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

Michael Malisoff
email: malisoff@lsu.edu

H Özbay

Stability Analysis of Switched Systems with Time-Varying Discontinuous Delays *

Keywords: switched system, time delay, time-varying system, asymptotic stability

A new technique is proposed for the stability analysis of nonlinear switched time-varying systems with time-varying discontinuous delays. It is based on an adaptation of Halanay's inequality to switched systems, combined with a recent trajectory based stability analysis technique. The result is applied to a family of the linear time-varying systems with time-varying delays. In particular, it is shown that this approach can be used for stabilization of LTV systems with time varying delays by a set of properly designed switched controllers.

Introduction

Switched systems in continuous-time are systems with discrete switching events. They use a switching signal that indicates which subsystem operates at each instant; see, e.g., [START_REF] Liberzon | Switching in Systems and Control[END_REF] and [START_REF] Wang | Stability of switched nonlinear systems with delay and disturbance[END_REF], for more details. They are fundamental systems which are encountered in a wide range of applications, including communication networks; see [START_REF] Cheng | An extension of LaSalle's Invariance Principle and its application to multi-agent consensus[END_REF][START_REF] Lee | Uniform asymptotic stability of nonlinear switched systems with an application to mobile robots[END_REF][START_REF] Sun | Stability Theory of Switched Dynamical Systems[END_REF][START_REF] Xie | Stabilisation of time-delay switched systems with constrained switching signals and its applications in networked control systems[END_REF][START_REF] Yang | On stabilization of switched nonlinear systems with unstable modes[END_REF]. In addition, delays are frequently present in models describing engineering processes. They also appear in the control inputs (actuator delays) or the outputs (measurement delays) in the feedback loops. Moreover, in some applications time delays can be time-varying and discontinuous: prime example is control over a network, where congestion and failures in certain links of the network leads to sudden changes in the routing, that causes the return-trip-time to change abruptly.

Even for systems without switches, the problem of analyzing the stability of systems with time-varying delays is difficult in general, especially when the delay is discontinuous. For systems with discontinuous delays, a strategy for stability analysis consists of representing the systems as switched systems, and then applying techniques from the theory of the switched systems; this is explained in [START_REF] Sun | Delay-hybrid-dependent stability for systems with time-varying delay[END_REF] and [START_REF] Yan | Robust Stabilization of Parameter Varying Time Delay Systems by Switched Controllers[END_REF]. These facts motivate the recent works [START_REF] Wang | Stability of switched nonlinear systems with delay and disturbance[END_REF] and [START_REF] Yan | Stability analysis of switched time-delay systems[END_REF], where switched nonlinear systems having lumped delays are studied, and many other contributions, such as [START_REF] Caliskan | Dwell-time computation for stability of switched systems with time delays[END_REF] and [START_REF] Vu | Stability of time-delay feedback switched linear systems[END_REF]. See also [START_REF] Xie | Periodical stabilization of switched linear systems[END_REF], for stability results for non-delayed switched systems, based on assuming the existence of a stable convex combination of their subsystems.

In the present paper, we will study switched systems whose switching signal only depends on time. Our main result is in Section 2, and is a new result that is based on combining Halanay's inequality [START_REF] Halanay | Differential Equations, Stability, Oscillations, Time Lags[END_REF] with the main result of [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF]. The key advantage of the technique is that it applies to broad classes of systems, including time-varying systems with switchings and discontinuous delays. This contrasts with much of the literature, since in many cases, the time-varying delays are assumed to be continuously differentiable [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays -a LPV approach[END_REF][START_REF] Briat | Memory-resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays[END_REF][START_REF] Fridman | On robust stability of linear neutral systems with time-varying delays[END_REF][START_REF] Wu | Delay-dependent criteria for robust stability of time-varying delay systems[END_REF] or the discontinuous part of the delay is assumed to be small in a suitable sense [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF]. Our key assumption here is a condition on the switching signal. This is basically a limitation on the number of switchings on all intervals of a certain length, but a key feature of our work is that we do not impose an upper bound on the size of the variation of the delay. Hence, our work contrasts with other works that involve delays and switching such as [START_REF] Yan | Robust Stabilization of Parameter Varying Time Delay Systems by Switched Controllers[END_REF] (which constructs a switching rule that yields stability properties, instead of proving stability properties for large classes of switching rules as we do here) and [START_REF] Wang | Stability of switched nonlinear systems with delay and disturbance[END_REF], which introduces upper bound on the delays that makes it possible to use Lyapunov-Krasovskii functionals (or LKFs) to prove stability properties.

We do not assume that all of the subsystems of the switched system are stable, nor we assume that a common Lypunov function for the subsystems is available. This is a valuable feature because there are several techniques for systems with strong or even weak common Lyapunov functions (such as those in [START_REF] Hespanha | Uniform stability of switched linear systems: extensions of La Salle's invariance principle[END_REF] and [START_REF] Mancilla-Aguilar | An extension of LaSalle's invariance principle for switched systems[END_REF]), but, in many cases, common Lyapunov functions do not exist, and the presence of a time-varying delay precludes the application of standard invariance principles. In many cases, the existence of a common strict Lyapunov function implies stability for any switching signal, and we aim to establish stability results under restricted switching signals because only this type of result makes it possible to solve problems for time-varying systems with time-varying delays; see Section 2.1. In Section 3, we apply the nonlinear result from Section 2 to a crucial family of systems, namely, linear time-varying systems with time-varying pointwise delays. We show that their stability can be analyzed under assumptions that can be verified by finding LKFs for time-invariant systems with constant pointwise delays. All our results rely on an assumption on the dwell time. We present illustrative examples in Section 4. Finally, concluding remarks are made in Section 5.

Notation. We will use the following notation and conventions. We omit arguments of functions when the arguments are clear from the context. We set N = {1, 2, . . .} and Z ≥0 = N ∪ {0}. For any dimensions k and n in N, the k × n matrix all of whose entries are 0 will also be denoted by 0. The usual Euclidean norm of vectors, and the induced norm of matrices, of any dimensions are denoted by | • |, and I is the identity matrix in the dimension under consideration. The floor function

E : [0, +∞) → Z ≥0 is defined by E(x) = k when k ∈ Z ≥0 is such that x ∈ [k, k + 1)
. We let f (t -) denote the limit from the left of functions f at points t in their domain where the left limit is defined. Given any constant τ b > 0, we let C in denote the set of all continuous functions φ : [-τ b , 0] → R n , which we call the set of all initial functions. We define Ξ t ∈ C in by Ξ t (s) = Ξ(t + s) for all choices of Ξ, s, and t for which the equality is defined. Let K ∞ be the set of all continuous functions g : [0, +∞) → [0, +∞) such that g(0) = 0, g is strictly increasing, and lim s→+∞ g(s) = +∞.

A Fundamental Result for Nonlinear Switched Systems with

Time Delay

Problem Definition and Preliminary Remarks

In this section we consider a nonlinear time-varying switched system with a delay, τ (t) ≥ 0, represented by

ẋ(t) = f σ(t) (t, x(t -τ (t)) (1)
where x is valued in R n , σ : [0, +∞) → {1, ..., k} is called the switching signal, k ∈ N and n ∈ N are arbitrary, and each f i is locally Lipschitz with respect to its last argument and piecewise continuous with respect to its first argument t for all i ∈ {1, ..., k}. We assume that the delay is bounded by a constant τ b ≥ τ (t) for all t ≥ 0 and the initial condition φ is in C in . The system (1) includes the important special case of dynamics with a time-varying piecewise continuous delay τ : [0, +∞) → [0, +∞).

Let {t i } be the switching sequence of non-negative real numbers associated with σ, i.e., the times σ changes to a new value, with t 0 = 0 and σ is such that σ(t) = σ(t i ) for all t ∈ [t i , t i+1 ). Moreover, we assume that there are two constants T 1 and T 2 such that

0 < T 1 < t i+1 -t i ≤ T 2 for all i ∈ Z ≥0 . (2)
The constant T 1 is usually called the dwell time, and the sequence {t i } is called a partition of [0, +∞). See [START_REF] Pepe | Stabilization in the sample-and-hold sense of nonlinear retarded systems[END_REF], and [START_REF] Koru | On dwell time minimization for switched delay systems: Freeweighting matrices method[END_REF] for design methods minimizing dwell times to ensure stability of delay systems. These papers develop stability results in the form "if T 1 < t i+1 -t i for all i, then the system is stable" under certain technical assumptions (usually requiring all subsystems to be stable), [START_REF] Caliskan | Dwell-time computation for stability of switched systems with time delays[END_REF]. In recent years there has been a considerable effort in trying to find the smallest possible T 1 for a given switched system, see e.g. [START_REF] Koru | On dwell time minimization for switched delay systems: Freeweighting matrices method[END_REF] and its references. In this paper, the stability condition to be derived depends not only on τ b and T 2 (see Assumption 3 below) but also on some other parameters to be defined below.

We next state our technical assumptions; see the end of this subsection for their motivations.

Assumption 1. There are k absolutely continuous functions V j : [0, +∞)×C in → [0, +∞) for j = 1, 2, . . . , k, real numbers α 1 , . . . , α k , nonnegative constants β 1 , . . . , β k , a continuous function W : R n → [0, +∞), and class K ∞ functions χ 1 and χ 2 such that

χ 1 (|φ(0)|) ≤ V j (t, φ) ≤ χ 2 (|φ| ∞ ) (3)
hold for all φ ∈ C in , t ∈ [0, +∞), and j ∈ {1, 2, . . . , k} and such that

Vσ(ti) (t) ≤ α σ(ti) V σ(ti) (t, x t ) + β σ(ti) sup ℓ∈[t-τ b ,t] W (x(ℓ)) (4)
holds almost everywhere along all trajectories of ẋ(t) = f σ(ti) (t, x t ) for almost all t ∈ [t i , t i+1 ) and all i ∈ Z ≥0 .

Assumption 2. The functions V j and W from Assumption 1 admit a constant µ > 1 such that

W (φ(0)) ≤ V 1 (t, φ) (5)
and

V i (t, φ) ≤ µV j (t, φ) (6)
hold for all φ ∈ C in , all i and j in {1, ..., k}, and all t ∈ [0, +∞).

Assumption 3. There are constants T ≥ τ b + T 2 and λ(T ) > 0 such that the inequality

t t-T α σ(ℓ) dℓ ≤ -λ(T ) (7)
holds for all t ≥ 0.

With the above notation we can now find a constant ν > 0 such that

t t-T β σ(s) e t s α σ(s) dℓ ds ≤ ν (8)
holds for all t ≥ 0. Note that ν exists because α σ(s) and β σ(s) take only finitely many possible values on bounded intervals of time. Then, we let N (r, t) denote the number of switching instants t i on [t -r, t) for all r > 0 and t ≥ 0, and define L(r) := sup t≥0 N (r, t) (which is finite, because of (2)). The next assumption is the stability condition of the main result to be stated in the next section.

Assumption 4. The inequality

µ L(T )+1 e -λ(T ) + µ L(T )+2 -1 µ -1 -L(T ) -1 νµ < 1 ( 9
)
is satisfied.

The following remarks summarize the notable features of the preceding assumptions.

Remark 2.1. The constants α i in Assumption 1 can be positive or negative. In other words, some systems ż(t) = f i (t, z t ) for some i ∈ {1, ..., k} may be unstable. In this case, to have stability for [START_REF] Ariba | Stability interval for time-varying delay systems[END_REF], their instability should be compensated in a certain sense by the stability of other systems ż(t) = f i (t, z t ), which is why we cannot extend our result to the case of switchings without restriction. Assumption 1 does not make it possible to apply Razumikhin's theorem or its recent extensions in [START_REF] Mazenc | Extension of Razumikhin's theorem for time-varying systems with delay[END_REF] and [START_REF] Zhou | Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] or Halanay's inequality to prove global asymptotic stability, because several Lyapunov functions are involved.

Remark 2.2. When a common Lyapunov functional is available, i.e., V 1 = V j for all j ∈ {2, ..., k}, the problem solved below can be solved using [START_REF] Mazenc | Extension of Razumikhin's theorem for time-varying systems with delay[END_REF] and [START_REF] Zhou | Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF]. Recall that globally uniformly exponential stability of a switched system does not imply that its subsystems have a common Lyapunov function; see [START_REF] Liberzon | Switching in Systems and Control[END_REF]Section 2.1.5]. Also, if a switched system is input-to-state stable (or ISS) under arbitrary switching, then uniform (with respect to the switching signals) ISS is equivalent to the existence of a common ISS Lyapunov function; see [START_REF] Mancilla-Aguilar | On converse Lyapunov theorems for ISS and iISS switched nonlinear systems[END_REF]. Using [20, Lemma A.1], we can extend our result to cases where the α i 's and β i 's are time varying.

For simplicity, we do not present this extension. Setting U (t, φ) = V σ(t) (t, φ), Assumption 1 gives

U (t) ≤ a(t)U (t, x t ) + b(t) sup ℓ∈[t-τ b ,t] W (x(ℓ)) (10)
almost everywhere, where a(t) = α σ(t) and b(t) = β σ(t) . This inequality does not make it possible to conclude as in [START_REF] Mazenc | Extension of Razumikhin's theorem for time-varying systems with delay[END_REF], because U is discontinuous.

Remark 2.3. The inequalities [START_REF] Fridman | On robust stability of linear neutral systems with time-varying delays[END_REF] in Assumption 2 are a standard assumption that is imposed in switching contexts; see [START_REF] Liberzon | Switching in Systems and Control[END_REF]. The inequality [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF] is the main assumption on the constants α i and the switching signal σ which ensures the stability of the system (1). Assumption 4 pertains to σ as well, because L depends on the dwell time T 1 . In fact, Assumption 4 and ( 7) are the fundamental constraints that σ has to satisfy for ensuring asymptotic stability. Notice that the inequalities (2) ensure that the functions N and L are well-defined and that they imply that for all r ≥ 0, the inequality

L(r) ≤ E r T 1 + 1 (11)
is satisfied. Indeed, if we consider g successive switching instants t i , t i+1 , ... t i+g-1 , then the length of the interval [t i , t i+g-1 ] is equal to the sum of the lengths of the intervals [t i+s , t i+s+1 ], s = 0 to g -2 which is strictly larger than (g -1)T 1 . Any interval [t -r, t) contains N (r, t) switching instants, so it follows that r > [N (r, t) -1]T 1 . Since N (r, t) is an integer, we obtain N (r, t) -1 < E(r/T 1 ), which is equivalent to [START_REF] Koru | On dwell time minimization for switched delay systems: Freeweighting matrices method[END_REF].

Main Result

With the notation introduced above we are now ready to state our main result.

Theorem 1. Let the system (1) satisfy Assumptions 1-4. Then its origin is a globally uniformly asymptotically stable equilibrium point.

Proof. For the sake of brevity, we continue to use the notation

a(t) = α σ(t) , b(t) = β σ(t) , and
U (t, φ) = V σ(t) (t, φ) (12)
from the previous subsection. One can prove that Assumptions 1-2 ensure that the system is forward complete. This can be proved by induction, because if the solution is defined over [0, t j ), then by integrating (4), we can prove that the solution is defined over [0, t j+1 ). In fact, forward completeness on [0, t 1 ] follows by combining Assumption 1 with the bounds W (x(ℓ)) ≤ µV j (ℓ, x(ℓ)) from Assumption 2, and applying Gronwall's inequality to the function sup ℓ∈[0,t] V j (ℓ, x(ℓ)) for all t ∈ [0, t 1 ], and then arguing inductively.

We next analyze the behavior of U along the trajectories of (1). Observe that Assumption 1 implies that if t ∈ [t m , t m+1 ), t * ∈ [t m , t], and m ∈ Z ≥0 are such that σ(t m ) = j, then the inequality

V j (t, x t ) ≤ e αj(t-t * ) V j (t * , x t * ) + β j t t * e αj (t-s) sup ℓ∈[s-τ b ,s] W (x(ℓ))ds (13)
is satisfied. It follows that for all j ∈ Z ≥0 , all t ∈ [t j , t j+1 ), and all t * ∈ [t j , t], the inequality

U (t, x t ) ≤ e t t * a(s)ds U (t * , x t * ) + t t * b(s)e t s a(r)dr sup ℓ∈[s-τ b ,s] W (x(ℓ))ds (14)
is satisfied. Next, let t ≥ T , j ∈ Z ≥0 , and q ≥ 2 be such that t ∈ [t j , t j+1 ), j -1 ≥ q and t -T ∈ [t j-q-1 , t j-q ). Observe for later use that q ≤ L(T ). We introduce the simplifying notation

E(ℓ, t) = e t ℓ a(s)ds , G i = µE(t i-1 , t i ), and w(t) = sup ℓ∈[t-τ b ,t] W (x(ℓ)) . (15)
From ( 14) with the choice t * = t j , and using the simplifying notation U (t) to mean U (t, x t ), we deduce that

U (t) ≤ E(t j , t)U (t j ) + b(t) t tj E(ℓ, t)w(ℓ)dℓ ≤ µE(t j , t)U (t - j ) + t tj b(ℓ)E(ℓ, t)w(ℓ)dℓ , (16)
where the last inequality is a consequence of ( 6) in Assumption 2. From the first inequality in ( 16) (with the choice j = l -1, and by taking the limit t → t - l from the left in ( 16)), we deduce that for all integers l ≥ 1, the inequality

U (t - l ) ≤ E(t l-1 , t l )U (t l-1 ) + t l t l-1 b(ℓ)E(ℓ, t l )w(ℓ)dℓ (17)
is satisfied. Consequently, using (6) in Assumption 2 and the nonnegativity of the G j 's, we obtain

U (t - j ) ≤ G j U (t - j-1 ) + tj tj-1 b(ℓ)E(ℓ, t j )w(ℓ)dℓ G j U (t - j-1 ) ≤ G j G j-1 U (t - j-2 ) + G j tj-1 tj-2 b(ℓ)E(ℓ, t j-1 )w(ℓ)dℓ . . . j r=j-q+2 G r U (t - j-q+1 ) ≤ j r=j-q+1 G r U (t - j-q ) + j r=j-q+2 G r tj-q+1 tj-q b(ℓ)E(ℓ, t j-q+1 )w(ℓ)dℓ, (18)
where we used the fact that q ≥ 2. By summing all of the inequalities in [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF], we obtain

U (t - j ) ≤ j r=j-q+1 G r U (t - j-q ) + tj tj-1 b(ℓ)E(ℓ, t j )w(ℓ)dℓ + G j tj-1 tj-2 b(ℓ)E(ℓ, t j-1 )w(ℓ)dℓ +... + j r=j-q+2 G r tj-q+1 tj-q b(ℓ)E(ℓ, t j-q+1 )w(ℓ)dℓ . (19)
Next note that for all h ∈ Z ≥0 and j ∈ Z ≥0 such that j ≥ h, we have

j r=h G r = µ j-h+1 E(t h-1 , t j ) and E(ℓ, t j-1 ) = E(ℓ, t j )(E(t j-1 , t j )) -1 (20)
for all ℓ ∈ [0, t j-1 ]. We deduce that

U (t - j ) ≤ µ q E(t j-q , t j )U (t - j-q ) + tj tj-1 b(ℓ)E(ℓ, t j )w(ℓ)dℓ + µ tj-1 tj-2 b(ℓ)E(ℓ, t j )w(ℓ)dℓ +... + µ q-1 tj-q+1 tj-q b(ℓ)E(ℓ, t j )w(ℓ)dℓ . (21)
Also, using [START_REF] Louisell | Delay differential systems with time-varying delay: new directions for stability theory[END_REF] with the choice t * = t -T (and by letting t → t - j-q in ( 14)), we obtain

U (t - j-q ) ≤ E(t -T, t j-q )U (t -T ) + tj-q t-T b(ℓ)E(ℓ, t j-q )w(ℓ)dℓ , (22)
since t -T ∈ [t j-q-1 , t j-q ). Consequently,

U (t - j ) ≤ µ q E(t j-q , t j )E(t -T, t j-q )U (t -T ) + µ q E(t j-q , t j ) tj-q t-T b(ℓ)E(ℓ, t j-q )w(ℓ)dℓ + tj tj-1 b(ℓ)E(ℓ, t j )w(ℓ)dℓ + µ tj-1 tj-2 b(ℓ)E(ℓ, t j )w(ℓ)dℓ +... + µ q-1 tj-q+1 tj-q b(ℓ)E(ℓ, t j )w(ℓ)dℓ = µ q E(t -T, t j )U (t -T ) + tj tj-1 b(ℓ)E(ℓ, t j )w(ℓ)dℓ + µ tj-1 tj-2 b(ℓ)E(ℓ, t j )w(ℓ)dℓ +... + µ q-1 tj-q+1 tj-q b(ℓ)E(ℓ, t j )w(ℓ)dℓ + µ q tj-q t-T b(ℓ)E(ℓ, t j )w(ℓ)dℓ . ( 23
)
From the second inequality in [START_REF] Mancilla-Aguilar | An extension of LaSalle's invariance principle for switched systems[END_REF], it follows that

U (t) ≤ µ q+1 E(t -T, t)U (t -T ) + t tj b(ℓ)E(ℓ, t)w(ℓ)dℓ + µ tj tj-1 b(ℓ)E(ℓ, t)w(ℓ)dℓ +µ 2 tj-1 tj-2 b(ℓ)E(ℓ, t)w(ℓ)dℓ + ... + µ q tj-q+1 tj-q b(ℓ)E(ℓ, t)w(ℓ)dℓ + µ q+1 tj-q t-T b(ℓ)E(ℓ, t)w(ℓ)dℓ ≤ µ q+1 e -λ(T ) U (t -T ) + Λ(t) sup ℓ∈[t-τ b -T,t] W (x(ℓ)) , (24)
where

Λ(t) = t tj b(ℓ)E(ℓ, t)dℓ + µ tj tj-1 b(ℓ)E(ℓ, t)dℓ + µ 2 tj-1 tj-2 b(ℓ)E(ℓ, t)dℓ + ... +µ q tj-q+1 tj-q b(ℓ)E(ℓ, t)dℓ + µ q+1 tj-q t-T b(ℓ)E(ℓ, t)dℓ , (25)
and where the last inequality is a consequence of (7) in Assumption 3. Since µ > 1, we have

Λ(t) = t t-T b(ℓ)e t ℓ a(s)ds dℓ + (µ -1) tj tj-1 b(ℓ)E(ℓ, t)dℓ + (µ 2 -1) tj-1 tj-2 b(ℓ)E(ℓ, t)dℓ + ... +(µ q -1) tj-q+1 tj-q b(ℓ)E(ℓ, t)dℓ + (µ q+1 -1) tj-q t-T b(ℓ)E(ℓ, t)dℓ ≤ t t-T b(ℓ)e t ℓ a(s)ds dℓ + µ -1 + µ 2 -1 + ... + µ L(T )+1 -1 t t-T b(ℓ)e t ℓ a(s)ds dℓ ≤ µ L(T )+2 -1 µ-1 -L(T ) -1 ν (26)
where the last inequality is a consequence of our choice of the bound ν from (8) and the formula for the geometric sum.

Using [START_REF] Wang | Stability of switched nonlinear systems with delay and disturbance[END_REF] to upper bound Λ(t) in [START_REF] Sun | Delay-hybrid-dependent stability for systems with time-varying delay[END_REF], recalling that µ > 1 and q ≤ L(T ), and using Assumption 2 to upper bound sup ℓ∈[t-τ b -T,t] W (x(ℓ)), we obtain

U (t, x t ) ≤ µ L(T )+1 e -λ(T ) U (t -T, x t-T ) + µ L(T )+2 -1 µ -1 -L(T ) -1 νµ sup ℓ∈[t-τ b -T,t] U (ℓ, x ℓ ) (27)
for all t ≥ T + τ b . We can therefore use Assumption 4 and [17, Lemma 1] (with the choices T * = T + τ b and w(ℓ) = U (ℓ + T + τ b ) in the lemma) to obtain an exponential decay estimate on U (t, x t ) that is valid for all t ≥ T + τ ; see Appendix A below for a statement of [17, Lemma 1]. Also, we can use Gronwall's inequality (as we did to prove the forward completeness property at the beginning of the proof) to obtain an exponential decay estimate on U (t, x t ) that is valid for all t ∈ [0, T + τ b ] (by applying Gronwall's inequality on successive intervals of the form [t j , t j+1 )). The desired global asymptotic stability estimate now follows from the upper and lower bounds for V in (3) in Assumption 1, so this concludes the proof.

Remark 2.4. If the χ i 's from Assumption 1 are quadratic functions, then we can also deduce that x(t) converges exponentially to zero.

Application to Linear Time Varying Systems

This section is devoted to application of the main result to a fundamental family of systems, namely, linear time-varying systems with time-varying lumped delays. For the sake of simplicity, we do not assume that these systems have switches, but we illustrate connections with switched systems. Throughout this section, we adopt the notation of Section 2.

Problem Definition and Preliminary Remarks

In this section we consider the system

ẋ(t) = A(t)x(t) + B(t)x(t -τ (t)) (28)
with x valued in R n for any n ∈ N and t ≥ 0. The following assumptions are in effect. hold for all t ≥ 0. Assumption 6. There exist an integer k ∈ N, a switching signal σ : [0, +∞) → {1, 2, . . . , k} whose switching instants {t i } admit positive constants T 1 and T 2 that satisfy (2) for all i ∈ Z ≥0 , a finite set ), and nonnegative constants ā, b, and τ such that

{(A i , B i , τ i ) : i ∈ {1, 2, . . . , k}} in R n×n × R n×n × [0, +∞
|A(t) -A σ(t) | ≤ a , |B(t) -B σ(t) | ≤ b and |τ (t) -τ σ(t) | ≤ τ (30)
hold for all t ≥ 0.

Assumption 7.

For each i ∈ {1, . . . , k}, there exist an absolutely continuous function

V i : C in → [0, +∞),
a real constant α i , and a positive constant γ i such that the time derivative of V i along all trajectories of

ż(t) = A i z(t) + B i z(t -τ i ) + δ i (t) (31)
for all continuous functions

δ i : [0, +∞) → R n satisfies Vi ≤ α i V i (z t ) + γ i |δ i (t)| 2 (32)
for almost all t, where the triples (A i , B i , τ i ) are from Assumption 6. Moreover, there are constants Ψ 1 > 0 and Ψ 2 > 0 such that for all φ ∈ C in and all i ∈ {1, ..., k}, we have

Ψ 1 |φ(0)| 2 ≤ V i (φ) ≤ Ψ 2 sup ℓ∈[-τ b ,0] |φ(ℓ)| 2 . ( 33
)
Also, the α i 's satisfy Assumption 3 with σ as defined in Assumption 6. Finally, there is a constant µ > 1 such that V i (φ) ≤ µV j (φ) holds for all (i, j) ∈ {1, ..., m} 2 and for all φ ∈ C in .

Assumption 8. Assumption 4 is satisfied, where ν is a constant that satisfies (8) as before, and σ, µ, and the α i 's are defined as in Assumptions 6-7, and the β i 's defined by

β i = γ i a + b + b b τ (a b + b b ) 2 Ψ 1 ( 34
)
for i = 1, 2, . . . , k.

Before stating our main result of this section, we point out several notable features of Assumptions 5-8.

Remark 3.1. In Assumption 6, we can always choose

a = a b + max i |A i |, b = b b + max i |B i | and τ = τ b + max i τ i .
However, these choices will not in general get the best (non-conservative) values for (9) in Assumption 4. It is also possible to see the triplets (A i , B i , τ i ) as nominal LTI systems; time varying perturbations around these systems (along with switchings) determine the LTV system [START_REF] Xie | Periodical stabilization of switched linear systems[END_REF]. Therefore, stability of the LTI switched systems together with certain conditions on the bounds defined in Assumption 6 will guarantee stability of the LTV system. For similar arguments see [START_REF] Yan | Robust Stabilization of Parameter Varying Time Delay Systems by Switched Controllers[END_REF].

Remark 3.2. Assumption 6 covers the important special case where A, B, and τ all have some period P > 0, by choosing

t i = iP/k and (A i , B i , τ i ) = (A(iP/k), B(iP/k), τ (iP/k
)) for all i ∈ {1, 2, . . . , k}, in which case the system (31) is replaced by

ż(t) = A iP k z(t) + B iP k z t -τ iP k + δ i (t), (35)
and we take the switching signal σ defined by σ(t) = i for all t ∈ [iP/k, (i + 1)P/k) and i ∈ {1, . . . , k}. However, Assumption 6 is much more general, e.g., because we do not require periodicity of the coefficient matrices or periodicity of the delay, nor do we require the t i 's to be evenly spaced. Besides, even when A, B and τ are periodic, it is worth trying to find a switching sequence with less switching instants than iP/k. All these comments motivate Appendix B.

Remark 3.3. For systems [START_REF] Yan | Dwell Time Optimization in Switching Control of Parameter Varying Time Delay Systems[END_REF] classical results, notably using LMIs [START_REF] Gu | Stability of Time-Delay Systems[END_REF], make it possible to construct timeinvariant quadratic LKFs. In Appendix C, we explain how LKFs are designed such that (32) and ( 33) are satisfied can be constructed. Also, we do not require the constants α i to be non-positive. Thus, we can allow some of the systems (31) to be unstable.

Remark 3.4. Since we only require τ (t) to be piecewise C 1 and bounded (instead of being C 1 ), it follows that the case of sampling can be handled. Indeed, sampling can be represented as sawtooth shaped delay functions, whose values are 0 at the sampling times, see e.g. [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. In fact, the present set-up allows nonuniform sampling and drifts in the sampling clocks. This further motivates our work, because controllers are typically implemented using sampled observations of their state arguments.

Remark 3.5. Even if the matrix A(t) is Hurwitz for all t ≥ 0, it does not follow that ẏ = A(t)y is globally exponentially stable; see [START_REF] Sontag | Mathematical Control Theory[END_REF]Exercise 5.7.21]. The systems [START_REF] Yan | Dwell Time Optimization in Switching Control of Parameter Varying Time Delay Systems[END_REF] do not in general admit a common Lyapunov functional. As noted in [START_REF] Louisell | Delay differential systems with time-varying delay: new directions for stability theory[END_REF], the fact that a system ẏ = Ay + By(t -h) is globally exponentially stable to 0 for any constant h ∈ [0, h] does not imply that any system ż = Az + Bz(t -h(t)) is globally exponentially stable to 0 when h is a function such that h(t) ∈ [0, h] for all t ≥ 0. This motivates Assumption 8.

Stability of the LTV System

The main result of this section is the stability of the LTV system (28).

Theorem 2. If the system (28) satisfies Assumptions 5-8, then it is globally exponentially stable to 0.

Proof. For each j ∈ {1, 2, . . . , k}, we can rewrite [START_REF] Xie | Periodical stabilization of switched linear systems[END_REF] as

ẋ(t) = A j x(t) + B j x(t -τ j ) + [A(t) -A j ]x(t) + [B(t) -B j ]x(t -τ j ) + B(t)[x(t -τ (t)) -x(t -τ j )] . (36)
Consequently, the system (28) can be represented as the following system with switches:

ẋ(t) = A σ(t) x(t) + B σ(t) x(t -τ σ(t) ) + Λ(t, σ(t), x t ), where Λ(t, j, x t ) = [A(t) -A j ]x(t) + [B(t) -B j ]x(t -τ j ) + B(t) t-τ (t) t-τj [A(ℓ)x(ℓ) + B(ℓ)x(ℓ -τ (ℓ))] dℓ (37)
for all t ∈ [t j , t j+1 ) and j ∈ Z ≥0 . Assumption 6 implies that for all t ≥ 2τ b + max i τ i , we have

|Λ(t, σ(t), x t )| ≤ a|x(t)| + b|x(t -τ σ(t) )| + b b t-τ (t) t-τ σ(t) [a b |x(ℓ)| + b b |x(ℓ -τ (ℓ))|] dℓ ≤ a|x(t)| + b|x(t -τ σ(t) )| + b b τ (a b + b b ) sup ℓ∈[t-2τ b -maxi τi,t] |x(ℓ)| ≤ a + b + b b τ (a b + b b ) sup ℓ∈[t-2τ b -maxi τi,t] |x(ℓ)| . ( 38
)
We deduce from Assumption 7 that

Vσ(t) (t) ≤ α σ(t) V σ(t) (x t ) + γ σ(t) a + b + b b τ (a b + b b ) 2 Ψ 1 sup ℓ∈[t-2τ b -maxi τi,t] Ψ 1 |x(ℓ)| 2 (39)
for all t ≥ 2τ b + max i τ i . Hence, Assumptions 1-4 are satisfied by (37), with W (x) = Ψ 1 |x| 2 . Then Theorem 1 allows us to conclude that this system is uniformly globally asymptotically stable. In fact, the proof of Theorem 1 provides an exponential decay estimate on U (t, x t ) = V σ(t) (t, x t ). By using the features of the V i 's, we deduce that ( 28) is also uniformly globally exponentially stable to 0. This completes the proof.

State Feedback Design for LTV Systems with Input Delay

In this section, in the spirit of what is done in [START_REF] Yan | Robust Stabilization of Parameter Varying Time Delay Systems by Switched Controllers[END_REF], we show how a control law with switchings (and a piecewise constant gain) can be fruitfully used to stabilize a time-varying system, even when they have no switchings.

Consider the system ẋ(t) = A(t)x(t) + B 0 (t)u(t -τ (t)) (40) with x valued in R n for any n ∈ N, u valued in R p for any p ∈ N and t ≥ 0. Similar to the previous section, the following assumptions are made. hold for all t ≥ 0.

Assumption 10. There exist an integer k ∈ N, a switching signal σ : [0, +∞) → {1, 2, . . . , k} whose switching instants {t i } admit positive constants T 1 and T 2 that satisfy (2) for all i ∈ Z ≥0 , a finite set {(A i , B 0,i , τ i ) : i ∈ {1, 2, . . . , k}} in R n×n × R n×n × [0, +∞), and nonnegative constants ā, b0 , and τ such that

|A(t) -A σ(t) | ≤ a , |B 0 (t) -B 0,σ(t) | ≤ b 0 and |τ (t) -τ σ(t) | ≤ τ (42)
hold for all t ≥ 0.

Assumption 11. For each i ∈ {1, . . . , k}, there exist an absolutely continuous function V i : C in → [0, +∞), a matrix K i a real constant α i , and a positive constant γ i such that the time derivative of V i along all trajectories of

ż(t) = A i z(t) + B 0,i K i z(t -τ i ) + δ i (t) (43) for all continuous functions δ i : [0, +∞) → R n satisfies Vi ≤ α i V i (z t ) + γ i |δ i (t)| 2 (44)
for almost all t, where the triples (A i , B 0,i , τ i ) are from Assumption 10. Moreover, there are constants Ψ 1 > 0 and Ψ 2 > 0 such that for all φ ∈ C in and all i ∈ {1, ..., k}, we have

Ψ 1 |φ(0)| 2 ≤ V i (φ) ≤ Ψ 2 sup ℓ∈[-τ b ,0] |φ(ℓ)| 2 . ( 45
)
Also, the α i 's satisfy Assumption 3 with σ as defined in Assumption 10. Finally, there is a constant µ > 1 such that V i (φ) ≤ µV j (φ) holds for all (i, j) ∈ {1, ..., m} 2 and for all φ ∈ C in .

Assumption 12. Assumption 4 is satisfied, where ν is a constant that satisfies (8) as before, and σ, µ, and the α i 's are defined as in Assumptions 9-11, and the β i 's defined by

β i = γ i a + b + b b τ (a b + b b ) 2 Ψ 1 ( 46
)
with b = b 0 max i∈{1,...,k} {|K i |}, b b = b 0,b max i∈{1,...,k} {|K i |} for i = 1, 2, . . . , k.
We are ready to state the following result:

Theorem 3. If the system (28) satisfies Assumptions 9-12, then it is globally exponentially stabilized to 0 by the control law

u(t -τ (t)) = K σ(t) x(t -τ (t)). ( 47
)
Proof. The closed-loop system is

ẋ(t) = A(t)x(t) + B(t)x(t -τ (t)) (48) with B(t) = B 0 (t)K σ(t) . Let B i = B 0,i K i Then |B(t) -B σ(t) | = |B 0 (t)K σ(t) -B 0,σ(t) K σ(t) | ≤ |B 0 (t) -B 0,σ(t) ||K σ(t) | (49)
From Assumption 9, it follows that

|B(t) -B σ(t) | ≤ b 0 max i∈{1,...,k} {|K i |} (50)
Also, using Assumption 9, we can prove that

|B(t)| ≤ b 0,b max i∈{1,...,k} {|K i |} (51)
for all t ≥ 0. Now, one can check that Assumptions 9-12 imply that the system (48) satisfies Assumptions 5-8. Then Theorem 2 allows us to conclude.

Obviously, the key which makes the above result useful is the design of the feedback gains K i , once (A i , B 0,i , τ i ) are fixed. This can be done by taking into account stability robustness, considering the bounds defined in (42), and an optimality criterion as in [START_REF] Yan | Dwell Time Optimization in Switching Control of Parameter Varying Time Delay Systems[END_REF].

Examples

In this section two examples are presented to illustrate the above results.

A Periodic System Example

Consider the switched linear system

ẋ(t) = A σ(t) x(t) + Bx(t -τ (t)) (52) with x valued in R 2 , τ (t) ∈ [0, τ b ] for any bound τ b > 0, A 1 = -I, A 2 = -2 -1 -1 -2 , and B = b b b b (53)
with b ≥ 0 being a constant, and σ : [0, +∞) → {1, 2} being periodic of some period P > 0 and defined by σ(t) = 1 when t ∈ 0, P 2 and σ(t) = 2 when t ∈ P 2 , P . We provide conditions on the constants b and P ensuring that the origin of ( 52) is globally exponentially stable.

We use the positive definite quadratic functions

V 1 (x) = √ 3|x| 2 /2 and V 2 (x) = x 2 1 + x 1 x 2 + x 2 2 .
Then for all t ∈ iP, P 2 + iP and integers i ∈ Z ≥0 , we have

V1 (t) = - √ 3|x(t)| 2 + √ 3x(t) ⊤ Bx(t -τ (t)) = -2V 1 (x(t)) + √ 3b[x 1 (t) + x 2 (t)][x 1 (t -τ (t)) + x 2 (t -τ (t))] ≤ -2V 1 (x(t)) + √ 3b(|x(t)| 2 + |x(t -τ (t))| 2 ) ≤ -2V 1 (x(t)) + 2b(V 1 (x(t)) + V 1 (x(t -τ (t)))) ≤ -2V 1 (x(t)) + 4b sup ℓ∈[t-τ ,t] V 1 (x(ℓ)) (54)
and when t ∈ iP + P 2 , iP + P , we instead have

V2 (t) = -(2x 1 + x 2 ) 2 -(x 1 + 2x 2 ) 2 + [2x 1 + x 2 x 1 + 2x 2 ]Bx(t -τ (t)) = -(2x 1 + x 2 ) 2 -(x 1 + 2x 2 ) 2 + 3b[x 1 + x 2 ][x 1 (t -τ (t)) + x 2 (t -τ (t))] ≤ -2V 2 (x(t)) + 4 √ 3b sup ℓ∈[t-τ ,t] V 1 (x(ℓ)) , (55)
where we used the fact that (c

1 + c 2 )(c 3 + c 4 ) ≤ c 2 1 + c 2 2 + c 2 3 + c 2 4
holds for all nonnegative values c 1 , c 2 , c 3 , and c 4 . Also,

V 2 (x) ≤ √ 3V 1 (x) and V 1 (x) ≤ √ 3V 2 (x) hold for all x ∈ R 2 .
Then, with our general notation from Assumptions 1-2, we can choose µ = √ 3, α 1 = -2, α 2 = -2, β 1 = 4b, and β 2 = 4 √ 3b. Also, choosing T = 2P, we obtain t t-T a(ℓ)dℓ = 2 P 0 a(ℓ)dℓ = -4P for all t ≥ T . Thus, Assumption 3 also holds, with λ(T ) = 4P, so Assumptions 1-3 are satisfied with W = V 1 . Also, we can choose ν = 2 √ 3b(1 -e -4P ). Moreover, L(2P) = 4. From [START_REF] Halanay | Differential Equations, Stability, Oscillations, Time Lags[END_REF] in Assumption 4 and Theorem 1, we deduce that if

µ L(T )+1 e -λ(T ) + µ L(T )+2 -1 µ -1 -L(T ) -1 νµ = 9 √ 3e -4P + 27 √ 3 -1 √ 3 -1 -5 6b 1 -e -4P < 1 (56)
then the system (52) is globally asymptotically stable to 0. Thus, roughly speaking, this system is globally asymptotically stable to 0 if P is sufficiently large and b is sufficiently small. Notice that P being large means that the system is slowly varying, and that the condition (56) is independent of τ b .

Switched State Feedback Design for a Periodic System

In this section, we illustrate Theorem 3 with the linear system with a pointwise time-varying delay

ẋ(t) = A(t)x(t) + B 0 u(t -τ (t)) ( 57
)
where x is valued in R 2 ,

B 0 = 0 1 (58)
with A and τ periodic of period P > 0 and defined by τ (t) = τ s when t ∈ [0, qP) and τ (t) = τ b > 0 when t ∈ [qP, P), and where τ s ∈ 0, 1 12 , q ∈ (0, 1) and τ b > 1 12 are constants and

A(t) = -1 0 0 0 when t ∈ [0, qP) and A(t) = 1 2 -2 -1 when t ∈ [qP, P) (59)
.

Step 1. We start our control design by choosing the following switching sequence: t 2j = jP, t 2j+1 = qP + jP and we define the switching signal σ by σ(t) = 1 when t ∈ [t 2j , t 2j+1 ) and σ(t) = 2 when t ∈ [t 2j+1 , t 2(j+1) ).

We select

u(t -τ (t)) = K σ(t) x(t -τ (t)). (60)
with K 1 = 0 -e -τs and K 2 = 0 Notice that we have A(t) = A σ(t) for all t ≥ 0. Moreover the eigenvalues of (A 1 + B 0 K 1 ) are {-1, -e -τs }, and those of (

A 2 + B 0 K 2 ) = A 2 are ±j √ 3.
So, the feedback system in mode σ(t) = 2 is not asymptotically stable, yet with proper switching scheme the system can be stabilized. Below computations derive a condition on admissible switching scheme

Step 2. We study the stability properties of the system

ż1 (t) = -z 1 + δ 11 (t) ż2 (t) = -e -τ1 z 2 (t -τ s ) + δ 12 (t) (62)
with δ 1 = (δ 11 , δ 12 ). Let us introduce

S(z 2,t ) = t t-τs e 2(m-t) z 2 (m) 2 dm + 6 z 2 (t) - t t-τs e m-t z 2 (m)dm 2 (63)
Elementary calculations give that the derivative of S(z 2,t ) along (62) satisfies:

Ṡ(t) = -2 t t-τs e 2(m-t) z 2 (m) 2 dm + z 2 (t) 2 -e -2τs z 2 (t -τ s ) 2 -12 t t-τs e m-t z 2 (m)dm -z 2 (t) 2 +12 z 2 (t) - t t-τs e m-t z 2 (m)dm [-e -τs z 2 (t -τ s ) + δ 12 (t) + e -τs z 2 (t -τ s )] ≤ -2 t t-τs e 2(m-t) z 2 (m) 2 dm + z 2 (t) 2 -12 t t-τs e m-t z 2 (m)dm -z 2 (t) 2 +12 z 2 (t) - t t-τs e m-t z 2 (m)dm δ 12 (t) (64) Now, observe that z 2 (t) 2 = t t-τs e m-t z 2 (m)dm -z 2 (t) - t t-τs e m-t z 2 (m)dm 2 ≤ 9 8 t t-τs e m-t z 2 (m)dm 2 + 3 z 2 (t) - t t-τs e m-t z 2 (m)dm 2 ≤ 9τs 8 t t-τs e 2(m-t) z 2 (m) 2 dm + 3 z 2 (t) - t t-τs e m-t z 2 (m)dm 2 (65)
where the last inequality is a consequence of Cauchy-Schwarz's inequality. An an immediate consequence, we have

Ṡ(t) ≤ -2 + 9τs 8 t t-τs e 2(m-t) z 2 (m) 2 dm -9 t t-τs e m-t z 2 (m)dm -z 2 (t) 2 +12 z 2 (t) - t t-τs e m-t z 2 (m)dm δ 12 (t) (66)
Since τ s ∈ 0, 1 12 , it follows that

Ṡ(t) ≤ - t t-τs e 2(m-t) z 2 (m) 2 dm -9 t t-τs e m-t z 2 (m)dm -z 2 (t) 2 +12 z 2 (t) - t t-τs e m-t z 2 (m)dm δ 12 (t) (67) Thus Ṡ(t) ≤ - t t-τs e 2(m-t) z 2 (m) 2 dm -3 t t-τs e m-t z 2 (m)dm -z 2 (t) 2 + 6δ 12 (t) 2 (68) Now, let V 1 (z t ) = S(z 2,t ) + 3z 1 (t) 2 (69)
Then

V1 (t) ≤ -3z 2 1 - t t-τs e 2(m-t) z 2 (m) 2 dm -3 t t-τs e m-t z 2 (m)dm -z 2 (t) 2 + 6|δ 1 (t)| 2 (70) It follows that V1 (t) ≤ - 1 2 V 1 (z t ) + 6|δ 1 (t)| 2 (71)
Step 3. We study the instability property of the system

ż1 (t) = z 1 + 2z 2 + δ 21 (t) ż2 (t) = -2z 1 -z 2 + δ 22 (t) (72)
Let us introduce the positive definite quadratic function

U 1 (z) = z 2 1 + z 2 2 + z 1 z 2 (73)
Elementary calculations give that the derivative of U 1 (z) along (72) satisfies U1 (t) = (2z

Consequently,

U2 (t) ≤ 1 2 z 2 (t) 2 + 1 4 (2z 1 (t) + z 2 (t)) 2 + 1 4 (2z 2 (t) + z 1 (t)) 2 + |δ 2 (t)| 2 = 1 4 [5z 1 (t) 2 + 7z 2 (t) 2 + 8z 1 (t)z 2 (t)] + |δ 2 (t)| 2 ≤ 1 4 [7U 1 (z(t)) + z 1 (t)z 2 (t)] + |δ 2 (t)| 2 ≤ 2U 1 (z(t)) + |δ 2 (t)| 2 ≤ 2U 2 (z t ) + |δ 2 (t)| 2 (77) Now, observe that V 1 (z t ) = t t-τs e 2(m-t) z 2 (m) 2 dm + 6 z 2 (t) - t t-τs e m-t z 2 (m)dm 2 + 3z 1 (t) 2 (78) Therefore V 1 (z t ) ≤ t t-
Since c ∈ (0, 1),

V 1 (z t ) ≥ t t-τs e 2(m-t) z 2 (m) 2 dm + 6 -6 c t t-τs e 2(m-t) z 2 (m) 2 dm + 3z 1 (t) 2 + 6(1 -c)z 2 (t) 2 = 7 -6 c t t-τs e 2(m-t) z 2 (m) 2 dm + 3z 1 (t) 2 + 6(1 -c)z 2 (t) 2 (82)
Choosing c = 13 14 , we obtain

V 1 (z t ) ≥ 7 13 t t-τs e 2(m-t) z 2 (m) 2 dm + 3z 1 (t) 2 + 3 7 z 2 (t) 2 ≥ 7e -2τs 13 t t-τs z 2 (m) 2 dm + 3 7 (z 1 (t) 2 + z 2 (t) 2 ) ≥ 7e -1 6 13 t t-τs z 2 (m) 2 dm + 3 7 (z 1 (t) 2 + z 2 (t) 2 ) (83)
We deduce that

U 2 (z t ) ≤ 3 2 (z 2 1 + z 2 2 ) + 1 2 t t-τs z 2 (m) 2 dm ≤ 7 2 V 1 (z t ) (84)
Now, from (79), we deduce that

V 1 (z t ) ≤ 6 t t-τs z 2 (m) 2 dm + 12z 2 1 + 12z 2 (t) 2 ≤ 6 t t-τs z 2 (m) 2 dm + 24[z 2 1 + z 2 2 + z 1 z 2 ] ≤ 24U 2 (z t ) (85) Let V 2 = gU 2 , where g > 0 is to be selected. Then V 1 (z t ) ≤ 24 g V 2 (z t ) V 2 (z t ) ≤ 7 2 gV 1 (z t ) (86)
So we choose g = 48 7 and obtain

V 1 (z t ) ≤ µV 2 (z t ) V 2 (z t ) ≤ µV 1 (z t ) (87)
with µ = 2 √ 21.

Step 4. Now, let us choose T = P. Now, observe that with the notation of Section 3.3, we have a = 0, b 0 = 0, τ = 0. It follows that β 1 = β 2 = 0 and ν = 0. We have α 1 = -1 2 (see (71)), α 2 = 2 (see (77)). We deduce that we can take λ(T ) = 1 2 qT -2(1 -q)T . In addition, notice that L(P) = 2. Clearly, V 1 and V 2 satisfy [START_REF] Yang | On stabilization of switched nonlinear systems with unstable modes[END_REF].

From Assumption 4, we deduce from Theorem 3 that the following is a sufficient condition for stability:

(2 √ 21) 3 < e [ 5 2 q-2]T , (88)
which is equivalent to 6 ln(2) + 3 ln( 21)

P + 4 5 < q (89)
This inequality is in accordance with what the intuition suggests: the interval where the system is unstable should be sufficiently small and the switchings should be not be too frequent.

Conclusion

We presented a new technique making it possible to establish the globally asymptotic stability of switched time-varying systems with time-varying delays. The result is used for switched controller design for the stability of a class of LTV systems. This technique relies on a trajectory based approach. It applies to a wide family of systems for which no other technique of stability analysis was available in the literature. We applied it to linear systems and have shown that it provides a result which only requires the construction of simple Lyapunov Krasovskii functionals. Much remains to be done: in particular, control designs under certain optimality and robustness conditions and ISS properties based on the results of the present paper will be subjects of future works.

Appendices A Key Lemma

We used the following key lemma:

0 = 0, v i+i -v i ∈ [v a , v b ]
for all i ≥ 0, w is continuous on each interval [v i , v i+1 ) for all i ≥ 0, and w(v - i ) exists and is finite for each i ∈ Z ≥0 . Let d : [0, +∞) → [0, +∞) be any piecewise continuous function, and assume that there is a constant ρ ∈ (0, 1) such that w

(t) ≤ ρ|w| [t-T * ,t] + d(t) (90)
holds for all t ≥ 0. Then w(t) ≤ |w| [-T * ,0] e ln(ρ)

T * t + 1 (1-ρ) 2 |d| [0,t] (91)
holds for all t ≥ 0.

The preceding lemma is [17, Lemma 1]. For its proof, see [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF].

B Construction of a special switching signal

When we adopt the technique of stability analysis exposed in Section 3, it is useful to minimize the number of switching instants associated with the function σ to diminish the conservatism of the stability conditions. This motivates the construction of the switching signal σ that we propose in this section. For the sake of simplicity, we consider the case where only A is time-varying, but the generalization to the case where A, B and τ are time-varying is straightforward.

Let A : [0, +∞) → R n×n be a continuous function such that |A(t)| ≤ a b for all t ≥ 0. Let ∆ > 0 be any constant.

The boundedness of A implies that one can establish the following fact: There are k a ∈ N constants matrices A i ∈ R n×n such that for all t ∈ [0, +∞), there is l ∈ E ka with E ka = {1, ..., k a } such that the inequality

|A(t) -A l | < ∆ (92)
holds.

Now, we assume that there is no j ∈ E ka such that there is an instant t c ≥ 0 such that for all t ≥ t c , |A(t) -A j | < 2∆. Indeed, this case is not interesting and trivial. Now, let us define an increasing sequence of nonnegative real numbers s i as follows. 1) Let s 0 = 0. Notice that the fact above implies that there is i

0 ∈ E ka such that |A(s 0 ) -A i0 | < ∆. 2) Let s 1 > 0 be such that, for all t ∈ [s 0 , s 1 ), |A(t) -A i0 | < 2∆ and |A(s 1 ) -A i0 | = 2∆. Then, the fact above implies that there is i 1 ∈ E ka such that |A(s 1 ) -A i1 | < ∆.
3) Now, we proceed by induction. Induction assumption: we assume we have s 0 , s 1 , ..., s l , l > 0 with 0 = s 0 < s 1 < ... < s l and i 0 ∈ E ka , ..., i l ∈ E ka such that for all m ∈ {1, ..., l},

|A(s m ) -A im-1 | = 2∆, |A(s m ) -A im | < ∆ and for all t ∈ [s m-1 , s m ), |A(t) -A im-1 | < 2∆.
4) According to our induction assumption, |A(s l ) -A i l | < ∆. Therefore there is s l+1 > s l such that, for all t ∈ [s l , s l+1 ), |A(t) -A i l | < 2∆ and |A(s l+1 ) -A i l | = 2∆. Now, we let i l+1 ∈ E ka be such that |A(s l+1 ) -A i l+1 | < ∆. The induction assumption is satisfied at the step i + 1. Now, we let σ be defined by σ(t) = i m when t ∈ [s m , s m+1 ). Then for all t ≥ 0, the inequality

|A(t) -A σ(t) | ≤ 2∆ (93)
is satisfied. Now, a question arises: does a constant T 1 > 0 such that

T 1 ≤ s i+1 -s i (94)
for all i ∈ N exist ? We present a case for which the answer is yes. Let us assume that A is of class C 1 and there is a constant d a > 0 such that for all t ∈ [0, +∞),

| Ȧ(t)| ≤ d a (95)
We have, for all m > 1,

|A(s m-1 ) -A(s m )| = |A(s m-1 ) -A im-1 + A im-1 -A(s m )| ≥ -|A(s m-1 ) -A im-1 | + |A im-1 -A(s m )| (96)
Also, for all q > 0, |A(s q ) -A iq-1 | = 2∆ and |A(s q ) -A iq | < ∆. We deduce that Thus, we can take T 1 = ∆ da . If there is no constant T 2 > 0 such that s i+1 -s i ≤ T 2 for all i ∈ N, then we can always replace s i by a new sequence t j which is composed of s i and additional switchings that are introduced to get a sufficiently large constant T 2 > 0 such that t i+1 -t i ≤ T 2 holds for all i ≥ 0. We can construct the t i 's as follows. We choose t 0 = 0 and t 1 = s 1 , and we set L i = Floor((s i -s i-1 )/s 1 ) for all integers i ≥ 2. If L 2 ∈ {0, 1}, then we choose t 2 = s 2 , and so we do not introduce any new switching instants on [0, s 2 ]. On the other hand, if L 2 ≥ 2, then we introduce the switching times t j+1 = s 1 + js 1 for j = 1, 2, . . . , L 2 -1, and we set t L2+1 = s 2 . By the definition of the floor function, we have t j ∈ (s 1 , s 2 ) for all j ∈ {2, 3, . . . , L 2 } and s 2 -L 2 s 1 ∈ [s 1 , 2s 1 ], so all of the switching times t i on [0, s 2 ] will satisfy T 1 ≤ t i -t i+1 ≤ 2s 1 . We continue in the same way, by introducing the switching times s p-1 + js 1 for j = 1, 2, . . . , L p -1 on the interval (s p-1 , s p ) if p ≥ 2 is such that L p ≥ 2, but not introducing any switching times in the interval (s p-1 , s p ) for integers p ≥ 2 for which L p ∈ {0, 1}. This produces a sequence t i such that T 1 ≤ t i -t i-1 ≤ 2s 1 for all integers i ≥ 1.

Then when a switching sequence σ 1 is associated with the sequence s i , we can choose the switching sequence σ 2 associated with t i by setting σ 1 (t) = σ 2 (t) for all t ∈ [0, +∞).

C Technical Result

Finding LKFs such that (32) and [START_REF] Yang | On stabilization of switched nonlinear systems with unstable modes[END_REF] hold is an easy task. We illustrate the key steps on an example involving two systems, this can be directly extended to any arbitrary number of systems. We easily deduce that (105) (107) are satisfied.

Assumption 5 .

 5 The functions A, B and τ are piecewise C 1 , and there are constants a b , b b , and τ b such that |A(t)| ≤ a b , |B(t)| ≤ b b , and τ (t) ≤ τ b (29)

Assumption 9 .

 9 The functions A, B 0 and τ are piecewise C 1 , and there are constants a b , b b , and τ b such that |A(t)| ≤ a b , |B 0 (t)| ≤ b 0,b , and τ (t) ≤ τ b (41)

Lemma A. 1 .

 1 Let T * > 0 be a constant. Let a piecewise continuous function w : [-T * , +∞) → [0, +∞) admit a sequence of real numbers v i and positive constants v a and v b such that v

-|A(s m- 1 )

 1 -A im-1 | + |A im-1 -A(s m )| = 2∆ -|A(s m-1 ) -A im-1 | ≥ 2∆ -∆(97)Combining (97) and (96), we obtain|A(s m-1 ) -A(s m )| ≥ ∆ (98)Now, from (95), we deduce that|A(s m-1 ) -A(s m )| ≤ d a (s m -s m-1 )(99)By combining (100) and (99), we deduce that ∆ ≤ d a (s m -s m-1 ) (100)

Lemma C. 1 .eeee

 1 Consider the systemsẊ(t) = A a X(t) + A b X(t -T a ) and Ż(t) = B a Z(t) + B b Z(t -T b )(101)where X and Z are valued in R n for any dimension n, and where T b and T a are any nonnegative values. Assume that the systems admit LKFs of the formQ a (X t ) = X(t) ⊤ P a X(t) + t t-Ta θ a (t -ℓ)X(ℓ) ⊤ R a X(ℓ)dℓ and Q b (Z t ) = Z(t) ⊤ P b Z(t) + t t-T b θ b (t -ℓ)Z(ℓ) ⊤ R b Z(ℓ)dℓ(102)respectively, whereP a ∈ R n×n , P b ∈ R n×n , R a ∈ Rn×n and R b ∈ R n×n are symmetric positive definite matrices and θ a and θ b are bounded, continuous and nonnegative values, such that their time derivatives along the corresponding systems in (101) satisfy Qa (t) ≤ -|X(t)| 2 and Qb (t) ≤ -|Z(t)| 2 ,(103)respectively. Then the functionalsQ * a (X t ) = Q a (X t ) -t+ℓ |X(ℓ)| 2 dℓ and Q * b (Z t ) = Q a (Z t ) -t+ℓ |Z(ℓ)| 2 dℓ (104) admit positive constants ̺ i > 0 for i = 1, 2, . . . , 7 such that Q * a (t) ≤ -̺ 1 Q * a (X t ) and Q * b (t) ≤ -̺ 2 Q * b (Z t ) (105)hold along all solutions of the corresponding systems. Moreover, we have that̺ 3 |φ(0)| 2 ≤ Q * a (φ) ≤ ̺ 4 sup ℓ∈[-T b ,0] |φ(ℓ)| 2 ̺ 5 |φ(0)| 2 ≤ Q * b (φ) ≤ ̺ 6 sup ℓ∈[-T b ,0] |φ(ℓ)| 2(106)andQ * a (φ) ≤ ̺ 7 Q * b (φ) and Q * b (φ) ≤ ̺ 7 Q * a (φ)(107)hold for all φ ∈ C in . Thus ̺ 7 plays the role of µ appearing in Assumption 2. Proof. The inequalities (106) are consequences of the definitions of Q a and Q b . Simple calculations give Q * a (t) ≤ --t+ℓ |X(ℓ)| 2 dℓ and Q * b (t) ≤ --t+ℓ |Z(ℓ)| 2 dℓ . (108)

  τs e 2(m-t) z 2 (m) 2 dm + 12 z 2 (t) 2 + (m) 2 dm + 3z 1 (t) 2 + 12z 2 (t)2

				t t-τs e m-t z 2 (m)dm	2	+ 3z 1 (t) 2
		≤ t-τs z 2 ≤ 6 t t-τs z 2 (m) 2 dm + 12 t t-τs e 2(m-t) dm t t t-τs z 2 (m) 2 dm + 3z 1 (t) 2 + 12z 2 (t) 2			(79)
	Also,
	V 1 (z t ) =	t t-τs e 2(m-t) z 2 (m) 2 dm + 6 z 2 (t) 2 + +3z 1 (t) 2	t t-τs e m-t z 2 (m)dm	2	-2z 2 (t)	t t-τs e m-t z 2 (m)dm
							(80)
	Using 2z 2 (t)	t t-τs e m-t z 2 (m)dm ≤ cz 2 (t) 2 + 1 c	t t-τs e m-t z 2 (m)dm	2	for any c ∈ (0, 1), we deduce that
	V 1 (z t ) ≥ =	t t-τs e 2(m-t) z 2 (m) 2 dm + 6 (1 -c)z 2 (t) 2 + 1 -1 c t t-τs e 2(m-t) z 2 (m) 2 dm + 6 -6 c t t-τs e m-t z 2 (m)dm t-τs e m-t z 2 (m)dm t 2 + 3z 1 (t) 2 + 6(1 -c)z 2 (t) 2 2 + 3z 1 (t) 2

* This work is supported by the PHC Bosphore 2016 France-Turkey (Mazenc and Özbay) under project numbers 35634QM (France) and EEEAG-115E820 (TUBITAK-the Scientific and Technological Research Council of Turkey), and by US National Science Foundation Grants 1408295 and 1436774 (Malisoff).


Olivier Godinot
email: olivier.godinot@agrocampus-ouest.fr

SEGAE: a serious game project for agroecology learning

Keywords: serious game, agroecology, systems thinking, active learning, transdisciplinarity, integrated crop livestock system

The main challenge of European agriculture is to provide food in sufficient quantity and quality while reducing its pressure on the environment. Agroecology seems a very relevant option to meet this challenge. The agroecological transition requires farmers, farm advisors and other agricultural professionals to have a holistic understanding of farming systems and agricultural practices, but most courses on agricultural science remain organized by discipline. There is thus an urgent need to develop more multidisciplinary and innovative approaches for agroecology teaching. This article presents a European project whose purpose is to develop an online serious game on agroecology. The main characteristics of the game are presented, as well as its intended uses. This serious game, as well as the other pedagogical resources produced during the project, is hoped to provide some useful material for a more multidisciplinary and experiential learning of agroecology in European Universities.

Introduction

European agriculture is facing many challenges, among which producing food and non-food products in sufficient quantity and quality, providing employment and profit for farmers and food chain actors, while reducing agricultural impacts on the environment [START_REF] Garnett | Sustainable intensification in agriculture: premises and policies[END_REF][START_REF] Ponisio | System-level approach needed to evaluate the transition to more sustainable agriculture[END_REF].

Agroecology can be defined as "the study of the interactions between plants, animals, humans and the environment within agricultural systems" [START_REF] Dalgaard | Agroecology, scaling and interdisciplinarity[END_REF]. Agroecological practices aim at increasing the level of ecosystem services provided to farming systems in order to sustain production while reducing environmental impacts through the decrease of anthropogenic inputs [START_REF] Bommarco | Ecological intensification: harnessing ecosystem services for food security[END_REF]. Agroecology is thus seen by several authors as a very pertinent option to reconcile the social, economic and environmental dimensions of agricultural sustainability [START_REF] Godfray | Food security: the challenge of feeding 9 billion people[END_REF][START_REF] Gliessman | Agroecology: the ecology of sustainable food systems[END_REF]. This is in line with a growing number of local, national and international regulations encouraging an agroecological transition of agriculture (support to investment, access to advice...), in order to decrease its negative environmental impacts while enhancing the ecosystem services it provides. It is thus necessary to build awareness and introduce the concepts of agroecology in the curricula of current and future professionals in the field of agriculture, i.e. students, future farmers and counsellors, as well as active professionals [START_REF] Valley | An emerging signature pedagogy for sustainable food systems education[END_REF]. However, higher education in most EU Universities is not yet fully adapted to train the future agricultural professionals on this topic. In particular, multidisciplinary approaches are not very developed in existing programs [START_REF] Francis | Transdisciplinary research for a sustainable agriculture and food sector[END_REF]. In their professional life, students will be facing agricultural challenges that can only be solved with a holistic view of farm management, encompassing animal and veterinary sciences, agronomy, ecology, environmental, economic and social sciences. The traditional, mostly discipline-oriented learning methods applied at university have often failed to help students understand the complex relationships between farming practices, agricultural production, environmental impacts and economic results.

Moreover, current pedagogical methods often lack interactive and experiential dimensions [START_REF] Lieblein | Becoming an agroecologist through action education[END_REF][START_REF] Schroeder-Moreno | Enhancing Active and Interactive Learning Online-Lessons Learned from an Online Introductory Agroecology Course[END_REF]. Theories on learning suggest that it is more effective when it is active, experiential, situated, problem-based and provides immediate feedback (Kiily, 2005). Only 20 % of oral information is usually memorized, whereas one understands and remembers 90 % of the information when experimented [START_REF] Davis | Applying Dale's Cone of Experience to increase learning and retention: A study of student learning in a foundational leadership course[END_REF]. Serious games appear to offer activities that exhibit these features. They offer an alternative pedagogy based on action and emotion. The learner is immersed in a virtual environment, where he can experiment different things, with no fear of doing mistakes. Then, reflecting on his actions, he will integrate good attitudes and new competences. Pieces of knowledge are disseminated throughout -or aside of -the game in the form of digital resources. It is an original and motivating learning experience adapted to the new generation of learners. Moreover, differentiated instruction can be implemented: different learning objectives can be set up to adapt to the different levels of knowledge of learners. This can explain why game-based learning has been rapidly growing in recent years [START_REF] Qian | Game-based Learning and 21st century skills: A review of recent research[END_REF].

However, few serious games are related to agriculture, and fewer address agroecological issues with a systemic view. Due to the strong incentives towards agroecology in Europe and the attractiveness of active learning tools, we think that a serious game on agroecology can be a very useful pedagogical tool. This paper presents an Erasmus+ project aimed at developing a serious game for digital learning of agroecology. The project is called SEGAE, which stands for Serious Game in AgroEcology. It is developed by six European Universities from Belgium, France, Italy and Poland. The main features of the game are presented, and some interesting features from existing serious games are identified to be reused in the game. Finally, limitations of serious games in general and this one in particular are identified.

Objectives of the SEGAE project

Targeted public

The serious game is mainly aimed at higher education students. Due to the consequent level of knowledge required to play the game, BSc students in the fields of agroecology, plant and animal sciences and MSc students from other fields related to agriculture (environmental sciences, economics, sociology) are our priority targets. However, the game will also be available for agricultural high schools with adapted pedagogical scenarios. The pedagogical activities proposed with the game will vary depending on the public. However, these should include (i) a presentation of the learning objectives and an overview of the game, (ii) a game play with the adapted scenario, and (iii) a discussion of results, methodology and the limitations of the game with the teacher.

Our contacts with agricultural co-operatives have also helped us identify a potential interest of agricultural counsellors for a better systemic vision of agriculture and information about agroecological practices. The game might in this case be part of a training on systems thinking or agroecology.

Targeting both students and current professionals will increase the number of users and potentially accelerate the diffusion of holistic agroecological thinking. It will however require developing pedagogical scenarios that are adapted to the learning needs and prior knowledge of those two different publics.

Main learning objectives

Our project plans to develop a serious game responding to three main objectives:

1. Help the players acquiring a systems thinking methodology, which is an essential skill for complex problem solving in agroecology [START_REF] Francis | Innovative education in agroecology: Experiential learning for a sustainable agriculture[END_REF][START_REF] Debruyne | Is the Current Educational Landscape Suited for Agroecology?[END_REF]. The players can learn this methodology by analyzing the farming system with indicators providing information on its technical features as well as systemic economic, environmental and social sustainability indicators. The game combines several scientific fields (plant science, animal and veterinary science, soil science, ecology, economics and social sciences) in a model whose properties cannot be anticipated by analyzing any one part of the system in isolation.

2. Find adapted solutions to solve a given agroecological problem. Putting the players in a situation of active learning and experimentation will allow them to test various options that a farmer could choose, and to instantly see their consequences on multiple aspects of the farm management. This inductive pedagogical approach helps the players evaluate the implemented actions, adapt their actions to their goals, and most importantly [START_REF] Francis | Innovative education in agroecology: Experiential learning for a sustainable agriculture[END_REF].

3. Provide practical knowledge about a large diversity of agroecological practices, their effects in different contexts and how to combine them in order to increase farming systems sustainability. This practical knowledge is also identified by De Bruyne et al.

(2017) as a critical skill for agroecology learning, and the serious game provides a tool to improve learning by "doing in silico".

Intended pedagogical activities

Several pedagogical uses will be possible with this game:

-"Systems thinking": the player will start with a typical farm from a chosen partner country (Belgium, France, Italy or Poland), and will have to improve as much as possible its economic, environmental and social results in a given virtual time (for instance 10 years).

Various agroecological practices will be available, given an annual budget and working time.

A set of steering indicators will help the player assessing the global impacts of his choices. This first game mode will enhance the player's comprehension of farming systems in their specific socio-ecological context.

-"Agroecological practices": the game will provide a large choice of agroecological practices. Detailed information will be available for each of these practices. The player will be able to get some scientific information on each practice, and to test its effects on the farming system. This game mode will provide scientific knowledge on agroecological practices and their systemic effects.

-"Teacher mode": The teacher will be able to set a starting situation for the farm (for instance, very productive with high environmental impacts). He will then define objectives for the player, and the means to do it (game length, farm budget and so on). This last game mode will be fully customizable. Depending on chosen settings, the level of difficulty will vary. This will allow teachers to use the game with high school, BSc, MSc students or agricultural professionals. Finally, game users will be able to propose their custom scenarios to the online community. This will enrich the game and offer a larger diversity of pedagogical opportunities to players.

Other project activities and outputs

The main activity throughout this three-year project is to build the serious game. After a review on agroecological practices, a conceptual model will be established, highlighting the main components that will be represented in the game as well as their interactions. The mathematical model will then be realized. An external software company will conduct the conception of the graphical interface.

A tutorial and pedagogical resources will be created during the project. The game and pedagogical resources will be tested with students during the course of the project. Training events for professionnals and teachers will also be organized. The serious game, as well as all project results, will be freely available online in four languages (English, French, Italian and Polish) at the end of the project.

Description of the serious game

Overall description

The serious game will allow the player to pilot a dairy and crop farm. We chose this farming system because combining crops and livestock offers various agroecological levers to improve the sustainability of farming systems [START_REF] Bonaudo | Agroecological principles for the redesign of integrated crop-livestock systems[END_REF]. Moreover, it is a usual farming system found in all partner countries. A typical dairy and crop farm will be developed for each partner country, which will lead to four different farms, thus providing a larger variety of pedagogical activities and players experience. The diversity of partner countries will give the game a truly European dimension, which will be interesting for its use outside the partner countries.

The player will have to make her/his farm evolve by implementing various agroecological practices, in order to attain a defined objective. The play screen will represent a dairy and crop farm landscape. The player will be able to click on each of the landscape elements (fields, animals, farm buildings, tractors, hedges...) and choose from a list of available agroecological practices. Information will be available for each practice, as well as a description of its main effects. The possibility to implement practices will be regulated by working time and budget constraints, as well as by some necessary developments to access novel techniques, in order to reflect real constraints encountered by farmers. Steering indicators (technical, social, environmental and economic indicators) will be displayed on screen, allowing the player to assess the relevance his strategy and the distance from his objectives.

Model architecture

The architecture of the serious game will be based on six modules: soil, crops, animals, ecosystem, socio-economic aspects and steering indicators (Fig. 1). All these modules will be coupled together to exchange information, in order to ensure the systemic dimension of the farm. They will represent the main biophysical, social and economic aspects of an integrated crop and dairy farm. Major agroecological practices found in the scientific literature will be implemented in the game. For each of these practices, the effects on each module will be described. Some important synergetic effects will also be described. Modules will be based on mathematical equations and/or expert knowledge but will remain simple, in order to ensure the game playability. The set of variables and parameters of each module as well as available farm practices and ecosystem services will be specific for each country.

The soil module will represent the main processes occurring in the soil under various agricultural practices. Water, nutrient and carbon cycles (including gaseous emissions, carbon storage and lixiviation) nutrient availability to plants, soil physical properties and soil biodiversity will be represented. They will be influenced by several practices, including fertilization and crop protection practices, organic matter recycling, soil tillage, as well as the diversity of crops in time and space.

The crop module will represent the major crops and forages, including grasslands, found in dairy farms of the four partner countries. It will calculate crop production and quality based on the soil and climate characteristics and farming practices (fertilizer use, crop rotations, weed, pest and disease management...), and the environmental impacts linked to crop production. Mixed crops, crop co-products such as straw, energy crops and agroforestry will be included in this submodel, as well as farm equipment for crops.

The animal module will represent the herd structure and dynamics (fertility, calving period and interval, culling rate…), feed requirements throughout the year, milk and meat production, animal health, manure production and use, animal housing and equipment, and environmental impacts from animals. Only dairy cows are envisaged for the moment, but different animal breeds will be available. Methanization will be part of this submodel.

The ecosystem module will represent ecosystem services other than production (which is represented in crop and animal submodels) and recycling of nutrients (which is represented in the soil submodel). It will thus represent the hedges, field margins, wetlands and other spaces of interest for ecosystem services. These services will include among others pollination, pests and disease control, genetic resources preservation and cultural services (recreation, landscape).

The economic module will represent the financial functioning of the farm: purchases, sales, investment capacity, loans. Prices will be influenced by product quality and labelling (organic for instance), as well as commercial outlets. Price variations will also occur to represent market volatility, which will provide an insight of the farming system resilience. The social submodel will estimate the workload of farmers, and represent the relationships between the farmer and other actors (consumers, neighbors, other farmers and counsellors…). Farmer's education, risk adversity, gender and age-based differences will not be explicitly considered are they are embedded in the player's strategy. However, this will represent an important topic of discussion with the teacher at the end of the simulation.

Finally, the indicator module will calculate a set of indicators related to technical management, as well as environmental, social and economic sustainability. It will use information from all other modules to provide a systemic view of the farming system and its evolutions.

Discussion

Insights from existing serious games

There is a wide body of scientific literature dealing with serious games (Connoly et al., 2012). However, a bibliographic review with the keywords "serious game" and "farm" or "farming" on Web of Sience, Google Scholar and Google did not allow us to identify any serious game allowing players to manage a realistic crop and livestock farming system with a technical and systemic (economic, social and environmental) approach. This is a strong point in favor of our project. Several interesting serious games about farming were nonetheless identified, and other serious games not related to farming were also studied. This chapter does not provide an exhaustive review on existing farming serious games, but rather highlights the features and that we considered interesting for our project.

Third World Farmer (https://3rdworldfarmer.org) is an online game at farm scale. It has been developed as a student project from the University of Copenhagen in 2005, and has been continuously improved since then. Its main goal is to to raise awareness about hunger and poverty-related issues in third world farming. The player experiences the difficulties of being a small African farmer seeking to improve his family's living standard while being confronted to bad harvests, corruption, guerrilla and other events. The agronomic and agroecological content of the game is very simple and cannot be a source of inspiration for SEGAE. The strengths of this game are its great playability and its wide use in academia, which has allowed millions of students to play the game over the years. We plan to get inspiration from its intuitive features (simple menu, clear pictograms, help in pop-up windows…) in our project.

Engele (http://www.rmtelevagesenvironnement.org/engele-rmt.htm) is a pig farm simulator created by French research centers, agricultural professionals and teachers. It has been developed in 2012 to help students and farmers understand the links between pig production and its environmental impacts [START_REF] Dourmad | A 3D serious game for teaching the environmental sustainability of pig farming systems[END_REF] and has been mostly used in French agricultural high schools since then. This game is based on a mathematical model linking pig feeding, meat production, excretion and manure management. The technical options of this game offer a level of detail that we would really like to reach in our project, but it is focused on pig production and thus crop management possibilities are limited. Moreover, there is no socio-economic dimension in the game. We plan to re-use the agile methodology that was used to develop this game with a software company, as well as the custom scenario approach that is also provided. The cost of the game was mostly driven by 3D graphic design, which does not seem an essential feature to other serious games, so we decided to put more emphasis on playability and gamification and less in graphic design.

The Forage Rummy [START_REF] Martin | Forage rummy: A game to support the participatory design of adapted livestock systems[END_REF] is a board game supported by a computer model built with Microsoft Excel. Its goal is to offer a discussion tool between farm advisors and farmers in order to test and evaluate modifications of their livestock system, feeding management and cropping and forage system. It is thus not exactly a serious game, but rather a decision support system. However, the conceptual model used in the tool is very inspiring: rather than a detailed and complex mechanistic biophysical model, it is based on a simple model relying on empirical data from experts. It thus requires very little parameterization data, and is easier to communicate to players [START_REF] Sautier | Participatory design of livestock systems adapted to new climatic conditions[END_REF].

The RESORTES board game (Speelman and García-Barrios, 2010b) deals with land-use planning in an agricultural landscape. It is a cooperative game aimed at fostering exchange and discussion between smallholder farmers and other community actors. Although the nature and goals of this serious game are very different from our project, it is important to note that the game combines economic, social and environmental evaluation of farmers strategies in a unique scoring system [START_REF] Speelman | Gaming for smallholder participation in the design of more sustainable agricultural landscapes[END_REF] that could be inspiring for our sustainability assessment system.

Finally, Clim'Way (http://climcity.cap-sciences.net) is an online serious game created in 2008 by a French science of scientific culture to raise awareness about energy consumption and greenhouse gas emissions reduction as well as adaptation to climate change in various sectors of activity, from industry to transport and agriculture. This serious game is inspired from the well known SimCity© video game. Although this game is not focused on agriculture, the graphic settings are intuitive, and the indicators are easy to follow. Moreover, each action is documented with a short pedagogical notice, highlighting the most important effects of the action and leading to more detailed information. We think that these features could be interesting in our game.

Limits of our games approach

There are several limitations to serious games approaches. The major limitation that we identified is to find the adequate equilibrium between simplification of the biophysical flows and interactions and the realism and reliability of the game simulations. This crucial issue led us to a conceptual model with a limited number of choices for each action. For instance, only 3 choices will be offered regarding tillage : systematic tillage, occasional tillage, and no-till. Some features and interactions will not be represented in our game : as an example, we decided to evaluate the interactions between farm management practices and functional biodiversity, but not on all biodiversity. By making these choices and giving this information to players, we hope to keep the model simple while realistic and attractive.

Another important limitation is the management of uncertainty. We plan to include some random events, such as climatic hazards, pest outbreaks and economic fluctuations that will introduce some uncertainty in the game. But for the sake of simplicity, we do not plan to include stochastic distributions of parameters: one action in one particular context will always lead to the same result. Yet it is well known that farm practices have very variable effects, and that interactions between system components can vary greatly, sometimes from positive to negative effects. This unsolved limitation highlights the fact that a session of serious game should also include a critical discussion of results with the teacher/trainer.

Another limitation in the model development is to adapt it to our different publics, from secondary students to agriculture professionals. We chose not to build a detailed decision support system for farm advisors, but rather a serious game about systemic thinking and agroecological practices. It is important to make this point very clear from the start with the players. Experienced professionals will probably spend less time individually playing the game, and more time on a discussion about how a systemic vision of the farm can modify their work practices. While students will be able to spend more time on the game and learn about the effects of various practices on the system indicators. So we are confident that the game will be usable with various publics with an adapted pedagogical program. This feeling is supported by the fact that Aqua Republica (http://aquarepublica.com/), a serious game dealing with water issues, includes a version for middle school children, a version for adult awareness raising and a version for stakeholder participation in real-life watershed management base on the same model [START_REF] Chew | Capacity building in water with serious games[END_REF].

Finally, a technical limitation is the cost and feasibility of maintenance and evolution of the game software to keep pace with rapidly changing technologies (smartphone, tablet) and computer software, the improvement in scientific knowledge and the development of new farming practices. To deal with this issue, we chose to give access (in an advanced user menu) to all game parameters in order to modify them according to new knowledge or in a different context. We also decided that the development of the program will be done with a common programming language, and that the code will be open-source in order to allow various users to propose improvements and modifications.

Conclusion and perspectives

Agroecology seems a relevant answer to many issues faced by today's agriculture.

Comprehensive tools for awareness-raising and education are needed for both the academic and professional stakeholders. Our serious game is designed in this perspective, based on several interesting examples. Its pedagogical possibilities will allow teachers to use it with a large diversity of publics and various pedagogical objectives throughout Europe. We hope that this innovative tool will help students and professionals better understanding the holistic approach needed to improve the sustainability of farming systems, learning about the effects of various agroecological practices, but also developing strategic and creative skills and a desire to keep on learning. Despite their pedagogical interests, it should be stressed that serious games also present important limitations that have to be identified and discussed with players.

This project was funded with the support of the European Commission. This publication is binding only on its author and the Commission is not responsible for any use which may be made of the information contained therein.

Figure 1 .

 1 Figure 1. Representation of the serious game structure

th European IFSA Symposium, 1-5 July 2018, Chania (Greece)


Michael Filhol
email: michael.filhol@limsi.fr

Annelies Braffort
email: annelies.braffort@limsi.fr

What constraints for representing multilinearity in Sign language?

In this paper, we present our current investigations on one of the main challenges in Sign Language modelling: multilinearity. We explain the way in which we extract grammatical rules from Sign Language corpus, and how we represent them with constraints. Two kinds are needed: time synchronisation and geometric constraints.

Introduction on Sign languages

Sign languages (SLs) are visual-gestural natural languages used by the deaf communities. SLs are less-resourced languages and have no writing system. In other words, there are very few reference books describing these languages (grammar rules, etc.), a limited number of dictionaries, and mainly small-sized corpora. Moreover, SL linguistics and processing research is quite recent, and a proper formal description system of the rules governing SL is still to be proposed.

In view of building SL processing systems (recognition, generation, translation...), we need a database of representations for SL elements such as lexical units or grammatical rules. Formal knowledge on these still being in its infancy, this requires thorough investigation beforehand, based on annotation and analysis of SL corpora.

The main challenges in SL modelling are: the relevant use of space in SL discourse, the iconicity that is present at various levels (iconic morphemes, spatial agreement...), and the multilinearity of the language. This paper gives an overview of our current investigations on this last property, started during the European project Dicta-Sign1 . Section 2 explains how we analyse corpora to build our rules, and section 3 presents our use of constraints to represent them.

Building rules by linking form to function

The first important fact about SL is that all body articulators from the waist up, whether manual, non-manual, skaletal or muscular, can be relevant as they may all-though they do not always-carry meaning. Therefore, all visible activity in a Sign production must be included in the study.

We have observed and partially annotated the French part of the DictaSign corpus [START_REF] Matthes | Elicitation tasks and materials designed for DictaSign's multi-lingual corpus[END_REF] and the older LS-Colin corpus [START_REF] Braffort | Analysis and Description of Blinking in French Sign Language for Automatic Generation[END_REF]. With as few articulators as eyebrows, eye-gaze and both left and right hand activity (and yet the signing body involving many more), it is already clear that articulators do not align in any trivial way. That is the second important specificity of SL to deal with.

With these two SL characteristics in mind, we proceed with the corpus study to find systematic links between:

surface form features, i.e. observable/annotated movements or positions of body articulators like a cheek puff, directed eyegaze, hand location change or shoulder line rotation; and linguistic function, i.e. the possibly language-dependant interpretation of the form, taken to be part of the signer's discourse intent.

Funtions can be identified on all layers of the language; none is linked to a specific one. Any linguistic operation purposely produced may be called a function, whether it inclines to a morphemic, lexical, syntactic or even discourse level. For example, turning a clause into an interrogative, using a known lexical sign, adding modality to a verb or outlining a full speech with intermediate titles are all possible functions. The point for our study is now to find features of the production form that are consistantly used over many occurrences of a linguistic function. Factoring out those form invariants raises a candidate set of necessary features for a given function. Describing this set of features under a label for the function raises what we call a production rule for the function.

For example, it is already well-documented that in LSF, space locations can be activated by pointing at the target in the signing space with the strong hand, immediately after directing the gaze towards the same target. Other elements are sometimes held on facial articulators or performed by the weak hand, but they do not occur consistantly while the eyegaze-then-point structure is invariant.

Formally describing the invariant above will create a production rule which can be labelled "point and activate locus". We explain how we handle those descriptions in the next section.

Constraints to represent production rules

There are two problems here to address. One is time synchronisation of parallel activity on a multi-linear signing score, in our example the simultaneity of gaze and index. The second is articulation of body articulators in space, e.g. the hand shape for the pointing gesture and the common direction for both eyegaze and index.

The next section explains the limits in existing models; the following two present the approach we use at LIMSI.

Traditional models

Most models in traditional Sign linguistics do not account for articulator synchronisation issues as fundamental elements of the SL discourse. They rather define elementary movements and/or sign units that can be concatenated [START_REF] Stokoe | Sign Language structure: an outline of the visual communication system of the American Deaf Studies[END_REF][START_REF] Prillwitz | HamNoSys version 2.0, an introductory guide[END_REF]. Some models [START_REF] Liddell | American Sign Language, the phonological base[END_REF] do promote time to a more essential place in their descriptions. Liddell and Johnson have shown that signs reveal (rather short) periods when all "features" of the hands align to form intentional postures, between which transitions take place to change from one established posture to the next (see fig. 1a further down). However, all of these models have only really been used (and even designed in the first place) for lexical units of SL, primarily even for "fully lexical signs" (i.e. non depicting). Though, SLs generally enrol more body articulators and even provide with productive ways of creating unlisted iconic units on the fly [START_REF] Cuxac | Langue des signes franaise, les voies de l'iconicit[END_REF][START_REF] Johnston | Australian Sign Language (Auslan): An Introduction to Sign Language Linguistics[END_REF].

Unfortunately, not much exists on SL formal modelling on the clause/sentence level. The most relevant progress is the ViSiCAST2 and eSign projects and their generative HPSG framework [START_REF] Marshall | Sign Language generation in an ALE HPSG[END_REF]. The basic timing unit in all generative approaches is the lexical item at the bottom of the syntax tree. Thus, any phonetic feature of the body must be borne by a lexical unit composing the clause and the productions burn down to a single tier carrying a lexical sequence. Articulator synchronisation is then only possible using the lexical boundaries.

A first account for multi-track description was given by the "P/C" formalism [START_REF] Huenerfauth | Generating American Sign Language classifier predicates for English-to-ASL machine translation[END_REF]. It defines two ways of conjoining parts of a signing activity: partitioning for parallel specification and constituting for sequence. But we argue [START_REF] Filhol | Combining two synchronisation methods in a linguistic model to describe Sign Language[END_REF] that it does not allow for abstract description of parts that overlap in time. Section 3.2 presents our approach to tackle this problem with free sets of constraints.

In addition to the timing issues, another question raised is how to specify the signing activity happening at a given time. The problem with L&J's representation is that every moment in time requires the same fixed set of parametric values, numbered a, b, c... in fig. 1a. But similarly to time constraints, we want this new specification to be flexible hence allow for under-specification if it is needed. Section 3.3 below deals with this issue.

Time synchronising constraints

While L&J produce good evidence that much of the signing activity does synchronise with postures, observation reveals that many constructs fall out of the scheme and require a more complex synchronisation technique. Many productions break the set of body articulators into subsets that act more or less independently, each in its own active period (a time interval, TI henceforth). Sometimes the full face acts as a group while the hands are moving; other times a few facial parts take on a different linguistic role and de-synchronise from rest of the face and hands; hands do not always work together; etc. Our proposition is to combine those two sync schemes into a same language for a better coverage of SL linguistic structures. We propose the AZee language, combination of the following two.

Zebedee is a language designed to specify the posture-transition patterns [START_REF] Filhol | Zebedee: a lexical description model for Sign Language synthesis[END_REF]. A description in Zebedee (or zebedescription) specifies:

1. an alternation of "key posture" (KP) and transition units (the horizontal axis in figure 1b below; 2. the articulation that takes place in each of those units (this is further explained in the next section). Azalee is a language designed to specify the free-hand (i.e. not necessarily aligned) synchronisation patterns for sets of separate time intervals (the TIs) on a timeline. Each Azalee specification (or azalisting) is a container for the two following parts, together specifying the intended rule fully:

1. all (usually overlapping) TIs are arranged on the timeline following temporal precedence constraints; 2. each TI specifies the signing activity taking place within.

In part [START_REF] Allen | Maintaining Knowledge about Temporal Intervals[END_REF], TIs are time-arranged with a minimal yet sufficient set of constraints in order not to over-specify the surface production and allow for any needed flexibility. The reason for preferring minimality for these sets is that we observe a lot of nested structures, all acting on the body articulators simultanously. Any superfluous constraint on an articulator in a rule will prevent other rules to use it freely, which may result in unsatisfactory productions.

The temporal precedence constraints can be of any form, whether strict or flexible, of precedence or duration, over full TIs using Allen logic operators [START_REF] Allen | Maintaining Knowledge about Temporal Intervals[END_REF] or over single boundaries. In part (2), a TI can either make use of a nested Azalee description, or use Zebedee for a simpler sync pattern. Figure 2 illustrates our example rule. Its Azalee pattern (outer box) is composed of 2 TIs, 'eye gaze' and 'pointing sign', which are zebedescriptions (KP-T alternation patterns). This flexible combination of two nestable timing patterns allows to account for complex synchronisation between body articulators.

Articulatory constraints

Fixed sets of parameters or features are not satisfactory to describe articulatory postures. As we have explained: To avoid both problems, we choose sets of necessary and sufficient constraints to specify both as much and as little as needed at any moment in time. In other words recalling the minimality of the sets of time constraints, we aim at minimal sets for the articulatory constraints. The constraints are mainly of two types: orientation (to align bones along vectors) and placement (to bring a body location to a point in space or in contact with another). For example, throughout the pointing gesture, we need to extend the index, which takes at least the following constraint, literally aligning the proximal index bone of the strong hand (side S below) in the same direction as the second:

Orient DIR!index(S,1) along DIR!index(S,2)

Placing the hand-the palm site (@PA) of the strong hand-in front of the body (forward fron the sternum site @ST) for the initial key posture will be written:

Place @WR(S) at @ST + <FWD | medium> Each timing unit holds a conjunction of such constraints, and the resulting set is as large as necessary and as small as sufficient. Figure 1b illustrates this approach and contrasts with previous figure 1a. The vertical depth needed to specify body articulation in a zebedescription may vary in time, according to the number of articulatory constraints. Note that they are fundamentally geometric; all typical geometric transformations (translations, symmetries, etc.) are supported.

Also, expressions can contain context references enclosed in square brackets to account for significant (often iconic) context variations. For our example above, the distal bone of the pointing finger !index(S,3) must be oriented to point at the target, which is a context dependency. The constraint will be something like3 : Orient DIR!index(S,3) along <ORIGIN!index(S,3), [target]>

We have presented the issues in formalising systematic links between form and function. Our methodology, based on SL corpus analysis, is applied regardless of the level, from the sub-lexical level to discourse structure. The formalisation of the production rules uses the AZee model, which allows to express Sign articulator synchronisation with time constraints, and body postures and movement with geometric constraints.

While the Zebedee part of this model is already well assessed (2,000+ LSF signs described), we have so far only really explored a dozen Azalee structures. We intend to continue this work to increase this number. A thorough evaluation will then be a starting point for implementation of AZee rules in Sign synthesis software applications, a longer-term prospect being text-to-SL translation.

In parallel and in order to collect 3d/spatial data, we will have to collect a new kind of SL corpus, built using a motion capture system. This will allow us to conduct similar studies on the spatial organisation of SL and then provide with a finer representation of the signing space and its use in SL discourse.

Fig. 1 .

 1 Fig. 1. (a) Liddell and Johnson's phonetic sequence with a fixed number of parameters; (b) Timing units in Zebedee: each unit is described with a variable number of elements

Fig. 2 .

 2 Fig. 2. AZee: an example of Azalee and Zebedee combination

http://www.dictasign.eu

http://www.visicast.cmp.uea.ac.uk

Operator <..., ...> creates a vector from two geometric points.