E Prulière
email: e.pruliere@i2m.u-bordeaux1.fr

S Metoui

A reduced numerical strategy based on PGD for composite shell structures simulations

Keywords: Shell structures, Composites, Proper Generalized Decomposition

This paper explores an alternative to shell computation. The proposed strategy uses the Proper Generalized Methods based on a separated representation. The idea is to solve the full 3D solid problem separating the in-plane and the out-of-plane spaces. This allows to represents complex fields in the thickness without the complexity and the computational cost of a solid mesh which is particularly interesting when dealing with multi-layer composite.

INTRODUCTION

The composite structures are more and more used because of the need in structures lightening especially in aeronautical applications. The damaging process in composite structures is of great complexity especially under dynamic solicitations. Organic matrix composites structures are generally composed of three characteristic scales. The microscopic scale is related to the fibres arrangement in the matrix, the mesoscopic scale is related to the plies and the macroscopic scale is related to the structures. Damage can occur at these three scales:

• fibres ruptures, fibres/matrix decohesion and matrix cracking at microscopic scale,

• delamination at mesoscopic scale,

• macroscopic ruptures.

Thus, the damaging of composites is a multidimensional phenomenon. An efficient model must be able to consider the different damaging types and their scales and the interaction between them. Another constrain to a good prediction of damages is that a good model must be able to be solved with numerical solver. With the classical numerical tools, it is definitely not conceivable to perform a complete multiscale simulation considering fibres and structures.

NUMERICAL STRATEGIES

Restriction of classic numerical strategies

The mainly used numerical method for mechanic simulations is the finite element method (FEM). For example, numerical simulations of composite structures are generally performed using multi-layered shell elements in the context of the finite elements method. This strategy has numerous advantages like a low computation time and the capability to reproduce the comportment of composites in most of cases. The main restriction of this approach is that it has only a coarse description of strain and stresses variations in the thickness. This approximation is no more valid when increasing the thickness, near the boundary and loading conditions and when non linear phenomena like delamination or other damaging processes occurs in the thickness. Consequently other strategies must be developed for damaging simulations.

The Proper Generalized Decomposition

Model reduction techniques are being developed in computational mechanics. This kind of strategies has prove its efficacy in many applications like thermal problems [START_REF] Chinesta | Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization[END_REF], stochastic simulations, rheology ([1], [START_REF] Ammar | A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Part II: Transient Simulation Using Space-Time Separated Representation[END_REF], [START_REF] Prulière | Recirculating flows involving short fiber suspensions: numerical difficulties and efficient micro-macro solvers[END_REF]) and structural mechanics [START_REF] Prulire | An efficient reduced simulation of residual stresses in composite forming processes[END_REF]. This techniques are based on a separated representation of the solution. Methods based on the proper orthognal decomposition (POD) needs a precomputation to build a reduced approximation basis and to find the solution. So it is needed to know the solution prior to compute the approximation basis. Others strategies likes the A priori Hyper Reduction Method, allows building the spacial approximation basis and the solution at the same time. Another method consist in searching the solution directly on a defined separated form. This method is called the Proper Generalized Decomposition (PGD). Details on the method can be found here [?]. A resolution with the PGD needs:

1. An equation to solve, i.e. a partial differential equation modelling the physical phenomena.

2. A separated decomposition of the unknown in order to build the solution.

3. An approximation of the solution on each subspace (a Finite Element approximation is often used). 4. Some boundary conditions defined on the subspaces (the application of boundary conditions may be a little complicated)

USING THE PGD ON SHELL STRUCTURES

Mechanical model

The model used is based on the classical momentum conservation equation:

divσ + f = ρΓ (1)
where σ denotes the stress tensor, div denotes the tensorial divergence,f is the volume force, ρ is the density and Γ is the acceleration. σ is linked to the deformation by the constitutive relation:

σ = Hε ( 2
)
where ε is the strain tensor and H is the fourth order rigidity tensor. ε is the symmetric gradient of the displacement u =(u, v, w).

The weak formulation of the equilibrium equation Eq. (??) without dynamic effect is:

Ω ε(u ⋆ ):(Hε(u)) = Ω u ⋆ .f + ∂Ω u ⋆ .(σ.n) ( 3
)
where Ω is the domain taken by the structure.

Separated representation

The idea is to use the PGD method separating a solid shell structure in two spaces: the mid-plane surface and the thickness. The shell structure occupies a region Ω=S×T where S describes the mid-plane surface and T =[ -e 2 ; e 2 ] is an interval containing all the positions in the thickness defined by a signed distance from the mid-plane. e denotes the maximum thickness of the shell.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(X,z) ≈ N i=1 F u i (X)G u i (z) v(X,z) ≈ N i=1 F v i (X)G v i (z) w(X,z) ≈ N i=1 F w i (X) G w i (z) ∀X ∈S (4)
X is the position vector and z is the local out-of-plane coordinate. The vector function

F i =( F u i ,F v i ,F w i
) is defined over the mid-plane surface S and the vector function

G i =( G u i ,G v i ,G w i
) is defined over the thickness of the shell structure.

Approximation on the subspace

The vector functions F i are approximated using shell or a plate finite elements. The shell mesh describe the position on the mid-plane. The G i functions are approximated by simple 1D finite elements describing the position on the thickness. It can be noticed that a 3D structured mesh of the shell structure can be derived from the two previous meshes (depicted in Figure 1).

References format

The bibliography should be the last section of the paper and use the layout as given in this example and 9pt font. Use an unnumbered heading 'References' in the primary heading format. For a book reference use example [?]. For a journal article use [?]. An article in proceedings should look like [?]. The references should be labeled in the order in which they appear in the text. Place references in the text using a number or a list of numbers between square brackets.

Figure 1 :

 1 Figure 1: Example of a triangular shell mesh for the mid plane surface (left) and a 1D mesh for the thickness (right).

CONCLUSIONSIn this brief article, a promising numerical strategy has been exposed. This strategy permits a full 3D modeling of structures with the computational complexity of a non linear shell modelling. This approach is interesting in order to simulate composites damaging with a low computational coast.


B Goeldel
email: benoit.goeldel@ensam.eu.

M El Mansori

D Dumur

Luca Settineri

Prof Luca Settineri

on Modeling of Machining Operations (CIRP CMMO) Simulation of roughness and surface texture evolution at macroscopic scale during cylinder honing process

Keywords: Honing, Grinding, Simulation

The honing process produces surface liners with specific functional properties. Engine performances and life expectancy are directly impacted by the quality of honed surface. The form quality, the roughness and the surface appearance manufactured by honing determine the friction of the piston in the liner. The process is however mechanically complex and the selection of the process parameters is currently based on empirical methods. The aim of this paper is thus to develop a macroscopic simulation environment which will help end-users during this setting-up stage. The development of this virtual tool is based on a space-time discretization and a macroscopic cutting model taking into account local contacts between the workpiece and the abrasive stones. The space-time discretization allows representing the machine environment including the tool, the workpiece and the machine kinematics. The cutting model allows converting kinematics and abrasive contacts in dynamic data and material removal rate by calculation. The cutting model is initially adjusted based on simple experiments. The stock removal equation is then extrapolated to the whole range of stone cutting conditions. This approximation allows simulating the real process and a whole honing cycle. Results are validated by comparison with industrial context experiments. The simulation of the whole honing cycle allows predicting the form quality, one of the roughness criteria and the surface appearance. Moreover, simulation results are represented by means of maps that allow looking at quality criteria for each point of the surface.

Introduction

The optimization of honing for mass production of cylinder blocks is a major challenge for designers of internal combustion engines. The energetic performance of the engine depends partly on this operation. Honing allows modifying the topography of the contact surface between the cylinder and the piston segments [START_REF] Santochi | A Study on the Functional Properties of a Honed Surface[END_REF]. The essential role of the interface is to ensure lubrication of the piston slides and the tightness of the combustion chamber. This strategy, which aims at reducing metallic contact, also allows oil consumption [START_REF] Haasis | New honing technique reduces oil consumption[END_REF] and greenhouse gas emissions to be reduced, and the engine's life to be increased. Currently, the setting method of the surface creation process is based on empirical knowledge concerning abrasion [START_REF] Klocke | Manufacturing processes 2 -Grinding, Honing, Lapping[END_REF]. It is complicated and costly in terms of time and equipment. Some models allow calculating the cutting force [START_REF] Ueda | Analytical investigation of the honing mechanism[END_REF] for a stone. Other studies model the creation of grooves by some grit [START_REF] Torrance | Modelling abrasive wear[END_REF]. Certain studies [START_REF] Gouskov | Influence of honing dynamics on surface formation[END_REF], [START_REF] Voronov | Modeling of bore honing[END_REF] simulate the vibratory behavior of the tool system. The goal of the current research is to endeavor to model honing abrasion in order to predict the quality of the surface obtained according to process parameters. The originality of the developed software is based on a linear model of the bearing ratio curve. Detailed analysis of this model has enabled us to predict the material removed, as well as the aspect of the honed surface (exact measurements, roughness, and texture). Experimental results, some coming from industrial series, validate our models and highlight certain unique honing properties. Finally, the characteristics of the final surface texture, which are the result of the study of simulated grooves, allow predicting surface aspect criteria.

The honing machine

The stones fixed on the honing tool are activated on 3 axes: a rotation (t), an axial translation z(t) and a radial expansion r(t). This axial expansion aimed at applying the stones to the surface of the cylinder bore can be f conducted according to position, using a position control with an electromechanical activator (EM), or through M M pressure using a hydraulic jack (HY ( ).

Y Y

Fig. 1. Axial stroke trajectory

In normal honing, honing rotation speed (t) is constant and axial translation is carried out at constant alternate speed, forming a stroke motion (figure 1). The length of the stroke motion is calculated so that the honing tool overlaps the height of the stones by a third at their highest and lowest inversion points. The groove angle and the cutting speed of the abrasive Vc are fixed to obtain the right texture and the best performance of the process [START_REF] Sabri | Process variability in honing of cylinder liner with vitrified bonded diamond tools[END_REF]. For the linear part of the movement, the peripheral speed Vp and the stroke speed Vb are calculated to fulfill the honing angle :

Vp Vb 2 tan and ² Vp Vb Vc ² (1) (1)
As the honing tool mass is considerable, every reversal of direction is carried out at a controlled acceleration. In t mass production, acceleration is set between 10 and 20m/s². Figure 1 illustrates the deterioration of the trajectory if it goes down to 5 m/s². A parabolic trajectory in stroke inversion, which may cause some defects on the honed surface, can be observed. Expansion of the stones is synchronized with the stroke motion and controlled by real time measurement of the diameter of the honed cylinder bore. These measurements are carried out via pneumatic nozzle fixed t on the honing tool [START_REF] Ning | Re-search on five sections measuring model in process pneumatic measurement system for inner hole honing[END_REF]. All of these elements will be considered during the simulation. The choice of abrasive is not the object of the simulation although it contributes to the quality of the resulting surface.

Simulation

Discretization

Studies of 3D modeling of cylinder bores have already been proposed [START_REF] Covington | Macroscopic cylinder honing process simulation[END_REF]. They allow studying the constraints on the honed cylinder bores, their deformation, but they are unable to model the amount of material removed to the same extent. Microscopic abrasion studies [START_REF] Tönshoff | Modeling and Simulation of Grinding Processes[END_REF], [START_REF] Szabó | Stability criteria and break out of grains of superhard of grinding tools[END_REF] are often limited to the study of the impact of a few abrasive particles and their results cannot really be applied to honing stones. Moreover, the influence of complex kinematics has never been studied. The proposed software is based on a surface representation of contact elements with square meshes of 1×1 mm. The kinematics of the m otion and of the honing tool is simulated in detail [START_REF] Goeldel | Macroscopic simulation of the liner honing process[END_REF]. Each of these meshes are linked to macroscopic criteria (shape, thickness, pressure), to microscopic criteria (roughness) and to mesoscopic criteria (aspect, texture). In spite of the evolution in the diameter of the cylinder bore and of the honing tool (by several hundreds of μm) during the honing process, the surface of the meshes is considered constant. The cylinder bore and the honing tool are therefore represented by matrices which have the same number of columns with a different number of lines (cylinder bore = remaining material; honing tool = stone thickness). Given the relative motion of the honing tool in relation to the cylinder bore, a time-domain discretization given by formula (2) allows obtaining at each iteration a single motion of a line or column.

Vb x x Vp x x t min

(2)

An overlaying model allows one to detect the cylinder bore meshes likely to be in contact with the honing (1) (1) at iteration and subsequently to calculate u interaction and to determine the values of the "inprocess measurements".

Kinematics and expansion

The evolution of the honing tool position in relation to the cylinder bore for a conventional honing machine is simulated in figure 1. The software also allows creating any motion z(t) linked to any given rotation speed (t) and expansion r(t) in real time. If the expansion is controlled in position (electromechanical system EM), the position evolves in a M M linear way with an speed Vexp programmed according to the stones contact expansion position. It is possible to impose a stronger impact. Once the expansion reaches an imposed residual material thickness, it remains constant during a number of additional strokes Nbsup, which can be programmed to carry out the "polishing stage", aimed at refining the roughness. Honing with two different expansion speeds is also planned.

Roughness, stock removal and cutting force for electromechanical expansion

Roughness is one of the most important properties, which quantifies the quality of the honed cylinder bore. It is usually measured according to criteria which result from the bearing ratio curve (Figure 2 (a)) and must respect specific norms [START_REF]Surface texture: profile method, surfaces having stratified functional properties[END_REF]. As in [START_REF] Corral | Use of roughness probability parameters to quantify the material removed in plateau-honing[END_REF], it has been decided to represent this curve as a straight line, which will be defined entirely by 2 points Fmin and Fmax. The maximum roughness of a surface constitutes the distance measured on a perpendicular level between the deepest groove in the micro geometry and the most protuberant peak on the surface. The linear model of the bearing ratio contains this information. Figure 2 (b) represents these notions for the cylinder bore and the honing tool with:

Fmin Fmax Rz and (3) (1) Rmin Rmax Tg
Rz represents the roughness of the cylinder bore and Tg models the indentations of the stone particles. In order to study the interaction between the abrasive and the cylinder bore, consider the contact of one single honing mesh with a cylinder bore mesh and denote U the depth of penetration of the honing tool into the cylinder bore material. 8 different cases of penetration according to the respective values of U, Tg and Rz have to be examined. Figure 2 (b) represents case A1 for which U<Tg<Rz. Figure 3 (a) presents the overlap of the bearing ratios for this case. Thus consider a reference (Ox) linked to the honing tool, originating from Rmin and turned towards the cylinder bore. In this base, the rate of presence of honing tool abrasive particles and the bearing ratio of the cylinder bore are expressed as follows:

Tg x Tg x T R % and Rz U Tg x x T F % (4)
During the honing tool's trajectory onto non-eroded zones, the bearing ratio remains unchanged. If one makes the assumption that the honing tool is extremely resistant and that it cannot be eroded; only the bearing ratio of the cylinder bore is affected. In the interaction zone, the new bearing ratio of the cylinder bore is as follows on the high part:

x T x T x T R F new F % 1 % % (5)
The new bearing ratio is parabolic in the high zone (figure 3 (b)). This calculated bearing ratio is not linear in whole roughness thickness range. The equivalent linear bearing replaces the previous. The equivalent roughness Rz i+1 is calculated to respect the quantity of material removed, as in the following equation:

dx Rz x dx x T ii Rz Rz i new F 00 1 1 % (6)
The minimum and maximum radial thicknesses of cylinder bore material are therefore updated using the following recurrence:

i 1 i Fmin Fmin ( 7
) i i i 1 i Fmin Fmax Tg U Fmax Fmax 3 3 (8)
According to our roughness model, the proportion of bearing surface in contact, for a radial position x, is written as follows:

x T x T x S R F % % % T x T x T x T x F (9) (1)
The strength of contact of a mesh will be proportional to the quantity of material in contact along the entire thickness of the interaction zone. This is therefore given by:

dx x S Contact c % (10) (1)
In case A1, one obtains:

i A c Rz Tg U 3 1 (11) ()
The ratio of orientation of the grooves, which corresponds to the "balance" defined by [START_REF] Sabri | Multiscale study of finish-honing process in mass production of cylinder liner[END_REF], is calculated by dividing the number of visible rising traces by the number of visible traces. A good surface texture is composed of half rising grooves and half descending (1) [START_REF] Santochi | A Study on the Functional Properties of a Honed Surface[END_REF] To obtain a suitable texture, the angles of the visible traces must to the set angle /2 with a tolerance of . The conformity of the texture angle is expressed by the ratio of the number of visible traces with a right angle, over the t otal number of visible traces.

Similar formulations giving Fmin, Fmax and x c are established for each case and are incorporated in the simulation. It is important to note that if U is higher than U Rz, Fmin will also be reduced during the iterations and that all the c constitute a continuous but markedly nonlinear curve according to U. U U

Texture, surface aspect

It is expected that as each stone motion leads to material removal; it leaves a trace on the honed surface. This trace is characterized by its direction, its relative depth quantified by U and by its absolute depth given by U Fmin on the cylinder bore indicator.

During the honing cycle, these three variables have

(1) (1) been for all the traces. Further to simulation of the cycle, only visible traces for which Fmin i s positioned between the final Fmax and Fmin are selected and compiled. Subsequently, the compilation of grooves is analyzed to establish the following criteria.

The first criterion is the number of visible traces, or d i [16]

Simulation results and experimental validation

By measuring the micro geometry of the surface of the abrasive stones, the value of the indentation of the abrasive particles Tg has been determined for 2 types of g regularly used stones: for IAS65/100, Tg=40 μm and g g SC500 Tg=20μm. This value is calculated as the g maximum roughness observed with the tactile profilometer along a 16mm measure.

Simulated honing cycles

A cycle generator identical to those existing in industrial honing machines was installed. Figure 4 (a) presents type 1 honing: expansion starts on the 120μm radial position when the honing tool has begun into the cylinder bore to 2/3rds. Initially, little power can be observed because the abrasive stone only affects the ridges of the roughness profile of the contact surface. f From the 900th iteration, expansion is equal to Fmin and the abrasive particles reach full material. In this case, the stabilized abrasion speed creates maximum force until the expansion ceases. On iteration 2400, the final measurement is reached and triggers the end of expansion. Subsequently, expansion is maintained in operation during the "polishing stage" and this leads to a reduction in the cutting force. Figure 4(b) presents type 2 honing: this method corresponds to the use of two successive expansion speeds. For this type of cycle, expansion was started using a very powerful starting impulse. This programming tip on real machines allows the regulators to reduce the cycle time. This strong initial advance results in a faster increase in the cutting force and therefore allows removing more material. The stable abrasion cycle for the first expansion speed is reached at the 800 th iteration. Switch from the first speed to the second is triggered by reaching an intermediate measurement before the final measurement. Reducing expansion speed disturbs cutting force, which is stable again at the 4500 th iteration. Finally, reaching the zero measurement triggers the polishing stage. Here, polishing is maintained long enough for the cutting power and the roughness to both reach zero at the same time. Diagrams (figure 4) are comparable to the data coming from the monitoring of the Nagel VS8 MS-U4 honing machine.

Roughness obtained after abrasion

Thanks to the model of material removal after abrasion [START_REF] Goeldel | Macroscopic simulation of the liner honing process[END_REF], the simulation allows one to observe local areas of roughness on each mesh during the honing cycle. The evolution of the mean roughness in the cylinder bore during the type 1 cycle was examined on a honing tool equipped with four stones using abrasives with an average grit indentation of 40 μm. The curves simulated for several constant expansion speeds are shown in Figure 5 (a) for abrasive stone IAS65/100. It is assumed that initial roughness is around 40 μm to better observe the decrease in roughness during the transitional cycle. It has been noticed that the stable cycle is reached more rapidly at an expansion speed of 16μm/s, but with the drawback of a roughness close to 9 μm. The final measurement, which triggers polishing, is obtained after 13 strokes whereas 23 are needed at an expansion speed of 8μm/s. An expansion speed of 8μm/s allows one to produce a roughness close to 7μm on a stabilized cycle. Starting the expansion via impulses is also simulated for an expansion speed of 8μm/s and one can observe that the stable regime is reached more rapidly and that consequently the cycle is shortened.

Figure 5 (b) represents the evolution of the mean roughness of the cylinder bore during the type 2 honing cycle for 2 abrasive stones whose average particle indentation is of 40 μm for the first stone, and 20 μm for the second. One can clearly identify the levels that correspond to the roughness produced on a stable regime. One can also note that the time constant is not affected by the grit size. A series of four cylinder bores (of the Renault K9K 1.5dCi engine) were honed according to the type 2 protocol, with two different abrasives. The maximum roughness of the honed cylinders was measured during the cycle after 10, 20 and 25 strokes. The points identified from these measurements are traced on Figure 5 (b). They confirm the validity of the abrasion model in stabilized cycles. The last measurement, after 25 strokes, is the furthest from the simulated roughness, because the phenomenon of abrasion during polishing is much too stochastic. This phenomenon is even more visible when the acceleration is reduced to 5 m/s² as in figure 6(d). In this case, up to 70% of non-compliant grooves can be observed. In reality, it is easy to identify these noncompliant zones with the naked eye, by observing the interior surface under a well oriented light. The software allows one to predict these texture defects. At present only the eye of a specialist can describe such defects on a honed surface.

Surface aspect

Conclusion

The honing simulation software presented in this article has been validated through experiment. It faithfully represents shape defects, roughness and gives accurate information about texture. The software allows one to predict the surface quality according to a maximum of process parameters. The impact of the kinematics on the roughness and texture has also been studied. This study has highlighted the fact that weak acceleration strongly deteriorates surface quality both during the simulation and in the experiment. In the lapping process, current cutting forces are very weak to limit the deformation of the cylinder. Consideration of more force requires the integration of macroscopic deformation of the piece. Knowledge of how surfaces are formed through abrasion is the first step in its tribological characterization. Digital surface models developed through honing simulation will be used shortly for a tribological study, which will allow linking functional surface performance with its production process.

Fig. 2 .

 2 Fig. 2. Roughness modeling (a) Bearing ratio curve (b) Maximum roughness

Fig. 3 .

 3 Fig. 3. Modification of the roughness profile (a) Superpositioning (b) New bearing ratio

Fig. 4 .

 4 Fig. 4. Evolution of trajectory, cutting diameter and force with abrasive stones IAS65/100, (a) Single expansion speed: type 1 cycle (b) Double expansion speed: type 2 cycle

Fig. 5 .

 5 Fig. 5. Roughness evolution during the honing process, (a) Influence of expansion in type 1 cycle with IAS65/100 (Tg=40μm), (b) Influence of grit size and number of stones in type 2

Following a type 2 Fig. 6 .

 26 honing cycle with IAS65/100 abrasive stones, the prediction of texture quality leads to the results shown in figure6. Mapping (a) allows one to observe that the visible final texture is composed of 50 to 100 traces made by the abrasive motion. Mapping (b) illustrates that, in general, the texture is composed by half of rising traces and by half of descending traces. Mapping (c) highlights total conformity in the middle of the cylinder bore. Non-compliant traces left by the abrasives during the periods of stroke motion inversion represent almost 20% of traces on the edges when the acceleration is worth 20 m/s². Mapping of surface aspect for 3 criteria


Guilherme Mazanti

Stabilization of Persistently Excited Linear Systems by Delayed Feedback Laws

Keywords: stabilization, switched systems, persistent excitation, delayed feedback

This paper considers the stabilization to the origin of a persistently excited linear system by means of a linear state feedback, where we suppose that the feedback law is not applied instantaneously, but after a certain positive delay (not necessarily constant). The main result is that, under certain spectral hypotheses on the linear system, stabilization by means of a linear delayed feedback is indeed possible, generalizing a previous result already known for non-delayed feedback laws.

Introduction

Consider a control system of the form ẋ(t) = Ax(t) + α(t)Bu(t),

x(t) ∈ R d , u(t) ∈ R m , α ∈ G, (1.1)
where x is the state variable, u is a control input, A and B are matrices of appropriate dimensions, and α belongs to a certain class G of measurable scalar signals α : R + → [0, 1]. This corresponds to the introduction on the linear control system ẋ = Ax + Bu of a certain signal α that determines when and how much the control u is active. Note that, when α takes its values on {0, 1}, (1.1) is actually a switched system between the dynamics of the uncontrolled system ẋ = Ax and the controlled one ẋ = Ax + Bu.

Several different phenomena may be modeled by signal α in (1.1), such as a failure in the transmission of the control u to the plant, a time-varying parameter affecting the control efficiency, or the allocation of control resources, among other possible phenomena. We are interested in general on robust control techniques of (1.1) with respect to α: we suppose that α is not precisely known and we wish our control strategy for (1.1) to be chosen independently of α and to be valid for any signal α in a certain class G.

The problem of controlling (1.1) by a suitable choice of u is obviously not interesting when α ≡ 0, or when α is zero for a large amount of time, since in this case the control u has a very limited effect on (1.1). The class G should thus ensure that the control input has a * CMAP, École Polytechnique and Team GECO, Inria Saclay, Palaiseau, France.

❡ • q ♠ ♣ ❛ ❤ ❡ ❡ ♦ ❝ ♥ ❡ ❛ ② r ❤ ♠ ❡ t ✉ ✉ ❅ t ✐ ❞ ③ ✳ ✐ ♥✐ ✳ ✉ • ❣
sufficient amount of action on the system. Among the possible choices for G, the class of (T, µ)-persistently exciting signals has attracted much interest recently (see, for instance, [8-11, 14, 20, 25, 27], and also [START_REF] Lovera | Global spacecraft attitude control using magnetic actuators[END_REF] for a similar condition) and, for T ≥ µ > 0, it consists on the signals α ∈ L ∞ (R + , [0, 1]) such that, for every t ∈ R + , t+T t α(s)ds ≥ µ.

(1.

2)

The class of these signals α is noted G(T, µ). Further examples of systems similar to (1.1) where the persistent excitation condition appears are given in [START_REF] Chaillet | Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases[END_REF][START_REF] Chitour | Stabilization of persistently excited linear systems[END_REF][START_REF] Loria | On the PE stabilization of timevarying systems: open questions and preliminary answers[END_REF], where the motivation for the use of persistently exciting signals is also more deeply discussed.

The condition of persistence of excitation (1.2) arises naturally in identification and adaptive control problems (see, e.g., [1-3, 7, 23]). In this context, we are led to study systems of the kind ẋ = -P(t)x, x ∈ R d , where P(t) is a symmetric non-negative definite matrix for every t. If P is bounded and has bounded derivative, it has been shown in [START_REF] Morgan | On the stability of nonautonomous differential equations ẋ = [A+B(t)]x with skew-symmetric matrix B(t)[END_REF] that the persistence of excitation of P, in the sense that α(t) = ξ T P(t)ξ is (T, µ)-persistently exciting for all unitary vectors ξ ∈ R d and for certain constants T ≥ µ > 0 independent of ξ , is a necessary and sufficient condition for the global exponential stability of ẋ = -P(t)x.

We consider the problem of stabilization of system (1.1) to the origin by means of a linear state feedback u = -Kx, where we require the choice of the gain matrix K not to depend on a particular signal α but instead on the class G(T, µ). In many practical situations, this feedback cannot be done instantaneously, for a certain state x(t) may not be available for measure before a certain delay τ has elapsed, and so the state measured in time t is actually x(t -τ(t)). Due to the several practical situations where time lags are introduced by sensors, actuators, or the transmission or processing of signals, the study of delayed systems in general is of much interest, and specially in the context of control systems [START_REF] Bellman | Differential-difference equations[END_REF][START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Hu | Dynamics of controlled mechanical systems with delayed feedback[END_REF][START_REF] Michiels | Stability and stabilization of time-delay systems: An eigenvalue-based approach[END_REF][START_REF] Niculescu | Delay effects on stability: A robust control approach[END_REF]. In several situations, the time-delay appearing in a system is not known exactly and may change with the time, and the literature usually classifies these delays in two types: slowly-varying delays, where its derivative satisfies | τ(t)| < 1, and fast-varying delays, without constraints on the derivative of the delay. In this paper, we take as possible delays τ measurable functions taking their values on a certain set T ⊂ R + , and we are thus in the framework of fast-varying delays.

This paper considers the problem of stabilization of (1.1) by a delayed feedback u(t) = -Kx(t -τ(t)), where the delay τ(t) may depend on t, and the closed-loop system becomes

ẋ(t) = Ax(t) -α(t)BKx(t -τ(t)), α ∈ G(T, µ), τ ∈ L ∞ (R + , T) (1.3)
where T ⊂ R + is the set where the delay τ takes its values. The goal of this paper is to present a stabilization result for system (1.3), showing that, under certain hypotheses on A and B, given T ≥ µ > 0 and τ 0 ≥ 0, there exist a neighborhood T of τ 0 in R + and K ∈ M m,d (R) such that, for any α ∈ G(T, µ) and any delay function τ ∈ L ∞ (R + , T), system (1.3) is exponentially stable, uniformly with respect to α and τ. This generalizes [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]Theorem 3.2], where the same result is given in the case of the non-delayed feedback u(t) = -Kx(t), corresponding thus to T = {0}.

Notice that (1.3) is related to switched linear systems with delays, since, when α(t) takes its values on {0, 1}, (1.3) becomes a switched system between the non-delayed uncontrolled dynamics ẋ = Ax and the delayed one ẋ(t) = Ax(t) -BKx(t -τ(t)), under the constraint of persistence of excitation given by (1.2). Several results exist concerning switched systems with delays, presented for instance in [START_REF] Kim | Stability of a class of linear switching systems with time delay[END_REF][START_REF] Liu | Stability theory of hybrid dynamical systems with time delay[END_REF][START_REF] Mahmoud | Switched time-delay systems[END_REF][START_REF] Sun | Stability and L 2 -gain analysis for switched delay systems: a delay-dependent method[END_REF][START_REF] Vu | Stability of time-delay feedback switched linear systems[END_REF][START_REF] Zhang | Stability of switched systems with time delay[END_REF]. Many of them apply Lyapunov function and functional techniques to obtain conditions on the systems, the delay and the switching law that guarantee stability under constrained or arbitrary switching, such as [START_REF] Kim | Stability of a class of linear switching systems with time delay[END_REF][START_REF] Liu | Stability theory of hybrid dynamical systems with time delay[END_REF][START_REF] Vu | Stability of time-delay feedback switched linear systems[END_REF]. The constraints on the switching law usually take the form of an average dwell time, as in [START_REF] Sun | Stability and L 2 -gain analysis for switched delay systems: a delay-dependent method[END_REF][START_REF] Vu | Stability of time-delay feedback switched linear systems[END_REF][START_REF] Zhang | Stability of switched systems with time delay[END_REF], or a strategy to design a switching rule, as in [START_REF] Kim | Stability of a class of linear switching systems with time delay[END_REF]. In this paper, we consider that α is an unknown signal satisfying the condition of persistence of excitation (1.2), which is different from the usual hypothesis of average dwell time used for switched systems since α may be active at arbitrarily small time intervals at each time. Our main technique consists on studying (1.3) through a timecontraction procedure and a limit system, which has been proved to be useful when studying persistently exciting systems in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] but, up to our knowledge, it has not been previously used to study delayed switched systems.

Let us comment briefly on the technique used in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] to consider the stabilizability of (1.3) in the non-delayed case. The main problem when dealing with the class G(T, µ) is that a signal α ∈ G(T, µ) may be zero on certain time intervals, and so the system follows its uncontrolled dynamics ẋ = Ax. On the other hand, for every ρ > 0, it is known by a result from [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] that one can choose a linear feedback u(t) = -Kx(t) that stabilizes (1.1) uniformly with respect to α ∈ L ∞ (R + , [ρ, 1]). The main idea in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] is to perform a change of variables corresponding to a time contraction by a factor ν > 0, which transforms a (T, µ)-signal α into a (T /ν, µ/ν)-signal α ν with α ν (t) = α(νt). It is possible to show that the family (α ν ) ν>0 admits a weak-⋆ convergent subsequence (α

ν n ) n∈N * in L ∞ (R + , [0, 1]
) with ν n → +∞ and that any weak-⋆ subsequential limit α ⋆ of (α ν ) ν>0 as ν → +∞ satisfies α ⋆ (t) ≥ µ /T almost everywhere. The idea is thus to study a certain limit system obtained as ν → +∞, for which stabilization can be obtained using the result from [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] mentioned above. It can then be shown by a limit procedure that the same feedback gain K also stabilizes a time-contracted system for a certain ν > 0 large enough, and one may finally adapt such a feedback gain K in order to obtain a stabilizer for the original system.

This time-contraction technique used in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] is well-adapted to deal with delays in the feedback, since a delay τ(t) in the original system will correspond to a delay τ(νt) ν in the timecontracted system. We may thus expect to obtain a non-delayed limit system as ν → +∞ similar to the one obtained in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] and to conclude the stabilizability of the original system by a similar argument. This intuition is actually true, as proved in Theorem 2.5 below, where we prove our stabilizability result by following the same time-contraction argument of the proof of [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]Theorem 3.2].

In their article [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF], the authors first prove their stabilization result in the particular case where the dynamics are given by the Jordan block J d (see (3.1) below), since it is a representative example containing most of the difficulties of the proof of the general case. We also treat the case of the Jordan block separately in this article (see Theorem 3.1), but in this particular case we have a stronger result, showing that stabilizability is possible for any bounded interval T ⊂ R + where the delay τ ∈ L ∞ (R + , T) may take its values, whereas in the general case we may only guarantee stabilizability for delays τ which are perturbations around a certain constant prescribed value τ 0 . This difference between the statements of our result in the general case and in the particular case of the Jordan block is more deeply discussed in Section 5.

The plan of the paper is the following. In Section 2, we present the notations and definitions used throughout this paper and recall the previous result of [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]. We then proceed to prove, in Section 3, the main theorem of this paper in the particular case of the Jordan block, which allows us to highlight the main ideas of the proof in a setting where the notations are much clearer than in the general case, and also leads to a stronger result than in the general case. The proof of our main theorem is presented in Section 4, and Section 5 discusses the results we obtained, and specially the difference in the statements of Theorems 3.1 and 2.5. The proofs of some technical lemmas used in this paper are given in the Appendices A and B.

Notations, Definitions and Previous Results

In this paper, M d,m (R) denotes the set of d × m matrices with real coefficients, which is denoted simply by M d (R) when d = m. As usual, we identify column matrices in M d,1 (R) with vectors in R d . The identity matrix in M d (R) is denoted by Id d and 0 d×m ∈ M d,m (R) denotes the matrix whose entries are all zero, the dimensions being possibly omitted if they are implicit. The block-diagonal matrix whose diagonal blocks are the square matrices a 1 , . . . , a d is denoted by diag(a 1 , . . . , a d ). The notation x indicates both the Euclidean norm of a vector x ∈ R d and the associated matrix norm. The real and imaginary parts of a complex number z are denoted by ℜ(z) and ℑ(z) respectively. The sets R + and N * denote, respectively, the sets of the nonnegative real numbers R + = [0, +∞) and the positive integers N * = {1, 2, 3, 4, . . . }. For two topological spaces X and Y , we denote by C 0 (X,Y ) the set of all continuous functions from X to Y .

Throughout this paper, we consider the system

ẋ(t) = Ax(t) + α(t)Bu(t), x(t) ∈ R d , u(t) ∈ R m , α ∈ G(T, µ), (2.1)
where

A ∈ M d (R), B ∈ M d,m (R)
, and we take persistently exciting signals α in the class G(T, µ) defined as follows.

Definition 2.1. Let T , µ be two positive constants with T ≥ µ. We say that a measurable function α : R + → [0, 1] is a (T, µ)-signal if, for every t ∈ R + , one has t+T t α(s)ds ≥ µ.

The set of (T, µ)-signals is denoted by G(T, µ). System (2.1) with α ∈ G(T, µ) is called a persistently excited system (PE system for short).

We shall consider the problem of stabilization of system (2.1) by means of a delayed linear state feedback u(t) = -Kx(t -τ(t)), where the delay τ is a function in L ∞ (R + , T) for a certain bounded set T ⊂ R + and K ∈ M m,d (R). With this feedback, system (2.1) takes the form

ẋ(t) = Ax(t) -α(t)BKx(t -τ(t)), α ∈ G(T, µ), τ ∈ L ∞ (R + , T).
(2.2)

Note that, for T ≥ µ > 0 and T ⊂ R + bounded, for every α ∈ L ∞ (R + , [0, 1]) and every τ ∈ L ∞ (R + , T), (2.2) satisfies the Carathéodory conditions for delayed equations (see, for instance, [13, Section 2.6 and Theorem 6.1.1]), and so, noting r = sup T, for any given initial condition

x 0 ∈ C 0 ([-r, 0], R d ), (2.
2) admits a unique continuous solution x defined on [-r, +∞), which is absolutely continuous on R + , coincides with x 0 on [-r, 0], and satisfies ẋ(t) = Ax(t)α(t)BKx(tτ(t)) for almost every t ∈ R + . In order to make explicit the dependence of the solution x on τ, x 0 , α and K, we denote x(t) = x(t; τ, x 0 , α, K).

In the context of delayed systems, stability is defined in terms of the uniform norm of the initial condition (see, for instance, [START_REF] Hale | Introduction to functional differential equations[END_REF]Chapter 5]), which motivates the following definition. Definition 2.2. Let T ≥ µ > 0 and T be a bounded subset of R + , and denote r = sup T. We say that K ∈ M m,d (R) is a (T, µ, T)-stabilizer for (2.2) if there exist constants C ≥ 1 and γ > 0 such that, for every α ∈ G(T, µ), every τ ∈ L ∞ (R + , T), and every initial condition x 0 ∈ C 0 ([-r, 0], R d ), the solution x(t; τ, x 0 , α, K) of (2.2) satisfies

x(t; τ, x 0 , α, K) ≤ Ce -γt sup s∈[-r,0] x 0 (s) , ∀t ≥ 0. Remark 2.3. If K is a (T, µ, T)-stabilizer for (2.
2), then, for every constant α ⋆ ∈ [ µ /T, 1] and every constant delay τ ⋆ ∈ T, the linear delayed system

ẋ(t) = Ax(t) -α ⋆ BKx(t -τ ⋆ ) (2.3)
is exponentially stable. This is an important remark, since the stability and stabilization of systems with a constant delay of the form (2.3) can be more easily studied (see, for instance, [START_REF] Michiels | Stability and stabilization of time-delay systems: An eigenvalue-based approach[END_REF][START_REF] Niculescu | Delay effects on stability: A robust control approach[END_REF]), giving rise to necessary conditions for K to be a (T, µ, T)-stabilizer. We shall use this approach later in Example 5.1.

Let us recall that a pair of matrices

(A, B) ∈ M d (R) × M d,m (R)
is said to be stabilizable if there exists a matrix K ∈ M m,d (R) such that A -BK is Hurwitz. This is equivalent to saying that there exists an invertible matrix

P ∈ M d (R) such that PAP -1 = A 1 A 3 0 A 2 , PB = B 1 0 ,
where A 2 is Hurwitz and (A 1 , B 1 ) is controllable. Stabilizability of a pair (A, B) means that the linear control system ẋ = Ax + Bu admits a linear state feedback u = -Kx such that the closedloop system ẋ = (A -BK)x is exponentially stable, and thus, in order to achieve the required stabilizability property for system (2.2), the stabilizability of (A, B) is a necessary condition when 0 ∈ T. This is what motivates us to consider only stabilizable pairs (A, B) in what follows.

The stabilizability of (2.2) by means of a non-delayed feedback law has been studied in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] in the case of a single-input system, i.e., when m = 1, and it has been generalized to the multiinput case in [START_REF] Chitour | Stabilization of persistently excited linear systems[END_REF]. In terms of Definition 2.2, this result can be stated as follows.

Theorem 2.4. Let (A, B) ∈ M d (R) × M d,m (
R) be a stabilizable pair and assume that the eigenvalues of A have non-positive real part. Then, for every T ≥ µ > 0, there exists a (T, µ, {0})stabilizer for (2.2).

The hypothesis that the eigenvalues of A have non-positive real part may seem restrictive, but it was shown in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] that Theorem 2.4 is not true for certain stabilizable pairs (A, B) and certain values of T, µ when A admits an eigenvalue with positive real part. This is actually an effect of the signal α in the dynamics of the system; note that, when α(t) ∈ {0, 1}, the closedloop system actually switches between the dynamics given by ẋ = Ax and ẋ = (A -BK)x, and the phenomena related to this switch, such as the overshooting phenomenon, may lead to the non-stabilizability of the switched system when A has an eigenvalue with positive real part, as detailed in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]. For more general information on the behavior of switched systems, we refer to [START_REF] Balde | A note on stability conditions for planar switched systems[END_REF][START_REF] Boscain | Stability of planar switched systems: the linear single input case[END_REF][START_REF] Liberzon | Switching in Systems and Control[END_REF][START_REF] Lin | Stability and stabilizability of switched linear systems: a survey of recent results[END_REF][START_REF] Margaliot | Stability analysis of switched systems using variational principles: an introduction[END_REF][START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF].

The main result of this paper is the following generalization of Theorem 2.4.

Theorem 2.5. Let (A, B) ∈ M d (R) × M d,m (R)
be a stabilizable pair and assume that the eigenvalues of A have non-positive real part. Then, for every T ≥ µ > 0 and every τ 0 ≥ 0, there exists a neighborhood T of τ 0 in R + and a (T, µ, T)-stabilizer for (2.2).

We prove this theorem here by generalizing the proof given in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] in the non-delayed case. The main point is that the time-contraction argument given in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF], when applied to a delayed system, reduces the effects of the delay in the system, in such a way that the limit system obtained by making the time-contraction parameter tend to infinity is essentially the same in the delayed and the non-delayed cases. In order to highlight these main ideas, we first consider a particular case of Theorem 2.5.

The d-Integrator

Before turning to the proof of Theorem 2.5, let us first consider the particular case where the dynamics of the system are given by the d-integrator, defined by the Jordan block

J d =            0 1 0 0 • • • 0 0 0 0 1 0 • • • 0 0 0 0 0 1 • • • 0 0 0 0 0 0 • • • 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 • • • 0 1 0 0 0 0 • • • 0 0            (3.1)
and by taking m = 1 and B = 0

• • • 0 1 T ∈ M d,1 (R).
This particular case will allow us to highlight the main ideas of the proof of Theorem 2.5, since it contains most of the difficulties of the general case. Furthermore, we can give in this case a stronger result, showing the existence of a (T, µ, T)-stabilizer for any bounded interval T ⊂ R + , and not only for perturbations around a certain value as in the general case of Theorem 2.5.

Theorem 3.1. Let A = J d , B = 0 • • • 0 1 T ∈ R d ,
and let T ≥ µ > 0 and r > 0 be given.

Then there exists a (T, µ,

[0, r])-stabilizer K ∈ M 1,d (R) for (2.2).
Proof. The proof follows the same idea of the proof of [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]Theorem 3.1]: we first perform a change of variables corresponding to a time contraction in order to relate (T, µ, [0, r])stabilizers to (T /ν, µ/ν, [0, r/ν])-stabilizers for ν > 0. We then study the stabilizability of a certain limit system, and this allows us to conclude the stabilizability of the original system for a certain ν > 0 large enough, thanks to the continuity result presented in the Appendix A.

Step 1. Time contraction

The system we consider is

ẋ(t) = J d x(t) -α(t)BKx(t -τ(t)), α ∈ G(T, µ), τ ∈ L ∞ (R + , [0, r]). (3.2)
For ν > 0, we define

D d,ν = diag(ν d-1 , . . . , ν, 1), (3.3)
which satisfies the relations

νD -1 d,ν J d D d,ν = J d , D d,ν B = B. (3.4)
Noting, for simplicity, x(t) = x(t; τ, x 0 , α, K), and defining

x ν (t) = D -1 d,ν x(νt), (3.5) x ν satisfies d dt x ν (t) = J d x ν (t) -α(νt)νBKD d,ν x ν t - τ(νt) ν (3.6)
and hence

x ν (t) = x t; τ(ν•) ν , D -1 d,ν x 0 (ν•), α ν , νKD d,ν with α ν (t) = α(νt), which is a (T /ν, µ/ν)-signal. Thus K is a (T, µ, [0, r])-stabilizer for (3.2) if and only if νKD d,ν is a (T /ν, µ/ν, [0, r/ν])-stabilizer.
This equivalence is crucial in what follows: instead of looking for a (T, µ, [0, r])-stabilizer for (3.2), we look for a (T /ν, µ/ν, [0, r/ν])-stabilizer for a certain ν > 0 large enough. The technique is thus to study a certain limit system obtained as ν → +∞, obtain a stabilizer for this non-delayed system and then show that this stabilizer is actually a (T /ν, µ/ν, [0, r/ν])-stabilizer for a certain ν > 0 large enough.

Step 2.

Limit system

We turn to the system ẋ(t)

= J d x(t) -α ⋆ (t)BKx(t), α ⋆ ∈ L ∞ (R + , [µ/T, 1]). (3.7)
It has been proved in [11, Theorem 3.1], using a result from [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], that one can find K ∈ M 1,d (R) and a positive definite matrix S ∈ M d (R), both independent of the particular signal

α ⋆ ∈ L ∞ (R + , [µ/T, 1]
), such that (3.7) is globally uniformly exponentially stable and V (x) = x T Sx decreases along all trajectories of (3.7), uniformly with respect to α ⋆ . In particular, there exists a time σ such that every trajectory of (3.7) starting in

B V 2 = {x ∈ R d |V (x) ≤ 2} at time 0 lies in B V 1 = {x ∈ R d |V (x) ≤ 1}
for every time larger than σ .

Step 3. Study of (3.6) through the limit system.

We wish to deduce from the conclusion obtained in the previous step that (3.2) admits a (T /ν, µ/ν, [0, r/ν])-stabilizer for a certain ν > 0 large enough. We claim that, for some ν > 0 large enough, every trajectory of

ẋ(t) = J d x(t) -α(t)BKx(t -τ(t)), α ∈ G(T /ν, µ/ν), τ ∈ L ∞ (R + , [0, r/ν]), with initial condition x 0 ∈ C 0 ([-r/ν, 0], B V 2 ) stays in B V
1 for every time larger than 2σ . In particular, by linearity, this will imply that K is a (T /ν, µ/ν, [0, r/ν])-stabilizer of (3.2) and thus ν -1 KD -1 d,ν is a (T, µ, [0, r])-stabilizer, concluding the proof. To prove this, assume, by contradiction, that for every n ∈ N * there exist

τ n ∈ L ∞ (R + , [0, r/n]), x (n) 0 ∈ C 0 ([-r/n, 0], B V 2 ), α n ∈ G(T /n, µ/n), and t n ∈ [2σ , 4σ ] such that, for every n ∈ N * , x t n ; τ n , x (n) 0 , α n , K / ∈ B V 1 . (3.8)
Up to the extraction of a subsequence, we can suppose that, as n → +∞,

t n → t ⋆ ∈ [2σ , 4σ ], x (n) 0 (0) → x ⋆ 0 ∈ B V 2 , and α n ⇀ α ⋆ ∈ L ∞ (R + , [0, 1
]) weakly-⋆; we also note that τ n (t) → 0

as n → +∞ uniformly on t ∈ R + . Then, applying Lemma A.1 proved in the Appendix A, we obtain that x t n ; τ n , x (n) 0 , α n , K converges to x(t ⋆ ; 0, x ⋆ 0 , α ⋆ , K) as n → +∞. We also note that, by [11, Lemma 2.5], α ⋆ (t) ≥ µ/T almost everywhere in R + , and so, by our previous study of (3.7), since t ⋆ ≥ 2σ , by linearity, we have

V (x(t ⋆ ; 0, x ⋆ 0 , α ⋆ , K)) ≤ 1 2 .
This contradicts (3.8), establishing the desired result.

Main Result

We now turn to the proof of our main result, Theorem 2.5. For a given stabilizable pair of matrices

(A, B) ∈ M d (R) × M d,m (R)
and for given T ≥ µ > 0 and τ 0 ≥ 0, we wish to find an interval T ⊂ R + of admissible perturbations around τ 0 and a (T, µ, T)-stabilizer for (2.2).

Proof of Theorem 2.5.

Step 1. Reduction to a canonical form Notice that we may reduce the theorem to the case where (A, B) is controllable, m = 1, and all the eigenvalues of A lie on the imaginary axis; this is detailed in Lemmas B.1, B.2, and B.3 in the Appendix B. We thus suppose from now on that (A, B) is controllable, m = 1, and ℜ(λ ) = 0 for every eigenvalue λ of A. We also reduce (A, B) to a normal form with which it shall be easier to work. Lemma 4.1. Suppose (A, B) ∈ M d (R) × R d is a controllable pair and ℜ(λ ) = 0 for every eigenvalue λ of A. Then, up to a linear transformation of coordinates, (2.1) can be written as

ẋ0 (t) = J r 0 x 0 (t) + α(t)b 0 u(t), x 0 (t) ∈ R r 0 , ẋ j (t) = (ω j A ( j) + J C r j )x j (t) + α(t)b j u(t), x j (t) ∈ R 2r j , j = 1, . . . , h, (4.1)
where the spectrum of A is σ (A) = {±iω j , j = j 0 , j 0 + 1, . . . , h} with all the ω j ≥ 0 distinct, j 0 = 1 if 0 / ∈ σ (A), j 0 = 0 and ω 0 = 0 otherwise; r j is the algebraic multiplicity of the eigenvalue iω j (with r 0 = 0 if 0 / ∈ σ (A)); J r 0 is the real Jordan block defined in (3.1);

J C n ∈ M 2n (R) is the Jordan block for complex eigenvalues, J C n =            0 2×2 Id 2 0 2×2 0 2×2 • • • 0 2×2 0 2×2 0 2×2 0 2×2 Id 2 0 2×2 • • • 0 2×2 0 2×2 0 2×2 0 2×2 0 2×2 Id 2 • • • 0 2×2 0 2×2 0 2×2 0 2×2 0 2×2 0 2×2 • • • 0 2×2 0 2×2 . . . . . . . . . . . . . . . . . . . . . 0 2×2 0 2×2 0 2×2 0 2×2 • • • 0 2×2 Id 2 0 2×2 0 2×2 0 2×2 0 2×2 • • • 0 2×2 0 2×2            , that is, J C n = J n ⊗ Id 2 in terms of the Kronecker product; A ( j) = diag(A 0 , . . . , A 0 ) ∈ M 2r j (R) with A 0 = 0 1 -1 0 ;
and b 0 and b j are respectively the vectors of R r 0 and R 2r j with all the coordinates equal to zero except the last one that is equal to one.

This lemma was proved in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] during the proof of Theorem 3.2 therein; for the sake of completeness, we present briefly its proof in the Appendix B.

Step 2. Time contraction

We work from now on with system (4.1). Given K ∈ M 1,d (R d ), we decompose K in blocks as

K = K 0 K 1 • • • K h with K 0 ∈ M 1,r 0 (R), K j ∈ M 1,2r j (R), j = 1, . . . , h, so that the feedback law u(t) = -Kx(t -τ(t)) is written as u(t) = -K 0 x 0 (t -τ(t)) -∑ h j=1 K j x j (t - τ(t)).
As in the proof of Theorem 3.1, we perform a change of time-space variables in the closed-loop system corresponding to a time contraction. Define

y 0 (t) = D -1 r 0 ,ν x 0 (νt), y j (t) = (D C r j ,ν ) -1 e -νtω j A ( j)
x j (νt), j = 1, . . . , h, with D n,ν as in (3.3), satisfying (3.4), and

D C n,ν = D n,ν ⊗ Id 2 = diag(ν n-1 , ν n-1 , . . . , ν, ν, 1, 1) ∈ M 2n (R), which satisfies ν(D C r j ,ν ) -1 J C r j D C r j ,ν = J C r j , D C r j ,ν b j = b j , j = 1, . . . , h.
Then y 0 , y 1 , . . . , y h satisfy

                   ẏ0 (t) = J r 0 y 0 (t) -α ν (t)b 0 K 0,ν y 0 t -τ(νt) ν + h ∑ ℓ=1 K ℓ,ν e (νt-τ(νt))ω ℓ A (ℓ) y ℓ t -τ(νt) ν , ẏ j (t) = J C r j y j (t) -α ν (t)e -νtω j A ( j) b j K 0,ν y 0 t -τ(νt) ν + + h ∑ ℓ=1 K ℓ,ν e (νt-τ(νt))ω ℓ A (ℓ) y ℓ t -τ(νt) ν , j = 1, . . . , h, (4.2) with α ν (t) = α(νt), K 0,ν = νK 0 D r 0 ,ν , K ℓ,ν = νK ℓ D C
r ℓ ,ν for ℓ = 1, . . . , h, and where we use that A ( j) D C r j ,ν = D C r j ,ν A ( j) and A ( j) J C r j = J C r j A ( j) for j = 1, . . . , h. This shows that the gain

K = K 0 K 1 • • • K h is a (T, µ, T)-stabilizer for (4.1) if and only if the gain K ν = K 0,ν K 1,ν • • • K h,ν is a (T /ν, µ/ν, T/ν)-stabilizer for (4.2), where T/ν = {t/ν | t ∈ T}.

Step 3. Choice of the feedback family

We turn to the problem of finding a neighborhood T of τ 0 in R + and a (T /ν, µ/ν, T/ν)stabilizer for (4.2) for a certain ν > 0, which will imply the theorem. We shall look for such a stabilizer K ν under a particular form. We write b 0 = 0 1 T and we take

K ν = K 0,ν K 1,ν • • • K h,ν with K 0,ν = K 0 , K 0 = k 0 1 • • • k 0 r 0 ∈ M 1,r 0 (R) K j,ν = K j ⊗ b T 0 e τ 0 ω j A 0 , K j = k j 1 • • • k j r j ∈ M 1,r j (R), j = 1, . . . , h. (4.3)
Now, since A (ℓ) = Id r ℓ ⊗A 0 , we have, for ℓ = 1, . . . , h, that

K ℓ,ν e (νt-τ(νt))ω ℓ A (ℓ) = K ℓ ⊗ b T 0 e (νt-τ(νt)+τ 0 )ω ℓ A 0 = = K ℓ ⊗ b T 0 e νtω ℓ A 0 + K ℓ ⊗ b T 0 e νtω ℓ A 0 e -(τ(νt)-τ 0 )ω ℓ A 0 -Id 2 .
Noting b j ∈ R r j the vector with all coordinates equal to zero except the last one that is equal to one, we have b j = b j ⊗ b 0 , and thus e -νtω j A ( j) b j = b j ⊗ e -νtω j A 0 b 0 . We finally write, for j, ℓ ∈ {1, . . . , h},

C (ν) 00 (t) = α ν (t), C (ν) 0 j (t) = α ν (t)b T 0 e νtω j A 0 , C (ν) j0 (t) = α ν (t)e -νtω j A 0 b 0 , C (ν) jℓ (t) = α ν (t)e -νtω j A 0 b 0 b T 0 e νtω j A 0 , P (ν) 00 (t) = P (ν) j0 (t) = 0, P (ν) 0 j (t) = α ν (t)b T 0 e νtω j A 0 e -(τ(νt)-τ 0 )ω j A 0 -Id 2 , P (ν) jℓ (t) = α ν (t)e -νtω j A 0 b 0 b T 0 e νtω ℓ A 0 e -(τ(νt)-τ 0 )ω ℓ A 0 -Id 2 , (4.4)
and thus system (4.2) can be written under the form

           ẏ0 (t) = J r 0 y 0 (t) - h ∑ ℓ=0 [b 0 K ℓ ⊗ (C (ν) 0ℓ (t) + P (ν) 0ℓ (t))]y ℓ t -τ(νt) ν , ẏ j (t) = J C r j y j (t) - h ∑ ℓ=0 [ b j K ℓ ⊗ (C (ν) jℓ (t) + P (ν) jℓ (t))]y ℓ t -τ(νt) ν , j = 1, . . . , h. (4.5)
We can arrange all the matrices

C (ν)
jℓ in a (2h + 1j 0 ) × (2h + 1j 0 ) symmetric matrix and all the matrices P (ν) jℓ in a (2h + 1j 0 ) × (2h + 1j 0 ) matrix respectively as

C (ν) (t) = C (ν) jℓ (t) j 0 ≤ j,ℓ≤h , P (ν) (t) = P (ν) jℓ (t) j 0 ≤ j,ℓ≤h . (4.6)
We take from now on T under the form T = [τ 0 -r, τ 0 + r] ∩ R + for a certain r > 0 to be chosen, and so

P (ν) jℓ (t) ≤ e -(τ(νt)-τ 0 )ω j A 0 -Id 2 = 2 1 -cos((τ(νt) -τ 0 )ω j ) ≤ (τ(νt) -τ 0 )ω j ≤ rΩ (4.7) with Ω = max{ω j | j = j 0 , . . . , h}.
Step 4.

Limit system

We wish to study (4.5) for large ν through a limit system approaching its behavior as ν → +∞, as we did with (3.6) in Theorem 3.1. The following lemma, proved later on in Appendix B, provides a stability result for the limit system that is used in the next step to study (4.5).

Lemma 4.2. Consider the system

           ẏ0 (t) = J r 0 y 0 (t) - h ∑ ℓ=0 [b 0 K ℓ ⊗ (C 0ℓ (t) + P 0ℓ (t))]y ℓ (t), ẏ j (t) = J C r j y j (t) - h ∑ ℓ=0 [ b j K ℓ ⊗ (C jℓ (t) + P jℓ (t))]y ℓ (t), j = 1, . . . , h, (4.8)
where y 0 ∈ R r 0 , y j ∈ R 2r j , J n and J C n are the Jordan blocks defined above, b 0 and b j are the vectors defined above, K j ∈ M 1,r j (R) are constant matrices, j = j 0 , . . . , h, C ⋆ , P ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) and the 2 × 2 time-dependent matrices C jℓ , P jℓ , 1 ≤ j, ℓ ≤ h, the (1j 0 ) × 2 time-dependent matrices C 0ℓ , P 0ℓ , the 2 × (1j 0 ) time-dependent matrices C j0 , P j0 and the signals C 00 , P 00 are defined by the relations

C ⋆ (t) = C jℓ (t) j 0 ≤ j,ℓ≤h , P ⋆ (t) = P jℓ (t) j 0 ≤ j,ℓ≤h , (4.9)
and we also assume that P jℓ (t) ≤ rΩ, for almost every t ∈ R + , ∀ j, ℓ ∈ { j 0 , . . . , h}. (4.10)

We write y = y T 0 y T 1 • • • y T h T .
Let ξ > 0. Then there exist C ≥ 1, γ > 0, r > 0, and K j ∈ M 1,r j (R), j = j 0 , . . . , h, such that, for every symmetric matrix C ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying C ⋆ (t) ≥ ξ Id 2h+1-j 0 almost everywhere, every P ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying (4.10) and every solution y of (4.8), we have y(t) ≤ Ce -γt y(0) , ∀t ≥ 0.

Step 5. Study of (4.5) through the limit system

To conclude the proof, we deduce the stability of (4.5) from that of (4.8) in the same way as we did in the proof of Theorem 3.1. Take T ≥ µ > 0 and τ 0 ≥ 0. By [11, Lemma 2.5], there exists ξ > 0 depending only on T, µ and ω j , j = j 0 , . . . , h, such that, for any α ∈ G(T, µ) and any ν > 0, the time-dependent matrix C (ν) constructed from α as in (4.4) and (4.6) is in L ∞ (R + , M 2h+1-j 0 (R)) and satisfies, for all t ≥ 0,

t+ T ν t C (ν) (s)ds ≥ ξ T ν Id 2h+1-j 0 .
For this ξ > 0, take C ≥ 1, γ > 0, r > 0, and

K j ∈ M 1,r j (R) as in Lemma 4.2. Set T = [τ 0 -r, τ 0 + r] ∩ R + and construct K = K 0 • • • K h from the K j , j = j 0 , . . . , h as in (4.3).
We want to show that, for ν > 0 large enough, K is a (T /ν, µ/ν, T/ν)-stabilizer for (4.2), and this will conclude the proof by the conclusion of Step 2. Note that, by Lemma 4.2, there exists a time σ > 0 depending only on C and γ such that, for every trajectory y of (4.8) starting in

B 2 = {x ∈ R d | x ≤ 2} at time 0 lies in B 1 = {x ∈ R d | x ≤ 1}
for every time larger than σ . We claim that, for some ν > 0 large enough, for every α ∈ G(T /ν, µ/ν), every τ ∈ L ∞ (R + , T/ν) and every initial condition y 0 ∈ C 0 ([-R/ν, 0], B 2 ), with R = sup T, the solution y of (4.5), with C (ν) and P (ν) given by (4.4) and (4.6), stays in B 1 for every time larger than 2σ . This will show, by linearity, that K is a (T /ν, µ/ν, T/ν)-stabilizer for (4.2). Assume, by contradiction, that for every n ∈ N * there exist τ n ∈ L ∞ (R + , T/n), y 0 n ∈ C 0 ([-R/n, 0], B 2 ), α n ∈ G(T /n, µ/n), and t n ∈ [2σ , 4σ ] such that, for every n ∈ N * , the solution y n of (4.5), with C (n) and P (n) given by (4.4) and (4.6), satisfies

y n (t n ) / ∈ B 1 . (4.11)
Up to the extraction of a subsequence, we can suppose that

lim n→∞ t n = t ⋆ ∈ [2σ , 4σ ], lim n→∞ y 0 n (0) = y 0 ⋆ ∈ B 2 , lim n→∞ C (n) = C ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R))
weakly-⋆,

lim n→∞ P (n) = P ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R))
weakly-⋆, and we also note that τ n (t) → 0 uniformly on t ∈ R + as n → +∞. Then, by Lemma A.1, y n converges to the solution y ⋆ of (4.8) associated to C ⋆ , P ⋆ and with initial condition y 0 ⋆ , uniformly on compact time intervals, and in particular y n (t n ) → y ⋆ (t ⋆ ). By [11, Lemma 2.5], we have C ⋆ (t) ≥ ξ Id 2h+1-j 0 for almost every t and, since P (n) jℓ (t) ≤ rΩ for every j, ℓ ∈ { j 0 , . . . , h} and almost every t ∈ R + , we have, by the lower semicontinuity of the norm of L ∞ (R + , M 2h+1-j 0 (R)), that P jℓ (t) ≤ rΩ for every j, ℓ ∈ { j 0 , . . . , h} and almost every t ∈ R + , where P jℓ is obtained from P ⋆ by (4.9). Thus we are under the hypotheses of Lemma 4.2, and so our previous discussion shows us that y ⋆ remains in B 1 for every time larger than σ ; by linearity, y ⋆ (t ⋆ ) ≤ 1 /2 since t ⋆ ≥ 2σ . This contradicts (4.11), establishing the desired result.

Further Discussion

We proved that persistently excited linear systems can be stabilized by a delayed feedback law when the uncontrolled dynamics of the system is given by a matrix A whose eigenvalues have all non-positive real part and when the delay varies in an interval around a constant value τ 0 , with the feedback matrix K depending on the matrices A, B, on the constants T and µ of the condition of persistence of excitation and on the reference delay τ 0 . This is a generalization of [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]Theorem 3.2], originally proved for the non-delayed case.

The technique of the proof consists on adapting the time-contraction argument of [11, Theorem 3.2] to the delayed case. Indeed, the time contraction also contracts the delay, reducing its effect, and the limit system obtained in the time-contraction procedure is the same as in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF], except for the new terms P jℓ , which are treated as perturbations of the limit system of [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF].

It is actually by treating these terms P jℓ as perturbations that we arrive to the construction of the delay neighborhood T around τ 0 where we can guarantee stabilizability. Note that the terms P jℓ do not appear in the limit system obtained when A = J d in Theorem 3.1, since they depend on the eigenvalues iω j , and this is the reason why we can obtain a (T, µ, T)-stabilizer for any bounded T ⊂ R + when A = J d in Theorem 3.1. This is a fundamental difference between Theorems 3.1 and 2.5 which we would like to highlight: in Theorem 3.1, stabilization can be achieved for any bounded set T ⊂ R + where the delay takes its values, whereas in Theorem 2.5 T is chosen as T = [τ 0 -r, τ 0 + r] ∩ R + , a perturbation around the constant value τ 0 .

A natural question is then to study if Theorem 2.5 might not be generalized for any bounded set T instead of considering only perturbations around τ 0 . This is actually not possible, as shown in the following example, where we take α identically equal to one, i.e., the control is completely active the whole time.

u(t) = -Kx(t -τ(t))
(5.2)

This control system does not depend on a persistently exciting signal α, but, in order to keep the notations we used previously, we shall consider it as a persistently excited system with constants T = µ, so that G(T, µ) = G(T, T ) reduces to the class containing only the constant signal identically equal to one. We want to prove that the conclusion of Theorem 3.1 does not hold for (5.1), that is, we want to show that there exists a bounded interval T for which (5.1) with the feedback (5.2) does not admit a (T, T, T)-stabilizer. Obviously, this also implies the non-existence of a (T, µ, T)-stabilizer for every µ ∈ (0, T ] since such a stabilizer would be in particular a (T, T, T)-stabilizer. We claim that (5.1) with the feedback (5.2) does not admit a (T, T, [0, 2π])-stabilizer. In order to simplify our analysis, we shall consider only constant-in-time delays in the interval [0, 2π], which allow us to apply the techniques of stability analysis for delayed systems presented in [START_REF] Michiels | Stability and stabilization of time-delay systems: An eigenvalue-based approach[END_REF].

The closed-loop system obtained from (5.1) with the feedback (5.2) and a constant delay τ ∈ [0, 2π] is ẋ(t) = Ax(t) -BKx(t -τ).

(5.3)

According to [START_REF] Michiels | Stability and stabilization of time-delay systems: An eigenvalue-based approach[END_REF]Proposition 1.6], the stability of (5.3) can be studied through the complex roots λ of the characteristic equation det λ Id 2 -A + BKe -λ τ = 0;

(5.4) the origin of (5.3) is exponentially stable if and only if all the roots λ of (5.4) satisfy ℜ(λ ) < 0, and exponential stability and asymptotic stability are also equivalent in this case. Writing K = k 1 k 2 , the characteristic equation (5.4) is

λ 2 + k 2 λ e -λ τ + 1 + k 1 e -λ τ = 0. (5.5)
We now want to show that, for every K ∈ M 1,2 (R), there exists τ ∈ [0, 2π] such that (5.5) admits a root λ with ℜ(λ ) ≥ 0. As remarked in [START_REF] Michiels | Stability and stabilization of time-delay systems: An eigenvalue-based approach[END_REF]Theorem 1.15], by the continuity of the real part of the largest eigenvalue with respect to the delay, this study is reduced to the problem of finding a delay τ ∈ [0, 2π] such that (5.5) admits a root λ with ℜ(λ ) = 0. The feedback K = 0 obviously does not stabilize the system to the origin, and so we suppose from now on that k 1 and k 2 are not simultaneously zero. We look for a certain τ ∈ [0, 2π] and a root λ = iω of (5.5) with ω ∈ R. We thus want ω to satisfy

1 -ω 2 + k 1 cos(τω) + k 2 ω sin(τω) = 0, -k 1 sin(τω) + k 2 ω cos(τω) = 0. This is equivalent to                sin θ = k 2 ω(ω 2 -1) k 2 2 ω 2 + k 2 1 , cos θ = k 1 (ω 2 -1) k 2 2 ω 2 + k 2 1 , θ = τω (5.6)
and such a system can only have a solution if sin 2 θ + cos 2 θ = 1, which is the case if and only if

(ω 2 -1) 2 = k 2 2 ω 2 + k 2 1 .
This last equation is a polynomial in ω 2 of degree 2, whose solutions can be computed explicitly as

ω 2 = 1 2 2 + k 2 2 ± (2 + k 2 2 ) 2 -4(1 -k 2 1 )
.

We consider from now on the solution

ω = 2 + k 2 2 + (2 + k 2 2 ) 2 -4(1 -k 2 1 ) 2 . Note that ω is well-defined in R since (2 + k 2 2 ) 2 > 4(1 -k 2 1 ) for any K ∈ M 1,2 (R) \ {0}
, and that ω ≥ 1. With this ω, we can thus find θ ∈ [0, 2π] such that (5.6) is satisfied, and so τ = θ /ω ∈ [0, 2π] since ω ≥ 1. Since the constructed (θ , τ, ω) satisfies (5.6), (5.5) is hence satisfied for τ and λ = iω, and thus (5.3) is not asymptotically stable. Hence (5.1) admits no (T, T, [0, 2π])-stabilizer.

Note that we could replace [0, 2π] in Example 5.1 for any other interval T ⊂ R + with length greater than or equal 2π, and so we conclude that (5.1) does not admit a (T, µ, T)-stabilizer if T contains an interval with length greater than or equal 2π.

The value 2π obtained in these computations comes from the fact that the dynamics given by the matrix A we chose correspond to rotations around the origin with unitary angular velocity, and 2π is the total time that a solution of ẋ = Ax takes to make a complete turn around the origin. If we choose A as A = 0 ω 0 -ω 0 0 for ω 0 = 0, then the same computations as in Example 5.1 show that no (T, µ, T)-stabilizer can exist for (5.1) if T contains an interval of length at least 2π ω 0 . In particular, this gives a link between an upper bound on the maximal length of an interval contained in T for which a (T, µ, T)-stabilizer exists and the eigenvalues of A on the imaginary axis.

This example shows that the fundamental difference in the statement of Theorems 3.1 and 2.5 concerning the choice of the set T actually comes from the dynamics of the system itself, and that no improvement of Theorem 2.5 as good as Theorem 3.1 can be obtained.

A Appendix: A Continuity Result for Delayed Systems

We show here a continuity result of the solution of a delayed system with respect to its parameters, in the spirit of [START_REF] Chaillet | Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases[END_REF]Proposition 21], which is used in the proof of Theorems 3.1 and 2.5. We place ourselves in a more general setting than (2.2), considering the system

ẋ(t) = Ax(t) + B(t)x(t -τ(t)), (A.1) where τ ∈ L ∞ (R + , [0, r]), and B ∈ L ∞ (R + , M d (R)
) is a time-dependent matrix. We remark that (A.1) satisfies the Carathéodory conditions for delayed equations, and so, for fixed τ and B and for any given initial condition x 0 ∈ C 0 ([-r, 0], R d ), (A.1) admits a unique continuous solution x defined on [-r, +∞), which we denote by x(t) = x(t; τ, x 0 , B); this solution is absolutely continuous on R + , coincides with x 0 on [-r, 0], and satisfies (A.1) for almost every t ∈ R + . Our continuity result can then be stated as follows.

Lemma A.1. Let (τ n ) n∈N * be a sequence on L ∞ (R + , [0, r]) such that τ n (t) → 0 as n → +∞ uniformly on R + . Suppose (x (n) 0 ) n∈N * is a sequence of functions in C 0 ([-r, 0], R d ) and (B n ) n∈N * a bounded sequence on L ∞ (R + , M d (R)) satisfying 1. lim n→+∞ x (n) 0 (0) = x ⋆ 0 for a certain x ⋆ 0 ∈ R d ;
2. there exists Λ > 0 such that x

(n) 0 (t) ≤ Λ for all n ∈ N * and all t ∈ [-r, 0];

3. B n ----⇀ n→+∞ B ⋆ weakly-⋆ for a certain B ⋆ ∈ L ∞ (R + , M d (R)). Then x(t; τ n , x (n) 0 , B n ) ----→ n→+∞ x(t; 0, x ⋆ 0 , B ⋆ ), uniformly on compact time intervals in R + .
Proof. We can extend B ⋆ outside R + to the whole real line in such a way that this extension is an element of L ∞ (R, M d (R)). We fix such an extension, so that x(•; 0, x ⋆ 0 , B ⋆ ) is absolutely continuous in R and satisfies (A.1) for almost every t ∈ R; note that this is possible since x(•; 0, x ⋆ 0 , B ⋆ ) is the solution of a non-delayed system. For simplicity, we shall note

x n (t) = x(t; τ n , x (n) 0 , B n ) and x ⋆ (t) = x(t; 0, x ⋆ 0 , B ⋆ ).
We also note by M an upper bound on B n L ∞ (R + ,M d (R)) and r n = sup t∈R + τ n (t), and, by the uniform convergence of τ n to 0, we have that r n → 0 as n → +∞.

Define e n (t) = x n (t)x ⋆ (t) for t ≥ -r. Then, for t ≥ 0, e n satisfies

ėn (t) = Ae n (t) + B n (t)e n (t -τ n (t)) + f n (t) (A.2) with f n given by f n (t) = B n (t)(x ⋆ (t -τ n (t)) -x ⋆ (t)) + (B n (t) -B ⋆ (t))x ⋆ (t).
Since x ⋆ is continuous, it follows from Lebesgue's Dominated Convergence Theorem that lim

n→+∞ t 0 B n (s)(x ⋆ (s -τ n (s)) -x ⋆ (s))ds =
F n (σ ) = 0 uniformly on t ∈ [0, T ].
Moreover, for s ∈ X n,σ , we have that

e n (s -τ n (s)) = x n (s -τ n (s)) -x ⋆ (s -τ n (s)) ≤ C,
where C = Λ + sup t∈ [-r,0] x ⋆ (t) , and so B Appendix: On the Proof of Theorem 2.5

sup σ ∈[t-r n ,t]∩[0,t] X n,σ
We prove here some of the results that were used in the proof of Theorem 2.5. The first three results, Lemmas B.1, B.2 and B.3, deal with the reduction of Theorem 2.5 to the case where (A, B) is controllable, m = 1 and all the eigenvalues of A lie on the imaginary axis. We begin by reducing the theorem to the case where (A, B) is controllable.

Lemma B.1. It suffices to prove Theorem 2.5 in the case where (A, B) is controllable.

Proof. Up to a linear change of variables, A and B can be decomposed on the controllable and uncontrollable parts according to Kalman decomposition as

A = A 1 A 3 0 A 2 , B = B 1 0 with A 1 ∈ M d ′ (R), A 2 ∈ M d-d ′ (R), B 1 ∈ M d ′ ,m (R),
ẋ1 (t) = A 1 x 1 (t) + A 3 x 2 (t) + α(t)B 1 u(t), ẋ2 (t) = A 2 x 2 (t), (B.1) with x 1 (t) ∈ R d ′ , x 2 (t) ∈ R d-d ′
, and x(t) = x 1 (t) T x 2 (t) T T . Now, suppose the theorem is proved for the controllable case and K ′ ∈ M m,d ′ (R) is a (T, µ, T)-stabilizer for (A 1 , B 1 ) for a certain neighborhood T of τ 0 in R + , associated with certain constants C 1 ≥ 1, γ 1 > 0 as in Definition 2.2. Take K = K ′ 0 ∈ M m,d (R), so that, with the feedback u(t) = -Kx(tτ(t)), (B.1) becomes

ẋ1 (t) = A 1 x 1 (t) -α(t)B 1 K ′ x 1 (t -τ(t)) + A 3 x 2 (t), ẋ2 (t) = A 2 x 2 (t). (B.2)
Let us note r = sup T. Take α ∈ G(T, µ), τ ∈ L ∞ (R + , T), and an initial condition x 0 ∈ C 0 ([-r, 0], R d ), written as x 0 (t) = x 0,1 (t) T x 0,2 (t) T T . Note by y(t) ∈ R d the solution of

ẏ(t) = A 1 y(t) -α(t)B 1 K ′ y(t -τ(t)), t > 0, y(t) = x 0,1 (t), t ∈ [-r, 0].
Then, by the hypothesis on K ′ , we have that

y(t) ≤ C 1 e -γ 1 t sup s∈[-r,0] x 0,1 (s) . (B.3)
The result on [13, Section 6.2] allows us to write the solution x(t) = x 1 (t) T x 2 (t) T T of (B.2) associated with α and τ and with initial condition x 0 as x 0 (s) , which proves that K is a (T, µ, T)-stabilizer for (A, B), as desired.

   x 1 (t) = y(t) + t 0 X(t, s)A 3 x 2 (s)ds, x 2 (t) = e A 2 t x 0,2 (0)
The following lemma shows that we may further reduce Theorem 2.5 to the single-input case. Its proof follows the same idea of [START_REF] Chitour | Stabilization of persistently excited linear systems[END_REF]Chapter 4,Theorem 4], where the original stabilization result for single-input systems of [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]Theorem 3.2] is generalized to the multi-input case by a recurrence on the number of inputs.

Lemma B.2. It suffices to prove Theorem 2.5 in the case where (A, B) is controllable and m = 1.

Proof. We may suppose (A, B) controllable by Lemma B.1. We suppose the theorem to be proved in the case m = 1 and we prove the general case by induction on m. Suppose the theorem has been proved for m -1, that is, for every d ∈ N * , for every A ∈ M d (R) and B ∈ M d,m-1 (R) such that (A, B) is a controllable pair and the eigenvalues of A have non-positive real part, for every T, µ with T ≥ µ > 0, and for every τ 0 ≥ 0, there exists a neighborhood T of τ 0 in R + and a (T, µ, T)-stabilizer for (2.2).

Take

A ∈ M d (R) and B ∈ M d,m (R) such that (A, B
) is a controllable pair and the eigenvalues of A have non-positive real part and fix T ≥ µ > 0 and τ 0 ≥ 0. Note by b ∈ R d the first column of B; we may suppose, without loss of generality, that b = 0, for otherwise the first input does not influence the system and it may thus be excluded, reducing the system to the case with m -1 inputs. We consider the pair (A, b), which may not be controllable, but can be decomposed according to Kalman decomposition: there exists an invertible P ∈ M d (R) such that

PAP -1 = A 1 A 3 0 A 2 , Pb = b 1 0 , with A 1 ∈ M d ′ (R), b 1 ∈ R d ′ ,
= A 1 A 3 0 A 2 z + α(t) b 1 B 3 0 B 2 u (B.7) with B 2 ∈ M d-d ′ ,m-1 (R) and B 3 ∈ M d ′ ,m-1 (R).
By the controllability of (A, B) and

(A 1 , b 1 ), it follows that (A 2 , B 2 ) is also controllable. Now B 2 ∈ M d-d ′ ,m-1 (R)
, and so, by the induction hypothesis, (A 2 , B 2 ) admits a (T, µ, T 2 )stabilizer K 2 ∈ M m-1,d-d ′ (R) for a certain neighborhood T 2 of τ 0 in R + . If Theorem 2.5 is proved in the controllable case with m = 1, then we can take a (T, µ, T 1 )-stabilizer

K 1 ∈ M 1,d ′ (R) for (A 1 , b 1 ) for a certain neighborhood T 1 of τ 0 in R + . We claim that K ∈ M m,d (R) given by K = K 1 0 0 K 2 is a (T, µ, T)-stabilizer for (A, B) for the neighborhood T = T 1 ∩ T 2 .
Indeed, with this feedback, system (B.7) becomes

ż(t) = A 1 A 3 0 A 2 z(t) -α(t) b 1 K 1 B 3 K 2 0 B 2 K 2 z(t -τ(t)). Noting z = z T 1 z T 2 T with z 1 ∈ R d ′ and z 2 ∈ R d-d ′ , we can thus write ż1 (t) = A 1 z 1 (t) -α(t)b 1 K 1 z 1 (t -τ(t)) + A 3 z 2 (t) -α(t)B 3 K 2 z 2 (t -τ(t)), ż2 (t) = A 2 z 2 (t) -α(t)B 2 K 2 z 2 (t -τ(t)). (B.8)
We denote by X(t, s) the fundamental matrix solution of ẋ(t) = A 1 x(t) -α(t)b 1 K 1 x(tτ(t)); by construction of K 1 and by [13, Lemma 6.5.3], we can find C 0 ≥ 1 and γ 0 > 0, both independent of α ∈ G(T, µ) and τ ∈ L ∞ (R + , T), such that X(t, s) ≤ C 0 e -γ 0 (t-s) , ∀t ≥ s ≥ 0.

Note r = sup T. Given an initial condition z T 0,1 z T 0,2 T ∈ C 0 ([-r, 0], R d ), note by y 1 and y 2 the solutions to ẏ1 (t) = A 1 y 1 (t) -α(t)b 1 K 1 y 1 (t -τ(t)), y 1 (t) = z 0,1 (t) for t ∈ [-r, 0], ẏ2 (t) = A 2 y 2 (t) -α(t)B 2 K 2 y 2 (t -τ(t)), y 2 (t) = z 0,2 (t) for t ∈ [-r, 0]. (B.9)

By construction of K 1 and K 2 , there exist C 1 ,C 2 ≥ 1 and γ 1 , γ 2 > 0 such that y j (t) ≤ C j e -γ j t sup s∈[-r,0] z 0, j (s) , j = 1, 2.

We can now write the solution of (B. Proof of Lemma 4.1. Up to a linear change of variables in (2.1), we may suppose that A is in its real Jordan normal form. A has a unique Jordan block associated with each {-iω j , iω j }, j = j 0 , . . . , h, for, otherwise, the rank of the matrix Aiω j Id d B would be strictly smaller than d, contradicting the Hautus test for controllability. Thus, up to a permutation of variables on R d , we can write A = diag(J r 0 , ω 1 A (1) + J C r 1 , . . . , ω h A (h) + J C r h ), and B ∈ R d is such that (A, B) is controllable. Now, take b ∈ R d as b = (b 0 ) T (b 1 ) T • • • (b h ) T T with b 0 and b j , j = 1, . . . , h, as defined in the statement of the lemma. It follows from Hautus test for controllability that (A, b) is controllable. But all controllable linear control systems associated with a pair (A, B) that have in common the eigenvalues of A, counted according to their multiplicity, are state-equivalent, since they can be transformed by a linear transformation of coordinates into the same system under controller form (see, e.g., [START_REF] Sontag | Mathematical control theory[END_REF]), and so (A, B) can be transformed into (A, b) by a linear transformation of coordinates, leading to the desired result.

Finally, to complete the proof of Theorem 2.5, we prove Lemma 4.2, which gives the uniform exponential stability of the limit system considered in the proof of Theorem 2.5.

Proof of Lemma 4.2. We consider the matrices P jℓ as a perturbations in (4.8), and so we consider first the non-perturbed system Let ξ > 0. It has been proved in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF]Theorem 3.2] that, for a given ξ > 0, one can find a gain K = K 0 K 1 • • • K h and a positive definite matrix S ∈ M d (R) such that, for every symmetric C ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying C ⋆ (t) ≥ ξ Id 2h+1-j 0 for almost every t ≥ 0, (B.12) is globally uniformly exponentially stable and V (y) = y T Sy decreases exponentially along all trajectories of (B.12), uniformly with respect to C ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying C ⋆ (t) ≥ ξ Id 2h+1-j 0 almost everywhere; i.e., there exist C ≥ 1 and γ > 0 such that, for every symmetric C ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying C ⋆ (t) ≥ ξ Id 2h+1-j 0 almost everywhere and every solution y of (B.12), we have y(t) ≤ Ce -2γt y(0) . We denote by X(t, s) the fundamental matrix solution of (B.12), i.e., for any y 0 ∈ R d , y(t) = X(t, s)y 0 is the unique solution to (B.12) with y(s) = y 0 . Hence we have the estimate X(t, s) ≤ Ce -2γ(t-s) .

           ẏ0 (t) = J r 0 y 0 (t) - h ∑ ℓ=0 [b 0 K ℓ ⊗C 0ℓ (t)]y ℓ (t), ẏ j (t) = J C
(B.13)

We now turn to the perturbed system (4.8). For a given ξ > 0, we take C ≥ 1, γ > 0 and K j as before. For every symmetric matrix C ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying C ⋆ (t) ≥ ξ Id 2h+1-j 0 almost everywhere, and every P ⋆ ∈ L ∞ (R + , M 2h+1-j 0 (R)) satisfying (4.10), we set A = diag(J r 0 , J C r 1 , . . . , J C r h ) ∈ M d (R), B(t) = b j K ℓ ⊗C jℓ (t) j 0 ≤ j,ℓ≤h , P(t) = b j K ℓ ⊗ P jℓ (t) j 0 ≤ j,ℓ≤h with b0 = b 0 . System (4.8) can thus be written under the form ẏ(t) = Ay(t) -B(t)y(t) -P(t)y(t)

and, using the fundamental matrix X of (B.12), we can write its solution for a given initial condition y 0 as y(t) = X(t, 0)y 0 -t 0 X(t, s)P(s)y(s)ds.

By (4.10), we can write P(t) ≤ C ′ rΩ for a certain constant C ′ > 0, and thus, up to increasing C, we have, by (B.13), y(t) ≤ Ce -2γt y 0 +CrΩ t 0 e -2γ(t-s) y(s) ds.

Applying Gronwall's Lemma to e 2γt y(t) , we thus obtain y(t) ≤ Ce -(2γ-CrΩ)t y 0 .

We choose r > 0 small enough so that 2γ -CrΩ ≥ γ, and so y(t) ≤ Ce -γt y 0 , which gives us the desired result.

Example 5 . 1 .

 51 Consider the control system ẋ = Ax + Bu (

  e n (s -τ n (s)) ds ≤ Cr n ----→ n→+∞ 0 uniformly on t ∈ [0, T ]. Hence ϕ n (t) ----→ n→+∞ 0 uniformly on [0, T ], from where we get, together with (A.4), that E n (t) ----→ n→+∞ 0 uniformly on [0, T ]. So e n (t) ----→ n→+∞ 0 uniformly on [0, T ], and, since T > 0 is arbitrary, this gives the desired result.

8 )

 8 in terms of the initial condition z T 0[-r, 0], R d ) using the variation-of-constants formula in [13, Section 6.2] as   z 1 (t) = y 1 (t) + t 0 X(t, s)(A 3 z 2 (s) -α(s)B 3 K 2 z 2 (s -τ(s)))ds, z 2 (t) = y 2 (t).

  r j y j (t) -h ∑ ℓ=0 [ b j K ℓ ⊗C jℓ (t)]y ℓ (t), j = 1, . . . , h. (B.[START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]

  This limit is uniform on compact time intervals in R + . Indeed, let T > 0 and X ⋆ = sup t∈[-r,T ] x ⋆ (t) ; we thus see that f n (t) ≤ 2MX ⋆ and so F n (t) ≤ 2MX ⋆ T for every t ∈ [0, T ]. Furthermore, for 0 ≤ t 1 < t 2 ≤ T , we haveF n (t 2 ) -F n (t 1 ) ≤ 2MX ⋆ (t 2t 1 ) ,and hence (F n ) is equicontinuous. Thus, by Arzelà-Ascoli Theorem, the closure of{F n | n ∈ N * } is a compact subset of C 0 ([0, T ], R d )with the topology of the uniform convergence, and so this set has at least one limit point; it has exactly one, for, if it had two distinct limit points, this would contradict the fact that (F n (t)) n∈N * tends pointwise to 0, and so the sequence(F n ) n∈N * converges uniformly to 0 in [0, T ]. = {s ∈ [0,t] | s -τ n (s) < 0}.This set is measurable and, since 0 ≤ τ n (t) ≤ r n for all t ∈ R + , we have that X n,t ⊂ [0, r n ], so that λ (X n,t ) ≤ r n for all t ∈ R + , where λ denotes the Lebesgue measure. Define also

	0 0 f n (s)ds, this shows that F n (t) ----→ for every t ≥ 0. By the weak-⋆ convergence of (B n ), we have that lim n→+∞ t 0 (B n (s) -B ⋆ (s))x ⋆ (s)ds = 0, and so f n satisfies lim n→+∞ t 0 f n (s)ds = 0 for every t ≥ 0. Letting F n (t) = t n→+∞ 0. t 2 t 1 e n (t) = e n (0) + F n (t) + t 0 Ae n (s)ds + t 0 B n (s)e n (s -τ n (s))ds, 0 for every t ≥ which gives us the estimate e n (t) ≤ e n (0) + F n (t) + t 0 A e n (s) ds + M t 0 e n (s -τ n (s)) ds. (A.3) Define X n,t E n (t) = sup s∈[t-r n ,t]∩[0,t] e n (s) and M ′ = A + M. From (A.3), we obtain e n (t) ≤ e n (0) + F n (t) + M X n,t e n (s -τ n (s)) ds + M ′ t 0 E n (s)ds, so that, for t ≥ 0, E n (t) ≤ ϕ n (t) + M ′ t 0 E n (s)ds, with ϕ n given by ϕ n (t) = e n (0) +sup σ ∈[t-r n ,t]∩[0,t] F n (σ ) + M X n,σ e n (s -τ n (s)) ds . Applying Gronwall's Lemma, we get E n (t) ≤ ϕ n (t) + M ′ t 0 ϕ n (s)e M ′ (t-s) ds (A.4) for t ≥ 0. Fix T > 0. Since lim n→+∞ F n (t) = 0 uniformly on [0, T ], we have that f Integrating (A.2) from 0 to t ≥ 0, we obtain lim sup n→+∞ σ ∈[t-r n ,t]∩[0,t]

n (s) ds ≤

  the other matrices having appropriate dimensions, and where (A 1 , B 1 ) is controllable (see, for instance, [29, Theorem 13.1]); since (A, B) is stabilizable, A 2 is Hurwitz. The open-loop system (2.1) can thus be written after the change of variables as

  ∈ M d ′ (R) is the fundamental matrix solution associated with the delayed system ż(t) = A 1 z(t) -α(t)B 1 K ′ z(t -τ(t)) (see[START_REF] Hale | Introduction to functional differential equations[END_REF] Section 6.1]). Our choice of K ′ guarantees that this last system is exponentially stable, uniformly with respect to α ∈ G(T, µ) and τ ∈ L ∞ (R + , T), and so, by[START_REF] Hale | Introduction to functional differential equations[END_REF] Lemma 6.5.3], there exist constants C 0 ≥ 1, γ 0 > 0 indepen- dent of α and τ such that Lemma 6.5.3] is proved only for the case of uniformity with respect to the initial time, but the same proof also applies for the case of uniformity with respect to other parameters. Note also that we do not need to consider uniformity with respect to the initial time since the classes G(T, µ) and L ∞ (R + , T) are invariant with respect to positive time translations and a non-zero initial time may be translated into terms of a different choice of α and τ.Since A 2 is Hurwitz, there exist C 2 ≥ 1, γ 2 > 0 such that e A 2 t ≤ C 2 e -γ 2 t .

	where X(t, s) X(t, s) ≤ C 0 e -γ 0 (t-s)	for all t ≥ s ≥ 0;	(B.5)
	note that [13,
			(B.4)
	,

(B.6)

Using the estimates (B.3), (B.5) and (B.6) in (B.4), we can find C ≥ 1 and γ > 0, depending only on C 0 , C 1 , C 2 , γ 0 , γ 1 , γ 2 , and thus independent of α and τ, such that x(t) ≤ Ce -γt sup s∈

[-r,0]

  all the other matrices have appropriate dimensions, and (A 1 , b 1 ) is controllable. Now, performing the change of variables z = Px in (2.1), the open-loop system becomes ż

Acknowledgment

The author would like to thank Y. Chitour and M. Sigalotti for having suggested the study of this problem and for the helpful discussions that followed.

It is thus easy to see that [-r,0] z 0,1 (s) +C ′ e -γ ′ t sup s∈ [-r,0] z 0,2 (s) , z 2 (t) ≤ C 2 e -γ 2 t sup s∈ [-r,0] z 0,2 (s) , for certain constants C ′ ≥ 1, γ ′ > 0, and so K is a (T, µ, T)-stabilizer for (B.7), as we wanted to prove. The result is thus established by induction.

We further reduce our proof of Theorem 2.5 to the case where all the eigenvalues of A lie on the imaginary axis.

Lemma B.3. It suffices to prove Theorem 2.5 in the case where (A, B) is controllable, m = 1, and ℜ(λ ) = 0 for every eigenvalue λ of A.

Proof. We may suppose (A, B) controllable and m = 1 by Lemma B.2. Up to a linear change of variables, A and B can be written as

, and x(t) = x 1 (t) T x 2 (t) T T . Now, suppose the theorem is proved for the case stated above and take

, and an initial condition x 0 ∈ C 0 ([-r, 0], R d ), written as x 0 (t) = x 0,1 (t) T x 0,2 (t) T T . By the hypothesis on K ′ , we have that the solution x(t) = x 1 (t) T x 2 (t) T T of (B.11) associated with α and τ and with initial condition

Applying the variation-of-constants formula to (B.11) and using an exponential estimate on e A 1 t , it is immediate to verify that K is a (T, µ, T)-stabilizer for (A, B).

Let us now present a proof of Lemma 4.1, which was originally done in [START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] and that we recall here for the sake of completeness.