Delphine Bresch-Pietri

Frederic Mazenc

Nicolas Petit

Robust compensation of a chattering time-varying input delay with jumps

Keywords: delay systems, distributed parameter systems, reduction method, prediction-based control, time-varying delay, hybrid systems

We investigate the design of a prediction-based controller for a linear system subject to a time-varying input delay, not necessarily First-In/First-Out (FIFO). This means that the input signals can be reordered. The feedback law uses the current delay value in the prediction. It does not exactly compensate the delay in the closed-loop dynamics but does not require to predict future delay values, contrary to the conventional prediction technique. Modeling the input delay as a transport Partial Differential Equation, we prove asymptotic stabilization of the system state, that is, robust delay compensation, providing that the average L 2 -norm of the delay time-derivative over some time-window is sufficiently small and that the average time between two discontinuities (average dwell time) is sufficiently large.

Introduction

Time-delays are ubiquitous in engineering systems. They can take the form of communication lags or a physical deadtime and, in all cases, reveal troublesome in the design and tuning of feedback control laws. Delays are a central concern for numerous systems. When delay stems from transportation of material, as is observed in mixing plants for liquid or gaseous fluids [START_REF] Chèbre | Feedback control and optimization for the production of commercial fuels by blending[END_REF] [START_REF] Petit | Motion planning for two classes of nonlinear systems with delays depending on the control[END_REF], automotive engine and exhaust line [START_REF] Depcik | One-dimensional automotive catalyst modeling[END_REF] or heat collector plant [START_REF] Sbarciog | Nonlinear Predictive Control of processes with variable time delay. A temperature control case study[END_REF], the dead-time satisfies the First-In/First-Out (FIFO) principle by definition, i.e., the delay D is such that Ḋ(t) < 1 for all time. However, this is not always the case. For example, communication delays can be subject to sudden variations and not satisfy the FIFO principle. This feature, sometimes refered to as fast-varying delay (see [START_REF] Seuret | Stability of systems with fast-varying delay using improved wirtinger's inequality[END_REF][START_REF] Shustin | On delay-derivative-dependent stability of systems with fast-varying delays[END_REF]), can also be exhibited for state-or inputdependent input delay systems [START_REF] Dieulot | Tracking control of a nonlinear system with input-dependent delay[END_REF], in which the delay variations can be related to the input in a very intricate manner, like, e.g., for crushing mill devices [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF].

We investigate the design of a prediction-based control law [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF] [START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF] [21] [START_REF] Smith | A controller to overcome dead time[END_REF], which is state-of-the-art for constant input delay [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF] [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF] [17] [START_REF] Mazenc | Generating positive and stable solutions through delayed state feedback[END_REF] [26] [START_REF] Moon | Robust stabilization of uncertain input-delayed systems using reduction method[END_REF] but has only been more recently used for time-varying delays (see [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF] Email addresses: delphine.bresch-pietri@mines-paristech.fr (Delphine Bresch-Pietri), frederic.mazenc@l2s.centralesupelec.fr. (Frederic Mazenc), nicolas.petit@mines-paristech.fr (Nicolas Petit).

or [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]). To compensate a varying input delay, the prediction has to be calculated over a time window the length of which matches the value of the future delay. In other words, one needs to predict the future variations of the delay to compensate it. This is the approach followed in [START_REF] Witrant | Stabilisation des systèmes commandés par réseaux[END_REF] for a communication time-varying delay, the variations of which are provided by a given known model. It has also been used in [START_REF] Bekiaris-Liberis | Compensation of time-varying input and state delays for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF] for a state-dependent delay or in [START_REF] Bekiaris-Liberis | Nonlinear control under delays that depend on delayed states[END_REF] for a delay depending on delayed state, where variations are anticipated by a careful prediction of the system state. However, in many cases, it is not possible to model the delay and, even if so, to predict the future delay values. For this reason, in this paper, in lieu of seeking exact delay compensation, we consider a prediction horizon equal to the current delay value, which is assumed to be known. This relaxed assumption is realistic. The delay itself can vary to a large extent, can be discontinuous and is not necessarily FIFO. By contrast with previous works accounting explicitly for the delay (that is, without recasting it as a disturbance) and assuming that Ḋ(t) ≤ 1 for t ≥ 0 (see [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF] [14] [START_REF] Yue | Delayed feedback control of uncertain systems with time-varying input delay[END_REF]), we allow the delay to be such that Ḋ(t) > 1 on some interval of time. A delay of this type, considered for the first time in preliminary study [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] in a prediction design context, is also considered in [START_REF] Mazenc | Lyapunov-Krasovskii Functionals and Application to Input Delay Compensation for Linear Time-Invariant Systems[END_REF] and [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach[END_REF], but, in these papers the delay is supposed to be equal to a function of class C 1 plus a small discontinuous part, treated as a disturbance. We do not impose such an assumption; in other words, we consider delays with more general types of discontinuities, covering the case where they have large discontinuous jumps.

We follow our preliminary study [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] which, as a first step, considered the delay function to be continuously differentiable, which is a demanding assumption from a practical point of view, and apply the novel time-varying version of Halany inequality proposed in [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach[END_REF] to address delay jumps. In this paper, as a result, the delay is only assumed to be piecewise continuously differentiable, encompassing potential sudden delay jumps and discontinuities, which are quite common, e.g., in the context of networks and communication protocols. Recasting the problem as an Ordinary Differential Equation (ODE) cascaded with a transport Partial Differential Equation (PDE), we use a backstepping transformation recently introduced in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF] to analyze the closed-loop stability. We prove asymptotic convergence of the system state providing that the delay time-derivative is sufficiently small in average, in the sense of an average L 2 -norm, and that the delay non-differentiability times are sufficiently sparse in time, in the sense of the average dwell time [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF].

The paper is organized as follows. In Section 2, we introduce the problem at stake, before designing our control strategy and stating our main result. The latter is proven in Section 3. Section 4 presents an illustrative simulation example.

Notations. In the following, a function f is said to be piecewise continuous on an interval [a, b] ⊂ R if the interval can be partitioned by a finite number of points a = t 0 < t 1 < . . . < t n = b so that f is continuous on each subinterval (t i-1 ,t i ) and f admits finite right-hand and left-hand limits at t i , i ∈ {0, . . . , n}. A function f is said to be piecewise continuous on R if the restriction of f to any interval is piecewise continuous. A function f is said to be piecewise continuously differentiable on R if both f and f are piecewise continuous on R. Standardly, we denote C pw (I, R) (resp. C pw (R, R)) the set of real-valued piecewise continuous function on an interval I ⊂ R (resp. on R) and f (t + ) (resp. f (t -)) the righthand (resp. left-hand) limit of f at point t, if it exists.

| • | is the usual Euclidean norm and, for a signal u(x, •) for x ∈ [0, 1], u(•) denotes its spatial L 2 -norm, i.e., u(t) = 1 0 u(x,t) 2 dx (1)
In the sequel, integrals should be understood in the Riemann integrability sense, that is, when the signal x → u(x, •) is not defined on a set S ⊂ [0, 1] of measure zero, we write

u(t) = 1 0 u(x,t) 2 dx = [0,1]\S u(x,t) 2 dx (2)
and similarly for time signals. Finally, for a matrix M the eigenvalues of which are all real numbers, λ (M) and λ (M) refer to the minimal and maximal eigenvalues of M.

Problem statement and control design

We consider the following (potentially) unstable linear dynamics

Ẋ(t) = AX(t) + BU(t -D(t)) (3)
in which X ∈ R n , U is scalar and the delay D satisfies the following assumption.

Assumption 1

The delay D is a piecewise continuously differentiable function with set of time instants of nondifferentiability

T = {t i , i ∈ N} (4)
and which satisfies (i) D(t) ∈ [D, D] for t ≥ 0, with 0 < D ≤ D (ii) there exists ∆ > 0 such that t it j ≥ ∆, (t j ,t i ) ∈ T 2 , i > j (iii) there exist T > 0 and δ > 0 such that, for all i ∈ N,

1 T t+T t Ḋ(s) 2 ds ≤ δ , t ∈ (t i ,t i+1 -T ), t i ∈ T (5)
Note that no assumption is made a priori on the timederivative of D. In particular, it is possible that Ḋ(t) > 1 for certain intervals of time. Also, it is worth observing that D is not necessarily continuous at time t i ∈ T .

In the sequel, we consider that the current value of the delay is known.

Our control objective is to design a prediction-based controller stabilizing the plant (3), using the knowledge of the current value of the delay D(t) at time t ≥ 0. With this aim in view, consider the following control law

U(t) =K ï e AD(t) X(t) + t t-D(t)
e A(t-s) BU(s)ds ò [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] in which the feedback gain K is such that A +BK is Hurwitz.

This controller approximately forecasts value of the state over a time window of varying length D(t). Note that this controller does not exactly match the predicted system state on a time-horizon D(t). Indeed, using the variation of constant formula

X(t + D(t)) =e AD(t) X(t) (7)
+ t t-D(t)
e A(t-s) BU(s + D(t) -D(s))ds

However, the integral in this last expression may not be implementable as it is not necessarily causal 1 . Further, even if one can implement this prediction, the involved integral can be approximated by the one used in [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] if D(t) -D(s) ≈ 0 for "most" instants t, i.e., under the assumption that the variations of the delay are sufficiently small in average. As this assumption is the one which is required later in Theorem 1 to robustly compensate the delay, we rather use the prediction form [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] which is easier to implement but also, and, more importantly, always causal.

Of course, exact compensation of the delay is not achieved with this controller, as it is based on the (incorrect) assumption that D(t ) = D(t) for t > t. To do so, one would need to consider a time window of which length would exactly match the value of the future delay, as performed in [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF] and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]. In details, defining ϕ(t) = t -D(t) and assuming that its inverse exists, exact delay-compensation is obtained with the feedback law U(t) = KX(ϕ -1 (t)). Yet, implementing this relation requires to predict the future variations of the delay via ϕ -1 (t). This may not be achieved in practice, when no delay model is available. More importantly, note that the inverse function ϕ -1 (t) may not exist for all time, if Ḋ(s) > 1 for some instants as ϕ may then be nonmonotonically increasing. This motivates our choice of the prediction-based controller [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF].

Theorem 1 Consider the closed-loop system consisting of the dynamics (3) and the control law [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] in which the delay D : R → [D, D] satisfies Assumption 1. Define the functional

Γ(t) = |X(t)| 2 + t t-D U(s) 2 ds ( 8
)
and a chatter bound N 0 and an average dwell time τ D ≥ 0 such that

N D (t, τ) ≤ N 0 + t -τ τ D (9)
in which N D (t, τ) denotes the number of discontinuities of Ḋ in the interval (τ,t). There exist δ * ∈ (0, 1) and τ * D > 0 such that, if δ < δ * and if τ D > τ * D , then there exist two constants γ, R > 0 such that

Γ(t) ≤ R max s∈[-D,0] Γ(s) e -γt , t ≥ 0 ( 10
)
Equation ( 5) along with the condition δ < δ * allows the delay time-derivative to be quite large for some time instants. However, to guarantee stability, it requires it to be sufficiently small in average, in the sense of the average L 2 -norm given in [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF]. In particular, the delay function can be non-FIFO for some time instants, as long as it is most 1 In details, if there exists s ∈ [t -D(t),t] such that s -D(s) ≥ t -D(t), i.e., if the delay D(t) is suddenly high and the information received at time t older than some previously received) while the one employed in (6) always is.

t -Ù D t t -Ù D t -Ù D t t v(0, t) u(1, t) v(1, t) = u(0, t) v(0, t) v(1, t) = u(0, t) u(1, t) v(0, t) u(1, t) v(1, t) = u(0, t) τ1 τ2 τ3 t = τ1 t = τ2 t = τ3 t -D(t) t -D(t) t -D(t) Direction of convection t1 t2 t3 D(t)
Fig. 1. Schematic views of the distributed variables introduced in ( 11)-( 12) for delay variations pictured in the left plot.

of the time (i.e. as δ * < 1). Second, delay jumps can occur provided that the average dwell time is large enough to guarantee stability.

Note that, as our prediction employs the current delay value D(t) instead of the time horizon ϕ -1 (t) to estimate the future system state, it can be highly inaccurate when the delay is fast varying. In this context, the requirement δ < δ * with δ introduced in ( 5) can also be interpreted as a condition for robust delay compensation achievement: if the delay varies sufficiently slowly most of the time, its current value D(t) used for prediction will remain, sufficiently often, close enough to its future values for the corresponding prediction to guarantee closed-loop stabilization.

We now detail the proof of this theorem.

3 Proof of Theorem 1

Reformulation of the plant as a cascade with a transport PDE

As a first step in our analysis, we define Û D > D and introduce the two distributed actuators

u(x,t) =U(t + D(t)(x -1)) (11) v(x,t) =U(t -Û D + x( Û D -D(t))) (12)
to reformulate the plant (3) into a PDEs-ODE cascade. In details, the variable u represents the history of the input on the (moving) horizon [t -D(t),t] while v completes it by the history of the input over the (moving) horizon [t -Û D,t -D(t)]. These variables are pictured in Figure 1.

Note that, from (6) and as the delay is piecewise continuously differentiable, the control law is also piecewise continuously differentiable and so are the distributed inputs u and v with respect to x and t. To clarify this point, we first formulate the following intermediate results.

Lemma 1

The control law defined in (6) is continuously differentiable on the union of intervals R\T .

Proof Consider the Dini derivative of U

D + U(t) = lim sup h→0 + U(t + h) -U(t) h =K ñ A ñ e AD(t) X(t) + t t-D(t) e A(t-s) BU(s)ds ô +BU(t) + Ḋ(t)e AD(t) [AX(t) + BU(t -D(t))] ó (13)
One can observe that the right-hand term of this equation is well-defined and continuous as long as t / ∈ T .

Lemma 2 For t i ∈ T , consider the sets

D i = {t ∈ R | t ≥ t i and t -D(t) ≤ t i } (14) Di = ¶ t ∈ R | t -D(t) ≥ t i and t -Û D ≤ t i © ( 15
)
and the variables

x i (t) =1 + t i -t D(t) , for t ∈ D i (16) xi (t) = t i + Û D -t Û D -D(t) , for t ∈ Di (17) Define X (t) = {x i (t) |D i t , i ∈ N} (18) X (t) = xi (t) | Di t , i ∈ N (19)
Then, the distributed variable u (resp. v) is continuously differentiable on the set

D u = {(x,t) | t / ∈ T , x / ∈ X (t)} (20) (resp. D v = (x,t) | t / ∈ T , x / ∈ X (t) ) (21)
Further, the function x i (resp. xi ) is continuously differentiable for t ∈ D i \T (resp. for t ∈ Di \T ) and satisfies

1 + Ḋ(t)(x i (t) -1) + D(t) ẋi (t) = 0 , t ∈ D i \T (22) (resp. 1 + ẋi (t)( Û D -D(t)) -x i (t) Ḋ(t) = 0 , t ∈ Di \T ) (23)
Before, providing a proof of this lemma, it is useful to make a few comments on the definitions introduced above and illustrated on Fig. 2. First, Û D > D has been introduced only to guarantee the well-posedness of (17) (otherwise, for Û D = D, this variable could be undefined for some time instants such that D(t) = D). Second, the subset of time D i (resp.

Di ) is a union of intervals which gathers times for which a spatial non-differentiability point in u (resp. v) related to the delay non-differentiability time t i exists. Conversely, for a given t ∈ R, the space set X (t) (resp. X (t)) is a finite union of singletons which gathers, at a given time t, all existing points of spatial non-differentiability of u (resp. v).

t 1 t 2 t 3 D(t) (a) Delay variations. 1 0 t 1 t 2 t 3
x 2 (t)

x 1 (t) t X (t) = {x 1 (t), x 2 (t)} D 1 = [t 1 , t 1 ] ∪ [t 2 , t 3 ] x t t 1 x 2 x 1 x 2 x 1
x 3 (b) Schematic view of the set of differentiability of u. Dasheddotted curves represent the sets where u is not differentiable. In particular, this union is indeed finite because the delay non-differentiability times are separated by at least ∆ due to Assumption 1.

0 1 t 1 t 2 t 3 x 1 (t 1 ) D 1 = [t 1 , t 3 ] t x x 2 x 1 x 1 (c) Schematic view of the set of differentiability of v. Dashed-dotted curves represent the sets where v is not differentiable. One can note that X (t) = / 0 while X (t) = {x 1 (t), x 2 (t)}.
Proof By definition, the time-derivative of the feedback u defined in [START_REF] Ruth | An introduction to infinitedimensional linear systems theory[END_REF] is given as

u t (x,t) =(1 + Ḋ(t)(x -1)) U(t + D(t)(x -1)) (24)
It is straightforward to see the u t is not well-defined for t ∈ T . Now, consider t / ∈ T , then u t is well-defined provided that x ∈ [0, 1] is such that t + D(t)(x -1) / ∈∈ T . This last condition can be formulated as x / ∈ X (t). Similar considerations can be made regarding the space-derivative of u. The result follows.

Finally, x i (t) ∈ X (t) implies that t + D(t)(x i (t) -1) = t i . Taking a time-derivative of this expression, one obtains [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF]. Similar conclusions can be obtained for the distributed variable v defined in [START_REF] Depcik | One-dimensional automotive catalyst modeling[END_REF] following the same steps, which concludes the proof. This allows to rewrite the plant (3) as the following PDEs-ODE cascade

Ẋ(t) = AX(t) + Bu(0,t) (25) D(t)u t = (1 + Ḋ(t)(x -1))u x , (x,t) ∈ D u (26) u(1,t) = U(t) (27) u(x,t + ) = U(t + + D(t + )(x -1)) , t ∈ T , x / ∈ X (t) (28)
( Û D -D(t))v t = (1 -x Ḋ(t))v x , (x,t) ∈ D v (29) v(1,t) = u(0,t) (30)
v(x,t + ) = U(t + -D + x( Û D -D(t + ))) , t ∈ T , x / ∈ X (t) (31)
This system is well-posed, in the sense of the following lemma, the proof of which is provided in Appendix.

Lemma 3 For any initial data (u, v) ∈ L 2 (0, 1) × L 2 (0, 1), the system (26)-( 31) has a unique weak solution in L 2 (0, 1) × L 2 (0, 1).

In details, the input delay is now represented as the cascade of an ODE [START_REF] Mazenc | Lyapunov-Krasovskii Functionals and Application to Input Delay Compensation for Linear Time-Invariant Systems[END_REF] fed by the output of a transport PDE ( 26)-( 28), with time-and space-varying propagation velocity, which can potentially be locally equal to zero or negative. This output also feeds a second transport PDE ( 29)- [START_REF] Sbarciog | Nonlinear Predictive Control of processes with variable time delay. A temperature control case study[END_REF] with time-and space-varying propagation velocity. It is worth mentioning that one needs to take into account both distributed variables (u, v) in the analysis to account for all potential values of the delay and the entire history of the input over the time interval [t -Û D,t].

Backstepping transformation and target system

To analyse this closed-loop system, following [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF], we define the backstepping transformation

w(x,t) =u(x,t) (32)
-K ï e AD(t)x X(t) + D(t)

x 0 e AD(t)(x-y) Bu(y,t)dy ò As previously, before starting our analysis, we investigate the well-posedness of this distributed variable.

Lemma 4

The backstepping transformation defined in [START_REF] Seuret | Stability of systems with fast-varying delay using improved wirtinger's inequality[END_REF] is continuously differentiable for (x,t) ∈ D u , as introduced in (20).

Proof Taking a space-derivative of (32), one gets

w x (x,t) =u x (x,t) -K ï AD(t)e AD(t)x X(t) + D(t)Bu(x,t) + AD(t) 2
x 0 e AD(t)(x-y) Bu(y,t)dy

ò ( 33
)
which is well-defined for (x,t) ∈ D u , following Lemma 2.

Similarly, taking a time-derivative of (32), one gets

w t (x,t) = u t (x,t) (34)
-K Ḋ(t) ï e AD(t)x AxX(t) +

x 0 e AD(t)(x-y) Bu(y,t)dy

ò -K ï e AD(t)x (AX + Bu(0,t)) + D(t) d dt ï (x i ,x i+1 )∈X [0,x] (t) 2 , x i <x i+1 x - i+1
x + i e AD(t)(x-y) Bu(y,t)dy òò in which we have introduced, for (a, b)

∈ [0, 1] 2 , X [a,b] (t) = {a} ∪ {x i ∈ X (t) | x i ≤ b} ∪ {b} (35)
and where

d dt ï (x i ,x i+1 )∈X [0,x] (t) 2 , x i <x i+1 x + i+1 x - i e AD(t)(x-y) Bu(y,t)dy ò = (x i ,x i+1 )∈X [0,x] (t) 2 , x i <x i+1 ï x - i+1 x + i d dt Å e AD(t)(x-y) Bu(y,t) ã dy + ẋi+1 e AD(t)(x-x i+1 ) Bu(x - i+1 ,t) -ẋi e AD(t)(x-x i ) Bu(x + i ,t) ò ( 36
)
with the convention ẋi = 0 if x i = 0 or x i = x. From these expressions, using Lemma 2, one can deduce that w t is wellposed for (x,t) ∈ D u .

Lemma 5

The infinite-dimensional backstepping transformation (32) together with the control law (6) transform the plant (3) into the target system

Ẋ(t) = (A + BK)X(t) + Bw(0,t) (37) D(t)w t = (1 + Ḋ(t)(x -1))w x -D(t) Ḋ(t)g(x,t) , (x,t) ∈ D u (38) w(1,t) = 0 (39) w(x,t + ) = u(x,t + ) -K ï e AD(t + )x X(t) (40)
+ D(t + )

x 0 e AD(t + )(x-y) Bu(y,t)dy

ò , t ∈ T ( Û D -D(t))v t = (1 -x Ḋ(t))v x , (x,t) ∈ D u (41) v(1,t) = w(0,t) + KX(t) (42) v(x,t + ) = U(t -D + x(D(t + ) -Û D)) , t ∈ T (43) with g(x,t) =Ke AD(t)x (AX + Bu(0,t)) (44)
Proof As previously, taking time-and space-derivatives of (32), one gets, using integration by parts for the second equation,

w t = u t -K Ḋ(t) ï e AD(t)x AxX(t) (45)

+

x 0 e AD(t)(x-y) (I + AD(t)(xy))Bu(y,t)dy ò -K ï e AD(t)x (AX + Bu(0,t)) + D(t)

x 0 e AD(t)(x-y) Bu t (y,t)dy ò -KD(t)

x i ∈X (0,x) (t) ẋi e AD(t)(x-x i (t)) B(u(x - i ,t)u(x + i ,t))

w x = u x -K ï e AD(t)x AD(t)X + D(t)Bu(x,t) + D(t)
x 0 e AD(t)(x-y) Bu x (y,t)dy -D(t)

(x i ,x i+1 )∈X [0,x] (t) 2 , x i <x i+1 [e AD(t)(x-x i+1 (t)) Bu(x - i+1 ,t) -e AD(t)(x-x i (t)) Bu(x + i ,t)] ò (46)
in which we have introduced

X (a,b) (t) = {x i ∈ X (t) | x i ∈ (a, b)} , (a, b) ∈ [0, 1] 2 (47)
Matching those two expressions and using ( 22) and ( 26), one obtains (38) with g(x,t) = K ï e AD(t)x AxX(t) + (1x)e AD(t)x (AX + Bu(0,t))

+
x 0 e AD(t)(x-y) (I + AD(t)(xy))Bu(y,t)dy ò + K

x 0 e AD(t)(x-y) B(yx)u x (y,t)dy (48)

+ K x i ∈X (0,x)(t) (x -x i )e AD(t)(x-x i ) B[u(x - i ,t) -u(x + i ,t)]
which, using the following integration by parts

x 0 e AD(t)(x-y) B(xy)u x dy = -xe ADx Bu(0,t)

x i ∈X (0,x)(t) (x -x i )e AD(t)(x-x i ) B[u(x - i ,t) -u(x + i ,t)] + x 0 e AD(x-y) (I + AD(t)(x -y))Bu(y,t)dy (49)
can be expressed as (44). The boundary condition (39) directly follows from the choice of the control law ( 6) and the backstepping transformation definition [START_REF] Seuret | Stability of systems with fast-varying delay using improved wirtinger's inequality[END_REF]. Finally, the boundary condition (42) follows from [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and the backstepping transformation (32) for x = 0.

As the target system presents the suitable boundary condition w(1,t) = 0, this is the one which is used in the Lyapunov analysis.

Stability analysis -Proof of Theorem 1

Consider the following Lyapunov functional candidate 2 , we deduce that there are µ 1 , µ 2 > 0 such that

V (t) =X(t) T PX(t) + b 1 D(t) 1 0 (1 + x)w(x,t) 2 dx + b 2 ( Û D -D(t)) 1 0 (1 + x)v(x,
t-D(t) t-Û D U(s) 2 ds = ( Û D -D(t)) v(t)
µ 1 Γ 0 (t) ≤ V (t) ≤ µ 2 Γ 0 (t) (55)
Now, consider t i ∈ T . Taking a time-derivative of (50) for t ∈ (t i ,t i+1 ) and using integrations by parts jointly with ( 22)-( 23) and Lemma 4, one gets

V (t) = -X T QX + 2X T PBw(0,t) + b 1 Å -(1 -Ḋ(t))w(0,t) 2 -w(t) 2 -2 Ḋ(t) 1 0 xw(x,t) 2 dx + x i ∈X (0,1) (t) (1 + x i (t)) × [D(t) ẋi (t) + 1 + Ḋ(t)(x i (t) -1)] =0 [w(x - i ) 2 -w(x + i ) 2 ] ã -2D(t) Ḋ(t) 1 0 (1 + x)w(x,t)g(x,t)dx ã + Ḋ(t) 1 0 (1 + x)[b 1 w(x,t) 2 -b 2 v(x,t) 2 ]dx + b 2 Å 2v(1,t) 2 -v(0,t) 2 -v(t) 2 -2 Ḋ(t)v(1,t) 2 + Ḋ(t) 1 0 (1 + 2x)v(x,t) 2 dx + x i ∈X (0,1) (t) (1 + x i (t)) (56) × [ ẋi (t)( Û D -D(t)) + 1 -x i Ḋ(t)] =0 [v(x - i ,t) 2 -v(x + i ,t) 2 ]
ã in which, from [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and [START_REF] Seuret | Stability of systems with fast-varying delay using improved wirtinger's inequality[END_REF],

2v(1,t) 2 ≤ 4(w(0,t) 2 + |K| 2 |X(t)| 2 ) (57)
Using the fact that, from [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] with Young and Cauchy-Schwarz inequalities,

u(0,t) 2 =U(t -D(t)) 2 ≤ M(|X(t -D(t))| 2 + u(t -D(t)) 2 ) , t ≥ D (58)
for a given positive constant M, together with (53) and Young and Cauchy-Schwarz inequalities, one obtains the existence of a constant M > 0 such that

2D(t) 1 0 (1 + x)w(x,t)g(x,t)dx ≤ M Å max s∈[-D,0] |X(t + s)| 2 + max s∈[-D,0] w(t + s) 2 ã (59) 2v(1,t) 2 ≤ M Å max s∈[-D,0] |X(t + s)| 2 + max s∈[-D,0] w(t + s) 2 ã (60) w(0,t) 2 ≤ M Å max s∈[-D,0] |X(t + s)| 2 + max s∈[-D,0] w(t + s) 2 ã ( 61
)
for t ≥ D. Therefore, with (57), ( 59)-( 61) and applying Young inequality, one gets f.a.a. . We now consider (63) for t ∈ (t i ,t i+1 ) and introduce W such that

t ≥ D V (t) ≤ - Å λ (Q) 2 -4b 2 |K| 2 ã |X(t)| 2 -b 1 w(t) 2 (62) -b 2 v(t) 2 - Å b 1 -4b 2 - 2|PB| 2 λ (Q) ã w(0,t) 2 + b 0 | Ḋ(t)| Å max s∈[-D,0] |X(t + s)| 2 + max s∈[-D,0] w(t + s) 2 ã in which b 0 = b 1 (4 + 2M) + b 2 (5 + M). Consequently, choosing b 2 = λ (Q) 16|K| 2 , b 1 > 4b 2 + 2|PB| 2 λ (Q) , it follows V (t) ≤ -ηV (t) + η 0 | Ḋ(t)| max s∈[-D,0] V (t + s) , f.a.a. t ≥ D (
Ẇ (t) = -ηW (t) + b(t) max s∈[-D,0] W (t + s) , f.a.a. t ∈ (t i ,t i+1 ] (64) W (t) = ® V (t) if t ∈ [t i -D,t i ) max V (t + i ),V (t - i ) if t = t i (65) in which b is a function such that b(t) = η 0 | Ḋ(t)| for t ∈ (t i ,t i+1
). Such a solution is well-defined, as Assumption 1-(ii) guarantees that the initial condition is piecewise continuous and according to Lemma 7. Applying Lemma 6, one concludes that there exists δ * ∈ (0,

Ä η η 0
ä 2 ) such that, for δ < δ * , there exist two constants r, γ > 0 (independent of t i+1t i ) such that, for t ≥ D,

W (t) ≤ r max s∈[-D,0] W (t i + s) e -γ(t-t i ) , t ∈ [t i ,t i+1 ) (66)
Considering z = W -V , with a contradiction argument (similarly to the one employed in the proof of Lemma 6), one can conclude that z(t) ≥ 0 for t ∈ [t i ,t i+1 ) and thus that

V (t) ≤ r max s∈[-D,0] W (t i + s) e -γ(t-t i ) , t ∈ [t i ,t i+1 ) (67)
Hence, for δ < δ * , using (55), the definition (65) and the fact that Γ 0 is continuous, one deduces the existence of R, γ > 0 such that, for t ≥ D,

Γ 0 (t) ≤ R max s∈[-D,0] Γ 0 (t i + s) e -γ(t-max{t i ,D}) , t ∈ [t i ,t i+1 ) (68)
Consequently, as Γ is a continuous functional, one gets

max s∈[-D,0] Γ 0 (t i+1 + s) ≤ Re -γ(∆t i -D) max s∈[-D,0] Γ 0 (t i + s)
in which, potentially, ∆t i ≤ D. Hence, with N D (t, τ) the number of discontinuities in the interval (τ,t), it follows that

Γ 0 (t) ≤ RN D (t,D) e -γ(t-D-N D (t,D)D) max s∈[-D,0] Γ 0 (D + s)
or equivalently, using (9)

Γ 0 (t) ≤ e N 0 (ln( R)+ γD) e - Ä γ- ln( R)+ γD τ D ä (t-D) max s∈[-D,0] Γ 0 (D + s) Consequently, if τ D > τ * D ∆ = ln( R)
γ + D and as Û D can be chosen arbitrarily close to D, there exist two constants R, γ > 0 such that the exponential stability result in terms of Γ of Theorem 1 holds.

Illustrative toy example: communication delay

To illustrate the relevance of the proposed prediction-based control law, we consider the unstable second-order dynamics

Ẋ(t) = 0 1 -1 1 X(t) + 0 1 U(t -D(t)) (69)
Controller Plant

X(t) U (t) U (t -D(t))
Network Fig. 3. Schematic view of the considered system with communication delay. The controller and the plant exchange information through a network in which a supervisor orchestrates the data routing by choosing between a family of candidates paths. in which D(t) is a piecewise continuously differentiable delay function as considered throughout this paper. It can be subject to large variations. A schematic view of the system at stake is given in Fig. 3. We consider that the controller sends orders to the plant through a network in which a supervisor orchestrates the data routing by choosing between a family of candidate paths. This routing tries to keep the data queuing lines below some acceptable value. If it increases too much, a new route is chosen, causing delay jumps. We assume that the communication channels between the plant and the controller are not symmetric, resulting into a sole input delay2 .

We first consider the delay function pictured in Fig. 4(b). In this case, data were routed in such a way that reordering occurs periodically, resulting in periodic jumps and non-FIFO delay variations ( Ḋ exhibits values larger than one periodically). The control law ( 6) is applied with the feedback gain K = -[9 10] and implemented with a trapezoidal discretization of the integral. Closed-loop simulation results are reported in Fig. 4. One can observe that the plant asymptotically converges, as Theorem 1 guarantees it can be the case. Indeed, the intervals during which the delay derivative is larger than one are reduced enough compared to the dwell time (constant in this example) to guarantee that the condition δ < δ * required by Theorem 1 holds. On the other hand, a similar controller but using a prediction based on a constant average delay value (D ≈ 0.22) fails to stabilize the plant. In all likelihood, this result could be explained by the selection of a relatively high feedback gain value K = -[910]. This illustrates the interest of using the current delay value as prediction horizon rather than an average value of it, grounding in a delay-robustness property of the prediction-based controller (as studied in a FIFO context in [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF]).

To evaluate our controller performance in a more challenging context, we consider now that the delay is a discrete-time random process, (D(n)) n∈N = (D(nT s )) n∈N with T s = 0.3s and D(n) a uniform random variable on [0.8, 1.2]. The control law ( 6) is now applied with the feedback gain K = - Fig. 5 and exhibit the same convergence property as previously. Similarly, a prediction-based controller using the expected delay value E(D(n)) = 1 as prediction horizon fails to stabilize the plant. Of course, this framework does not fit into the mathematical formalism considered throughout this paper, which could be considered as a first step to address this case. This is a direction of future works, the interest of which is strengthen by the simulation results of Fig. 5.

Conclusion

This paper presents a prediction-based control for a timevarying input delay, the variations of which are not assumed to satisfy a FIFO property and are not assumed to be continuous or with sufficiently small jumps. We propose to use the current delay as a prediction horizon and proved that the closed-loop system exponentially converges, provided that the delay time-derivative is sufficiently small in the sense of an average L 2 -norm and that the delay discontinuities are sufficiently sparse in the sense of the average dwell time. This result is very promising as it enables to alleviate the very limiting assumption Ḋ(t) < 1, t ≥ 0 and to consider delays with strong discontinuities.

Much more remains to be done. Extension of this work to more general dynamics, such as time-varying or nonlinear ones, is a first path to explore. This will likely require the design of new Halanay-type Delay Differential Inequalities. Further, the interest of Lyapunov techniques standardly used in the field of sampled-data systems (looped functionals) should be investigated in this context.

Appendix A: Well-posedness of (26)-( 31) -Proof of Lemma 3

We start this proof by noticing that ( 11)-( 12) is a solution to the system ( 26)- [START_REF] Sbarciog | Nonlinear Predictive Control of processes with variable time delay. A temperature control case study[END_REF]. We now wish to prove that this solution is unique and depends continuously on its initial condition. With this aim in view, consider (u, v) a solution to ( 26)-( 31) and introduce the following distributed variable

ū(x,t) =        u Å D D(t) (x -1) + 1,t ã , x ∈ ï D -D(t) D , 1 ò v Å D D -D(t) x,t ã , x ∈ ï 0, D -D(t) D ò (70)
which can be inverted as

       u(x,t) = ū Å 1 + D(t) D (x -1),t ã v(x,t) = ū Å D -D(t) D x,t ã (71)
Note that the variable (70) is well-defined as ū Ä D-D(t) D ä = u(0,t) = v(1,t) according to [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]. Furthermore, taking timeand space-derivatives of ū and using ( 26)- [START_REF] Sbarciog | Nonlinear Predictive Control of processes with variable time delay. A temperature control case study[END_REF], one proves that it satisfies

D ūt = ūx (72) ū(1,t) =U(t) (73)
which has a unique weak solution in L 2 (0, 1) (see [START_REF] Ruth | An introduction to infinitedimensional linear systems theory[END_REF]). Thus, from its inverse (71), this implies that (u, v) is the unique weak solution of ( 26)-( 31) and depends continuously on its initial condition.

Appendix B: Time-varying Halanay inequality

We use the following result, the proof of which is inspired from [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF].

Lemma 6 Consider a nonnegative piecewise continuously differentiable function x with only one discontinuity at time t = 0 and such that

ẋ(t) ≤ -ax(t) + b(t) max s∈[-D,0] x(t + s) , f.a.a. t > 0 x 0 = ψ ∈ C pw ([-D, 0], R) (74)
in which D ≥ 0, a ≥ 0 and b : R + → R + is a piecewise continuous function with only one discontinuity at time t = 0 and which satisfies for some T > 0, δ > 0 and T

0 > D + T 1 T t+T t b(s) 2 ds ≤ δ , t ∈ [0, T 0 -T ] (75)
There exists δ * ∈ (0, a 2 ) (independent of T 0 ) such that, if δ < δ * , then there exist two constants γ, r ≥ 0 (independent of T 0 ) such that

∀t ≥ t 0 x(t) ≤ r max ψe -γt , t ∈ [0, T 0 ] (76)
Proof We start by proving the existence of r 0 > 0 such that max

s∈[-T -D,0] x(T + s) ≤ r 0 max ψ (77)
Let k > max ψ and y such that y(t) = k for t ∈ [-D, 0] and Finally, one can note that δ * is a decreasing function with respect to T which tends to a 2 as T tends to zero. This gives the expected result.

y(t) = k exp Ä t 0 b(s)ds ä for t ∈ [0, T ].
In the above, we also needed the following result.

f e (s) (84)
Finally, applying the maximum theorem [START_REF] Berge | Topological Spaces: including a treatment of multi-valued functions, vector spaces, and convexity[END_REF], one concludes that t → max s∈[t-D,t i+1 -D] f e (s) and t → max s∈[t i ,t] f e (s) are continuous functions as f e is continuous over the intervals under consideration. Therefore, it follows that h is continuous over (t i ,t i+1 ) by composition of continuous functions. One also concludes that h admits a finite left-hand side limit at t i+1 and right-hand site limit at t i , which concludes the proof.

Fig. 2 .

 2 Fig. 2. Schematic views of the sets introduced in Lemma 2 for a given delay function pictured in the left-hand plot 2(a).

  [3 4]. Corresponding simulation results are pictured in (a) State and control evolution.

  Delay and functionals evolution.

Fig. 4 .

 4 Fig. 4. Simulation results with a feedback gain K = -[9 10] and initial conditions X(0) = [1 0] T , U 0 = 0. The controller proposed in this paper is compared with a prediction-based controller using a constant delay D ≈ 0.22.

  (a) State and control evolution.

  Delay and functionals evolution.

Fig. 5 .

 5 Fig. 5. Simulation results with a feedback gain K = -[3 4] and initial conditions X(0) = [1 0] T , U 0 = 0. The delay is a discretetime random process, (D(n)) n∈N = (D(nT s )) n∈N with T s = 0.3s and D(n) a uniform random variable on [0.8, 1.2]. The controller proposed in this paper is compared with a prediction-based controller using a constant delay D = 1.

  (t-s) b(s)ds ã max s∈[-T -D,0]x(t + s) in which the right-hand side is well-defined according to Lemma 7. Using Cauchy-Schwarz inequality and from (75), one obtainsx(t) ≤ e -aT + 1e -2aT 2a t t-T b(s) 2 ds max s∈[-T -D,0]x(t + s)≤c max s∈[-T -D,0] x(t + s) in which c =e -aT + 1e -2aT 2a √ T δ(80) Thus, with (77), if c < 1, the result holds. With straightforward calculations, one obtains that c < 1 if and only if δ ≤ 2a T 1e -aT 1 + e aT ∆ = δ *

Lemma 7

 7 Consider f ∈ C pw (R, R) and D ≥ 0, then h : t ∈ R → max s∈[t-D,t] f (s) ∈ C pw (R, R)(81)Proof The main idea of this proof is to construct a new grid for h which gathers the grid corresponding to f and the same one but delayed by D-units of time.Consider I = [a, b]. Define Ĩ = [a-D, b] and a finite sequence (t n ) 0≤n≤ Ñ partitioning Ĩ and corresponding to the piecewise continuous function f . Define

  Consider z = yx and, by contradiction, assume there exists t 1 such that z(t 1 ) < 0. Then, by continuity, there existst 2 such that ) ≥ b(t 2 )y(t 2 )b(t 2 ) max x(t 2 ) > 0in which the last inequality follows from the definition of t 2 and is in contradiction with (79). Therefore, z(t) ≥ 0, t ∈ [0, T ]. As the previous considerations hold for all k > max ψ, using Cauchy-Schwarz inequality, one concludes that As t ∈ [0, T ], using (75), it follows that (77) holds with r 0 = e Now, consider t ≥ T . Integrating (74) between t -T and t, one gets x(t) ≤e -aT x(t -T ) +

	√ δ T	√ T .
					Thus, y is an increas-
		ing function which satisfies
		ẏ(t) =b(t) max	y(t + s) , t > 0	(78)
					s∈[-D,0]
			z(t) > 0 for t ∈ [0,t 2 ) z(t 2 ) = 0
			ż(t 2 ) ≤ 0	(79)
		However, from (74) and (78), it follows that
		ż(t 2 x(t) ≤ exp	Å t 0	ã b(s)ds	max ψ
		≤ exp		0	t	b(s) 2 ds √ t max ψ , t ∈ [0, T ]

  By construction, this sequence is finite anda = t 0 < t 1 < . . . < t N = b (for a given N ∈ N). Consider t ∈ (t i ,t i+1 ). By definition, as [t -D,t] = [t -D,t i+1 -D) ∪ [t i+1 -D,t i ] ∪ (t i ,t],

						h(t) = max	sup	f (s), max	f (s), sup	f (s)
							s∈[t-D,t i+1 -D)	s∈[t i+1 -D,t i ]	s∈(t i ,t]	(83)
						or, denoting f e the continuity extension of f (which exists
						as f is piecewise continuous),
						®
						h(t) = max	max s∈[t-D,t i+1 -D]	f e (s), max s∈[t i+1 -D,t i ]	f (s), max s∈[t i ,t]
	t 0 =a , t i+1 = min min 0≤n≤	Ñ	tn , D + min 0≤n≤	Ñ	tn	(82)
	t i <t n		t i -D<t n

An even more general and representative modeling could also include a time-varying output delay (to account for the fact that the plant sends outputs to the controller through a similar network). To handle this additive complexity, our control strategy should be extended to handle linear time-varying dynamics.


Saeed Ahmed
email: saeed.ahmed@bilkent.edu.tr

Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Hitay Özbay

Dynamic Output Feedback Stabilization of Switched Linear Systems with Delay via a Trajectory Based Approach ⋆

Keywords: Switched systems, delay, output feedback, observer, stabilization

A new technique is proposed to construct observers and to achieve output feedback stabilization of a class of continuous-time switched linear systems with a time-varying delay in the output. The delay is a piecewise continuous bounded function of time and no constraint is imposed on the delay derivative. For stability analysis, an extension of a recent trajectory based approach is used; this is fundamentally different from classical Lyapunov function based methods. A stability condition is given in terms of the upper bound on the time-varying delay to ensure global uniform exponential stability of the switched feedback system. The main result applies in cases where some of the subsystems of the switched system are not stabilizable and not detectable.

Introduction

Switched systems have extensive applications in networks, automotive control, power systems, aircraft and air traffic control, process control, mechanical systems, and many other domains; see [START_REF] Lin | Stability and stabilizability of switched linear systems: a survey of recent results[END_REF], and the references therein. Due to this strong motivation, many questions related to switched systems such as stability [START_REF] Liberzon | Switching in systems and control[END_REF]; [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]; [START_REF] Sun | Stability theory of switched dynamical systems[END_REF]), controllability [START_REF] Liu | Structural controllability of switched linear systems[END_REF]; [START_REF] Sun | Controllability and reachability criteria for switched linear systems[END_REF]), observability and reachability [START_REF] Hespanha | Nonlinear norm-observability notions and stability of switched systems[END_REF]; [START_REF] Ji | A constructive approach to reachability realization of discrete-time switched linear systems[END_REF]; [START_REF] Sun | Controllability and reachability criteria for switched linear systems[END_REF]; [START_REF] Tanwani | Observability for switched linear systems: characterization and observer design[END_REF]), and synthesis [START_REF] Pettersson | Synthesis of switched linear systems[END_REF]; [START_REF] Sun | Analysis and synthesis of switched linear control systems[END_REF]), have been extensively studied in various contributions. Stability and stabilization are challenging problems pertaining to switched systems due to their hybrid nature and they are the main topic of the present paper.

There are mainly two approaches used in the literature for establishing the stability of switched systems: (i) It is shown in [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF] that existence of a common strict Lyapunov function is a necessary and sufficient condition for the switched system to be stable under arbitrary switching. On the other hand, when such a Lyapunov function exists, finding it may be a difficult task because it is an NP-hard problem; see [START_REF] Blondel | NP-hardness of some linear control design problems[END_REF]. (ii) [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF] also showed that even if a switched system does not possess a common strict Lyapunov function, it may be stable under a dwell-time requirement, typically derived using multiple strict Lyapunov functions. It is worth mentioning that multiple Lyapunov functions may lead to an undesirable attenuation property which can only be mitigated by imposing some strong assumptions; see [START_REF] Zhai | Disturbance attenuation properties of time-controlled switched systems[END_REF]. Both of the above mentioned approaches are mainly developed for non-delayed systems. But measurement delays are present in many practical applications, such as chemical processes, aerodynamics and communication networks, and they are time-varying (see for instance [START_REF] Wu | LPV systems with parameter-varying time delays: analysis and control[END_REF]; [START_REF] Yan | Robust controller design for AQM and H ∞ -performance analysis[END_REF]). Therefore, the problem of stabilizing switched systems when a time-varying delay is present in the output is strongly motivated. State feedback stabilization of delayed switched linear systems is proposed in [START_REF] Vu | Stability of timedelay feedback switched linear systems[END_REF] using a combi-nation of the multiple Lyapunov functions approach and the merging switching signal technique. An online and offline state feedback controller design for delayed switched linear systems in the detection of the switching signal are discussed in [START_REF] Xie | Stabilization of switched linear systems with time-delay in detection of switching signal[END_REF]. Moreover, [START_REF] Koru | Dwell time-based stabilisation of switched delay systems using free-weighting matrices[END_REF] and [START_REF] Yan | Robust stabilization of parameter varying time delay systems by switched controllers[END_REF] present state feedback designs for delayed switched systems using a dwell-time based stability analysis approach. Note that [START_REF] Koru | Dwell time-based stabilisation of switched delay systems using free-weighting matrices[END_REF], [START_REF] Vu | Stability of timedelay feedback switched linear systems[END_REF], [START_REF] Xie | Stabilization of switched linear systems with time-delay in detection of switching signal[END_REF], and [START_REF] Yan | Robust stabilization of parameter varying time delay systems by switched controllers[END_REF] assume that all of the subsystems of the switched system are controllable. Finally, a state feedback stabilization problem for a class of delayed switched systems is studied in [START_REF] Kim | Stability of a class of linear switching systems with time delay[END_REF] and [START_REF] Sun | Stability analysis for linear switched systems with time-varying delay[END_REF] under the assumption that the subsystems satisfy a certain Hurwitz convex combination condition. A common Lyapunov function approach is used in [START_REF] Kim | Stability of a class of linear switching systems with time delay[END_REF] and [START_REF] Sun | Stability analysis for linear switched systems with time-varying delay[END_REF] to carry out stability analysis.

Contributions of this study:

We propose a new technique to design observers and stabilizing dynamic output feedbacks offering robust stability results with respect to the presence of a time-varying pointwise delay in the output of the switched linear system. To establish the stability of the closed-loop switched system, we develop an extension of the trajectory based stability result recently proposed in [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], and [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach[END_REF]. We wish to point out that the new extension of the trajectory based approach we state and prove in the present paper is of interest by itself: it can be applied to a wide range of systems, notably to families of systems with time-varying delays wider than those invoked in [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], and [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach[END_REF], and therefore it is one of the important contributions of our work. We think that our main result can be regarded as an extension of [START_REF] Kim | Stability of a class of linear switching systems with time delay[END_REF], [START_REF] Koru | Dwell time-based stabilisation of switched delay systems using free-weighting matrices[END_REF], [START_REF] Sun | Stability analysis for linear switched systems with time-varying delay[END_REF], [START_REF] Vu | Stability of timedelay feedback switched linear systems[END_REF], [START_REF] Xie | Stabilization of switched linear systems with time-delay in detection of switching signal[END_REF], [START_REF] Yan | Robust stabilization of parameter varying time delay systems by switched controllers[END_REF][START_REF] Zhai | Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach[END_REF], offering new advantages because, (i) our study does not assume that all the states are available for feedback, (ii) it is not limited to systems whose all subsystems are stabilizable and detectable, (iii) we use a new extension of trajectory based approach for stability analysis which circumvents the serious obstacle presented by the search for appropriate Lyapunov functions, (iv) the application of our results is not restricted to the class of delayed switched systems where all the convex combinations of the subsystems in the absence of control must be Hurwitz, (v) we allow the delay to be time-varying and piecewise continuous function of time, and we do not impose any constraint on the upper bound of the delay derivative. Now, we point out that the present paper is a continuation of our conference paper Mazenc et al. (2017). We propose a significant extension of it by including, (i) dynamic output feedback stabilization, (ii) a new extension of trajectory based approach of [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] to produce less conservative results, (iii) a sys-tematic way to compute an explicit value for the lower bound on the largest admissible delay for a broad family of switched systems so that when the delay is smaller than this bound, global uniform exponential stability (GUES) of the feedback switched systems is guaranteed. Moreover, we do not assume that the systems have synchronous switching sequences. Organization of the paper: An extension of the trajectory based approach is given in Section 2. Section 3 is devoted to the main result of the paper. Section 4 discusses computational issues related to the delay bound. The results are illustrated by a numerical example in Section 5. Finally, we summarize and highlight our contributions in Section 6. Notation: The notation will be simplified whenever no confusion can arise from the context. I denotes the identity matrix of any dimension. The usual Euclidean norm of vectors, and the induced norm of matrices, are denoted by | • |. Given any constant τ > 0, we let C([-τ, 0], R n ) denote the set of all continuous R n -valued functions that are defined on [-τ, 0]. We abbreviate this set as C in , and call it the set of all initial functions. Also, for any continuous function x : [-τ, ∞) → R n and all t ≥ 0, we define x t by x t (θ) = x(t + θ) for all θ ∈ [-τ, 0], i. e., x t ∈ C in is the translation operator. A vector or a matrix is nonnegative (resp. positive) if all of its entries are nonnegative (resp. positive). We write M ≻ 0 (resp. M 0) to indicate that M is a symmetric positive definite (resp. negative semi-definite) matrix. For two vectors

V = (v 1 ...v n ) ⊤ and U = (u 1 ...u n ) ⊤ , we write V ≤ U to indicate that for all i ∈ {1, ..., n}, v i ≤ u i .

Extension of the trajectory based approach

We now provide with an extension of the trajectory based approach given in [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF].

Lemma 1 Let us consider a constant T > 0 and l functions z g : [-T, +∞) → [0, +∞), g = 1, ..., l. Let Z(t) = (z 1 (t) ... z l (t)) ⊤ and, for any θ ≥ 0 and t ≥ θ, define V θ (t) = sup s∈[t-θ,t] z 1 (s) ... sup s∈[t-θ,t] z l (s) ⊤ . Let Υ ∈ R l×l be a nonnegative Schur stable matrix. If for all t ≥ 0, the inequalities Z(t) ≤ ΥV T (t) are satisfied, then lim t→+∞ z g (t) = 0 ∀ g = 1, . . . , l.
Proof. Since Υ is Schur stable, there is an integer q > 1 such that

|Υ q | √ l < 1 . (1)
From Lemma 4 of the Appendix A, we deduce that

Z(t) ≤ Υ q V qT (t) (2) for all t ≥ qT . Consequently, |Z(t)| ≤ |Υ q ||V qT (t)|. Using |V qT (t)| ≤ √ l sup s∈[t-qT,t]
|Z(s)|, we obtain

|Z(t)| ≤ |Υ q | √ l sup s∈[t-qT,t] |Z(s)| .
This inequality, in combination with the inequality (1) and [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], Lemma 1), allows us to conclude the result.

Observer and control design

We introduce a range dwell-time condition, i.e. a sequence of real numbers t k such that there are two positive constants δ and δ such that t 0 = 0 and for all k ∈ Z ≥0 ,

t k+1 -t k ∈ [δ, δ] .
(3)

Definition 1 Let π = {(i 0 , t 0 ), . . . , (i k , t k ), . . . , |i k ∈ Ξ, k ∈ Z ≥0 } be a switching sequence. The function σ : [0, ∞) → Ξ = {1, ..., n} such that σ(t) = i k when t ∈ [t k , t k+1
) is called an associated switching signal.

We consider the continuous-time switched linear system:

ẋ(t) = A σ(t) x(t) + B σ(t) u(t) y(t) = C σ(t) x(t -τ (t)) (4) with x ∈ R dx , u ∈ R du , y ∈ R dy , for all t ≥ 0, τ (t) ∈ [0, τ ]
with τ > 0 and an initial condition in C in . The delay τ (t) is supposed to be a piecewise continuous function.

For any i ∈ Ξ, A i , B i , and C i are real and constant matrices of compatible dimensions and σ is a switching signal. We introduce an assumption which pertains to the stabilizability and the detectability of the system (4), but does not imply that all the pairs (A i , B i ) are stabilizable and all the pairs (A i , C i ) are detectable.

Assumption 1 There are matrices K i and L i for all i ∈ Ξ and constants T ≥ τ , a ∈ [0, 1), b ≥ 0, c ∈ [0, 1) and d ≥ 0 such that the solutions of the system

α(t) = M σ(t) α(t) + ζ(t) (5
)

with M i = A i + B i K i and ζ being a piecewise continuous function, satisfy |α(t)| ≤ a|α(t -T )| + b sup ℓ∈[t-T,t] |ζ(ℓ)| (6)
for all t T . Similarly, the solutions of the system

β(t) = N σ(t) β(t) + η(t) (7)
with N i = A i + L i C i and η being a piecewise continuous function, satisfy the following inequality for all t ≥ T

|β(t)| ≤ c|β(t -T )| + d sup ℓ∈[t-T,t] |η(ℓ)| . ( 8
)
Theorem 1 Let the system (4) satisfy Assumption 1 and, s 1 , s 2 and s 3 be defined by

s 1 = sup i∈Ξ |B i K i | , s 2 = sup i∈Ξ |L i C i | , s 3 = sup i∈Ξ |M i | . (9) If τ (t) ≤ τ < τu (10) for all t ≥ 0, where τu = (1 -a)(1 -c) ds 1 s 2 ((1 -a) + bs 3 ) , ( 11
)
then the origin of the following feedback system is GUES:

       ẋ(t) = A σ(t) x(t) + B σ(t) K σ(t) x(t) ẋ(t) = A σ(t) x(t) + B σ(t) K σ(t) x(t) +L σ(t) [C σ(t) x(t) -y(t)] . (12)
Proof. Let us introduce x(t) = x(t) -x(t). Then,

ẋ(t) = A σ(t) x(t) + L σ(t) [C σ(t) x(t) -C σ(t) x(t -τ (t))] .
As an immediate consequence, using the definitions of the matrices M i and N i , we obtain

ẋ(t) = M σ(t) x(t) + B σ(t) K σ(t) x(t) ẋ(t) = N σ(t) x(t) + L σ(t) C σ(t) [x(t) -x(t -τ (t))] .
From Assumption 1 and the equality

x(ℓ)-x(ℓ-τ (ℓ)) = ℓ ℓ-τ (ℓ) [M σ(m) x(m) + B σ(m) K σ(m) x(m)]dm, it follows that, for all t ≥ T + τ , |x(t)| ≤ a|x(t -T )| + b sup ℓ∈[t-T,t] |B σ(ℓ) K σ(ℓ) x(ℓ)| , (13) |x(t)| ≤ c|x(t -T )| + d sup ℓ∈[t-T,t] L σ(ℓ) C σ(ℓ) × ℓ ℓ-τ (ℓ) [M σ(m) x(m) + B σ(m) K σ(m) x(m)]dm . ( 14
)
Using the constants defined in (9), we deduce from ( 13) and ( 14) that (x(t), x(t)) satisfies:

|x(t)| ≤ a|x(t -T )| + bs 1 sup ℓ∈[t-T -τ,t] |x(ℓ)| , |x(t)| ≤ ds 2 s 3 τ sup ℓ∈[t-T -τ ,t] |x(ℓ)| +(c + ds 1 s 2 τ ) sup ℓ∈[t-T -τ,t] |x(ℓ)| .
Lemma 1 ensures that the origin of ( 12) is GUES if

a bs 1 ds 2 s 3 τ ds 1 s 2 τ + c
is Schur stable, which is equivalent to

a+c+ds1s2 τ 2 + a+c+ds1s2 τ 2 2 -ac -ds 1 s 2 (a -bs 3 ) τ < 1 ,
from which we derive the simpler condition (10).

Parameters of the delay bound

In this section, we illustrate a method to determine the constants a, b, c, and d appearing in Assumption 1. Consider a continuous-time switched linear system

ξ(t) = Ω σ(t) ξ(t) + ϑ(t) , (15)
where ξ ∈ R d ξ , the switching signal σ is associated to a sequence t k of the type of those introduced in Section 3 and ϑ is a piecewise continuous function.

Lemma 2 Let the system (15) be such that there are real numbers d 1 > 0, d 2 > 0, µ ≥ 1, γ > 0 and symmetric positive definite matrices Q l , l ∈ Ξ, such that the LMIs

d 1 I Q i d 2 I , ( 16
) Q i µQ j , (17) Ω ⊤ i Q i + Q i Ω i -γQ i ( 18
)
are satisfied for all i, j ∈ Ξ. Moreover, the constant µ △ = µe -γδ is such that

µ △ < 1 . ( 19
)
Then, along the trajectory of ( 15), the inequality

|ξ(t)| ≤ d 2 d 1 µµ ρ △ e γδ |ξ(t-T )|+ µ d 2 γd 1 T sup ℓ∈[t-T,t] |ϑ(ℓ)|
holds for all t ≥ T where T > 0 and ρ is a positive integer depending on the choice of T such that for all t ∈ [t k , t k+1 ), we have t -T ∈ [t k-ρ-1 , t k-ρ ). Moreover, we have d2 d1 µµ ρ △ e γδ < 1 when ρ > 1 ln(µ∆) ln d1 d2µ -γδ .

For the proof of Lemma 2, see Appendix B.

Remark 1 1. Note that ( 19) holds if and only if δ > ln(µ) γ , which defines a minimum dwell-time condition.

2. Conditions of Lemma 2 are always satisfied when the matrices Ω i , ∀ i ∈ Ξ, are Hurwitz; i.e., one can always find symmetric positive definite matrices Q i , i ∈ Ξ, and real numbers d 1 > 0, d 2 > 0, µ ≥ 1, γ > 0 satisfying the LMIs ( 16), ( 17) and ( 18). In the next section we illustrate an alternative approach for the case where some of Ω i 's are not Hurwitz.

Illustrative Example

Consider the continuous-time switched linear system (4)

with x ∈ R 2 , τ ∈ [0, τ ), σ(t) = 1 if 4ℓκ ≤ t < (4ℓ + 3)κ 2 if (4ℓ + 3)κ ≤ t < 4(ℓ + 1)κ , ( 20
)
where κ > 0 is to be determined, ℓ = 0, 1, 2, . . ., and

A 1 = 0 -1/2 2/5 0 , B 1 = 0 0 0 0 , C 1 = 1 0 0 1 , A 2 = 0 -2/5 1/2 0 , B 2 = 1 0 0 1 , C 2 = 0 0 0 0 .
Let us observe that the subsystem (A 1 , B 1 , C 1 ) is not stabilizable but it is detectable whereas the subsystem (A 2 , B 2 , C 2 ) is stabilizable but not detectable. Moreover, in the absence of control, no convex combination of the A 1 and A 2 is Hurwitz. Furthermore, the subsystems cannot be stabilized by a static output feedback u = K i y. In this example, we have δ = κ and δ = 3κ and the switchings are periodic with a period of 4κ. We will determine a set of parameters for the delay bound depending on κ.

Preliminary result

First, we provide a preliminary result which shows how Assumption 1 can be satisfied in this particular example where some of the subsystems of the switched systems are not stabilizable and not detectable.

Lemma 3 Consider the switched linear system

ż(t) = Γ σ(t) z(t) + ̺(t) (21)
with σ defined by ( 20), and let

Γ 1 ∈ R 2×2 , Γ 2 ∈ R 2×2
and κ > 0 be such that the matrix S κ := e Γ2κ e 3Γ1κ is Schur stable. Let Φ ⋆ be the state transition matrix of the system ( 21) with ̺ = 0:

∂Φ ⋆ ∂t (t, s) = Γ σ(t) Φ ⋆ (t, s) , Φ ⋆ (s, s) = I ,
for all t ∈ R and s ∈ R. Then, for all s ≥ 0, t ≥ s

|Φ ⋆ (t, s)| ≤ p 1 e -p2(t-s) ( 22
)
with p 1 = e 8κ max{|Γ1|,|Γ2|} c κ e 2dκ and p 2 = d κ /4κ, where c κ > 1 and d κ > 0 are such that for all m ∈ N,

|S m κ | ≤ c κ e -dκm . ( 23
)
Moreover, for all T > 0,

|z(t)| ≤ p 1 e -p2T |z(t -T )| + p1(1-e -p 2 T ) p2 sup ℓ∈[t-T,t] |̺(ℓ)| . (24)
For the proof of Lemma 3, see Appendix C.

Remark 2 Since p 2 > 0, then p 1 e -p2T < 1 when T > ln(p1)
p2 , which determines a lower bound for T .

Output feedback stabilization

Let us choose the gain matrices as

K 2 = -1/2 0 0 -4/7 , L 1 = -3/5 0 0 -4/5 . Setting Γ 1 = M 1 = A 1 and Γ 2 = M 2 = A 2 + B 2 K 2 ,
one can easily corroborate that ( 23) is satisfied with the choice of κ = 0.1, c k = 1.01, and

d k = 0.001 for all m ∈ N. Setting z = α, Ω i = Γ i = M i for i ∈ {1, 2}
, and ̺ = ζ, it can be easily verified that ( 22) is satisfied by (5) with p 1 = e 8κ max{|Γ1|,|Γ2|} c κ e 2dκ = 1.7142 and p 2 = d κ /4κ = 1.0025. Using Lemma 3 with T = 6, one can observe that the solutions of system (5) satisfy ( 6) with a = p 1 e -p2T = 0.0042, b = (p 1 /p 2 ) 1 -e -p2T = 1.7057. A similar analysis shows that the solutions of system ( 7) satisfy ( 8) with c = 0.0052, d = 2.1156 and T = 6. Therefore, we conclude that the switched delay system satisfies Assumption 1. Finally, application of Theorem 1 with s 1 = 0.5714, s 2 = 0.8, s 3 = 0.7611, and with the preceding choices of the parameters yields τu = 0.4465. Fig. 1 shows the simulation of system (12) for this particular example for a piecewise continuous sawtooth function τ (t) of a fundamental frequency of 1 Hz described by τ (t) = 0.2(t-⌊t⌋) where the switching signal σ(t) is given by ( 20) with κ = 0.1. The initial conditions are chosen to be x 1 (0) = -0.5, x 2 (0) = -1, x1 (0) = 0.5, and x2 (0) = 1, and the sample rate is 1 kHz.

It is worth emphasizing here that [START_REF] Vu | Stability of timedelay feedback switched linear systems[END_REF] assumes that all of the modes of the delayed switched system are controllable and [START_REF] Sun | Stability analysis for linear switched systems with time-varying delay[END_REF] requires the derivative of the delay to be bounded which makes it impossible to apply their results to this example; and it also seems to us that there is no direct way to extend them to the output feedback case considered in this paper.

Conclusions

We presented dynamic output feedback stabilization results for systems with switches in the difficult case where a time-varying pointwise delay in the output is present. The technique of proof we proposed is based on the recent trajectory based approach. To solve the conservatism problem we encountered in Mazenc et al. (2017), we developed an extension of the main result of [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], which is of interest for its own sake. Many extensions of the results of the present paper are possible, pertaining for instance to design of K i and L i for maximization of the delay bound, robustness issues with respect to disturbances, the presence of a delay in the input, the design of reduced order observers and extensions to families of nonlinear systems.

A Technical lemma

Lemma 4 Let R ∈ R m×m be a nonnegative matrix. Let us consider functions w j : [0, +∞) → [0, +∞), j = 1, . . . , m, and a constant h > 0 such that for all t ≥ h, w = (w 1 ... w m ) ⊤ satisfies

w(t) ≤ Rζ(t) (A.1) with ζ(t) = sup ℓ∈[t-h,t] w 1 (ℓ) . . . sup ℓ∈[t-h,t] w m (ℓ) ⊤ .
Then, for all integer k larger than 1, and all t ≥ kh, we have

w(t) ≤ R k Ψ k (t) with Ψ k (t) = sup ℓ∈[t-kh,t] w 1 (ℓ) . . . sup ℓ∈[t-kh,t] w m (ℓ) ⊤ .
Proof. We prove the lemma by induction: Induction Assumption: There is l ∈ N, l > 0 such that the result of Lemma 4 holds for all k ∈ {1, ..., l}.

Step 1:

The assumption is satisfied at the step 1.

Step l: Let us assume that it is satisfied at the step l ≥ 1. Then the inequalities

w(t) ≤ R l Ψ l (t) (A.2)
hold for all t ≥ lh. From (A.1), we deduce that for all t ≥ (l + 1)h and ℓ ∈ [t -lh, t], the inequalities

     w 1 (ℓ) . . . w m (ℓ)      ≤ R        sup s∈[ℓ-h,ℓ] w 1 (s) . . . sup s∈[ℓ-h,ℓ] w m (s)        ≤ RΨ l+1 (t) hold. It follows that Ψ l (t) ≤ RΨ l+1 (t) . (A.3)
By combining (A.2) and(A.3), we deduce that

w(t) ≤ R l+1 Ψ l+1 (t)
for all t ≥ (l + 1)h. Thus the induction assumption is satisfied at the step l + 1. This concludes the proof.

B Proof of Lemma 2

Let us define Lyapunov functions:

V i (ξ) = ξ ⊤ Q i ξ, ∀i ∈ Ξ .
We deduce from (18) that when σ(t) = i, then the derivative of V i along the trajectories of (15) satisfies

Vi (ξ(t)) ≤ -2γV i (ξ(t)) + 2ξ(t) ⊤ Q i ϑ(t) ≤ -γV i (ξ(t)) + 1 γ ϑ(t) ⊤ Q i ϑ(t) (B.1)
where the last inequality is deduced from the Young's inequality. Now, let us integrate (B.1) between two instants s and t, t ≥ s, belonging to the same sampling interval where σ(t) = l. Then

V l (ξ(t)) ≤ e γ(s-t) V l (ξ(s)) + 1 γ t s e γm-γt ϑ(m) ⊤ Q l ϑ(m)dm ≤ e γ(s-t) V l (ξ(s)) + d2 γ t s e γm-γt |ϑ(m)| 2 dm , (B.
2) where the last inequality is a consequence of ( 16). Now, let us consider

T > 0, t ≥ T such that t ∈ [t k , t k+1 ) for some k ∈ Z ≥0 and let ρ ∈ N be such that t -T ∈ [t k-ρ-1 , t k-ρ ). From (B.2), we deduce that V σ(t k ) (ξ(t)) ≤ e -γ(t-t k ) V σ(t k ) (ξ(t k )) + d2 γ t t k e γm-γt |ϑ(m)| 2 dm ≤ µe -γ(t-t k ) V σ(t k-1 ) (ξ(t k )) + d2 γ t t k |ϑ(m)| 2 dm , (B.3)
where the last inequality is a consequence of (17). For similar reasons, Since (19) holds and T is arbitrarily large, one can choose T such that the corresponding ρ is so that d2 d1 µµ ρ △ e γδ < 1. This concludes the proof.

V σ(t k-1 ) (ξ(t k )) ≤ µ △ V σ(t k-2 ) (ξ(t k-1 )) + d2 γ t k t k-1 |ϑ(m)| 2 dm . . . V σ(t k-ρ ) (ξ(t k-ρ+1 )) ≤ µ △ V σ(t k-ρ-1 ) (ξ(t k-ρ )) + d2 γ t k-ρ+1 t k-ρ |ϑ(m)| 2 dm (B.4) V σ(t k-ρ-1 ) (ξ(t k-ρ )) ≤ e γ(t-T -t k-ρ ) V σ(t k-ρ-1 ) (ξ(t -T )) + d2

C Proof of Lemma 3

Let us introduce a sequence: g ℓ = 4ℓκ. Then for all integer n > 0, z(g ℓ ) = S n κ z(g ℓ-n ). Thus Φ ⋆ (g ℓ , g ℓ-n ) = S n κ . Let t ∈ R and s ∈ R be such that t > s ≥ t -4κ. Then |Φ ⋆ (t, s)| ≤ e 4κ max{|Γ1|,|Γ2|} .

(C.1) Now, let t ∈ R and s ∈ R be such that t + 4κ > s. Then there is ℓ such that t ∈ [g ℓ , g ℓ+1 ) and r ∈ N, r > 0 such that s ∈ [g ℓ-r-1 , g ℓ-r ). Then

|̺(ℓ)|

where the last inequality is a consequence of ( 22).

Fig. 1 .

 1 Fig. 1. Simulation results.

  .3), (B.4) and (B.5), and then using the definition of range dwell-time condition from (3), we getV σ(t k ) (ξ(t)) ≤ µ µ ρ △ e γδ V σ(t k-ρ-1 ) (ξ(t -T ))

  |Φ ⋆ (t, s)| ≤ e 8κ max{|Γ1|,|Γ2|} |Φ ⋆ (g ℓ , g ℓ-r )| .Since S κ is Schur stable, there are c κ > 1 andd κ > 0 such that for all m ∈ N, |S m κ | ≤ c κ e -dκm . Thus |Φ ⋆ (t, s)| ≤ e 8κ max{|Γ1|,|Γ2|} c κ e -dκr . Now, notice that r ≥ t-s 4κ -2. Consequently, |Φ ⋆ (t, s)| ≤ e 8κ max{|Γ1|,|Γ2|} c κ e 2dκ e -dκ t-s 4κ . (C.2) From (C.1) and (C.2), we deduce that for all t ≥ s, |Φ ⋆ (t, s)| ≤ e 8κ max{|Γ1|,|Γ2|} c κ e 2dκ e -dκ t-s 4κ . (C.3) This allows us to conclude that (22) is satisfied. Now, by integrating (21), we obtain that for all t ≥ T , |z(t)| = Φ ⋆ (t, t -T )z(t -T ) + t t-T Φ ⋆ (t, ℓ)̺(ℓ)dℓ ≤ p 1 e -p2T |z(t -T )| + t t-T p 1 e -p2(t-ℓ) dℓ sup ℓ∈[t-T,t]

	It follows that
	|Φ

⋆ (t, s)| ≤ e 8κ max{|Γ1|,|Γ2|} |S r κ | .

This work is supported by the PHC Bosphore 2016 France-Turkey under Project No. 35634QM and the Scientific and Technological Research Council of Turkey (T ÜB İTAK) under Project No. EEEAG-115E820.