Jijju Thomas
email: j.thomas@tue.nl

Christophe Fiter
email: christophe.fiter@univ-lille.fr

Laurentiu Hetel
email: laurentiu.hetel@centralelille.fr

Nathan Van De Wouw

Jean-Pierre Richard
email: jean-pierre.richard@centralelille.fr

Dissipativity-based Framework for Stability Analysis of Aperiodically Sampled Nonlinear Systems with Time-varying Delay ⋆

Keywords: Nonlinear sampled-data systems, Dissipativity, Time-varying delay, Stability analysis

In this paper, we provide novel conditions for stability analysis of aperiodically sampled nonlinear control systems subjected to time-varying delay. The proposed approach can also deal with cases in which delay is larger than the sampling interval. It is applicable to a general class of nonlinear systems and provides sufficient criteria for stability that aid in making trade-offs between control performance and the bounds on sampling interval and delay. As a stepping stone, a preliminary and generic result based on dissipativity, is introduced to analyse the exponential stability of a class of feedback-interconnected systems. The nonlinear sampled-data system is remodelled to consider the effects of sampling and delay in the dissipativity framework, as perturbations to the nominal closed-loop system. This leads to constructive stability conditions for a continuous time closed-loop system given by the feedback interconnection of the nominal closed-loop system and an operator(s) that captures the effects of sampling and delay. For Linear Time-Invariant (LTI) systems, we recover simple Linear Matrix Inequality (LMI) and frequency domain conditions previously proposed in the robust control framework.

Introduction

Currently, almost all sampled-data control systems are implemented numerically, and embedded in a networked environment where data is exchanged between sensors, controllers and actuators through digital communication channels [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF][START_REF] Zhang | Stability of networked control systems[END_REF]. Examples include mobile sensor networks, smart grids, highway systems, etc., see [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF].

However, in such control configurations, perturbing effects such as sampling jitter, data-packet dropouts, delays, etc., are often introduced in the network and this impacts the overall stability of the system [START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF][START_REF] Zhang | Stability of networked control systems[END_REF][START_REF] Hespanha | A survey of recent results in networked control systems[END_REF][START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. From the point of view of control theory, such phenomena are considered as sampled-data systems with aperiodic sampling and/or time-varying delay, or more generally, as Networked Control Systems (NCS) [START_REF] Zhang | Stability of networked control systems[END_REF]. In this paper, we focus on the stability analysis problem for aperiodically sampled nonlinear systems subjected to time-varying delay.

Existing literature provides various methods that deal with the stability analysis of sampled-data systems, with or without delay. An overview of different approaches in the case of aperiodic sampled-data systems can be found in [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. These approaches are broadly classified into four categories, i.e., the Time-delay approach, the Discretetime approach, the Hybrid systems approach, and the Input-output approach. The Time-delay approach, has been largely used in the context of Linear Time Invariant (LTI) systems [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. One of the advantages of this approach is that it can easily handle situations in which delay is greater than sampling period [START_REF] Van De Wouw | Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays[END_REF]. However, it is usually difficult to make a differentiation between sampling induced delay and actuation induced delay. The approach has also been extended to nonlinear systems [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF][START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]. The Discrete-time approach, has been used for stability analysis of LTI systems [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF][START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF][START_REF] Van De Wouw | Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays[END_REF] and in some cases, nonlinear systems [START_REF] De | A discrete-time framework for stability analysis of nonlinear networked control systems[END_REF][START_REF] Pepe | On Lyapunov Krasovskii characterizations of stability notions for discrete-time systems with uncertain timevarying time delays[END_REF]. Since it is based on the exact system discretization, it leads to very accurate numerical tools for stability analysis. However, intersampling behaviour has been taken into account only in the case of LTI systems, see for example, [START_REF] Cloosterman | Robust stability of networked control systems with time-varying network-induced delays[END_REF]. Additionally, the application of such discretization-based approach is challenging for general nonlinear systems and for the large-delay case, see [START_REF] Mattioni | Nonlinear discrete-time systems with delayed control: A reduction[END_REF][START_REF] Ilia | Multirate versions of sampled-data stabilization of nonlinear systems[END_REF]. The Hybrid system approach, was developed based on the fact that systems with sampling-and-hold in control and sensor signals can be modelled using impulsive systems [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. In the LTI systems case, by using Impulsive Delay Differential Equations, situations when delay is greater than the sampling interval were also studied [START_REF] Liu | Network-based control via a novel analysis of hybrid systems with time-varying delays[END_REF]. However, for nonlinear systems, the analysis has only been done for cases in which delay is less than the sampling interval [START_REF] Borgers | Tradeoffs between quality-of-control and quality-of-service in large-scale nonlinear networked control systems[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF].

The Input-output approach, treats the error induced by sampling and/or delay as a perturbation to the continuous-time control system and captures its effects using an operator [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF][START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF]. This approach is intuitively simple to develop and the stability analysis problem is related to the classical robust control framework [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF][START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. A primary advantage of this approach is that it can easily include perturbations as well as nonlinearities. However, in the case of LTI systems, this approach has been used for stability analysis in the presence of sampling, and delay, only separately. The existing results only provide L 2 -stability criteria for LTI systems. Generally, it can be shown that this implies asymptotic stability of the LTI sampled-data system. However, in such cases, it is difficult to describe the system performance, even in terms of the transient decay-rate. In the case of nonlinear systems, this approach has been employed to analyse stability only in the case of aperiodic sampling in the absence of delay [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF]. Providing constructive conditions for stability of nonlinear systems with aperiodic sampling and time-varying delay is largely an open problem.

In this paper, we provide a novel framework to analyse the stability of aperiodically sampled nonlinear systems subjected to time-varying delay, using an approach inspired from the notion of dissipativity [START_REF] Willems | Dissipative dynamical systems[END_REF]. The main contributions of this paper are as follows. We introduce a constructive approach that is applicable to a general class of aperiodically sampled nonlinear systems with time-varying delays, even in the scenario when delay is greater than the sampling interval. We provide two tractable exponential stability conditions by taking into account the specific discontinuities in delay, as well as inter-sampling and inter-actuation behaviour. The dissipativity-based approach proposed in this paper leads to conditions in terms of dissipativity type properties of the associated continuous-time system, for which many results for classes of nonlinear systems exist in literature. Additionally, the approach provides bounds on operator(s) characterizing sampling, hold and delay effects. The proposed results also aid in deciding the trade-off between system decay-rate, and the bounds on sampling interval and delay. As a stepping stone, we introduce a primary result that provides exponential stability conditions for a class of feedback interconnected systems, which bear relevance to a range of problems in the robust control framework. The first criterion caters to the so-called 'large delay case', which delineates the situation arising often in information transmission over shared networks, where the delay introduced to the data packet exceeds the sampling interval of the sensors. The second criterion, a less conservative one, deals with the 'small delay case' where delay is less than the sampling period. This scenario has been studied in numerous theoretical as well as practical settings (see [START_REF] Zhang | Stability of networked control systems[END_REF][START_REF] Xiao | Robustness analysis of asynchronous sampled-data multiagent networks with time-varying delays[END_REF][START_REF] Cloosterman | Robust stability of networked control systems with time-varying network-induced delays[END_REF]). For example, in [START_REF] Cloosterman | Robust stability of networked control systems with time-varying network-induced delays[END_REF], it was shown that in the case of a single sensor sampling periodically, when the sampled-data experienced delays less than sampling-interval, the system was rendered unstable. The problem becomes much more complex when the sensors and actuators involved have aperiodic sampling and actuation frequencies. In our analysis for the small-delay case, two separate operators are used to capture the effects of sampling and delay. In the case of LTI systems, we recover simple LMI and frequency domain conditions previously proposed in the robust control framework [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF][START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF].

The outline of this paper is as follows. In Section 2, we introduce the problem setting which comprises of a generic aperiodically sampled nonlinear system subjected to time-delay. In Section 3, a preliminary stability result in the exponential dissipativity framework is provided, for a class of feedback interconnected systems. Section 4 deals with the stability analysis of the nonlinear sampled-data system under the large-delay case. It begins with a model reformulation of the problem setting in terms of the feedback interconnection introduced in Section 3. Next, the remodelled system properties are exploited to formulate a required supply function that will be used to provide a stability criterion by employing the result introduced in Section 3. Section 5 introduces the stability analysis of the nonlinear sampled-data system in the small-delay case, and follows a similar outline as Section 4. In Section 6, examples are provided to corroborate the effectiveness of the proposed results in the nonlinear as well as linear case. Finally, conclusions and an insight into possible future work are given in Section 7. The proofs of the results introduced in this paper, if not given in the main body of the paper, are given in the appendices. Notations: Throughout the paper, we denote R + = {x ∈ R ∶ x ≥ 0}. The Euclidean norm of a vector x ∈ R n is denoted by x . For a time-varying vector z(t) ∈ R n , ż(t) is the Dini derivative given by ż(t) ≜ lim h→0 + sup z(t+h)-z(t) h . We denote W n as the set of all piecewise continuous n-dimensional functions over R + . The notation N ⋆ is used to denote the set {N {0}}. The set of all continuously differentiable functions is denoted by C 1 , and the set of all continuous functions are denoted by C 0 . The maximum and minimum eigen values of a matrix M ∈ R n×n are denoted by δ max and δ min , respectively. The Euclidean norm of a matrix M is given by M 2 = δ max (M T M ).

Problem Statement

Consider the nonlinear system

ẋp (t) = f (x p (t)) + g(x p (t))u(t), ∀t ≥ 0, (1)
with the nonlinear sampled-data control

u(t) = 0, ∀t ∈ [0, a 0 ), κ(x p (s k )), ∀t ∈ [a k , a k+1 ), k ∈ N, (2)
where x p (t) ∈ R np is the system state, and u(t) ∈ R mp is the control input based on the continuous time signal

u c (t) = κ(x p (t)), ∀t ≥ 0, (3)
subjected to sampling and delay. It is assumed that in the absence of sampling and delay, the origin of system [START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF] with

u(t) = u c (t), is exponentially stable. The functions f ∶ R np ↦ R np with f (0) = 0, g ∶ R np ↦ R np×mp are glob-
ally Lipschitz, and the function κ ∶ R np ↦ R mp belongs to C 1 . The time instants s k and a k specify the sampling instants (when sensors send the measured state value to the controller) and actuation instants (when the control input is updated at the actuator level) respectively. We consider a sampling sequence {s k } k∈N satisfying

s k+1 = s k + h k , ∀k ∈ N, (4)
where the time-varying sampling interval h k satisfies 0 < h ≤ h k ≤ h, ∀k ∈ N. Similarly, we consider the actuation sequence {a k } k∈N such that

a k = s k + τ k , ∀k ∈ N, (5)
where τ k is the time-varying delay between sampling and actuation instants and satisfies 0 ≤ τ ≤ τ k ≤ τ , ∀k ∈ N. Hypothesis 1: The actuation instants satisfy

a k < a k+1 , ∀k ∈ N. (6)
This assumption allows the bound on delay, τ , to be greater than the bound on sampling interval, h, but under the constraint that the actuation instants occur in an order corresponding to the sampling instants. Without loss of generality, we consider that the first actuation occurs at time a 0 = τ + h, while the first sampling instant is s 0 = a 0τ 0 . This assumption can also be ensured with a time-scale shift. Throughout the paper, P denotes the nonlinear closed-loop sampled-data system defined by ( 1), ( 2), ( 4)- [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF]. The objective of this paper is to analyse the exponential stability of the system P.

Preliminary Generic Stability Result

In this paper, we will use the fact that system P can be remodelled as the feedback-interconnection given by

Σ ∶ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ẋ(t) = f0 (x(t)) y(t) = h0 (x(t)) ∀t ∈ [0, a 0 ), ẋ(t) = f (x(t)) + ḡ(x(t))ω(t) y(t) = h(x(t)) + l(x(t))ω(t) ∀t ≥ a 0 , (7)
with

x(t) ∈ R n , ω(t) ∈ R m , y(t) ∈ R p , x(0) = x 0 , and the operator ∆ ∶ W p ↦ W m such that ω = ∆y. ( 8
)
The function f0 in ( 7) is considered to be globally Lipschitz, with a Lipschitz constant k 0 and f0 (0) = 0. Additionally, we consider that the functions f , ḡ, h and l are sufficiently smooth. We assume that solutions exist for the feedback interconnection Σ -∆. We shall denote the feedback interconnection ( 7)-( 8) by Σ -∆. Such interconnection models will be introduced in Sections 4 and 5, wherein the functions introduced in (7) will also be detected. This will also establish the relation between the dimensions n introduced in (7) and n p introduced in (1). Prior to presenting such models, we will formulate, a technical result concerning exponential stability of Σ-∆. This result will serve as a stepping stone for the stability analysis of systems of the form (1), ( 2), ( 4)- [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF].

Theorem 1 Consider the feedback interconnection Σ -∆ and the following assumptions: Assumption 1: There exists a supply function

S ∶ R + × R p × R m ↦ R continuous in all parameters satisfying the integral constraint ∫ t 0 S θ, φ(θ), (∆φ)(θ) dθ ≤ 0, ∀t ≥ 0, φ ∈ W p . ( 9
)
Assumption 2: There exists a continuously differentiable storage function V ∶ R n ↦ R + and scalars 0 < c 1 < c 2 , and q > 0 such that

c 1 x q ≤ V (x) ≤ c 2 x q . ( 10
)
Assumption 3: There exist scalars λ ∈ R and ρ > 0 such that the inequalities

-S (t, y(t), ω(t)) ≤ ρV (x(t)), ∀t ∈ [0, a 0 ), (11)
V (x(t)) ≥ λV (x(t)), t ∈ [0, a 0 ), (12)
V (x(t)) + αV (x(t)) ≤ e -α(t-a0) S (t, y(t), ω(t)) , ∀t ≥ a 0 , (13) are satisfied for some α > 0, along the solutions of the system Σ -∆. Then Σ -∆ is exponentially stable with a decay-rate of at least α q, i.e., ∃δ > 0 ∶ ∀t ≥ 0, x(t) ≤ δe -α q t x(0) . Inequality ( 13) is motivated from the notion of exponential dissipativity introduced in [START_REF] Chellaboina | Exponentially dissipative nonlinear dynamical systems: a nonlinear extension of strict positive realness[END_REF], wherein exponentially weighted storage and supply functions were used to establish exponential stability conditions for nonlinear dynamical systems. The aforementioned theorem is a general result for stability analysis of feedback interconnected systems of the form Σ -∆. However, it also applies to the robustness analysis of systems subjected to various perturbations that can be modelled by an operator of the form [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. Remark: If the assumptions in Theorem 1 only hold locally, the results can be extended easily in a manner similar to the one shown in [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF], so that the conditions hold in a compact set containing the origin. Note that the result provided in [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF] holds only for scenarios with aperiodic sampling alone. Theorem 1 generalizes the result in [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF] by taking into account a general class of perturbation characterizing the effects of sampling and delay. The following sections explain how Theorem 1 allows for building robust stability critera for the nonlinear sampled-data system P. In Section 4, we consider the large delay case given by Hypothesis 1, i.e. [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF]. Similarly, in Section 5, we provide stability conditions for the small delay case, given by τ k < h k , ∀k ∈ N.

Stability Analysis for the Large Delay Case

In this section, we provide a constructive approach for applying Theorem 1 to analyse the stability of system P introduced in Section 2. The term 'large delay' signifies Hypothesis 1, which implies that the delay τ k can indeed be greater than the sampling interval h k , under the constraint that the actuation instants occur in order. Theorem 1 can be used in this scenario by reformulating the system P as an interconnection of the form Σ -∆ given by ( 7)- [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF], so that the effects of sampling and delay are included as a perturbation. In order to do so, we define the perturbation induced by sampling and delay as

e(t) = 0, ∀t ∈ [0, a 0 ), κ(x p (s k )) -κ(x p (t)), ∀t ∈ [a k , a k+1 ), k ∈ N. (14)
For all t ≥ a 0 , e(t) can be interpreted as the 'error' on the control action when compared to a continuous time controller as given in (3). We will introduce an operator ∆ that helps in expressing the error e(t) in an alternate manner. Additionally, we provide the functions introduced in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], so that the dynamics of the interconnection Σ -∆ and the sampled-data system P are equivalent.

System Model Reformulation

In this section, we introduce a particular case of operator ∆ in [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF], with m = p = m p , that captures the perturbation [START_REF] Karafyllis | Global stability results for systems under sampled-data control[END_REF]. Subsequently, the system P given by ( 1), ( 2), ( 4)-( 6) is reformulated in terms of a feedback interconnection of the form Σ -∆ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF].

Lemma 2 Consider the operator ∆ ∶ W mp ↦ W mp defined for any signal z ∈ W mp as

(∆z)(t) = 0, ∀t ∈ [0, a 0 ), -∫ t s k z(s)ds, ∀t ∈ [a k , a k+1 ), k ∈ N, (15)
and the derivative of the continuous control in (3),

uc (t) = d dt κ(x p (t)). (16)
Then, the sampling and delay induced error e defined in ( 14) can be expressed as e = ∆ uc .

We show next how the sampled-data system P can be remodelled in the format Σ-∆ given by ( 7), [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. This formulation in conjunction with Lemma 2 is used to prove the equivalence between the sampled-data system P and the interconnection Σ -∆.

Lemma 3 Consider the system Σ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], with

f0 (x) = f (x), h0 (x) = ∂κ(x) ∂x f0 (x), f (x) = f (x) + g(x)κ(x), ḡ(x) = g(x), h(x) = ∂κ(x) ∂x f (x), l(x) = ∂κ(x) ∂x ḡ(x), (17)
n = n p , m = p = m p , and the operator ∆ in [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF], defined by [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF]. Then, system P can be expressed as the feedback interconnection Σ -∆ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF], with x = x p . Remark : Modelling system (1), [START_REF] Borgers | Tradeoffs between quality-of-control and quality-of-service in large-scale nonlinear networked control systems[END_REF] in the form of ( 7), [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF] implies adding an artificial output y, that will correspond to the derivative of the continuous-time control input, as given in [START_REF] Liu | Network-based control via a novel analysis of hybrid systems with time-varying delays[END_REF]. Lemmas 2 and 3 will be used to provide constructive stability conditions for the system P. In the following section, as a prerequisite for this development, the properties of ∆ in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF] are exploited to provide a supply function S that satisfies the assumptions in Theorem 1.

Stability Analysis

In this section, we characterize the properties of ∆ by a supply function S satisfying assumption [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. Lemma 4 Consider ∆ defined in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF], α ∈ R + and R ∈ R mp×mp with R = R T > 0. Then, for all z ∈ W mp , t 0 S (θ, z(θ), (∆z)(θ)) dθ ≤ 0, ∀t ≥ 0, [START_REF] Mattioni | Nonlinear discrete-time systems with delayed control: A reduction[END_REF] where the function

S ∶ R + × R mp × R mp ↦ R is defined by S ∶ (θ, v, w) ↦ e α(θ-a0) w T Rw -γ 2 v T Rv , (19)
with γ 2 = ( h + τ ) 2 e α( h+τ ) .

The result presented in Lemma 4 holds for any symmetric positive definite matrix R characterizing the supply function. The following Theorems 5 and 6, provide tools to tune the matrix R. The supply function given by ( 19), together with Lemmas 2 and 3, can now be used to provide stability conditions for the sampled-data system P.

Theorem 5 Consider system P in (1), ( 2), ( 4)-( 6), the interconnection Σ -∆ given by ( 7), ( 8), ( 15) and [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF].

If there exists a supply function S of the form [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] and a storage function V ∶ R n ↦ R + that satisfy assumptions [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], ( 11), ( 12) and (13), then system P is exponentially stable with a decay-rate α q.

Proof First, we exploit Lemma 3 to show the equivalence between P in (1), ( 2), ( 4)-( 6) and Σ -∆ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. Then, by Lemma 4, Assumption 1 in Theorem 1 is satisfied for the operator ∆ defined by [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF]. Under the conditions of the theorem, Assumptions 2 and 3 of Theorem 1 are satisfied. Applying Theorem 1, Σ -∆ is proved to be exponentially stable and therefore, so is system P.∎

Remark:

The aforementioned theorem provides (only) sufficient stability conditions based on the existence of a storage function. In the following sections, we will present how this can be used in a constructive manner based on LMI and Sum of Squares (SOS) criteria. In Section 6, we will illustrate with examples, how Theorem 5 can be used to provide stability conditions for nonlinear sampled-data systems of the form given by P. In Section 6.1, for an exemplary nonlinear system, we will show how the matrix R characterizing the supply function, can be tuned using standard MATLAB routines.

Stability Criterion for Linear Systems

Consider the linear sampled-data system P L given by

ẋ(t) = Ax(t) + Bu(t), ∀t ≥ 0, (20)
with

u(t) = 0, ∀t ∈ [0, a 0 ), Kx(s k ), ∀t ∈ [a k , a k+1 ), k ∈ N, (21)
where x(t) ∈ R n , A ∈ R n×n , B ∈ R n×m , and K ∈ R m×n . Now, we provide a stability criterion for the linear sampled-data system P L in the form of tractable LMI.

Theorem 6 Consider α ∈ R + . The linear sampled-data system P L is exponentially stable with a decay-rate α 2 if there exists

P = P T > 0 and R = R T > 0 such that ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ĀT P + P Ā + αP P B B T P 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ γ 2 R 0 0 -R ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, (22)
with Ā = A + BK, and γ 2 = ( h + τ ) 2 e α( h+τ ) .

Remark: Applying the Kalman-Yakubovich-Popov Lemma, we can infer that the LMI given by ( 22) is equivalent to the frequency-domain criterion G ∞ < 1 γ, where G is the operator defined by the transfer function

G(s) = K Ā(sI -Ā -α 2 I) -1 B + KB.
This result is in fact a generalization of the results provided in [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] and [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF]. We have extended the results in [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF][START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF] by providing stability conditions for non-linear sampled-data systems while guaranteeing an exponential decay-rate. If α = 0, and h = 0, we recover the result in [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF]. Similarly, if α = 0, and τ = 0, we recover the result provided in [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF]. In Section 6.2, we will demonstrate how matrices P and R can be tuned numerically using standard LMI solvers.

Stability Analysis for the Small Delay Case

The large-delay case studied in Section 4 is more generic to processes communicating via a shared network, where traffic flow can increase considerably. However, in some cases, it has been shown that it is desirable to have delay less than sampling interval since sampled data arriving in a non-chronological order at the actuator can be hazardous from a control point of view [START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF]. Consequentially, this would make the implementations of algorithms and analysis much more complex. In this section, we will demonstrate how considering sampling and delay separately in the small-delay case, gives a less conservative stability criterion. The following assumption is considered throughout the section.

Hypothesis 2:

The actuation based on the sampled state x(s k ) is implemented before the next sampling instant s k+1 , i.e.,

τ k < h k , ∀k ∈ N. (23)
Next, we re-formulate the sampled-data model for system P in order to include the effects of sampling and delay using two separate errors, denoted by e s (t) and e d (t), respectively. Consider the continuous-time control

u c (t) = κ(x p (t)). The sampled version of this signal is u s (t) = κ(x p (s k )), ∀t ∈ [s k , s k+1 ), k ∈ N. The sampling- induced error e s (t) is e s (t) = u s (t) -u c (t). Without loss
of generality, we consider that e s (t) = 0, ∀t < s 0 . Then,

e s (t) = 0, ∀t ∈ [0, s 0 ), κ(x p (s k )) -κ(x p (t)), ∀t ∈ [s k , s k+1 ), k ∈ N. (24)
The delayed version of u s (t) is the control signal u(t) applied at the level of the actuator. We introduce another error e d (t), which can be given by u(t)-u s (t). Note that we can define the error e d (t) = 0, ∀t < a 0 , since it bears no relevance. Formally, e d (t) is given by

e d (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ∀t ∈ [0, a 0 ), 0, ∀t ∈ [a k-1 , s k ), k ∈ N ⋆ , κ(x p (s k-1 )) -κ(x p (s k )), ∀t ∈ [s k , a k ), k ∈ N ⋆ . ( 25
)
Using this formulation for e s (t) and e d (t), given by ( 24) and ( 25), respectively, we proceed to reformulate the sampled-data system P in the form of Σ -∆.

System Model Reformulation

In this section, we introduce two different operators ∆ s and ∆ d , which capture the errors induced by sampling and delay given in ( 24) and ( 25), respectively. In an approach similar to the one used in Section 4.1, system P under Hypothesis 2, i.e. [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF], can be represented as a feedback interconnection of the form Σ -∆.

Lemma 7 Consider the operator ∆ ∶ W 2mp ↦ W 2mp ∆ ∶ φ = ⎛ ⎝ v w ⎞ ⎠ → (∆φ) = ⎛ ⎝ ∆ s v ∆ d w ⎞ ⎠ , ∀v ∈ W mp , w ∈ W mp , (26)
under Hypothesis 2, i.e. [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF], where

(∆ s v)(t) = 0, ∀t ∈ [0, s 0 ), -∫ t s k v(θ)dθ, ∀t ∈ [s k , s k+1 ), k ∈ N, (27) and
(∆ d w)(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ∀t ∈ [0, a 0 ), 0, ∀t ∈ [a k-1 , s k ), k ∈ N ⋆ , -∫ s k s k-1 w(θ)dθ, ∀t ∈ [s k , a k ), k ∈ N ⋆ . ( 28
) Then, the sampling and delay induced errors defined in ( 24) and ( 25), respectively, can be expressed as

⎛ ⎝ e s e d ⎞ ⎠ = ⎛ ⎝ ∆ s uc ∆ d uc ⎞ ⎠ , (29)
with uc given by [START_REF] Liu | Network-based control via a novel analysis of hybrid systems with time-varying delays[END_REF].

Analogous to the approach used in Section 4, we now proceed to reformulate the sampled-data system P under Hypothesis 2, i.e. [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF], in the format Σ -∆ given by ( 7), [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. In the following lemma, by using such a model reformulation along with Lemma 7, we provide the equivalence between the sampled-data system P under Hypothesis 2, and the feedback interconnection Σ -∆.

Lemma 8 Consider the system Σ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], with

f0 (x) = f (x), h0 (x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∂κ(x) ∂x f0 (x) ∂κ(x) ∂x f0 (x) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , f (x) = f (x) + g(x)κ(x), ḡ(x) = g(x) g(x) , h(x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∂κ(x) ∂x f (x) ∂κ(x) ∂x f (x) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , l(x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∂κ(x) ∂x ḡ(x) ∂κ(x) ∂x ḡ(x) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (30)
n = n p , m = p = 2m p , and the operator ∆ in (8), defined by ( 26), ( 27) and (28) under Hypothesis 2, i.e. [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF].

Then, the sampled-data system P can be expressed as the feedback interconnection Σ -∆, with x = x p .

Lemmas 7 and 8 are used to provide constructive stability criterion for sampled-data system P under Hypothesis 2. To this end, the supply function S given in Theorem 1 needs to be formulated. We proceed in this direction by studying the properties of operators ∆ s and ∆ d .

Stability Analysis

In this section, we characterize the properties of ∆ s and ∆ d , by functions S s and S d , respectively. Consequently, we formulate the supply function S = S s + S d .

Lemma 9 Consider the operator ∆ s defined in [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF],

β ∈ R + and R s ∈ R mp×mp with R s = R T s > 0. Then, ∫ t 0 S s (θ, v(θ), (∆ s v)(θ)) dθ ≤ 0, ∀t ≥ 0, v ∈ W mp , ( 31
)
where the function

S s ∶ R + × R mp × R mp ↦ R is defined as S s ∶ (θ, v, µ) → e β(θ-a0) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ v µ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ 2 s R s γ 2 s β 2 R s γ 2 s β 2 R s (1 -γ 2 s β 2 4 )R s ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ v µ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (32) with γ s = 2 h π . Lemma 10 Consider ∆ d defined in (28) under Assump- tion 2, β ∈ R + and R d ∈ R mp×mp with R d = R T d > 0. Then, for all w ∈ W mp , t 0 S d (θ, w(θ), (∆ d w)(θ)) dθ ≤ 0, ∀t ≥ 0, ( 33
)
where the function

S d ∶ R + × R mp × R mp ↦ R is defined as S d ∶ (θ, w, ε) → e β(θ-a0) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ w ε ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ d R d 0 0 R d ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ w ε ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (34)
with γ d = hτ e β( h+τ ) .

The functions S s and S d given in Lemmas 9 and 10, provide the sampling and delay component, respectively, of the supply function S = S s + S d . As follows, we use the supply function S = S s + S d to provide a general, more accurate stability criterion for the sampled-data system P under Hypothesis 2, i.e., when delay is less than sampling interval.

Theorem 11 Consider system P, the interconnection Σ-∆ given by ( 7), ( 8), ( 26), ( 27), ( 28) and [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF]. If there exist functions S = S s + S d defined using [START_REF] De | A discrete-time framework for stability analysis of nonlinear networked control systems[END_REF] and [START_REF] Xiao | Robustness analysis of asynchronous sampled-data multiagent networks with time-varying delays[END_REF], and V ∶ R np ↦ R + that satisfy assumptions ( 10), ( 11), ( 12) and (13), then system P is exponentially stable with a decay-rate α q.

Proof We exploit Lemma 8 to establish the equivalence between system P under Hypothesis 2 and Σ -∆ in ( 7), [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF]. Then, by Lemmas 9 and 10, Assumption 1 in Theorem 1 is satisfied for the operator ∆ defined by ( 26), ( 27) and [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. Under the conditions of the theorem, Assumptions 2 and 3 of Theorem 1 are satisfied. Applying Theorem 1, Σ -∆ is proved to be exponentially stable and by equivalence, so is system P. ∎

The result presented in Theorem 6 holds for any positive symmetric definite matrices R s and R d characterizing the supply function. In Section 6.1, we will illustrate how Theorem 11 provides less conservative results for the sampled-data system P under Hypothesis 2, i.e., for the small delay case. The usage of numerical tools to tune matrices R s and R d , will also be shown.

Stability Criterion for Linear Systems

In this section, we recall the linear sampled-data system P L described in Section 4.3 by [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF]. Based on the Lemmas 9 and 10, we provide the following stability criterion for system P L under Hypothesis 2.

Theorem 12 Consider a scalar α ∈ R + and Hypothesis 2. The linear sampled-data system P L is exponentially stable with a decay-rate α 2 if there exist

P = P T > 0, R s = R s T > 0, and R d = R d T > 0 such that ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ĀT P + P Ā + αP P B BT P 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T Φ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, ( 35
)
with Ā = A + BK, B = B B , and

Φ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ γ 2 s R s + γ d R d -γ 2 s α 2 R s 0 -γ 2 s α 2 R s (γ 2 s α 2 4 -1)R s 0 0 0 -R d ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ( 36
)
where γ s = 2 h π and γ d = hτ e α( h+τ ) .

Remark: When α = 0, τ = 0 (implying no delay component in S), the LMI [START_REF] Zhang | Stability of networked control systems[END_REF] translates to a form similar to LMI [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Consequentially, by virtue of the Kalman-Yakubovich-Popov lemma, we can recover the frequency domain condition introduced in [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF], i.e., G ∞ < π 2 h , where G is the operator defined by the transfer function

G(s) = K Ā(sI -Ā -α 2 I) -1 B + K B.
In Section 6.2, we will illustrate with examples, how the LMI [START_REF] Zhang | Stability of networked control systems[END_REF] provides less conservative results for LTI systems under Hypothesis 2, i.e., for the small delay case.

Illustrative Examples

In this section, we illustrate the effectiveness of our proposed results via examples. The provided examples highlight the difference between the single-error approach and the separate-error approach in terms of conservativeness and trade-offs between control performance and the bounds on sampling interval and delay. The result presented in this paper provides a foundation for deciding the trade-off between maximum delay τ , maximum sampling period h, and decay-rate α. By fixing one of the parameters, the trade-off between the remaining parameters can be obtained. For example, by fixing τ , and gridding over h and α, a trade-off between the decayrate and the maximum allowable sampling interval can be obtained. In a similar manner, fixing h will give the trade-off between α and τ , and so on.

Nonlinear System Example

We consider the following example [START_REF] Karafyllis | Global stability results for systems under sampled-data control[END_REF][START_REF] Nesic | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF][START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF],

ẋ(t) = dx(t) 2 -x(t) 3 + u(t), (37)
with a bounded time-varying parameter d(t) ≤ 1, and a stabilizing control u(t) = κ(x(t)) = -2x(t) subjected to both sampling and delay. Since the function f (x) = x 2x 3 is locally Lipschitz, our results will only hold locally on any compact set containing the origin.

Large-delay Case

Using the definition in [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], we reformulate the system model in the form Σ -∆, where Σ is given by

ẋ(t) = dx 2 (t) -x 3 (t) y(t) = -2(dx 2 (t) -x 3 (t)) ∀t ∈ [0, a 0 ), ẋ(t) = dx 2 (t) -x 3 (t) -2x(t) + w(t) y(t) = -2(dx 2 (t) -x 3 (t) -2x(t) + w(t)) ∀t ≥ a 0 .
(38) We use a storage function of the form V (x) = ax 2 +bx 4 as given in [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF]. Using [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF], we obtain the supply function Fig. 2. Trade-off between desired decay-rate α and h + τ for the nonlinear system (37), in the large-delay case.

S(θ, y, w) = e α(θ-a0) Rw 2 (θ)γ 2 Ry 2 (θ) , with γ 2 = ( h+τ ) 2 e α( h+τ ) . For this case, from condition (13), we can infer that the values of ( h + τ ) satisfying the inequality

(2ax + 4bx 3 )(dx 2 -x 3 -2x + w) + α(ax 2 + bx 4 ) -Rw 2 +4( h + τ ) 2 e α( h+τ ) R(dx 2 -x 3 -2x + w) 2 ≤ 0, (39)
will guarantee exponential stability. If (39) can be expressed as a Sum of Squares (SOS) for all the values of (d, d 2 ) ∈ {(1, 0), (1, 1), (-1, 0), (-1, 1)}, then it will be SOS for any time-varying d(t) ≤ 1. Using SOSTOOLS [START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF], Figure 1 provides the feasible values of h and τ (in red) for α = 0.1, and all values of (d, d 2 ). It can be seen from Figure 1 that, for α = 0.1, h and τ satisfy a maximum bound h + τ ≤ 0.45, with a = 0.7079, b = 0.1890 and R = 0.4268. The parameters a, b and R are optimized using SOSTOOLS. Additionally, the trade-off between the desired decay-rate α 2 and h + τ is shown in Figure 2.

Small-delay Case

Now, we shall provide bounds on h and τ in the smalldelay case, i.e., τ k < h k . In this case, the system model is reformulated in the form Σ, given by ẋ(t) = dx 2 (t)x 3 (t)

y(t) = y 1 (t) y 2 (t) T ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ∀t ∈ [0, a 0 ), (40)
with y 1 (t) = y 2 (t) = -2(dx 2 (t)x 3 (t)) and

ẋ(t) = dx 2 (t) -x 3 (t) -2x(t) + w s (t) + w d (t), y(t) = y 1 (t) y 2 (t) T , ∀t ≥ a 0 , (41)
where y 1 (t) = y 2 (t) = -2(dx 2 (t)x 3 (t) -2x(t) + w s (t) + w d (t)). Using ( 32) and ( 34), we get the supply function

S(θ, y, w) = S s (θ, y 1 , w s ) + S d (θ, y 2 , w d ), = e α(θ-a0) -γ 2 s R s y 2 1 (θ) -γ d R d y 2 2 (θ) +γ 2 s αR s y 1 (θ)w s (θ) + (1 -γ 2 s α 2 4 )R s w 2 s +R d w 2 d , (42)
where γ s = 2 h π and γ d = hτ e β( h+τ ) . Therefore, by using the supply function (42) in condition ( 13), we must deduce the values of h and τ satisfying the inequality

(2ax + 4bx 3 )(dx 2 -x 3 -2x + w s + w d ) + α(ax 2 + bx 4 ) +4(γ 2 s R s + γ d R d )(dx 2 -x 3 -2x + w s + w d ) 2 +2γ 2 s αR s (dx 2 -x 3 -2x + w s + w d )w s -(1 -γ 2 s α 2 4 )R s w 2 s -R d w 2 d ≤ 0, (43)
in order to guarantee exponential stability of the system (37), with α > 0, τ k < h k . For the sake of comparison with the feasibility region obtained in the large-delay case, we choose α = 0.1. In a similar manner as shown in the large-delay case, we use SOSTOOLS to obtain the feasible values of h and τ satisfying inequality (43), for all values of (d, d 2 ), while optimizing the values of a, b, R s and R d . The feasibility plot in the small-delay case is given in Figure 1 (in blue). In Figure 1, it can be seen that the red feasibility plot (indicating feasibility for the large-delay case) and the blue feasibility plot (indicating the small-delay case) overlap. This overlapping region represents the feasible values of h and τ obtained when the criterion (39) provided for the large-delay case, is applied to the small-delay case. In such scenarios, Theorem 11 and Theorem 12 always provide better results in comparison to the results given by Theorem 5 and Theorem 6, respectively. In Figure 1, when τ → 0, we can see that bounds on h upto 0.72 are feasible while using the tools presented in the small-delay analysis. The tool presented in the large-delay case, on the other hand, accommodates h upto 0.45, thereby implying an improvement of about 60% while using the result provided in the small-delay case. Additionally, when h = 0.27, the feasible values of τ in the large and small-delay cases, are approximately upto 0.17 and 0.27, respectively, showing an improvement of about 59%. Using these numerical arguments, it can be concluded that for the small-delay case, capturing the effects of sampling and delay using two separate errors gives less conservative results. However, the amount of improvement in the small-delay case over the large-delay case depends on the parameter α. We illustrate this in the following section for a linear system example. Remark: The less-conservative nature of the results proposed in the small-delay case can also be justified from a theoretical perspective. In the large-delay case, the supply function was formulated using Jensen's inequality, which introduces conservativeness [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -Application to time-delay and sampled-data systems[END_REF]. On the other hand, in the small-delay case, Wirtinger's inequality has been used. For this case, the improvement over Jensen's inequality is well known in the literature [START_REF] Seuret | Jensen's and Wirtinger's inequalities for time-delay systems[END_REF]. For the same example in the absence of delay, in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and [START_REF] Karafyllis | Global stability results for systems under sampled-data control[END_REF], upper-bounds of 0.368 and 0.143, respectively, were obtained for the sampling intervals without any performance guarantee. This is comparable to the small-delay case we have considered, with τ = 0. Additionally, in [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF], an upper-bound of 0.72 was proposed for the system (37) without delay, with α = 0.1. From the results proposed for the small-delay case, by setting τ = 0, indicating sampling without any delay, we can see in Figure 1 that we obtain the same upper-bound of 0.72 on the sampling intervals, as proposed in [START_REF] Omran | Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control[END_REF], with a = 2.9153 × 10 -6 , b = 7.29 × 10 -7 , R s = 1.6964 × 10 -6 and R d = 1.2465. However, our results have an added advantage that we provide tractable stability conditions for the nonlinear sampled-data system in the presence of time-varying delay.

Linear System Example

Consider the system (20) characterized by the parameters [36]

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 3 2 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0.6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , K = -1 6 . ( 44
)
By virtue of Theorem 6, we can compute the maximum allowable values of h + τ with respect to α from the LMI [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. The LMI ( 22) is solved using YALMIP, by optimizing parameters P and R, for different values of α and h + τ . The feasibility region thus obtained will aid in deciding the trade-off between a desired decay rate while taking into account the maximum bounds on sampling interval and delay. Considering α ∈ {0.01, 1, 2}, we obtain the bounds on h and τ as shown in Figure 3 (in red solid, dashed and dotted lines). For the LTI system (44), if α = 0 and h = 0, we recover the bound on τ as given in [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF]. For the chosen values of α ∈ {0.01, 1, 2}, we also compute the bounds on h and τ in the small-delay case (as shown in Figure 3 in blue solid, dashed and dotted lines). Following a similar explanation as given in Section 6.1.2, we can conclude that for the small-delay case, differentiating the effects of sampling and delay using two separate errors, the LMI in [START_REF] Zhang | Stability of networked control systems[END_REF] introduced in Theorem 12 provides less conservative results in comparison to the criterion provided in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] (applied to the smalldelay case). Figure 3 Fig. 3. Bounds on h and τ for the LTI system (44) in the large-delay case (in red), and in the small-delay case (in blue), for α = 0.01 (solid line), α = 1 (dashed line) and α = 2 (dotted line). The analytical stability bounds on constant h k and τ k are given by the green line [START_REF] Zhang | Stability of networked control systems[END_REF].

amount of improvement in the small-delay case over the large-delay case, on the parameter α. If α = 0 and τ = 0, we recover the bound on h as proposed in [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF]. Therefore, we can conclude that by applying our generic nonlinear tools to the linear case, we provide bounds on h and τ that are not more conservative in comparison to the bounds provided in [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF][START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF]. Also, it has to be noted that despite the fact that the condition in ( 22) is more conservative when applied to the small-delay case, the result is still important since it is applicable to the more generic large-delay case.

Conclusion

In this paper, novel approaches for stability analysis of aperiodically sampled nonlinear systems with timevarying delay are provided. The framework introduced in this paper holds for a general class of nonlinear systems and provides tools that help in deciding required tradeoffs between the system decay-rate and the bounds on sampling interval and delay. As a preliminary result, an approach inspired from the notion of exponential dissipativity is used to provide stability conditions for a class of feedback interconnected systems, while guaranteeing a desired decay-rate. The nonlinear sampled-data system is remodelled as a feedback interconnection of the nominal closed-loop system and an operator that captures the effects of sampling and delay, thereby leading to constructive stability conditions. The proposed approach leads to conditions on dissipativity properties of the system, for which many results exist in literature. When applying the results to LTI case, we see that they generalize existing frequency domain and LMI conditions in the robust stability framework. For the case when delay is less than sampling interval, a less conservative stability criterion is obtained by considering two separate operators to capture the effects of sampling and delay. The effectiveness of the proposed theoretical results have been corroborated via simulation results for an exemplary nonlinear system. We foresee numerous extensions. For example, a more realistic scenario would involve multiple sensors and actuators, each with unique bounds on sampling interval and delay [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF][START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF].

A Proof of Theorem 1

Let us first upper-bound the response x(t) for all t ≥ a 0 . Consider the function

W (t) = e α(t-a0) V (x(t))- t a0 S θ, y(θ), ω(θ) dθ, ∀t ≥ a 0 , (A.1)
From condition (13), we have Ẇ (t) ≤ 0, for all t ≥ a 0 and therefore W (t) ≤ W (a 0 ), for all t ≥ a 0 , which can be stated as e α(t-a0) V (x(t)) -∫ t a0 S θ, y(θ), ω(θ) dθ ≤ V (x(a 0 )). Therefore, for all t ≥ a 0 , we obtain V (x(t)) ≤ e -α(t-a0) -∫ a0 0 S θ, y(θ), ω(θ) dθ + ∫ t 0 S θ, y(θ), ω(θ) dθ + V (x(a 0 )) , (A.2) and by using [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], for all θ ≥ 0, we have

V (x(t)) ≤ e -α(t-a0) -∫ a0 0 S θ, y(θ), ω(θ) dθ +V (x(a 0 ))] . (A.3)
By integrating condition ( 12) for all t ∈ [0, a 0 ), we have

V (x(a 0 )) ≥ e λ(a0-t) V (x(t)), ∀t ∈ [0, a 0 ). (A.4)
Then, by integrating condition (11) and using (A.4) for all t ∈ [0, a 0 ), we have -∫

a0 0 S (θ, y(θ), ω(θ)) dθ ≤ ρ ∫ a0 0 V (x(θ))dθ ≤ ρ ∫ a0 0 e λ(θ-a0) V (x(a 0 ))dθ = ηV (x(a 0 )), where η ∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ρe -λa 0 λ (e λa0 -1), if λ ≠ 0, ρa 0 , if λ = 0. Consequently, from (A.3), V (x(t)) ≤ e -α(t-a0) (1 + η)V (x(a 0 )), = e -α(t-a0) CV (x(a 0 )), ∀t ≥ a 0 (A.5)
with C ∶= η + 1 > 1. Then, from [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], we obtain for all t ≥ a 0 , c 1 x(t) q ≤ V (x(t)) ≤ e -α(t-a0) CV (x(a 0 )) ≤ e -α(t-a0) Cc 2 x(a 0 ) q , and thus

x(t) ≤ q C c 2 c 1 e -α q (t-a0) x(a 0 ) , ∀t ≥ a 0 . (A.6)
Now, let us analyse the response in the interval t ∈ [0, a 0 ). Using the definition of system Σ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF] for all t ∈ [0, a 0 ), we have ẋ(t) = f0 (x(t)), where f0 is globally Lipschitz continuous with some constant k 0 and f0 (0) = 0. Hence, we have that x(t)x(0) = ∫ t 0 f0 (x(s))ds, implying, using the Triangular Inequality, that

x(t) ≤ x(0) + t 0 f0 (x(s)) ds. (A.7)
Since f0 is Lipschitz continuous and f0 (0) = 0, we have f0 (x(s)) = f0 (x(s)) -f0 (0) ≤ k 0 x(s) -0 = k 0 x(s) . Consequently, (A.7) leads to x(t) ≤ x(0) + k 0 ∫ t 0 x(s) ds. By virtue of Gronwall's inequality, we obtain

x(t) ≤ x(0) e k0t , ∀t ∈ [0, a 0 ), (A.8)
implying x(a 0 ) ≤ e k0a0 x(0) . From (A.6), we obtain

x(t) ≤ q C c 2 c 1 e -α
q (t-a0) e k0a0 x(0) , ∀t ≥ a 0 . (A.9)

Additionally, for all t ∈ [0, a 0 ), we can upper-bound inequality (A.8) by x(t) ≤ e k0t x(0) ≤ e k0a0 x(0) ≤ q C c2 c1 e -α q (t-a0) e k0a0 x(0) , ∀t ∈ [0, a 0 ), since C > 1 (see (A.5)), c 2 ≥ c 1 (see ( 10)), and -α q (ta 0 ) ≥ 0. Consequently, using (A.9), we obtain x(t) ≤ q C c2 c1 e -α q (t-a0) e k0a0 x(0) = δe -αt q x(0) , for all t ≥ 0, with δ ∶= e (k0+ α q )a0 q C c2 c1 , thereby implying that the system Σ -∆ is exponentially stable with a decay-rate of at least α q.

B Proof of Lemma 2

(1) For all t ∈ [0, a 0 ): As per the definition of e(t) in ( 14) and ∆ in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF], we have e(t) = 0 = (∆ uc )(t), ∀t ∈ [0, a 0 ).

(2) For all t ∈ [a k , a k+1 ), k ∈ N: We have e(t) = κ(x p (s k ))κ(x p (t)), which can be reformulated as e(t) = -∫ t s k d ds κ(x p (s))ds = -∫ t s k uc (s)ds. Therefore, using the definition of ∆ in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF], it can be concluded that indeed e(t) = (∆ uc )(t).

C Proof of Lemma 3

Consider the system P in (1), ( 2), ( 4)- [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF]. Moreover, consider the following notational relations:

y(t) = uc (t), (C.1)
with uc (t) given by ( 16), and ω(t) = e(t), with e(t) defined by [START_REF] Karafyllis | Global stability results for systems under sampled-data control[END_REF]. By virtue of Lemma 2, we have, ω(t) = e(t) = (∆ uc )(t) = (∆y)(t), ∀t ≥ 0.

(1) For all t ∈ [0, a 0 ): As per the definition of system P, we have ẋp (t) = f (x p (t)), (C.2) and using [START_REF] Liu | Network-based control via a novel analysis of hybrid systems with time-varying delays[END_REF],

y(t) = uc (t) = d dt κ(x p (t)) = ∂κ(xp(t)) ∂xp f (x p (t)). (C.3)
Using ( 17), (C.2) and (C.3) this is equivalent to ẋp (t) = f0 (x p (t)), y(t) = h0 (x p (t)). This expresses the dynamics of system Σ for t ∈ [0, a 0 ), given by ( 7), with f0 and h0 as defined in [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], i.e, for all t ∈ [0, a 0 ), x(t) = x p (t). Additionally, using f0 (x) = f (x), it can be concluded from the definition of system P that the function f0 is globally Lipschitz continuous with f0 (0) = 0.

(2) For all t ∈ [a k , a k+1 ), k ∈ N: The dynamics of system P is given by ẋp

(t) = f (x p (t)) + g(x p (t))u(t) = f (x p (t)) + g(x p (t))κ(x p (s k )) = f (x p (t)) + g(x p (t))κ(x p (t)) + g(x p (t)) [κ(x p (s k )) -κ(x p (t))].
Using [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], and recalling the definition of e(t) in ( 14), we obtain ẋp (t) = f (x p (t)) + ḡ(x p (t))e(t). This is equivalent to the dynamics of system Σ for t ≥ a 0 , given by ( 7), with ω(t) = e(t) and the functions f and ḡ defined by [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], i.e., for all t ≥ a 0 , we have x = x p . Additionally, from (C.1) and ( 16) we have,

y(t) = d dt κ(x p (t)) = ∂κ(xp(t)) ∂xp f (x p (t)) + ḡ(x p (t))e(t)
. Once again, using notation ( 17) and e(t) = ω(t), we have, y(t) = h(x p (t)) + l(x p (t))ω(t), which is the same as y defined in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], for t ≥ a 0 , since we have already shown x = x p . Therefore, it can be seen that system P can be expressed as the feedback interconnection Σ -∆, with the functions f0 , h0 , f , ḡ, h, and l defined by [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF].

D Proof of Lemma 4

(1) For t ∈ [0, a 0 ): Using the definition of ∆ in ( 15), we have (∆z)(θ) = 0, for all θ ∈ [0, t) and clearly [START_REF] Mattioni | Nonlinear discrete-time systems with delayed control: A reduction[END_REF] holds in this case since S(θ, z(θ), (∆z

)(θ)) = -γ 2 z T (θ)Rz(θ) ≤ 0.
(2) For t ≥ a 0 : Let w(t) denote

w(t) = (∆z)(t) = - t s k z(ζ)dζ, ∀t ∈ [a k , a k+1 ), k ∈ N.
(D.1) Using Jensen's inequality, we obtain w

T (t)Rw(t) ≤ (t - s k ) ∫ t s k z T (ζ)Rz(ζ)dζ ≤ ( h + τ ) ∫ t s k z T (ζ)Rz(ζ)dζ. Using the change of variable s = ζ -t, we obtain w T (t)Rw(t) ≤ ( h + τ ) ∫ 0 s k -t z T (t + s)Rz(t + s)ds ≤ ( h + τ ) ∫ 0 -( h+τ ) z T (t + s)Rz(t + s)ds. Therefore, ∫ t a0 e α(θ-a0) w T (θ)Rw(θ)dθ ≤ ( h + τ ) ∫ t a0 e α(θ-a0) ∫ 0 -( h+τ ) z T (θ + s)Rz(θ + s)ds dθ. Substituting u = θ + s, we have that ∫ t a0 e α(θ-a0) w T (θ)Rw(θ)dθ ≤ ( h + τ ) ∫ 0 -( h+τ ) ∫
t+s a0+s e α(u-s-a0) z T (u)Rz(u)du ds.

(D.2) Since the inner integral in the right-hand side of the inequality in (D.2) is always positive, we can upper bound the left-hand side in (D.2) using the limits of s and obtain

∫ t a0 e α(θ-a0) w T (θ)Rw(θ)dθ ≤ ( h + τ ) ∫ 0 -( h+τ ) ∫ t+0 a0-( h+τ ) e α(u+( h+τ )-a0) z T (u)Rz(u)du ds ≤ ( h + τ )e α( h+τ ) ∫ 0 -( h+τ ) ∫ t 0 e α(u-a0) z T (u)Rz(u)du ds = ( h + τ ) 2 e α( h+τ ) ∫ t 0 e α(u-a0) z T (u)Rz(u)du.
As per definition [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF], we have w(t) = 0 for all 0 ≤ t < a 0 and, consequently, ∫ t 0 e α(θ-a0) w T (θ)Rw(θ)dθ = ∫ t a0 e α(θ-a0) w T (θ)Rw(θ)dθ ≤ ( h + τ ) 2 e α( h+τ ) ∫ t 0 e α(u-a0) z T (u)Rz(u)du. Hence, using the definition of w(t) in (D.1), we have that ∫ t 0 e α(θ-a0) ((∆z) T (θ)R(∆z)(θ) -( h + τ ) 2 e α( h+τ ) z T (θ)Rz(θ))dθ ≤ 0, which proves the integral inequality [START_REF] Mattioni | Nonlinear discrete-time systems with delayed control: A reduction[END_REF], thereby concluding the proof.

E Proof of Theorem 6

Comparing the sampled-data systems P L and P, we have, f (x(t)) ∶= Ax(t), g(x(t)) ∶= B, κ(x(s k )) ∶= Kx(s k ). Hence, the sampling and delay induced error is given by e(t) = 0, ∀t ∈ [0, a 0 ), Kx(s k ) -Kx(t), ∀t ∈ [a k , a k+1 ), k ∈ N, thereby implying that using the operator ∆ defined in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF], we can state e = ∆(K ẋ). Using the inequality in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], we proceed to prove that the assumptions introduced in Theorem 1 will hold for V (x) = x T P x and S (t, y(t), w(t)) defined by [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]. For the LTI system P L , the functions given in [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF] are given by f0 (x(t)) ∶= Ax(t), h0 (x(t)) ∶= KAx(t), f (x(t)) ∶= Āx(t), ḡ(x(t)) ∶= B, h(x(t)) ∶= K Āx(t), and l(x(t)) ∶= KB, where Ā = (A + BK).

(1) Satisfying Assumption 1, i.e., [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF]: By virtue of Lemma 4, we can see that the supply function S (t, y(t), w(t)) defined by [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] satisfies assumption [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] in Theorem 1, i.e., ∫ t 0 S θ, y(θ), (∆y)(θ) dθ ≤ 0, ∀t ≥ 0.

(2) Satisfying Assumption 2, i.e., [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]: With V (x) = x T P x, P = P T > 0 and x ∈ R n , we have that δ min (P ) x 2 ≤ x T P x ≤ δ max (P ) x 2 , implying Assumption 2 is satisfied with q = 2, c 1 = δ min (P ) and c 2 = δ max (P ).

(3) Satisfying Assumption 3, inequality [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]: Consider the function S (t, y(t), w(t)) defined by [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]. For all t ∈ [0, a 0 ), since y(t) = h0 (x(t)) = KAx(t) and ω(t) = 0, we have that for all t ∈ [0, a 0 ), -S(t, y(t), ω(t)) = -S(t, h0 (x(t)), 0) = e α(t-a0) γ 2 x T (t)(KA

) T R(KA)x(t) ≤ max θ∈[0,a0]
δ max e α(θ-a0) γ 2 (KA) T R(KA) x(t) 2

≤ ρV (x(t)), with ρ = δmax γ 2 (KA) T R(KA) δmin(P ) . (4) Satisfying Assumption 3, inequality (12): We have V (x(t)) = x(t) T P x(t) for all t ≥ 0. For all t ∈ [0, a 0 ),

ẋ(t) = f0 (x(t)) = Ax(t), V (x(t)) = x(t) T A T P + P A x(t) ≥ δmin A T P +P A δmax(P ) V (x(t)). Therefore, inequality (12)
is satisfied for any λ ≤ δmin A T P +P A δmax(P ) . ( 5) Satisfying Assumption 3, inequality (13): Consider the function W (t) = V (x(t)) + αV (x(t))e -α(t-a0) S t, y(t), e(t) , defined for all t ≥ a 0 with V (x) = x T P x, and the function S t, y(t), e(t) defined by [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]. Clearly, inequality [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] in Assumption 3 holds if W (t) ≤ 0, for all t ≥ a 0 . We have, S (t, y(t), e(t)) = e α(t-a0)

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ y(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ 2 R 0 0 R ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ y(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = e α(t-a0) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T N ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , where N = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ 2 R 0 0 R ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
.

Therefore, we have that for all t ≥ a 0 ,

W (t) = V (x(t)) + αV (x(t)) -e -α(t-a0) S t, y(t), e(t) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ĀT P + P Ā P B B T P 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + α ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ P 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ -N ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T Γ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) e(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , with Γ ∶= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ĀT P + P Ā + αP P B B T P 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ -N . Substituting N in the expression for Γ gives Γ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ĀT P + P Ā + αP P B B T P 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ -N = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ĀT P + P Ā + αP P B B T P 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ γ 2 R 0 0 -R ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
. A sufficient condition for W (t) ≤ 0 for all t ≥ a 0 will therefore be given by Γ ≤ 0, which is guaranteed by [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Consequently, we have proved that inequality [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] in Assumption 3 is satisfied for the chosen storage and supply functions.

We have shown that all the assumptions of Theorem 1 hold for V (x) = x T P x and S (t, y(t), e(t)) defined by [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] and, therefore, the exponential stability of system P is guaranteed with a decay rate greater than or equal to α 2.

F Proof of Lemma 7

(1) Expressing e s using ∆ s : Recalling the definition of e s (t) in ( 24), and by using the operator definition for ∆ s in ( 27), we can state using ( 16) that for all t ∈ [0, s 0 ), e s (t

) = 0 = (∆ s uc )(t). Similarly, for all t ∈ [s k , s k+1 ), k ∈ N, e s (t) = κ(x p (s k )) -κ(x p (t)) = -∫ t s k d ds κ(x p (s))ds = -∫ t s k uc (s)ds = (∆ s uc )(t)
. Hence, we have e s (t) = (∆ s uc )(t), ∀t ≥ 0.

(2) Expressing e d using ∆ d : In a similar manner, using the definition of e d (t) in [START_REF] Pepe | On Lyapunov Krasovskii characterizations of stability notions for discrete-time systems with uncertain timevarying time delays[END_REF] and the operator definition for ∆ d defined in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], we have, for all t ∈ [0,

a 0 ) ∪ [a k-1 , s k ) k∈N ⋆ , e d (t) = 0 = (∆ d uc )(t). Similarly, for all t ∈ [s k , a k ), k ∈ N ⋆ , e d (t) = κ(x p (s k-1 )) -κ(x p (s k )) = -∫ s k s k-1 d ds κ(x p (s))ds = -∫ s k s k-1 uc (s)ds = (∆ d uc )(t).
Hence, we obtain e d (t) = (∆ d uc )(t), ∀t ≥ 0.

G Proof of Lemma 8

Consider system P, the notations y(t) = y 1 (t) y 2 (t)

T = uc (t) uc (t)
T , with uc defined by ( 16), and ω(t) = e s (t) e d (t) T , with e s (t) and e d (t) given by ( 24) and (25), respectively. By virtue of Lemma 7, we have

ω(t) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ e s (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (∆ s uc )(t) (∆ d uc )(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = (∆y)(t), ∀t ≥ 0, (G.1)
with ∆ s and ∆ d given in ( 27) and [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], respectively. In order to establish the equivalence between system P and the feedback interconnection Σ -∆, we begin by reformulating the dynamics of system P for all t ∈ [0, a 0 ), t ∈ [a k , s k+1 ) k∈N , and t ∈ [s k+1 , a k+1 ) k∈N , i.e., for all t ≥ 0.

(1) For all t ∈ [0, a 0 ): Consider the system P. We have that ẋp (t) = f (x p (t)), and using [START_REF] Liu | Network-based control via a novel analysis of hybrid systems with time-varying delays[END_REF],

y 1 (t) = y 2 (t) = uc (t) = d dt κ(x p (t)) = ∂κ(xp(t)) ∂xp ẋp (t) = ∂κ(xp(t)) ∂xp f (x p (t)
). Therefore, using the notation in [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF], we obtain

ẋp (t) = f0 (x p (t)), y(t) = y 1 (t) y 2 (t) T = h0 (x p (t)).
Note that this is the dynamics of system Σ for t ∈ [0, a 0 ), given by [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], with the functions f0 and h0 as defined in [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF], i.e., for all t ∈ [0, a 0 ), x(t) = x p (t). Additionally, as per the notation in [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], since f0 (x) = f (x), it can be concluded from the definition of system P that the function f0 is globally Lipschitz continuous with f0 (0) = 0.

(2) For all t ∈ [a k , s k+1 ), k ∈ N: The dynamics of system P is given by ẋp

(t) = f (x p (t)) + g(x p (t))u(t) = f (x p (t)) + g(x p (t))κ(x p (s k )) = f (x p (t)) + g(x p (t)) κ(x p (t)) + g(x p (t)) [κ(x p (s k )) -κ(x p (t))].
Using the definitions of sampling and delay induced errors in ( 24) and ( 25), respectively, we have e s (t) = κ(x p (s k ))-κ(x p (t)), ∀t ∈ [a k , s k+1 ), and e d (t) = 0 for all t ∈ [a k , s k+1 ). Therefore, we can reformulate the dynamics of system P for all t ∈ [a k , s k+1 ) as ẋp

(t) = f (x p (t)) + g(x p (t))κ(x p (t)) + g(x p (t))e s (t) + g(x p (t))e d (t) = f (x p (t)) + g(x p (t))κ(x p (t)) + g(x p (t)) g(x p (t))
e s (t) e d (t) T . Using the notation in [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF], this can be written as ẋp (t) = f (x p (t)) + ḡ(x p (t))ω(t), with ω(t) as in (G.1). This is the same as dynamics of system Σ for t ∈ [a k , s k+1 ), k ∈ N, given by ( 7), with ω defined in (G.1), and the functions f and ḡ defined by [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF] (G.2) Therefore, using the notation in [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF] once again, we obtain y(t) = y 1 (t) y 2 (t) T = h(x p (t)) + l(x p (t))ω(t), which is the same as y defined in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], for t ∈ [a k , s k+1 ), with x = x p .

(3) For all t ∈ [s k+1 , a k+1 ), k ∈ N: Once again, we proceed to reformulate the dynamics of system P given by ẋp

(t) = f (x p (t)) + g(x p (t))u(t) = f (x p (t)) + g(x p (t))κ(x p (s k )) = f (x p (t)) + g(x p (t))κ(x p (s k )) + g(x p (t))κ(x p (t)) -g(x p (t))κ(x p (t)) + g(x p (t)) κ(x p (s k+1 )) -g(x p (t))κ(x p (s k+1 )) = (f (x p (t)) + g(x p (t))κ(x p (t))) + g(x p (t)) [κ(x p (s k+1 )) -κ(x p (t))] + g(x p (t)) [κ(x p (s k )) -κ(x p (s k+1 ))].
Using the definitions in [START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF] and [START_REF] Pepe | On Lyapunov Krasovskii characterizations of stability notions for discrete-time systems with uncertain timevarying time delays[END_REF] and considering Hypothesis 2, we have that e s (t) = κ(x p (s k+1 ))κ(x p (t)), ∀t ∈ [s k+1 , s k+2 ) ⊃ [s k+1 , a k+1 ), and e d (t) = κ(x p (s k ))κ(x p (s k+1 )), ∀t ∈ [s k+1 , a k+1 ). Therefore, using the notation in [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF], we can reformulate the dynamics of system P for all t ∈ [s k+1 , a k+1 ) as ẋp (t) = f (x p (t)) + g(x p (t))e s (t) + g(x p (t))e d (t) = f (x p (t)) + ḡ(x p (t))ω(t). This is the same as dynamics of system Σ in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF] for all t ∈ [s k+1 , a k+1 ], k ∈ N, with ω(t) given by (G.1), and the functions f and ḡ defined by [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF], with x(t) = x p (t). Additionally, using the notation y 1 (t) = y 2 (t) = uc (t), and following the reasoning given in (G.2), we get,y(t) = y 1 (t) y 2 (t) T = h(x p (t)) + l(x p (t))ω(t), which is the same as y defined in [START_REF] Fiter | Stability of LTI systems with distributed sensors and aperiodic sampling[END_REF], for t ∈ [s k+1 , a k+1 ), k ∈ N, with x = x p . Therefore, system P can be expressed in the form of the interconnection Σ -∆, with the functions f0 , h0 , f , ḡ, h, and l defined by [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF].

H Proof of Lemma 9

(1) For t ∈ [0, s 0 ): As per the definition of ∆ s in ( 27), we have that (∆ s v)(t) = 0, ∀t ∈ [0, s 0 ). Therefore, for all θ ∈ [0, s 0 ), we have S s (θ, v(θ),

(∆ s v) (θ)) = -γ 2 s v T (θ)R s v(θ), implying that indeed ∫ t 0 S s (θ, v(θ), (∆ s v)(θ)) dθ ≤ 0. (2) For t ∈ [s k , s k+1 ), k ∈ N: We have that ∫ t s k e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ = ∫ t s k √ e β(θ-a0) (∆ s v) T (θ)R s √ e β(θ-a0) (∆ s v)(θ)dθ. Since (∆ s v)(s k ) = 0,
by applying Wirtinger's inequality [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], we obtain -a0) . As per the definition of ∆ s in [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], we have that

∫ t s k e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ ≤ 4(t-s k ) 2 π 2 ∫ t s k d dθ √ e β(θ-a0) (∆ s v)(θ) T R s d dθ √ e β(θ-a0) (∆ s v)(θ) dθ, (H.1) with d dθ √ e β(θ-a0) (∆ s v)(θ) = √ e β(θ-a0) d dθ (∆ s v)(θ) + (∆ s v)(θ) β 2 √ e β(θ
(∆ s v)(θ) = -∫ θ s k v(ψ)dψ, ∀θ ∈ [s k , s k+1 ), k ∈ N, implying that d dθ (∆ s v)(θ) = -v(θ). Therefore, d dθ ( √ e β(θ-a0) (∆ s v)(θ)) = √ e β(θ-a0) (-v(θ) + β 2 (∆ s v)(θ)), implying that d dθ √ e β(θ-a0) (∆ s v) T (θ) T R s d dθ √ e β(θ-a0) (∆ s v)(θ) = e β(θ-a0) -v(θ) + β 2 (∆ s v)(θ) T R s -v(θ) + β 2 (∆ s v)(θ) = e β(θ-a0) (v T (θ)R s v(θ) -β 2 v T (θ)R s (∆ s v)(θ) - β 2 (∆ s v) T (θ)R s v(θ)+ β 2 4 (∆ s v) T (θ)R s (∆ s v)(θ)). Hence, d dθ √ e β(θ-a0) (∆ s v) T (θ) T R s d dθ √ e β(θ-a0) (∆ s v)(θ) = e β(θ-a0) ξ (v(θ), (∆ s v)(θ)) , (H.2)
where

ξ (v(θ), (∆ s v)(θ)) ∶= v T (θ)R s v(θ) -β 2 v T (θ)R s (∆ s v)(θ) -β 2 (∆ s v) T (θ)R s v(θ) + β 2 4 (∆ s v) T (θ)R s (∆ s v)(θ) . (H.3)
Substituting (H.2) into inequality (H.1), we have for all

t ∈ [s k , s k+1 ), k ∈ N, ∫ t s k e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ ≤ 4 h2 π 2 ∫ t s k e β(θ-a0) ξ (v(θ), (∆ s v)(θ)
) dθ, where we have used that (ts k ) ≤ h for all t ∈ [s k , s k+1 ). Now, for any t ∈ [s k , s k+1 ), since (∆ s v)(t) = 0, ∀t < s 0 (see ( 27)), we can state that

∫ t 0 e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ = ∫ t s0 e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ = ∑ k-1 i=0 ∫ si+1 si e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ + ∫ t s k e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ ≤ 4 h2 π 2 [ ∑ k-1 i=0 ∫ si+1 si e β(θ-a0) ξ (v(θ), (∆ s v)(θ)) dθ + ∫ t s k e β(θ-a0) ξ (v(θ), (∆ s v)(θ)) dθ]. Therefore, ∫ t 0 e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ ≤ 4 h2 π 2 ∫ t s0 e β(θ-a0) ξ (v(θ), (∆ s v)(θ)) dθ .
(H.4) Since (∆ s v)(t) = 0 for all t < s 0 , and v(t) ∈ W mp , using the definition of ξ (v(θ), (

∆ s v)(θ)) in (H.3), ∀t ∈ [0, s 0 ), 4 h2
π 2 ∫ s0 0 e β(θ-a0) ξ (v(θ), (∆ s v)(θ)) dθ = 4 h2 π 2 ∫ s0 0 e β(θ-a0) v T (θ)R s v(θ)dθ ≥ 0.
(H.5) Therefore, by adding (H.5) and (H.4), we obtain

∫ t 0 e β(θ-a0) (∆ s v) T (θ)R s (∆ s v)(θ)dθ ≤ γ 2 s ∫ t 0 e β(θ-a0) ξ (v(θ), (∆ s v)(θ)) dθ, ∀t ≥ 0, (H.6)
with γ s = 2 h π . Substituting ξ (v(θ), (∆ s v)(θ)) from (H.3) in (H.6), we arrive at ∫ t 0 S s (θ, v(θ), (∆ s v)(θ)) dθ ≤ 0, where S s is given by [START_REF] De | A discrete-time framework for stability analysis of nonlinear networked control systems[END_REF].

I Proof of Lemma 10

From the definition of ∆ d in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], we have that

(∆ d w)(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ∀t ∈ [0, a 0 ), 0, ∀t ∈ [a k , s k+1 ), k ∈ N -∫ s k s k-1 w(θ)dθ, ∀t ∈ [s k , a k ), k ∈ N ⋆ . (I.1) (1
) For all t ∈ [0, s 1 ): We have (∆ d w)(θ) = 0 for all θ ∈ [0, s 1 ), thereby giving S d (θ, w(θ), (∆ d w)(θ)) = -e β(θ-a0) w T (θ)R d w(θ)dθ ≤ 0, ∀θ ∈ [0, s 1 ), which implies ∫ t 0 S d (θ, w(θ), (∆ d w)(θ)) dθ ≤ 0, ∀t ∈ [0, s 1 ).

(2) For all t ≥ s 1 : If t ∈ [s k , a k ) k∈N ⋆ , by virtue of Jensen's inequality, and using (I.1), we have that

e β(t-a0) (∆ d w) T (t)R d (∆ d w)(t) ≤ he β(t-a0) ∫ s k s k-1 w T (θ)R d w(θ)dθ, (I.
2) as here we used that s k -

s k+1 ≤ h, ∀k ∈ N ⋆ . Let t ∈ [s N , s N +1 ) N ∈N ⋆ , which implies that ∫ t s1 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ = ∑ N -1 k=1 ∫ a k s k e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ + ∫ s k+1 a k e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ + ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∫ t s N e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ, t ∈ [s N , a N ) ∫ a N s N e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ + ∫ t a N e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ, t ∈ [a N , s N +1 ). (I.
3) We know that for t ∈ [a k , s k+1 ) k∈N , (∆ d z)(t) = 0. Additionally, using the upper bound in (I.2), we have that

∫ t s1 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ ≤ ∑ N -1 k=1 h ∫ a k s k e β(θ-a0) ∫ s k s k-1 w T (η)R d w(η)dη dθ + ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h ∫ t s N e β(θ-a0) ∫ s N s N -1 w T (η) R d w(η)dη) dθ, t ∈ [s N , a N ) h ∫ a N s N e β(θ-a0) ∫ s N s N -1 w T (η) R d w(η)dη) dθ, t ∈ [a N , s N +1 ).
(I.4) Next, we simplify each of the integrals present in the right side of the inequality above. First, consider the term h ∫

a k s k e β(θ-a0) ∫ s k s k-1 w T (η)R d w(η)dη dθ = he -βa0 ∫ a k s k e βθ ∫ s k s k-1 w T (η)R d w(η)dη dθ. Let θ = s k + s ⇒ dθ = ds. This leads to h ∫ a k s k e β(θ-a0) ∫ s k s k-1 w T (η)R d w(η)dη dθ ≤ he -βa0 ∫ τ 0 e β(s k +s) ∫ s k s k-1 w T (η)R d w(η)dη ds.
(I.5) Since s ∈ [0, τ ] in (I.5), it can be stated that e β(s k +s) ≤ e β(s k +τ ) . Hence, he -βa0 ∫ τ 0 e β(s k +s) ( ∫

s k s k-1 w T (η)R d w(η)dη)ds ≤ he β(-a0+τ ) ∫ τ 0 e βs k ( ∫ s k s k-1 w T (η)R d w(η)dη)ds ≤ he β(-a0+τ ) ∫ τ 0 ( ∫ s k s k-1 e βs k w T (η)R d w(η)dη)ds. Here, η ∈ [s k-1 , s k ],
which allows us to make the upper bounding e βs k ≤ e β(η+ h) , thereby resulting in

h e β(-a0+τ ) ∫ τ 0 ∫ s k s k-1 e βs k w T (η)R d w(η)dη ds ≤ he β(τ + h) ∫ τ 0 ∫ s k s k-1 e β(η-a0) w T (η)R d w(η)dη ds ≤ hτ e β(τ + h) ∫ s k s k-1 e β(η-a0) w T (η)R d w(η)dη. (I.6)
Thus, by combining (I.5)-(I.6), we have that h ∫

a k s k e β(θ-a0) ∫ s k s k-1 w T (η)R d w(η)dη dθ ≤ hτ e β(τ + h) ∫ s k s k-1 e β(η-a0) w T (η)R d w(η)dη. Substituting this in (I.4) gives ∫ t s1 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ ≤ ∑ N -1 k=1 hτ e β(τ + h) ∫ s k s k-1 e β(η-a0) w T (η)R d w(η)dη + hτ e β(τ + h) ∫ s N s N -1 e β(η-a0) w T (η)R d w(η)dη ≤ hτ e β(τ + h) ∫ s N s0 e β(η-a0) w T (η)R d w(η)dη. Therefore, ∫ t s1 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ ≤ hτ e β(τ + h) ∫ s N s0 e β(η-a0) w T (η)R d w(η)dη. (I.7)
Since (∆ d w)(t) = 0 for t < s 1 (see ( 28)), we have that

∫ t s1 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ = ∫ t 0 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ. Additionally, since w ∈ W mp , we can state e β(η-a0) w T (η)R d w(η) ≥ 0, ∀η ≥ 0, thereby implying that hτ e β(τ + h) ∫ s N s0 e β(η-a0) w T (η)R d w(η)dη ≤ hτ e β(τ + h) ∫ t 0 e β(η-a0) w T (η)R d w(η)dη. Consequently, we can rewrite (I.7) as ∫ t 0 e β(θ-a0) (∆ d w) T (θ)R d (∆ d w)(θ)dθ ≤ hτ e β(τ + h) ∫ t 0 e β(η-a0) w T (η)R d w(η)dη. By rearrang- ing the terms, we have ∫ t 0 S d (θ, w(θ), (∆ d w)(θ)) dθ ≤ 0, ∀t ≥ 0, where S d (θ, w(θ), (∆ d w)(θ)
) is given by [START_REF] Xiao | Robustness analysis of asynchronous sampled-data multiagent networks with time-varying delays[END_REF].

J Proof of Theorem 12

Let us recall the linear sampled-data system P L described in Section 4.3 by [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF]. The sampling-induced error is given by

e s (t) = 0, ∀t ∈ [0, s 0 ), Kx(s k ) -Kx(t), ∀t ∈ [s k , s k+1 ), k ∈ N, = (∆ s (K ẋ)) (t)
, where ∆ s is given by [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF]. Similarly, the delay-induced error is given by

e d (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ∀t ∈ [0, a 0 ), 0, ∀t ∈ [a k-1 , s k ), k ∈ N ⋆ , Kx(s k-1 ) -Kx(s k ), ∀t ∈ [s k , a k ), k ∈ N ⋆ , = (∆ d (K ẋ)) (t)
, where ∆ d is given by [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. Additionally, the functions defined in [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF] are given by f0

(x(t)) = Ax(t), h0 (x(t)) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ KAx(t) KAx(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , f (x(t)) = Āx(t), ḡ(x(t)) = B B , h(x(t)) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Āx(t) K Āx(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , l(x(t)) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ KB KB ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (J.1)
Let us consider that condition [START_REF] Zhang | Stability of networked control systems[END_REF] holds. Then, we proceed to prove that the assumptions introduced in Theorem 1 will hold for the storage function V (x) = x T P x and the supply function where S s and S d are defined by ( 32) and ( 34), respectively, with β = α. Additionally, based on the functions given in (J.1), we have y 1 (t) = y 2 (t) = K ẋ(t). Let us now show that the assumptions in Theorem 1 are validated.

S ∶ R + × R 2n × R 2n → R
(1) Satisfying Assumption 1, i.e., [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF]: By virtue of Lemmas 9 and 10, we have that ∫ t 0 S s (θ, y 1 (θ), (∆ s y 1 )(θ)) dθ ≤ 0, ∀t ≥ 0, and ∫ t 0 S d (θ, y 2 (θ), (∆ d y 2 )(θ)) dθ ≤ 0, ∀t ≥ 0. Consequently, as per the definition of the supply function in (J.2), we obtain ∫ t 0 S (θ, y(θ), ω(θ)) dθ ≤ 0, ∀t ≥ 0.

(2) Satisfying Assumption 2, i.e., [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]: With V (x) = x T P x, P = P T > 0 and x ∈ R n , we have δ min (P ) x 2 ≤ x T P x ≤ δ max (P ) x 2 , implying Assumption 2 is satisfied with q = 2, c 1 = δ min (P ) and c 2 = δ max (P ).

(3) Satisfying Assumption 3, inequality [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]: Consider the function S (t, y(t), ω(t)) given in (J.2). Then, we need to prove that -S (t, y(t), ω(t)) ≤ ρV (x(t)) , ∀t ∈ [0, a 0 ). We proceed to prove this inequality by considering the time intervals [0, s 0 ) and [s 0 , a 0 ) separately. For all t ∈ [0, s 0 ): Using the definition of system Σ in ( 7), [START_REF] Thomas | L 2 -stability criterion for systems with decentralized asynchronous controllers[END_REF], and the operator ∆ defined in ( 8), [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] we have that y

(t) = h0 (x(t)) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ KAx(t) KAx(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
, and

ω(t) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ e s (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
, ∀t ∈ [0, s 0 ). Hence, for all t ∈ [0, s 0 ), -S (t, y(t), ω(t)) = -S t, h0 (x(t)), 0 = -(S s (t, KAx(t), 0) For all t ∈ [s 0 , a 0 ): From (J.

+ S d (t, KAx(t), 0)) = e α(t-a0) x T (t)(KA) T γ 2 s R s + γ d R d (KA)x(t) . Therefore, -S (t, y(t), ω(t)) ≤ ρ 1 V (x(t)), (
⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ 2 s R s γ 2 s α 2 R s γ 2 s α 2 R s (1 -γ 2 s α 2 4 )R s ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ y 1 (t) e s (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
, (J.7) and thus, from (J.5) and (J.6), we get S s (t, y 1 (t), e s (t)) = S s t, KAx(t), K e A(s0-t) -I x(t) = x T (t)M(t)x(t), ∀t ∈ [s 0 , a 0 ), where M(t) = e α(t-a0) = e α(t-a0)

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ KA K e A(s0-t) -I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ 2 s R s γ 2 s α 2 R s γ 2 s α 2 R s (1 -γ 2 s α 2 4 )R s ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ KA K e A(s0
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ y 2 (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γ d R d 0 0 R d ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ y 2 (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
, (J.8) and thus, from (J.5) and (J.6), S d (t, y 2 (t), e d (t)) = S d (t, KAx(t), 0) = x T (t)N (t)x(t), ∀t ∈ [s 0 , a 0 ), with N (t) = -γ d e α(t-a0) (KA) T R d (KA). Therefore, we have the total supply function S satisfying -S (t, y(t), ω(t)) = -S s (t, y 1 (t), e s (t))-S d (t, y 2 (t), e d (t)) = x T (t)M (t)x(t), where M (t) = -M(t) -N (t). Hence, for all t ∈ [s 0 , a 0 ), we can state that -S (t, y(t), ω(t)) ≤ ρ 2 V (x(t)), (J.9)

where

ρ 2 = max θ∈[s0,a0]
{δ max [M (θ)]} δ min (P ) . (J.10)

Then, from (J.3) and (J.9), we have -S (t, y(t), ω(t)) ≤ ρV (x(t)), ∀t ∈ [0, a 0 ), where ρ = max{ρ 1 , ρ 2 } with ρ 1 and ρ 2 given by (J.4) and (J.10), respectively. (4) Satisfying Assumption 3, inequality (12): We have V (x(t)) = x(t) T P x(t) for all t ≥ 0. For all t ∈ [0, a 0 ), since ẋ(t) = f0 (x(t)) = Ax(t), it holds that V (x(t)) =

x(t) T A T P + P A x(t) ≥ δmin A T P +P A δmax(P )

V (x(t)). Therefore, it is clear that inequality [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] is satisfied for any λ ≤ Consider the function W (t) = V (x(t)) + αV (x(t))e -α(t-a0) S t, y(t), e(t) , defined for all t ≥ a 0 with V (x) = x T P x, and the function S defined by (J.2). Clearly, the inequality in (13) holds if W (t) ≤ 0, for all t ≥ a 0 . Using the definitions of S s (t, y 1 (t), e s (t)) and S d (t, y 2 (t), e d (t)) in (J.7) and (J.8), respectively, and from (J.1), since y 1 (t) = y 2 (t) = K ẋ(t), for all t ≥ 0, we have that S (t, y(t), ω(t)) = e α(t-a0)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ K ẋ(t) e s (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ T Ψ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ K ẋ(t) e s (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (J.11) with, Ψ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -γ 2 s R s -γ d R d γ 2 s α 2 R s 0 γ 2 s α 2 R s (1 -γ 2 s α 2 4 )R s 0 0 0 R d ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
. From the system dynamics defined by ( 7) and (J.1), we have that

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ K ẋ(t)
e s (t)

e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā KB KB 0 I 0 0 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x(t)
e s (t) ,

e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x(
where

N = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T Ψ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
. Therefore, ∀t ≥ a 0 , we have that W (t) = V (x(t)) + αV (x(t))e -α(t-a0) S t, y(t), e(t) =

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -N ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x(
⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ T Φ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K Ā K B 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
, with Ā = A + BK, B = B B , and Φ = -Ψ described in [START_REF] Zhang | Stability Analysis of Networked Control Systems[END_REF]. A sufficient condition for W (t) ≤ 0, ∀t ≥ a 0 is Γ ≤ 0, and guaranteed by [START_REF] Zhang | Stability of networked control systems[END_REF]. Consequently, we have proved that the inequality (13) is satisfied.

We have shown that all the assumptions of Theorem 1 hold for V (x) = x T P x and S(t, y(t), ω(t)) defined by (J.2) and hence, using Theorem 1, system P is exponentially stable with a decay rate greater than or equal to α 2.

Fig. 1 .

 1 Fig.1. Feasible values of h and τ for the nonlinear system (37) with α = 0.1, in the large-delay case (in red), and in the small-delay case (in blue).

J. 3 ) with ρ 1 where γ s = 2 hπ

 312 = e -ατ0 δ max (KA) T γ 2 s R s + γ d R d (KA) δ min (P ), (J.4) and γ d = hτ e α( h+τ ) .

  δmin A T P +P A δmax(P ) . (5) Satisfying Assumption 3, inequality (13):

  also gives the dependence of the

		0.2				α = 0.01, large-delay case
						α = 0.01, small-delay case
		0.15				α = 1, large-delay case α = 1, small-delay case
						α = 2, large-delay case
	τ	0.1				α = 2, small-delay case Analytical bounds for
						constant h k and τ k
		0.05
		0	0	0.1	0.2	0.3	h	0.4	0.5	0.6	0.7

  , i.e., for all t ∈ [a k , s k+1 ), with x = x p .

			Additionally, we have
	uc (t) = d dt κ(x p (t)) and hence,
	y 1 (t) = y 2 (t) =	∂κ(xp(t)) ∂xp	f (x p (t)) + ḡ(x p (t))ω(t) .

  given by S (t, y(t), ω(t)) = S s t, I 0 y(t), I 0 ω(t) +S d t, 0 I y(t), 0 I ω(t) = S s (t, y 1 (t), e s (t)) + S d (t, y 2 (t), e d (t))

	(J.2)

  open loop for all t ∈ [s 0 , a 0 ), x(s 0 ) = e A(s0-t) x(t). Therefore, we have that , ∀t ∈ [s 0 , a 0 ). (J.6) Now, consider the function S s defined in[START_REF] De | A discrete-time framework for stability analysis of nonlinear networked control systems[END_REF]. Since we have already shown in Lemma 7 that (∆ s y 1 )(t) = e s (t), we have that S s (t, y 1 (t), (∆ s y 1 )(t)) = S s (t, y 1 (t), e s (t))

	with and ω(t) = system is in ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ e s (t) e d (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ K e A(s0-t) -I x(t) 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = e α(t-a0) ⎡ ⎢ ⎢ y 1 (t) ⎢ ⎢ ⎣ e s (t)	1), we have y(t) = y 1 (t) = y 2 (t) = KAx(t), e s (t) e d (t) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Kx(s 0 ) -Kx(t) 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . Since the ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ y 1 (t) y 2 (t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (J.5)

  -t) -I

	considering the function S	⎤ ⎥ ⎥ ⎥ . ⎥ ⎦	Similarly,

d defined by

[START_REF] Xiao | Robustness analysis of asynchronous sampled-data multiagent networks with time-varying delays[END_REF]

, we have that

S d (t, y 2 (t), (∆ d y 2 )(t)) = S d (t, y 2 (t), e d (t))

Acknowledgements

This work was supported by project UCoCoS, funded by the European Union's EU Framework Programme for Research and Innovation, Horizon H2020, Grant Agreement No: 675080.