Gökhan Göksu
email: goksug@itu.edu.tr

Antoine Chaillet
email: antoine.chaillet@l2s.centralesupelec.fr

Analysis of Integral Input-to-State Stable time-delay systems in cascade

Keywords: Time-delay systems, integral input-to-state stability, Lyapunov-Krasovskii functional, cascade interconnections

We consider the cascade interconnection of two nonlinear time-delay systems, each of which being integral input-to-state stable (iISS). We provide an explicit growth rate condition on the dissipation rate of the driving system and the input rate of the driven system under which the overall cascade is globally asymptotically stable in the absence of inputs (0-GAS), and its solutions are bounded in response to any input with a suitable bounded-energy assumption.

INTRODUCTION

The input-to-state stability (ISS) notion has been introduced three decades ago in [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] to analyze dynamical systems affected by external disturbances. In few words, the ISS property imposes that, in the absence of disturbances, the dynamical system evolves properly (namely, its equilibrium is globally asymptotically stable: 0-GAS) and that, in the presence of perturbations, this nominal behavior is preserved up to a steady-state error proportional to the magnitude of the applied disturbance. Although the ISS framework was first introduced in a finitedimensional context, it has more recently been extended to time-delay systems (TDS), see for instance [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]; [START_REF] Karafyllis | Stability and stabilization of nonlinear systems[END_REF]; [START_REF] Mironchenko | Input-to-state stability of time-delay systems: criteria and open problems[END_REF].

An important and well-known relaxation of ISS is integral input-to-state stability (iISS). It was originally introduced in Sontag (1998) for finite-dimensional systems and extended to TDS in [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]. Instead of measuring the impact of the input magnitude on the steadystate behavior of the system, it rather focuses on its energy.

Beyond ensuring some robustness to exogenous disturbances, ISS and iISS constitute powerful tools to study the stability of interconnected systems. For feedback interconnected systems, small-gain arguments can be invoked both for simple loops and networks of interconnected systems [START_REF] Jiang | A Lyapunov formulation of nonlinear small gain theorem for interconnected systems[END_REF][START_REF] Dashkovskiy | Small gain theorems for large scale systems and construction of ISS Lyapunov functions[END_REF][START_REF] Ito | State-dependent scaling problems and stability of interconnected iISS and ISS systems[END_REF]. In the simpler case of cascade interconnections, in which the influence of one subsystem on the other is unidirectional, ISS is known to be naturally preserved [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF][START_REF] Chaillet | Strong iISS is preserved under cascade interconnection[END_REF]. On the contrary, cascades of iISS subsystems may not be iISS [START_REF] Arcak | A unifying integral ISS framework for stability of nonlinear cascades[END_REF]: additional growth-rate requirements are needed [START_REF] Angeli | A tight small gain theorem for not necessarily ISS systems[END_REF][START_REF] Chaillet | Integral input to state stable systems in cascade[END_REF][START_REF] Ito | A Lyapunov approach to cascade interconnection of Integral Input-to-State Stable systems[END_REF]. See also [START_REF] Panteley | Growth rate conditions for stability of cascaded time-varying systems[END_REF] and its TDS extension based on Razumikhin approach in [START_REF] Sedova | The global asymptotic stability and stabilization in nonlinear cascade systems with delay[END_REF].

So far, the vast majority of the existing stability results about cascades of iISS systems are restricted to finitedimensional systems. Cascades can be seen as a particular case of feedback interconnection, thus allowing the use iISS small-gain results for time-delay systems such as [START_REF] Ito | A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals[END_REF]. However, this approach often leads to overly conservative stability conditions. In this paper, we provide explicit growth-rate conditions under which the cascade of two iISS systems preserves interesting stability and robustness features, namely global asymptotic stability in the absence of exogenous inputs (0-GAS) and bounded solutions in response to a class of inputs with bounded energy, which constitute the two main practical features of the iISS property. These growth rate conditions are based on the Lyapunov-Krasovskii functionals (LKF) associated to the two individual subsystems, and require that the dissipation rate of the driving subsystem has greater growth near zero than the input rate of the driven subsystem.

An appealing feature of the proposed results is that they rely on LKF that dissipate in a point-wise manner, meaning solely in terms of the current value of the solution's norm. This differs from more stringent LKF conditions for iISS [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF][START_REF] Lin | Lyapunov descriptions of integral-input-to-state-stability for systems with delays[END_REF], which require a dissipation rate that involves the whole LKF itself. See [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems[END_REF][START_REF] Kankanamalage | On Lyapunov-Krasovskii Characterizations of Inputto-Output Stability[END_REF] for more discussion on this aspect.

After recalling background notions in Section 2, we state our main result in Section 3 with an immediate corollary for input-free cascades. Proofs are provided in Section 4 and an illustrative example is given in Section 5. We conclude in Section 6 with possible future work directions.

PRELIMINARIES

Notation

A function α : R ≥0 → R ≥0 is said to be of class PD if it is continuous, and satisfies α(0) = 0 and α(s) > 0 for all s > 0. α ∈ K if α ∈ PD and it is increasing. α ∈ K ∞ if α ∈ K and it is unbounded. α ∈ L if it is continuous non-increasing and tends to zero as its argument tends to infinity. A function β : R ≥0 × R ≥0 → R ≥0 is of class KL if β(•, t) ∈ K for each t ∈ R ≥0 and β(s, •) ∈ L for each s ∈ R ≥0 . Given q 1 , q 2 ∈ PD, we say that q 1 has greater growth than q 2 in a neighborhood of zero (and we write q 2 (s) = O s→0 + (q 1 (s))) if there exists a constant k ≥ 0 such that lim sup s→0 + q 2 (s)/q 1 (s) ≤ k. Given x ∈ R n , |x| denotes its Euclidean norm. Given δ ≥ 0, C denotes the set of all continuous functions ϕ : [-δ; 0] → R. Given any φ ∈ C n , φ := sup τ ∈[-δ,0] |φ(τ )|. We denote by U the set of all measurable essentially bounded signals u : R ≥0 → R.

Definitions

Consider the nonlinear TDS ẋ(t) = f (x t , u(t)),

(1) where u ∈ U m is the input, x(t) ∈ R n is the current value of the solution and x t ∈ C n is the state history defined with the maximum delay δ ≥ 0 as

x t (s) := x(t + s), ∀s ∈ [-δ, 0]. (2) f : C n × R m → R n is
assumed to be Lipschitz on bounded sets and to satisfy f (0, 0) = 0.

We recall the definition of iISS, originally introduced in a delay-free context [START_REF] Sontag | Comments on integral variants of ISS[END_REF]. Definition 1. (iISS, [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]). The system (1) is said to be integral input-to-state stable (iISS) if there exists β ∈ KL and ν, σ ∈ K ∞ such that, for all x 0 ∈ C n and all u ∈ U m , its solution satisfies

|x(t)| ≤ β( x 0 , t) + ν t 0 σ(|u(s)|)ds , ∀t ≥ 0. (3)
A first consequence of this property is forward completeness: since t → t 0 σ(|u(s)|)ds is bounded on any bounded time interval, (3) ensures that no finite escape-time can occur which, in view of (Hale, 1977, Theorem 3.2, p. 43), ensures that solutions of (1) exists at all positive times.

Another consequence of iISS is that, in the absence of inputs, the system is globally asymptotically stable. Definition 2. (0-GAS). The TDS (1) is said to be globally asymptotically stable in the absence of inputs (0-GAS) if there exists β ∈ KL such that, for all x 0 ∈ C n , the solution of the input-free system ẋ(t) = f (x t , 0) satisfies |x(t)| ≤ β( x 0 , t), ∀t ≥ 0. Remark 1. The concept defined above is usually called 0-UGAS in infinite-dimensional ISS literature (see for instance the survey Mironchenko and Prieur ( 2019)), whereas 0-GAS is often defined as the combination of stability and global attractivity. For finite-dimensional systems both properties are equivalent, but this is not the case for infinite-dimensional systems. Nevertheless, we decide to stick to the 0-GAS acronym (without explicitly stressing that convergence is uniform in the initial state) for the sake of homogeneity with the finite-dimensional terminology. iISS actually goes beyond the internal stability property of 0-GAS, by ensuring some robustness properties known as bounded energy-bounded state and bounded-energy converging state. Definition 3. (BEBS, BECS). The TDS ( 1) is said to have the bounded energy-bounded state (BEBS) property if there exists ζ ∈ K ∞ such that, for all x 0 ∈ C n and all u ∈ U m , its solution satisfies

∞ 0 ζ(|u(s)|)ds < ∞ ⇒ sup t≥0 |x(t)| < ∞. (4)
It is said to have the bounded energy-converging state (BECS) property if there exists ζ ∈ K ∞ such that, for all x 0 ∈ C n and all u ∈ U m , its solution satisfies

∞ 0 ζ(|u(s)|)ds < ∞ ⇒ lim t→∞ |x(t)| = 0. ( 5
)
The strength of iISS lies in the fact that it not only guarantees 0-GAS, but also BEBS and BECS. Proposition 1. (iISS ⇒ 0-GAS, BEBS, BECS). If the TDS (1) is iISS, then it is BEBS and BECS. In particular, if (3) holds, then ( 4) and ( 5) hold with ζ = σ.

A similar result was proven in (Sontag, 1998, Proposition 6) for finite-dimensional systems. The proof of Proposition 1 follows along the same lines and is therefore omitted. Remark 2. For finite-dimensional systems, iISS is actually equivalent to 0-GAS combined with the weaker requirement:

∞ 0 ζ(|u(s)|)ds < ∞ ⇒ lim inf t→∞ |x(t)| < ∞,
as proved in [START_REF] Angeli | Separation principles for input-output and integralinput-to-state stability[END_REF]. We are not aware of any extension of this fact to TDS.

Sufficient condition for iISS

Another strength of iISS is that it can be established using a Lyapunov-Krasovskii functional. We call a Lyapunov-Krasovskii functional (LKF) candidate any functional V :

C n → R ≥0 , Lipschitz on bounded sets, for which there exist α, α ∈ K ∞ such that α(|φ(0)|) ≤ V (φ) ≤ α( φ ), ∀φ ∈ C n .
(6) Its upper-right Dini derivative along the solutions of ( 1) is then defined for all t ≥ 0 as

D + V (x t , u(t)) := lim sup h→0 + V (x t+h ) -V (x t ) h . ( 7
)
Proposition 2. (iISS LKF, [START_REF] Lin | Lyapunov descriptions of integral-input-to-state-stability for systems with delays[END_REF]). The TDS (1) is iISS if and only if there exists a LKF candidate V : C n → R ≥0 , α ∈ PD and γ ∈ K ∞ , such that, given any x 0 ∈ C n and any u ∈ U m , the following holds for all t ≥ 0:

D + V (x t , u(t)) ≤ -α(V (x t )) + γ(|u(t)|). ( 8
)
This result is reminiscent of the iISS characterization obtained for finite-dimensional systems [START_REF] Angeli | A characterization of integral input-to-state stability[END_REF]. In what follows, we will actually allow for a weaker dissipation rate which proves very handy in practice, as illustrated through an example in Section 5.

CASCADE INTERCONNECTION

Consider two nonlinear TDS in cascade:

ẋ1 (t) = f 1 (x 1t , x 2 (t -δ 1 )) (9a) ẋ2 (t) = f 2 (x 2t , u(t)),
(9b) where x 1 (t) ∈ R n1 and x 2 (t) ∈ R n2 are the current values of the state, x 1t ∈ C n1 , x 2t ∈ C n2 are the state histories, and u ∈ U m is the input. The functions f 1 :

C n1 × R n2 → R n1 and f 2 : C n2 × R m → R n2
are assumed to be Lipschitz on bounded sets and satisfying f 1 (0, 0) = 0 and f 2 (0, 0) = 0. The interconnection between the two systems is assumed to be through a discrete delay

δ 1 ∈ [0, δ].
The main purpose of this study is to investigate under which condition some stability and robustness properties of the cascade ( 9) can be ensured based on the assumption that each of the subsystems (9a) and ( 9b) is iISS.

Our main result, proved in Section 4.2, is the following. Theorem 1. Assume that there exist two LKF candidates

V i : C ni → R ≥0 and η i ∈ K ∞ , i ∈ {1, 2}, such that, along any solution of ẋ1 (t) = f 1 (x 1t , u 1 (t)) with u 1 ∈ U n2 , D + V 1 (x 1t , u 1 (t)) ≤ - α 1 (|x 1 (t)|) 1 + η 1 (V 1 (x 1t )) + γ 1 (|u 1 (t)|) (10)
and, along any solution of (9b), 11) for all t ≥ 0. Assume further that (9b) is iISS 1 and γ 1 (s) = O s→0 + (α 2 (s)). ( 12) Then the cascade ( 9) is 0-GAS and satisfies the BEBS property.

D + V 2 (x 2t , u(t)) ≤ - α 2 (|x 2 (t)|) 1 + η 2 (V 2 (x 2t )) + γ 2 (|u(t)|). (
Note that the dissipation rates required on the LKFs are less conservative than the one in Proposition 2. Indeed, for a LKF candidate V , (6) holds with some α, α ∈ K ∞ . For α ∈ PD, (Angeli et al., 2000, Lemma IV.1) ensures the existence of σ ∈ K ∞ and ∈ L such that α(s) ≥ σ(s) (s) = (0)σ(s) (s) (0) . Pick any η ∈ K ∞ such that η(s) ≥ (0) (s) -1 (such a function exists as the right-hand side of this expression is continuous, zero at zero, and nondecreasing). Then we have that α(V (x t ))

≥ (0)σ(V (xt)) 1+η(V (xt)) ≥ (0)σ•α -1 (|x(t)|) 1+η(V (xt))
. Hence, starting from a dissipation rate α(V (x t )) as in Proposition 2, we recover a dissipation rate as in ( 10) and ( 11).

The growth rate condition (12) requires that the increasing part α 2 of the driving subsystem's dissipation rate has greater growth than the input rate γ 1 of the driven subsystem around the origin. This condition is reminiscent of the one obtained in [START_REF] Chaillet | Integral input to state stable systems in cascade[END_REF] for finitedimensional systems. Nonetheless, it is worth stressing that the main result in [START_REF] Chaillet | Integral input to state stable systems in cascade[END_REF] goes beyond Theorem 1 by ensuring iISS of the overall cascade whereas, here, only 0-GAS and BEBS (key features of iISS) were obtained. This is due to the fact that, as recalled in 1 In [START_REF] Chaillet | Integral input-to-state stability of delay systems based on Lyapunov-Krasovskii functionals with point-wise dissipation rate[END_REF], it was claimed that (11) implies that the driving subsystem is iISS, but we have recently found a flaw in the proof of that result that we were not able to correct yet. For safety, we therefore assume explicitly that (9b) is iISS, although this is probably redundant with (11).

Remark 2, the result presented in [START_REF] Angeli | Separation principles for input-output and integralinput-to-state stability[END_REF], which allows to recover iISS from 0-GAS plus (a relaxed version of) BEBS has not yet been extended to TDS.

It is worth mentioning that the small-gain results for iISS TDS provided in [START_REF] Ito | A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals[END_REF] can also be used to study cascade interconnections. However, the resulting requirement turns out to involve the upper and lower bounds on V 1 and V 2 , thus leading to a more conservative stability condition. More crucially, that result imposes that the dissipation rates α i in (8) for the driving and driven subsystems are of class K (rather than PD), meaning that both subsystems are required to have an ISS-like behavior for small inputs. In particular, the results in Ito et al. ( 2010) cannot be used for the example of Section 5.

A straightforward consequence of Theorem 1 is that, under the above growth-rate condition, the cascade composed of an iISS subsystem driven by a GAS one is itself GAS. To state this, consider the following input-free cascade:

ẋ1 (t) = f 1 (x 1t , x 2 (t -δ 1 )) (13a) ẋ2 (t) = f 2 (x 2t ), (13b)
under similar regularity assumptions as the ones on ( 9). Then we have the following.

Corollary 1. Assume that there exist two LKF candidates

V 1 : C n1 → R ≥0 and V 2 : C n2 → R ≥0 , α i , α i , η i ∈ K ∞ , α i ∈ K, i ∈ {1, 2}, and γ 1 ∈ K ∞ such that, along any solution of ẋ1 (t) = f 1 (x 1t , u 1 (t)) with u 1 ∈ U n2 , D + V 1 (x 1t , u 1 (t)) ≤ - α 1 (|x 1 (t)|) 1 + η 1 (V 1 (x 1t )) + γ 1 (|u 1 (t)|),
and, along any solution of (13b),

D + V 2 (x 2t ) ≤ - α 2 (|x 2 (t)|) 1 + η 1 (V 2 (x 2t ))
, for all t ≥ 0. Assume further that

γ 1 (s) = O s→0 + (α 2 (s)).
Then the cascade ( 13) is globally asymptotically stable.

This result complements existing delay-free results [START_REF] Panteley | Growth rate conditions for stability of cascaded time-varying systems[END_REF][START_REF] Arcak | A unifying integral ISS framework for stability of nonlinear cascades[END_REF][START_REF] Angeli | A tight small gain theorem for not necessarily ISS systems[END_REF][START_REF] Chaillet | Integral input to state stable systems in cascade[END_REF] as well as the Razumikhin approach developed in [START_REF] Sedova | The global asymptotic stability and stabilization in nonlinear cascade systems with delay[END_REF] for TDS.

The proof of Theorem 1 relies on the following change of dissipation rate result, proved in Section 4.1. Lemma 2. Let f : C n → R n be Lipschitz on bounded sets and V : C n → R ≥0 be a LKF candidate satisfying, along any solution of the TDS ẋ(t) = f (x t ),

D + V (x t ) ≤ - α(|x(t)|) 1 + η(V (x t )) , (14)
for some α ∈ PD and η ∈ K ∞ . Let α be any PD function satisfying

α(s) = O s→0 + (α(s)). ( 15
)
Then there exists a continuously differentiable function ρ ∈ K ∞ such that the functional Ṽ := ρ • V satisfies

D + Ṽ (x t ) ≤ -α(|x(t)|).

PROOFS

Proof of Lemma 2

First recall that, since V is a LKF candidate, there exist α, α ∈ K ∞ such that, for all φ ∈ C n , α(|φ(0)|) ≤ V (φ) ≤ α( φ ). ( 16) We will show that there exists a continuous non-decreasing function q : R ≥0 → R ≥0 satisfying q(s) > 0 for all s > 0 such that the requested function ρ can be written as ρ(s) = s 0 q(r)dr for all s ≥ 0. Note that, with this choice, ρ is a continuously differentiable K ∞ function and the function Ṽ = ρ • V is then Lipschitz on bounded sets. Moreover, in view of ( 16), Ṽ is a LKF candidate. Furthermore, its Dini derivative along the solutions of ẋ(t) = f (x t ) reads, in view of ( 14) and (Ito et al., 2010, Lemma 7),

D + Ṽ (x t ) ≤ -q(V (x t )) α(|x(t)|) 1 + η(V (x t ))
.

So we need to find a continuous non-decreasing function q satisfying q(s) > 0 for all s > 0 such that

q(V (x t )) α(|x(t)|) 1 + η(V (x t )) ≥ α(|x(t)|). ( 17
)
Let µ : R ≥0 → R ≥0 be defined as

µ(s) := sup r∈[0,s] α(r) α(r) , ∀s ≥ 0. ( 18
)
The function s → α(s)/α(s) is continuous on (0, +∞).

In addition, condition (15) ensures that it is bounded on any interval of the form [0, a], a ≥ 0. µ being clearly nondecreasing, it follows that it admits a limit at zero. Hence, σ is continuous and non-decreasing on R ≥0 . We claim that a possible choice of q to fulfill ( 17) is then q(s) := µ • α -1 (s) 1 + η(s) , ∀s ≥ 0. This function q is indeed a continuous non-decreasing function, positive out of zero. With this choice, and invoking ( 16) and ( 18), we have that

q(V (x t )) α(|x(t)|) 1 + η(V (x t )) ≥ σ • α -1 (V (x t ))α(|x(t)|) ≥ σ(|x(t)|)α(|x(t)|) ≥ α(|x(t)|), thus establishing (17).

Proof of Theorem 1 i) Forward completeness

We start by showing that solutions of the cascade ( 9) exist at all positive times. This is rather straightforward for the driving subsystem as it is assumed iISS (hence, forward complete). This in turn implies that x 2 (•) is continuous on R ≥-δ as the solution of a forward complete TDS. (10) then implies that, over the maximal domain of existence of x 1t ,

D + V 1 (x 1t , x 2 (t -δ 1 )) ≤ γ 1 (|x 2 (t -δ 1 )|)
. This ensures that x 1t is bounded over any finite time interval, thus impeding finite escape times. Forward completeness follows (see the remark after (3)).

ii) 0-GAS We next show that the cascade ( 9) is 0-GAS, meaning that the input-free system

ẋ1 (t) = f 1 (x 1t , x 2 (t -δ 1 )) (19a) ẋ2 (t) = f 2 (x 2t , 0) (19b
) is globally asymptotically stable. Note that, in view of (11), the LKF V 2 satisfies, along the solutions of (19b),

D + V 2 (x 2t ) ≤ - α 2 (|x 2 (t)|) 1 + η 2 (V 2 (x 2t ))
.

Observe that the growth rate condition (12) ensures that 2γ 1 (s) = O s→0 + (α 2 (s)). It follows from Lemma 2 that there exists a continuously differentiable function ρ ∈ K ∞ such that the functional

Ṽ2 := ρ • V 2 (20) satisfies D + Ṽ2 (x 2t ) ≤ -2γ 1 (|x 2 (t)|). ( 21
)
We next modify the LKF Ṽ2 in such a way that its dissipation rate 2γ 1 involves not only the current value of the driving state |x 2 (t)| but also its delayed value |x 2 (tδ 1 )|. To that aim, consider the LKF defined as

V 2 (φ 2 ) := Ṽ2 (φ 2 ) + 0 -δ1 γ 1 (|φ 2 (τ )|)dτ, ∀φ 2 ∈ C n2 . Since V 2 is a LKF candidate, it satisfies α 2 (|φ 2 (0)|) ≤ V 2 (φ 2 ) ≤ α 2 ( φ 2 ), ∀φ 2 ∈ C n2 . (22) for some α 2 , α 2 ∈ K ∞ . It follows from (20) that α2 (|φ 2 (0)|) ≤ V 2 (φ 2 ) ≤ α2 ( φ 2 ), (23)
where α2 :

= ρ • α 2 ∈ K ∞ and α2 := ρ • α 2 + δ 1 γ 1 ∈ K ∞ .
Moreover, along the solutions of ( 19), it holds that

V 2 (x 2t ) := Ṽ2 (x 2t ) + t t-δ1 γ 1 (|x 2 (τ )|)dτ.
In view of (21), its Dini derivative therefore reads

D + V 2 (x 2t ) ≤ D + Ṽ2 (x 2t ) + γ 1 (|x 2 (t)|) -γ 1 (|x 2 (t -δ 1 )|) ≤ -γ 1 (|x 2 (t)|) -γ 1 (|x 2 (t -δ 1 )|).
(24) Furthermore (10) ensures that

D + V 1 x 1t , x 2 (t-δ 1 ) ≤- α 1 (|x 1 (t)|) 1 + η 1 (V 1 (x 1t )) +γ 1 (|x 2 (t -δ 1 )|).
Summing this with (24), we get that

D + V(x t ) ≤ - α 1 (|x 1 (t)|) 1 + η 1 (V 1 (x 1t )) -γ 1 (|x 2 (t)|) ≤ - α 1 (|x 1 (t)|) + γ 1 (|x 2 (t)|) 1 + η 1 (V(x t )) , where V(x t ) := V 1 (x 1t ) + V 2 (x 2t ). The function z = (z T 1 , z T 2 ) T → α 1 (|z 1 |) + α 2 (|z 2 |)
being continuous, positive definite and non-vanishing as |z| → ∞, there exists α ∈ K such that α 1 (|z 1 |) + α 2 (|z 2 |) ≥ α(|z|): see (Khalil, 2002, Lemma 4.3). Therefore

D + V(x t ) ≤ - α(|x(t)|) 1 + η 1 (V(x t )) . ( 25
)
In addition, V 1 being a LKF candidate, there exist

α 1 , α 1 ∈ K ∞ such that α 1 (|φ 1 (0)|) ≤ V 1 (φ 1 ) ≤ α 1 ( φ 1 ), ∀φ 1 ∈ C n1 .
(26) Combining ( 26) and ( 23), it holds that

α 1 (|φ 1 (0)|) + α2 (|φ 2 (0)|) ≤ V(φ) ≤ α 1 ( φ ) + α2 ( φ ). The function z = (z T 1 , z T 2 ) T → α 1 (|z 1 |) + α2 (|z 2 |
) being continuous, positive definite and radially unbounded, (Khalil, 2002, Lemma 4.3)

ensures the existence of α ∈ K ∞ such that α 1 (|z 1 |) + α2 (|z 2 |) ≥ α(|z|). Letting α := α 1 + α2 ∈ K ∞ , it follows that α(|φ(0)|) ≤ V(φ) ≤ α( φ ).
(27) GAS of (19) then follows from ( 25) and ( 27) by invoking [START_REF] Chaillet | Integral input-to-state stability of delay systems based on Lyapunov-Krasovskii functionals with point-wise dissipation rate[END_REF], Proposition 1).

iii) BEBS

We finally proceed to establishing the BEBS property. Since the driving subsystem is iISS by assumption, there exists β 2 ∈ KL, ν 2 and σ 2 ∈ K ∞ such that, given any u ∈ U m and any

x 20 ∈ C n2 , |x 2 (t)| ≤ β 2 ( x 20 , t) + ν 2 t 0 σ 2 (|u(s)|)ds , ∀t ≥ 0.
(28) Assume that the following bounded energy holds:

∞ 0 max{γ 2 (|u(τ )|), σ 2 (|u(τ )|)}dτ ≤ c ( 29
)
for some c ≥ 0. Proposition 1 then ensures that lim t→∞ |x 2 (t)| = 0. Note that this ensures in particular the existence of a finite time T ≥ 0 (possibly depending on x 20 and u) such that x 2t ≤ 1, ∀t ≥ T, (30) which, in view of ( 22), guarantees that V 2 (x 2t ) ≤ α 2 (1), ∀t ≥ T.

(31) Consequently, by integrating the dissipation inequality (11) of V 2 , we have that, for all t ≥ T ,

V 2 (x 2t ) -V 2 (x 20 ) ≤ - t 0 α 2 (|x 2 (τ )|) 1 + η 2 (V 2 (x 2τ )) dτ + t 0 γ 2 (|u(τ )|)dτ ≤ - ∞ T α 2 (|x 2 (τ )|) η2 dτ + ∞ 0 γ 2 (|u(τ )|)dτ,
where η2 := 1 + η 2 • α 2 (1). Hence, from ( 22) and ( 29),

∞ T α 2 (|x 2 (τ )|)dτ ≤ (α 2 ( x 20 ) + c) η2 . (32)
From the growth rate condition (12) and the continuity of γ 1 and α 2 , there exists k > 0 such that γ

1 (s) ≤ kα 2 (s) for all s ∈ [0, 1]. Since (30) implies that |x 2 (t)| ≤ 1 for all t ≥ T , it follows that ∞ -δ1 γ 1 (|x 2 (τ )|)dτ = T -δ1 γ 1 (|x 2 (τ )|)dτ + ∞ T γ 1 (|x 2 (τ )|)dτ ≤ T -δ1 γ 1 (|x 2 (τ )|)dτ + ∞ T kα 2 (|x 2 (τ )|)dτ. Letting x( x 0 ) := k η2 (α 2 ( x 20 ) + c), we get from (32) ∞ -δ1 γ 1 (|x 2 (τ )|)dτ ≤ T -δ1 γ 1 (|x 2 (τ )|)dτ + c( x 0 ). (33)
Similarly, by integrating the dissipation inequality (10) with u 1 (t) = x 2 (t -δ 1 ) and using (26), we have that

α 1 (|x 1 (t)|) ≤ α 1 ( x 10 ) - t 0 α 1 (|x 1 (τ )|) 1 + η 1 (V 1 (x 1t )) dτ + t 0 γ 1 (|x 2 (τ -δ 1 )|)dτ ≤ α 1 ( x 10 ) + t-δ1 -δ1 γ 1 (|x 2 (τ )|)dτ.
In particular, it holds from (33) that

α 1 (|x 1 (t)|) ≤ α 1 ( x 10 ) + T 0 γ 1 (|x 2 (τ )|)dτ + c( x 0 ).
We conclude from this and (30) that, under the bounded energy assumption (29), the solutions of ( 9) are bounded, meaning that the cascade owns the BEBS property.

ILLUSTRATIVE EXAMPLE

We illustrate the applicability of our result through an academic example. Consider the following cascade TDS involving both discrete and distributed delays:

ẋ1 (t) = -sat(x 1 (t)) + 1 4 sat(x 1 (t -1)) + x 1 (t)x 2 (t -2) 2 (34a) ẋ2 (t) = - 3 2 x 2 (t) + x 2 (t -1) + u(t) t t-1 x 2 (τ )dτ, ( 34b
)
where sat(s) := sign(s) min{|s|, 1} for all s ∈ R. This system is in the form (9) with n 1 = n 2 = 1, m = 1, and δ 1 = δ = 2. Consider the LKF candidates defined as

V 1 (φ 1 ) := ln 1+φ 1 (0) 2 + 1 2 0 -1 φ 1 (τ )sat(φ 1 (τ ))dτ (35a) V 2 (φ 2 ) := ln 1 + φ 2 (0) 2 + 0 -1 φ 2 (τ ) 2 dτ (35b)
for all φ 1 , φ 2 ∈ C. The derivative of V 2 along the solutions of the driving subsystem (34b) reads

D + V 2 (x 2t , u(t)) = - 1 1 + x 2 (t) 2 + 0 -1 x 2 (t + τ ) 2 dτ × 2x 2 (t) - 3 2 x 2 (t) + x 2 (t -1) + u(t) t t-1 x 2 (τ )dτ + x 2 (t) 2 -x 2 (t -1) 2 . Observing that x 2 (t)x 2 (t -1) ≤ 1 2 (x 2 (t) 2 + x 2 (t -1) 2 ) and x 2 (t) t t-1 x 2 (τ )dτ ≤ 1 2 x 2 (t) 2 + t t-1 x 2 (τ )dτ 2 ≤ 1 2 x 2 (t) 2 + t t-1
x 2 (τ ) 2 dτ and defining η 2 (s) := e s -1, we get that

D + V 2 (x 2t , u(t)) ≤ - x 2 (t) 2 1 + η 2 (V 2 (x 2t )) + |u(t)|.
On the other hand, the derivative of V 1 along the solutions of the driven subsystem (34a) reads

D + V 1 (x 1t , x 2t ) = 1 1 + x 1 (t) 2 + 1 2 t t-1 x 1 (τ )sat(x 1 (τ ))dτ
× 2x 1 (t) -sat(x 1 (t))+ 1 4 sat(x 1 (t -1))+x 1 (t)x 2 (t -2) 2 + 1 2 x 1 (t) 2 -x 1 (t -1) 2 .

Observe that x 1 (t)sat(x 1 (t -1)) ≤ x 1 (t)sat(x 1 (t)) + x 1 (t -1)sat(x 1 (t -1)). Defining η 1 (s) := e s -1, we obtain that

D + V 1 (x 1t , x 2t ) ≤ - x 1 (t)sat(x 1 (t)) 1 + η 1 (V 1 (x 1t )) + 2x 2 (t -2) 2 .
In other words, the assumptions of Theorem 1 are fulfilled with α 1 (s) = sat(s)s, α 2 (s) = s 2 , η 1 (s) = η 2 (s) = e s -1, γ 1 (s) = 2s 2 and γ 2 (s) = s. With these rates, condition ( 12) is fulfilled, so we conclude that the cascade ( 35) is 0-GAS and owns the BEBS property.

CONCLUSION

We have provided conditions under which the cascade of two iISS TDS is 0-GAS and has the BEBS property. Like in the finite-dimensional case, these conditions take the form of growth restrictions on the input rate of the driven subsystem and the dissipation rate of the driving one. An academic example illustrates the applicability of the result.

A limitation of our main result lies in the way the two subsystems are interconnected: although a pure delay is allowed in our setup, we were not able to extend the result to a more generic interconnection (meaning for a driven subsystem of the form ẋ1 (t) = f 1 (x 1t , x 2t )). Another significant limitation is that we were not able to guarantee that the overall cascade is itself iISS. This is due to the fact that, contrarily to the finite-dimensional case, it has not yet been proved that 0-GAS combined with the BEBS property is enough to guarantee iISS for time-delay systems (see Remark 2). Future work will aim at solving these issues and at allowing the input u to impact directly the driven subsystem as well.

Gökhan

  Göksu is supported by The Scientific and Technological Research Council of Turkey (T ÜB İTAK) 2214-A (Application No: 1059B141801351).

ACKNOWLEDGEMENTS

The authors are thankful to Prof. Willie Wong and Prof. Yacine Chitour for fruitful discussions on technical aspects.


Antoine Chaillet
email: antoine.chaillet@centralesupelec.fr.

Jakub Orlowski
email: jakub.orlowski@l2s.centralesupelec.fr.

Pierdomenico Pepe
email: pierdomenico.pepe@univaq.it

Jakub Orłowski

A relaxed Lyapunov-Krasovskii

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

A powerful tool to study the stability of nonlinear timedelay systems is the Lyapunov-Krasovkii approach. This method requires to find a proper functional that decreases along the solutions of the system [START_REF] Krasovskii | Problems of the theory of stability of Motion[END_REF], [START_REF] Hale | Theory of functional differential equations[END_REF], [START_REF] Karafyllis | Stability and stabilization of nonlinear systems[END_REF], [START_REF] Niculescu | Delay effects on stability: A robust control approach[END_REF]. The key difference with the Lyapunov approach for delay-free systems is that the object of study is no longer a function, but rather a functional (as its argument is the whole state history over a bounded time interval).

In order to ensure asymptotic stability, the Lyapunov-Krasovskii approach requires that the derivative of the functional V dissipates in a point-wise manner, in the sense that its derivative along solutions needs to be negative as long as x(t) = 0. This light requirement turns out to be very handy in practice. However, when it comes to exponential stability, the only existing results require a dissipation rate that involves the whole functional itself (of the form V ≤ -εV , see [START_REF] Krasovskii | Problems of the theory of stability of Motion[END_REF], [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF], [START_REF] Yeganefar | Input-to-State Stability of time-delay systems: a link with exponential stability[END_REF]): we call this a strict dissipation. In a delay free context, strict and point-wise dissipations are the same, as the Lyapunov function is upper and lower bounded by the state norm. However, for time-delay systems, these two bounds often differ (the lower bound involves the point-wise value of the solution's norm, whereas the upper bound involves the supremum norm of the state history).

The question of what stability properties can be guaranteed with a point-wise dissipation was already the subject of [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems[END_REF] and [START_REF] Chaillet | Integral input-to-state stability of delay systems based on Lyapunov-Krasovskii functionals with point-wise dissipation rate[END_REF] for input-to-state stability properties and some questions remain open in that respect. Here, we address systems without input and show that a point-wise dissipation is indeed enough to show global exponential stability for systems with globally Lipschitz dynamics. Our proof is constructive: based on a functional with point-wise dissipation, we explicitly construct a functional that dissipates in terms of the whole state history norm. For globally Lipschitz dynamics, we show that point-wise, strict, and history-wise dissipations are the same from the point of view of global exponential stability.

We briefly describe how this result can be used to study robustness to exogenous inputs. We also illustrate its applicability with an example taken from neuroscience literature.

II. PROBLEM STATEMENT

We consider nonlinear time-delay systems of the form:

ẋ(t) = f (x t ). (1)
The state x t is defined as x t (s) := x(t+s) for all s ∈ [-δ; 0], where δ ≥ 0 denotes the maximum delay of the system. For each t ≥ 0, x t ∈ C n where C := C([-δ; 0], R), the set of continuous functions from [-δ; 0] to R. The vector field f : C n → R n is assumed to be Lipschitz on bounded sets (it will actually be assumed globally Lipschitz in most of the paper, but the introductory results below do not require this) and to satisfy f (0) = 0.

We make use of two different norms. Given a vector z ∈ R n , |z| denotes its Euclidean norm, namely:

|z| := n i=1 z 2
i . Given a function φ ∈ C n , φ := max s∈[-δ;0] |φ(s)|. We will also make use of comparison functions: α : R ≥0 → R ≥0 is of class K if it is zero at zero, continuous, and increasing; α ∈ K ∞ if α ∈ K and it is unbounded; β ∈ KL if, for each t ≥ 0, β(•, t) ∈ K and, for each s ≥ 0, β(s, •) is continuous, non-increasing, and tends to zero as its argument tends to infinity.

A sufficient condition for global asymptotic stability of (1) dates back to the fifties [START_REF] Krasovskii | Problems of the theory of stability of Motion[END_REF]. It makes use of Lyapunov-Krasovskii functionals (LKF), which have become a classical tool for the stability analysis of time-delay systems. It states that, if there exist a functional V : C n → R ≥0 and K ∞ functions α, α, and α such that, for all φ ∈ C n ,

α(|φ(0)|) ≤ V (φ) ≤ α( φ ) D + (1) V (φ) ≤ -α(|φ(0)|)
, then the origin of (1) is globally asymptotically stable. In this result, formally stated in e.g. [5, Theorem 2.1, p. 105], D +

(1) V denotes the upper-right Dini derivative of V along the solutions of (1), namely:

D + (1) V (φ) := lim sup h→0 + V (x h (φ)) -V (φ) h .
Several aspects are worth noticing. First, the upper and lower bounds on the LKF V differ: the upper bound involves the whole norm φ of the state history whereas the lower bound involves solely the current value of the solution's norm |φ(0)|. Using the terminology of e.g. [START_REF] Mironchenko | Global converse Lyapunov theorems for infinite-dimensional systems[END_REF], V does not need to be coercive (which would require a lower bound involving the whole φ ). This turns out to be useful in some applications. More crucially, the Dini derivative of V is required to dissipate only in terms of the current value of the solution's norm, rather than the LKF itself. This key property, which we name "point-wise dissipation", often simplifies the analysis, as will be illustrated in Section V. A particular form of global asymptotic stability is the following.

Definition 1 (GES) The origin of system (1) is globally exponentially stable (GES) if there exists η, γ > 0 such that, for each x 0 ∈ C n , its solution satisfies

|x(t; x 0 )| ≤ η x 0 e -γt , ∀t ≥ 0.
GES is a powerful stability property, as it ensures both an exponential decay to the origin and a transient overshoot proportional to the initial state norm x 0 . Moreover, under regularity conditions on the vector field, this property is known to ensure robustness to exogenous disturbances in an input-to-state stability (ISS) sense [START_REF] Yeganefar | Input-to-State Stability of time-delay systems: a link with exponential stability[END_REF]. LKF characterizations of exponential stability also exist for a long time. Nevertheless, quite surprisingly, none of them allow for a point-wise dissipation rate. In order to make this more precise, we introduce the following terminology.

Definition 2 (GES LKF) Let V : C n → R ≥0 be a functional, Lipschitz on bounded sets of C n , for which there exist k, k > 0 such that, for all φ ∈ C n ,

k|φ(0)| 2 ≤ V (φ) ≤ k φ 2 .
Then V is said to be:

• a GES LKF with history-wise dissipation for (1) if there exists k > 0 such that

D + (1) V (φ) ≤ -k φ 2 , ∀φ ∈ C n ,
• a strict GES LKF for (1) if there exists k > 0 such that

D + (1) V (φ) ≤ -kV (φ), ∀φ ∈ C n ,
• a GES LKF with point-wise dissipation for (1) if there exists k > 0 such that

D + (1) V (φ) ≤ -k|φ(0)| 2 , ∀φ ∈ C n .
These three types of GES LKF thus differ only in the way they dissipate along the solutions of (1). Clearly, any GES LKF with history-wise dissipation is a strict GES LKF, which is also a GES LKF with point-wise dissipation.

The fact that GES can be established with a strict LKF (hence, with an LFK with history-wise dissipation) is known [START_REF] Krasovskii | Problems of the theory of stability of Motion[END_REF], [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF], [START_REF] Yeganefar | Input-to-State Stability of time-delay systems: a link with exponential stability[END_REF]. However, obtaining a strict LKF on practical examples usually requires some tricks. For instance, for an LKF that involves only the sum of terms of the form |φ i (0)| 2 and 0 -δ |φ i (s)| 2 ds (as is often the case), we can add a term in the kernel of the integral (e.g. exponential) to get a dissipation involving the whole LKF [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF], [START_REF] Ito | A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals[END_REF], [START_REF] Mazenc | Construction of Lyapunov functionals for coupled differential and continuous time difference equations[END_REF].

Still, these tricks are not guaranteed to work with any LKF. Moreover, the use of GES LKF with point-wise dissipation would be much handier in practice and would homogenize GES theory with that for global asymptotic stability. Our main result establishes that these three notions of LKF are actually equivalent for systems defined by a globally Lipschitz vector field and ensure global exponential stability.

III. MAIN RESULT

Our main result, proved in Section IV, is the following.

Theorem 1 (GES characterizations) Let f : C n → R n be globally Lipschitz and satisfy f (0) = 0. Then the following statements are equivalent:

i) (1) admits a GES LKF with history-wise dissipation ii) (1) admits a strict GES LKF iii) (1) admits a GES LKF with point-wise dissipation iv) the origin of (1) is GES.

For globally Lipschitz systems, point-wise dissipation is thus sufficient to establish GES and the existence of a strict GES LKF, and even a GES LKF with history-wise dissipation, then come for free.

The above result can prove useful when studying robustness to exogenous disturbances. For this, the main tool is ISS, which was originally introduced for delay-free systems [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] and then extended to time-delay systems [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF], [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF], [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF]. It concerns systems of the form:

ẋ(t) = f (x t , u(t)), (2)
where f : C n × R m → R n is Lipschitz on bounded sets and satisfies f (0, 0) = 0. The input u is assumed to be in U m , the set of signals u : R ≥0 → R m that are measurable and locally bounded. We recall the definition of ISS below.

Definition 3 (ISS) The system (2) is input-to-state stable (ISS) if there exist β ∈ KL and µ ∈ K ∞ such that, for any x 0 ∈ C n and any u ∈ U m , its solution satisfies

|x(t; x 0 , u)| ≤ β( x 0 , t) + µ sup s∈[0;t] |u(s)| , ∀t ≥ 0.
We refer the reader to the survey [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] for more information about ISS. It was shown in [START_REF] Yeganefar | Input-to-State Stability of time-delay systems: a link with exponential stability[END_REF] that GES guarantees ISS under some regularity assumptions on f . Thus, the following is a direct consequence of Theorem 1.

Corollary 1 (GES & ISS)

Assume the input-free system ẋ(t) = f (x t , 0) admits a GES LKF with point-wise dissipation. Assume further that f ( • , 0) is globally Lispchitz and that there exists c > 0 and q ∈ [0; 1) such that, for all φ ∈ C n and all v ∈ R m , |f (φ, v) -f (φ, 0)| ≤ c max{ φ q ; 1}|v|.

(3

)
Then system (2) is ISS.

Proof: By Theorem 1, the origin of the input-free system ẋ(t) = f (x t , 0) is GES. ISS of (2) then follows from [START_REF] Yeganefar | Input-to-State Stability of time-delay systems: a link with exponential stability[END_REF]Theorem 3.2].

The regularity assumption (3) is fulfilled in particular if f is globally Lipschitz in both its arguments (in which case q = 0). This corollary slightly complements [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems[END_REF], which investigates whether a point-wise dissipation is enough to guarantee ISS (note that this question is still open).

IV. PROOF OF THEOREM 1

The implications i) ⇒ ii) and ii) ⇒ iii) are trivial. In [START_REF] Yeganefar | Input-to-State Stability of time-delay systems: a link with exponential stability[END_REF]Theorem 2.4], it is stated that iv) guarantees the existence of a globally Lipschitz functional V 0 :

C n → R ≥0 and κ, κ, κ > 0 such that, for all φ ∈ C n , κ φ ≤ V 0 (φ) ≤ κ φ and D + (1) V 0 (φ) ≤ -κ φ . Hence, the functional V := V 2 0 satisfies κ 2 φ 2 ≤ V (φ) ≤ κ 2 φ 2 and D + (1) V (φ) = 2V 0 (φ)D + (1) V 0 (φ) ≤ -κκ φ 2
. Thus, iv) implies i) (acually, with a coercive LKF). So it is sufficient to prove that iii) ⇒ iv), namely that the existence of a GES LKF with pointwise dissipation guarantees global exponential stability. To that aim, recall that iii) means that there exists a functional V : C n → R ≥0 , Lipschitz on bounded sets, and k, k, k > 0 such that, for all φ ∈ C n ,

k|φ(0)| 2 ≤ V (φ) ≤ k φ 2 (4) D + (1) V (φ) ≤ -k|φ(0)| 2 . (5)
Let

W (φ) := 0 -δ max s∈[τ ;0] |φ(s)| 2 dτ, ∀φ ∈ C n . (6)
The proof consists in showing that the functional

V(φ) := V (φ) + εW (φ),
with some ε > 0, has a history-wise dissipation rate along the solutions of (1). We start by showing the following.

Claim 1 Given any ε > 0, V is Lipschitz on bounded sets.

Proof: Since V is Lipschitz on bounded sets, it is sufficient to show that W enjoys the same property. First observe that, for all τ ∈ [-δ; 0] and for all φ, ψ ∈ C n ,

max s∈[τ ;0] |φ(s)| 2 -max s∈[τ ;0] |ψ(s)| 2 ≤ max s∈[τ ;0] |φ(s)| 2 -|ψ(s)| 2 . ( 7
)
To see this, consider first the case when max s∈

[τ ;0] |φ(s)| 2 ≥ max s∈[τ ;0] |ψ(s)| 2 and let s * ∈ [τ ; 0] be such that max s∈[τ ;0] |φ(s)| 2 = |φ(s * )| 2 . Then it holds that max s∈[τ ;0] |φ(s)| 2 -max s∈[τ ;0] |ψ(s)| 2 = |φ(s * )| 2 -max s∈[τ ;0] |ψ(s)| 2 ≤ |φ(s * )| 2 -|ψ(s * )| 2 ≤ max s∈[τ ;0] |φ(s)| 2 -|ψ(s)| 2 ≤ max s∈[τ ;0] |φ(s)| 2 -|ψ(s)| 2 .
The case when max s∈[τ ;0] |φ(s)| 2 ≤ max s∈[τ ;0] |ψ(s)| 2 can be addressed in the same way, which establishes [START_REF] Ito | A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals[END_REF]. Let X be any bounded set of C n and assume that φ, ψ ∈ X . Since the function x → x 2 is locally Lipschitz, there exists [START_REF] Ito | A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals[END_REF] that, for all τ ∈ [-δ; 0] and all φ, ψ ∈ X ,

> 0 such that |φ(s)| 2 -|ψ(s)| 2 ≤ |φ(s)| -|ψ(s)| ≤ |φ(s) -ψ(s)|. It follows from
max s∈[τ ;0] |φ(s)| 2 -max s∈[τ ;0] |ψ(s)| 2 ≤ max s∈[τ ;0] |φ(s) -ψ(s)| ≤ φ -ψ . (8)
Thus, in view of ( 6), it holds for all φ, ψ ∈ X that

|W (φ) -W (ψ)| = 0 -δ max s∈[τ ;0] |φ(s)| 2 dτ - 0 -δ max s∈[τ ;0] |ψ(s)| 2 dτ ≤ 0 -δ max s∈[τ ;0] |φ(s)| 2 -max s∈[τ ;0] |ψ(s)| 2 dτ ≤ δ φ -ψ ,
meaning that W is Lipschitz on bounded sets.

Using ( 4) and ( 6), we also have the following.

Claim 2 Given any ε > 0 it holds that

k|φ(0)| 2 ≤ V(φ) ≤ ( k + εδ) φ 2 , ∀φ ∈ C n .
We now proceed to computing the Dini derivative of V along the solutions of (1). To that aim, consider the solution t → x t of (1) starting from any x 0 ∈ C n at t = 0. Note that, since f is globally Lipschitz, this solution exists at all times t ≥ 0 and is unique [START_REF] Hale | Theory of functional differential equations[END_REF]. Hence, for all t ≥ 0, it holds that

W (x t ) = 0 -δ max s∈[τ ;0] |x(t + s)| 2 dτ.
Operating successively the changes of variables s ← t + s and τ ← τ + t, it follows that

W (x t ) = t t-δ max s∈[τ ;t] |x(s)| 2 dτ. (9)
The following statement constitutes the key idea of the paper: adding W to the GES LKF V with point-wise dissipation allows to obtain a history-wise dissipation term.

Claim 3 For all t ≥ 0, it holds that

D + (1) W (x t ) ≤ |x(t)| 2 -x t 2 + 2δ|x(t)| |f (x t )|.
Proof: If τ → max s∈[τ ;t] |x(s)| 2 were differentiable, then the derivative of W could be easily computed using Leibniz integral rule. Unfortunately, this may not be the case. This is why the proof of this claim is a bit technical. In order to lighten the notation, define:

m(τ, t) := max s∈[τ ;t] |x(s)| 2 , ∀t ≥ τ ≥ 0. ( 10
)
Then the function m satisfies the following:

P1 m(τ, t) ≥ 0 for all t ≥ τ ≥ 0 P2 τ → m(τ,
= m(t, t) -m(t -δ, t) = |x(t)| 2 -x t 2 ,
where the last equality comes from [START_REF] Krasovskii | Problems of the theory of stability of Motion[END_REF]. It follows that

D + (1) W (x t ) ≤ I(t) + |x(t)| 2 -x t 2 . ( 12
)
So all we need to show is that

I(t) ≤ 2δ|x(t)| |f (x t )|.
In view of [START_REF] Mazenc | Construction of Lyapunov functionals for coupled differential and continuous time difference equations[END_REF], I(t) can be written as

I(t) = lim sup h→0 + J(h, t) h , (13)
where

J
+ t+h t m(τ, t + h) -|x(t)| 2 dτ = t t-δ m(τ, t + h) -m(τ, t) dτ - t+h-δ t-δ m(τ, t + h) -m(τ, t) dτ + t+h t m(τ, t + h) -|x(t)| 2 dτ.
Observing that the second integral of this expression is nonnegative (due to P3), we get that

J(h, t) ≤ t t-δ m(τ, t + h) -m(τ, t) dτ + t+h t m(τ, t + h) -|x(t)| 2 dτ.
It follows from ( 13) that I(t) ≤ I 1 (t) + I 2 (t), where

I 1 (t) := lim sup h→0 + 1 h t t-δ m(τ, t + h) -m(τ, t) dτ I 2 (t) := lim sup h→0 + 1 h t+h t m(τ, t + h) -|x(t)| 2 dτ.
We start by computing I 2 (t): using P2, it holds that

I 2 (t) ≤ lim sup h→0 + 1 h t+h t m(t, t + h) -|x(t)| 2 dτ ≤ lim sup h→0 + m(t, t + h) -|x(t)| 2 = 0,
using [START_REF] Krasovskii | Problems of the theory of stability of Motion[END_REF] and the continuity of s → |x(s)|. Thus, I(t) ≤ I 1 (t). Moreover, using P4, it holds for all t ≥ τ that m(τ, t+ h) = max{m(τ, t); m(t, t + h)}. It follows that

I 1 (t) = lim sup h→0 + 1 h t t-δ m(τ, t + h) -m(τ, t) dτ = lim sup h→0 + 1 h t t-δ max{0; m(t, t + h) -m(τ, t)}dτ ≤ lim sup h→0 + δ h max{0; m(t, t + h) -|x(t)| 2 }, since m(τ, t) = max s∈[τ ;t] |x(s)| 2 ≥ |x(t)| 2 . Furthermore, for all s ≥ t, x(s) = x(t) + s t f (x θ )dθ. It follows that m(t, t + h) = max s∈[t;t+h] x(s) T x(s) = max s∈[t;t+h] |x(t)| 2 + 2x(t) T s t f (x θ )dθ + s t f (x θ )dθ 2 ≤ |x(t)| 2 + 2|x(t)| t+h t |f (x θ )|dθ + t+h t |f (x θ )|dθ 2 .
Recalling that m(t, t) = |x(t)| 2 , we obtain:

I 1 (t) ≤ lim sup h→0 + δ h   2|x(t)| t+h t |f (x θ )|dθ + t+h t |f (x θ )|dθ 2   .
Now, observe that (5) ensures in particular that D +

(1) V (x t ) ≤ 0. This, combined with (4), guarantees that |x(t)| ≤ k k x 0 , which in turn implies that x t ≤ k k x 0 . Since f is globally Lipschitz and f (0) = 0, there exists a positive constant f such that |f (x t )| ≤ f x 0 . Consequently:

I(t) ≤ I 1 (t) ≤ 2δ|x(t)| lim sup h→0 + 1 h t+h t |f (x θ )|dθ + lim sup h→0 + h 2 2 f x 0 2 h ≤ 2δ|x(t)| |f (x t )|.
The claim then follows from [START_REF] Mironchenko | Global converse Lyapunov theorems for infinite-dimensional systems[END_REF].

We can now conclude the proof of Theorem 1. Since |f (x t )| ≤ f x t for some f > 0, it follows from Claim 3 that, for all t ≥ 0,

D + (1) W (x t ) ≤ |x(t)| 2 -x t 2 + 2δ f |x(t)| x t . (14)
Now, recall that V(φ) := V (φ) + εW (φ), with ε > 0 to be chosen. ( 5) and ( 14) ensure that, for any λ > 0,

D + (1) V(x t ) ≤ -k|x(t)| 2 +ε|x(t)| 2 -ε x t 2 +2εδ f |x(t)| x t ≤ -(k -ε)|x(t)| 2 -ε x t 2 +εδ f λ|x(t)| 2 + x t 2 λ ≤ -(k -ε -εδ f λ)|x(t)| 2 -ε 1 - δ f λ x t 2 .
Pick λ = 2δ f , then

D + (1) V(x t ) ≤ -k -ε -2εδ 2 2 f |x(t)| 2 - ε 2 x t 2 .
Thus, by picking ε := k/(1 + 2δ 2 2 f ), we finally obtain that

D + (1) V(x t ) ≤ - ε 2 x t 2 , ∀t ≥ 0. (15)
From this, a strategy to conclude GES is to use the bounds on V from Claim 2 to show the exponential decay of V(x t ) with the help of a comparison lemma by taking into account that V is locally Lipschitz (thus the problem of the absolute continuity is overcome, see [START_REF] Pepe | The problem of the absolute continuity for Lyapunov-Krasovskii functionals[END_REF]), and that the function t → W (x t ) is locally absolutely continuous. An alternative is to directly integrate [START_REF] Pasillas-Lépine | Delay-induced oscillations in Wilson and Cowan's model: An analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects[END_REF] to get from Claim 2 that

∞ 0 x t 2 dt ≤ 2 ε V (x 0 ) ≤ 2 k ε x 0 2 , ∀x 0 ∈ C n ,
and GES follows from the integral criterion proposed in [START_REF] Ichikawa | Equivalence of Lp stability and exponential stability for a class of nonlinear semigroups[END_REF].

V. EXAMPLE

In [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF], a model of two interconnected neuronal populations was proposed to explain a possible mechanism for parkinsonian brain oscillations onset. This model reads:

τ 1 ẋ1 =-x 1 +S 1 c 11 x 1 (t -δ 11 )-c 12 x 2 (t-δ 12 )+µ (16a) τ 2 ẋ2 =-x 2 +S 2 c 21 x 1 (t -δ 21 ) -c 22 x 2 (t -δ 22 ) , (16b)
where the argument t of the non-delayed variables have been omitted. In this model, x 1 (t) and x 2 (t) represent the neuronal activity of populations 1 and 2. τ 1 , τ 2 > 0 are time constants. S 1 , S 2 : R → R are Lipschitz function (with no loss of generality, we assume their Lipschitz constant is 1). For each i, j ∈ {1, 2}, c ij ≥ 0 represents the synaptic strength from population j to population i and the delays δ ij ≥ 0 take into account the finite velocity of action potentials propagation from population j to population i. µ(t) ∈ R represents an artificial stimulation signal that can be introduced through implanted electrodes in order to disrupt pathological oscillations.

With appropriate parameter values, this system exhibits sustained pathological oscillations as a result of instability caused by strong synaptic connectivity between the two populations [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF], [START_REF] Pasillas-Lépine | Delay-induced oscillations in Wilson and Cowan's model: An analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects[END_REF]. This model has been studied in [START_REF] Haidar | A firing-rate regulation strategy for closed-loop deep brain stimulation[END_REF], using linearization around equilibrium, to show that a proportional feedback strategy from Population 1 to itself is able to locally stabilize the system. This was confirmed in a spatio-temporal extension of this model [START_REF] Chaillet | Robust stabilization of delayed neural fields with partial measurement and actuation[END_REF]. The following result states that this feedback globally exponentially stabilizes the system.

Proposition 1 For each i, j ∈ {1, 2}, let c ij , δ ij ≥ 0 and τ i > 0. Assume that functions S i are globally Lipschitz with Lipschitz constant 1, non-decreasing, and such that S i (0) = 0. Then there exists θ * ≥ 0 such that, for all θ ≥ θ * , the system (16) in closed loop with µ(t) = -θx 1 (t) is GES.

The proof consists in finding a GES LKF with point-wise dissipation. For this particular example, one could find a strict LKF (by putting exponential terms in the kernels of (18) below), but at the price of more involved computations. Note that if additive disturbances act in the dynamics [START_REF] Pepe | The problem of the absolute continuity for Lyapunov-Krasovskii functionals[END_REF], whether inside or outside the functions S i , ISS can easily be derived based on Corollary 1.

Proof: For any θ ≥ 0, the closed-loop dynamics reads:

τ 1 ẋ1 =-x 1 +S 1 c 11 x 1 (t-δ 11 )-c 12 x 2 (t -δ 12 )-θx 1 (17a) τ 2 ẋ2 =-x 2 +S 2 c 21 x 1 (t -δ 21 ) -c 22 x 2 (t -δ 22 ) . (17b) Let V (φ) = V 1 (φ 1 ) + ρV 2 (φ 2 ) where, for each j ∈ {1, 2}, V j (φ) = 1 2 τ j φ(0) 2 + 2 i=1 0 -δij λ ij φ(s) 2 ds , (18)
with ρ > 0 and λ ij > 0 to be chosen.

+ (17) V=-φ 1 (0) 2 -ρφ 2 (0) 2 +φ 1 (0)S 1 c 11 φ 1 (-δ 11 )-c 12 φ 2 (-δ 12 ) -θφ 1 (0) + ρφ 2 (0)S 2 c 21 φ 1 (-δ 21 ) -c 22 φ 2 (-δ 22 ) + 2 i=1 λ i1 2 φ 1 (0) 2 -φ 1 (-δ i1 ) 2 + 2 i=1 ρλ i2 2 φ 2 (0) 2 -φ 2 (-δ i2 ) 2 .
Since each S i is globally Lipschitz with Lipschitz constant 1, nondecreasing, and satisfies S i (0) = 0, it holds that |S i (s)| ≤ |s| and S i (s)s ≥ 0 for all s ∈ R. Based on this, the examination of two cases, whether or not the argument of S 1 dominates the control signal -θφ 1 (0),

shows that φ 1 (0)S 1 c 11 φ 1 (-δ 11 )-c 12 φ 2 (-δ 12 )-θφ 1 (0) ≤ 4 θ c 2 11 x 1 (t -δ 11 ) 2 + c 2 12 x 2 (t -δ 12 ) 2 . In the same way, φ 2 (0)S 2 (c 21 φ 1 (-δ 21 -c 22 φ 2 (-δ 22 )) ≤ c 21 2 ξ 1 φ 2 (0) 2 + φ 1 (-δ 21 ) 2 ξ 1 + c 22 2 ξ 2 φ 2 (0) 2 + φ 2 (-δ 22 ) 2 ξ 2 .
for any ξ 1 ; ξ 2 > 0. Putting it all together, it follows that

D + (17) V (φ) ≤ -φ 1 (0) 2 -ρφ 2 (0) 2 + 4 θ c 2 11 φ 1 (-δ 11 ) 2 + c 2 12 φ 2 (-δ 12 ) 2 + ρc 21 2 ξ 1 φ 2 (0) 2 + 1 ξ 1 φ 1 (-δ 21 ) 2 + ρc 22 2 ξ 2 φ 2 (0) 2 + 1 ξ 2 φ 2 (-δ 22 ) 2 + 1 2 2 i=1 λ i1 φ 1 (0) 2 -φ 1 (-δ i1 ) 2 +ρλ i2 φ 2 (0) 2 -φ 2 (-δ i2 ) 2 .
Regrouping terms, it follows that By Theorem 1, ( 17) is GES for these values.

D + (17) V (φ) ≤ -1-c22 2 |φ(0)| 2 ,

VI. CONCLUSIONS AND PERSPECTIVES

In this paper, we have shown that GES can be established with a functional that dissipates only point-wisely, if the dynamics are globally Lipschitz. We have quickly evoked some consequences in terms of robustness to exogenous inputs and illustrated its applicability through an example taken from neuroscience literature.

The restriction to globally Lipschitz dynamics is of course the main drawback of the approach: further work is needed to investigate whether the proposed LKF construction can be extended to merely locally Lipschitz systems. Also, the LKF considered here have quadratic bounds: deeper investigations are required to check whether bounds in the power p ≥ 1 can also be employed. The proof presented here suggests that this extension is feasible for p > 1, but linear bounds (p = 1) seem less straightforward. This is a pity as converse results for GES do make use of linear bounds [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF] and such an extension would probably allow to construct a globally Lipschitz LKF with history-wise dissipation (which would be expected in view of the strong regularity requirements imposed on the fector field). We also hope that the proposed LKF construction will be of some help to solve the conjecture posed in [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems[END_REF] about ISS based on point-wise dissipation.

θλ12 c 2 12 = 16c 2 12 θ( 1 -θ c 2 11 and λ 21 = ρc21 ξ1 = 2ρc 2 21 1 - 2 .+ 4c 2 21 c 2 12 ( 1 -

 121212121 = c22 ξ2 , (f ) is fulfilled and (c) reads c 21 ξ 1 + c 22 (ξ 2 + 1/ξ 2 ) + λ 12 < 2.For positive arguments, the function x → x + 1/x has a minimum for x = 1, so we set ξ 2 = 1. By settingλ 12 = 1-c22 2 and ξ 1 = λ12 c21 = 1-c22 2c21 ,(c) is fulfilled provided that c 22 < 1, which we have by assumption. The choice ρ = 8 c22) validates (d). Moreover, by letting λ 11 = 8 c22 , (e) and (b) are fulfilled and (a) reads 8 Replacing ρ by its value, possible choices of θ are: θ ≥ θ * := 8 c 2 11 c 22 ) 2 .

  V is Lipschitz on bounded sets. Moreover, it satisfies bounds of the form (4) by setting k = τ 1 +ρτ 2 +λ 11 δ 11 +λ 21 δ 21 +ρλ 12 δ 12 +ρλ 22 δ 22 and k = min{τ 1 , τ 2 } min{1, ρ}/2. Moreover, for all φ ∈ C 2 ,

	D

  provided that: (a) : λ 11 + λ 21 < 2, (b) : ρc21 ξ1 ≤ λ 21 , (c) : c 21 ξ 1 + c 22 ξ 2 + λ 12 + λ 22 < 2, (d) : 8

	(e) : 8 θ c 2 11 ≤ λ 11 ,	12 ≤ λ 12 , θρ c 2 (f ) : c22 ξ2 ≤ λ 22 .
	By setting λ 22

VII. ACKNOWLEDGMENT

The authors wish to thank M. Malisoff, P. Mason, and M. Sigalotti for fruitful discussions about this question.


Antoine Chaillet

Pierdomenico Pepe

Integral input-to-state stability of delay systems based on Lyapunov-Krasovskii functionals with point-wise dissipation rate

We show that a Lyapunov-Krasovskii functional whose dissipation rate involves solely the current instantaneous value of the state norm is enough to guarantee integral inputto-state stability (iISS). This result generalizes existing sufficient conditions for iISS, where the dissipation rate involves the whole Lyapunov-Krasovskii functional itself, and simplifies their applicability. Moreover, it provides a more natural bridge with the classical condition for global asymptotic stability of input-free systems. The proof strategy we employ relies on a novel characterization of global asymptotic stability, which may be of interest on its own.

I. INTRODUCTION

The input-to-state stability (ISS) framework, standard tool in the analysis and control of finite-dimensional systems [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF], [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], has more recently extended to infinite-dimensional dynamics, including time-delay systems. A handy tool to establish ISS of time-delay systems is to rely on Lyapunov-Krasovskii functionals (LKF) [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]. More precisely, just like in the finite-dimensional case, ISS is guaranteed provided that the derivative of such functional V along the system's solutions satisfies

V ≤ -α(V ) + γ(|u|), (1)
where α, γ are K ∞ functions and u is the system's input.

In this differential inequality, the dissipation rate α involves the whole LKF V . In [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems[END_REF], we have recently conjectured that ISS would hold also under the less stringent condition that

V ≤ -α(|x(t)|) + γ(|u|), (2)
meaning with a dissipation rate that involves only the current instantaneous value of the state norm (what we call a point-wise dissipation rate). Such a formulation would simplify the ISS analysis of time-delay systems. It would also homogenize ISS theory with the Lyapunov-Krasovskii theory of input-free systems, as global asymptotic stability of such systems is guaranteed under a point-wise dissipation rate, namely V ≤ -α(|x(t)|) [START_REF] Hale | Theory of functional differential equations[END_REF]. Although this simpler characterization of ISS has been established for particular classes of systems in [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems[END_REF], the conjecture has not yet been proved or disproved.

In the present paper, we prove this conjecture for a weaker robustness property, namely integral input-to-state stability (iISS). This property was introduced in [START_REF] Sontag | Comments on integral variants of ISS[END_REF] for finitedimensional systems. It imposes that solutions are bounded A. Chaillet is with Laboratoire des Signaux et Systèmes, CentraleSupélec, Univ. Paris Saclay, Gif sur Yvette, France and with Institut Universitaire de France antoine.chaillet@centralesupelec.fr

P.
Pepe is with University of L'Aquila, Italy pierdomenico.pepe@univaq.it by a decaying term of the initial state plus a term depending on the energy brought by the input. iISS of time-delay systems can be established in a similar way as ISS: it was shown in [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] that if a dissipation inequality like (1) holds with a positive definite α then iISS holds. A similar result was proposed in [START_REF] Mironchenko | Characterizations of integral input-tostate stability for bilinear systems in infinite dimensions[END_REF] in a wider infinite-dimensional context.

Here, we show that a point-wise dissipation rate like in (2) (with α merely positive definite) is enough to establish iISS. The proof of this result requires generalizations, that may be of interest on their own, of existing finite-dimensional results to time-delay systems. More precisely, we show that if a time-delay system is globally asymptotically stable in the absence of inputs (0-GAS), then a dissipation inequality like (1) holds (with α positive definite), but with a LKF V that may not be radially unbounded. We also show that 0-GAS implies iISS provided that the system is zero-output dissipative. These two properties were originally established in [START_REF] Angeli | A characterization of integral input to state stability[END_REF] for finite-dimensional systems. Finally, we show that iISS holds under the even less conservative assumption that

V ≤ -σ(|x(t)|, x t )+γ(|u|), where σ ∈ KL and γ ∈ K ∞ .
The outline of the paper is as follows. We start by presenting the class of systems considered here and by recalling iISS for time-delay systems and the existing LKF sufficient conditions to establish it. Then, in Section III, we state our main results. An academic example is provided in Section IV. All proofs are provided in Section V. We conclude with some perspective for future work.

Notations. Given x ∈ R n , |x| denotes its Euclidean norm. Given a set I ⊂ R and a measurable signal u : I → R m , u := ess sup t∈I |u(t)|. U denotes the set of all signals u : R ≥0 → R m that are measurable and locally essentially bounded. Given θ > 0 and n ∈ N ≥1 , the set

C([-θ; 0], R n ) of all continuous functions φ : [-θ; 0] → R n is denoted by X . A function α : R ≥0 → R ≥0 is said to be of class PD if is continuous and positive definite. α ∈ K if α ∈ PD and it is increasing. α ∈ K ∞ if α ∈ K and it is unbounded. A function β : R ≥0 × R ≥0 → R ≥0 is of class KL if β(•, t) ∈ K for each t ∈ R ≥0
and, for each s ∈ R ≥0 , β(s, •) is continuous, non-increasing and tends to zero as its argument tends to infinity.

II. CONTEXT AND DEFINITIONS

A. Time-delay systems

The paper considers nonlinear delayed systems of the form

ẋ(t) = f (x t , u(t)), ∀t ≥ 0 a.e. (3)
x(t) ∈ R n is the instantaneous value of the state at time t ≥ 0. x t ∈ X denotes its history over the time interval [t -θ; t], where θ is larger than, or equal to, the maximum delay involved in the dynamics. On its domain of existence, x t ∈ X is defined as

x t : [-θ; 0] → R n s → x(t + s).
The input u is assumed to be in the set U. The function f : X × R m → R n is assumed to be Lipschitz on any bounded subset of X ×R m → R n . These assumptions ensure that, given any x 0 ∈ X , system (3) admits a unique and locally absolutely continuous solution on a maximal time interval [0, b), b ∈ (0; +∞]. Moreover, if b < +∞, then the solution is unbounded on [0, b). See [START_REF] Hale | Theory of functional differential equations[END_REF]. We also assume that f (0, 0) = 0.

B. iISS

iISS of such time-delay systems was introduced in [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]. It is a natural extension the iISS property that was originally introduced in a finite-dimensional context [START_REF] Sontag | Comments on integral variants of ISS[END_REF].

Definition 1 (iISS, [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]) The delay system (3) is integral input-to-state stable (iISS) if there exist β ∈ KL and δ 1 , δ 2 ∈ K ∞ such that, given any x 0 ∈ X and any u ∈ U, the corresponding solution x(•)

:= x( • ; x 0 , u) satisfies |x(t)| ≤ β( x 0 , t) + δ 1 t 0 δ 2 (|u(τ )|)dτ , ∀t ≥ 0.
Note iISS imposes that the system is forward complete for each u ∈ U, meaning that x t exists for all t ≥ 0, as the above state estimate impedes any finite escape time. Moreover, just like its finite-dimensional counterpart, iISS guarantees not only that the origin of the input-free system ẋ(t) = f (x t , 0) is globally asymptotically stable (GAS), but also induces some robustness with respect to the disturbance u. In particular, it can be seen that, in response to any input with finite energy (as measured through the function δ 2 ), the state eventually converges to the origin:

∞ 0 δ 2 (|u(τ )|)dτ < ∞ ⇒ lim t→∞ x t = 0.

C. iISS functionals

We consider two kinds of iISS Lyapunov-Krasovskii functionals: those whose dissipation rate involves the functional itself (here referred to as "strict iISS LKF") and those whose dissipation rate involves merely the present value of the state norm ("relaxed iISS LKF").

We start by recalling Driver's derivative [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF]. Given any continuous function V : X → R, its Driver's derivative along (3) is defined, for all φ ∈ X and all v ∈ R m , as

D + (3) V (φ, v) := lim sup h→0 + V (φ * h,v ) -V (φ) h , (4)
where, for all h ∈ (0; θ) and all v ∈ R m , φ * h,v ∈ X is defined as

φ * h,v (s) := φ(s + h) if s ∈ [-θ; -h) φ(0) + f (φ, v)(s + h) if s ∈ [-h; 0]. (5)
It was shown in [START_REF] Pepe | On Liapunov-Krasovskii functionals under Caratheodory conditions[END_REF] that, if V is Lipschitz on any bounded set of X , then, under the assumptions made above on the vector field f , Driver's derivative of V computed at (x t , u(t)) coincides almost everywhere (on the domain of existence of x t ) with the upper-right Dini derivative of t → V (x t ):

D + (3) V (x t , u(t)) = lim sup h→0 + V (x t+h ) -V (x t ) h , ∀t ≥ 0 a.e.
Definition 2 (Strict/Relaxed iISS LKF) Let V : X → R ≥0 be Lipschitz on any bounded subset of X . Assume there exists α, α ∈ K ∞ such that

α(|φ(0)|) ≤ V (φ) ≤ α( φ ), ∀φ ∈ X . ( 6
)
Then V is said to be a strict iISS Lyapunov-Krasovskii functional (LKF) for (3) if there exist α ∈ PD and γ ∈ K ∞ such that, for all φ ∈ X and all v ∈ R m ,

D + (3) V (φ, v) ≤ -α(V (φ)) + γ(|v|). (7)
It is called a relaxed iISS LKF for (3) if it satisfies (6) and, for all φ ∈ X and all v ∈ R n ,

D + (3) V (φ, v) ≤ -α(|φ(0)|) + γ(|v|). (8)
In both cases, α and γ are respectively referred to as a dissipation rate and a supply rate.

The key difference between these two iISS LKF lies in the nature of the dissipation: a strict iISS LKF dissipates in terms of the whole LKF itself, whereas a relaxed iISS LKF dissipates only in terms of the current instantaneous value of the state norm. In view of (6), any strict iISS LKF is also a relaxed iISS. Note that this distinction between relaxed and strict iISS LKF is irrelevant in a finite-dimensional context.

D. Problem statement

It is known from [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] that, if the system (3) admits a strict iISS LKF, then it is iISS. That result extended the Lyapunov sufficient condition for iISS of finite-dimensional systems. For non-delayed systems, this Lyapunov condition turns out to be also necessary for iISS 1 [START_REF] Angeli | A characterization of integral input to state stability[END_REF].

The sufficient condition for iISS in [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] provides a handy way to establish iISS of time-delay systems. However, in order to obtain a dissipation rate that involves the whole LKF, some mathematical tricks are often needed that complicate the analysis. More crucially, such a strict LKF is not required in the absence of inputs. To make this more precise recall that global asymptotic stability of time-delay systems is defined as follows.

Definition 3 (GAS) The input-free system

ẋ(t) = f (x t ) (9)
is said to be globally asymptotically stable (GAS) if there exists β ∈ KL such that, for all x 0 ∈ X , its solution satisfies

|x(t)| ≤ β( x 0 , t), ∀t ≥ 0. (10)
The classical Lyapunov-Krasovskii condition, recalled in [4, Theorem 2.1, p.105], states that global asymptotic stability (GAS) of the input-free system (9) is guaranteed if there exists a functional V : X → R ≥0 , Lipschitz on all bounded subsets of X , such that, for all φ ∈ X ,

α(|φ(0)|) ≤ V (φ) ≤ α( φ ) (11) D + (9) V (φ) ≤ -α(|φ(0)|), (12)
where α, α ∈ K ∞ , and α ∈ PD.

It is clear from ( 12) that a point-wise dissipation rate is enough to show GAS. In order to simplify the analysis and to homogenize theory with input-free systems, we here address the question whether a point-wise dissipation rate (namely, a relaxed iISS LKF) is also enough to establish iISS.

MAIN RESUTS

A. A relaxed condition for GAS

We start by observing that GAS can be established by a weaker requirement than ( 11)- [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF].

Proposition 1 (GAS characterization) The system (9) is GAS if and only if there exist a functional V : X → R ≥0 , Lipschitz on all bounded subsets of X , K ∞ functions α and α, and a KL function σ such that, for all φ ∈ X ,

α(|φ(0)|) ≤ V (φ) ≤ α( φ ) (13) D + (9) V (φ) ≤ -σ(|φ(0)|, φ ). (14)
The proof lines of this result are provided in Section V-A. As we will see through an example in Section IV, this characterization may be handy for instance in situations where the employed LKF is of the form V = ln(1 + Ṽ ), where Ṽ denotes another LKF.

B. iISS by point-wise dissipation

Our main result establishes that a point-wise dissipation rate is enough to guarantee iISS.

Theorem 1 (Relaxed iISS LKF ⇒ iISS) If the system (3) admits a relaxed iISS LKF, then it is iISS and admits a strict iISS LKF.

As detailed in Section V-E, this result is actually a corollary of the following more general statement.

Theorem 2 (Sufficient condition for iISS) Assume that there exist a functional V : X → R ≥0 , Lipschitz on each bounded subset of X , α, α ∈ K ∞ , and σ ∈ KL such that, for all φ ∈ X and all v ∈ R m ,

α(|φ(0)|) ≤ V (φ) ≤ α( φ ) (15) D + (3) V (φ, v) ≤ -σ(|φ(0)|, φ ) + γ(|v|). ( 16
)
Then the system (3) is iISS and admits a strict iISS LKF.

The proof is provided in Section V-D. It is made of two steps. The first one characterizes the internal stability of the system in terms of a semi-proper iISS LKF. The second one shows that this characterization combined with a dissipativity condition is enough to establish iISS. Each of these two steps lead to results that may be of interest on their own: we treat them in the following two sections.

C. 0-GAS characterization

Recall that the system ẋ(t) = f (x t , u(t)) is said to be 0-GAS if the input-free system ẋ(t) = f (x t , 0) is GAS. The following result establishes that 0-GAS is equivalent to the existence of a semi-proper iISS LKF. The finite-dimensional counterpart of this result was originally established in [1, Proposition II.5].

Proposition 2 (0-GAS characterization) The delay system (3) is 0-GAS if and only if there exist a functional V : X → R ≥0 , Lipschitz on each bounded subset of X , a continuously differentiable class K function π satisfying π (s) > 0 for all s ∈ R ≥0 , α, α, γ ∈ K ∞ , and α ∈ PD such that

α( φ ) ≤ V (φ) ≤ α( φ ), ∀φ ∈ X , (17)
and the functional W := π • V satisfies, for all φ ∈ X and all v ∈ R m ,

D + (3) W (φ, v) ≤ -α( φ ) + γ(|v|). (18)
The proof of this result is provided in Section V-B. It is worth stressing that, in the above statement, W might not be radially unbounded (as π might not be a K ∞ function), which is why (18) does not readily guarantee iISS (otherwise any 0-GAS system would be iISS, which is not true even in a finite-dimensional context [START_REF] Angeli | A characterization of integral input to state stability[END_REF]).

It is also important to notice that the dissipation rate in (18) involves the supremum norm of the state history and that V is both upper and lower bounded by functions of φ . In particular, using the terminology of e.g. [START_REF] Mironchenko | Global converse Lyapunov theorems for infinite-dimensional systems[END_REF], V is a coercive LKF. These two features are instrumental in establishing Theorem 1.

D. Link with dissipativity

It turns out that Theorems 1 and 2 are direct consequences of the result presented next, which establishes iISS based on the internal stability of the system and a dissipativity property with respect to the input. Definition 4 (Zero-output dissipativity) The system (3) is said to be zero-output dissipative if there exist a functional V : X → R ≥0 , Lipschitz on each bounded subset of X , and class K ∞ functions α, α, and µ such that, for all φ ∈ X and all v ∈ R m ,

α(|φ(0)|) ≤ V (φ) ≤ α( φ ) D + (3) V (φ,v) ≤ µ(|v|).
This property is a natural extension of its finitedimensional counterpart, originally introduced in [START_REF] Angeli | A characterization of integral input to state stability[END_REF]. It imposes that V grows at most linearly in time for any bounded input u. We underline the fact that, in the above definition, we do not impose coerciveness on V : it just needs to be lower bounded by a function of |φ(0)| rather than the whole supremum norm of the state history φ . Here again, this allows to make use of standard LKF employed in stability analysis of autonomous systems.

It was shown in [START_REF] Angeli | A characterization of integral input to state stability[END_REF] that, for finite-dimensional systems, iISS is guaranteed by zero-output dissipativity, provided that the system is 0-GAS 2 . The following result extends this to time-delay systems.

Theorem 3 (0-GAS + dissipativity ⇒ iISS) If the system (3) is 0-GAS and zero-output dissipative, then it is iISS and admits a strict iISS LKF.

The proof is provided in Section V-C. As compared to Theorem 1, this result allows to conclude iISS for systems whose internal stability is established by other means than the construction of a Lyapunov-Krasovskii functional, such as through Krasovskii-LaSalle invariance principle arguments [4, Theorem 3.1, p. 119].

IV. EXAMPLE We provide an academic example in order to illustrate our findings. Consider the scalar system

ẋ(t) = -x(t) + c x(t + θ) + x(t)u(t), (19)
with c ≥ 0. In the absence of an input, the system is linear:

ẋ(t) = -x(t) + c x(t + θ) (20)
and its stability can be assessed by the following LKF:

W (φ) = φ(0) 2 + c 0 -θ φ(τ ) 2 dτ,
which can be bounded as in [START_REF] Karafyllis | Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations[END_REF]:

|φ(0)| 2 ≤ W (φ) ≤ (1 + c θ) φ 2 .
Moreover, using the fact that ab ≤ (a 2 + b 2 )/2 for all a, b ∈ R, it holds that

D + (20) W (φ, 0) = 2φ(0) -φ(0) + c φ(-θ) + c φ(0) 2 -φ(-θ) 2 ≤ -2(1 -c)φ(0) 2 .
If c ∈ [0; 1), 0-GAS of (19) follows from [4, Theorem 2.1, p.105]. However, this LKF is not a strict iISS LKF, as it dissipates in a point-wise manner, and therefore cannot be used to invoke existing iISS tools such as [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]. Nevertheless, it can easily be employed to construct a relaxed iISS LKF. Indeed, in the presence of an input, it holds that

D + (19) W (φ, v) ≤ -2(1 -c)φ(0) 2 + φ(0) 2 |v|. Consequently, the functional V := ln(1 + W ) satisfies D + (19) V (φ, v) = D + (19) W (φ, v) 1 + W (φ) ≤ -2(1 -c) φ(0) 2 1 + (1 + c) φ 2 + |v|.
2 iISS is actually equivalent to 0-GAS plus zero-output dissipativity for delay-free systems [START_REF] Angeli | A characterization of integral input to state stability[END_REF].

Theorem 2 then ensures that (19) is iISS and admits a strict (hence, a relaxed) iISS LKF.

V. PROOFS A. Proof of Proposition 1 (sketch)

The necessity part of this statement follows readily from converse LKF results such as [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF]Lemma A.8] 3 . So we now focus on showing that ( 13)-( 14) implies GAS. First observe that stability and boundedness readily hold since D + (9) V (φ) ≤ 0, which ensures from (13) that, for all x 0 ∈ X ,

|x(t)| ≤ α -1 • α( x 0 ), ∀t ≥ 0. (21)
In view of Lemma 2.4 and Corollary 2.6 in [START_REF] Karafyllis | Lyapunov theorems for systems described by retarded functional differential equations[END_REF], GAS as defined in terms of a KL estimate (Definition 3) follows if we manage to show that, for all ε > 0 and all R ≥ 0, there exists a time τ ≥ 0 such that x t ≤ ε for all t ≥ τ provided that x 0 ≤ R (this is the formulation of Property P3 in [START_REF] Karafyllis | Lyapunov theorems for systems described by retarded functional differential equations[END_REF] for autonomous disturbance-free systems). This attractiveness property can be established following the lines of the proof for GAS using Lyapunov-Krasovskii approach: see [4, Theorem 2.1, p. 105]. We do not include details here due to space limitations.

B. Proof of Proposition 2

We start by showing that (17)-( 18) guarantee 0-GAS. This is done by showing that V is a proper LKF for the input-free system

ẋ(t) = f (x t , 0). (22)
In view of ( 4) and (18), it holds that

D + (22) W (φ) = lim sup h→0 + W (φ * h ) -W (φ) h = lim sup h→0 + π • V (φ * h ) -π • V (φ) h ≤ -α( φ ).
This shows in particular that V (φ * h ) ≤ V (φ) for h small enough. Since π is continuously differentiable, the mean value theorem guarantees the existence of c(φ, h) ∈

[V (φ * h ); V (φ)] such that π • V (φ * h ) -π • V (φ) = π c(φ, h) V (φ * h ) -V (φ) . In view of (5), lim h→0 + V (φ * h ) = V (φ). Consequently, lim h→0 + π c(φ, h) = π • V (φ). It follows that D + (22) W (φ) = π • V (φ) lim sup h→0 + V (φ * h ) -V (φ) h ≤ -α( φ ).
Recalling that π (s) > 0 for all s ∈ R ≥0 , we conclude that

D + (22) V (φ) = lim sup h→0 + V (φ * h ) -V (φ) h ≤ - α( φ ) π • V (φ) ≤ - α(|φ(0)|) π • α( φ )
for all φ ∈ X , where we used (17). Let ξ(s) := 1 + s + max r∈[0;s] π • α(r) for all s ∈ R ≥0 . Then ξ is a positive continuous unbounded function satisfying ξ(•) ≥ π • α(•).

It follows that (r, s) → α(r)/ξ(s) is a KL function and

D + (22) V (φ) ≤ - α(|φ(0)|) ξ( φ ) .
Unlike W , V is radially unbounded (the functions α and α in (17) are of class K ∞ ). GAS of (22) (hence, 0-GAS of (3)) then follows from Proposition 1.

We now proceed to establish the converse. Assume the delay system (3) is 0-GAS, meaning that the input-free system (22) is GAS. Then, it holds from [START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF]Lemma A.8] that there exists a functional V : X → R ≥0 , Lipschitz on each bounded subset of X such that, for all φ ∈ X ,

α( φ ) ≤ V (φ) ≤ α( φ ) (23) D + (22) V (φ) ≤ -η( φ ). (24)
for some α, α, η ∈ K ∞ . Proceeding as in the proof of the main result in [START_REF] Yeganefar | Input-to-state stability of time-delay systems: a Link with exponential stability[END_REF], the derivative of this functional along the solutions of the perturbed system (3) reads

D + (3) V (φ, v) = lim sup h→0 + 1 h V (φ * h,v ) -V (φ) = lim sup h→0 + 1 h V (φ * h,0 ) -V (φ) + V (φ * h,v ) -V (φ * h,0 ) ≤ D + (22) V (φ) + lim sup h→0 + 1 h V (φ * h,v ) -V (φ * h,0 ) ≤ -η( φ ) + lim sup h→0 + 1 h V (φ * h,v ) -V (φ * h,0 ) . ( 25
)
As observed in e.g. [START_REF] Karafyllis | Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations[END_REF], since f and V are Lipschitz on bounded sets, there exist continuous nondecreasing functions V , f : R ≥0 → R ≥0 such that, for all φ, ϕ ∈ X and all

u, v ∈ R m , |V (φ) -V (ϕ)| ≤ V ( φ + ϕ ) φ -ϕ (26) |f (φ, v) -f (ϕ, u)| ≤ f ( φ + ϕ + |u| + |v|) × ( φ -ϕ + |v -u|). (27)
Note that f and V can be chosen greater than 1 with no loss of generality. Based on this, we have the following:

V (φ * h,v )-V (φ * h,0 ) ≤ V φ * h,v + φ * h,0 φ * h,v -φ * h,0 .
In view of (5), φ * h,v ≤ φ + |f (φ, v)|h. Consequently, from (27) and the fact that f (0, 0) = 0, it holds that φ

* h,v ≤ φ + f ( φ + |v|)( φ + |v|)h. It follows that V (φ * h,v ) -V (φ * h,0 ) ≤ (28) V 2 φ + 2 f ( φ + |v|)( φ + |v|)h φ * h,v -φ * h,0 .
In view of [START_REF] Karafyllis | Lyapunov theorems for systems described by retarded functional differential equations[END_REF] Plugging this into (25), we obtain that

D + (3) V (φ, v) ≤ -η( φ ) + V (2 φ ) f (2 φ + |v|)|v|.
Since f is nondecreasing, it holds that f (a+b) ≤ f (2a)+ f (2b) for all a, b ∈ R ≥0 . Consequently, recalling that f was chosen not smaller than 1,

D + (3) V ≤ -η( φ ) + V (2 φ ) f (4 φ ) + f (2|v|) |v| ≤ -η( φ ) + V (2 φ ) f (4 φ ) + 1 f (2|v|)|v|.
For future reference, we summarize these findings in the following statement 4 , which is an extension of [1, Lemma IV.10] to time-delay systems.

Proposition 3 (0-GAS characterization) The system (3) is 0-GAS if and only if there exist a functional V : X → R ≥0 , Lispchitz on each bounded subset of X , a nondecreasing continuous function : R ≥0 → R ≥0 , and α, α, η, γ ∈ K ∞ such that, for all φ ∈ X and all v ∈ R m , α( φ ) ≤ V (φ) ≤ α( φ ) ,

D + (
where α ∈ K ∞ is the lower bound on V (see ( 23)). Note that π (s) > 0 for all s ∈ R ≥0 . Letting W := π • V , we get

D + (3) W (φ, v) = π (V (φ))D + (3) V (φ, v) ≤ -η( φ ) + ( φ )γ(|v|) 1 + • α -1 (V (φ)) ≤ -α( φ ) + γ(|v|),
where the functions α is defined as α(s) := η(s)

1 + • α -1 • α(s)
.

The conclusion follows since α ∈ PD and γ ∈ K ∞ .

3 ) 0 dr 1 +

 301 V (φ, v) ≤ -η( φ ) + ( φ )γ(|v|).Now, consider the continuously differentiable class K function defined asπ(s) := s • α -1 (r)

  (2 φ + |v|)|v|(h + s) (29) ≤ f (2 φ + |v|)|v|h. ) ≤ V (2 φ ) f (2 φ +|v|)|v|.

	Using again (27), we get that
	φ * h,v -φ * h,0 ≤ sup
			s∈[-h;0]
			(30)
	It then follows from (28) that
	lim sup h→0 +	1 h	V (φ * h,v )-V (φ * h,0

, φ * h,v and φ * h,0 coincide on [-θ; -h). Hence,

φ * h,v -φ * h,0 = sup s∈[-h;0] φ * h,v (s) -φ * h,0 (s) ≤ sup s∈[-h;0] |f (φ, v) -f (φ, 0)| (h + s).

f

We are not aware of any extension of such converse results for timedelay systems, except for particular classes of systems[START_REF] Mironchenko | Characterizations of integral input-tostate stability for bilinear systems in infinite dimensions[END_REF].

[START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF] deals with neutral delay systems, but includes systems like (22) as special cases.

C. Proof of Theorem 3

The zero-output dissipativity assumption means that there exist a functional V : X → R ≥0 , Lipschitz on all bounded subsets of X , and class K ∞ functions α, α, and µ such that, for all φ ∈ X and all v ∈ R m ,

Moreover, since the system is assumed to be 0-GAS, Proposition 2 guarantees that there exist η ∈ PD, η, η ∈ K, γ ∈ K ∞ , and a functional W : X → R ≥0 , Lipschitz on all bounded subsets of X , such that, for all φ ∈ X and all v ∈ R m ,

Let Ṽ := V + W . Then, Ṽ is Lipschitz on bounded sets of X and we get from (31) and (33) that

where α := η + α. Moreover, (32) and (34) give that

Observing that α, α, and γ + µ are K ∞ functions and that η • α-1 ∈ PD, we conclude from ( 35)-( 36) that Ṽ is a strict iISS LKF for (3) and iISS follows from [10, Theorem 3.7].

D. Proof of Theorem 2

By assumption, there exist a functional V : X → R ≥0 , Lipschitz on all bounded subsets of X , α, α, γ ∈ K ∞ , and σ ∈ KL such that, for all φ ∈ X and all v ∈ R m ,

For v = 0, it holds in particular that D + (3) V (φ, 0) ≤ -σ(|φ(0)|, φ ). In view of Proposition 1, we conclude that the input-free system ẋ(t) = f (x t , 0) is GAS; in other words, the system (3) is 0-GAS. Moreover, it readily holds from (37) that D +

(3) V (φ, v) ≤ γ(|v|), so the system (3) is zero-ouptut dissipative. The conclusion then follows from Theorem 3.

E. Proof of Theorem 1

Let V : X → R ≥0 be a relaxed iISS LKF, meaning that there exist α, α, γ ∈ K ∞ , and α ∈ PD such that, for all φ ∈ X and all v ∈ R m ,

Let σ(r, s) := α(r) 1+s for all r, s ≥ 0. Then σ ∈ KL and

We can then invoke Theorem 2 to conclude.

VI. CONCLUSION AND PERSPECTIVES

We have shown that a point-wise dissipation rate is enough to guarantee iISS of time-delay systems. We believe that this result will simplify the iISS analysis in specific scenarios. Moreover, it makes the iISS framework more conform to the existing theory of disturbance-free systems. In establishing this result, we have in turn provided a relaxed Lyapunov-Krasovskii characterization of global asymptotic stability and extended the link between iISS and 0-GAS plus zero-output dissipativity to time-delay systems.

To the best of our knowledge, the possibility to establish ISS through a point-wise dissipation rate remains an open question. Moreover, we are not aware of any result stating iISS guarantees the existence of a (relaxed) iISS LKF. These two points would deserve further exploration.