Chaouki Aouiti
email: chaouki.aouiti@fsb.rnu.tn

Qing Hui
email: qing.hui@unl.edu

Hediene Jallouli
email: jalouli.hedienne@fsb.rnu.tn

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Keywords: Sliding mode control, inertial neural networks, fixed-time stability and synchronization, time-varying delay

HAL is

Sliding mode control based fixed-time stabilization and synchronization of inertial neural networks with time-varying delays 1 Introduction Artificial Neural Networks (ANNs) are a system whose design is originally inspired by the operation of biological neurons and which has subsequently moved closer to statistical methods. In the last decade, the use of ANNs has been developed in many disciplines such as economic sciences, ecology and environment or biology and medicine [START_REF] Huang | Neural networks in finance and economics forecasting[END_REF][START_REF] Malmgren | Artificial Neural Networks in Medicine and Biology: Proceedings of the ANNIMAB-1 Conference[END_REF][START_REF] Papik | Application of neural networks in medicine-a review[END_REF][START_REF] Zhang | Computational ecology: artificial neural networks and their applications[END_REF]. They are applied in particular to solve problems of classification, prediction, categorisation, optimization, pattern recognition, associative memory or secure communication [START_REF] Alimi | Finite-time and fixed-time synchronization of a class of inertial neural networks with multi-proportional delays and its application to secure communication[END_REF][START_REF] Aouiti | The design of beta basis function neural network and beta fuzzy systems by a hierarchical genetic algorithm[END_REF][START_REF] Aouiti | Stability analysis for a class of impulsive high-order hopfield neural networks with leakage time-varying delays[END_REF][START_REF] Forti | Generalized neural network for nonsmooth nonlinear programming problems[END_REF][START_REF] Huang | Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders[END_REF][START_REF] Aquino | Novel nonlinear hypothesis for the delta parallel robot modeling[END_REF][START_REF] De | SOFMLS: online self-organizing fuzzy modified least-squares network[END_REF][START_REF] Chiang | Wavelet-based eeg processing for epilepsy detection using fuzzy entropy and associative petri net[END_REF][START_REF] De Rubio | Stability analysis of the modified levenberg-marquardt algorithm for the artificial neural network training[END_REF][START_REF] Meda-Campaña | On the estimation and control of nonlinear systems with parametric uncertainties and noisy outputs[END_REF][START_REF] Hernández | Hybrid neural networks for big data classification[END_REF]. In data processing, ANNs are a method of approximation of complex systems particularly useful when systems are difficult to model using conventional statistical methods. In recent years, numerous outcomes on Lyapunov stability, periodicity analysis and synchronization of various types of ANNs (Hopfield, Cohen-Grossberg, Competitive, BAM, MAM) have been obtained for instance in [START_REF] Alimi | Dynamics and oscillations of generalized high-order hopfield neural networks with mixed delays[END_REF][START_REF] Aouiti | The design of beta basis function neural network and beta fuzzy systems by a hierarchical genetic algorithm[END_REF][START_REF] Aouiti | A genetic-designed beta basis function neural network for multi-variable functions approximation[END_REF][START_REF] Aouiti | Stability analysis for a class of impulsive high-order hopfield neural networks with leakage time-varying delays[END_REF][START_REF] Aouiti | Global exponential convergence of neutral-type competitive neural networks with multi-proportional delays, distributed delays and time-varying delay in leakage delays[END_REF][START_REF] Aouiti | Existence and global exponential stability of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms[END_REF][START_REF] Aouiti | Finite time boundedness of neutral high-order Hopfield neural networks with time delay in the leakage term and mixed time delays[END_REF][START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF][START_REF] Aouiti | Impulsive generalised high-order recurrent neural networks with mixed delays: stability and periodicity[END_REF][START_REF] Aouiti | Finite Time Stabilization of Uncertain Delayed Hopfield Neural Networks with a Time-varying Leakage Delay via Non-Chattering Control[END_REF][START_REF] Mhamdi | Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays[END_REF].

Wheeler and Schieve were the first to publish the article [START_REF] Wheeler | Stability and chaos in an inertial two-neuron system[END_REF] in 1997 about INNs which are more complex than lots of kinds of NNs. INNs have been widely developed and many important results about asymptotic stability, robust stability, synchronization, stabilization and existence of periodic solutions have been established [START_REF] Cao | Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays[END_REF][START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF][START_REF] Guo | Finite-time synchronization of inertial memristive neural networks with time delay via delay-dependent control[END_REF][START_REF] Li | Nonlinear measure approach for the robust exponential stability analysis of interval inertial cohen-grossberg neural networks[END_REF][START_REF] Liu | Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation[END_REF][START_REF] Wheeler | Stability and chaos in an inertial two-neuron system[END_REF][START_REF] Yunquan | Stability and existence of periodic solutions in inertial BAM neural networks with time delay[END_REF][START_REF] Zhang | Global exponential stability via inequality technique for inertial BAM neural networks with time delays[END_REF]. In recent studies, the FTSt of INNs has been intensively considered where the closed-loop system solutions reach the equilibrium point in finite-time. The function giving the time for the solutions to reach the equilibrium is called the settling-time. The finite-time stability has a great importance in real applications like robotics, optimization problems, pattern recognition, vehicle system or spacecrafts [START_REF] Du | Finite-time attitude tracking control of spacecraft with application to attitude synchronization[END_REF][START_REF] Graham | A neural network approach for safety and collision avoidance in robotic systems[END_REF][START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF][START_REF] Kosmatopoulos | High-order neural network structures for identification of dynamical systems[END_REF][START_REF] Lee | Translation, rotation and scale invariant pattern recognition using spectral analysis and hybrid genetic-neural-fuzzy networks[END_REF][START_REF] Savković-Stevanović | A neural network model for analysis and optimization of processes[END_REF]. It is a common knowledge that the settling-time estimation in finite-time stabilization problems depends on the initial values of the dynamical systems [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF]. This results in disadvantages for real applications because in many practical systems it is difficult or impossible to obtain in advance the initial conditions [START_REF] Cao | Fixed-time synchronization of delayed memristor-based recurrent neural networks[END_REF]. The initial conditions of many practical systems are difficult to obtain accurately or even impossible to know in advance [START_REF] Cao | Fixed-time synchronization of delayed memristor-based recurrent neural networks[END_REF]. To settle this problem, Polyakov has developed the concept of fixed-time stability in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF].

It is well known that there are rare and inconsistent observations in measurements with the largest part of the population of observations, called outliers. The influence of external disturbances, modeling errors, and various uncertainties in the real systems, as well as the robust and filtering techniques to cope with these problems, are best shown in the following recent papers [START_REF] Chen | Event-based fuzzy control for ts fuzzy networked systems with various data missing[END_REF][START_REF] Cheng | Asynchronous fault detection filtering for piecewise homogenous markov jump linear systems via a dual hidden markov model[END_REF][START_REF] Stojanovic | State and parameter joint estimation of linear stochastic systems in presence of faults and non-gaussian noises[END_REF][START_REF] Zhang | Finite-time asynchronous dissipative filtering of conic-type nonlinear markov jump systems[END_REF][START_REF] Zhou | Pd-type iterative learning control for uncertain spatially interconnected systems[END_REF]. Stabilization control of NNs has attracted more and more attention, and there are different types of stabilization: asymptotic stabilization; exponential stabilization; finite-time stabilization; fixed-time stabilization [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF][START_REF] Aouiti | Fixed-time stabilization of fuzzy neutraltype inertial neural networks with time-varying delay[END_REF], as well as different types of control strategies: feedback control [START_REF] Alimi | Finite-time and fixed-time synchronization of a class of inertial neural networks with multi-proportional delays and its application to secure communication[END_REF][START_REF] Aouiti | Fixed-time stabilization of fuzzy neutraltype inertial neural networks with time-varying delay[END_REF]; feedback tracking control [START_REF] Tong | Robust adaptive fuzzy backstepping output feedback tracking control for nonlinear system with dynamic uncertainties[END_REF][START_REF] Tong | Observer-based adaptive fuzzy tracking control for strictfeedback nonlinear systems with unknown control gain functions[END_REF]; adaptive control [START_REF] Li | Command Filter Based Adaptive Fuzzy Finite-Time Control for Switched Nonlinear Systems Using State-Dependent Switching Method[END_REF][START_REF] Li | Adaptive fuzzy control of switched nonlinear timevarying delay systems with prescribed performance and unmodeled dynamics[END_REF]; hybrid control [START_REF] Rajchakit | Hybrid control scheme for projective lag synchronization of Riemann-Liouville sense fractional order memristive BAM neural networks with mixed delays[END_REF]; intermittent control [START_REF] Yang | Finite-time synchronisation of neural networks with discrete and distributed delays via periodically intermittent memory feedback control[END_REF]; pinning control [START_REF] Wu | Cluster synchronization of linearly coupled complex networks under pinning control[END_REF]; impulsive control [START_REF] Zhang | Synchronization of stochastic dynamical networks under impulsive control with time delays[END_REF], or SMC [START_REF] Ni | Fast fixed-time nonsingular terminal sliding mode control and its application to chaos suppression in power system[END_REF]. As an application of fixed-time stability, FTSt of NNs has received tremendous interest in recent years since it can make system stabilized within a bounded time even if the initial conditions are unknown in advance [START_REF] Aouiti | Fixed-time stabilization of fuzzy neutraltype inertial neural networks with time-varying delay[END_REF][START_REF] Li | Finite-time and fixed-time stabilization control of delayed memristive neural networks: Robust analysis technique[END_REF][START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF]. The fixed-time controller makes it stabilized at a fixed-time and the settling-time is independent on the initial conditions.

The authors of [START_REF] Li | Finite-time and fixed-time stabilization control of delayed memristive neural networks: Robust analysis technique[END_REF] discuss the finite-time and fixed-time stabilization for delayed memristive NNs based on Filippov solutions, and discontinuous controller. The study of FTSt for a general class of NNs (Cohen-Grossberg BAM) was investigated in [START_REF] Li | Fixed-time stabilization of impulsive Cohen-Grossberg BAM neural networks[END_REF] by using differential inequalities, some comparison techniques, and fixed-time stability theorem. In [START_REF] Aouiti | A New LMI Approach to Finite and Fixed Time Stabilization of High-Order Class of BAM Neural Networks with Time-Varying Delays[END_REF], the authors study the FTSt for a general class of BAM with time-varying delay using the finite time and fixed time stability theory, and linear matrix inequalities. In [START_REF] Yang | Finite-time synchronisation of neural networks with discrete and distributed delays via periodically intermittent memory feedback control[END_REF], the problem of finite-time synchronisation of NNks with mixed delays was investigated by using periodically intermittent memory feedback control, finite-time stability theory, linear matrix inequalities and differential inequality techniques. In [START_REF] Zhao | Sliding mode control in finite time stabilization for synchronization of chaotic systems[END_REF], the problem of finite-time stabilization for synchronization was investigated based on geometric homogeneity and integral sliding mode manifold. The robust finite-time synchronization of a class of chaotic systems via adaptive global SMC was considered in [START_REF] Xi | Robust finite-time synchronization of a class of chaotic systems via adaptive global sliding mode control[END_REF] based on new chattering-free control laws. The problem of improved SMC for finite-time synchronization of nonidentical delayed recurrent NNs was considered in [START_REF] Xiong | Improved sliding mode control for finite-time synchronization of nonidentical delayed recurrent neural networks[END_REF] by designing a suitable SMC based on Lyapunov stability theory. The integral SMC approach is presented in [START_REF] Shi | Projective synchronization of different chaotic neural networks with mixed time delays based on an integral sliding mode controller[END_REF] to investigate the projective synchronization of nonidentical chaotic NNs with mixed time delays by considering a proper sliding surface, constructing Lyapunov-Krasovskii functional and using linear matrix inequality techniques. In [START_REF] Gan | Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control[END_REF], the synchronization problems of chaotic fuzzy cellular NNs with timevarying delays were investigated based on SMC approach. In [START_REF] Zhang | Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller[END_REF], the authors study the projective synchronization for different chaotic time-delayed NNs by using Lyapunov-Krasovskii functional and linear matrix inequality techniques.

In this article, the problems of FTSt and FTSy are addressed for INNs under a new control strategy, namely SMC which is a nonlinear control technique that possesses remarkable properties of precision, robustness and easy implementation [START_REF] Utkin | Sliding mode control in electro-mechanical systems[END_REF]. SMC has two main advantages: first the system dynamics behaviour can be adapted to the choice of the sliding surface, secondly the closed-loop system is robust to uncertainties [START_REF] Decarlo | Variable structure control of nonlinear multivariable systems: a tutorial[END_REF][START_REF] Utkin | Variable structure systems with sliding modes[END_REF][START_REF] Young | A control engineer's guide to sliding mode control[END_REF]. Its concise design process and its ability to stabilize and synchronize the delayed INNs in a small fixed time are two advantages of the SMC.

The contributions of this article are described below:

-A novel control strategy for FTSt and FTSy of INNs with time-varying and distributed delays is introduced. We investigate the FTSt and FTSy of INNs with time-varying and distributed delays by using SMC with an adapted sliding surface. Moreover, by writing our system as a second-order multi-variable system, two different feedback control laws are built to obtain the FTSt and FTSy of INNs with time-varying and distributed delays. However, these controllers have singularities.

-By using some proprieties of a well defined sliding surface, new sufficient conditions for almost FTSt and almost FTSy of INNs with time-varying delay are obtained such that the controllers are not singular.

The paper is organized as follows. After some technical lemmas and definitions recalled in Section 2, the model description is given in Section 3. In Section 4, two sliding mode controllers are derived for the FTSt of INNs with time-varying delay, while the FTSy of INNs with time-varying delay is further considered in Section 5. In Section 6, illustrative examples are given to show the effectiveness of the proposed control design methods. The article ended with a conclusion in Section 7.

Preliminaries

Let R denote the set of real numbers, R + the set of positive real numbers, R n the n-dimensional Euclidean space, R n×n the set of squared matrices of size n, . the Euclidean norm such that x 2 = x T x, A T the transpose of the matrix A = (a ij ) 1≤i,j≤n , diag(a 1 , • • • , a n ) the diagonal matrix, |A| the matrix (|a ij |) 1≤i,j≤n and I n the identity matrix of size n.

Let us recall some results on finite-time stability and fixed-time stability. Consider the following ordinary differential equation

ẏ(t) = f (y(t)), y(t) ∈ R n (1)
y(0) = y 0 with f a continuous function such that f (0) = 0.

Definition 1 [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF] System (1) is globally finite-time stable if it is Lyapunov stable and for all y 0 ∈ R n there exists T(y 0 ) > 0 dependent on the initial conditions such that lim t→T(y0) y(t) = 0, i.e. y(t) ≡ 0 for all t ≥ T(y 0 ). The function T is called the settling-time.

Definition 2 [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] System (1) is globally fixed-time stable if:

(1) it is globally finite-time stable;

(2) the settling-time function T is upper bounded by a constant T max > 0, i.e. T(y 0 ) ≤ T max .

Definition 3 [START_REF] Zuo | Fixed-time stabilization of second-order uncertain multivariable nonlinear systems[END_REF] System (1) is globally almost fixed-time stable if it is globally finite-time stable and there exists a constant T max > 0 such that for all > 0 we have T(y 0 ) ≤ T max + T where T → 0 when → 0.

Remark 1 According to Definition 1 and Definition 2, we conclude that contrary to finite-time stability, the settling-time of fixed-time stability is independent on initial conditions. In many fields, such as pattern recognition or secure communication, the initial values are mostly hard to be acquired [START_REF] Zuo | An overview of recent advances in fixed-time cooperative control of multiagent systems[END_REF]. So fixed-time stability is more suitable and applicable than finite-time stability.

Lemma 1 [START_REF] Parsegov | Fixed-time consensus algorithm for multi-agent systems with integrator dynamics[END_REF] If there exists a continuously differentiable positive definite radially unbounded function V :

R n → R + such V(y(t)) ≤ -αV(y(t)) 1 -βV(y(t)) 2 (2)
where α > 0, β > 0 and 1 > 1 > 2 > 0 then system (1) is globally fixed-time stable and the settling-time T(y 0 ) satisfies

T(y 0 ) ≤ T max := 1 α( 1 -1) + 1 β(1 -2 ) (3)
The function V is called a Lyapunov function for system (1).

Remark 2 Without loss of generality, if 1 = 1 + 1 µ and 2 = 1 -1 µ are satisfied with µ > 1, the upper bound of the settling-time function T(y 0 ) of system (1) can be estimated by

T(y 0 ) ≤ µπ 2 α β . (4)
Lemma 2 [START_REF] Zuo | Fixed-time stabilization of second-order uncertain multivariable nonlinear systems[END_REF] The following system

ż(t) = -α z(t) 1-1 z(t) -β z(t) 2-1 z(t), (5)
where z ∈ R n , α, β > 0, 1 > 1 and 0 < 2 < 1 is globally fixed-time stable and the settling-time is given by (3).

Model description

The following INN with time-varying and distributed delays is considered

d 2 r (t) dt 2 = -λ dr (t) dt -a r (t) + n k=1 c k g k (r k (t)) + n k=1 d k g k (r k (t -τ (t))) + n k=1 b k t -∞ k k (t -σ)g k (r k (σ))dσ + u (t), t > 0 (6)
where

-= 1, • • • , n with n the number of neurons to consider; -r(.) = r 1 (.), • • • , r n (.)
T is the neuron state vector;

-d 2 r (t) dt 2
denotes the inertial term of system (6); λ > 0, a > 0; -k k stands for the delay kernel; τ (.) stands for the time-varying transmission delay; f k (.), g k (.) denotes the activation functions with

f k (0) = g k (0) = 0 for k = 1, • • • , n; -u (.) : R → R denotes the control input;
the initial conditions of system (6) are given by

r (s) = ϕ (s), dr (s) ds = ξ (s); s ∈ [-∞, 0] (7)
where = 1, • • • , n, ϕ and ξ are bounded and continuous functions.

Letting h (t) = dr (t) dt , then system ( 6) is written as

   dr (t) dt = h (t) dh (t) dt = -λ h (t) -a r (t) + n k=1 c k f k (r k (t)) + n k=1 d k g k (r k (t -τ (t))) + n k=1 b k t -∞ k k (t -σ)g k (r k (σ))dσ + u (t) (8)
that is

   ṙ(t) = h(t) ḣ(t) = -Λh(t) -Ar(t) + CF (r(t)) + DG(r(t -τ (t))) +B t -∞ K(t -σ)G(r(σ))dσ + u(t) (9)
where

r = (r 1 , • • • , r n ), h = (h 1 , • • • , h n ), A = diag(a 1 , • • • , a n ), Λ = diag(λ 1 , • • • , λ n ), C = (c k ) n×n , B = (b k ) n×n , D = (d k ) n×n , K(.) = diag(k 1 , • • • , k n ), F (.) = (f 1 (.), • • • , f n (.)) T , G(.) = (g 1 (.), • • • , g n (.)) T and u(.) = (u 1 (.), • • • , u n (.)) T .
So we can write system (6) as a multidimensional second-order system.

Definition 4 [START_REF] Liu | Fixed-time stabilization of second-order systems with unknown nonlinear inherent dynamics[END_REF] System ( 9) is globally finite-time stabilizable if there exists a feedback control u(r, h) such that the closed-loop system is finite-time stable, i.e. there exists a functional T such that for all initial condition r 0 , the solution r(t, r 0 ) of the closed-loop system (9) with controller u(r, h) is well defined and the settling-time T(r 0 ) satisfy lim t→ T(r0) r(t, r 0 ) = 0.

Moreover, system ( 9) is globally fixed-time stabilizable if:

it is globally finite-time stabilizable; the settling-time functional T(r 0 ) is upper bounded by a constant Tmax ≥ 0.

Now, let us introduce some assumptions useful in the following.

(A 1 ) All continuous functions

f k with k = 1, • • • , n satisfy the Lipschitz condi- tion f k (ν) -f k (κ) ≤ M k ν -κ where ν, κ ∈ R n and M k is a positive constant. We denote M = diag(M 1 , • • • , M n ). (A 2 ) The delay Kernel k k (.) with k = 1, • • • , n are positive bump functions such that +∞ 0 k k (σ)dσ = k k . (A 3 ) For all k = 1, • • • , n, there exists constants N k such that |g k (κ)| ≤ N k , κ ∈ R. We denote N = diag(N 1 , • • • , N n ).

Fixed time stabilization of time-delayed INNs

In this section, the FTSt of INNs with time-varying delay is investigated. To perceive the FTSt of system (9), we start with the following sliding variable:

s(r(t), h(t)) = h(t) + α2 r(t) r(t) p2-1 + β2 r(t) r(t) q2-1 (10)
where α2 , β2 > 0, p 2 > 1 and 1 2 < q 2 < 1 and the sliding surface

S = (r, h) ∈ R 2n : s(r, h) = 0 .
Moreover, we propose the following feedback control design:

u(r(t), h(t)) = -α 2 I n + (p 2 -1) r(t)r T (t) r(t) r(t) p2-1 h(t) -β2 I n + (q 2 -1) r(t)r T (t) r(t) r(t) q2-1 h(t) -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ r(t) s(t) s(t) -Ψ h(t) s(t) s(t) -Q s(t) s(t) , (11)
where α1 , β1 > 0, p 1 > 1 and

1 2 < q 1 < 1, Φ = (φ k ) n×n with Φ ≥ A + M |C|, Ψ = diag(ψ 1 , • • • , ψ n ) with ψ ≥ Λ and Q ≥ N (|B|K + |D|). Let us remark that r(t)r T (t) belongs to R n×n .
Theorem 1 Under assumptions (A 1 ), (A 2 ) and (A 3 ), system ( 9) is global fixed-time stabilizable under the proposed controller [START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF] and the settling-time satisfies

T(r 0 ) ≤ 2 =1 1 α (p -1) + 1 β (1 -q ) . ( 12
)
Proof By computing the derivative of the sliding variable s(t) along the trajectory of system (9), we obtain

ṡ(t) = -Λh(t) -Ar(t) + CF (r(t)) + DG(r(t -τ (t))) +B t -∞ K(t -σ)G(r(σ))dσ + u(t)
+α 2 I n + (p 2 -1) r(t)r T (t) r(t) r(t) p2-1 h(t) + β2 I n + (q 2 -1) r(t)r T (t) r(t) r(t) q2-1 h(t). (13)
Applying the controller (11) into (13) leads to

ṡ(t) = -Λh(t) -Ar(t) + CF (r(t)) + DG(r(t -τ (t))) + B t -∞ K(t -σ)G(r(σ))dσ -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ r(t) s(t) s(t) -Ψ h(t) s(t) s(t) -Q s(t) s(t) . ( 14
)
Now consider the Lyapunov function V (s(t)) = 1 2 s(t) T s(t). We obtain

V (s(t)) = s(t) T -Λh(t) -Ar(t) + CF (r(t)) + DG(r(t -τ (t))) +B t -∞ K(t -σ)G(r(σ))dσ -α1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ r(t) s(t) s(t) -Ψ h(t) s(t) s(t) -Q s(t) s(t) So, it leads to V (s(t)) ≤ Λ s(t) h(t) + A s(t) r(t) + M |C| s(t) r(t) + N |D| s(t) +N |B|K s(t) -α1 s(t) p1+1 -β1 s(t) q1+1 -Φ r(t) s(t) -Ψ h(t) s(t) -Q s(t) , that is V (s(t)) ≤ (A + M |C| -Φ) s(t) r(t) + (Λ -Ψ ) s(t) h(t) +(N (|D| + |B|K) -Q) s(t) -α1 s(t) p1+1 -β1 s(t) q1+1 ≤ -α 1 s(t) p1+1 -β1 s(t) q1+1 ≤ -α 1 (2V (s(t))) p 1 +1 2 -β1 (2V (s(t))) q 1 +1 2 . ( 15
)
Letting

y(t) = 2V (t) then ẏ(t) = V (t) √ 2V (t)
and we get from ( 15)

ẏ(t) ≤ -α 1 y p1 (t) -β1 y q1 (t). (16)
Then, from Lemma 1 we conclude that s(t) converges to zero in fixed time and the settling-time is bounded by

T1 := 1 α1 (p 2 -1) + 1 β1 (1 -q 1 ) . ( 17
)
It implies that s(t) = 0 for all t ≥ T1 . From [START_REF] Aouiti | A New LMI Approach to Finite and Fixed Time Stabilization of High-Order Class of BAM Neural Networks with Time-Varying Delays[END_REF], we obtain

ṙ(t) = -α 2 r(t) r(t) p2-1 -β2 r r(t) q2-1 . ( 18
)
From Lemma 2, we conclude that system (18) achieved the fixed-time stability and the settling-time is upper bounded by

T2 := 1 α2 (p 2 -1) + 1 β2 (1 -q 2 ) . ( 19
)
As a conclusion, r(t) = 0 for all t ≥ T1 + T2 , which it implies that h(t) = ṙ(t) = 0 for all t ≥ T1 + T2 .

The algorithm of the proposed SMC is presented as follow: Algorithm 1:

Step 1: Choose α2 , β2 , p 2 , and q 2 . Then, we construct the sliding variable.

Step 2: Since f k (.), k k (.), and g k (.

) (k = 1, • • • , n) are known, we calculate M k , k k , and N k .
Then, we choose ψ , φ k , and

q k ( = 1, • • • , n).
Step 3: Select the design parameters α1 , β1 , p 1 , and q 1 . Subsequently, we design the sliding mode controller.

It is clear that controller [START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF] is undefined in the set {(r(t), h(t)) : r(t) = 0} due to the fact that 1 2 < q 2 < 1. To get rid of the singularity of controller ( 11), we use the saturation function given by

sat (z(t)) = sign(z(t)) min{|z(t)|, } (20)
with > 0 a threshold parameter and r ∈ R. It results a new controller written as follows

u(t) = -α 2 I n + (p 2 -1) r(t)r T (t) r(t) r(t) p2-1 h(t) -β2 I n + (q 2 -1) r(t)r T (t) r(t) r(t) q2 sat h(t) r(t) -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ r(t) s(t) s(t) -Ψ h(t) s(t) s(t) -Q s(t) s(t) . ( 21
)
Let us now state a new theorem with the non singular controller (21) whose proof is inspired by [START_REF] Zuo | Fixed-time stabilization of second-order uncertain multivariable nonlinear systems[END_REF].

Theorem 2 Under assumptions (A 1 ), (A 2 ) and (A 3 ), system (9) is globally almost fixed-time stable under the non singular controller [START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF] and the settlingtime verifies

T(r 0 ) ≤ 2 k=1 1 αk (p k -1) + 1 βk (1 -q k ) + T ( 22
)
where T denote the small time-margin accounting for the saturation function.

Proof Consider the sliding variable [START_REF] Aouiti | A New LMI Approach to Finite and Fixed Time Stabilization of High-Order Class of BAM Neural Networks with Time-Varying Delays[END_REF] and substituting the feedback control [START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF] into the time derivative of the sliding variable given by ( 13) leads to

ṡ(r(t), h(t)) = -Λh(t) -Ar(t) + CF (r(t)) + DG(r(t -τ (t))) +B t -∞ K(t -σ)G(r(σ))dσ -α1 s(t) p1-1 s(t) + β2 I n + (q 2 -1) r(t)r T (t) r(t) r(t) q2 h(t) r(t) -sat h(t) r(t) -β1 s(t) q1-1 s(t) -Φ r(t) s(t) s(t) -Ψ h(t) s(t) s(t) -Q s(t) s(t) .( 23
)
To show that the sliding surface is globally attractive we define the following subsets:

S k = (r, h) ∈ R 2n : |h k | r ≥ ; k = 1, • • • , n (24)
and

Y = S 1 ∪ S 2 ∪ • • • ∪ S n , X = R 2n \ Y.
Case 1: the state of system ( 9) is in X. Then we have sat

h(t) r(t) = h(t) r(t)
and ( 23) rewrites as [START_REF] Aquino | Novel nonlinear hypothesis for the delta parallel robot modeling[END_REF]. Using the same technique as in Theorem 1, we obtain that s(t) = 0 for all t > T1 where T1 is given by ( 17), or the system trajectory enters the set Y in finite time bounded by T1 .

Case 2: the state of system ( 9) is in Y. Then there exists

k ∈ {1, • • • , n} such that sat h k (t) r(t)
= which implies that due to the small value of r(.), there exists at least one element of the control input u which is saturated. From (9), the solution of r k can be calculated as follows

r k (t) = r k (0) + t 0 h k (σ)dσ.
As on each set S k we have h k > 0 or h k < 0, it implies that r k will be monotonically decreasing or increasing until (r(t), h(t)) leaves the set S k . Consequently, for all > 0 there exists a finite-time T k such that the state system (r(t), h(t))

leaves S k and |h k (t)| r(t) < so that sat h k (t) r(t) = h k (t) r(t)
. The same holds for all

sat h (t) r(t) = h (t) r(t) with = 1, • • • , n.
Thus, for a given > 0, there exists a finite time T ≤ n k=1 T k such that the state of system (9) reaches X. Case 3: the state of system (9) satisfies s(r(t), h(t)) = 0. So similar to the proof of Theorem 1, system (18) achieves the fixed-time stability and the settlingtime is upper bounded by [START_REF] Cheng | Asynchronous fault detection filtering for piecewise homogenous markov jump linear systems via a dual hidden markov model[END_REF]. As a conclusion, system (9) is globally fixed-time stable under the feedback control [START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF] and the settling-time is bounded by T1 + T2 + T .

Remark 3 If a sufficiently large threshold parameter > 0 is chosen, then r is allowed to be very small and thus the domain in which the control input is saturated is very limited. This implies that the time margin T can be made very small. Remark 4 Similar to Remark 2, if p k = 1 + 1

µ k and q k = 1 + 1 µ k with µ k > 1 for k = 1, 2, then less conservative estimates for the upper bound of the settlingtime in Theorem 2 can be calculated as

T(r 0 ) ≤ 2 =1 µ π 2 α β + T . ( 25
)
5 Fixed-time synchronization of time-delayed INNs

In this section, the synchronization of time-delayed INNs between master and slave systems is investigated. Let us take the following systems as the master system:

d 2 x (t) dt 2 = -λ dx (t) dt -a x (t) + n k=1 c k f k (x k (t)) + n k=1 d k g k (x k (t -τ (t))) + n k=1 b k t -∞ k k (t -σ)g k (x k (σ))dσ + I (26)
and as the slave system:

d 2 y (t) dt 2 = -λ dy (t) dt -a y (t) + n k=1 c k f k (y k (t)) + n k=1 d k g k (y k (t -τ (t))) + n k=1 b k t -∞ k k (t -σ)g k (y k (σ))dσ + I + ū (t) (27)
where ū (.) denotes the controller. Now, define the error between the master system and the salve system as e (t) = y (t) -x (t) for = 1, • • • , n. Then the error system can be described as follow

d 2 e (t) dt 2 = -λ dy (t) dt - dx (t) dt -a (y (t) -x (t)) + n k=1 c k (f k (y k (t)) -f k (x k (t))) + n k=1 d k (g k (y k (t -τ (t))) -g k (x k (t -τ (t)))) + n k=1 b k t -∞ k k (t -σ)(g k (y k (σ)) -g k (x k (σ)))dσ + ū (t). ( 28
)
That is

d 2 e (t) dt 2 = -λ de (t) dt -a e (t) + n k=1 c k f k + n k=1 d k g k + n k=1 b k t -∞ k k (t -σ) g k dσ + ū (t). ( 29
)
where

f k = f k (y k (t)) -f k (x k (t)), g k = g k (y k (t -τ (t))) -g k (x k (t -τ (t))), g k = g k (y k (s) -g k (x k (s).
Letting z (t) = de (t) dt , then system (29) is written as:

     de (t) dt = z (t) dz (t) dt = -λ z (t) -a e (t) + n k=1 c k f k + n k=1 d k g k + n k=1 b k t -∞ k k (t -σ) g k dσ + ū (t) (30) that is de(t) dt = z(t) dz(t) dt = -Λz(t) -Ae(t) + C F + D G + B t -∞ K(t -σ) Gdσ + ū(t) (31)
where e = (e

1 , • • • , e n ) T , z = (z 1 , • • • , z n ) T , ū = (ū 1 , • • • , ūn ), F = f 1 , • • • , f n T , G = g 1 , • • • , g n T , G = ( g 1 , • • • , g 1 ) T .
Definition 5 The master system (26) and the salve system ( 27) are said to be fixed-time synchronized if there exists a functional T such that for all initial conditions e 0 of system [START_REF] Huang | Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders[END_REF], the solution e(t, e 0 ) is well defined and the settling-time T(e 0 ) satisfies lim t→ T(e0) e(t, e 0 ) = 0.

For the FTSy of the master system (26) and the salve system (27), we define the following sliding variable s(t) = z(t) + α2 e(t) e(t) p2-1 + β2 e(t) e(t) q2-1 [START_REF] Huang | Neural networks in finance and economics forecasting[END_REF] where α2 , β2 > 0, p 2 > 1 and 1 2 < q 2 < 1. Then, the feedback control design is given by

ū(t) = -α 2 I n + (p 2 -1)
e(t)e T (t) e(t) e(t) p2-1 z(t)

-β2 I n + (q 2 -1) e(t)e T (t) e(t) e(t) q2-1 z(t) -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ e(t) s(t) s(t) -Ψ z(t) s(t) s(t) - Q s(t) s(t) , (33)
where α1 , β1 > 0,

p 1 > 1, 1 2 < q 1 < 1, Φ = (φ k ) n×n with Φ ≥ A + M |C|, Ψ = diag(ψ 1 , • • • , ψ n ) with ψ ≥ Λ and Q ≥ 2N (|B|K + |D|).
Theorem 3 Under assumptions (A 1 ), (A 2 ) and (A 3 ), the master system ( 26) is fixed-time synchronized with the salve system [START_REF] Gan | Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control[END_REF] under the proposed controller [START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF] and the upper bound of the settling-time is given by [START_REF] Aouiti | Impulsive generalised high-order recurrent neural networks with mixed delays: stability and periodicity[END_REF].

Proof By computing the derivative of the sliding variable s along the trajectory of system [START_REF] Huang | Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders[END_REF], we obtain

ṡ(t) = -Λz(t) -Ae(t) + C F + D G + B t -∞ K(t -σ) Gdσ +ū + α2 I n + (p 2 -1)
e(t)e T (t) e(t) e(t) p2-1 z(t)

+ β2 I n + (q 2 -1) e(t)e T (t) e(t) e(t) q2-1 z(t). (34)
Applying the controller ( 33) into (34) leads to

ṡ(t) = -Λz(t) -Ae(t) + C F + D G + B t -∞ K(t -σ) Gdσ -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ e(t) s(t) s(t) -Ψ z(t) s(t) s(t) -Q s(t) s(t) . ( 35
)
Now, considered the Lyapunov function

V (s(t)) = 1 2 s(t) T s(t) we obtain V (s(t)) ≤ s(t) T -Λz(t) -Ae(t) + C F + D G + B t -∞ K(t -σ) Gdσ -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ e(t) s(t) s(t) -Ψ z(t) s(t) s(t) - Q s(t) s(t) that is V (s(t)) ≤ Λ s(t) z(t) + A s(t) e(t) + M |C| s(t) e(t) + 2N |D| s(t) +2N |B|K s(t) -α1 s(t) p1+1 -β1 s(t) q1+1 -Φ e(t) s(t) -Ψ z(t) s(t) -Q s(t) . It yields V (s(t)) ≤ (A + M |C| -Φ) s(t) e(t) + (Λ -Ψ ) s(t) z(t) +(2N (|D| + |B|K) -Q) s(t) -α1 s(t) p1+1 -β1 s(t) q1+1 ≤ -α 1 s(t) p1+1 -β1 s(t) q1+1 ≤ -α 1 (2V ) p 1 +1 2 -β1 (2V ) q 1 +1 2 Letting x(t) = 2V (t) and ẋ(t) = V (t) √ 2V (t)
, then we get from ( 36)

ẋ(t) ≤ -α 1 x p1 (t) -β1 x q1 (t). ( 37
)
From Lemma 1 we conclude that s(t) converges to zero and the settling-time is bounded by [START_REF] Cao | Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays[END_REF]. It implies that s(t) = 0 for all t ≥ T 1 . From [START_REF] Huang | Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders[END_REF], we obtain ė(t) = -α 2 e(t) e(t) p2-1 -β2 e e(t) q2-1 .

From Lemma 2, we conclude that system [START_REF] Li | Finite-time and fixed-time stabilization control of delayed memristive neural networks: Robust analysis technique[END_REF] converges to zero and the settling-time is upper bounded by [START_REF] Cheng | Asynchronous fault detection filtering for piecewise homogenous markov jump linear systems via a dual hidden markov model[END_REF]. As a conclusion, e(t) = 0 for all t ≥ T1 + T2 which implies that z(t) = ė(t) = 0 for all t ≥ T1 + T2 .

To get rid of the singularity of the controller [START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF], we use the saturation function and the new controller is expressed as follows:

ū(t) = -α 2 I n + (p 2 -1)
e(t)e T (t) e(t) e(t) p2-1 z(t)

-β2 I n + (q 2 -1) e(t)
-α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ e(t) s(t) s(t) -Ψ z(t) s(t) s(t) - Q s(t) s(t) . ( 39
)
Theorem 4 Under assumptions (A 1 ), (A 2 ) and (A 3 ), the master system ( 26) is globally almost fixed-time synchronized with the salve system ( 27) under the non-singular controller [START_REF] Li | Nonlinear measure approach for the robust exponential stability analysis of interval inertial cohen-grossberg neural networks[END_REF], and the settling-time verifies

T(e 0 ) ≤ 2 k=1 1 αk (p k -1) + 1 βk (1 -q k ) + T ( 40
)
where T denotes the small time margin accounting for the saturation function.

Proof Consider the sliding variable [START_REF] Huang | Neural networks in finance and economics forecasting[END_REF] and substituting the controller ( 39) into (34) leads to

ṡ(t) = -Λz(t) -Ae(t) + C F + D G + B t -∞ K(t -σ) Gdσ + β2 [I n + (q 2 -1) e(t)e T (t) e(t) ] e(t) q2 z(t) e(t) -sat z(t) e(t) -α 1 s(t) p1-1 s(t) -β1 s(t) q1-1 s(t) -Φ e(t) s(t) s(t) -Ψ z(t) s(t) s(t) -Q s(t) s(t) . ( 41
)
We define the following subsets:

S k = (e, z) ∈ R 2n : |z k (t)| e(t) ≥ ; k = 1, • • • , n. ( 42
) Let Y = S 1 ∪ S 2 ∪ • • • ∪ S n , X = R 2n \ Y.
Case 1: the state of system ( 31) is in X. Then we have sat

z(t) e(t) = z(t) e(t)
and ( 41) rewrites as [START_REF] Kosmatopoulos | High-order neural network structures for identification of dynamical systems[END_REF]. Using the same technique as in Theorem 3, we obtain that s(t) = 0 for all t > T1 where T1 is given by ( 17), or the system trajectory enters the set Y in finite time bounded by T1 .

Case 2: the state of system ( 31) is in

Y. So there exists k ∈ {1, • • • , n} such that sat z k (t) e(t)
= which implies that due to the small value of e(.), there exists at least one element of the vector input control u which is saturated. From ( 31), the solution of e k can be calculated as follows

e k (t) = e k (0) + t 0 z k (s)ds.
As on each set S k , we have z k > 0 or z k < 0, it implies that e k is monotonically decreasing or increasing until (e(t), z(t)) leaves the set S k . Consequently, for all > 0, there exists a finite time T k such that the state system (e(t), z(t))

leave s k and |z k (t)| e(t)
< so sat z k (t)

e(t)
= z k (t) e(t) . The same holds for all

sat z (t) e(t)
= z (t) e(t) with = 1, • • • , n. Thus, for a given > 0, there exists a finite time T ≤ n j=1 T j such that the state system reaches X. Case 3: the state of system (31) satisfies s(e(t), z(t)) = 0. So, similar to the proof of Theorem 3, system (31) converges to zero and the settling-time is upper bounded by [START_REF] Cheng | Asynchronous fault detection filtering for piecewise homogenous markov jump linear systems via a dual hidden markov model[END_REF]. As a conclusion, the master system ( 26) is globally almost fixed-time synchronized with the salve system [START_REF] Gan | Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control[END_REF] under the feedback control [START_REF] Li | Nonlinear measure approach for the robust exponential stability analysis of interval inertial cohen-grossberg neural networks[END_REF] and the settling-time is bounded by T1 + T1 + T .

Remark 5

The controller [START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF] and [START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF] are discontinuous ones because they use the sign function which could produce the undesired chattering, while in some cases continuity is necessary. Under these circumstances, we can use the sat function in the controllers [START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF] and [START_REF] Li | Nonlinear measure approach for the robust exponential stability analysis of interval inertial cohen-grossberg neural networks[END_REF] to decrease this effect.

Remark 6 It should be noted that the synchronization is the stabilization of the error between the transmission and the receiver systems. The synchronization phenomenon is often encountered in nature, which is considered as an adjustment of rhythms of oscillating objects due to their internal weak couplings [START_REF] Blekhman | Self-synchronization and controlled synchronization: general definition and example design[END_REF]. It has a great importance in real application. For example, fireflies in flocks give off flashes of light with the same frequency on a summer night or the synchronous motion of space is established between the moon's orbital motion and its rotational motion. In order to achieve self-synchronization in the asynchronous communication techniques between the transmitter and the receiver, the two sides must simultaneously send and receive information over the same frequency.

Remark 7

In this work, the FTSt and the FTSy are investigated for a class of INNs with time-varying and distributed delays. For the first time, new SMC schemes are proposed in our paper for the FTSt and the FTSy of master-slave INNs with delays where the settling-time is bounded independently from the initial conditions by constructing Lyapunov functionals and by using analytical techniques. The main limitation of SMC in practical applications is the chattering problem. The chattering is the natural price that the SMC pays in order to eliminate completely unknown matched external disturbances.

Remark 8 Compared with the results of [START_REF] Gan | Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control[END_REF][START_REF] Shi | Projective synchronization of different chaotic neural networks with mixed time delays based on an integral sliding mode controller[END_REF][START_REF] Zhang | Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller[END_REF], the improved SMC approach of this brief provide better results for the following reasons. Firstly, the integral sliding mode surface is novel and needs less information, which has the performance that once the sliding mode variable satisfies s(t) → 0, the FTSy will be guaranteed independent of the initial conditions. Secondly, the presented control approach can be conveniently verified without solving any LMIs. Finally, the FTSy between drive-response systems ( 26) and ( 27) can be obtained via the presented SMC approach, however, it cannot be guaranteed in the articles cited above.

Numerical Examples

In this section, we present four examples to illustrate the effectiveness of the results presented in this article.

Example 1 Consider the following INN with time-varying delay

d 2 r (t) dt 2 = -λ dr (t) dt -a r (t)+ n k=1 c k f k (r k (t))+ n k=1 d k g k (r k (t-τ (t)))+u (t) (43)
and the transformation dr (t) dt = h (t) leading to the system: Choosing α = β = 4. For µ 1 = µ 2 = 4, we have p 1 = p 2 = 1 + 1 µ = 1.25 and q 1 = q 2 = 1 -1 µ1 = 0.75. The other parameters of the feedback control [START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF] are designed as follows:

   dr (t) dt = h (t) dh (t) dt = -λ h (t) -a r (t) + n k=1 c k f k (r k (t)) + n k=1 d k g k (r k (t -τ (t))) + u (t) (44) with n = 2, = 1, 2, a 1 = a 2 = 3, λ 1 = λ 2 =
ψ 1 = ψ 2 = 6 ≥ a 1 = 3, ψ 2 = 6 ≥ a 2 = 3, φ 11 = 4 ≥ a 1 + M 1 |c 11 | = 3.3, φ 12 = 3.5 ≥ a 1 + M 1 |c 12 | = 3.15, φ 21 = 3.5 ≥ a 2 + M 2 |c 21 | = 3.25, φ 22 = 4 ≥ a 2 + M 2 |c 22 | = 3.4, ψ 1 = 7 ≥ λ 1 , ψ 2 = 7 ≥ λ 2 , q 11 = 1 ≥ N |d 11 | = 0.6, q 12 = 0.6 ≥ N |d 12 | = 0.2, q 21 = 1 ≥ N |d 21 | = 0.3, q 22 = 0.6 ≥ N |d 22 | = 0.
3 and the initial condition are set as r 1 (0) = -0.5, r 2 (0) = 0.5, h 1 (0) = 1, h 2 (0) = -1. From Theorem 1 we obtain that system (44) is globally fixed-time stabilized under controller [START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF]. In addition, the settling-time satisfies the following inequality

T(0) ≤ 2 =1 µ π 2 α β = π 2 + π 2 ≤ 3.1416.
The state trajectories of system [START_REF] Malmgren | Artificial Neural Networks in Medicine and Biology: Proceedings of the ANNIMAB-1 Conference[END_REF] under controller [START_REF] Aouiti | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF] Taking α1 = α2 = β1 = β2 = 1, µ 1 = 1.5, µ 2 = 2.5, = 0.1, p 1 = 1 + 1 1.5 , p 2 = 1 + 1 2.5 , q 1 = 1 -1 1.5 and q 2 = 1 -1 2.5 . The parameters of the feedback control [START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF] are designed as follow:

φ 11 = 3 ≥ a 1 + M 1 |c 11 | = 2.5, φ 12 = 3.5 ≥ a 1 + M 2 |c 12 | = 2.35, φ 21 = 3.5 ≥ a 2 + M 1 |c 21 | = 2.5, φ 22 = 4 ≥ a 2 + M 2 |c 22 | = 2.7, ψ 1 = 7 ≥ λ 1 = 3, ψ 2 = 7 ≥ λ 2 = 3, q 11 = 1 ≥ N 1 |d 11 | = 0.6, q 12 = 0.6 ≥ N 2 |d 12 | = 0.2, q 21 = 1 ≥ N 1 |d 21 | = 0.3, q 22 = 0.6 ≥ N 2 |d 22 | = 0.2.
The initial values are taken as r 1 (0) = 0.5, r 2 (0) = -0.5, h 1 (0) = -0.7, h 2 (0) = 0.6. Theorem 2 guarantees not only that system [START_REF] Mhamdi | Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays[END_REF] is globally fixedtime stabilized under the controller [START_REF] Cui | Global asymptotic and robust stability of inertial neural networks with proportional delays[END_REF] but also that the settling-time satisfies the following inequality

T(0) ≤ 2 =1 µ π 2 α β + T = 1.5π 2 + 2.5π 2 + 0.001 ≤ 6.2842
with T = 0.001. We see on Figure 2 that system (46) is globally almost fixedtime stable.

Example 3 We consider the following INN master system

d 2 x (t) dt 2 = -λ dx (t) dt -a x (t)+ n k=1 c k f k (x k (t))+ n k=1 d k g k (x k (t-τ (t))) (47
d 2 y (t) dt 2 = -λ dy (t) dt -a y (t)+ n k=1 c k f k (y k (t))+ n k=1 d k f k (y k (t-τ (t)))+u (t) (48
) where the parameters of system (48) are same as the system [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF]. as follows:

Choosing α1 = α2 = β1 = β1 = 2, µ 1 = µ 2 = 3, p 1 = 1 + 1 µ1 , p 2 = 1 + 1 µ2 , q 1 = 1 -1 µ1 and q 2 = 1 -1 µ2 .
φ 11 = 13 ≥ a 1 + M 1 |c 11 | = 12.4, φ 12 = 13 ≥ a 1 + M 1 |c 12 | = 12.2, φ 21 = 13 ≥ a 2 + M 2 |c 21 | = 12.5, φ 22 = 13 ≥ a 2 + M 2 |c 22 | = 12.5, ψ 1 = 8 ≥ λ 1 , ψ 2 = 8 ≥ λ 1 , q 11 = 3 ≥ 2N 1 |d 11 | = 1.4, q 12 = 1.5 ≥ 2N 1 |d 12 | = 0.6, q 21 = 2 ≥ 2N 2 |d 21 | = 0.8, q 22 = 3 ≥ 2N 2 |d 22 | = 0.8.
The initial values are taken as x 1 (0) = 1, x 2 (0) = 0.5, y 1 (0) = -0.5, y 2 (0) = 0.5. From Theorem 3, the master system (47) and the slave system (48) are globally fixed-time synchronized under controller [START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF]. In addition, the settling-time satisfy the following inequality

T(0) ≤ 2 =1 µ π 2 α β = 3π 2 ≤ 4.7124.
After a short time, the master system (47) tracks the slave system (48) as illustrated in Figure 3 and Figure 4. The synchronization errors are depicted in Figure 5 and Figure 6.

Example 4 We consider the following INN master system:

d 2 x(t) dt 2 = -Λ dx(t) dt -Ax(t)+CG(x(t))+DG(x(t-τ (t)))+B t -∞ K(t-s)G(x(s))ds (49)
where Fig. 6 Trajectories of e 1 (t) and e 2 (t) tanh(.). The salve system is given as follows:

k 1 (x) = k 2 (x) = e -x , τ = 2, A = 5 0 0 5 , Λ = 6 0 0 6 , C = 0.6 0.2 -0.3 0.3 , B = 0.1 -0.2 -0.
d 2 y (t) dt 2 = -λ dy (t) dt -a y (t) + n k=1 c k f k (y k (t)) + n k=1 d k f k (y k (t -τ (t))) + n k=1 b k t -∞ k k (t -σ)g k (x k (σ))dσ + u (t) (50)
where the parameters of system (50) are same as the system [START_REF] Papik | Application of neural networks in medicine-a review[END_REF].

Choosing µ 1 = 1.5, µ 2 = 2.5, α1 = β1 = α2 = β2 = 1, = 0.1, p 1 = 1+ 1 µ1 = 1.6667, p 2 = 1+ 1 µ2 = 1.4, q 1 = 1-1 µ1 = 0.3333 and q 2 = 1-1 µ2 = 0.6 The parameters of the feedback control (39) are given by: φ The initial values are taken as x 1 (0) = -0.6, x 2 (0) = 0.6, y 1 (0) = 0.8, y 2 (0) = -0.8. From Theorem 4, the master system (49) and the slave system (50) are globally fixed-time synchronized under controller [START_REF] Li | Nonlinear measure approach for the robust exponential stability analysis of interval inertial cohen-grossberg neural networks[END_REF]. After a short time, the master system (49) tracks the slave system (50) as illustrated in Figures 7 and8 with T = 0.001. Remark 9

0
The study of the fixed-time stability of neural networks is of major interest for many applications such as spacecraft attitude dynamics [START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF] and secure communication [START_REF] Yang | A survey of chaotic secure communication systems[END_REF]. Secure communication using synchronization between chaotic systems is a new concept of secure communication. In [START_REF] Alimi | Finite-time and fixed-time synchronization of a class of inertial neural networks with multi-proportional delays and its application to secure communication[END_REF], the authors present the secure communication synchronization problem as real application of inertial neural networks.

Conclusion

In this article, the fixed-time stabilization and fixed-time synchronization of inertial neural networks with time-varying delay are inspected. By using sliding variables, two sliding mode controllers are obtained. Moreover, to get rid of the singularities two more sliding mode controllers are designed leading to almost fixed-time stabilization and almost fixed-time synchronization. Our results are novel and complementary to the existing results about neural networks. There exist several ideas for future research. The first topic is related to the activation function. For processing of complex processes with delays the speed and performance of neural network in such case will depend on quality of activation function chosen. For this reason, a scaled polynomial constant unit activation function SPOCU [START_REF] Kisel'ák | SPOCU": scaled polynomial constant unit activation function[END_REF] is going to be tested with respect to its performance. The second one focuses on the fixed-time stabilization and fixed-time synchronization under sliding mode control of:

1. a more general model of inertial neural networks such as inertial BAM neural networks, high-order inertial neural networks, quaternion-valued inertial Cohen-Grossberg neural networks, Clifford-valued inertial neural networks; 2. a more general model of fractional-order time-delayed inertial neural network.

  4, c 11 = 0.3, c 12 = 0.15, c 21 = 0.25, c 22 = -0.4, d 11 = 0.6, d 12 = 0.2, d 21 = -0.3, d 22 = 0.3, τ = 1, the activation functions f (s) = 0.5[|s + 1| -|s -1|] and g (.) = tanh(.). We have |f k (ν) -f k (κ)| ≤ |ν -κ| and |g k (κ)| ≤ 1, then Assumption (A 1 ) and (A 2 ) hold with M k = 1 and N k = 1.

t-∞

  K(t -s)G(r(s))ds + u[START_REF] Mhamdi | Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays[END_REF] wherek 1 (x) = k 2 (x) = e -x , τ = 2, functions F (.) = G(.) = tanh(.). We have |f k (ν) -f k (κ)| ≤ |ν -κ| and |g k (κ)| ≤ 1, then Assumption (A 1 ) and (A 2 ) hold with M k = N k = 1.We have ∞ 0 e -s ds = 1, then assumption (A 3 ) holds.

  ) where n = 2, τ = 1, a 1 = a 2 = 12, λ 1 = λ 2 = 7, c 11 = 0.4, c 12 = 0.2, c 21 = 0.5, c 22 = -0.5, d 11 = 0.7, d 12 = 0.3, d 21 = -0.4, d 22 = 0.4, the activation functions f (s) = 0.5[|s + 1| -|s -1|]. It is clear that Assumption (A 1 ) and (A 2 ) hold and M k = N k = 1. The salve system is given by

  11 = 9 ≥ a 1 + M 1 |c 11 | = 5.6, φ 12 = 9 ≥ a 1 + M 1 |c 12 | = 5.2, φ 21 = 9 ≥ a 2 + M 2 |c 21 | = 5.3, φ 22 = 9 ≥ a 2 + M 2 |c 22 | = 5.3, ψ 1 = 7 ≥ λ 1 , ψ 2 = 7 ≥ λ 1 , q 11 = 1.9 ≥ 2N 1 (|b 11 | + |d 11 |) = 1.2, q 12 = 1.2 ≥ 2N 1 (|b 12 | + |d 12 |) = 1.1, q 21 = 2 ≥ 2N 2 (|b 21 | + |d 21 |) = 2, q 22 = 3 ≥ 2N 2 (|b 22 | + |d 22 |) = 2.2.

  . The synchronization errors are depicted in Figures9 and 10

Fig. 9 Fig. 10

 910 Fig.9Trajectories of z 1 (t) and z 2 (t)

  The parameters of the feedback control[START_REF] Jiang | Fixed-time attitude control for rigid spacecraft with actuator saturation and faults[END_REF] are

		0.6 0.8					h 1 (t)		h 2 (t)	r 1 (t)	r 2 (t)
		0.4
	h i (t) and r i (t), i=1,2	-0.4 -0.2 0 0.2
		-0.6
		-0.8
		-1
		0	1	2	3	4	5	6	7	8	9	10
							Time (t)
	Fig. 2 Trajectories of system (46) with controller (21)

  Trajectories of z 1 (t) and z 2 (t)

		1.2										x 1 (t)		0.6								x 2 (t)
			1										y 1 (t)		0.4								y 2 (t)
		0.8
															0.2
	x 1 (t) and y 1 (t)	0 0.2 0.4 0.6											x 2 (t) and y 2 (t)	-0.2 0
															-0.4
		-0.2
		-0.4												-0.6
		-0.6												-0.8
			0	1	2	3	4	5	6	7	8	9	10		0	1	2	3	4	5	6	7	8	9	10
							Time (t)												Time (t)
	Fig. 3 Trajectories of x 1 (t) and y 1 (t)		Fig. 4 Trajectories of x 2 (t) and y 2 (t)
			3										z 1 (t)		1.5								e 1 (t)
			2										z 2 (t)		1								e 2 (t)
	z 1 (t) and z 2 (t)	-1 0 1											e 1 (t) and e 2 (t)	-0.5 0 0.5
			-2
			-3												-1
			-4												-1.5
	0 Fig. 5 0 1 2 3 4 5 6 7 8 9 10 Time (t)	1	2	3	4	5 Time (t)	6	7	8	9	10
						5 -0.4	, D =		0.5 0.35 -0.5 -0.7	, the activation function G(.) =

  Trajectories of x 1 (t) and y 1 (t) Trajectories of x 2 (t) and y 2 (t)

		1	x 1 (t)	0.6								x 2 (t)
		0.8	y 1 (t)	0.4								y 2 (t)
		0.6		0.2
	x 1 (t) and y 1 (t)	0 0.2 0.4	x 2 (t) and y 2 (t)	-0.4 -0.2 0
		-0.2		-0.6
		-0.4		-0.8
		-0.6		-1
	1 Fig. 7 0 2 3 4 5 6 7 8 9 10 Time (t) 1 2 3 4 5 6 7 8 9 10 -3 -2 -1 0 1 3 z 1 (t) and z 2 (t) z 1 (t) Fig. 8 0 2 z 2 (t)	1	2	3	4	5 Time (t)	6	7	8	9	10
		Time (t)

Conflict of interest:

The authors declare that they have no conflict of interest.


Pr Eric

Pr Danièle

Human and animal cells as factories for therapeutic molecules production: applications and culture processes

Introduction

Animal and human cells have progressively become a versatile platform to produce various therapeutic biomolecules. This category of expression systems mainly includes mammalian (essentially hamster, mouse and human cells) and insect cells. From a general point of view, nonhuman mammalian cells are still the most used at the industrial scale (production of glycosylated recombinant proteins and of adenoviral vectors among others) but some continuous human cell lines can be used in vitro in a bioreactor to obtain a supernatant enriched in a specific protein or in extracellular vesicles after selective isolation. Moreover, human stem cells can be used in vitro as a source of differentiated cells, as with induced pluripotent stem cells (iPS). Considering that stem cells can be the source of more differentiated cells, their use in vivo make them cell factories. The use of human cells as an in vivo factory has been performed for decades with hematopoietic stem cells (HSC) for the reconstitution of hematopoietic blood cells and immune cells. On the same model, specific T cells like Viral specific T cells (VST) can proliferate in vivo and differentiate into cytotoxic anti-viral T cells. Mesenchymal stromal cells (MSC) when used in regenerative medicine, can also be considered as cell factory able to produce differentiated cells in vivo. From a process engineering point of view, the success of the development and strengthening of biotherapies using animal and human cells should be based on an advanced knowledge of the cell behavior in their culture system, namely the deciphering of the couplings between cells and their culture environment. In view of the above, we will first present hereafter the operative and expected applications of animal and human cells as in vitro factories but also as in vivo factories, focusing on mesenchymal stromal cells. Then, the current and future strategies of optimization of the interactions between the cells and their biochemical and hydromechanical environments will be discussed.

2. Human and animal cell culture for the production of therapeutic biomolecules: main applications

Industrial applications of continuous cell lines

Compared to primary cells, which are derived from extraction and/or digestion processes of biological tissues, continuous cell lines theoretically have an unlimited capacity to divide, thus allowing for a major expansion of the use of animal cells in industrial biotechnology processes. These continuous lines are either cells that have undergone a mutation of genes involved in the cell cycle, or cells transformed by an oncogene, or cancer cells taken from an organism. Thus, as early as 1975, the fusion of rodent lymphocytes and myeloma cells allowed the creation of hybridomas and the production of monoclonal antibodies, which are mainly used today in diagnosis.

Gradually, in the 1980s, the use of these cells to produce therapeutic biomolecules was accepted by the regulatory authorities; the scaling up of culture processes to ensure a significant reduction in production costs was facilitated by obtaining (i) cell lines that were robust to agitated conditions and (ii) optimized culture media. The technology of cloning a gene coding for a protein of interest into an expression vector introduced into the cell has opened broad horizons for the production of customized recombinant proteins, making mammalian cells sophisticated platforms for the industrial production of these biomolecules. For example, in 1987, tissue plasminogen activator (tPA) was the first therapeutic molecule produced by Chinese Hamster Ovary (CHO) cells to be approved for marketing. Since these breakthrough innovations, the development and marketing authorizations of molecules produced by animal cells and in particular by mammalian cells have continued to rapidly increase (Figure 1). The market for recombinant proteins dedicated to therapeutic use, and especially that of monoclonal antibodies, is the major application sector driving the industrial use of animal cells. Monoclonal antibodies are used in the treatment of cancer, cardiovascular diseases, respiratory diseases, infections, ophthalmic or inflammatory diseases. They are currently finding new uses in the treatment of severe infections such as SARS-CoV2 (antibodies bamlanivimab and etesevimab).

The most common cell line is the CHO cell, which alone seems to account for approximately 75 % of recombinant protein production. In addition to mammalian cells, insect cells, infected with a baculovirus carrying the gene coding for the protein of interest, are also attractive tools for the transient production of proteins on a smaller scale. A list of the main mammalian cell lines and their recognized applications is given in Table 1.

More recently, new lines of human origin are emerging for the production of adenoviral vectors or recombinant proteins (HEK293, PER.C6)1 . They present a higher advantage as they allow the production of a non-immunogenic glycan. The main drawback is that they are susceptible to human virus contaminations requiring viral inactivations. HEK293 (Human Embryo Kidney 293) and its different variants, transfected with viral DNA, is the most prominent human cell line used for protein expression. Other cell-lines, including HT-1080 (human fribrosarcoma), CAP (from human amniocytes), PER.C6 (from human retinoblast), and HuH-7 (from human hepatoma) are used for the generation of factor VII, adenoviral vectors, immunoglobulins, or factor IX, respectively2 .

Human cells for the production of extracellular vesicles

Extracellular vesicles (EV), including exosomes, micro-vesicles, and apoptotic bodies display different sizes (from 30 nm for exosomes to 5 µm for apoptotic bodies), derived from most of cell types, and implicated in intercellular communication participating in physiological but also pathological processes like initiation and progression of tumor3 . Among EV, exosomes are generated secondary to the invagination of the endosomal membrane, harboring a unique composition, and are further released from the cell by membrane fusion4 . Exosomes have been described as micro-carriers releasing proteins and lipids, non-coding and coding RNAs, micro-RNAs (miRNA) for the regulation of biological processes, genomic DNA and transference of antigens or immunogenic molecules for the activation of T-cells during immune responses5 . They exhibit many characteristics from the originated cells with which they share properties, making them relevant (i) as biomarkers for the diagnosis of a wide variety of health disorders and (ii) for cell-free therapeutics as they could present a better safety profile than the original cell therapy6 . For example, they can be generated from cells interfering with immune response like dendritic cells, macrophages and MSC inducing either immunostimulating or immunomodulatory effects. Monocyte-derived-exosomes appear to be able to escape immune phagocytosis, allowing their increased circulation and efficacy compared to other exosome types7 . Interestingly, they can also be derived from tumor cells, and they can elicit an antitumor cellular immune response and easily target the tumor due to their membrane homology with tumor cells. However, there are controversial reports showing that tumor cell-EV promote cancer progression through different mechanisms like induction of angiogenesis, escape to apoptosis, and intensification of tumor cell proliferation 3 ….

Thereby, due to their high biocompatibility and ability to penetrate through biological barriers, their enhanced stability and limited immunogenicity, exosomes are very promising tools for targeted drugdelivery going beyond the limits of liposomes 3 . Many kinds of chemotherapeutics drugs have been loaded into exosomes like paclitaxel or doxorubicin, allowing an increased uptake by tumor cells compared to the liposomal form of the drug and even overcoming drug resistance mechanisms. The cargo can also be therapeutic RNA or proteins.

Despite the advantages and promising interests of EV, many challenges remain for bringing exosomes into the clinic8 . First, cell source selection is a critical step depending on the targeted tissue and on the elucidation of potential deleterious effects, as mentioned previously for tumor cell-EV.

Exosome generation from standardized cell lines would be of great interest. The scale-up of the culture systems of cells must take into account serum-free medium, stirred tank bioreactors. Second, the definition of a purification process for achieving high yield and purity is a limiting milestone. Ultracentrifugation is the well-known method for exosome isolation. Improvements are needed to reach a large-scale manufacturing of exosomes and new methods are currently emerging like sizeexclusion chromatography, ciliated micropillars nano-traps, acoustic wave separation and flow fieldflow fraction9 . Third, specific quality controls (QC), including potency assays, have to be sought and developed. New technologies are necessary to allow an on-line fast QC. Finally, while overcoming those production concerns, a key issue will be to determine the optimal dose, the maximum tolerated dose, the minimum efficacy dose and the route of administration.

Different early phase clinical trials have been initiated and, for some of them, completed in different indications like tissue repair, cancer, Graft versus Host Disease (GVHD) or chemotherapy delivery, confirming the great interest for this promising therapy and that there is still a long way to move on and bring them to the bed side 9 .

2.3. Induced Pluripotent stem cells: a cell factory for the generation of more mature derived cells

Induced pluripotent stem cells (iPSCs) were first obtained by Yamanaka and colleagues while reprogramming adult cells, transferring 4 limiting transcription factors (Oct3/4, Sox2, c-Myc, and Klf4) 10 . Compared to embryonic stem cells (ESC), iPSC could present molecular anomalies (epigenic traces of their somatic origin, genomic instability possibly due to the reprogramming process, or alteration of mitochondrial DNA) but also different advantages, mainly the capacity to be generated in an autologous setting avoiding immunological problem. Those pluripotent stem cells can then be differentiated in many types of cells (neurons, MSC, …). However, although the autologous setting is attractive to overcome immune conflicts, a large-scale production of stable and controlled iPSC lines is only possible in an allogeneic setting. Some iPSC banks for research or clinical trials are currently developed in different countries but many challenges remain to meet the medical needs11 : (i) overcoming immune rejection is an issue by developing large HLA typed banks to allow HLA matching with the recipient or by gene editing, (ii), implementing QC to ensure safety and efficiency of cells (especially genomic sequencing). Finally, although iPSCs are considered devoid of ethical problems compared to embryonic stem cells, there are currently raising ethics proceedings in France considering iPSC could be potentially differentiated into gametes (Ethic law in project).

MSC: a cell factory producing soluble factors depending to the inflammatory environment

Discovered in 1970s in bone marrow (BM), Mesenchymal Stromal/Stem Cells (MSC) have since been found in multiple tissues like adipose tissue, dental pulp, menstrual blood or extra-embryonic source such as Wharton's Jelly (WJ) or placenta 12 . In 2006, the International Society for Cellular Therapy (ISCT) defined Mesenchymal stem/stromal cells as (1) adherent to the cell culture plastic;

(2) able to differentiate into osteocytes, chondrocytes, and adipocytes;

(3) with a phenotype that is positive for the CD73 CD90 CD105 mesenchymal markers and negative for the CD34 CD45 HLA-DR hematopoietic markers (MSC can modulate both innate and adaptive immunity by cell contacts, secretion of cytokines, chemokines, growth factors, and release of extracellular vesicles 13 . Briefly, immune properties of MSC rely on two main mechanisms: a direct interaction of MSC with immune cells and a paracrine effect of MSC. The main factors expressed by MSC in response to immune cell interactions are: microRNAs, PD-L1, HLA-G, Prostaglandin E2 (PGE2), cytokines (transforming growth factor-β (TGFβ), Interleukin (IL)6, IL10, HGF, VEGF…), and one enzyme (Indoleamine 2,3-dioxygenase: IDO). Immune properties of adult MSC have been widely reported 14 . Two MSC phenotypesimmunomodulatory or immunostimulatory-depending on the inflammatory context and the stimulation of their TLR were described 15 . In a cell anergy context, they could exhibit a proinflammatory phenotype, MSC1, making it possible to reduce apoptosis and promote T-cell survival.

On the other hand, in case of inflammation, they could adopt an immunosuppressive and antiinflammatory phenotype, called MSC2. This dual phenotype according to environment brings new opportunities for therapeutic uses. However, these immunomodulatory cells are slightly impacted by the host immune system (lack of expression of MHC class II antigens and CD80/CD86 costimulatory molecules, weak expression of MHC class I antigens) allowing there use in an allogeneic setting without considering HLA compatibility. Current data suggest also that MSC exert strong antimicrobial effects both indirectly, across their role in the host immune response against pathogens and directly, by the secretion of antimicrobial peptides and proteins (AMPs), but also by the expression of molecules such as IDO and IL17 16 .

Up to now, more than 800 clinical trials have been performed with MSC from different sources in various indications, highlighting their safety and, in some of high phase clinical trials, their efficacy. Market authorization has been obtained in USA and Europe in GVHD treatment or in Crohn's disease fistulas.

In an allogeneic setting, high numbers of MSC have to be generated. In academic laboratories, MSC production is mainly performed in 2-dimension (2D) culture flasks which is time consuming and yields not compatible with the medical need. Currently, industrialization of 2D models begins with the marketing of bioreactors. Bioreactor increases cell culture surface and limits steric hindrance. A more versatile and scalable technology of culture has also arisen last recent years: the 3-dimensional model, consisting in using microcarriers as adherence support, in stirred tank bioreactors, while reducing the operating cost of the process. Regardless of differences due to MSC source, different challenges remain: (i) the recovery of MSC from microcarriers to achieve a good yield, (ii) the maximum volume of culture while keeping stable MSC characteristics (most of the cultures in stirred bioreactors operate today at a scale of few liters 17,18 (iii) and on-line monitoring of culture parameters.

As previously mentioned, exosomes are generated from MSC as they exhibit anti-inflammatory and immunomodulatory capacities and would allow a cell-free therapy. However, although the production of standardized homogeneous batches would be an improvement compared to whole MSC, the main advantage of the whole cells would be lost. The strength of MSC relies on their main drawback: to be a "living drug". The use of EV instead of cells induces the loss of adaptability capacities (dual phenotype MSC1 and MSC2). In multifactorial immune pathologies with concomitant inflammation and lymphopenia as described in sepsis, could it be possible to do without this main benefit?

Adoptive T cell immunotherapy: T cell factories allowing in vivo cell expansion and differentiation into cytotoxic T cells

The recovery of virus-specific T-cell immunity is crucial for patients after hematopoietic stem cell transplantation (HSCT) to avoid viral infections or reactivations. Based on donor lymphocyte infusions experiments, ex vivo virus-specific T cells (VST) were generated for patients with viral infection, refractory to anti-viral drugs 19 . VST can be expanded ex-vivo but also isolated in vitro from a donor leukapheresis using an immunomagnetic separation system (Miltenyi Biotec). In this last situation, a very small number of isolated specific T cell is obtained and infused into the patient, requiring an in vivo expansion when T cells interact with antigen-presenting-cell presenting viral peptides. As mentioned before, such VST are generated for a dedicated patient. Thus, large scale manufacturing is not an issue. The Prodigy platform, a closed, automated system, compliant with GMP guidelines developed by Miltenyi Biotec, allows bringing this medicine to the bed side. On a clinical point of view, randomized controlled studies are still missing to conclude on efficacy and safety of these allogeneic VST. Such a phase III clinical trial is currently including patients on a European scale (sponsor: Munich university).

Comparison with other expression systems

Today, most of recombinant glycoproteins produced by biotechnology are produced in mammalian cells. Other expression systems can be used, such as bacteria (Escherichia coli), yeast (Pichia pastoris, Saccharomyces cerevisiae), insect cells (Sf9, Sf21, BTI 5B1-4) or plants. These systems have clear advantages as they allow higher product concentrations; their production processes are considered as more easily scalable with reduced production costs (particularly in terms of culture media costs) compared to mammalian cell production processes. However, they lack the ability to implement complete post-translational modifications of the protein, notably glycosylation. Thus, their use is generally restricted to the production of small proteins or to the vaccine industry. Notable examples are the production of human insulin by the yeast Pichia Pastoris, the production of pseudo-viral particles for the design of the vaccine against the Papilloma virus (GARDASIL9) by Saccharomyces cerevisiae or viral vectors by Sf9 insect cells. The main advantages and disadvantages of mammalian and non-mammalian cells as expression systems are reported in Table 2.

Industrial processes at the service of cell factories

A production facility needs to interact with its socio-economic environment, with infrastructures allowing the routing of raw materials and people, and the management of flows leaving the factory (waste, products). The optimization of internal flows (communication, goods, etc.) is also one of the keys to the efficient running of the production means. Like an industrial plant, the animal cell can be described as a "factory" not so much because of the production it generates but essentially because of the complexity of the extra-and intracellular exchanges and interactions observed. To describe this complexity of flows, modelling tools are used, such as metabolic modelling, which represents the cell factory in the form of a "road map" and quantifies the distribution of flows within this mesh. For instance, a recent study modelled a CHO cell with 2101 genes and 4527 metabolites interconnected by 7436 reactions 21 ! Even if, in the near future, we can hope that this type of model will be used to control the production process, the approach favored in recent years by the engineering sciences to size and optimize culture processes is a systemic approach, which incorporates the intracellular machinery without precisely describing it. Thus, the determination of optimal culture conditions is based on the empirical or semi-empirical control of extracellular environments (that we will call the inlets of the cell factory) promoting the targeted performances of the cell (that we will call the outflows). By extension, the successful scale-up of animal cell culture processes to an industrial scale implies tacitly, but resolutely, that the maintenance of bioproduction performance from one scale to another can be guaranteed by the conservation of the input flows into the cell factory (the same causes promoting the same effects), provided that the intrinsic biological characteristics of the cell are also maintained (Figure 2). With the progress of knowledge and modelling tools, this systemic approach is progressively moving towards a multi-scale modelling that integrates all the length and time scales of the production process: metabolites/reactions, cell/physiological state, cell populations, bioreactor/hydrodynamics, process/pilot (see paragraph 4). Controlling the microenvironments of the cell factory relies on the deployment of experimental and numerical methodologies that will aim to (i) strengthen the understanding of cell/environment couplings and (ii) predict, in a sufficiently anticipated manner, the temporal evolutions of cellular responses, knowing the dynamics of these at a given time. Among the challenges taken up or to be taken up, let us mention in particular:

Control and optimization of substrate and nutrient concentrations. To achieve this goal, the culture processes are or should be based on the use of culture media, if possible, chemically defined, the composition of which will have been optimized during preliminary screening phases; this strategy makes it possible, in particular, to eliminate possible batch-to-batch variability in the culture medium. While the use of such media is gradually standardized for the industrial production of recombinant proteins (for these applications the use of compounds of plant origin or recombinant compounds in place of animal or human sources is also very frequent), the addition of compounds of animal or human origin (fetal calf serum, human platelet lysate, growth factors) remains classic for the culture of primary human cells (MSC). However, as the kinetics of nutrient consumption or generation of metabolic products modifies this initial biochemical environment, it is also necessary to have methods for measuring, as far as possible "on-line", the concentrations of substrates and metabolites. Thus, the 2010s have seen the emergence and concretization of spectroscopic sensor technologies implemented in bioreactors (dielectric, RAMAN, near infrared, etc.) including on industrial production lines. Once combined, these sensors can not only measure in real time the concentrations of glucose, lactate, and glutamine but also the concentration and quality of the monoclonal antibody produced 18 . They are also required to improve the performance of fed-batch or continuous-perfused modes of culture.

Knowledge of hydromechanical environments. Most of the time, the use of agitated bioreactors imposes a turbulent, multiphase (gas-liquid, liquid-solid or gas-liquid-solid) hydrodynamics, notably characterized by its temporal variability and spatial heterogeneity. The description of the local hydrodynamics imposed on the cells during their culture makes it possible to anticipate the appearance of cellular damage in case of too strong agitation, to predict modifications in the size of cellular aggregates, or to generate mechanical environments suitable for the differentiation of stem cells, for example 18 . It can also contribute to a better understanding of the phenomena of mixing and mass transfer (oxygenation, desorption of CO 2 ) or even to optimize bioreactor designs for a targeted application 22 . However, it requires the deployment of sophisticated experimental and numerical methodologies (numerical fluid mechanics, particle image velocimetry) which still have limitations in the case of complex multiphase flows. To overcome these limitations, the use of macroscopic scaleup criteria (mechanical power, mean Kolmogorov scale, volumetric oxygen transfer coefficient, etc.)

Figure 1 .

 1 Figure 1. Number of monoclonal antibodies approved by FDA.

Figure 2 .

 2 Figure 2. The concept of cell factory: the key integrated scale for process optimization.

Table 1 .

 1 Most common mammalian cell lines and related applications.

	Cell	Biological source	Some common uses
	CHO	Chinese Hamster Ovary cells	Recombinant proteins
	VERO	Kidney epithelial cells extracted from an African green monkey	Human and veterinary viral vaccines (rotavirus, rabies, poliovirus)
	BHK	Baby Hamster Kidney fibroblasts	Factor VIII, veterinary viral vaccines
	HEK 293	Human embryonic kidney	Adenoviral vectors (SARS-CoV2)
	Hybridomas	Murine hybrid cell line	Monoclonal antibodies
	PER.C6	Derived from human embryonic retinal cells	Recombinant proteins, adenoviral vectors (SARS-CoV2)
	NS0	Derived from the non-secreting murine myeloma	Recombinant proteins
	MDCK	Madin-Darby canine kidney cells	Viral vaccines (flu)
	MRC5	Human fetal lung fibroblast cells	Human viral vaccines (flu)

Table 2 .

 2 Advantages and drawbacks of mammalian and non-mammalian expression systems adapted from O'Flaherty et al. (2020) 20 .

	Cell source	Advantages	Drawbacks
	Mammalian (human or animal)	Excretion of proteins; growth in suspension or adhered on microcarriers in mixed bioreactors; scale-up robustness	Risk of viral contaminations; high culture costs; moderate yields and cell concentrations (except in perfused or optimized fed-batch modes of culture)
		Higher yields; fast growth;
	Non-mammalian	less risk of viral	Partial post-translational modifications of
	(yeasts, bacteria, plant,	contaminations; lower	therapeutic proteins; risk of protein
	insect)	production costs; scale-up	aggregation and inclusion body formation
		robustness

O'Flaherty et al (2020). Biotech Adv, 107552.

Dumont et al. (2015). Crit. Rev. Biotechnol. 18:1.

Meng et al. (2020), Drug Delivery, 27 :585.

Bunggulawa et al. (2018) J Nanobiotechnology 16:81

Gurunathan et al. (2019) Cells 8 :307.

Donoso-Quezada et al. (2020) Crit Rev Biotechnol 40:804.

Luan et al. (2017). Acta Pharmacol Sin 38 :754.

Song et al. (2020) Am J Reprod Immunol 85:e13329.

Willis et al. (2017) Front Cardiovasc Med 4 :63.

Takahashi and Yamanaka et al. (2006). Cell 126 :663.

Umekage et al. (2019). Inflamm Regen 39:17.

Ferrari et al. (2014). Appl. Biochem. Biotechnol 172:1004.

Sion, C. et al. (2020). Biochem Eng J, 157, 107521.

Qian et al. (2018). Bone Marrow Transplant 53:114.

O'Flaherty et al. (2020). Biotech Adv, 107552.

Maillot et al. (2021). Biotechnol Adv, doi.org/10.1016/j.biotechadv.2021.107765.

makes it possible to maintain the hydromechanical stresses within acceptable limits or to dissolve the oxygen at a controlled rate.

Strengthen (further!) the interactions between the cell factory and the culture processes

The versatility and industrial operability of continuous cell lines or primary cells will be all the more strengthened that the constraints of cell factory operation are integrated early into the culture process optimization strategies and/or by using advanced modelling approaches. Thus, the prediction of the evolution of the physical cell environments during the transition to an industrial production scale (mechanical constraints, concentration heterogeneity) must be carried out during the process development phases, using mechanistic simulation approaches (CFD, compartments) and, if necessary, dedicated scale-down methodologies (two-stage systems, bioreactor screening). The construction of predictive multiscale models, helping not only to predict cell performance under given culture conditions but also to optimize them, is currently underway and should be strengthened in the coming years (Figure 3). It requires the use of online measurement technologies to characterize cell dynamics and predict the actions needed to maintain cell productivity through implementation of optimal control strategies. On-line monitoring of cell functionality is also an important challenge to be met in the coming years to allow the application of these advanced understanding approaches. In this general framework of Quality By Design applied to animal and human cell cultures 23 , the contribution of Artificial Intelligence and in particular Deep Learning by Neural Networks, numerical simulation of flows coupled to cell kinetics and new generations of online sensors will be essential to "close the loop".

The authors would like to thank Dr Loïc Reppel (UTCT), Dr Caroline Laroye (UTCT), Dr Bruno Ebel (LRGP), Dr Céline Loubière (MTINOV) and Dr Latifa Chebil (LRGP) for the article revision.