Wenjie Mei
email: wenjie.mei@inria.fr

Denis Efimov

Rosane Ushirobira

| Alexander Aleksandrov

On convergence conditions for generalized Persidskii systems

Keywords: convergence conditions, generalized Persidskii systems, LMI

The convergence conditions for a class of generalized Persidskii systems and their discretized dynamics are introduced, which can be checked through linear (matrix) inequalities. The case of almost periodic convergence for this class of systems with almost periodic input is also studied. The proposed results are applied to a Lotka-Volterra model and opinion dynamics.

INTRODUCTION

A dynamical system is convergent if it admits a unique and bounded asymptotically stable solution [START_REF] Pavlov | Convergent Systems: Analysis and Synthesis[END_REF][START_REF] Pavlov | Convergent Systems: Nonlinear Simplicity[END_REF] . So computing only one trajectory is sufficient to determine the asymptotic behavior of all others. This property is similar to the incremental stability [START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF][START_REF] Rüffer | Convergent systems vs. incremental stability[END_REF] (without the boundedness requirement) and the contraction theory is frequently used for its analysis [START_REF] Jouffroy | Tutorial on Incremental Stability Analysis using Contraction Theory[END_REF] . The convergence property has received increased attention due to its usefulness in system analysis and design, including the case of several co-existing solutions for nonlinear output regulation [START_REF] Pavlov | Uniform Output Regulation of Nonlinear Systems[END_REF] , synchronization [START_REF] Pogromsky | Passivity Based Design of Synchronizing Systems[END_REF][START_REF] Cai | Synchronization of complex dynamical networks by the incremental ISS approach[END_REF][START_REF] Stan | Analysis of Interconnected Oscillators by Dissipativity Theory[END_REF] , steady-state analysis of nonlinear systems [START_REF] Pavlov | Frequency Response Functions for Nonlinear Convergent Systems[END_REF] and many other control problems.

It is evident that not all nonlinear systems are convergent: the trajectories may be convergent for some inputs (e.g., zero input), and an example of such a behavior is demonstrated by Duffing's model [START_REF] Liu | Convergent systems[END_REF] , in which multiple limit cycles phenomena and subharmonic oscillations may occur. Thus, for nonlinear systems, it is required to consider additional conditions for investigating the convergence property.

We may additionally ask for preservation conditions for the attracting solution of the (almost) periodicity of the input [START_REF] Kosov | Investigation of convergence of large scale almost periodic systems by means of comparison vector functions with components as forms of even degrees[END_REF][START_REF] Kosov | On the convergence phenomenon in complex almost periodic systems[END_REF] . Note that in nonlinear systems, the frequency of entertainment is not necessarily kept by the output 14 , and a typical example is again the Duffing's model, which can generate chaotic trajectories under harmonic excitation. The relations between the existence of almost periodic solutions and properties of the right-hand sides of dynamical systems were intensively studied during the last century (see, e.g., Markov's theorem [START_REF] Levitan | Almost periodic funcitons and differential equations[END_REF] ).

The convergence property itself also has a rich history of investigation. For example, Pliss [START_REF] Pliss | Nonlocal problems of the theory of oscillations[END_REF] studied convergence of the system ẋ = 𝑓 (𝑡, 𝑥) with the periodic right-hand side. Later, Demidovich [START_REF] Pavlov | Convergent dynamics, a tribute to Boris Pavlovich Demidovich[END_REF] extended the particular case analyzed by Pliss to a more general 𝑓 . In the same vein, Zubov [START_REF] Zubov | Oscillations in Nonlinear and Controlled Systems[END_REF][START_REF] Zubov | Theory of Oscillations[END_REF] established a criterion of periodic/almost periodic convergence for nonlinear systems. Further exploration of convergence properties for nonlinear systems includes convergence-like properties by Lyapunov's second method [START_REF] Yoshizawa | Stability theory by Lyapunov's second method[END_REF] , sufficient conditions for the convergence of a system with a scalar nonlinearity satisfying an incremental sector condition [START_REF] Yakubovich | Method of matrix unequalities in theory of nonlinear control systems stability. I. Forced oscillations absolute stability[END_REF] , the convergence of perturbed nonlinear systems [START_REF] Hallam | Convergence of solutions of perturbed nonlinear differential equations[END_REF][START_REF] Angeli | A Lyapunov approach to incremental stability properties[END_REF] , and convergence conditions for a class of nonlinear dynamics [START_REF] Aleksandrov | Convergence conditions for some classes of nonlinear systems[END_REF] .

Digital revolution highlighted importance and originated discrete-time modeling for many phenomena. In addition, for realworld systems having the models formulated in continuous time, for implementation or simulation, these models have to be discretized. For strongly nonlinear systems the obtained discrete-time representations may loose the stability and convergence properties of the continuous-time counterparts [START_REF] Efimov | Realization and Discretization of Asymptotically Stable Homogeneous Systems[END_REF][START_REF] Polyakov | Consistent Discretization of Finite-Time and Fixed-Time Stable Systems[END_REF] . This motivates development of conditions of similar quality performance characteristics for the systems functioning in discrete time.

In this paper, the convergence property and existence of almost periodic trajectories will be studied in continuous and discrete time for a class of Persidskii systems, which was introduced in [START_REF] Barbashin | On construction of Lyapunov functions for nonlinear systems[END_REF][START_REF] Persidskii | Concerning problem of absolute stability[END_REF] as ẋ = 𝐴 0 𝑥 + 𝐴 1 𝑓 (𝑥), and recently its generalized version was investigated in [START_REF] Efimov | Robust stability analysis and implementation of Persidskii systems[END_REF][START_REF] Mei | Feedback synchronization in Persidskii systems[END_REF] by taking into account multiple nonlinearities: ẋ = 𝐴 0 𝑥+ ∑ 𝑀 𝑖=1 𝐴 𝑖 𝑓 𝑖 (𝑥) (more detailed definition is given below). The advantage of this class of models is that despite its important nonlinear nature (with 𝑀 ≥ 1), a Lyapunov function is suggested in the literature such that the analysis of its properties can be performed by solving linear matrix inequalities (LMIs), which is a constructive approach.

The organization of this paper is as follows. In Section 2, the preliminaries and the definitions of used stability properties are presented. The class of considered systems and the imposed conditions for nonlinearities are introduced in Section 3. The basic conditions of convergence for generalized Persidskii systems, and of almost periodic convergence for those systems with Metzler coefficient matrices are given in Section 4. In Section 5 convergence conditions for two types of discretizations of the considered systems are studied. The Lotka-Volterra model and opinion dynamics are considered to illustrate the efficiency of the proposed results in Section 6.

NOTATION

• ℕ, ℤ and ℝ represent the sets of natural numbers, integers and real numbers, respectively, ℝ + = {𝑠 ∈ ℝ ∶ 𝑠 ≥ 0}. ℝ 𝑛 (ℝ 𝑛 + ) and ℝ 𝑚×𝑛 denote the vector spaces of 𝑛-tuples of real numbers (nonnegative real numbers) and 𝑚 × 𝑛 real matrices, respectively. The set of 𝑛 × 𝑛 diagonal matrices with nonnegative elements on the diagonal is denoted by 𝔻 𝑛 + . • The symbol ‖⋅‖ refers to the Euclidean norm on ℝ 𝑛 (and the induced matrix norm ‖𝐴‖ for a matrix 𝐴 ∈ ℝ 𝑚×𝑛 ).

• The transpose of a matrix 𝐴 ∈ ℝ 𝑛×𝑛 is denoted by 𝐴 ⊤ .

• For 𝑝, 𝑛 ∈ ℕ with 𝑝 ≤ 𝑛, the notation 𝑝, 𝑛 is used to represent the set {𝑝, … , 𝑛}. For all 𝑖, 𝑗 ∈ 𝑝, 𝑛, let (𝐵 𝑖,𝑗 ) 𝑛 𝑖,𝑗=𝑝 denote the block matrix

[ 𝐵 𝑝,𝑝 ⋯ 𝐵 𝑝,𝑛 ⋮ ⋱ ⋮ 𝐵 𝑛,𝑝 ⋯ 𝐵 𝑛,𝑛 ] .
• The 𝑝 × 𝑛 block diagonal matrix with matrices 𝑣 𝑖 ∈ ℝ 𝑝 𝑖 ×𝑛 𝑖 (𝑝 = ∑ 𝑁 𝑖=1 𝑝 𝑖 , 𝑛 = ∑ 𝑁 𝑖=1 𝑛 𝑖 ) along the main diagonal is denoted by diag(𝑣 1 , ..., 𝑣 𝑁 ). Let 𝐼 𝑛 stand for the 𝑛 × 𝑛 identity matrix, and denote the vector of dimension 𝑛 with all elements equal 1 by 𝟏 𝑛 .

• For a vector 𝑣 ∈ ℝ 𝑛 , 𝑣 > 0 (𝑣 < 0) represents 𝑣 𝑖 > 0 (𝑣 𝑖 < 0) for 𝑖 ∈ 1, 𝑛. For 𝑟 ∈ ℝ, let |𝑟| denote its absolute value.

• For a Lebesgue measurable function 𝑢

∶ ℝ → ℝ 𝑞 , define the norm ‖𝑢‖ (𝑡 1 ,𝑡 2 ) = ess sup 𝑡∈(𝑡 1 ,𝑡 2 ) ‖𝑢(𝑡)‖ for (𝑡 1 , 𝑡 2 ) ⊆ ℝ. We denote by ℒ 𝑞 ∞ the space of functions 𝑢 with ‖𝑢‖ ∞ ∶=‖𝑢‖ (-∞,+∞) < +∞. For a bounded input sequence { ũ(𝑘)} 𝑘∈ℤ , ũ(𝑘) ∈ ℝ 𝑞 , define the norm ‖ ũ‖ 𝑘 1 ,𝑘 2 = sup 𝑘 ′ ∈𝑘 1 ,𝑘 2 ‖ ũ(𝑘 ′ )‖ for 𝑘 1 , 𝑘 2 ∈ ℤ. Let 𝒰 𝑞 ∞ denote the space of functions ũ with ‖ ũ‖ ∞ ∶=‖ ũ‖ 𝑘 1 ,𝑘 2 < +∞ for 𝑘 1 → -∞, 𝑘 2 → +∞. • A continuous function 𝜎 ∶ ℝ + → ℝ + belongs to class 𝒦 if it is strictly increasing and 𝜎(0) = 0; it belongs to class 𝒦 ∞ if it is also unbounded. A continuous function 𝛽 ∶ ℝ + × ℝ + → ℝ + belongs to class 𝒦 ℒ if 𝛽(⋅, 𝑟) ∈ 𝒦 and 𝛽(𝑟, ⋅
) is a decreasing to zero function for any fixed 𝑟 > 0.

• For a continuously differentiable function 𝑉 ∶ ℝ 𝑛 → ℝ, denote by ∇𝑉 (𝜈)𝑓 (𝜈) the Lie derivative of 𝑉 along the vector field 𝑓 evaluated at point 𝜈 ∈ ℝ 𝑛 .

PRELIMINARIES

Stability properties in continuous time

Let the differential equation

ẋ(𝑡) = 𝐹 (𝑥(𝑡), 𝑢(𝑡)), 𝑡 ≥ 𝑡 0 ∈ ℝ, 𝑦(𝑡) = 𝐻(𝑥(𝑡)) (1)
be given, where 𝑥(𝑡) ∈ ℝ 𝑛 is the state vector; 𝑢(𝑡) ∈ ℝ 𝑞 is the external input, 𝑢 ∈ ℒ 𝑞 ∞ ; 𝑦(𝑡) ∈ ℝ 𝑝 is the output vector. Moreover, 𝐹 ∶ ℝ 𝑛+𝑞 → ℝ 𝑛 is a nonlinear continuous mapping and 𝐻 ∶ ℝ 𝑛 → ℝ 𝑝 is continuously differentiable. For an initial state 𝑥 0 ∈ ℝ 𝑛 , initial time 𝑡 0 ∈ ℝ and 𝑢 ∈ ℒ 𝑞 ∞ we denote a corresponding solution of the system (1) by 𝑥(𝑡, 𝑥 0 , 𝑡 0 , 𝑢) for the values of 𝑡 ≥ 𝑡 0 the solution exists, so the corresponding output is 𝑦(𝑡, 𝑥 0 , 𝑡 0 , 𝑢) = 𝐻 ( 𝑥(𝑡, 𝑥 0 , 𝑡 0 , 𝑢) ) . In the case that 𝑡 0 = 0 (i.e., 𝑡 ∈ ℝ + ), we simplify the corresponding denotations to: 𝑥(𝑡, 𝑥 0 , 𝑢) and 𝑦(𝑡, 𝑥 0 , 𝑢), which are also applicable in the sequel. Definition 4 ( 31 ). The system (1) is called forward complete if for all 𝑥 0 ∈ ℝ 𝑛 and 𝑢 ∈ ℒ 𝑞 ∞ , the solution 𝑥(𝑡, 𝑥 0 , 𝑢) is uniquely defined for all 𝑡 ∈ ℝ + . Definition 5 [START_REF] Sontag | Lyapunov Characterizations of Input to Output Stability[END_REF]. A forward complete system (1) is said to be 1. input-to-output stable (IOS) if there exist 𝛽 ∈ 𝒦 ℒ and 𝛾 ∈ 𝒦 such that

‖𝑦(𝑡, 𝑥 0 , 𝑢)‖ ≤ 𝛽 ( ‖𝑥 0 ‖, 𝑡 ) + 𝛾(‖𝑢‖ ∞ ), ∀𝑡 ∈ ℝ +
for any 𝑥 0 ∈ ℝ 𝑛 and 𝑢 ∈ ℒ 𝑞 ∞ . In the special case when 𝑦 = 𝑥, the IOS property is called input-to-state stability (ISS).

2. state-independent input-to-output stable (SIIOS) if there exist 𝛽 ∈ 𝒦 ℒ , 𝛾 ∈ 𝒦 such that ‖𝑦(𝑡, 𝑥 0 , 𝑢)‖ ≤ 𝛽(‖𝐻(𝑥 0 )‖, 𝑡) + 𝛾(‖𝑢‖ ∞ ), ∀𝑡 ∈ ℝ + for any 𝑥 0 ∈ ℝ 𝑛 and 𝑢 ∈ ℒ 𝑞 ∞ .

3. robustly output stable (ROS) if there exist a smooth function 𝛼 ∈ 𝒦 ∞ and 𝛽 ∈ 𝒦 ℒ such that the system

ẋ = F (𝑥, 𝜅) ∶= 𝐹 (𝑥, 𝜅𝛼(‖𝐻(𝑥)‖)) (2)
is forward complete, and the estimate

‖𝑦 𝛼 (𝑡, 𝑥 0 , 𝜅)‖ ≤ 𝛽(‖𝑥 0 ‖, 𝑡), ∀𝑡 ∈ ℝ + is satisfied for all 𝑥 0 ∈ ℝ 𝑛 , 𝜅 ∈ ℒ 𝑞
∞ with ‖𝜅‖ ∞ ≤ 1 and 𝑦 𝛼 (𝑡, 𝑥 0 , 𝜅) = 𝐻(𝑥(𝑡, 𝑥 0 , 𝜅)) denotes the output function of the system (2). Definition 6 [START_REF] Sontag | Lyapunov Characterizations of Input to Output Stability[END_REF]. A forward complete system (1) is said to be uniformly bounded-input-bounded-state stable (UBIBS) if there exists 𝛽 ∈ 𝒦 such that ‖𝑥(𝑡, 𝑥 0 , 𝑢)‖ ≤ max{𝛽(‖𝑥 0 ‖), 𝛽(‖𝑢‖ ∞ )}, ∀𝑡 ∈ ℝ + for all 𝑥 0 ∈ ℝ 𝑛 and 𝑢 ∈ ℒ 𝑞 ∞ .

Convergence definitions

Definition 7 ( 1 ). The system (1) is convergent if it admits a unique bounded solution for 𝑡 ∈ ℝ that is globally asymptotically stable.

Definition 8 [START_REF] Zubov | Theory of Oscillations[END_REF]. The system (1) possesses almost periodic convergence property, if it is convergent and the admitted solution is almost periodic.

Stability properties in discrete time

Consider a discrete-time system:

𝑥(𝑘 + 1) = F (𝑥(𝑘), 𝑢(𝑘)), 𝑦(𝑘) = 𝐻(𝑥(𝑘)), (3)
where 𝑥(𝑘) ∈ ℝ 𝑛 , 𝑢(𝑘) ∈ ℝ 𝑞 and 𝑦(𝑘) ∈ ℝ 𝑝 are defined as before; 𝑘 0 ≤ 𝑘 ∈ ℤ with 𝑘 0 ∈ ℤ; F ∶ ℝ 𝑛+𝑞 → ℝ 𝑛 is a continuous mapping; 𝑢 ∈ 𝒰 𝑞 ∞ , the mapping 𝐻 and the solution for (3) are defined as in the system (1). Definition 9 ( 32 ). The system (3) is called input-to-state stable (ISS) if there exist 𝛽 ∈ 𝒦 ℒ and 𝛾 ∈ 𝒦 such that ‖𝑥(𝑘, 𝑥 0 , 𝑢)‖ ≤ 𝛽

( ‖𝑥 0 ‖, 𝑘 ) + 𝛾(‖𝑢‖ ∞ )
for any 𝑘 ∈ ℕ, 𝑥 0 ∈ ℝ 𝑛 and 𝑢 ∈ 𝒰 𝑞 ∞ .

Lyapunov characterizations

Definition 10 ( 31 ). For the system (1), a smooth function

𝑉 ∶ ℝ 𝑛 → ℝ + is 1. an ISS-Lyapunov function if there exist 𝛼 1 , 𝛼 2 , 𝛼 3 ∈ 𝒦 ∞ and 𝜒 ∈ 𝒦 such that 𝛼 1 (‖𝑥‖) ≤ 𝑉 (𝑥) ≤ 𝛼 2 (‖𝑥‖), (4)
‖𝑥‖ ≥ 𝜒(‖𝑢‖) ⇒ ∇𝑉 (𝑥)𝐹 (𝑥, 𝑢) ≤ -𝛼 3 (‖𝑥‖) for all 𝑥 ∈ ℝ 𝑛 and 𝑢 ∈ ℝ 𝑞 . 2. an IOS-Lyapunov function if there exist 𝛼 1 , 𝛼 2 ∈ 𝒦 ∞ , 𝜒 ∈ 𝒦 and 𝛼 3 ∈ 𝒦 ℒ such that 𝛼 1 (‖𝐻(𝑥)‖) ≤ 𝑉 (𝑥) ≤ 𝛼 2 (‖𝑥‖), (5)
𝑉 (𝑥) ≥ 𝜒(‖𝑢‖) ⇒ ∇𝑉 (𝑥)𝐹 (𝑥, 𝑢) ≤ -𝛼 3 (𝑉 (𝑥), ‖𝑥‖)
for all 𝑥 ∈ ℝ 𝑛 and 𝑢 ∈ ℝ 𝑞 .

3. an SIIOS-Lyapunov function if there exist 𝛼 1 , 𝛼 2 ∈ 𝒦 ∞ and 𝜒, 𝛼 3 ∈ 𝒦 such that

𝛼 1 (‖𝐻(𝑥)‖) ≤ 𝑉 (𝑥) ≤ 𝛼 2 (‖𝐻(𝑥)‖), (6)
𝑉 (𝑥) ≥ 𝜒(‖𝑢‖) ⇒ ∇𝑉 (𝑥)𝐹 (𝑥, 𝑢) ≤ -𝛼 3 (𝑉 (𝑥))
for all 𝑥 ∈ ℝ 𝑛 and 𝑢 ∈ ℝ 𝑞 .

a ROS-Lyapunov function if there exist 𝛼

1 , 𝛼 2 ∈ 𝒦 ∞ , 𝜒 ∈ 𝒦 and 𝛼 3 ∈ 𝒦 ℒ such that 𝛼 1 (‖𝐻(𝑥)‖) ≤ 𝑉 (𝑥) ≤ 𝛼 2 (‖𝑥‖), (7)
‖𝐻(𝑥)‖ ≥ 𝜒(‖𝑢‖) ⇒ ∇𝑉 (𝑥)𝐹 (𝑥, 𝑢) ≤ -𝛼 3 (𝑉 (𝑥), ‖𝑥‖) for all 𝑥 ∈ ℝ 𝑛 and 𝑢 ∈ ℝ 𝑞 . Definition 11 ( 32 ). For the system (3), a smooth function 𝑉 ∶ ℝ 𝑛 → ℝ + is an ISS-Lyapunov function if there exist 𝛼 1 , 𝛼 2 , 𝛼 3 ∈ 𝒦 ∞ and 𝜒 ∈ 𝒦 such that 𝛼 1 (‖𝑥‖) ≤ 𝑉 (𝑥) ≤ 𝛼 2 (‖𝑥‖), (8)
‖𝑥‖ ≥ 𝜒(‖𝑢‖) ⇒ 𝑉 ( F (𝑥, 𝑢)) -𝑉 (𝑥) ≤ -𝛼 3 (‖𝑥‖) for all 𝑥 ∈ ℝ 𝑛 and 𝑢 ∈ ℝ 𝑞 .
Theorem 1 [START_REF] Sontag | On characterizations of input-to-state stability with respect to compact sets[END_REF][START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] ). The system (1) or ( 3) is ISS if and only if it admits an ISS-Lyapunov function.

Theorem 2 ( 31 ). A UBIBS system (1) is IOS (ROS) if and only if it admits an IOS (ROS)-Lyapunov function.

Theorem 3 [START_REF] Sontag | Lyapunov Characterizations of Input to Output Stability[END_REF]. A forward complete system (1) is SIIOS if and only if it admits an SIIOS-Lyapunov function.

Remark 1. The UBIBS requirement can be dropped in the sufficient conditions of IOS or ROS if the system (1) is forward complete (or it possesses the unboundedness observability property [START_REF] Angeli | Forward completeness, unboundedness observability, and their Lyapunov characterizations[END_REF] ) and admits an IOS-Lyapunov function satisfying the conditions (5), with

𝑉 (𝑥) ≥ 𝜒(‖𝑢‖) ⇒ ∇𝑉 (𝑥)𝐹 (𝑥, 𝑢) ≤ -𝛼 3 (𝑉 (𝑥)) (9)
or a ROS-Lyapunov function with the conditions (7), and

‖𝐻(𝑥)‖ ≥ 𝜒(‖𝑢‖) ⇒ ∇𝑉 (𝑥)𝐹 (𝑥, 𝑢) ≤ -𝛼 3 (𝑉 (𝑥)) (10)
for all 𝑥 ∈ ℝ 𝑛 and 𝑢 ∈ ℝ 𝑞 , some 𝛼 1 , 𝛼 2 ∈ 𝒦 ∞ and 𝜒, 𝛼 3 ∈ 𝒦 , respectively.

In the rest of the paper, to lighten the notation the time-dependency of variables might remain implicitly understood, for instance we will write 𝑥 for 𝑥(𝑡) or 𝑥(𝑘).

PROBLEM STATEMENT

Consider the following system in generalized Persidskii form [START_REF] Persidskii | Concerning problem of absolute stability[END_REF] :

ẋ(𝑡) = 𝐴 0 𝑥(𝑡) + 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑥(𝑡)) + 𝜑(𝑡), 𝑡 ≥ 𝑡 0 , ( 11
)
where 𝑥(𝑡) ∈ ℝ 𝑛 is the state vector; 𝑡 0 ∈ ℝ denotes the initial time and

𝑥(𝑡 0 ) = 𝑥 0 ; 𝐴 𝑠 ∈ ℝ 𝑛×𝑛 for 𝑠 ∈ 0, 𝑀 (𝑀 ∈ ℕ ⧵ {0}); 𝐹 𝑗 (𝑥(𝑡)) = [𝑓 1 𝑗 (𝑥 1 (𝑡)) … 𝑓 𝑛 𝑗 (𝑥 𝑛 (𝑡))] ⊤ for 𝑗 ∈ 1
, 𝑀 are the functions ensuring the existence of the solutions of the system (11) at least locally in time; the disturbance input

𝜑 = [ 𝜑 1 … 𝜑 𝑛 ] ⊤ ∈ ℒ 𝑛 ∞ .
In this paper, the corresponding term (in sum or a sequence) must be omitted if for an index the upper bound is smaller than the lower one.

The sector restrictions on In such a case 𝑚 = 0 implies that all nonlinearities are bounded (at least in positive or negative direction for a component of the state vector).

𝑓 𝑖 𝑗 ∶ ℝ → ℝ, 𝑗 ∈ 1, 𝑀, 𝑖 ∈ 1,
Assumption 2. Assume that the functions 𝑓 𝑖 𝑗 are continuous and strictly increasing for all 𝑖 ∈ 1, 𝑛 and 𝑗 ∈ 1, 𝑀. Note that 𝜇 = 𝑀 under Assumption 2. Along with (11), consider the system

ẏ(𝑡) = 𝐴 0 𝑦(𝑡) + 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑦(𝑡)) + 𝜑(𝑡) ( 12
)
with the same input but with different initial conditions 𝑦(𝑡 0 ) ∈ ℝ 𝑛 . Define 𝑧 = 𝑦 -𝑥, then the error dynamics of ( 11), ( 12) is

ż = 𝐴 0 𝑧 + 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐺 𝑗 (𝑥, 𝑧), ( 13
)
where

𝐺 𝑗 (𝑥, 𝑧) = 𝐹 𝑗 (𝑧 + 𝑥) -𝐹 𝑗 (𝑥) = [ 𝑔 1 𝑗 (𝑥 1 , 𝑧 1 ) … 𝑔 𝑛 𝑗 (𝑥 𝑛 , 𝑧 𝑛 ) ] ⊤ ∈ ℝ 𝑛 , 𝑔 𝑟 𝑗 (𝑥 𝑟 , 𝑧 𝑟 ) ∶= 𝑓 𝑟 𝑗 (𝑧 𝑟 + 𝑥 𝑟 ) -𝑓 𝑟 𝑗 (𝑥 𝑟 ), ∀𝑗 ∈ 1, 𝑀, 𝑟 ∈ 1, 𝑛.
Note that for any 𝑥 ∈ ℝ 𝑛 the functions 𝐺 𝑗 in the variable 𝑧 ∈ ℝ 𝑛 satisfy the properties formulated in assumptions 1 and 2.

We then impose the following assumption used for the stability analysis of the system (13) in Section 4:

Assumption 3. Assume that there exist 𝑆 𝑗 0 , 𝑆 𝑗 1 , 𝑆 𝑗 2 , 𝑆 𝑗,𝑗 ′ 3 , 𝐻 𝑗 0 , 𝐻 𝑗 1 , 𝐻 𝑗 2 , 𝐻 𝑗,𝑗 ′ 3 ∈ 𝔻 𝑛 + with 𝑗, 𝑗 ′ ∈ 1, 𝑀 such that 𝐺 𝑗 (𝑥, 𝑧) ⊤ 𝐺 𝑗 (𝑥, 𝑧) ≤ 𝑧 ⊤ 𝑆 𝑗 0 𝑧 + 2𝑧 ⊤ 𝑆 𝑗 1 𝐺 𝑗 (𝑥, 𝑧) + 2𝑧 ⊤ 𝑆 𝑗 2 𝐹 𝑗 (𝑧) + 2 𝑀 ∑ 𝑗 ′ =1 𝐺 𝑗 (𝑥, 𝑧) ⊤ 𝑆 𝑗,𝑗 ′ 3 𝐹 𝑗 ′ (𝑧)
and

𝐹 𝑗 (𝑧) ⊤ 𝐹 𝑗 (𝑧) ≤ 𝑧 ⊤ 𝐻 𝑗 0 𝑧 + 2𝑧 ⊤ 𝐻 𝑗 1 𝐺 𝑗 (𝑥, 𝑧) + 2𝑧 ⊤ 𝐻 𝑗 2 𝐹 𝑗 (𝑧) + 2 𝑀 ∑ 𝑗 ′ =1 𝐺 𝑗 (𝑥, 𝑧) ⊤ 𝐻 𝑗,𝑗 ′ 3 𝐹 𝑗 ′ (𝑧)
for all 𝑥, 𝑦 ∈ ℝ 𝑛 and 𝑧 = 𝑥 -𝑦.

Any globally Lipschitz function 𝐹 𝑗 satisfies such a condition (moreover, in our setting, the local Lipschitz continuity is enough, as it will be explained in Remark 2 below).

Note that for analysis of the convergence property in (11) we need to consider the system with 𝑡 ∈ ℝ. However, the time dependence of the right-hand side of (11) comes through the input 𝜑(𝑡) only, then using IOS/ISS framework we can investigate the global stability of that system for 𝑡 ∈ ℝ + with 𝜑 ∈  𝑛 ∞ , which allows us next to extend the obtained results to 𝑡 ∈ ℝ.

MAIN RESULTS

The conditions of (almost periodic) convergence of ( 11) are given in this section for the general case and for the case of positive systems (all matrices 𝐴 𝑠 , 𝑠 ∈ 0, 𝑀 are Metzler).

General convergence conditions

Theorem 4. Let assumptions 1, 2 and 3 be satisfied. If there exist 0

≤ 𝑃 = 𝑃 ⊤ , P = P ⊤ ∈ ℝ 𝑛×𝑛 ; {Λ 𝑗 = diag(Λ 𝑗,1 , … , Λ 𝑗,𝑛 )} 𝑀 𝑗=1 , { Λ𝑗 = diag( Λ𝑗,1 , … , Λ𝑗,𝑛 )} 𝑀 𝑗=1 , {Ξ 𝑠 } 𝑀 𝑠=0 , {Υ 𝑠,𝑟 } 0≤𝑠<𝑟≤𝑀 , { Υ𝑗,𝑗 ′ } 𝑀 𝑗,𝑗 ′ =1 , {Γ 𝑗 } 𝑀 𝑗=1 , {Ω 𝑗 } 𝑀 𝑗=1 ⊂ 𝔻 𝑛 + ; Ξ0 ∈ 𝔻 𝑛 + ; 0 < Φ = Φ ⊤ ∈ ℝ 𝑛×𝑛 ; 𝛾, 𝜃 > 0 such that 𝑃 + 𝜇 ∑ 𝑗=1 Λ 𝑗 > 0; 𝑄 = 𝑄 ⊤ ≤ 0, (14)
P + 𝜇 ∑ 𝑗=1 Λ𝑗 > 0; Q = Q⊤ ≤ 0, (15)
Ξ 0 + 𝑀 ∑ 𝑗=1 Υ 0,𝑗 + 𝑚 ∑ 𝑠=1 Ξ 𝑠 + 𝑚 ∑ 𝑠=1 𝑚 ∑ 𝑟=𝑠+1 Υ 𝑠,𝑟 > 0, (16)
Ξ0 -𝛾 𝑀 ∑ 𝑗=1 𝑆 𝑗 0 -𝜃 𝑀 ∑ 𝑗=1 𝐻 𝑗 0 ≥ 0, Γ 𝑗 -𝛾𝑆 𝑗 1 -𝜃𝐻 𝑗 1 ≥ 0, Ω 𝑗 -𝛾𝑆 𝑗 2 -𝜃𝐻 𝑗 2 ≥ 0, Υ𝑗,𝑗 ′ -𝛾𝑆 𝑗,𝑗 ′ 3 -𝜃𝐻 𝑗,𝑗 ′ 3 ≥ 0, Ξ0 -𝛾 𝑀 ∑ 𝑗=1 𝑆 𝑗 0 -𝜃 𝑀 ∑ 𝑗=1 𝐻 𝑗 0 + 𝑀 ∑ 𝑗=1 ( Γ 𝑗 -𝛾𝑆 𝑗 1 -𝜃𝐻 𝑗 1 + Ω 𝑗 -𝛾𝑆 𝑗 2 -𝜃𝐻 𝑗 2 ) + 𝑀 ∑ 𝑗=1 𝑀 ∑ 𝑗 ′ =1 Υ𝑗,𝑗 ′ -𝛾𝑆 𝑗,𝑗 ′ 3 -𝜃𝐻 𝑗,𝑗 ′ 3 > 0, (17)
where

𝑄 = (𝑄 𝑎, 𝑏 ) 𝑀+2 𝑎, 𝑏=1 , Q = ( Q𝑎, 𝑏 ) 3 𝑎, 𝑏=1
with

𝑄 1,1 = 𝐴 ⊤ 0 𝑃 + 𝑃 𝐴 0 + Ξ 0 , 𝑄 𝑗+1,𝑗+1 = 𝐴 ⊤ 𝑗 Λ 𝑗 + Λ 𝑗 𝐴 𝑗 + Ξ 𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 1,𝑗+1 = 𝑃 𝐴 𝑗 + 𝐴 ⊤ 0 Λ 𝑗 + Υ 0,𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 𝑠+1,𝑟+1 = 𝐴 ⊤ 𝑠 Λ 𝑟 + Λ 𝑠 𝐴 𝑟 + Υ 𝑠,𝑟 , 𝑠 ∈ 1, 𝑀 -1, 𝑟 ∈ 𝑠 + 1, 𝑀, 𝑄 1, 𝑀+2 = 𝑃 , 𝑄 𝑀+2, 𝑀+2 = -Φ, 𝑄 𝑗+1, 𝑀+2 = Λ 𝑗 , 𝑗 ∈ 1, 𝑀, Q1,1 = 𝐴 ⊤ 0 𝑃 + 𝑃 𝐴 0 + Ξ0 , Q2,2 = -𝛾𝐼 𝑛𝑀 , Q1,2 = 𝑃 𝐴 + Γ, Q1,3 = 𝐴 ⊤ 0 Λ + Ω, Q2,3 = 𝐴 ⊤ Λ + Υ, Q3,3 = -𝜃𝐼 𝑛𝑀 , 𝐴 = [ 𝐴 1 … 𝐴 𝑀 ] , Γ = [ Γ 1 … Γ 𝑀 ] , Λ = [ Λ1 … Λ𝑀 ] , Ω = [ Ω 1 … Ω 𝑀 ] , Υ = ( Υ𝑗,𝑗 ′ ) 𝑀 𝑗, 𝑗 ′ =1 ,
then a forward complete system (11) is convergent.

Proof. Consider a candidate Lyapunov function

𝑉 (𝑥) = 𝑥 ⊤ 𝑃 𝑥 + 2 𝑀 ∑ 𝑗=1 𝑛 ∑ 𝑖=1 Λ 𝑗,𝑖 𝑥 𝑖 ∫ 0 𝑓 𝑖 𝑗 (𝜏)𝑑𝜏, (18)
which is positive definite and radially unbounded due to Finsler's lemma under the condition ( 14) and Assumption 1 (the first inclusion property in ( 4) is verified). Then, consider the time derivative of 𝑉 :

V = ẋ⊤ 𝑃 𝑥 + 𝑥 ⊤ 𝑃 ẋ + 2 𝑀 ∑ 𝑗=1 ẋ⊤ Λ 𝑗 𝐹 𝑗 (𝑥) = 𝑥 ⊤ ( 𝐴 ⊤ 0 𝑃 + 𝑃 𝐴 0 ) 𝑥 + ( 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥) ⊤ 𝐴 ⊤ 𝑗 ) 𝑃 𝑥 + 𝑥 ⊤ 𝑃 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑥) + 2𝑥 ⊤ 𝑃 𝜑 +2 𝑀 ∑ 𝑗=1 ( 𝑥 ⊤ 𝐴 ⊤ 0 Λ 𝑗 𝐹 𝑗 (𝑥) + 𝜑 ⊤ Λ 𝑗 𝐹 𝑗 (𝑥) + ( 𝑀 ∑ 𝑠=1 𝐹 𝑠 (𝑥) ⊤ 𝐴 ⊤ 𝑠 ) Λ 𝑗 𝐹 𝑗 (𝑥)
) .

Therefore, under (14) we obtain

V = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥 𝐹 1 (𝑥) ⋮ 𝐹 𝑀 (𝑥) 𝜑 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⊤ 𝑄 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥 𝐹 1 (𝑥) ⋮ 𝐹 𝑀 (𝑥) 𝜑 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -𝑥 ⊤ Ξ 0 𝑥 - 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥) ⊤ Ξ 𝑗 𝐹 𝑗 (𝑥) -2 𝑀 ∑ 𝑗=1 𝑥 ⊤ Υ 0,𝑗 𝐹 𝑗 (𝑥) -2 𝑀-1 ∑ 𝑠=1 𝑀 ∑ 𝑟=𝑠+1 𝐹 𝑠 (𝑥) ⊤ Υ 𝑠,𝑟 𝐹 𝑟 (𝑥) + 𝜑 ⊤ Φ𝜑 ≤ -𝑥 ⊤ Ξ 0 𝑥 - 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥) ⊤ Ξ 𝑗 𝐹 𝑗 (𝑥) -2 𝑀 ∑ 𝑗=1 𝑥 ⊤ Υ 0,𝑗 𝐹 𝑗 (𝑥) -2 𝑀-1 ∑ 𝑠=1 𝑀 ∑ 𝑟=𝑠+1 𝐹 𝑠 (𝑥) ⊤ Υ 𝑠,𝑟 𝐹 𝑟 (𝑥) + 𝜑 ⊤ Φ𝜑,
where the right-hand side of the last inequality is radially unbounded as a function of 𝑥 under the condition (16) and Assumption 1, then the second relation in (4) can be recovered and the ISS property of the system (11) is guaranteed. Therefore, all solutions of the system are bounded. Next, we need to establish the asymptotic stability of the dynamics (13) that implies existence of a globally attracting solution in (11), for which a Lyapunov function

Ṽ (𝑧) = 𝑧 ⊤ P 𝑧 + 2 𝑀 ∑ 𝑗=1 𝑛 ∑ 𝑖=1 Λ𝑗,𝑖 𝑧 𝑖 ∫ 0 𝑓 𝑖 𝑗 (𝜏)𝑑𝜏
is considered, and its positive definiteness and radial unboundedness in the variable 𝑧 follows the same argumentation as before, whose time derivative admits the following representation:

̇Ṽ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑧 𝐺 1 (𝑥, 𝑧) ⋮ 𝐺 𝑀 (𝑥, 𝑧) 𝐹 1 (𝑧) ⋮ 𝐹 𝑀 (𝑧) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⊤ Q ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑧 𝐺 1 (𝑥, 𝑧) ⋮ 𝐺 𝑀 (𝑥, 𝑧) 𝐹 1 (𝑧) ⋮ 𝐹 𝑀 (𝑧) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + 𝛾 𝑀 ∑ 𝑗=1 𝐺 𝑗 (𝑥, 𝑧) ⊤ 𝐺 𝑗 (𝑥, 𝑧) + 𝜃 𝑀 ∑ 𝑗=1 𝐹 ⊤ 𝑗 (𝑧)𝐹 𝑗 (𝑧) -𝑧 ⊤ Ξ0 𝑧 -2 𝑀 ∑ 𝑗=1 𝑧 ⊤ Γ 𝑗 𝐺 𝑗 (𝑥, 𝑧) -2 𝑀 ∑ 𝑗=1 𝑧 ⊤ Ω 𝑗 𝐹 𝑗 (𝑧) -2 𝑀 ∑ 𝑗=1 𝑀 ∑ 𝑗 ′ =1
𝐺 𝑗 (𝑥, 𝑧) ⊤ Υ𝑗,𝑗 ′ 𝐹 𝑗 ′ (𝑧).

Since Q ≤ 0 and applying Assumption 3, we get

̇Ṽ ≤ -𝑧 ⊤ ( Ξ0 -𝛾 𝑀 ∑ 𝑗=1 𝑆 𝑗 0 -𝜃 𝑀 ∑ 𝑗=1 𝐻 𝑗 0 ) 𝑧 -2 𝑀 ∑ 𝑗=1 𝑧 ⊤ ( Γ 𝑗 -𝛾𝑆 𝑗 1 -𝜃𝐻 𝑗 1 ) 𝐺 𝑗 (𝑥, 𝑧) -2 𝑀 ∑ 𝑗=1 𝑧 ⊤ ( Ω 𝑗 -𝛾𝑆 𝑗 2 -𝜃𝐻 𝑗 2 ) 𝐹 𝑗 (𝑧) -2 𝑀 ∑ 𝑗=1 𝐺 𝑗 (𝑥, 𝑧) ⊤ 𝑀 ∑ 𝑗 ′ =1 ( Υ𝑗,𝑗 ′ -𝛾𝑆 𝑗,𝑗 ′ 3 -𝜃𝐻 𝑗,𝑗 ′ 3 ) 𝐹 𝑗 ′ (𝑧).
The second and the fourth terms in the above estimate are nonnegative due to Assumption 2. The SIIOS characteristics in (6) are verified due to (17). Therefore, we can substantiate that the system (11), ( 13) is SIIOS with respect to 𝑧, then there exists 𝛽 ∈ 𝒦 ℒ such that ‖𝑧(𝑡, 𝑧 0 , 0)‖ ≤ 𝛽(‖𝑧 0 ‖, 𝑡), ∀𝑡 ∈ ℝ + for any 𝑧 0 ∈ ℝ 𝑛 . Finally, (11) has all solutions bounded and converging to a unique globally attracting one, which describes the convergence property due to Definition 7.

Remark 2. Note that Assumption 3 is used only in the second part of the proof of Theorem 4, where the ISS property of the system (11) has been already established. It means that despite the assumption is formulated for all 𝑥, 𝑦 ∈ ℝ 𝑛 , the variable 𝑥 (that represents the limit motion) can be assumed to be bounded, which enlarges greatly the class of admissible nonlinearities verifying Assumption 3. The current formulation of the assumption is chosen for brevity of exposition. Remark 3. To prove Theorem 4, in the first part, it is enough to substantiate global stability of trajectories for the system (11), then conditions of ISS can be relaxed. For example, all bounded nonlinearities can be treated as a part of the input 𝜑, or some cross-terms including such nonlinearities, as 𝜑 ⊤ Λ 𝑗 𝐹 𝑗 (𝑥) for instance, can be hidden in 𝜑, which may sparse the matrix 𝑄.

Remark 4. It is also possible to use IOS/ROS properties to show the convergence for the system (13) (see Theorem 2 and Remark 1 for proof). Let 𝐻(𝑧) = 𝑧 (19) be the output of ( 13) and consider IOS or ROS property of the system (11), ( 13), (19). We see that the system (11), ( 13), ( 19) is with zero input, the conditions ( 7), ( 9) and ( 10) can be verified. Thus, all solutions of (13) are bounded by

‖𝑧(𝑡, 𝑧 0 , 0)‖ ≤ 𝛽 ( ‖𝑧 0 ‖, 𝑡 )
for some 𝛽 ∈ 𝒦 ℒ , all 𝑧 0 ∈ ℝ 𝑛 and 𝑡 ≥ 0.

Corollary 1.

Let 𝜑 ∈ ℒ 𝑛 ∞ be an almost periodic function for 𝑡 ∈ ℝ, the functions 𝐹 𝑗 , 𝑗 ∈ 1, 𝑀 be locally Lipschitz continuous and all conditions of Theorem 4 be verified, then the system (11) possesses almost periodic convergence property.

Proof. The claim repeats the arguments given in [START_REF] Zubov | Oscillations in Nonlinear and Controlled Systems[END_REF][START_REF] Zubov | Theory of Oscillations[END_REF] for a convergent system (11).

Case of Metzler matrices

In this subsection, we consider convergence and almost periodic convergence conditions for the system (11) with coefficient matrices satisfying the following assumption:

Assumption 4. Assume that the matrices 𝐴 𝑠 ∈ ℝ 𝑛×𝑛 , 𝑠 ∈ 0, 𝑀 are Metzler and there exists a vector 𝜈 > 0 in ℝ 𝑛 with 𝜈 ⊤ 𝐴 𝑠 = -𝑐 ⊤ 𝑠 ≤ 0 for 𝑠 ∈ 0, 𝑀 and ∑ 𝑚 𝑠 ′ =0 𝑐 𝑠 ′ > 0. Note that we do not require that all matrices 𝐴 𝑠 with 𝑠 ∈ 0, 𝑀 be Hurwitz, but only its total composition, evaluated by ∑ 𝑚 𝑠 ′ =0 𝑐 𝑠 ′ , has to provide the stability. The proofs of the results of this subsection can be found in [START_REF] Mei | Convergence conditions for Persidskii systems[END_REF] . Theorem 5. If assumptions 1, 2 and 4 are fulfilled, then the system (11) is convergent.

In this case, Assumption 4 replaces all LMIs given in the formulation of Theorem 4.

Corollary 2. Let 𝜑 ∈ ℒ 𝑛

∞ be an almost periodic function for 𝑡 ∈ ℝ, the functions 𝐹 𝑗 , 𝑗 ∈ 1, 𝑀 be locally Lipschitz continuous and assumptions 1, 2 and 4 be fulfilled, then the system (11) possesses almost periodic convergence property.

CONVERGENCE CONDITIONS FOR DISCRETIZED SYSTEMS

In this section, the convergence conditions for two kinds of discretized systems are given, which are obtained from application of two discretization approaches to the system (11), namely the explicit and the implicit Euler methods are studied. Only the convergence property is investigated, the almost periodic one can be obtained by adapting the conditions of [START_REF] Ataeva | Property of convergence for difference systems[END_REF] and corollaries 1 or 2 (it is skipped for brevity). Assumption 5. Assume that for every 𝑖 ∈ 1, 𝑛, 𝑗 ∈ 1, 𝑀, the function 𝑓 𝑖 𝑗 is globally Lipschitz continuous, i.e., there exists a number

𝐿 𝑖 𝑗 > 0 such that | | | 𝑓 𝑖 𝑗 (𝑟 1 ) -𝑓 𝑖 𝑗 (𝑟 2 ) | | | ≤ 𝐿 𝑖 𝑗 | | 𝑟 1 -𝑟 2 | | for all 𝑟 1 , 𝑟 2 ∈ ℝ.

Explicit Euler method

Consider the difference system

𝑥(𝑘 + 1) = 𝑥(𝑘) + ℎ𝐴 0 𝑥(𝑘) + ℎ 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑥(𝑘)) + ℎ𝜑(𝑘), ( 20
)
where ℎ > 0 is a discretization step; 𝑘 0 ∈ ℤ and 𝑘 0 ≤ 𝑘 ∈ ℤ; 𝜑 ∈ 𝒰 𝑛 ∞ ; 𝑥(𝑘) and 𝐹 𝑗 , 𝑗 ∈ 1, 𝑀 are defined as in the system (11). It is clear that system (20) is the result of discretization of (11) with the aid of the explicit Euler method. Theorem 6. Let assumptions 1, 2, 3 and 5 be satisfied. If there exist 0 ≤ 𝑃 = 𝑃 ⊤ , P

= P ⊤ ∈ ℝ 𝑛×𝑛 ; {Λ 𝑗 = diag(Λ 𝑗,1 , … , Λ 𝑗,𝑛 )} 𝑀 𝑗=1 , { Λ𝑗 = diag( Λ𝑗,1 , … , Λ𝑗,𝑛 )} 𝑀 𝑗=1 , {Ξ 𝑠 } 𝑀 𝑠=0 , {Υ 𝑠,𝑟 } 0≤𝑠<𝑟≤𝑀 , { Υ𝑗,𝑗 ′ } 𝑀 𝑗,𝑗 ′ =1 , {Γ 𝑗 } 𝑀 𝑗=1 , {Ω 𝑗 } 𝑀 𝑗=1 ⊂ 𝔻 𝑛 + ; Ξ0 ∈ 𝔻 𝑛 + ; 0 < Φ = Φ ⊤ ∈ ℝ 𝑛×𝑛
, 𝛾, 𝜃 > 0 such that the properties ( 14)-( 17) are verified for

𝑄 = (𝑄 𝑎, 𝑏 ) 𝑀+2 𝑎, 𝑏=1 , Q = ( Q𝑎, 𝑏 ) 3 𝑎, 𝑏=1
with

𝑄 1,1 = 𝐴 ⊤ 0 𝑃 + 𝑃 𝐴 0 + ℎ𝐴 ⊤ 0 Π𝐴 0 + Ξ 0 , 𝑄 𝑗+1,𝑗+1 = ℎ𝐴 ⊤ 𝑗 Π𝐴 𝑗 + Λ 𝑗 𝐴 𝑗 + 𝐴 ⊤ 𝑗 Λ 𝑗 + Ξ 𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 1,𝑗+1 = 𝑃 𝐴 𝑗 + 𝐴 ⊤ 0 Λ 𝑗 + ℎ𝐴 ⊤ 0 Π𝐴 𝑗 + Υ 0,𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 𝑠+1,𝑟+1 = ℎ𝐴 ⊤ 𝑠 Π𝐴 𝑟 + 𝐴 ⊤ 𝑠 Λ 𝑟 + Λ 𝑠 𝐴 𝑟 + Υ 𝑠,𝑟 , 𝑠 ∈ 1, 𝑀 -1, 𝑟 ∈ 𝑠 + 1, 𝑀, 𝑄 𝑀+2, 𝑀+2 = ℎΠ -Φ; 𝑄 𝑗+1, 𝑀+2 = ℎ𝐴 ⊤ 𝑗 Π + Λ 𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 1, 𝑀+2 = 𝑃 + ℎ𝐴 ⊤ 0 Π, Π = 𝑃 + 2 𝑀 ∑ 𝑗=1 𝐿 𝑗 Λ 𝑗 , 𝐿 𝑗 ∶= diag(𝐿 1 𝑗 , ..., 𝐿 𝑛 𝑗 ), 𝑗 ∈ 1, 𝑀, Q1,1 = 𝐴 ⊤ 0 𝑃 + 𝑃 𝐴 0 + ℎ𝐴 ⊤ 0 Π𝐴 0 + Ξ0 , Q2,2 = ℎ𝐴 ⊤ Π𝐴 -𝛾𝐼 𝑛𝑀 , Q1,2 = ℎ𝐴 ⊤ 0 Π𝐴 + 𝑃 𝐴 + Γ, Q1,3 = 𝐴 ⊤ 0 Λ + Ω, Q2,3 = 𝐴 ⊤ Λ + Υ, Q3,3 = -𝜃𝐼 𝑛𝑀 , Π = 𝑃 + 2 𝑀 ∑ 𝑗=1 𝐿 𝑗 Λ𝑗 , 𝐴 = [ 𝐴 1 … 𝐴 𝑀 ] , Γ = [ Γ 1 … Γ 𝑀 ] , Λ = [ Λ1 … Λ𝑀 ] , Ω = [ Ω 1 … Ω 𝑀 ] , Υ = ( Υ𝑗,𝑗 ′ ) 𝑀 𝑗, 𝑗 ′ =1 ,
then the system (20) is convergent.

Proof. Consider a candidate Lyapunov function

𝑉 (𝑥(𝑘)) = 𝑥(𝑘) ⊤ 𝑃 𝑥(𝑘) + 2 𝑀 ∑ 𝑗=1 𝑛 ∑ 𝑖=1 Λ 𝑗,𝑖 𝑥 𝑖 (𝑘) ∫ 0 𝑓 𝑖 𝑗 (𝜏)𝑑𝜏
for the system (20), so the first condition in ( 8) is verified as before. Then, consider the difference of 𝑉 :

Δ𝑉 ∶= 𝑥(𝑘 + 1) ⊤ 𝑃 𝑥(𝑘 + 1) -𝑥(𝑘) ⊤ 𝑃 𝑥(𝑘) + 2 𝑀 ∑ 𝑗=1 𝑛 ∑ 𝑖=1 Λ 𝑗,𝑖 𝑥 𝑖 (𝑘+1) ∫ 𝑥 𝑖 (𝑘) 𝑓 𝑖 𝑗 (𝜏)𝑑𝜏 = [ 𝑥(𝑘 + 1) ⊤ 𝑃 𝑥(𝑘 + 1) -𝑥(𝑘) ⊤ 𝑃 𝑥(𝑘) ] + 2ℎ
[ 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥(𝑘)) ⊤ Λ 𝑗 𝑇 (𝑘) ] +2 [ 𝑀 ∑ 𝑗=1 𝐺 𝑗 (𝑥(𝑘), 𝜍(𝑘) -𝑥(𝑘)) ⊤ Λ 𝑗 (𝑥(𝑘 + 1) -𝑥(𝑘)) ] ≤ [ 𝑥(𝑘 + 1) ⊤ 𝑃 𝑥(𝑘 + 1) -𝑥(𝑘) ⊤ 𝑃 𝑥(𝑘) ] + 2ℎ
[ 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥(𝑘)) ⊤ Λ 𝑗 𝑇 (𝑘) ] +2ℎ 2 [ 𝑇 (𝑘) ⊤ ( 𝑀 ∑ 𝑗=1 𝐿 𝑗 Λ 𝑗 ) 𝑇 (𝑘) ] = ℎ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥(𝑘) 𝐹 1 (𝑥(𝑘)) ⋮ 𝐹 𝑀 (𝑥(𝑘)) 𝜑(𝑘) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⊤ 𝑄 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥(𝑘) 𝐹 1 (𝑥(𝑘)) ⋮ 𝐹 𝑀 (𝑥(𝑘)) 𝜑(𝑘) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -ℎ𝑥(𝑘) ⊤ Ξ 0 𝑥(𝑘) -ℎ 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥(𝑘)) ⊤ Ξ 𝑗 𝐹 𝑗 (𝑥(𝑘)) -2ℎ 𝑀 ∑ 𝑗=1 𝑥(𝑘) ⊤ Υ 0,𝑗 𝐹 𝑗 (𝑥(𝑘)) -2ℎ 𝑀-1 ∑ 𝑠=1 𝑀 ∑ 𝑟=𝑠+1 𝐹 𝑠 (𝑥(𝑘)) ⊤ Υ 𝑠,𝑟 𝐹 𝑟 (𝑥(𝑘)) + ℎ𝜑(𝑘) ⊤ Φ𝜑(𝑘)
by Assumption 5 and the Mean Value Theorem, where 𝑇 (𝑘) = 𝐴 0 𝑥(𝑘) + ∑ 𝑀 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑥(𝑘)) + 𝜑(𝑘) and 𝜍 𝑖 (𝑘) ∈ [min(𝑥 𝑖 (𝑘), 𝑥 𝑖 (𝑘 + 1)), max(𝑥 𝑖 (𝑘), 𝑥 𝑖 (𝑘 + 1))], 𝑖 ∈ 1, 𝑛 for all 𝑘 ≥ 𝑘 0 . Therefore, with the condition ( 14) we obtain

Δ𝑉 ≤ -ℎ𝑥 ⊤ Ξ 0 𝑥 -ℎ 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥) ⊤ Ξ 𝑗 𝐹 𝑗 (𝑥) -2ℎ 𝑀 ∑ 𝑗=1 𝑥 ⊤ Υ 0,𝑗 𝐹 𝑗 (𝑥) -2ℎ 𝑀-1 ∑ 𝑠=1 𝑀 ∑ 𝑟=𝑠+1 𝐹 𝑠 (𝑥) ⊤ Υ 𝑠,𝑟 𝐹 𝑟 (𝑥) + ℎ𝜑 ⊤ Φ𝜑,
so that under assumptions 1, 2 and the condition (16) the system (20) is ISS.

In a similar way, we then study the asymptotic stability of

𝑧(𝑘 + 1) = 𝑧(𝑘) + ℎ𝐴 0 𝑧(𝑘) + ℎ 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐺 𝑗 (𝑥(𝑘), 𝑧(𝑘)) (21)
with 𝑦(𝑘 + 1) = 𝑦(𝑘) + ℎ𝐴 0 𝑦(𝑘) + ℎ ∑ 𝑀 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑦(𝑘)) + ℎ𝜑(𝑘) and 𝑧(𝑘) = 𝑦(𝑘) -𝑥(𝑘) for all 𝑘 ≥ 𝑘 0 , for which we consider a Lyapunov function

Ṽ (𝑧(𝑘)) = 𝑧(𝑘) ⊤ P 𝑧(𝑘) + 2 𝑀 ∑ 𝑗=1 𝑛 ∑ 𝑖=1 Λ𝑗,𝑖 𝑧 𝑖 (𝑘) ∫ 0 𝑓 𝑖 𝑗 (𝜏)𝑑𝜏
for the dynamics (21). Under the repetition of previous development in the proof for Theorem 4, it follows that under assumptions 1, 2, 3, 5 and the conditions ( 15), ( 17)

Δ Ṽ ≤ -ℎ𝑧 ⊤ ( Ξ0 -𝛾 𝑀 ∑ 𝑗=1 𝑆 𝑗 0 -𝜃 𝑀 ∑ 𝑗=1 𝐻 𝑗 0 ) 𝑧 -2ℎ 𝑀 ∑ 𝑗=1 𝑧 ⊤ ( Γ 𝑗 -𝛾𝑆 𝑗 1 -𝜃𝐻 𝑗 1 ) 𝐺 𝑗 (𝑥, 𝑧) -2ℎ 𝑀 ∑ 𝑗=1 𝑧 ⊤ ( Ω 𝑗 -𝛾𝑆 𝑗 2 -𝜃𝐻 𝑗 2 ) 𝐹 𝑗 (𝑧) -2ℎ 𝑀 ∑ 𝑗=1 𝐺 𝑗 (𝑥, 𝑧) ⊤ 𝑀 ∑ 𝑗 ′ =1 ( Υ𝑗,𝑗 ′ -𝛾𝑆 𝑗,𝑗 ′ 3 -𝜃𝐻 𝑗,𝑗 ′ 3 ) 𝐹 𝑗 ′ (𝑧) < 0
for all 𝑧 ≠ 0 and 𝑥 ∈ ℝ 𝑛 . Therefore, ( 21) is globally asymptotically stable (GAS). This completes the proof.

Note that if Assumption 5 holds only locally, i.e., the nonlinear functions in (20) are locally Lipschitz, then local convergence can be established in the same way.

The result of Theorem 6 can be reformulated as follows:

Corollary 3. Let all conditions of Theorem 4 be satisfied with Assumption 5. Then there exists ℎ 0 > 0 such that for all ℎ ∈ (0, ℎ 0 ] the system (20) is convergent.

Proof. It is easy to check that the difference between matrices 𝑄 and Q given in theorems 4 and 6 is proportional to ℎ, and all other conditions are the same, except Assumption 5 that is additionally imposed in the last theorem. This substantiates the claim.

Implicit Euler method

In this subsection the convergence conditions for a discretization of (11) with the aid of implicit Euler method is investigated. The considered discretized system is

𝑥(𝑘 + 1) = 𝑥(𝑘) + ℎ𝐴 0 𝑥(𝑘 + 1) + ℎ 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑥(𝑘 + 1)) + ℎ𝜑(𝑘 + 1), (22)
where 𝑥(𝑘), ℎ, 𝑘, 𝐹 𝑗 , 𝑗 ∈ 1, 𝑀 and 𝜑(𝑘 + 1) are defined as in the system (20), and it has to be resolved with respect to 𝑥(𝑘 + 1).

Remark 5. In the sequel we assume that for any ℎ > 0 and 𝜁, 𝜑 ∈ ℝ 𝑛 , the equation

(𝐼 𝑛 -ℎ𝐴 0 )𝜉 -ℎ 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝜉) = 𝜁 + ℎ𝜑 ( 23
)
admits a unique solution 𝜉 = 𝜉(ℎ, 𝜁, 𝜑) that continuously depends on ℎ, 𝜁, 𝜑. Hence, for any 𝑘 0 ∈ ℤ and 𝑥(𝑘 0 ), the system (22) admits a unique solution 𝑥(𝑘, 𝑥(𝑘 0 ), 𝑘 0 , 𝜑) that is defined for all 𝑘 0 ≤ 𝑘 ∈ ℤ. Sufficient conditions of such a property (under assumptions 1 and 2 with 𝑀 = 1) were obtained in [START_REF] Sandberg | Some Theorems on Properties of DC Equations of Nonlinear Networks[END_REF][START_REF] Sandberg | Global inverse function theorems[END_REF] .

Theorem 7. Let assumptions 1 and 2 be satisfied. If there exist 0

< 𝑃 = 𝑃 ⊤ ∈ ℝ 𝑛×𝑛 ; {Ξ 𝑠 } 𝑀 𝑠=0 , {Υ 𝑠,𝑟 } 0≤𝑠<𝑟≤𝑀 ⊂ 𝔻 𝑛 + and 0 < Φ = Φ ⊤ ∈ ℝ 𝑛×𝑛 such that 𝑄 = 𝑄 ⊤ ≤ 0, (24)
Ξ 0 + 𝑀 ∑ 𝑗=1 Υ 0,𝑗 + 𝑚 ∑ 𝑠=1 Ξ 𝑠 + 𝑚 ∑ 𝑠=1 𝑚 ∑ 𝑟=𝑠+1 Υ 𝑠,𝑟 > 0, (25)
where

𝑄 = (𝑄 𝑎, 𝑏 ) 𝑀+2 𝑎, 𝑏=1
with

𝑄 1,1 = 𝐴 ⊤ 0 𝑃 + 𝑃 𝐴 0 -ℎ𝐴 ⊤ 0 𝑃 𝐴 0 + Ξ 0 , 𝑄 𝑗+1,𝑗+1 = -ℎ𝐴 ⊤ 𝑗 𝑃 𝐴 𝑗 + Ξ 𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 1,𝑗+1 = 𝑃 𝐴 𝑗 -ℎ𝐴 ⊤ 0 𝑃 𝐴 𝑗 + Υ 0,𝑗 , 𝑗 ∈ 1, 𝑀, 𝑄 𝑠+1,𝑟+1 = -ℎ𝐴 ⊤ 𝑠 𝑃 𝐴 𝑟 + Υ 𝑠,𝑟 , 𝑠 ∈ 1, 𝑀 -1, 𝑟 ∈ 𝑠 + 1, 𝑀, 𝑄 1, 𝑀+2 = 𝑃 -ℎ𝐴 ⊤ 0 𝑃 , 𝑄 𝑀+2, 𝑀+2 = -ℎ𝑃 -Φ; 𝑄 𝑗+1, 𝑀+2 = -ℎ𝐴 ⊤ 𝑗 𝑃 , 𝑗 ∈ 1, 𝑀,
then the system ( 22) is convergent.

Proof. Consider a candidate Lyapunov function

𝑉 (𝑥(𝑘)) = 𝑥(𝑘) ⊤ 𝑃 𝑥(𝑘)
for (22), again the first condition in ( 8) is recovered. Then, the increment of such a 𝑉 for (22) takes the form: for all 𝑧(𝑘 + 1) ≠ 0 and 𝑥(𝑘 + 1) ∈ ℝ 𝑛 . Thus, the dynamics (26) is GAS. Therefore, we conclude that the system ( 22) is convergent.

Δ𝑉 = 𝑥(𝑘 + 1) ⊤ 𝑃 𝑥(𝑘 + 1) -𝑥(𝑘) ⊤ 𝑃 𝑥(𝑘) = ℎ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥(𝑘 + 1) 𝐹 1 (𝑥(𝑘 + 1)) ⋮ 𝐹 𝑀 (𝑥(𝑘 + 1)) 𝜑(𝑘 + 1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⊤ 𝑄 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥(𝑘 + 1) 𝐹 1 (𝑥(𝑘 + 1)) ⋮ 𝐹 𝑀 (𝑥(𝑘 + 1)) 𝜑(𝑘 + 1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -ℎ𝑥(𝑘 + 1) ⊤ Ξ 0 𝑥(𝑘 + 1) -ℎ 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥(𝑘 + 1)) ⊤ Ξ 𝑗 𝐹 𝑗 (𝑥(𝑘 + 1)) -2ℎ

EXAMPLES

Application to a modified Lotka-Volterra model

In this subsection, a modified Lotka-Volterra (LV) dynamics is considered, which has been widely investigated in infectious disease, biology, finance, to mention a few subjects [START_REF] Holt | Infectious Disease and Species Coexistence: A Model of Lotka-Volterra Form[END_REF] . However, the earliest basic model does not reflect some important phenomena, e.g., stable coexistence and predator preference. Therefore, there are many modified LV models, among them the following one considers additionally the mutualistic interactions in different species [START_REF] Holland | Population Dynamics and Mutualism: Functional Responses of Benefits and Costs[END_REF] :

ẋ(𝑡) = diag(𝑥(𝑡))(𝑟 0 + 𝑟(𝑡) + 𝐴𝑥(𝑡) + 𝐴 2 𝜙(𝑥(𝑡))), 𝑡 ∈ ℝ, (27)
where 𝑥(𝑡) ∈ ℝ 𝑛 + is the population sizes of 𝑛 species; 𝑟 0 ∈ ℝ 𝑛 denotes the birth or mortality rates of the species and 𝑟(𝑡) ∈ ℝ 𝑛 is introduced to model the deviations of the rates from the nominal quantities 𝑟 0 ; 𝐴 ∈ ℝ 𝑛×𝑛 represents the community matrix and 𝐴 2 ∈ ℝ 𝑛×𝑛 refers to the net effect of the mutualism (𝐴 and 𝐴 2 are Metzler); the function 𝜙 ∶ ℝ 𝑛 + → ℝ 𝑛 + is diagonal, where 𝜙 𝑖 (𝑥 𝑖 ) = 𝑥 𝑖 𝑏 𝑖 +𝑥 𝑖 with constants 𝑏 𝑖 > 0 for 𝑖 ∈ 1, 𝑛. Assuming existence of a unique equilibrium point 𝑥 𝑒 ∈ ℝ 𝑛 + ⧵ {0} for (27) with 𝑟(𝑡) = 0, and defining 𝜌(𝑡) = ln(𝑥(𝑡)) -ln(𝑥 𝑒 ),

we obtain ρ(𝑡) = 𝐴 1 𝐹 1 (𝜌(𝑡)) + 𝐴 2 𝐹 2 (𝜌(𝑡)) + 𝑟(𝑡), (28)
where

𝐴 1 ∶= 𝐴 diag(𝑥 𝑒 ), 𝐹 1 (𝜌) = 𝑒 𝜌 -𝟏 𝑛 , 𝐹 2 (𝜌) = 𝜙 ( diag(𝑥 𝑒 )𝑒 𝜌 ) -𝜙(𝑥 𝑒 ).
Note that for such functions 𝐹 1 and 𝐹 2 the requirement of Assumption 1 for 𝜈 → -∞ is not satisfied. However, as in [START_REF] Aleksandrov | Convergence conditions for some classes of nonlinear systems[END_REF] , due to assumed existence of the global equilibrium 𝑥 𝑒 it is possible to show that for 𝑟 0 + 𝑟(𝑡) ≥ 𝑟 min > 0 all trajectories converge to a neighborhood of the steady state, where the analysis can be next performed without taking into account the unbounded deviations of the state. For illustration, let

𝐴 = [ -0.8 0.3 0.4 -0.6 ] , 𝐴 2 = [ -0.5 0.2 0.3 -0.9 ] , 𝑏 = [ 0.1 0.4 ] , 𝑟 0 = [ 0.2 0.9 ] , 𝑟(𝑡) = [ sin(𝑡) cos(𝑡) ] ,
where 𝑟(𝑡) is periodic and represents the external input for (28), then the value of 𝑥 𝑒 can be obtained as previously mentioned. Assumptions 3 and 5 are satisfied for By choosing a periodic and oscillating input signal, we can see a convergence phenomenon that is not caused by asymptotic stability of the considered system. The trajectories with distinct initial states

𝑆 1 0 = 𝑆 2 0 = [ 0.0041 0 0 0.0070 ] , 𝑆 1 1 = [ 0.1177 0 0 0.1617 ] , 𝑆 2 1 = [ 0.2171 0 0 0.2907 ] , 𝑆 1 2 = 𝑆 2 2 = 10 -5 × [ 0.0109 0 0 0.0117 ] , 𝑆 1,1 3 = 𝑆 1,2 3 = [ 0.1092 0 0 0.2105 ] , 𝑆

Application to opinion dynamics

For opinion dynamics among a network of agents, the following model can be considered [START_REF] Altafini | Consensus Problems on Networks With Antagonistic Interactions[END_REF][START_REF] Baumann | Modeling Echo Chambers and Polarization Dynamics in Social Networks[END_REF][START_REF] Baumann | Emergence of Polarized Ideological Opinions in Multidimensional Topic Spaces[END_REF] :

ẋ(𝑡) = -𝑥(𝑡) + 𝑀 ∑ 𝑗=1 𝐾 𝑗 𝐴 𝑗 tanh(𝛼 𝑗 𝑥(𝑡)) + 𝐵𝑢(𝑡) + 𝜑(𝑡), (29)
where 𝑥(𝑡) ∈ ℝ 𝑛 is the opinion variable of 𝑛 agents, and the sign of 𝑥 𝑖 (𝑡), 𝑖 = 1, … , 𝑛 describes the qualitative stance toward a binary choice (the larger |𝑥 𝑖 (𝑡)|, more extreme is the opinion of the agent 𝑖); 𝑀 ≥ 1 is the number of social networks connecting the agents; 𝐾 𝑗 > 0 denotes the social interaction strength among agents in the network 𝑗 ∈ 1, 𝑀, 𝐴 𝑗 ∈ ℝ 𝑛×𝑛 is the adjacency matrix in that network, and 𝛼 𝑗 > 0 characterizes controversialness of the issue for 𝑗 th media (for small 𝛼 𝑗 , the social influence of moderate individuals on others is weak, while for a large 𝛼 𝑗 , everybody have a strong social impact); 𝐵 ∈ ℝ 𝑛×𝑞 ; 𝑢(𝑡) ∈ ℝ 𝑞 is a controlling input that can be applied to modify the network connections among the agents; 𝜑(𝑡) ∈ ℝ 𝑛 can be used to model the off-network influences on orientations of agents (e.g., governmental communication). The detailed motivation for this model (for the case 𝑀 = 1 and time-varying matrix 𝐴 1 ) can be found in [START_REF] Baumann | Modeling Echo Chambers and Polarization Dynamics in Social Networks[END_REF][START_REF] Baumann | Emergence of Polarized Ideological Opinions in Multidimensional Topic Spaces[END_REF] . Clearly, the system under feedback control is presented in the form of (11), and assumptions 1, 2, 3 and 5 are verified. If Assumption 4 is also valid, then in the resulting closed-loop system, the opinion of an agent can be changed by the external manipulation 𝜑 only. Since all nonlinearities are bounded and 𝐴 0 = -𝐼 𝑛 , the system always has bounded solutions for bounded inputs. Hence, the conditions of Theorem 4 can be simplified, as mentioned in Remark 3, by focusing only on the asymptotic stability of (13).

Let 𝑛 = 4, 𝑀 = 2, The trajectories with different initial conditions are presented in Fig. 2, and the LMIs in Theorem 4 are verified. As we can conclude from these results, the opinions of all agents are directed to the predetermined by 𝜑 direction. Simulations of the same system with 𝑢 = 0 and 𝜑 = 0 demonstrate pluralism of opinions in the uncontrolled network depending on the initial conditions.

𝐴 1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐴 2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐵 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2

FIGURE 2

The state trajectories of (29) versus the time 𝑡

CONCLUSION

In this paper, broad convergence conditions for generalized Persidskii systems and their discretizations, and the auxiliary results of convergence and almost periodic convergence conditions for those systems, with Metzler coefficient matrices, were proposed. The formulated conditions were obtained in the form of LMIs (linear inequalities in the Metzler case), which can be constructively verified. The simulations of Lotka-Volterra models and opinion dynamics were presented for the examination of the proposed results. Future directions of research include the analysis of more sophisticated Lyapunov functions and the investigation of robust convergence properties, and the influence of delays as well.

Assumption 1 .

 1 𝑛, are imposed as: Assume that for any 𝑖 ∈ 1, 𝑛 and 𝑗 ∈ 1, 𝑀, 𝜈𝑓 𝑖 𝑗 (𝜈) > 0, ∀𝜈 ∈ ℝ∖{0}. Under Assumption 1, there exists an index 𝑚 ∈ 0, 𝑀 such that for all 𝑖 ∈ 1, 𝑛, 𝑎 ∈ 1, 𝑚 lim 𝜈→±∞ 𝑓 𝑖 𝑎 (𝜈) = ±∞, and there exists 𝜇 ∈ 𝑚, 𝑀 such that for all 𝑖 ∈ 1, 𝑛, 𝑏 ∈ 1, 𝜇 lim 𝜈→±∞ 𝜈 ∫ 0 𝑓 𝑖 𝑏 (𝑟)𝑑𝑟 = +∞.

  (𝑥(𝑘 + 1)) ⊤ Υ 𝑠,𝑟 𝐹 𝑟 (𝑥(𝑘 + 1+ 1) ⊤ Υ 0,𝑗 𝐹 𝑗 (𝑥(𝑘 + 1)) + ℎ𝜑(𝑘 + 1) ⊤ Φ𝜑(𝑘 + 1).Therefore, under(24) we haveΔ𝑉 ≤ -ℎ𝑥(𝑘 + 1) ⊤ Ξ 0 𝑥(𝑘 + 1) -ℎ 𝑀 ∑ 𝑗=1 𝐹 𝑗 (𝑥(𝑘 + 1)) ⊤ Ξ 𝑗 𝐹 𝑗 (𝑥(𝑘 + 1(𝑥(𝑘 + 1)) ⊤ Υ 𝑠,𝑟 𝐹 𝑟 (𝑥(𝑘 + 1)) -2ℎ 𝑀 ∑ 𝑗=1 𝑥(𝑘 + 1) ⊤ Υ 0,𝑗 𝐹 𝑗 (𝑥(𝑘 + 1)) +ℎ𝜑(𝑘 + 1) ⊤ Φ𝜑(𝑘 + 1),hence, with the condition (25) the system(22) is ISS (recall Remark 5).Similarly, for the dynamics𝑧(𝑘 + 1) = 𝑧(𝑘) + ℎ𝐴 0 𝑧(𝑘 + 1) + ℎ 𝑀 ∑ 𝑗=1 𝐴 𝑗 𝐺 𝑗 (𝑥(𝑘 + 1), 𝑧(𝑘 + 1))(26)with 𝑦(𝑘 + 1) = 𝑦(𝑘) + ℎ𝐴 0 𝑦(𝑘 + 1) + ℎ ∑ 𝑀 𝑗=1 𝐴 𝑗 𝐹 𝑗 (𝑦(𝑘 + 1)) + ℎ𝜑(𝑘 + 1) and 𝑧(𝑘) = 𝑦(𝑘) -𝑥(𝑘) for all 𝑘 ≥ 𝑘 0 , by considering Ṽ (𝑧) = 𝑧 ⊤ P 𝑧 and following the previous proof lines we obtain Δ Ṽ ≤ -ℎ𝑧(𝑘 + 1) ⊤ Ξ 0 𝑧(𝑘 + 1) -ℎ 𝑀 ∑ 𝑗=1 𝐺 𝑗 (𝑥(𝑘 + 1), 𝑧(𝑘 + 1)) ⊤ Ξ 𝑗 𝐺 𝑗 (𝑥(𝑘 + 1), 𝑧(𝑘 + 1+ 1) ⊤ Υ 0,𝑗 𝐺 𝑗 (𝑥(𝑘 + 1), 𝑧(𝑘 + 1(𝑥(𝑘 + 1), 𝑧(𝑘 + 1)) ⊤ Υ 𝑠,𝑟 𝐺 𝑟 (𝑥(𝑘 + 1), 𝑧(𝑘 + 1)) < 0

𝑥 1 7 ]

 17 are shown in Fig.1. These results demonstrate the convergence phenomenon in(28).

6 FIGURE 1

 61 FIGURE 1The state trajectories of (27) versus the time 𝑡

  ). A continuous function 𝜓 ∶ ℝ → ℝ 𝑛 is called almost periodic if, for every 𝜀 > 0 there exists 𝜏 𝜀 > 0 such that every interval [𝑟, 𝑟 + 𝜏 𝜖 ] for any 𝑟 ∈ ℝ contains at least one number 𝜂 for which

	‖𝜓(𝑡 + 𝜂) -𝜓(𝑡)‖ < 𝜀
	for all 𝑡 ∈ ℝ.

  , 𝐾 1 = 2, 𝐾 2 = 1, 𝛼 1 = 0.5, 𝛼 2 = 0.1, 𝜑(𝑡) = = 𝑍 1 tanh(𝛼 1 𝑥(𝑡)) + 𝑍 2 tanh(𝛼 2 𝑥(𝑡)),

		1 1 0	⎤		⎡ 0.25 ⎤
		-1 0.5 0 -0.1 4 -0.2 3 1 ⎥ ⎥ ⎥		⎢ ⎢ ⎢	0.6 0.3	⎥ ⎥ ⎥	,
		2 0.1 4 2	⎥		⎢	0.8	⎥
				⎦		⎣	⎦
	⎡ 𝑢(𝑡) 𝑍 1 = 1.1107 1.2711 -0.3044 -1.0206 ⎤ ⎢ ⎢ ⎢ 3.1045 0.7848 -0.4515 -2.8485 -5.4459 -3.667 0.0203 3.4097 ⎥ ⎥ ⎥	, 𝑍 2 =	⎡ -1.3417 0.5825 1.0519 -0.1570 ⎤ ⎢ ⎢ ⎢ -2.8894 1.922 2.2015 0.3864 ⎥ 4.6529 -2.2669 -3.6353 0.5077 ⎥ ⎥	.
	⎢	10.6159 6.4136 -0.4136 -6.6364 ⎥			⎢ -8.2295 4.0852 6.2886 -0.9977 ⎥
	⎣	⎦			⎣	⎦

ACKNOWLEDGMENT

This work was partially supported by the China Scholarship Council (CSC) Scholarship (CSC No. 201908050104), and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

DATA AVAILABILITY STATEMENT

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.


Wilson R Lourenço
email: wilson.lourenco@mnhn.fr

The genus Ananteris Thorell, 1891 (Scorpiones: Buthidae) in the state of Amapá Brazil, and description of a new species

Keywords: Scorpion, taxonomy, Ananteris, Amapá state, new species, vicariant element

A new species of the genus Ananteris Thorell is described from Monte Karupina in the Serra Lombarda, State of Amapá, representing the first record for this state and the most northern location for one Ananteris in Brazil. The scorpion was collected in the base of Monte Karupina (an Inselberg), which presents a typical rainforest vegetation. The description of this new species brings further evidence about some particularities in the biogeographic patterns of distribution presented by the majority of Ananteris species. The species are highly endemic and can inhabit in sympatry with other species of the genus. In the present case the new species may represent a vicariant element recently isolated and distributed along a corridor going from the Brazilian Guiana (Amapá) to the Coastal region of Suriname.

Introduction

After the creation of the genus Ananteris by [START_REF] Thorell | Nova species Brasiliana ordinis Scorpionum[END_REF] this group remained inconspicuous and for almost a century only three species were known. In recent years, the number of species knew a significant growing, starting with the revision by [START_REF] Lourenço | Révision du genre Ananteris Thorell, 1891 (Scorpiones, Buthidae) et description de six espèces nouvelles[END_REF] which was followed by the description of numerous species from almost all regions of South America, in particular from Brazil, Venezuela and the countries of the Guayana region (e. g. González-Sponga, 2006;[START_REF] Lourenço | Comments on the Ananterinae Pocock, 1900 (Scorpiones: Buthidae) and description of a new remarkable species of Ananteris from Peru[END_REF][START_REF] Lourenço | A propos de la localité d'origine d'Ananteris balzanii Thorell, 1891, espèce type du genre Ananteris Thorell, 1891 (Scorpiones: Buthidae)[END_REF][START_REF] Lourenço | Some additional comments on the Ananterinae or 'Ananteris group' and description of a new species of Ananteris Thorell, 1891 from Central Brazil (Scorpiones: Buthidae)[END_REF][START_REF] Lourenço | The genus Ananteris Thorell, 1891 (Scorpiones: Buthidae) in Central Brazil, and description of one new species[END_REF][START_REF] Lourenço | Some additional comments on the Ananterinae or 'Ananteris group' and description of a new species of Ananteris Thorell, 1891 from Central Brazil (Scorpiones: Buthidae)[END_REF]Lourenço et al., 2020a;[START_REF] Ythier | A synopsis of the genus Ananteris Thorell, 1891 (Scorpiones: Buthidae) in French Guiana, with description of four new species[END_REF]. A recent synopsis proposed by Dupré (2020) suggests 92 described species up to 2020.

With such numbers, Ananteris occupies the third rank among scorpion genera, being only surpassed by genera Centruroides Marx (≈ 95 species) and Tityus C. L. Koch (≈ 220 species) species. The validity of some species can probably be questioned, such as those described from Venezuela by González-Sponga (2006). This country holds on its own about half of the known species. Nevertheless, the biogeographic patterns of distribution presented by a majority of Ananteris species seem to support this concentration in some limited zones of distribution. The species are highly endemic and can be observed in sympatry with other species of the genus [START_REF] Lourenço | Révision du genre Ananteris Thorell, 1891 (Scorpiones, Buthidae) et description de six espèces nouvelles[END_REF][START_REF] Lourenço | The genus Ananteris Thorell, 1891 (Scorpiones: Buthidae) in Central Brazil, and description of one new species[END_REF][START_REF] Lourenço | Some additional comments on the Ananterinae or 'Ananteris group' and description of a new species of Ananteris Thorell, 1891 from Central Brazil (Scorpiones: Buthidae)[END_REF]. Convergent morphological patterns also suggest the possible existence of vicariant species, a model already observed for other buthid genera such as Grosphus Simon in Madagascar (Lourenço et al., 2020b).

It is quite possible that a significant revision of this genus may bring more precise details on its pattern of distribution and differentiation. Recently some information about the existence of a possibly revisionary work become available; in fact a doctoral thesis produced by Santos da Silva (2019). The access to this publication was however limited to a simplified version of this monograph. From what was presented in the simplified version, it seems, once again, that it corresponds mainly to one more phylogenetic study which attempt to change most aspects of the classical classification, but in most cases based only on theoretical evidence but without the necessary study of the available type material. Only the full publication of this study will reveals which final conclusions are reached by the author.

In this note one new species is described from low altitude rain forests located at the base of Monte Karupina in the Serra Lombarda, in fact an Inselberg. This Mountain is located in the State of Amapá which is also part of the Guayana region [START_REF] Mori | The Guayana lowland floristic province[END_REF]. The type locality of the new species represents the most northern location ever confirmed for an Ananteris species in Brazil.

Methods

Measurements and illustrations were produced using a Wild M5 stereo-microscope with a drawing tube and an ocular micrometre. Measurements follow [START_REF] Stahnke | Scorpion nomenclature and mensuration[END_REF] and are given in mm. Trichobothrial notations are those developed by [START_REF] Vachon | Etude des caractères utilisés pour classer les familles et les genres de Scorpions (Arachnides). 1. La trichobothriotaxie en arachnologie. Sigles trichobothriaux et types de trichobothriotaxie chez les Scorpions[END_REF], morphological terminology mostly follows [START_REF] Vachon | Etudes sur les scorpions[END_REF] and [START_REF] Hjelle | Anatomy and morphology[END_REF], and chelicerae dentition follows [START_REF] Vachon | De l'utilité, en systématique, d'une nomenclature des dents des chélicères chez les Scorpions[END_REF].

Taxonomic treatment

Family Buthidae C. L. Koch, 1837

Genus Ananteris [START_REF] Thorell | Nova species Brasiliana ordinis Scorpionum[END_REF] Ananteris karupina sp. n. (6)(7)(8)(9)(10) Type material. Brazil, State of Amapá, Serra Lombarda, Monte Karupina (Inselberg), in low altitude rain forest (J.-M. Betsch leg.), IV/2001. One male holotype. The holotype will be deposited in the collections of the Museu Nacional, Rio de Janeiro, Brazil.

Etymology. The specific name is placed in apposition to the generic name and refers to the type locality, Karupina Mountain.

Diagnosis. Species of small size compared to the average size of the other species within the genus (total length 17.2 mm for male holotype; see morphometric values after the description). General coloration yellow, marked with brownish spots over the body and appendages. Chelicerae yellow without variegated spots over their entire surface; fingers strongly marked with blackish spots. Chela fingers with six rows of granules. Pectines of male holotype with 17-16 teeth. Telson with a globular shape, not elongate and with some conspicuous granules on ventral carina; subaculear tubercle spinoid. Carinae and granulation moderately marked. Metasomal segments with 10-8-8-8-5 carina; dorsal on segments II-IV with some spinoid granules.

Trichobothriotaxy, type A-. Femur trichobotrium e1 proximal in relation to d5; fixed finger trichobothria db and est at the same level; trichobothrium eb2 on external aspect of patella in a more proximal position than eb1. The type specimen showed a weak reaction to UV light.

Description based on male holotype

Coloration. Generally yellow with brownish variegated pigmented zones on the body and appendages. Carapace yellow with brownish spots, better marked centrally;

anterior and posterior edges with white spots; median ocular tubercle and lateral eyes dark, almost blackish. Mesosoma with confluent dark brown spots on tergites I to VI; tergite VII with a triangular central dark brown spot and square shaped dark brown spots on lateral sides. Sternites yellow to pale yellow without spots; pectines pale yellow; genital operculum, sternum and coxapophysis yellow, without spots.

Metasoma yellow; dorsal aspect of segments I to IV with a triangular brown spot; ventral and lateral sides of all segments with dark brown spots on their posterior region. Telson with vesicle yellow; base of subaculear tooth with diminutive brownish spots; base of aculeus reddish yellow, tip reddish. Chelicerae yellow without any spots over their entire surface; fingers yellow but strongly marked with blackish; teeth reddish. Pedipalp femur with dorsal side almost entirely marked with blackish-brown spots, with few yellowish zones posteriorly; patella yellow with several conspicuous brown spots; chela hand yellow, strongly marked with brown spots; fingers dark brown. Legs yellow intensely marked with brown spots.

Morphology.

Carapace with moderately marked granulation over the entire surface except on the anterior and distal edges; anterior margin almost straight, with a small convexity; carinae moderately to weakly marked; median ocular carinae forming an interocular furrow; lateral ocular carinae and posterior median carinae weak or vestigial; median ocular tubercle distinctly anterior to the centre of carapace; median eyes separated by a little less than one ocular diameter; three pairs of lateral eyes.

Tergites with moderately marked granulation, similar to that of carapace, better marked posteriorly; axial carina well marked on all tergites; tergite VII pentacarinate, with axial carina incomplete, median and lateral pairs of carinae complete. Sternum subpentagonal. Pectines large, with pectinal tooth count 17-16 for male holotype; fulcra absent. Sternites almost smooth; spiracles linear, slightly elongated. Metasomal segment I with 10 carinae; segments II-IV with 8 complete carinae; segment V with 5 complete carinae; intercarinal spaces weakly granular; segment V rounded without granules; dorsal and latero-dorsal carinae of segments I to IV with minute spinoid granules on their posterior region. Telson slightly globular in shape and weakly elongated and smooth; ventral median carina well marked with some strong granules; aculeus with a subaculear tooth strong and spinoid. Pedipalp femur with five carinae almost complete; patella with carinae much less marked, sometimes incomplete; chela with carinae weak to vestigial, almost always incomplete, made of scattered granules; dorso-internal carina of patella with 4-5 better marked spinoid granules; fixed and movable fingers with 6-6 longitudinal rows of granules, almost straight or slightly oblique, separated by bigger granules. Legs with tibial spurs well developed.

Cheliceral dentition characteristic of family Buthidae [START_REF] Vachon | De l'utilité, en systématique, d'une nomenclature des dents des chélicères chez les Scorpions[END_REF].

Trichobothriotaxy of type A- [START_REF] Vachon | Etude des caractères utilisés pour classer les familles et les genres de Scorpions (Arachnides). 1. La trichobothriotaxie en arachnologie. Sigles trichobothriaux et types de trichobothriotaxie chez les Scorpions[END_REF][START_REF] Vachon | Sur l'utilisation de la trichobothriotaxie du bras des pédipalpes des Scorpions (Arachnides) dans le classement des genres de la famille des Buthidae Simon[END_REF]. Voir diagnosis for details.

Morphometric values (in mm) of the male holotype of Ananteris sipilili and the male holotype of Ananteris karupina sp. n. Total length including telson, 24.4/17.2.

Carapace: length, 2.9/2.2; anterior width, 1.8/1.5; posterior width, 2.5/2.0. Mesosoma length, 5.6/4.3. Metasomal segments. I: length, 1.5/1.1; width, 1.5/1.2; II: length, 1.7/1.2; width, 1.4/1.1; III: length, 2.0/1.4; width, 1.4/1.1; IV: length, 2.7/1.7; width, 1.4/1.0; V: length, 4.3/2.8; width, 1.3/1.0; depth, 1.5/1.0. Telson: length, 3.7/2.5; width, 0.8/0.7; depth, 0.8/0.7. Pedipalp: femur length, 2.5/2.0, width, 0.7/0.5; patella length, 3.2/2.4, width, 0.8/0.6; chela length, 3.9/3.1, width, 0.6/0.4, depth, 0.5/0.4; movable finger length, 3.0/2.1.

Relationships. Ananteris karupina sp. n. seems to be related to Ananteris sipilili [START_REF] Ythier | A synopsis of the genus Ananteris Thorell, 1891 (Scorpiones: Buthidae) in French Guiana, with description of four new species[END_REF], species known from the north-west range of French Guiana. Both species are similar for a number of features and could even represent vicariant species, recently isolated, distributed along a corridor going from the Brazilian Guiana (Amapá) to the Coastal region of Suriname. The two species can however be readily distinguished by a number of characters: (i) A global smaller size for the new species (see measurements), (ii) Telson shorter and more to oval in the new species (see figs.

  Figures captions

Fig. 11 .

 11 Fig. 11. The Monte Karupina (Inselberg) in the Serra Lombarda surrounded by the low

Fig. 12 .

 12 Fig. 12. Map of the Guayana region, showing the type localities of Ananteris sipilili


Carole Pflumio

Idriss Troussier

Shan Xu

Julia Sun

Claire Salleron

Matthieu Petit

Arnaud Caubet

Valentin Beddok

Stéphanie Calugaru

Joël Servagi-Vernat

Castelli

Sun

Julia Salleron

Claire Petit

Matthieu Caubet

Arnaud Beddok

Valentin Calugaru

Stéphanie Servagi-Vernat

Joël Castelli

Jessica Miroir

Marco Krengli

Paul Giraud

Edouard Romano

Jonathan Khalifa

Mélanie Doré

Nicolas Blanchard

Alexandre Coutte

Charles Dupin

Shakeel Sumodhee

Yoann Pointreau

Samir Patel

Amel Rehailia-Blanchard

Ludivine Catteau

René-Jean Bensadoun

Yungan Tao

Vincent Roth

Lionnel Geoffrois

Jean-Christophe Faivre

Prof Juliette Thariat
email: jthariat@gmail.com

Unilateral or bilateral irradiation in cervical lymph node metastases of unknown primary? A retrospective cohort study

Keywords: neoplasms, unknown primary -head and neck neoplasms, radiotherapy, neck dissection, chemotherapy

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

Introduction

Head and neck cancer of unknown primary (CUP) represents 1-4% of head and neck tumours [START_REF] Galloway | Management of Squamous Cancer Metastatic to Cervical Nodes With an Unknown Primary Site[END_REF][START_REF] Grau | Cervical lymph node metastases from unknown primary tumours. Results from a national survey by the Danish Society for Head and Neck Oncology[END_REF]. Their diagnostic work-up includes fine-needle aspiration of the node(s), PET-CT and panendoscopy usually with tonsillectomy and/or mucosectomy [START_REF] Kwee | PET and PET/CT for unknown primary tumors[END_REF][START_REF] Rusthoven | The role of fluorodeoxyglucose positron emission tomography in cervical lymph node metastases from an unknown primary tumor[END_REF][START_REF] Troussier | Advances in the management of cervical lymphadenopathies of unknown primary; advances in diagnostic imaging and surgical modalities and new international staging system[END_REF][START_REF] Waltonen | Metastatic carcinoma of the neck of unknown primary origin: evolution and efficacy of the modern workup[END_REF][START_REF] Chen | Value of fused positron emission tomography CT in detecting primaries in patients with primary unknown cervical lymph node metastasis[END_REF][START_REF] De Ridder | Unknown primary head and neck squamous cell carcinoma in the era of fluorodeoxyglucose-positron emission tomography/CT and intensity-modulated radiotherapy[END_REF][START_REF] Arosio | Neck lymph node metastases from unknown primary[END_REF][START_REF] Mani | Role of 18-Fludeoxyglucose positron emission tomography-computed tomography and subsequent panendoscopy in head and neck squamous cell carcinoma of unknown primary[END_REF][START_REF] Smith | Cost-effectiveness analysis of PET-CT-guided management for locally advanced head and neck cancer[END_REF], as well as human papilloma virus (HPV) and Epstein Barr Virus (EBV) testing since the 2017 TNM classification [START_REF] Motz | Changes in Unknown Primary Squamous Cell Carcinoma of the Head and Neck at Initial Presentation in the Era of Human Papillomavirus[END_REF]. Neck dissection is used both as a diagnostic and therapeutic modality.

Irradiation aims to prevent regional relapse (≈10% of patients) [START_REF] De Ridder | Unknown primary head and neck squamous cell carcinoma in the era of fluorodeoxyglucose-positron emission tomography/CT and intensity-modulated radiotherapy[END_REF][START_REF] Hamoir | The role of neck dissection in the setting of chemoradiation therapy for head and neck squamous cell carcinoma with advanced neck disease[END_REF][START_REF] Thariat | Prediction of neck dissection requirement after definitive radiotherapy for head-and-neck squamous cell carcinoma[END_REF] and metachronous mucosal failure of the upper aerodigestive tract (≈5-15%) [START_REF] Troussier | Advances in the management of cervical lymphadenopathies of unknown primary; advances in diagnostic imaging and surgical modalities and new international staging system[END_REF][START_REF] Arosio | Neck lymph node metastases from unknown primary[END_REF][START_REF] Chen | Improved dosimetric and clinical outcomes with intensity-modulated radiotherapy for head-and-neck cancer of unknown primary origin[END_REF]. A current area of controversy is whether selective or extensive irradiation of nodal areas should be performed, and whether de-escalation of mucosal irradiation can be performed based on the low relapse rates, toxicity of extensive irradiation and presumed rates of HPV-related carcinomas. On the other hand, intensity-modulated radiation therapy (IMRT) has improved the tolerance to extensive nodal and mucosal irradiation to the point where it may prevent more locoregional relapses than elective irradiation while minimizing toxicity [START_REF] Grau | Cervical lymph node metastases from unknown primary tumours. Results from a national survey by the Danish Society for Head and Neck Oncology[END_REF][START_REF] Madani | Intensity-modulated radiotherapy for cervical lymph node metastases from unknown primary cancer[END_REF][START_REF] Nutting | Parotid-sparing intensity modulated versus conventional radiotherapy in head and neck cancer (PARSPORT): a phase 3 multicentre randomised controlled trial[END_REF]. Due to the rarity of CUP, however, the level of evidence is currently based only on retrospective studies of less than 200 patients [START_REF] Galloway | Management of Squamous Cancer Metastatic to Cervical Nodes With an Unknown Primary Site[END_REF][START_REF] Glynne-Jones | Metastatic carcinoma in the cervical lymph nodes from an occult primary: a conservative approach to the role of radiotherapy[END_REF][START_REF] Marcial-Vega | Cervical metastases from unknown primaries: radiotherapeutic management and appearance of subsequent primaries[END_REF][START_REF] Perkins | Radiotherapeutic management of cervical lymph node metastases from an unknown primary site[END_REF][START_REF] Reddy | Metastatic carcinoma in the cervical lymph nodes from an unknown primary site: results of bilateral neck plus mucosal irradiation vs. ipsilateral neck irradiation[END_REF][START_REF] Weir | Radiation treatment of cervical lymph node metastases from an unknown primary: an analysis of outcome by treatment volume and other prognostic factors[END_REF][START_REF] Colletier | Postoperative radiation for squamous cell carcinoma metastatic to cervical lymph nodes from an unknown primary site: outcomes and patterns of failure[END_REF][START_REF] Ligey | Impact of target volumes and radiation technique on loco-regional control and survival for patients with unilateral cervical lymph node metastases from an unknown primary[END_REF]. To date, no prospective randomized trial has ever been completed to advocate for or against either strategy, as the sole randomized trial (NCT00047125; unpublished) started was terminated early due to insufficient accrual.

We aimed to assess whether bilateral and unilateral nodal neck irradiation resulted in different outcomes in terms of local and regional control and of toxicities.

Material and methods

This institutional review board-and ethical committee-approved retrospective, multicentre and international study included patients irradiated for CUP between 2000 and 2015. Patients with squamous cell CUP were included after proper diagnostic work-up showing absence of distant metastases and a histology-proven diagnosis of carcinoma, and were treated with curative external beam radiotherapy. The diagnostic work up has changed over time. For example, the use of PET-CT has become more systematic after 2008 after demonstration of its performances in the detection of mucosal head and neck primaries [START_REF] Johansen | Prospective study of 18FDG-PET in the detection and management of patients with lymph node metastases to the neck from an unknown primary tumor. Results from the DAHANCA-13 study[END_REF]. Apart from PET-CT, the diagnostic work up of CUPs included FNA then panendoscopy and head neck and chest CT. Patients with adenocarcinomas (or non-squamous cell carcinomas), lymphomas, melanomas or sarcomas, or previous head and neck irradiation were excluded. Data were collected on https://www.easy-crf.com/ambicup/ (encrypted secured website) and included age, gender, imaging, nodal stage, extranodal spread, nodal diameter, histology, differentiation and HPV/EBV status. Treatment-related data included neck dissection, radiotherapy technique (three-dimensional (3D) or IMRT), total dose and fractions, interruption of radiotherapy, and target volumes: uni or bilateral nodal irradiation and their risk-dependent dose levels, pan-mucosal or elective or no mucosal irradiation, chemotherapy (neoadjuvant or concomitant).

We refer to microscopic mucosal disease turning into a macroscopic primary tumor if left untreated at the time of diagnosis of CUP. Of note, a second primary is usually defined as a primary tumor occurring in another site compared to first primary event. However, by definition CUP do not exhibit a primary. Another aspect of the definition for second primaries is time to occurrence later than 5 years after first event.

Patients underwent follow-up visits according to standards at their institutions and their physician's discretion. Acute and late toxicities were based on the National Cancer Institute Common Terminology Criteria for Adverse Events, Version 3.0 (from descriptions in charts).

Statistics

Quantitative parameters were described by median, mean and standard deviation, qualitative parameters by frequency and percentage. Missing data were not computed in the percentages.

Regional failure was defined as the persistence or recurrence of tumoral lymph node(s) and local failure as emergence of primary in the mucosae of the upper aerodigestive tract. Local, regional and locoregional relapses were described with the Fine and Gray model, to take into account competing risks such as emergence of metastases or death whatever the cause. For CUP-specific survival, we only considered death due to head and neck cancer, and the Fine and Gray model was also computed to consider death due to other causes as a competing risk.

The Kaplan-Meier method was performed to describe overall survival (OS) defined as the time lapse between the date of diagnosis and the date of death, whatever the cause. The prognostic value of each factor was studied using the bivariate Gray model, and the results were expressed with the hazard ratio (HR) and its 95% confidence intervals. The parameters with a p-value less than 0.1 in bivariate analysis were introduced in a multivariate Gray model, with backward selection. All statistical analyses were performed using SAS software (SAS Institute Inc., Cary, NC 25513). P-values <0.05 were considered statistically significant.

Results

From 2000 to 2015, 377 patients were irradiated for CUP, of whom 27 were excluded due to other histology (n=2), no radiotherapy (n=1) or insufficient follow-up data (n=20). Patient and tumour characteristics of the 350 patients treated in 20 institutions are presented in Table 1.

Patients with N2a/b disease represented the majority of the population, but N3 disease was also frequently observed. A majority (74.5%) of patients had unilateral nodal disease, while 82 (25.5%) patients had N2c or bilateral N3 disease. Fifty-eight (70.7%) patients with bilateral disease had N3 presentation. Conventional squamous cell carcinomas accounted for 97.7% of all carcinomas. Human papilloma status was tested in only 58 patients and was positive in 18 of them. Before 2005, 15% of patients had a PET (or PET CT), in 2005 50% and after 2006, 95%.

Treatment characteristics are presented in Table 2. A majority of patients underwent neck dissection (74.4%), while the other patients were either inoperable or had unresectable disease. All had nodal irradiation and 304 (87.6%) had mucosal (elective or extended) irradiation. A majority of patients underwent concomitant chemotherapy (64.8%) and 9.8% had neoadjuvant chemotherapy. Among 297 patients with available data on disease and irradiation side, sixty-one (20.5%) patients had unilateral disease and underwent unilateral irradiation, 155 (52.2%) had unilateral disease and underwent bilateral irradiation and eightyone (27.3%) patients had bilateral disease and bilateral irradiation. In 306 patients for whom target volume side was reported, there was 1% unilateral irradiation until 2008 (1/89). In contrast, after 2009, 29% of the patients underwent bilateral irradiation (63/216), p<0.001.

Equal proportions of patients received 3D irradiation or IMRT. The oropharynx was the most commonly irradiated primary site (84.9%), while the nasopharynx, larynx and hypopharynx were irradiated in two-thirds of the patients and the oral cavity in less than a quarter of patients. Those N1 patients who underwent RT were included, there were 39 (11.5%) patients with N1 disease presented in Table 1. One patient underwent radiochemotherapy exclusively, others underwent neck dissection then radiochemotherapy.

The median follow-up was 37 months (IQR: 24; 63). Out of 256 patients living at last followup, 64 (25.0%) patients had less than 24 months follow-up but at least three months followup. Crude failure rates are presented in Table 3. Ninety-three (26.6%) patients had an isolated or combined relapse at a median time of 12 months. Of these, there were 26 (7.4%) local relapses, 41 (11.7%) regional relapses and 46 (13.1%) metastatic relapses. Details of the patterns of failure are presented in Figure 1 (supplementary data). Among the five patients with bilateral nodal disease at diagnosis, four had bilateral relapse and one had unilateral relapse in the neck. Among the 36 patients with unilateral disease at diagnosis, 23 had unilateral relapse, seven had contralateral relapse and six patients had bilateral relapse. Of those patients with unilateral relapse, 19 had extended nodal irradiation (three patients without detailed nodal volume irradiation), with a median of five and a minimum of four nodal levels irradiated, suggesting that nodal relapse occurred in field. The median dose at the site of relapse was 54Gy (IQR 30; 60). Sixty-two (17.7%) patients died of head and neck cancer. At last follow-up, 64.3% (225) patients were alive without disease. Cumulative threeyear incidence of local (Figure 2a), regional (Figure 2b) and locoregional (Figure 2c Prognostic factors of local, regional relapse and CUP-specific death are presented in Table 4.

In multivariate analysis, mucosal irradiation was the only independent prognostic factor associated with better local control. There was no statistical difference between pan-mucosal and selective mucosal irradiation (HR 1.36 [0.48;3.86], p=0.55) among the 304 patients undergoing mucosal irradiation. There was no significant association between irradiation of mucosal site (oral cavity, oropharynx, nasopharynx, larynx or hypopharynx) and mucosal relapse (Table 4). In multivariate analysis, advanced (N2b/c and N3) or early (N1 and N2a) disease, no neck dissection and interruption of radiotherapy for more than four days were prognostic factors of regional relapse.

In multivariate analysis, absence of PET-CT at diagnosis, largest nodal diameter, N2c/N3 disease, neoadjuvant chemotherapy and interruption of radiotherapy were prognostic factors of CUP-specific death. Metastatic relapse was less frequent in patients with a PET-CT at diagnosis or than in those without (data not shown). There were no toxic deaths, therefore toxicity does not explain the more frequent CUP-specific deaths associated with neoadjuvant chemotherapy (data not shown).

Unilateral or bilateral nodal irradiation resulted in statistically similar outcomes (Table 4) for 297 patients with available data on disease and irradiation side (Figure 4). However, in patients with unilateral disease, the cumulative three-year incidence of local relapse (Figure 3a) was 4.3% [95%CI 0.9-7.6] for those undergoing bilateral irradiation, while it was 11.1% [95%CI 2.3-19.2] in patients undergoing unilateral irradiation (p=0.32, HR 0.61 [95 CI 0.23-1.63]). Similarly, the cumulative incidence of regional failure (Figure 3b) was 7.7% [95%CI 3.2-11.9] for those undergoing bilateral irradiation, while it was 16.9% [95%CI 6.1-26.4] in patients undergoing unilateral irradiation (p=0.17

44]).

There was no significant difference between <=2008 versus >2008 in terms of local relapse, regional relapse or CUP-related death (data not shown).

Acute and late toxicities

Severe (grade 3-4) acute and late toxicities are presented in Table 5. They were assessed in 301 (86.0%) patients. There were no grade 5 (lethal) toxicities. Acute toxicities mostly consisted of dysphagia, mucositis and pain. Severe dysphagia and pain were more frequent in cases of bilateral nodal irradiation (both p<0.01). Late toxicities, which occurred in less than 15% of all patients, mainly consisted of severe xerostomia, dysphagia and fibrosis. Severe xerostomia and dysphagia were more frequent after bilateral nodal irradiation (both p<0.01).

Toxicities were responsible for treatment interruption of four consecutive days or more in 23 (6.6%) patients.

Bilateral irradiation was performed with 3D in 52% of patients (127/242), while unilateral irradiation was performed with 3D in 16% of cases (10/64, p<0.001) only. There was a trend for more toxicities with bilateral 3D irradiation vs IMRT in case in bilateral disease. Patients undergoing bilateral irradiation (n=242) with 3D irradiation vs IMRT had similar rates of severe acute toxicities but more late fibrosis (12.1% (15) vs 0.9% (1); p<0.01), xerostomia (25.8% (32) vs 6.3% (7); p<0.01) and dysphagia (15.3% (19) vs 2.7% (3); p<0.01).

Discussion

With 350 patients, the present study is the largest to date in a rare subgroup of head and neck cancers, and it specifically addressed "standard" bilateral extended nodal volume irradiation versus de-escalation with unilateral (often elective) nodal irradiation in patients with CUP.

Most patients underwent bilateral irradiation; 52.2% of them had bilateral irradiation for unilateral nodal disease, while 20.5% of them had unilateral irradiation for unilateral disease.

Of note, IMRT became a standard of care in head and neck cancers in 2011 [START_REF] Grau | Cervical lymph node metastases from unknown primary tumours. Results from a national survey by the Danish Society for Head and Neck Oncology[END_REF]. While some institutions have been advocating unilateral irradiation since around 1995 because of concerns around rare locoregional events and radiation toxicities, others have moved toward IMRTbased bilateral irradiation to decrease the rate of toxicities while maintaining excellent locoregional control rates. As a result, half the patients of this series were treated with IMRT.

Our results suggest that some late toxicities after bilateral 3D irradiation can be avoided with IMRT. Thus, toxicities following bilateral or unilateral irradiation should be investigated in larger IMRT studies.

The present study shows that the regional control rate and occurrence of mucosal primaries did not differ between patients who had unilateral irradiation and those who had bilateral irradiation. However, as observed on curves of cumulative events in patients with unilateral disease at diagnosis, patients with bilateral irradiation appeared to do better compared to patients undergoing unilateral irradiation. Consistent with other series, the number of events was low, as 11.7% of patients had a regional relapse, and 7.4% had a mucosal failure during follow-up but the median follow-up was limited to 37 months (IQR: 24; 63). For Ligey et al, the nodal relapse rate was 34% after unilateral neck irradiation and 25% after bilateral radiotherapy (p = 0.21) after a median follow-up of 3.3 years. A primary head and neck tumour occurred in 12% after unilateral irradiation and 6% after bilateral radiotherapy (difference not significant) [START_REF] Weir | Radiation treatment of cervical lymph node metastases from an unknown primary: an analysis of outcome by treatment volume and other prognostic factors[END_REF][START_REF] Colletier | Postoperative radiation for squamous cell carcinoma metastatic to cervical lymph nodes from an unknown primary site: outcomes and patterns of failure[END_REF][START_REF] Ligey | Impact of target volumes and radiation technique on loco-regional control and survival for patients with unilateral cervical lymph node metastases from an unknown primary[END_REF]. The original hypothesis was that unilateral irradiation would be responsible for 15% more relapses than bilateral irradiation. However, a quarter of patients had bilateral disease at diagnosis and half the patients underwent 3D irradiation. Thus, we will investigate whether the benefit of bilateral irradiation in patients with unilateral disease might become significant (with a power of 80%) in a larger study which includes 591 additional patients, with 272 patients undergoing unilateral IMRT. We will also assess the ongoing trend to de-escalate nodal and mucosal radiotherapy volume, and ultimate disease control after salvage treatment of nodal and/or mucosal failures. On the other hand, this present study shows that both selective and pan-mucosal irradiation, the latter extending from the nasopharynx to the hypopharynx and larynx, helped to avoid mucosal failures and allowed a significant CUP-specific survival benefit compared to no mucosal irradiation. Further data are needed to investigate whether elective mucosal irradiation yields similar local control to, and fewer late severe toxicities than, pan-mucosal irradiation. Altogether, our observations favour bilateral nodal irradiation and mucosal irradiation.

As for nodal control, advanced stage and no neck dissection were associated with poorer regional control. As most patients were French, they underwent upfront neck dissection per national CUP policy. Thus, patients undergoing non-surgical options upfront and no neck dissection afterwards [START_REF] Mehanna | PET-CT Surveillance versus Neck Dissection in Advanced Head and Neck Cancer[END_REF] were an unfavourable group, and neoadjuvant chemotherapy did not compensate for their poorer prognosis. Moreover, neck dissection improves locoregional control but not survival in the era of chemoradiation for CUP. For example, in a meta-analysis by Balaker et al, patients who underwent neck dissection with either postoperative radiation or chemoradiation had a 5-year survival of 52.4% compared to 46.6% for those treated with chemoradiation alone; however, this difference was not statistically significant [START_REF] Amsbaugh | Neck dissection for unknown cancer of the head and neck in the era of chemoradiation[END_REF][START_REF] Al Kadah | Cancer of unknown primary (CUP) of the head and neck: retrospective analysis of 81 patients[END_REF][START_REF] Balaker | Cancer of unknown primary: does treatment modality make a difference?[END_REF].

Omission of neck dissection, which is responsible for shoulder, neural (XI) and swallowing morbidity, was not our study aim and was not evaluated in our series due to neck dissection policy.

Interestingly, in addition to advanced nodal stage and size [START_REF] Al Kadah | Cancer of unknown primary (CUP) of the head and neck: retrospective analysis of 81 patients[END_REF][START_REF] Balaker | Cancer of unknown primary: does treatment modality make a difference?[END_REF], neoadjuvant chemotherapy and interruption of radiotherapy, the fact that absence of PET-CT at diagnosis had a negative effect on CUP-specific survival is intriguing. It is possible that patients not undergoing PET-CT at diagnosis were more likely to have subclinical metastases, and so died of symptomatic metastases later in follow-up, than those with no metastases on PET-CT [START_REF] Johansen | Prospective study of 18FDG-PET in the detection and management of patients with lymph node metastases to the neck from an unknown primary tumor. Results from the DAHANCA-13 study[END_REF].

Another hypothesis is that PET-CT improves the definition of nodal target volumes before neck dissection and irradiation [START_REF] Lonneux | Positron emission tomography with [18F]fluorodeoxyglucose improves staging and patient management in patients with head and neck squamous cell carcinoma: a multicenter prospective study[END_REF]. In contrast to neoadjuvant chemotherapy, concomitant chemotherapy was not associated with poorer prognosis. Most patients received cisplatin where poor prognostic factors, as defined in other head and neck cancers [START_REF] Bernier | Postoperative irradiation with or without concomitant chemotherapy for locally advanced head and neck cancer[END_REF][START_REF] Al-Sarraf | Chemoradiotherapy versus radiotherapy in patients with advanced nasopharyngeal cancer: phase III randomized Intergroup study 0099[END_REF], were identified following evaluation of the neck dissection specimen.

Study limitations include the lack of systematic HPV testing. However, to date, HPV testing is only recommended in oropharyngeal cancers and EBV for nasopharyngeal cancers only.

Reporting of HPV or EBV status has not been standard practice in participating institutions.

In this series, all patients had unknown primaries (T0) after thorough diagnostic locoregional and distant procedures. While recent retrospective studies suggest that HPV testing should be systematic [START_REF] Zhou | Squamous cell carcinoma of unknown primary of the head and neck: Favorable prognostic factors comparable to those in oropharyngeal cancer[END_REF] to advocate treatment de-escalation [START_REF] Strojan | Squamous cell carcinoma of unknown primary tumor metastatic to neck nodes: role of elective irradiation[END_REF][START_REF] Schroeder | Human papillomavirus as prognostic marker with rising prevalence in neck squamous cell carcinoma of unknown primary: A retrospective multicentre study[END_REF][START_REF] Chen | Oropharynx-directed ipsilateral irradiation for p16-positive squamous cell carcinoma involving the cervical lymph nodes of unknown primary origin[END_REF], such data may be premature if the unknown (yet microscopic) primary indeed resides in the larynx or hypopharynx. In our series, only five out the 26 mucosal relapses occurred in the oropharynx only. Whether HPVguided de-escalation of radiotherapy volumes is relevant regardless of the involved neck level(s) is questionable given the results of our study. Such a strategy might better apply to cystic nodes and/or levels 2 and 3, and should be investigated with more stringent methodology. The TNM 2017 classification might be overemphasizing the value of HPV testing. As suggested by the landmark Lindberg study in 1972, the risk for nodal involvement can be estimated based on the primary location. The reverse may be applied for CUPs.

There could be an effect of time and that was indeed our initial hypothesis but there was however no significant difference between <=2008 versus >2008 in terms of local relapse, regional relapse or CUP-related death in our study. We had observed a progressive switch in practice despite 1/ no or very limited level of evidence in favour of unilateral irradiation rather than bilateral irradiation 2/ no major event in favour of unilateral irradiation 3/ the possibility to limit the morbidity of irradiation (and in particular bilateral irradiation) with IMRT. To investigate the latter hypothesis, the length of the study rather appears as a strength as we could collect data from patients with similar disease presentation but undergoing either 3D or IMRT.

Trends in PET CT have clearly changed dramatically over years. We analyzed rates more specifically. Before 2005, 15% of patients had a PET (or PET CT), in 2005 50% and after 2006, 95%. There was however no impact of PET CT on locoregional control. It was related to CUP-specific death. Our hypothesis is that metastatic patients were excluded while some may have been included in the study if they had had no PET CT due to undiagnosed metastases.

Missing data are clearly a weakness as in many retrospective studies, but it is indicated and even with incomplete patient data for certain items, this remains a large study compared to other CUP publications (297 patients with available data on disease and irradiation sides out of 350 patients=84%). Of note, the 53 patients with missing data on disease and irradiation sides had comparable characteristics than the others (data not show). Consequently, the 297 patients are representative for the whole population. Unfortunately, the location of the initial nodal desease is missing but N-stage and unilateral/bilateral desease are clearly specified. By being multicentric, we may consider that this study allowed for investigating the impact of dose and technique, in contrast to a monocentric with single practice.

16 000 new head and neck cases are diagnosed in France; ie about 600 cases with CUPs.

It is difficult to get exhaustivity in retrospective studies but we tried to have a representation of differents kinds of health care institutions (private, public, tertiary versus regional centers etc). Thus the number achieved is representative and relevant to investigate radiation practice thus it is a full overview.

Conclusion

This large study of cervical lymphadenopathies of unknown primary suggests that unilateral neck irradiation may not yet be the treatment standard, as it may result in slightly worse rates of mucosal and nodal relapse. Severe toxicities were, however, more frequent after bilateral irradiation than unilateral irradiation. Molecular biomarkers are probably necessary to better predict the primary site of origin in a way that is adapted for the neck levels involved.

However, not all CUPs are HPV-positive. Thus, de-escalation of the volumes of nodal and/or mucosal irradiation with IMRT should be investigated further. The prognostic impact of the 8 th TNM 2017 classification, which takes into account EBV and HPV in CUP, should also be assessed. We are continuing this study so as to collect enough patients to reach sufficient power.

Tables and Figures

Figures: no color necessary

  ) failures were 5.6% [95%CI 3.1-8.1], 11.8% [95%CI 8.2-15.2] and 15.0% [95%CI 1.0-18.8], respectively. Three-year OS was 80.6% [95%CI 75.5-84.8] and cumulative incidence of CUP-specific death was 15.3% [95% CI 11.0-19.3]. Details of nodal and mucosal relapses are presented in supplementary data.

  , HR 0.56 [95 CI 0.25-1.27]). Locoregional incidence is shown in Figure 3c. Again, the cumulative incidence of CUP-specific deaths (Figure 3d) was 9.2% [95%CI 4.1-14.0] for those patients undergoing bilateral irradiation, while it was 15.5% [95%CI 4.1-25.6] in patients undergoing unilateral irradiation (p=0.92, HR 1.04[95% CI 0.45-2.41]). The third group of patients, i.e. those patients with bilateral disease at diagnosis who underwent bilateral irradiation, had a cumulative incidence of CUPspecific deaths of 26.9% [95%CI 15.1-37.0] (p=0.06 HR=2.28[95% CI 0.95-5.

Figure 1 (Figure 4 :Table 1 :Table 2 :Table 3 : 4 :

 141234 Figure 1 (supplementary data): patterns of failure

Table 5 :

 5 Acute and late adverse events grade 3-4. Footnotes: Nc, Not calculated; RTN, Nodal radiotherapy. Results expressed with frequency and percentages

Table 6 (

 6 supplementary data): Sides of mucosal and regional relapses

Table 1 :

 1 Patient and tumour characteristics Extranodal spread = 37, HPV = 292. Totals account for missing data, percentages are calculated with known data only

	Characteristics
	Age (years)

Table 2 :

 2 Characteristics of irradiation, surgery and other antineoplastic treatments

	Characteristics
	Surgery
	Tonsillectomy

Table 3 :

 3 Description of crude rates of each outcome

	Any relapse

Table 4 :

 4 Prognostic factors of local and regional relapse and cause-specific death in bivariate and multivariate analyses using the Gray model for competing risk data

			Local relapse				Regional relapse			CUP Specific death
		Bivariate analyses		Multivariate analysis		Bivariate analyses		Multivariate analysis		Bivariate analyses		Multivariate analysis
		HR 95% CI	p	HR 95% CI	p	HR 95% CI	p	HR 95% CI	p	HR 95% CI	p	HR 95% CI	p
	Patients and tumours
	Male gender	2.55 [0.62 ;10.60]	0.20			1.52 [0.60 ;3.85]	0.37			3.15 [1.15 ;8.67]	0.03*
	Age	1.04 [1.0 ;1.09]	0.07*			1.02 [1.00 ;1.05]	0.11			1.00 [0.97 ;1.02]	0.84
	18FDG PET	0.63 [0.28 ;1.43]	0.27			0.84 [0.40 ;1.79]	0.66			0.41 [0.24 ;0.71]	<0.01*	0.43 [0.23 ;0.80]	<0.01
	Neck dissection	0.87 [0.36 ;2.08]	0.76			0.39 [0.21 ;0.73]	<0.01*	0.43 [0.23 ;0.83]	0.01	0.70 [0.40 ;1.23]	0.22
	Diameter of largest node (cm)	0.99 [0.95 ;1.03]	0.65			1.02 [0.99 ;1.06]	0.17			1.06 [1.04 ;1.08]	<0.01*	1.06 [1.03 ;1.09]	<0.01
	TNM
	N1+N2a	1				1		1		1		1
	N2b	0.69 [0.24 ;2.00]	0.50			2.13 [0.82 ;5.53]	0.12	2.34 [0.92 ;5.96]	0.07	1.54 [0.74 ;3.19]	0.25	1.07 [0.46 ;2.47]	0.87
	N2c+N3	1.24 [0.49 ;3.13]	0.65			3.94 [1.61 ;9.66]	<0.01*	3.49 [1.43 ;8.49]	<0.01	3.67 [1.87 ;7.21]	<0.01*	2.68 [1.32 ;5.43]	<0.01
	Extranodal spread	1.03 [0.41 ;2.59]	0.95			1.69 [0.74 ;3.87]	0.21			2.24 [1.06 ;4.73]	0.04*
	SCC	0.64 [0.16 ;2.68]	0.55			0.71 [0.23 ;2.23]	0.56			0.82 [0.30 ;2.25]	0.70
	Differentiation
	Well differentiated	1				1				1
	Moderate+poor+undifferentiated	1.76 [0.65 ;4.75]	0.27			0.53 [0.28 ;0.99]	0.047*			0.70 [0.41 ;1.20]	0.20
	Nodal irradiation
	Radiotherapy technique
	3D	1				1				1
	IMRT	1.91 [0.88 ;4.12]	0.10			1.18 [0.63 ;2.19]	0.61			0.64 [0.37 ;1.12]	0.12
	Group
	Unilateral disease & unilateral irradiation	1				1				1
	Unilateral disease & bilateral irradiation	0.61 [0.23 ;1.63]	0.32			0.56 [0.25 ;1.27]	0.166			1.04 [0.45 ;2.41]	0.92
	Bilateral disease & bilateral irradiation	0.58 [0.19 ;1.80]	0.35			1.16 [0.50 ;2.67]	0.74			2.28 [0.95 ;5.44]	0.06*
	Total dose (Gy)
	≤ 56	1				1				1
	> 56 and ≤ 63	1.96 [0.33 ;11.82]	0.46			0.68 [0.12 ;3.86]	0.66			2.30 [0.74 ;7.17]	0.15
	> 63	1.73 [0.40 ;7.58]	0.46			1.49 [0.52 ;4.30]	0.46			1.74 [0.68 ;4.48]	0.25
	Nodal high-risk level irradiation	1.52 [0.46 ;5.05]	0.49			2.57 [0.77 ;8.57]	0.13			4.23 [1.29 ;13.84]	0.02*
	Nodal medium-risk level irradiation	1.31 [0.59 ;2.91]	0.51			0.67 [0.34 ;1.32]	0.25			1.58 [0.92 ;2.71]	0.10*
	Nodal low-risk level irradiation	0.82 [0.24 ;2.84]	0.76			1.47 [0.45 ;4.80]	0.52			0.97 [0.42 ;2.28]	0.95
	RT interruption ≥ 4 days	2.45 [0.81 ;7.42]	0.11			3.48 [1.47 ;8.21]	0.0045*	3.39 [1.46 ;7.88]	<0.01	3.23 [1.57 ;6.63]	<0.01*	3.81 [1.71 ;8.50]	<0.01
	Mucosal irradiation
	Mucosal irradiation	0.30 [0.13 ;0.69]	<0.01*	0.30 [0.13 ;0.69]	<0.01	0.70 [0.31 ;1.57]	0.39			0.64 [0.33 ;1.26]	0.20
	Total dose > 50 Gy	2.10 [0.84 ;5.28]	0.11
	Nasopharynx	0.56 [0.21 ;1.50]	0.25
	Oropharynx	0.33 [0.04 ;2.74]	0.31
	Hypopharynx	1.09 [0.25 ;4.74]	0.91
	Larynx	2.19 [0.51 ;9.48]	0.30
	Oral cavity	2.31 [0.90 ;5.96]	0.08
	Chemotherapy
	Neoadjuvant	1.14 [0.34 ;3.86]	0.83			2.78 [1.33 ;5.80]	<0.01*			2.53 [1.27 ;5.06]	<0.01*	2.52 [1.19 ;5.33]	0.02
	Concomitant	0.48 [0.22 ;1.03]	0.06*			0.98 [0.52 ;1.83]	0.94			0.85 [0.52 ;1.39]	0.51

*Variables included in multivariate analysis with backward selection HR 95% CI, Hazard ratio 95% confidence interval; 18 FDG PET, 18Fluorodeoxyglucose positron emission tomography; SCC, Squamous cell carcinoma; IMRT, Intensity-modulated radiation therapy; 3D, Threedimensional; Uni-D Uni-I, unilateral disease at diagnosis undergoing unilateral irradiation; Uni-D Bi-I, unilateral disease at diagnosis undergoing bilateral irradiation; Bi-D Bi-I, Bilateral disease at diagnosis undergoing bilateral irradiation; Gy, Gray; NRT, Nodal radiotherapy; HR, High risk level; IR, Intermediate risk level; LR, Low risk level; RT, radiotherapy

Table 5 :

 5 Acute and late adverse events grade 3-4 for the 297 patients with an available evaluation according to the side of irradiation and technique.Nc, Not calculated; NRT, Nodal radiotherapy. Results expressed with frequency and percentages Totals account for missing data, percentages are calculated with known data only

		Global	Unilateral NRT	Bilateral NRT	p	Unilateral NRT		Bilateral NRT
						2D or 3D	IMRT	p	2D or 3D	IMRT	p
	Number of available data	297	64	242		10	54		127	115
	Acute toxicities
	Dysphagia	78 (26.2%)	8 (12.7%)	70 (29.8%)	<0.01	0% (0)	15.1% (8)	0.33	33.1% (41)	26.1% (29)	0.25
	Mucositis	69 (23.3%)	10 (16.4%)	59 (25.1%)	0.15	11.1% (1)	17.3% (9)	1	29.0% (36)	20.7% (23)	0.14
	Pain	45 (15.0%)	4 (6.3%)	41 (17.3%)	0.03	0% (0)	7.4% (4)	Nc	15.9% (20)	18.9% (21)	0.54
	Dermatitis	0 (0%)	0 (0%)	0 (0%)	Nc	0	0		0	0
	Other acute toxicity	11 (3.6%)	2 (3.2%)	9 (3.8%)	1	10% (1)	1.96% (1)		6(4.8%)	3(2.7%)	0.51
	Late toxicities
	Xerostomia	40 (13.5%)	1 (1.6%)	39 (16.5%)	<0.01	10% (1)	0% (0)	Nc	25.8% (32)	6.3% (7)	<0.01
	Dysphagia	22 (7.4%)	0 (0%)	22 (9.3%)	<0.01	0	0		15.3% (19)	2.7% (3)	<0.01
	Fibrosis	18 (6.1%)	2 (3.3%)	16 (6.8%)	0.54	10% (1)	1.96% (1)	Nc	12.1% (15)	0.9% (1)	<0.01
	Pain	8 (2.7%)	1 (1.6%)	7 (3.0%)	1	0% (0)	1.96% (1)	Nc	4.84% (6)	0.9% (1)	0.13
	Osteonecrosis	4 (1.3%)	0 (0%)	4 (1.7%)	Nc	0	0		3.23% (4)	0% (0)	0.12
	Second cancer	4 (1.3%)	0 (0%)	4 (1.7%)	Nc	0	0		3.2%(4)	0	0.12
	Oesophageal stricture	3 (1.0%)	0 (0%)	3 (1.3%)	Nc	0	0		2.42% (3)	0% (0)	0.26
	Trismus	1 (0.3%)	0 (0%)	1 (0.4%)	Nc	0	0		0.81% (1)	0% (0)	Nc
	Other	8 (2.7%)	1 (1.6%)	7 (3.0%)	1	1(10.0%)	0	Nc	4.0% (5)	1.8%(2)	0.45

Acknowledgements:

The manuscript has been revised for the English by an independent scientific language editing service (Angloscribe).

Conflict of interest statement:

The authors have declared no conflicts of interest with respect to the study Side of regional relapse for bilateral disease according to radiotherapy target volume