domain: math spi

text:
Kaïs Ïs Ammari
email: kais.ammari@fsm.rnu.tn

Islam Boussaada
email: islam.boussaada@l2s.centralesupelec.fr

AND Silviu-Iulian Niculescu
email: silviu.niculescu@l2s.centralesupelec.fr

Sami Tliba
email: sami.tliba@l2s.centralesupelec.fr

Multiplicity manifolds as an opening to prescribe exponential decay : auto-regressive boundary feedback in wave equation stabilization

Keywords: 2010 Mathematics Subject Classification. 46, N20, 35B05, 93D15, 93D20 Multiplicity-induced-dominancy (MID), Partial pole placement, quasipolynomials, PDE control, boundary delay (model), wave equation, Exponential decay KA ÏS AMMARI, ISLAM BOUSSAADA, SILVIU-IULIAN NICULESCU, 4. P3δ Software 31

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

Introduction

Since the work of d'Alembert published more than 275 years ago and devoted to the vibrating string problem, the one-dimensional (two-way) wave equation is a subject of recurring interest, and there is a large literature on the qualitative and quantitative properties of its solutions. Surprisingly, such a simple model can be derived in a large variety of physical settings, and represents the simplest example of second-order hyperbolic partial differential equations (PDEs) model. It plays more than an important role in continuum/quantum mechanics (disturbances space-traveling over time), geophysics (e.g., modeling seismic waves), oceanography physics (dynamics of the ocean waves), electrical engineering (see, for instance, models representing lossless propagation phenomena through transmission lines) and control engineering (e.g. vibration control problems), to cite only a few.

In this context, the existing links between hyperbolic PDEs and delay-differential equations of neutral type should be pointed out * . More precisely, as shown by Abolinia and Myshkis [START_REF] Abolinia | Mixed problem for an almost linear hyperbolic system in the plane (in russian)[END_REF] and Cooke and Krumme [START_REF] Cooke | Differential-difference equations and nonlinear partial-boundary value problems for linear hyperbolic partial differential equations[END_REF] in the 60s, integrating along the characteristics of the boundary value problems for hyperbolic PDE leads to such neutral DDEs-based models. Interestingly, such results seem to insist (for the first time) on the functional character of the corresponding (neutral delay-differential) equations generated by the DAlembert method in the sense that they are considered as legitimate but autonomous mathematical objects. Furthermore, the properties of solutions of the mixed initial-boundary value problem can be established by using the results obtained for the functional equation and a representation formula of the solutions of the mixed problem in terms of the solutions of the associated equation.

In the sequel, we will exploit both model representations in the context of boundary control of the wave equation, and we will explicitly emphasize the existing links between the model parameters and the largest multiplicity of the roots of the corresponding characteristic function. * see, e.g., [START_REF] Hale | Introduction to functional-differential equations[END_REF] and the references therein for appropriate definitions and classification of DDEs In both retarded and neutral delay-differential equations (DDEs) cases, the generic maximal allowable multiplicity of a characteristic root induces dominancy, that is, it simply represents the rightmost characteristic root. The characterization of such a property called multiplicity induced dominancy (MID) can be found in [START_REF] Mazanti | Multiplicity-induced-dominancy for delay-differential equations of retarded type[END_REF][START_REF] Boussaada | The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of kummer functions[END_REF].

In controlling the wave equation, the proposed control law includes appropriate past information from both input and output signals and we called it an auto-regressive system due to the way the signals appear in the system's dynamic. From the control point of view, the corresponding controller called QPB (from quasipolynomial-based) is easy to implement. Furthermore, it belongs to the class of low-complexity infinite-dimensional controllers since on the one hand it includes a reduced number of parameters, and it is infinite-dimensional due to the presence of the delay terms in the model, on the other hand. Such a (linear) controller needs to be well-posed and formally stable † . In fact, it was emphasized in [START_REF] Loiseau | Neutral-type time-delay systems that are not formally stable are not bibo stabilizable[END_REF] that neutral systems that are not formally stable cannot be stabilized in the sense of bounded-input/boundedoutput. As a matter of fact, in the 70s, [START_REF] Barman | L2-instability of linear time-invariant distributed feedback systems perturbed by a small delay in the loop[END_REF] pointed out the existence of such a sensitivity phenomenon ‡ with respect to "small" delay perturbations, and such a problem was largely covered in the open literature during the period 80s-90s. For instance, amongst others, conditions for guaranteeing stability robustness of the closed-loop systems can be found in [START_REF] Logemann | The effect of small delays in the feedback loop on the stability of neutral systems[END_REF][START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF], where the authors proved the robustness in terms of spectral radius of some appropriate well-posed and regular transfer function.

As expected, if the dynamics of the closed-loop system include more than one delay, the problem becomes hard to deal with. For example, some strange and unexpected phenomena can appear and, in particular, the sensitivity of the (exponential) stability with respect to the delays ratio is one of the interesting cases treated in the open literature is of utmost importance within the control framework and amply treated in the open literature. By analogy with the interference phenomenon in Physics § , such a phenomenon is called delay interference, and it was first observed and discussed by MacDonald in the 80s [START_REF] Macdonald | An interference effect of independent delays[END_REF].

In the retarded case, the characterization of such a property and its links with the delays-independent stability is proposed in [START_REF] Michiels | Characterization of delay-independent stability and delay-interference phenomena[END_REF]. Finally, although the extension to neutral systems is not explicitly discussed, [START_REF] Michiels | Characterization of delay-independent stability and delay-interference phenomena[END_REF] addresses also the delay interference problem in the case of the continuous-time delay-difference equation including multiple delays, and a few examples are presented. Surprisingly, such a phenomenon also appears in control theory as the controller structure may induce delay interference for the closed-loop system. For instance, in controlling the wave equation, [START_REF] Wang | Wave equation stabilization by delays equal to even multiples of the wave propagation time[END_REF] proposed a boundary control law including a single delay block, and despite the simplicity of the controller, the closed-loop system is subject to delay interference, and it lacks robustness with respect to the delays. † that is all the poles of its transfer function have negative real part ‡ an infinitesimal "small" delay can destabilize a dynamical system § In Physics, it represents the combination of two or more wave motions to form a resultant wave in which the displacement is reinforced or canceled.

The contribution of the paper is threefold. First, the use of a QPB controller combined with the idea of forcing the multiplicity allows for avoiding the delay sensitivity phenomena mentioned above (with respect to "small" delays or/and delays ratio). In other words, the controller depicts robustness properties with respect to the QPB delay parameter. Second, by exploiting the delay as a control parameter, an explicit estimation of the decay rate is derived and such an estimation corresponds to the generic MID case for some values of the delay parameter. It should be mentioned that due to the structure of the continuous-time delay-difference equation, forcing multiplicity does not necessarily lead to the dominance but to some explicit bounds regarding the location of the rightmost characteristic root. Third, the QPB controller is explicitly applied to the longitudinal vibration control of a long drill pipe transmission line and a numerical study shows the effectiveness of the method. If the use of QPB in vibration control was already proposed in the literature by some of the authors of this paper [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system[END_REF][START_REF] Tliba | Active vibration control through quasi-polynomial based controller[END_REF], it should be mentioned that its application to the wave equation represents a novelty.

Finally, to the best of the authors' knowledge, such an approach was not considered in the literature and, as such, it also represents a novelty. We believe that the underlying ideas of the proposed methodology are useful in the study of the dynamical behavior of other classes of infinite-dimensional systems.

The remaining of the paper is organized as follows. Section 2 provides a brief overview on the MID property, as well as a comprehensive example illustrating the benefits of a multiplicity manifold in stability analysis and stabilization even when the MID fails. Section 3 is dedicated to recalling the recent results on the control of the wave equation with delay in the boundary and ends with the problem statement.

For the sake of rigor, the well-posedness of the considered problem is provided in Section 4. Section 5 exhibits conditions under which the corresponding semigroup decays exponentially to the null steady state. A spectral formulation via DAlemberts formula is carried out in Section 6 where the system of partial differential equations is transformed into a functional difference equation. In Section 7, we present a summary of standard results on the stability of such difference equations and unfold the main results of the present paper. Namely, the MID property is investigated: in particular, values of the time delay which is multiple of the wave propagation time are addressed. Next, when the MID fails, we propose a general strategy to prescribe a lower bound for the solution's exponential decay rate. Finally, as an illustration of the proposed methodology, in Section 8 we provide a numerical study of the problem of longitudinal vibration control of a long drill pipe transmission line. The paper is concluded by a discussion on software perspectives.

On the control structure and strategy: MID property and on beyond

Since the 30s and the theory of servomechanisms proposed by Hazen [START_REF] Hazen | Theory of servo-mechanisms[END_REF], it is commonly accepted that, in most of practical situations, the presence of time-delays in dynamical processes is synonymous with instability and bad behaviors. However, there are cases where the delay in the input can induce a stabilizing effect on the dynamics of the system, and the idea of exploiting the delay as a control parameter is not new (see, e.g., [START_REF] Niculescu | Delay Effects on Stability. A Robust Control Approach[END_REF][START_REF]Stability, Control, and Computation for Time-Delay Systems. An Eigenvalue-Based Approach[END_REF], and the references therein). For instance, at the end of the 70s, [START_REF] Suh | Proportional minus delay controller[END_REF] introduced the so-called "proportional-minus-delay" (PMD) controller and showed that such a controller can replace the well-known PD-controller (quick responses to input changes) with less sensitivity to highfrequency noise. Such a controller was further implemented in various case studies including, amongst others, the well-known inverted pendulum (see, e.g., [START_REF] Sieber | Extending the permissible control loop latency for the controlled inverted pendulum[END_REF][START_REF] Boussaada | Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains[END_REF], to cite only a few).

In this work, we are exploiting a low-complexity controller structure called QuasiPolynomial-Based controller (QPB) which has been recently introduced in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system[END_REF], the idea of which is close to the aforementioned PMD control in the sense that the controller structure is of PMD type however in both output and input signals, the delay being used as a design parameter to improve the overall behavior of the closed-loop system. In other words, given a generic dynamical system to control, modeled by the input-to-output mathematical model y(t) = G {v(t)} where G {•} denotes the appropriate input-to-output mathematical operator of this system, y(t) ∈ R is its measured output and v(t) ∈ R its controlled input. For constructing the controlled input v at a given time value t, we need an appropriate past information from both the input v and the output y in addition to the present information given by the output. Hence, the QPB controller consists in the generic output feedback control law which is defined by the following continuous-time delay-difference equation relating y to v in the following fashion:

(2.1) v(t) = -α v(t -τ) + β y(t) + γ y(t -τ),
where α, β and γ are some real scalar parameters with the constraint γ α β and τ is a positive scalar corresponding to the delay parameter of the QPB controler. Relation (2.1) is nothing but an auto-regressive equation, defining a dynamical system with its own actual and past behavior.

Lemma 2.1. Given the control law in (2.1) for any τ ∈ R. If γ = α β, then (2.1) is equivalent to

(2.2) v(t) = β y(t). Proof. Suppose γ = α β for α, β ∈ R. Then (2.1) is equivalent to v(t) -β y(t) = -α (v(t -τ) -β y(t -τ)) .
By defining f (t) := v(t) -β y(t), the previous equation reads as

f (t) = -α f (t -τ).
Since this last relation holds for any t, τ ∈ R, the only solution for the above difference equation is f (t) ≡ 0, leading so to (2.2). Indeed, if α = 0, then f (t) = 0 for all t ∈ R. Now, for α 0, the previous equation is equivalent to having f (t 1 ) = α f (t 2 ) for all t 1 , t 2 ∈ R. For any t 3 t 1 , t 2 one also has f (t 1 ) = α f (t 3 ), so that

f (t 3 ) = f (t 2 ) =: f 0 ∈ R. Since we also must have f 0 = α f 0 with α 0, then f 0 = 0. Reciprocally, if (2.2)
is true for all t ∈ R, then for any τ ∈ R,

v(t -τ) = β y(t -τ).
By combining this last relation with (2.2), one gets (2.1) by defining γ := α β.

Despite its simplicity in terms of construction, such an infinite-dimensional controller induces good behaviors in closed loop and has proven its effectiveness through various applications in active vibration damping. Indeed, in some previous works, the authors have already considered the active vibration control of some specific mechanical structures described by the PDEs of continuous mechanical structures, via the same controller as the one proposed in this work. It was the case for a beam-like one in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system[END_REF] and an axisymmetric membrane, both equipped with a piezoelectric sensor and a piezoelectric actuator that are collocated. However, the PDEs have been first discretized via the Finite Element Method in order to transform the systems' model into some finite-dimensional ones. Then, the same controller has been designed to achieve the vibrations' damping by using the MID paradigm that is described below. The results have proven the efficiency of such a control structure, motivating its use's extension directly to the PDE framework.

Having to adjust the four QPB parameters for achieving some control objectives may be seen as an undue difficulty. However, it offers degrees of freedom for performance purposes with the low cost of adding an auto-regressive term. Notice that the QPB parameters' tuning has always been done through an analytical qualitative/quantitative framework called partial pole placement (PPP) consisting in the stabilization of the closed-loop solutions with a prescribed admissible decay rate, which is a great advantage in practical applications. As a matter of fact, in all generality, assigning the exact location of a spectrum composed of infinitely many eigenvalues with only finitely many parameters is impossible. Nevertheless, the strategy of PPP consists in tuning the parameters in order to assign only finitely many eigenvalues, while guaranteeing that the rightmost eigenvalue is among them. Hence, the negativity of the assigned rightmost eigenvalue certifies the exponential stability. While tuning parameters to assign some eigenvalues is typically a trivial interpolation question, guaranteeing dominance of the chosen eigenvalues is a difficult problem.

Exploring some erstwhile ideas, present for instance in [START_REF] Pinney | Ordinary difference-differential equations[END_REF], the seminal works [START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems: a link with Vandermonde and Birkhoff incidence matrices[END_REF][START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF] highlighted the fact that roots of quasipolynomials of high enough multiplicity tend to be dominant, in what came to be known as the multiplicity-induced-dominancy (MID) property. When available, this property can be helpful in the stabilization of time-delay systems since it suffices to select the system's free parameters in order to guarantee the existence of such a root of high multiplicity with a negative real part, and the MID property will ensure its dominance. While early works verified the MID property for some classes of time-delay systems, typically of low order, more recent papers, such as [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system[END_REF][START_REF] Fueyo | On the pole placement of scalar linear delay systems with two delays[END_REF][START_REF]The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions[END_REF][START_REF] Mazanti | Multiplicity-induced-dominancy for delay-differential equations of retarded type[END_REF][START_REF]Padé approximation and hypergeometric functions: A missing link with the spectrum of delay-differential equations[END_REF][START_REF] Boussaada | MID property for delay systems: Insights on spectral values with intermediate multiplicity[END_REF][START_REF] Balogh | Conditions for stabilizability of time-delay systems with realrooted plant[END_REF][START_REF] Ma | PID control design for first-order delay systems via MID pole placement: performance vs. robustness[END_REF][START_REF] Boussaada | Multiplicity-induced-dominancy in parametric second-order delay differential equations: analysis and application in control design[END_REF][START_REF] Benarab | Multiplicity-induced-dominancy property for second-order neutral differential equations with application in oscillation damping[END_REF], have extended those results to more general situations.

Since the multiplicity of spectral values is not important as such, instead of assigning a single root of high enough multiplicity, some recent works such as [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of timedelay systems: Vandermonde matrices and exponential decay[END_REF] have considered the assignment of several simple real roots and shown that, in several situations, the rightmost root among those assigned is dominant, consisting in what has been named the coexisting-real-roots-induced-dominancy (CRRID) property. Assigning several simple real roots instead of a single root of large multiplicity allows weaker constraints in the control design and less sensitivity to parameters' variations or uncertainties, as explored in the applications in [START_REF] Boussaada | New features of P3δ software. Insights and demos[END_REF] or [START_REF] Tliba | Active vibration control through quasi-polynomial based controller[END_REF].

The main ingredient behind most of the proofs of the MID and the CRRID properties described in the aforementioned references is an integral representation of the corresponding quasipolynomial when the roots are assigned. In the case of the MID, when assigning roots of maximal possible multiplicity, such an integral representation turns out to be nothing but the well-known Kummer confluent hypergeometric function, as proved in [START_REF] Mazanti | Multiplicity-induced-dominancy for delay-differential equations of retarded type[END_REF]. In the case of the MID property with multiplicity smaller than the maximal one, the integral representation involves linear combinations of Kummer functions [START_REF] Boussaada | MID property for delay systems: Insights on spectral values with intermediate multiplicity[END_REF], while, for the CRRID properties, generalizations of hypergeometric functions are involved, see for instance [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of timedelay systems: Vandermonde matrices and exponential decay[END_REF].

Proofs of dominance then rely on results pertaining to the locations of roots of such special functions, such as those from [START_REF]Some remarks on the location of non-asymptotic zeros of Whittaker and Kummer hypergeometric functions[END_REF][START_REF] Wynn | On the zeros of certain confluent hypergeometric functions[END_REF][START_REF] Saff | On the zeros and poles of Padé approximants to e z . III[END_REF].

Let us define the quasipolynomial

Q : C × R * + → C as (2.
3) Q(s, τ) := P 0 (s) + P τ (s) e -τ s ,

where P 0 and P τ are polynomials with real coefficients given by (2.4)

P 0 (s) = s n + n-1 k=0 a k s k , P τ (s) = m k=0 α k s k .
The degree of Q is defined as the sum of the degrees of the involved polynomials plus the number of delays, i.e., deg (Q) := n + m + 1, see for instance [START_REF]The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions[END_REF].

Definition 2.2. We say that a characteristic root s 0 of Q satisfies the MID property if the following holds:

(i): its algebraic multiplicity (denoted by M(s 0 )) is larger than one;

(ii): it is dominant in the sense that all the characteristic roots s ∈ C of (2.3) are located to the left of s 0 in C -, i.e., s satisfies the condition (s) ≤ (s 0 ).

In other words, s 0 is the rightmost root of the spectrum and defines the spectral abscissa of the quasipolynomial Q.

In the case where M(s 0 ) = deg(Q), it was shown in [START_REF] Mazanti | Multiplicity-induced-dominancy for delay-differential equations of retarded type[END_REF] (in the case where m = n -1) and in [START_REF]The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions[END_REF] (in the general case m ≤ n) that s 0 satisfies the MID property. This "limit" case is also called generic MID or GMID for short. In the case where M(s 0 ) < deg(Q) and the MID hold, those cases refer to the Intermediate MID or IMID for short.

Notice that even if these remarkable properties are not valid, the existence of multiple roots or the coexistence of negative roots may consist in breaches for the location of the spectral abscissa as can be seen in the following example borrowed from [START_REF] Boussaada | Multiplicity-induced-dominancy in parametric second-order delay differential equations: analysis and application in control design[END_REF]. Let us revisit the following control problem where the aim is to establish a delayed-state-feedback controller capable of fast stabilizing and damping oscillations of solutions of a given second-order system:

(2.5)

         ü(t) + a 1 u(t) + a 0 u(t) = v(t), where v(t) = -α 0 u(t -τ) -α 1 u(t -τ).
More precisely, one is concerned with an open-loop system admitting one oscillating (possibly unstable) mode when the discriminant ∆ = a 2 1 -4 a 0 of the open-loop characteristic polynomial satisfies ∆ < 0. Notice that, by looking for the closed-loop system solution in the form u(t) = e s t or simply by employing the Laplace transform, one gets the corresponding characteristic equation in the complex domain which reads:

(2.6)

         Q(s, τ) = P 0 (s) + P 1 (s) e -τ s , where P 0 (s) = s 2 + a 1 s + a 0 and P 1 (s) = α 1 s + α 0 .
It has been shown in [START_REF] Boussaada | Multiplicity-induced-dominancy in parametric second-order delay differential equations: analysis and application in control design[END_REF]Theorem 4.1] that the only admissible quadruple spectral value is achieved at

s = s * where (2.7) s * = - a 1 + √ -2 ∆ 2 ,
with τ = τ * and the system parameters satisfy

τ * = 2 -2 ∆ , α 0 = 5 ∆-a 1 √ -2 ∆ 4 e s * τ * , α 1 = - √ -2 ∆ 2 e s * τ * .
In this case, s * corresponds to the spectral abscissa of the closed-loop (2.5) and consequently, it is said to satisfy the GMID property, see also [START_REF] Mazanti | Multiplicity-induced-dominancy for delay-differential equations of retarded type[END_REF][START_REF]The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions[END_REF]. Further, in the same work, it has been shown in [START_REF] Boussaada | Multiplicity-induced-dominancy in parametric second-order delay differential equations: analysis and application in control design[END_REF]Theorem 4.2] that the only admissible triple spectral values are (2.8)

s ± = -τ a 1 -4 ± √ 8 + τ 2 ∆ 2 τ ,
which exist only for ∆ ≥ -8/τ 2 and are achieved when the control parameters satisfy:

( ± )                  α 0 = a 1 s ± + a 1 2 2 - ∆ 2 + 6 a 1 + 10 s ± τ + 6 τ 2 e s ± τ , α 1 = 2 s 0 + a 1 + 2 τ e s ± τ .
Interestingly, it has been shown that if, ( + ) is satisfied then s = s + is the spectral abscissa corresponding to (2.6) that is the IMID is satisfied. Nonetheless, if ( -) is satisfied then s -cannot be the spectral abscissa corresponding to (2.6) as it is always dominated by a single real root s 0 . Further, in the particular case of ∆ * = -8 τ 2 , the triple root at s -merges with the aforementioned s 0 giving rise to the quadruple spectral value s * which is proven to be dominant. So that, as illustrated in Figure 1 in the study of the delayed PD controller in the problem of stabilization and quenching oscillations of the harmonic oscillator (no damping) with natural frequency ω 0 = √ 2

(2.9)

         ü(t) + 2 u(t) = v(t) where v(t) = -α 0 u(t -τ) -α 1 u(t -τ),
and controllers gains satisfying ( -), thanks to the continuity of the spectrum location with respect to parameters variation, for instance by decreasing delay τ = τ *with ∈ (0, 0.255), one is able to monitor for the delay value τ = τ crit for which the rightmost spectral value is s 0 = 0 (loss of the exponential stability). This gives a sensitivity chart of the spectrum distribution with respect to delay variation illustrating a remarkable splitting of the abscissa s * on the real axis into a breakaway branch s 0 , which becomes itself the rightmost spectral value, moving towards the origin of the complex plane and a dominated locked triple spectral value s -branch moving in the opposite direction.

the closed-loop system solution's decay rate s 0 (spectral abscissa location) for a delay range τ ∈ (τ crit , τ * ) where τ crit corresponds to the delay value achieving s 0 = 0. This provides a comprehensive example for which one is able to assign a lower-bound of the closed-loop solution's decay rate, even though the multiple spectral root is not the spectral abscissa.

Problem Statement

We study the boundary stabilization of a wave equation in (0, 1) ⊂ R including auto-regressive boundary conditions given by:

(3.10)

                             u tt (x, t) -u xx (x, t) = 0, x ∈ (0, 1), t > 0, u(0, t) = 0, t > 0, u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ (0, 1), u x (1, t) = -αu x (1, t -τ) + βu t (1, t) + γu t (1, t -τ), t > 0, u t (1, t -τ) = 0, u x (1, t -τ) = 0, t ∈ (0, τ).
The constant τ > 0 is a time delay, α, β and γ are real numbers corresponding to the QPB control law's gains and the initial data u 0 , u 1 are given functions belonging to suitable spaces that will be precised later.

It is worth noticing that any choice of the gains such that γ = α β must be rejected because it amounts to having a boundary condition (3.10) that would be reduced to

(3.11) u x (1, t) = β u t (1, t).
Such a case has already been considered previously by several works, as for example in [START_REF] Chen | Control and stabilization for the wave equation in a bounded domain[END_REF][START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF][START_REF] Lagnese | Note on boundary stabilization of wave equations[END_REF], and is known as a damping injection term.

The problem of boundary stabilization of the wave equation with time delay in the output feedback loop has been considered in several works; see for instance [START_REF] Auriol | Comparing advanced control strategies to eliminate stick-slip oscillations in drillstrings[END_REF] for the problem of damping torsional vibrations in a drilling system and [START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF] for a special emphasis on time delay equal to a multiple of the wave propagation time. More precisely, in [START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF], the system (3.10) has been considered with α = β = 0. Some remarkable and unexpected properties have been pointed out. In particular, a stabilizing effect of even multiples of the wave propagation time has been exhibited. However, odd multiples induce unstable closed-loop system dynamics. Furthermore, a lack of robustness to small perturbations in time delay has been reported.

The purpose of this paper is to study problem (3.10), in which two delayed terms act in the boundary conditions. To our best knowledge, the auto-regressive term weighted by coefficient α is the main novelty in such a problem. In recent years, one very active area of mathematical control theory has been the investigation of the delay effect in the stabilization of hyperbolic systems. Many authors have shown that delays can destabilize a system that is asymptotically stable in the absence of delays (see [START_REF] Benhassi | Feedback stabilization of a class of evolution equations with delay[END_REF][START_REF]Exponential energy decay of some coupled second order systems[END_REF][START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF][START_REF]Stabilization of the wave equation with boundary or internal distributed delay[END_REF][START_REF] Nicaise | Stabilization of second order evolution equations with unbounded feedback with delay[END_REF] for more details).

As it has been proved by Datko [39, Example 3.5], systems of the form (3.12)

           w tt (x, t) -w xx (x, t) -aw xxt (x, t) = 0, x ∈ (0, 1), t > 0, w (0, t) = 0, w x (1, t) = -kw t (1, t -τ) , t > 0,
where a, k and τ are positive constants become unstable for arbitrarily small values of τ and any values of a and k. In (3.12) and even in the presence of the strong damping -aw xxt , without any other damping, the overall structure can be unstable. This was one of the main motivations for considering problem (3.10).

Of course, the structures of problems (3.10) and (3.12) are different due to the nature of the boundary conditions in each problem.

Subsequently, Datko et al [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF] treated the following one-dimensional problem:

(3.13)

                       u tt (x, t) -u xx (x, t) + 2a u t (x, t) + a 2 u(x, t) = 0, 0 < x < 1, t > 0, u(0, t) = 0, t > 0, u x (1, t) = -k u t (1, t -τ), t > 0,
which models the vibrations of a string clamped at one end and free at the other, where u(x, t) is the displacement of the string. Also, the string is controlled by a boundary control force (with a delay) at the free end. They showed that, if the positive constants a and k satisfy

k e 2a + 1 e 2a -1 < 1,
then the delayed feedback system (3.13) is stable for all sufficiently small delays. On the other hand, if

k e 2a + 1 e 2a -1 > 1,
then there exists a dense open set D ⊂ (0, ∞) such that for each τ ∈ D, system (3.13) admits exponentially unstable solutions.

It is well known that in the absence of delay in (3.13), that is, for τ = 0, the system is uniformly asymptotically stable under the condition a 2 + k 2 > 0 and the total energy of the solution satisfies for all t > 0,

(3.14) E(t, u) := 1 0 u 2 t + u 2 x + a 2 u 2 dx ≤ C E (0, u) e -αt
for some positive constant α; see [START_REF]Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain[END_REF] for more details.

In [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF], the authors examined a wave equation system with a linear boundary damping term exhibiting a delay. Namely, they considered the following system

(3.15)                                              u tt -∆u = 0, x ∈ Ω, t > 0 , u(x, t) = 0, x ∈ γ 0 , t > 0 , ∂u ∂ν (x, t) = µ 1 u t (x, t) + µ 2 u t (x, t -τ) x ∈ γ 1 , t > 0 , u(x, 0) = u 0 (x), x ∈ Ω , u t (x, 0) = u 1 (x), x ∈ Ω , u t (x, t -τ) = g 0 (x, t -τ), x ∈ Ω, τ ∈ (0, 1) ,
and proved, under the assumption

(3.16) µ 2 < µ 1
(which means that the weight of the feedback with delay is smaller than the one without), that the null stationary state is exponentially stable. On the contrary, if (3.16) does not hold, they produced a sequence of delays for which the corresponding solution of (3.15) is unstable. The main approach used in [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF], is an observability inequality obtained by means of a Carleman estimate.

The case of time-varying delay (i.e. τ = τ(t) is a function depending on t) has been studied by Nicaise,

Valein and Fridman [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF] in one space dimension. In their work, an exponential stability result was given under the condition:

(3.17)

µ 2 < √ 1 -d µ 1 ,
where d is a constant satisfying

(3.18) τ (t) ≤ d < 1, ∀t > 0.
Delay effects arise in many applications and practical problems and it is well known that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay (e.g. [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF][START_REF] Lagnese | Note on boundary stabilization of wave equations[END_REF][START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF]).

Let us also cite the recent work of Ammari and Nicaise, [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF], in which the authors performed a complete study of the stabilization of elastic systems by collocated feedback with or without delay, see also [3,

.

Well-posedness of problem (3.10)

In this section, we shall first transform the delay boundary conditions by introducing a new unknown.

Then, a semigroup approach and the Lumer-Phillips' theorem shall be used to prove the existence and the uniqueness of the solution of the problem (3.10).

Note that the well-posedness of the evolution equations with delay is not always obtained. Recently, Dreher, Quintilla and Racke have shown some ill-posedness results for a wide range of evolution equations with a delay term [START_REF] Dreher | Ill-posed problems in thermomechanics[END_REF].

In To overcome the problem of the boundary delay, we introduce the new variables:

(4.19) z (ρ, t) := u t (1, t -τρ) , w(ρ, t) := u x (1, t -τρ), ρ ∈ (0, 1) , t > 0.
Then, we have

τz t (ρ, t) + z ρ (ρ, t) = 0, in (0, 1) × (0, +∞) , (4.20) τw t (ρ, t) + w ρ (ρ, t) = 0, in (0, 1) × (0, +∞) . (4.21)
Hence, problem (3.10) is equivalent to:

(4.22)                                                            u tt (x, t) -u xx (x, t) = 0, x ∈ (0, 1), t > 0, u(0, t) = 0, t > 0, u x (1, t) + αw(1, t) = βu t (1, t) + γz(1, t), t > 0 τz t (ρ, t) + z ρ (ρ, t) = 0, ρ ∈ (0, 1) , t > 0 τw t (ρ, t) + w ρ (ρ, t) = 0, ρ ∈ (0, 1) , t > 0 z(0, t) = u t (1, t), w(0, t) = u x (1, t), t > 0 u(x, 0) = u 0 (x),
x ∈ (0, 1),

u t (x, 0) = u 1 (x), x ∈ (0, 1), z(ρ, 0) = 0, w(ρ, 0) = 0, ρ ∈ (0, 1).
The first natural question is the existence of solutions for the problem (4.22). In the following, we shall give a sufficient condition guaranteeing that the above problem is well-posed.

For this purpose, we use a semigroup formulation of the initial-boundary value problem (4.22). If we denote V := (u, u t , z, w) T , we define the energy space:

H := H 1 (0, 1) × L 2 (0, 1) × L 2 (0, 1) × L 2 (0, 1),
where H 1 (0, 1) := u ∈ H 1 (0, 1)/ u(0) = 0 , and (., .) denotes the scalar product in L 2 (0, 1), i.e. (u, v) =

1 0 u(x)v(x)dx.
Clearly, H is a Hilbert space with respect to the inner product

(4.23) V 1 , V 2 H = 1 0 u 1 x u 2 x dx + 1 0 v 1 v 2 dx + ξ 1 1 0 z 1 z 2 dρ + ξ 2 1 0 w 1 w 2 dρ for V 1 = (u 1 , v 1 , z 1 , w 1 ) T , V 2 = (u 2 , v 2 , z 2 , w 2 )
T and ξ 1 , ξ 2 > 0 nonnegative real numbers to be defined later.

Therefore, if V 0 ∈ H and V ∈ H , the problem (4.22) is formally equivalent to the following abstract evolution equation in the Hilbert space H :

(4.24)          V (t) = A V(t), t > 0, V (0) = V 0 ,
where denotes the derivative with respect to time t, V 0 := (u 0 , u 1 , 0, 0) T and the operator A is defined by:

A                               u v w z                               =                             v u xx -τ -1 z ρ -τ -1 w ρ                             .
The domain of A is the set of V = (u, v, z, w) T such that:

(u, v, z, w) T ∈ H 1 (0, 1) ∩ H 2 (0, 1) × H 1 (0, 1) × H 1 (0, 1) × H 1 (0, 1), (4.25)

v(1) = z(0), u x (1) = w(0), u x (1) + αw(1) = βv(1) + γz(1). (4.26)
If (α, β, γ, ξ 1 , ξ 2 ) satisfies the following conditions:

(4.27) 0 < α < 1, β < γ 1 + 2α 2 1 -α 2 < 0, 2τ α 2 γ 2 β(α 2 -1) - γ 2 2β < ξ 1 < -βτ, ξ 2 = - τ β ,
then, the well-posedness of problem (4.22) is ensured by: Theorem 4.1. Let V 0 ∈ H , then there exists a unique solution V ∈ C (R + ; H ) of problem (4.24). More-

over, if V 0 ∈ D (A ), then V ∈ C (R + ; D (A )) ∩ C 1 (R + ; H ) .
Proof. The proof of the existence and the uniqueness of the solution of problem (4.24) relies on the semigroup approach and the Lumer-Phillips' theorem.

Indeed, let V = (u, v, w, z) T ∈ D (A )
. By definition of the operator A and the scalar product of H , we have:

A V, V H = 1 0 u x v x dx + 1 0 vu xx dx - ξ 1 τ 1 0 zz ρ dρ - ξ 2 τ 1 0 ww ρ dρ.
By Green's formula we obtain:

(4.28) A V, V H = u x (1)v(1) - ξ 1 2τ z 2 (1) -z 2 (0) - ξ 2 2τ w 2 (1) -w 2 (0) .
Hence, equation (4.28) becomes:

(4.29) A V, V H = (-αw(1) + βz(0) + γz(1))z(0) -ξ 1 2τ z 2 (1) -z 2 (0) -ξ 2 2τ w 2 (1) -w 2 (0) = -ξ 1 2τ z 2 (1) -ξ 2 2τ w 2 (1) + β + ξ 1 2τ z 2 (0) + ξ 2 2τ w 2 (0) -αw(1)z(0) + γz(1)z(0) = -ξ 1 2τ z 2 (1) -ξ 2 2τ w 2 (1) + β + ξ 1 2τ z 2 (0) + ξ 2 2τ (-αw(1) + βz(0) + γz(1)) 2 -αw(1)z(0) + γz(1)z(0) = -ξ 1 2τ z 2 (1) -ξ 2 2τ w 2 (1) + β + ξ 1 2τ z 2 (0) + ξ 2 2τ α 2 w 2 (1) + ξ 2 2τ β 2 z 2 (0) + ξ 2 2τ γ 2 z 2 (1) -ξ 2 αγ
τ w(1)z(1). To treat the last terms in the above equation,Young's inequality gives, for all ε > 0:

(4.30) A V, V H ≤ -ξ 1 2τ + ξ 2 2τ γ 2 + ε ξ 2 τ |αγ| z 2 (1) + β + ξ 1 2τ + ξ 2 2τ β 2 z 2 (0)+ -ξ 2 2τ + ξ 2 2τ α 2 + ξ 2 2ε |αγ| τ w 2 (1).
According to condition (4.27) and by choosing αγ

α 2 -1 < ε < ξ 1 2τ + γ 2 2β β αγ , we obtain (4.31) A V, V H ≤ 0 .
Thus the operator A is dissipative. Now, we want to show that ∀λ > 0, λI -A is surjective. To this end, it is clear that it suffices to show that λI -A is surjective for some λ > 0.

For

F = ( f 1 , f 2 , f 3 , f 4 ) T ∈ H , let V = (u, v, z, w) T ∈ D (A ) be a solution of (λI -A ) V = F,
where

λu -v = f 1 , (4.32) λv -u xx = f 2 , (4.33) λz + 1 τ z ρ = f 3 , (4.34) λw + 1 τ w ρ = f 4 . (4.35)
To find V = (u, v, z, w) T ∈ D (A ) solution of the system (4.32)-(4.35), we suppose that u is determined with the appropriate regularity. Then, from (4.32), we get:

(4.36) v = λu -f 1 .
Therefore, from the compatibility condition on γ 1 AND equations (4.26), we determine z(0) and w(0) by: (4.37)

z(0) = v(1) = λu(1) -f 1 (1), w(0) = u x (1).
Thus, from equations (4.34)-(4.35), z and w are given, respectively, by: (4.38)

         z(ρ) = (λu(1) -f 1 ( 1
)) e -λρτ + e -λρτ ρ 0 e λρs f 3 (s) ds , ρ ∈ (0, 1), w(ρ) = u x (1) e -λρτ + e -λρτ ρ 0 e λρs f 4 (s) ds , ρ ∈ (0, 1).

Consequently, knowing u, we may deduce v by equation (4.36), and z and w by system (4.38).

We recall that since V = (u, v, z, w) T ∈ D (A ), from equations (4.33) and (4.36), u must satisfy:

(4.39) λ 2 u -u xx = f 2 + λ f 1 , in (0, 1)
with the boundary conditions (4.40)

               u(0) = 0, 1 + αe -λτ u x (1) -λ β + γe -λτ u(1) = -(β + γe -λτ ) f (1) + 1 0 e -λ(τ-s) (-α f 4 (s) + γ f 3 (s)) ds.
Using the previous expression, we get:

(4.41) u(x) =                      -(β+γe -λτ ) f (1)+ 1 0 e -λ(τ-s) (-α f 4 (s)+γ f 3 (s)) ds λ(1+αe -λτ )ch(λ)-λ(β+γe -λτ )sh(λ) + - 1 0 (1+αe -λτ )+(β+γe -λτ )sh(λ(1-y)) ( f 2 (y)+λ f 1 (y)) dy λ(1+αe -λτ )ch(λ)-λ(β+γe -λτ )sh(λ) sh(x)+ 1 λ x 0 sh(λ(x -y)) ( f 2 (y) + λ f 1 (y)) dy, x ∈ (0, 1).
As a result, u ∈ H 2 (0, 1) ∩ H 1 (0, 1) verifies (4.39)-(4.40), so we recover u and v. Next, by (4.38) we obtain z, w. Thus, we have found V = (u, v, z, w) T ∈ D (A ) solution of (λI -A ) V = F.

The operator A generates a C 0 semigroup of contractions e tA on H . Owing to the Lumer-Phillips' theorem, there exists a unique solution V ∈ C (R + ; H ) of the problem (4.24). This completes the proof of Theorem 4.1.

Asymptotic behavior

In this section, we show that under condition (4.27), the semigroup e tA decays exponentially to the null steady state. To this end, our technique is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

Theorem 5.1. Suppose that condition (4.27) holds. Then, there exist constants C, ω > 0 such that, for all V 0 ∈ H, the semigroup e tA satisfies the following estimate

(5.42) e tA V 0 H ≤ C e -ωt V 0 H , ∀ t > 0.
Proof of theorem 5.1. We shall use the following frequency domain theorem for uniform stability from Huang-Prüss [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]

(iδI -L) -1 L(H) < ∞,
where ρ(L) denotes the resolvent set of the operator L.

First, we consider the point spectrum of A .

Lemma 5.3. The spectrum of A contains no point on the imaginary axis.

Proof. Since A admits a compact resolvent, its spectrum σ(A ) consists only of eigenvalues of A . We shall show that the equation Taking the inner product of (5.45) with Z, inequality (4.30) entails : z(0) = 0, w(1) = 0 and z(1) = 0. Thus, we have z = 0 and w = 0, and since u x (1) = w(0), v(1) = z(0), we also obtain u = 0 and v = 0. As a consequence, the only solution of (5.45) is the trivial one.

The following lemma shows that (5.44) holds with L = A .

Lemma 5.4. The resolvent operator of A satisfies condition (5.44).

Proof. Suppose that condition (5.44) is false. By the Banach-Steinhaus Theorem (see [START_REF] Brezis | Analyse fonctionnelle. Théorie et applications[END_REF]), there exist a sequence of real numbers δ n → +∞ and a sequence of vectors

Z n = (u n , v n , z n , w n ) t ∈ D(A ) with Z n H = 1 such that (5.50) ||(iδ n I -A )Z n || H → 0 as n → ∞, i.e.,
iδ n u nv n ≡ f n → 0 in H 1 (0, 1), (5.51) iδ n v nu n ≡ g n → 0 in L 2 (0, 1), (5.52)

iδ n z n + 1 τ ∂ ρ z n ≡ h n → 0 in L 2 (0, 1), (5.53) iδ n w n + 1 τ ∂ ρ w n ≡ k n → 0 in L 2 (0, 1). (5.54)
Our goal is to derive from (5.50) that ||Z n || H converges to zero, in order to obtain a contradiction.

First step. Notice that we have

(5.55) ||(iδ n I -A )Z n || H ≥ | ( (iδ n I -A )Z n , Z n H ) |.
Then, by inequality (4.30) and limit (5.50), we derive (5.56)

z n (1) → 0, v n (1) = z n (0) → 0, w n (1) → 0.
This further leads, by limit (5.51) and the trace theorem, to (5.57) δ n u n (1) → 0.

Moreover, since Z n ∈ D(A ), by limits (5.56) we get (5.58) u n (1) → 0.

Recalling that z n (ρ) = z n (0) e -iδ n τρ + τ ρ 0 e -iδ n τ(ρ-s) h n (s) ds, (5.59)

w n (ρ) = w n (1) e -iδ n τ(ρ-1) -τ 1 ρ
e -iδ n τ(ρ-s) k n (s) ds, (5.60) limits (5.56) imply that (5.61) z n → 0 in L 2 (0, 1), w n → 0 in L 2 (0, 1).

Second step. Now, let us express v n as a function of u n from equation (5.51) and substitute it into (5.52)

to get

(5.62) -δ 2 n u n -u n = iδ n f n + g n .
Next, we take the inner product of the above equation (5.62) with xu in L 2 (0, 1), we obtain that

1 0 (-δ 2 n u n (x) -u n (x))xu (x) dx = 1 0 (iδ n f n (x) + g n (x)) xu (x) dx = (5.63) 1 0 g n (x)xu n (x) dx - 1 0 iδ n f n (x)xu n (x) dx - 1 0 iδ n f n (x)u n (x) dx + iδ n u n (1) f n (1).
It is clear that the right-hand side of equality (5.63) converges to zero since f n , g n converge to zero in H 1 and L 2 , respectively. By a straightforward calculation, we deduce that

1 0 -δ 2 n u n (x)xu n (x) dx = 1 2 1 0 δ 2 n u 2 n (x) dx - δ 2 n 2 u 2 n (1) and 1 0 -u n (x)xu n (x) dx = 1 2 1 0 (u n ) 2 (x) dx - (u n (1)) 2 2 .
This leads to

1 0 δ 2 n u 2 n (x) dx + 1 0 (u n ) 2 (x) dx -δ 2 n u 2 n (1) -(u n (1)) 2 -→ 0.
So, limits (5.57), (5.58) and (5.61), obviously contradict

Z n H = 1.
The two hypotheses of Lemma 5.2 are proved by Lemma 5.4. Then inequality (5.42) holds which ends the proof of Theorem 5.1.

A spectral formulation via D'Alembert's formula

In order to express the problem into a form more suited for control purposes with a spectral approach, the wave equation in system (3.10) is now considered through the D'Alembert formula which allows the solution of the wave equation to be written as:

(6.64) u(x, t) = 1 2 (u 0 (x + t) + u 0 (x -t)) + 1 2 x+t x-t u 1 (ξ) dξ.
Introducing the variable θ as follows:

(6.65)

               θ(x) = - 1 2 u 0 (-x) + 1 2 -x 0 u 1 (s) ds for x ∈ (-1, 0), θ(x) = 1 2 u 0 (x) + 1 2
x 0 u 1 (s) ds for x ∈ (0, 1), enables express u in terms of θ :

u(x, t) = θ(x + t) -θ(t -x), (6.66) so that u t (x, t) = θ (x + t) -θ (-x + t), (6.67) u x (x, t) = θ (x + t) + θ (-x + t). (6.68)
Thus, the control function on the boundary x = 1, say v(t) := u x (1, t), is then given by (6.69)

v(t) = θ (1 + t) + θ (-1 + t).
By defining η := 1 + t, one gets:

v(η -1) = θ (η) + θ (η -2), (6.70) so that v(η -1 -τ) = θ (η -τ) + θ (η -2 -τ), (6.71)
Furthermore, the feedback control law's expression in (3.10) becomes:

(6.72) v(η -1) + α v(η -1 -τ) = β θ (η) -θ (η -2) + γ θ (η -τ) -θ (η -2 -τ) .
Hence, one can consider the expression of v in equations (6.70) and (6.71) and substitute them into equation (6.72). As a consequence, in the case where β 1, we get (6.73)

θ (η) + 1 + β 1 -β θ (η -2) + α -γ 1 -β θ (η -τ) + α + γ 1 -β θ (η -2 -τ) = 0.
which is a continuous time-difference equation that is exclusively expressed in terms of θ (x).

Remark 6.1. Note that the case β = 1 is not consistent with the dissipativity conditions established in (4.27) in Section 4. Therefore, β 1 is the only case considered in the sequel.

A parametric stability analysis via a spectral approach

Let s ∈ C be the Laplace variable of the Laplace Transform operator and the delay such that τ ∈ R * + . Applying the Laplace transform to equation (6.73), one gets the following characteristic function:

(7.74) Q(s, τ) := 1 + 1 + β 1 -β e -2s + α -γ 1 -β e -τs + α + γ 1 -β e -(τ+2)s .
The exponential stability of the closed-loop system in (6.73) can be achieved if, and only if, the roots of (7.74) have a strictly negative real part. Moreover, a decay-rate of σ, with σ > 0, for θ (η) converging to zero can be obtained if, and only if, -σ is an upper-bound on the real part of any root of (7.74). In fact, the well-known Hale-Silkowski criterion completely characterizes the exponential stability of (6.73), see for instance [46, Chapter 9, Theorem 6.1] and for further refinement and generalization of the above result see [START_REF] Chitour | Stabilization of persistently excited linear systems[END_REF]. As a matter of fact, since the three involved delays in equation (6.73) (2, τ, and τ + 2) are rationally dependent, then one can transform (6.73) into an equivalent matrix equation involving only two delays 2 and τ. By denoting

(7.75) f (t) :=          θ (t) θ (t -2)          , A :=          1+β β-1 0 α+γ β-1 0          , B :=          α-γ β-1 α+γ β-1 0 0          , equation (6.73
) is equivalent to the system:

(7.76) f (t) = A f (t -2) + B f (t -τ).
Furthermore, if the delays τ and 2 are rationally independent, then using [45, Chapter 9, Theorem 6.1] one concludes that (7.76) is stable locally in the delays if, and only if, sup

ϕ 1 , ϕ 2 ∈[0, 2 π] µ A e i ϕ 1 + B e i ϕ 2 < 1,
where µ(•) designates the spectral radius of a given square matrix.

This entails the following necessary and sufficient stability conditions:

               1 + 1 + β β -1 > 2 α β -1 , 1 - 1 + β β -1 > 2 γ β -1 ,
which reduce for β < 0 to the following set of necessary and sufficient conditions:

         |γ| < 1, |α| < -β.
However, if the delays τ and 2 are rationally dependent, then one can further transform (7.76) into an equivalent augmented matrix equation and again using [45, Chapter 9, Theorem 6.1] one can obtain explicit necessary and sufficient conditions.

Despite the importance of the result recalled above from a qualitative point of view, it does not unfortunately provide any information on the solution decay rate.

In the following section, we shall exploit the manifold of spectral values' multiplicities to get some insights on the solutions' decay rates.

7.1. On the MID paradigm . Notice that, in the case of a quasipolynomial with more than one exponential term, such as the one under consideration in (7.74), the characterization of the MID property remains an open problem, to which we partially contribute in the present work.

As emphasized in Section 2, the MID property consists in conditions under which a multiple spectral value is dominant. More precisely, in this section, we shall provide some configurations in which the GMID applies; this corresponds to the dominancy of spectral values with a multiplicity which is equal to the degree of the considered quasipolynomial. Notice that such a degree may vary when some coefficients are set to be zero or when some delays are set to be equal. In particular, the case τ = 2 will be considered separately since it allows to decrease the degree of the quasipolynomial Q.

In this case, the quasipolynomial Q reads

(7.77) Q(s, 2) = 1 + (1 + β + α -γ) e -2s 1 -β + (α + γ) e -4s 1 -β
which admits a degree equal to two for γ α and γ α + β + 1. We are able to enunciate the following:

Theorem 7.1. Consider the quasipolynomial Q(•, τ) given by (7.74) and let τ = 2.

A given real number s 0 is a double root of (7.77) if, and only if,

(7.78)                  α = γ 2 e -2 s 0 -e -4 s 0 -1 + 2 e -4 s 0 -2 e -2 s 0 -1 , β =
(2 γ -1) e -4 s 0 -2 e -2 s 0 -1 e -4 s 0 -2 e -2 s 0 -1 .

If (7.78) is satisfied then the GMID holds, that is, s 0 corresponds to the spectral abscissa of the quasipolynomial Q(•, 2) given by (7.77). Furthermore, all zeros of (7.77) are double and lie on the vertical axis

(s) = s 0 .
Proof. Since the degree of the quasipolynomial (7.77) is equal to 2, let us investigate its admissible double roots. Let s 0 ∈ R be a double root of (7.77); that is Q(s 0 , 2) = Q s (s 0 , 2) = 0. Thanks to the linearity of the two equations obtained in the parameters α and β, one easily deduces conditions (7.78).

By substituting the resulting conditions into the expression of Q(•, 2) one gets for arbitrary γ:

Q(s, 2) = e -2 (s-s 0 ) -1 2 ,
which means that each root of Q(•, 2) is necessarily double. Furthermore, let s = x + iω s 0 be a root of Q(•, 2) then x and ω satisfy:

         e 2 (x-s 0 ) sin (2 ω) = 0 e -2 (x-s 0 ) cos (2 ω) = 1.
From the first equation one gets that ω = k 2 π with k ∈ Z * and the second equation yields x = s 0 , which concludes the proof. Now, let us consider again the quasipolynomial (7.74) where α |γ| and |β| 1, i.e., the case where the quasipolynomial's degree is equal to three. Theorem 7.2. Consider the quasipolynomial Q(•, τ) given by (7.74) and let τ 2.

A given real number s 0 is a triple root of (7.74) if, and only if,

(7.79)                              α = (τ -2) e 2 s 0 -(τ + 2) (τ + 2) e 2 s 0 -(τ -2) e τ s 0 , β = (τ + 2) e 2 s 0 + (τ -2) (τ + 2) e 2 s 0 -(τ -2) , γ = (τ -2) e 2 s 0 + (τ + 2) (τ + 2) e 2 s 0 -(τ -2) e τ s 0 .
If (7.79) is satisfied and τ is an even integer strictly greater than 2, then the GMID holds, that is, s 0 corresponds to the spectral abscissa of the quasipolynomial Q(•, 2 k) given by (7.74).

To prove the above Theorem, one needs the following two results: the first one which is a classical Theorem of A. Cohn [START_REF] Cohn | Über die anzahl der wurzeln einer algebraischen gleichung in einem kreise[END_REF] and the second is due to Eneström-Kakeya [START_REF] Kakeya | On the limits of the roots of an algebraic equation with positive coefficients[END_REF][START_REF] Eneström | Remarque sur un théorème relatif aux racines de l'équation {a n}{xˆn}+{an-1}{xn-1}++{a 1} x+{a 0}= 0 où tous les coefficientes a sont réels et positifs[END_REF]; see also [START_REF] Dieudonné | La théorie analytique des polynômes d'une variable (à coefficients quelconques)[END_REF][START_REF] Marden | The Geometry of the Zeros of a Polynomial in the Complex Plane[END_REF][START_REF] Lakatos | Polynomials with all zeros on the unit circle[END_REF][START_REF] Chen | On the polynomials with all their zeros on the unit circle[END_REF] for further insights on polynomials with all zeros on the unit circle.

Theorem 7.3 ([36]

). All zeros of the polynomial

P(s) = n k=0 a k s k ∈ C[s]
of degree n lie on the unit circle if, and only if,

i) P is self-inversive, i.e. (7.80) a n-k = āk (k = 0, • • • , n) where ∈ C, | | = 1;
ii) all zeros of P lie inside or on this circle.

If the coefficients a k (k = 0, • • • , n) are real, then in equalities (7.80), the coefficient should also be real, thus these polynomials are self-inversive if, and only if, [START_REF] Kakeya | On the limits of the roots of an algebraic equation with positive coefficients[END_REF][START_REF] Eneström | Remarque sur un théorème relatif aux racines de l'équation {a n}{xˆn}+{an-1}{xn-1}++{a 1} x+{a 0}= 0 où tous les coefficientes a sont réels et positifs[END_REF]). The absolute value of zeros of the polynomial

j) either = 1 and a n-k = a k (k = 0, • • • , n) i.e. P is reciprocal, jj) or = -1 and a n-k = -a k (k = 0, • • • , n) i.e. P is anti-reciprocal. Theorem 7.4 ([
P(s) = n k=0 a k s k ∈ R[s]
whose coefficients are positive, lie between the greatest and the least of the n quotients

a n-1 a n , a n-2 a n-1 , • • • , a 1 a 2 , a 0 a 1 .
Proof. The first claim is shown in the same manner as the proof of Theorem 7.1. Next, let us introduce the change of variable e -2 (s-s 0 ) = X. One can show that for any τ = 2 k where k is an integer greater than 2, one has

Q(s, 2 k) = R k (X) with: R k (X) = k+1 l=0 a k X k = -X k+1 + k + 1 k -1 X k - k + 1 k -1 X + 1.
Interestingly, the R k coefficients satisfy:

a k+1-l = -a l for l = 0, • • • , k + 1, which entails that R k is self- inversive, anti-reciprocal and skew-adjoint polynomial since R k (X) = -X k+1 R k ( 1 X ). Notice that X = 1 is a triple root of R k .
Furthermore, the first derivative of R k can be written:

R k (X) = -X k (k + 1) + (k + 1) X k-1 k k -1 - k + 1 k -1 ,
for which X = 1 is a double root. Let us investigate the zeros of R k or equivalently the following polynomial:

1 -k k + 1 R k (X) = (k -1) X k -k X k-1 + 1
One easily shows by recurrence that:

1 -k k + 1 R k (X) = (X -1) 2 k-2 l=0 (l + 1) X l .
Hence, Theorem 7.4 immediately incurs that beyond the double root at X = 1 the roots of R k satisfy

1 2 ≤ |X| ≤ k -2 k -1 < 1.
Finally, since R k is self-inversive and all zeros of R k lie inside or on the unit circle, then Theorem 7.3 asserts that all roots of R k lie on the unit circle, which concludes the proof.

Remark 7.5. It should be pointed out that apart from delay values τ = 2 k the GMID property holds for τ = 1 as it will be shown in the sequel. As a matter of fact, if (7.79) is satisfied and τ = 1, then one is in the following particular case:

Q(s, 1) = -e -(s-s 0 ) -1 3 ,
showing that the GMID obviously holds, i.e. s 0 corresponds to the spectral abscissa of the above quasipolynomial.

Remark 7.6. From a control theory viewpoint, the MID property can be exploited by tuning the control parameters as emphasized above after prescribing a negative number s 0 which corresponds to the closedloop system solution's decay rate.

When the MID property fails, one can prescribe a lower bound for the decay rate as will be discussed in the next section.

7.2.

Beyond the MID property in output feedback stabilization of the wave equation in (3.10) . By substituting the expressions of α, β and γ given in system (7.79) into the expression of Q in expression (7.74) and by introducing the variable change (7.81)

p := s -s 0 , it comes (7.82) Q(p, τ) := 1 - τ + 2 τ -2 e -2 p + τ + 2 τ -2 e -τ p -e -(τ+2) p ,
and we have now to examine the roots of Q with respect to τ. It is well-known that zeros of quasipolynomials with real coefficients are symmetric with respect to the real axis, that is, zeros are either real or appear in complex conjugate pairs. The following lemma underlines an additional symmetry structuring the distribution of zeros of Q with respect to the imaginary axis, that is a central symmetry.

Lemma 7.7. Let p ∈ C be a zero of Q defined by expression (7.82). Then, -p is also a zero of Q.

Proof. Let us evaluate Q at -p where p is a zero of Q.

(7.83) Q(-p, τ) = 1 - τ + 2 τ -2 e 2 p + τ + 2 τ -2 e τ p -e (τ+2) p
Multiplying both sides by e -(τ+2) p , one gets

(7.84) e -(τ+2) p Q(-p, τ) = e -(τ+2) p - τ + 2 τ -2 e -τ p + τ + 2 τ -2 e -2 p -1
which yields e -(τ+2) p Q(-p, τ) = -Q(p, τ). Since for all p ∈ C one has e -(τ+2) p 0, then necessarily Q(-p, τ) = 0.

In order to locate the zeros of (7.82), we require the following settings and results from [START_REF] Henry | Linear autonomous neutral functional differential equations[END_REF], see also [START_REF] Avellar | On the zeros of exponential polynomials[END_REF] for further refinements. Consider the quasipolynomial

(7.85) Θ(p, κ, h) := N k=0 κ k e -p χ k .h where κ = (κ 1 , . . . , κ N ) T ∈ R N , h = (h 1 , . . . , h M ) T ∈ R M + , χ j = (χ j,1 , . . . , χ j,M ), χ j,k ∈ N * ( j ∈ 1, N , k ∈ 1, M ) and χ j .h = M k=1 χ j,k h k .
We also adopt the notations κ 0 = 1 and χ 0 = (0, . . . , 0). Define Z Θ (κ, h) := (p) : Θ(p, κ, h) = 0 and denote its closure by ZΘ (κ, h). Lemma 7.8 ([48]). If the equation Θ(x + iω, κ, h) = 0 is satisfied for some reals x and ω, then the lengths |κ j |e -xχ j .h , j ∈ 0, N can form a closed polygon; that is, none of these lengths is larger than the sum of the others: |κ j |e -xχ j .h ≤ k j |κ k |e -x χ k .h for j ∈ 0, N . Also, following Henry [START_REF] Henry | Linear autonomous neutral functional differential equations[END_REF], let us define ρ j = ρ j (κ, h) ( j ∈ 0, N ), if they exist, by the relation (7.86) |κ j |e -ρ j χ j .h = k j |κ k |e -ρ j χ k .h for j ∈ 0, N .

If χ N .h ≥ χ j .h > 0 for j ∈ 1, N -1 , then ρ N and ρ 0 are uniquely defined and ρ N < ρ 0 for N ≥ 2.

Lemma 7.9

([48]). If χ N .h ≥ χ N-1 .h > . . . > χ 1 .h > 0, then (7.87) ZΘ (κ, h) ⊆ [ρ N , ρ 0 ].
The following lemma provides a vertical strip in the complex plane, which is symmetric with respect to the imaginary axis and contains the set of zeros of Q:

Lemma 7.10.

(7.88) Z Q(κ, h) ⊆ [-ρ * , ρ * ],
where ρ * is the unique positive zero of

(7.89) Q(ρ, τ) := 1 - τ + 2 τ -2 e -2ρ - τ + 2 τ -2 e -τρ -e -(τ+2)ρ .
Proof. The symmetry of Z Q(κ, h) with respect to the imaginary axis is a direct consequence of Lemma 7.9 and Lemma 7.7. Furthermore, Lemma 7.9 asserts that, if it exists, ρ * is a positive zero of Q given by (7.89). Finally, independently from the sign of τ -2, using simple real analysis arguments, one easily checks that Q(0, τ) < 0 and lim ρ→∞ Q(ρ, τ) = 1 for all τ ∈ R * + \{2} and shows by computing Qρ , the first derivative of Q with respect to ρ, that Q is strictly increasing. Hence, ρ * exists as a function of τ and it is unique. Now, ρ * is a root of (7.89) if, and only if,

(7.90) e -(τ+2)ρ * = 1 - τ + 2 τ -2 e -2ρ * - τ + 2 τ -2 e -τρ *
Substituting the above expression into Qρ , the first derivative of Q with respect to ρ, and evaluating the obtained expression at ρ * entails:

Qρ (ρ * , τ) = - (τ + 2) e -2ρ * τ -|τ -2| + 2 e -τρ * |τ -2| .
which never vanish for any τ ∈ R * + \{2}. Indeed, for τ < 2, Qρ is of constant sign and strictly decreasing with respect to ρ * . In addition, if we assume that Qρ vanishes at ρ * for τ > 2, we get (7.91) τ = -2 e -τρ * -2 e -2ρ * -1 .

However, by eliminating e -τρ * from the above expression of Qρ and substituting it into the expression of Q, we obtain

Q(ρ * , τ) = τ 2 -4 e -2ρ 2|τ -2| -e -(τ+2)ρ - τ 2 = 0, that is, (7.92) τ = 2 e -(τ+2)ρ * -2 e -2ρ * e -2ρ * -1 ,
which is inconsistent with (7.91) and the fact that ρ * > 0. Consequently, the Implicit Function Theorem is then applicable to (7.89) and asserts that ρ * = ρ * (τ) with

ρ * (τ) = 4 + -τ 2 + 4 ρ * (τ) e -2ρ * (τ) + 4 e -τρ * (τ) + ρ * (τ) |τ -2| (τ -2)
e -2ρ * (τ) τ -|τ -2| + 2 e -τρ * (τ) τ 2 -4 .

Lemma 7.11. Consider the quasipolynomial Q given by (7.89) with τ 2. Then the spectral abscissa σ of Q is lower-bounded by s 0 + ρ(τ) where ρ is given by

(7.93) ρ(τ) := 1 min{τ, 2} ln 1 + 2 τ + 2 |τ -2| .
Proof. It is easy to observe that for ρ > 0 and τ 2

Q(ρ, τ) ≥ 1 -1 + 2 τ + 2 |τ -2| e -τρ .
We remark that the right-hand side of this last inequality admits a single root at ρ given by (7.93). In conclusion, for any τ 2, one has Q( ρ(τ), τ) > 0, which asserts that ρ * (τ) ≤ ρ(τ) from the Intermediate Value Theorem.

Thanks to the above results and lemmas, the proof of the following theorem, which gives a certified decay rate's lower-bound for the closed-loop system's solution, is immediate.

Theorem 7.12. Consider the output feedback stabilization of the wave equation in (3.10) with an arbitrary positive delay τ > 0 then the following assertions hold:

• If τ = 2, then the control parameter tuning prescribed in system (7.78) allows to assign the solution's exponential decay rate at an arbitrary -s 0 ;

• If τ = 2 k
where k is an integer greater than one, then the control parameter tuning prescribed in system (7.79) allows to assign the solution's exponential decay rate at an arbitrary -s 0 ;

• If τ 2 k, then the control parameter tuning prescribed in system (7.79) allows a closed-loop solution decaying exponentially faster than -s 0ρ(τ), where ρ is defined by expression (7.93).

Proof. The D'Alembert transformation of the wave equation (3.10) turns it into the difference equation (6.73). Next, applying the Laplace transform in the frequency-domain to the latter, yields the characteristic quasipolynomial function (7.74). Finally, using the normalization (7.81), we end up with expression (7.82). The first assertion is a direct consequence of Theorem 7.1. The second assertion is a direct consequence of Theorem 7.2. The third assertion follows directly from Lemma 7.11.

Remark 7.13. Figure 2 shows the locus of ρ given by (7.93) (the proposed upper-bound on the real parts of the zeros of the quasipolynomial Q) as a function of the delay τ. Thanks to the linear change of variables (7.81), this enables the selection of an appropriate pair (s 0 , τ) in the filled gray region, providing an upper-bound on the spectral abscissa of the quasipolynomial Q given in (7.74). As asserted in Theorem 7.12, the desired decay rate towards the steady state equilibrium is greater than

) (i ∈ N) is such that (s i ) < s 0 + ρ * < s 0 + ρ(τ)
. By prescribing a minimal decay rate σ > 0, s 0 is chosen such that s 0 + ρ(τ) < -σ < 0 to ensure the asymptotic stability, i.e., s 0 < -σρ(τ).

Application: Longitudinal vibration control of a long drill pipe transmission line

Drilling is an important operation for the production of oil and gas. It is a key process in this production and exploration industry. According to us, a phenomenon that can cause poor operational performances, and even develop into the catastrophic failure of the well, is the presence of vibrations in the drill-string, which is mainly composed of long pipes, due to torsional, axial and also lateral vibrations. This is a spatially distributed system that is modeled fundamentally by PDEs. Such a system represents a very challenging modeling and control problem, see for instance [START_REF] Márquez | Analysis and control of oilwell drilling vibrations: a time-delay systems approach[END_REF][START_REF] Saldivar | A control oriented guided tour in oilwell drilling vibration modeling[END_REF] and references therein.

In this section, we propose to illustrate the main result of this work through the example of the longitudinal vibrations of a long steal drill pipe, for which we aim to reduce the vibrations thanks to the QPB controller set as above.

8.1. Description of the problem. Figure 3 shows a sketch of the considered drill pipe, clamped at one end and controlled by an axial force at the free-end. Such a situation can occur when the end of the drill-string is blocked at the bottom of the well. The drill pipe is submitted to an axial force d(t) located at position l, which disturbs it and provokes axial vibrations. Table 1. Material and geometrical parameters of the pipe.

F(t) L U(x, t) x x = 0 x = l d(t)
The parameters of the considered pipe are given in Table 1. The pipr is assumed to be cylindrical, steel The control is made via the free end by exerting a force F(t) that is related to the strain U x (L, t) by the relation F(t) = E A U x (L, t). For x ∈ (0, L), the initial conditions are U(x, 0) = 0, U t (x, 0) = 0. Finally, for x > 0, the denormalized control law in equation (3.10) is To perform numerical simulations of the closed-loop system (3.10), we propose to adopt the mechanical engineering approach which consists in handling numerically the system's dynamics separately from the control law's dynamics. Both of them are then combined once the numerical schemes are established.

U x (L, t) = -α U x (L, t -d τ) + β c U t (L, t) + γ c U t (L, t -d τ), ( 8
Concerning the wave equation, there are usually two numerical methods: the finite difference and the finite element methods. In this section, to address this issue, we propose to use the Finite Element Method in order to approximate the wave equation's PDE, because this approach is nowadays very well mastered by scientists; several books are devoted to this method [START_REF] Shabana | Vibration of discrete and continuous systems[END_REF][START_REF] Imbert | Analyse des structures par éléments finis[END_REF][START_REF] Komzsik | What Every Engineer Should Know About Computational Techniques of Finite Element Analysis, What every engineer should know[END_REF]. Moreover, this approach requires a lower amount of computations than the one required by the finite difference approach for a given computation precision.

Finite element modeling.

The long pipe of Figure 3 is divided into N ∈ N * basic parts called elements, as in Figure 4. This step is called the meshing of the structure. Each element connects two consecutive points of the meshing that are the nodes, with a total of N + 1 nodes. A basic element e for the wave equation case is well described by a truss element, which is assumed to carry only tension or compression loads, resulting in axial displacements only. In this case, the displacement field within an element e is given, in the element's local coordinate x ∈ [0, l e i ], by

U i ( x, t) = (1 -ξ i ( x)) ξ i ( x)          U i 1 (t) U i 2 (t)          =: S i ( x) q e i (t),
where ξ i ( x) := x/l e i is the normalized space variable inside the element i (connecting the nodes n i and n i+1 ), S i ( x) is the consistent interpolating (shape) function, q e i (t) ∈ R 2 is the nodal displacement vector of the i th element and U i 1 (t), U i 2 (t) ∈ R are the i th element's nodal coordinates corresponding to its nodes' displacements.

n i n i-1 n i+1 x = 0 l e i-1 U i 1 (t) U i 2 (t) = U i+1 1 (t)
Figure 4 The next step consists in computing the kinetic T i and strain U i energies at time t within a given element i, which are defined by:

T i (t) := 1 2 l e i 0 ρ (∂ t U i ( x, t)) T (∂ t U i ( x, t)) A d x, U i (t) := 1 2 l e i 0 E (∂ xU i ( x, t)) T (∂ xU i ( x, t)) A d x.
F i (t) S i ( x) q e i (t). The whole structure's meshing is now considered by computing the global kinetic and strain energies, as well as the global virtual work (the assembling step):

(8.98) T := N i=1 T i , U := N i=1 U i , W := N i=1 W i ,
where N is the number of elements. Let us define now the Generalized Coordinates displacement vector q(t) := q N+1 (t) q N (t) • • • q 2 (t) q 1 (t)

T ∈ R N+1 , where q i (t) denotes the i th node's axial displacement which is, according to the notations of Figure 4, equal to U i-1 2 (t) ≡ U i 1 (t). This last relation transcribes the connectivity of elements i -1 and i. By establishing the relation between q e i (t) and q(t), q e i (t) = B i q(t) where B i is an appropriate matrix derived from the previously defined element connectivity relations (B i ∈ R 2×(N+1) ), one can deduce from relations (8.98) the matrix motion equation given by (8.99) M q(t) + K q(t) = F (t),

where

M := N i=1 B i T M i B i (M ∈ R (N+1)×(N+1) ) is the structure's mass matrix, K := N i=1 B i T K i B i (K ∈ R (N+1)×(N+1)
) is the structure's stiffness matrix, and F (t) ∈ R (N+1)×(N) is the Generalized Forces' vector given by F (t) := N i=1 B i T S i ( x) T F i (t), assuming only one axial force F i (t) applied to the i th element at the local position x. In our case, the boundary conditions are taken into account by letting first q 1 (t) ≡ 0 (the fixed end at x = 0). Moreover, the control force F(t) applied at x = L, i.e., the N th element, is described by the generalized force vector e 1 F(t), where e N+1 is the first column vector of the identity matrix of size N + 1. The disturbance force d(t) can be addressed in the same way once the exact drill pipe meshing is performed.

8.3. Numerical developments and physical intuitive interpretations. The considered drill pipe described by Figure 3 and Table 1 has been meshed with N = 60 elements of equal lengths, l e i = 50 (m), for i = 1 to N. Even if this number may seem insufficient to correctly discretize the wave length of the fastest wave, it appropriately describes its low frequencies' behavior, which is what we aim to analyze with a low-frequency spectrum for the considered disturbance. The time simulation of the motion equation (8.99) is performed with a fixed time-step of 10 -4 (sec) thanks to an ODE 3 numerical method (the Runge-Kutta of order 3 proposed by Bogacki and Shampine [START_REF] Shampine | The matlab ode suite[END_REF]). This time-step's choice is made to cope with the delay's numerical value considered in the QPB's control law.

The left side of Figure 5a shows a simulation of the drill pipe elongation, supposed to be at rest at the initial time, which is executed to illustrate the QPB controller's performances for the active vibration damping of the drill pipe. The pipe is enduring an impact happening at x = l modeled by a force d(t)

that corresponds to half a period of a sinus function of magnitude 10000 (N) with an impulse width of 0.1 second displayed to the right. The time propagation of the shock-wave to the free-end can be seen on this figure and computed thanks to the wave propagation speed c, by t p := (Ll)/c ≈ 0.395 seconds.

Since the drill pipe model of (8.94) does not include any dissipating term, the pipe begins to vibrate persistently until the output feedback control applies. The latter is designed to implement the following specifications. A nominal decay rate of σ = 0.8 has been fixed in the normalized framework. By using (7.79) combined with Theorem 7.12, a pair (s 0 , τ) is selected according the chart given in Fig. Remark 8.2. It is worth mentioning that, with the QPB controller's four degrees of freedom and the results gathered in Theorem 7.12, the pair (s 0 , τ) can be arbitrarely selected. This enables the control designer to make a choice for the performances that is robust to numerical approximations , e.g., the delay parameter. tablished in [START_REF] Boussaada | Partial pole placement via delay action: A Python software for delayed feedback stabilizing design[END_REF][START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of timedelay systems: Vandermonde matrices and exponential decay[END_REF]. Initially, P3δ has been established for the control design of retarded differential equations. Based on the results of this work as well as the references [START_REF] Mazanti | Effects of roots of maximal multiplicity on the stability of some classes of delay differential-algebraic systems: The lossless propagation case[END_REF][START_REF] Ma | PID control design for first-order delay systems via MID pole placement: performance vs. robustness[END_REF], the software has been updated and is now able to treat linear neutral functional differential equations as well. P3δ is freely available for download on https://cutt.ly/p3delta, where installation instructions, video demonstrations, and the user's guide are also available. Interested readers may also directly contact one of the authors of the paper. Since its inception, P3δ is committed to be available to the greatest number and on all available platforms. The current version of the software exists as a local executable version as well as an online version ready to use in one click. The online version of P3δ is hosted on servers thanks to the Binder service [START_REF] Jupyter | Binder 2.0 -Reproducible, interactive, sharable environments for science at scale[END_REF]. Binder allows to create instances of personalized computing

Figure 1 .

 1 Figure 1. The spectrum distribution of (2.5) with a 0 = 2, a 1 = 0 and control parameters given by ( -) with respect to delay variation (from red to blue) by continuously decreasing the delay value from τ * = 1, where the nominal spectrum distribution is represented by filled diamonds ( ) and achieved for the delay value τ = τ * = 1 for which the quadruple spectral value s * corresponds to the spectral abscissa, to τ crit , where the end spectrum distribution is represented by the cross symbol (×) which is achieved

  order to prove the global existence and the uniqueness of the solution of problem (3.10), we shall first transform the problem (3.10) to the problem (4.22) by making the change of variables (4.19), and then we shall use the semigroup approach to prove the existence of the unique solution of problem (4.22).

(5. 45 )

 45 A Z = iδZ with Z = (u, v, z, w) T ∈ D(A ) and δ ∈ R admits only the trivial solution. Equation (5.45) reads as follows : iδuv = 0, (5.46) iδvu xx = 0, (5.47) iδz + τ -1 z ρ = 0, (5.48) iδw + τ -1 w ρ = 0 . (5.49)

Figure 2 .

 2 Figure 2. The behavior of ρ given by expression (7.93) as a function of the delay τ.

Figure 3 Young

 3 Figure 3

  based, and homogeneous, i.e., the density ρ and Young modulus E are identically distributed along the pipe's length. Let us denote by U(x, t) the longitudinal displacement of a point (cross-section) of the pipe located at position x and given at time t. The wave equation that describes the longitudinal vibrations of the pipe is given by (8.94) U tt (x, t) = c 2 U xx (x, t), for x ∈ (0, L), where c is the wave propagation speed given by the relation c := E ρ . The long pipe is fixed at one end, which is expressed by the boundary condition U(0, t) = 0. (8.95)

  .96) where d := L c is the time scaling factor weighting the normalized time variable t to give the true time variable t. This control law is assumed to be initiated byU t (L, td τ) = 0, U x (L, td τ) = 0 for t ∈ (0, d τ).Remark 8.1. The wave equation problem modeled by equation (3.10) has used some normalized (i.e., dimensionless) time t and position x variables, in such a way that their corresponding relations to the considered true variables are x = L x and t = d t. In this case, the normalized longitudinal displacement is related to the true one by the relation (8.97) u(x, t) := U(x, t), = U(L x, d t).

  2. The resulting controller's coefficients are reported within Figure5ain the normalized form. One can observe that the dissipativity conditions in (4.27) of Section 4 are satisfied. The first ten seconds show the natural behavior in blue of the undamped pipe. At t = 10 seconds, the output feedback control is applied and one can compare the closed-loop behavior in red with the natural one in blue. From the closing loop time t = 10, the settling time to zero within an interval of ±5% of the value U(L, 10), can be observed roughly equal to 2.2 seconds. It corresponds to three times the time constant computed with the prescribed decay rate (3 /(-s 0 -σ) ≈ 2.2). Moreover, whatever the choice of (s 0 , τ) is made, an incompressible duration of roughly 1.18 second is observed between the closing loop time and the moment the controlled damping is effective. It corresponds to 2 d, where 2 is the normalized delay due to the forward-backward duration of the wave propagation.

  Figure 6a illustrates this through time simulations with the same configuration as those in Figures 5a & 5b

  .

8. 4 .

 4 P3δ Software. P3δ, whose name stands for Partial pole placement via delay action, is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time delays, thanks to properties of the distribution of the quasipolynomials' zeros. It exploits mainly the MID/GMID property as well as the coexisting real roots-induced-dominancy, or CRRID for short, es-

  Free-end's time response to a shock-like force d(t) -with (red) & without (blue) feedback control. The feedback control loop is closed at t = 10 (sec).

  Time response of the controlled force F(t) to a shock-like force d(t) happening at the initial time t = 0. The feedback control loop is closed at t = 10 (sec). On the right, a zoom on the time interval where the closed-loop control signal is activated.

Figure 5 .

 5 Figure 5. Time responses without & with output feedback control.

  Free-end's time response to a shock-like force d(t) -with (various colours) & without (blue) feedback control. The feedback control loop is closed at t = 10 (sec). The wave propagation time is represented by the vertical dot-dashed line, at t ≈ 11.18 seconds. Different delay parameters τ have been implemented around its nominal computed value within an interval of ± 20% (with a 5% increment), without changing the nominal values for the other parameters α, β and γ. This emphasizes the robustness of the QPB controller with respect to the implemented delay.

  Free-end's time response to a shock-like force d(t) -with (red) & without (blue) feedback control with an other choice for the pair (s 0 , τ) (see the normalized values within the figure). The same time response of the configuration depicted in Figure5a is superimposed (dashed red curve) on this new one (red), emphasizing a better exponential decay rate.[START_REF]Further results on the long-time behavior of a 2D overhead crane with a boundary delay: exponential convergence[END_REF]. Free-end's time responses through various situations. environment directly from a GitHub repository that can be employed and shared by users. The Binder service is free to use and is powered by BinderHub, an open-source tool that deploys the service in the cloud. The online version of P3δ is written in Python and structured as a Jupyter Notebook, an open document format which can contain live code, equations, visualizations, and text. The Jupyter Notebook is completed by a friendly user interface built using interactive widgets from Python's ipywidgets module. Future development of P3δ will also provide a certified upper-bound of the decay rate when the MID property fails.

Acknowledgement

We warmly thank Jean Jacques Loiseau (LS2N, CNRS) and Guilherme Mazanti (L2S, Inria Saclay) for valuable discussions on the stability of continuous-time difference equations. Regarding the drilling control, we address particular thanks to Jean Auriol (L2S, CNRS) and Hugues Mounier (L2S, University Paris-Saclay). Last but not least, we thank Karim Trabelsi (IPSA Paris) for careful reading of the manuscript and for comments.