x	pu j w j q 1 psq ds ˇˇˇd x
	0				0		0
			ď	ż `8 0	ρ n j pxq	? x	› › pu j w j q 1 › › L 2 pR

˚q dx ď }pu j w j q 1 } L 2 pR ˚q ? n ż `8 0 ρ j pyq ? y dy. H À }R α,n pzq ´Rα pzq} LpH ´1pΓq,H 1 pΓqq `› › f 2 › › H ´1pΓq `}g} L 2 pΓq À }R α,n pzq ´Rα pzq} LpH ´1pΓq,H 1 pΓqq }F } H . Here the relation f À g means that there exists a constant C (independent of n) such that f ď Cg. By density of H 1 pΓq ˆL2 pΓq in H we get › › pW α ´zq ´1 ´pW α,n ´zq ´1› › LpH q À }R α,n pzq ´Rα pzq} LpH ´1pΓq,H 1 pΓqq , and we conclude with Proposition 4.2.

  .6) This is possible since detpM ´q " α ´N ‰ 0. We have ϕ P C 1 pR ˚q X 9H 1 pR ˚q X 9 H 2 pR ˚q.

	For s ą 0 we have
	ϕp´sq " ´M ´1 ´M`ψ psq Ý ÝÝ Ñ sÑ0	´M ´1 ´M`ψ p0q " 0 " ϕp0q,

and since pf, gq P DompW

Acknowledgments. The first author (D.K.) was supported by the EXPRO grant No. 20-17749X of the Czech Science Foundation. The second author (J.R.) was supported by the ANR LabEx CIMI (grant ANR-11-LABX-0040) within the French State Programme "Investissements d'Avenir".