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

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Foued Miaadi
email: foued.miaadi@yahoo.com

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

Finite time

Keywords: High-order neural networks, finite time boundedness, topological degree, stabilization, Lyapunov-Krasovskii functional, LMI, neutral systems, leakage delay

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Neural Networks (NNs) have been widely studied due to their practical applications in lots of areas such as model identification, signal processing, image processing, pattern recognition, optimization problems, associative memories [START_REF] Bishop | Neural networks for pattern recognition[END_REF][START_REF] Aouiti | A genetic-designed beta basis function neural network for multi-variable functions approximation[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 | New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays[END_REF]. The majority of these applications requires the stability of the designed NNs. It should be pointed out that the delay has a great effect on the system performances. Therefore, the stability analysis of delayed NNs has attracted the attention of many researchers and a lot of results have been obtained (see for instance [START_REF] Aouiti | Oscillation of impulsive neutral delay generalized high-order Hopfield neural networks[END_REF][START_REF] Aouiti | Pseudo almost automorphic solutions of recurrent neural networks with time-varying coefficients and mixed delays[END_REF][START_REF] Chen | Exponential stability criterion for interval neural networks with discrete and distributed delays[END_REF][START_REF] Huang | Global exponential stability of neutral highorder stochastic Hopfield neural networks with Markovian jump parameters and mixed time delays[END_REF][START_REF] Liu | Almost periodic solutions for Hopfield neural networks with continuously distributed delays[END_REF][START_REF] Peng | LMI-based global exponential stability of equilibrium point for neutral delayed BAM neural networks with delays in leakage terms via new inequality technique[END_REF][START_REF] Wang | On global asymptotic stability of neural networks with discrete and distributed delays[END_REF]). In [START_REF] Peng | LMI-based global exponential stability of equilibrium point for neutral delayed BAM neural networks with delays in leakage terms via new inequality technique[END_REF][START_REF] Huang | Global exponential stability and periodic solutions of recurrent neural networks with delays[END_REF], the authors discussed the case of constant delays and in [START_REF] Aouiti | Pseudo almost automorphic solutions of recurrent neural networks with time-varying coefficients and mixed delays[END_REF][START_REF] Liu | Almost periodic solutions for Hopfield neural networks with continuously distributed delays[END_REF][START_REF] Wang | On global asymptotic stability of neural networks with discrete and distributed delays[END_REF][START_REF] Ammar | Existence and uniqueness of pseudo almost-periodic solutions of recurrent neural networks with time-varying coefficients and mixed delays[END_REF] the authors analyzed the stability of NNs with continuously distributed delays.

Recently, another kind of delay, variously known as leakage delay, is investigated in [START_REF] Gopalsamy | Leakage delays in BAM[END_REF][START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF][START_REF] Li | Asymptotic stability analysis on nonlinear systems with leakage delay[END_REF][START_REF] Peng | Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms[END_REF]. It has been proved that this kind of delay tends to render the systems of NNs unstable. The effect of the leakage delay on stability is one of the important research topics in the field of the stability of NNs [START_REF] Aouiti | Piecewise pseudo almost periodic solution for impulsive generalised high-order Hopfield neural networks with leakage delays[END_REF]. Many researchers analyzed the effect of the leakage delay on the Lyapunov stability of various kinds of NNs such that bidirectional associative memory NNs [START_REF] Balasubramaniam | Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays[END_REF][START_REF] Lakshmanan | Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays[END_REF] or impulsive NNs [START_REF] Li | Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations[END_REF][START_REF] Li | Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations[END_REF]. However, it should be pointed out that manipulating this kind of delay is not easy. In addition, a kind of time delay systems, appointed neutral-type delay systems, is used by many authors due to their practical applications [START_REF] Wang | Reliable H ∞ control for discrete-time piecewise linear systems with infinite distributed delays[END_REF]. There are several results discussing the stability of neutral-type NNs, see for instance [START_REF] Feng | Delay-dependent stability of neutral type neural networks with distributed delays[END_REF][START_REF] Li | Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type[END_REF][START_REF] Aouiti | Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays[END_REF] and references therein.

In order to render the stability criteria less conservative with respect to the delays, several methods are developed in the literature [START_REF] Huang | State estimation of recurrent neural networks with time-varying delay: A novel delay partition approach[END_REF][START_REF] Lakshmanan | Delay-dependent state estimation of neural networks with mixed timevarying delays[END_REF][START_REF] Li | A new model transformation of discrete-time systems with time-varying delay and its application to stability analysis[END_REF][START_REF] Li | New delay-variation-dependent stability for neural networks with time-varying delay[END_REF][START_REF] Li | Improved stability criteria of neural networks with timevarying delays: An augmented LKF approach[END_REF][START_REF] Yue | A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay[END_REF][START_REF] Yan | Finite-time stability and stabilization of itô stochastic systems with Markovian switching: mode-dependent parameter approach[END_REF][START_REF] Qi | Finite-time passivity and passification for stochastic time-delayed Markovian switching systems with partly known transition rates[END_REF]. Most of these methods are based on a Lyapunov functional and associated LMIs because these inequalities can be numerically checked. However, it is well known that when a novel Lyapunov functional is designed for reducing the conservatism with respect to the delays a greater complexity in terms of inequalities and variables to be calculated can appear. Therefore, the problem of reducing simultaneously the number of decision variables and the conservatism with respect to the delays arises.

Most of the previous works on stability were mainly based on the classical Lyapunov stability which is associated with an infinite time interval. However, only a finite time interval is considered in practical applications [START_REF] Onori | Finite time stability design via feedback linearization[END_REF]. In 1953, Kamenkov has introduced in [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF] the concept of finite time boundedness (FTB). Dorato reported in [START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF] that FTB and Lyapunov stability are two independent concepts. Many studies have addressed the FTB problem of NNs, see for instance [START_REF] Ma | Finite time non-fragile dissipative control for uncertain TS fuzzy system with time-varying delay[END_REF][START_REF] Tan | Finite-time stability of neural networks with impulse effects and time-varying delay[END_REF][START_REF] Zhao | Finite-time boundedness analysis of memristive neural network with time-varying delay[END_REF][START_REF] Zhou | Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations[END_REF][START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF][START_REF] He | Finite-time boundedness of uncertain time-delayed neural network with Markovian jumping parameters[END_REF][START_REF] Jiang | Finite time stability of Cohen-Grossberg neural network with time-varying delays[END_REF][START_REF] Lin | Finite-time boundedness and L 2 -gain analysis for switched delay systems with norm-bounded disturbance[END_REF][START_REF] Shen | LMI-based finite-time boundedness analysis of neural networks with parametric uncertainties[END_REF][START_REF] Wang | Extended finite-time H ∞ control for uncertain switched linear neutral systems with time-varying delays[END_REF][START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF][START_REF] Zhang | Finite-time boundedness for uncertain discrete neural networks with time-delays and Markovian jumps[END_REF]. For neutral-type NNs, the FTB was studied in [START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF][START_REF] Wang | Extended finite-time H ∞ control for uncertain switched linear neutral systems with time-varying delays[END_REF]. In [START_REF] He | Finite-time boundedness of uncertain time-delayed neural network with Markovian jumping parameters[END_REF][START_REF] Zhang | Finite-time boundedness for uncertain discrete neural networks with time-delays and Markovian jumps[END_REF] and [START_REF] Zhou | Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations[END_REF], the class of uncertain NNs with Markovian jumps and the complex-valued NNs are studied respectively. Also the impulsive NNs is investigated in [START_REF] Tan | Finite-time stability of neural networks with impulse effects and time-varying delay[END_REF] and the authors of [START_REF] Zhao | Finite-time boundedness analysis of memristive neural network with time-varying delay[END_REF][START_REF] Ma | Finite time non-fragile dissipative control for uncertain TS fuzzy system with time-varying delay[END_REF] deal with the problem of FTB for Memristive NNs and TS-fuzzy system with time-varying delay respectively. Moreover, all the previous results on FTB are for lower order NNs. This class of lower order NNs has several limitations (see for instance [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF][START_REF] Kamp | Recursive neural networks for associative memory[END_REF]) which leads to consider the class of NNs with high-order connections. This class of high-order NNs has stronger approximation characteristics, important storage capacity, faster convergence rate and higher fault tolerance than lower-order NNs [START_REF] Zheng | Matrix measure based stability criteria for high-order neural networks with proportional delay[END_REF]. To the best of the authors' knowledge, the FTB and FTB-stabilization of neutral high-order Hopfield NNs (NHOHNNs) with a leakage delay and with both discrete and distributed delays, called mixed time delays, have not been fully investigated in the literature. Here is the main goal of our article. The delays considered in our article contain an infinite distributed delay which occurs in practice [START_REF] Wang | Reliable H ∞ control for discrete-time piecewise linear systems with infinite distributed delays[END_REF]. Indeed, there is no result for the FTB of NNs with infinite distributed delays. Despite important studies available yet to deal with the FTB concept [START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF][START_REF] He | Finite-time boundedness of uncertain time-delayed neural network with Markovian jumping parameters[END_REF][START_REF] Wang | Extended finite-time H ∞ control for uncertain switched linear neutral systems with time-varying delays[END_REF][START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF][START_REF] Zhang | Finite-time boundedness for uncertain discrete neural networks with time-delays and Markovian jumps[END_REF], there is no result for the FTB of NNs with non-differentiable time-varying delay. Here is also a novelty of our article.

In our article, the FTB and FTB-stabilization are analyzed for a general class of NHOHNNs with time delay in the leakage term and mixed time delays. The main contributions are as follows:

(i) the LKF and LMIs techniques are used to study the effect of the leakage delay on the FTB results of the concerned NHOHNNs;

(ii) simplified LMI conditions are established to reduce the conservatism with respect to the delays and provide a less computational load simultaneously;

(iii) by using the FTB analysis, some conditions are given for the FTB-stabilization of the concerned NHOHNNs with a leakage delay. Moreover, the NHOHNNs considered in this article are subjected to non-differentiable timevarying delays.

The article is organized as follows. In Section 2, some preliminaries useful for the study of the class of NHOHNNs are presented. The existence of equilibrium points is addressed in Section 3. Then, the FTB and the FTB-stabilization are studied in Section 4. Numerical examples are presented in Section 5 to illuminate the validity of the proposed results. Finally, some concluding remarks are drawn in Section 6.

Preliminaries

In this article, we use the following notations:

• R, Z + , R n and R n×n stand for the set of real numbers, the set of positive integers, the n-dimensional real space equipped with the euclidean norm . and the set of n×n real matrices respectively;

• A T , A -1 , A > 0, A < 0 and I means respectively the transpose of A, the inverse of A, the matrix A is positive definite, the matrix -A is positive definite and the identity matrix with appropriate dimensions;

• λ max (A), λ min (A) and * stand for the maximum eigenvalue of A, the minimum eigenvalue of A and the symmetric block in one symmetric matrix respectively;

• X + = |x i j | n×n
where X = x i j ) n×n ;

• for any interval

I ⊂ R and V ⊂ R k (1 ≤ k ≤ n), C(I, V )
and C 1 b (I, V ) stand respectively for the set {φ : I → V : φ is continuous} and {φ : I → V : φ is continuously differentiable and bounded}.

We consider the following neutral high-order Hopfield neural networks (NHOHNNs) with time delay in the leakage term and mixed time delays

                           ẋi (t) = -c i x i (t -σ ) + n ∑ j=1 a i j f j (x j (t -τ (t))) + n ∑ j=1 n ∑ k=1 T i jk f k (x k (t -τ(t))) f j (x j (t -τ(t))) + n ∑ j=1 b i j t -∞ k j (t -s) f j (x j (s)) ds + n ∑ j=1 d i j ẋ j (t -h(t)) + J i , t > 0, i = 1, . . . , n x(s) = φ (s), s ∈ (-∞, 0] (1)
where c i > 0, C = diag (c 1 , . . . , c n ), a i j , b i j and d i j are the interconnection weight coefficients of the neurons, T i jk the secondorder synaptic weights of the NNs, J i an external input vector, x(t) = (x 1 (t), . . . , x n (t)) T the neuron state vector of the NNs, ẋ the time derivative of the neuron state, f (x(.)) = ( f 1 (x 1 (.)) , . . . , f n (x n (.))) T the neuron activation function such that f i (0) = 0 for all 1 ≤ i ≤ n, σ ≥ 0 a constant which is the leakage delay,

K = diag (k 1 (.), . . . , k n (.))
the delay kernel, τ(.) and h(.) the time-varying transmission delays satisfying 0

≤ τ(t) ≤ τ, 0 ≤ h(t) ≤ h and ḣ(t) ≤ h * < 1. The initial condition satisfies φ (.) ∈ C 1 b ((-∞, 0], R n )
where the norm is defined by

φ h = max sup s≤0 φ (s) , sup -h≤s≤0 φ (s) .
The term of the right-hand side of System (1) involving the time derivative of the state renders the system of neutral-type (see for instance [START_REF] Hale | Introduction to functional differential equations[END_REF]). Several studies have been done on System (1), especially in [START_REF] Huang | Global exponential stability of neutral highorder stochastic Hopfield neural networks with Markovian jump parameters and mixed time delays[END_REF], where the global exponential stability is treated for σ = 0 and with Markovian jump parameters. In [START_REF] Li | Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales[END_REF], the authors studied the stability of the almost automorphic solutions for the same model but without taking into consideration infinite distributed delays. Note that such System (1) contains many of the well-known models as special cases.

Let us introduce the following three assumptions:

(H1) for all x, y ∈ R, there exist constants ω > 0, M - j and M + j such that

| f j (x)| ≤ ω j and M - j ≤ | f j (x) -f j (y)| x -y ≤ M + j , (
H2) for all j = 1, . . . , n, the functions k j : R + → R + are continuous and satisfy

+∞ 0 k j (u)du = k j and +∞ 0 u k j (u)du < ∞, (H3) C -A + M -B + K + M is a M-matrix.
Remark 2.1. Under assumptions (H1) and (H2), the existence of the solutions of System (1) is guaranteed by using [53, Theorem 2.1]. It should also be pointed out that the constants M + j and M - j can be negative or positive in the assumption (H1) which allows Lurie-type functions if we take M + j , M - j > 0 [START_REF] Wu | Lagrange stability of memristive neural networks with discrete and distributed delays[END_REF] or Lipschitz functions if we take M - j = -M + j < 0. Therefore (H1) is weaker than the assumptions used for instance in [START_REF] Cao | Exponential stability of high-order bidirectional associative memory neural networks with time delays[END_REF][START_REF]Dynamics of high-order Hopfield neural networks with time delays[END_REF][START_REF] Wang | Global exponential stability and periodic solutions of high-order bidirectional associative memory (BAM) neural networks with time delays and impulses[END_REF]. Finally, if we only consider Lipschitz activation functions as in [START_REF] Huang | Global exponential stability and periodic solutions of recurrent neural networks with delays[END_REF], the method used in [START_REF] Huang | Global exponential stability and periodic solutions of recurrent neural networks with delays[END_REF] for proving the existence of an equilibrium point does not work for System [START_REF] Bishop | Neural networks for pattern recognition[END_REF] with the high-order terms. Definition 2.1. ( [START_REF] Huang | Global exponential stability and periodic solutions of recurrent neural networks with delays[END_REF]) A nonsingular matrix X = (x i j ) n×n is a M-matrix if X = rIβ with β ≥ 0 and r > ρ (β ) where ρ (β ) stands for the spectral radius of the matrix β .

Let

Z n×n = A = (a i j ) ∈ R n×n : a ii > 0, a i j ≤ 0, for all i = j then we have the following lemma: Lemma 2.2. ( [START_REF] Huang | Global exponential stability and periodic solutions of recurrent neural networks with delays[END_REF]) The following claims are equivalent: (i) there exist δ j > 0 such that n ∑ j=1 a ji δ j > 0 for all i = 1, . . . , n;

(ii) A ∈ Z n×n is a M-matrix;

(iii) there exist δ j > 0 such that n ∑ j=1 a i j δ j > 0 for all i = 1, . . . , n.

Let us give the definition of the topological degree. Definition 2.3. ( [START_REF] Huang | Global exponential stability and periodic solutions of recurrent neural networks with delays[END_REF]) Assume that F : Ω → R n is a continuously differentiable function where Ω ⊂ R n is an open bounded set. If J F (u) = 0 for any u ∈ F -1 (p) and p / ∈ F (∂ Ω), where J F denotes the Jacobian determinant relative to F, then the topological degree relative to Ω and p is given by

deg (F, Ω, p) =    ∑ u∈F -1 (p) sign J F (u) if F -1 (p) = / 0 0 if F -1 (p) = / 0
Let us introduce the notion of finite time boundedness.

Definition 2.4. ( [START_REF] Bai | Finite-time stability analysis of discrete-time fuzzy Hopfield neural network[END_REF]) System (1) is said to be finite time bounded

(FT B) with respect to (c 1 , c 2 , R, T ) where 0 < c 1 ≤ c 2 , R > 0 and T > 0 if for all t ∈ [0, T ] we have max sup θ ≤0 {x T (θ )Rx(θ )}, sup -h≤θ ≤0 { ẋT (θ )R ẋ(θ )} ≤ c 1 implies that x T (t)Rx(t) < c 2 .
Let us give two lemmas useful to prove our first result. Lemma 2.6. ( [START_REF] Berman | Nonnegative matrices in the mathematical sciences[END_REF]) Let P ∈ R n×n be a symmetric matrix, then we have λ min (P) x T x ≤ x T Px ≤ λ max (P) x T x for any x ∈ R n .

Existence of equilibrium points

In [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF], the existence of equilibrium points for neutral lower order Hopfield NNs with time delay in the leakage term is discussed. Thanks to the high-order terms, the NNs given by (1) are more general. Compared with results in [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF], we extend the existence of equilibrium points to the more general class of NHOHNNs.

We introduce the following notations:

M j = max |M - j |, |M + j | , M = diag (M 1 , . . . , M n ) A = [a i j ] n×n , B = [b i j ] n×n , D = [d i j ] n×n .
We present a sufficient condition which guarantee the existence of equilibrium points for the NHOHNNs given by (1).

-c i x * i + n ∑ j=1 a i j f j (x * j ) + n ∑ j=1 n ∑ k=1 T i jk f k (x * k ) f j (x * j ) + n ∑ j=1 b i j t -∞ k j (t -s) f j (x * j )ds + n ∑ j=1 d i j ẋ * j + J i = 0 (2)
By using (H2), Equation ( 2) is equivalent to

-c i x * i + n ∑ j=1 a i j f j (x * j ) + n ∑ j=1 n ∑ k=1 T i jk f k (x * k ) f j (x * j ) + n ∑ j=1 b i j k j f j (x * j ) + J i = 0. Let l i (x) = c i x i - n ∑ j=1 a i j + b i j k j f j (x j ) - n ∑ j=1 n ∑ k=1 T i jk f k (x k ) f j (x j ) -J i . (3)
Clearly, a solution of l(x) = (l 1 (x), . . . , l n (x)) T = 0 is an equilibrium of System [START_REF] Bishop | Neural networks for pattern recognition[END_REF]. Now, we define the homotopy mapping

F(x, λ ) = (F 1 (x), . . . , F n (x)) T
where λ ∈ [0, 1] and

F i (x) = λ l i (x) + (1 -λ )x i . It follows from assumptions (H1) -(H2) that for all 1 ≤ i ≤ n |F i (x, λ )| = λ c i x i - n ∑ j=1 a i j f j (x j ) - n ∑ j=1 n ∑ k=1 T i jk f k (x k ) f j (x j ) - n ∑ j=1 b i j k j f j (x j ) -J i + (1 -λ )x i ≥ |λ c i x i + (1 -λ )x i | -λ n ∑ j=1 |a i j || f j (x j )| -λ n ∑ j=1 n ∑ k=1 T i jk | f k (x k )| f j (x j ) -λ n ∑ j=1 |b i j ||k j || f j (x j )| -λ |J i | ≥ |λ c i x i + (1 -λ )x i -λ n ∑ j=1 |a i j |M j |x j | -λ n ∑ j=1 |b i j ||k j |M j |x j | -λ |J i | + n ∑ j=1 n ∑ k=1 |T i jk |ω 2 . Since C -A + M -B + K + M is a M-matrix, Lemma 2.
2 implies that there exists constants δ i > 0 with i = 1, . . . , n such that

δ i c i - n ∑ j=1 δ j |a i j |M j - n ∑ j=1 δ j |b i j ||k j |M j > 0 or (4)
δ i c i - n ∑ j=1 δ j |a ji |M i - n ∑ j=1 δ j |b ji ||k i |M i > 0. For F(x) = (F 1 (x 1 ), . . . , F n (x n )) T , we have n ∑ i=1 δ i |F i (x, λ )| ≥ n ∑ i=1 δ i (1 -λ )|x i | + λ n ∑ i=1 δ i c i |x i | -δ i n ∑ j=1 |a i j |M j |x j | -δ i n ∑ j=1 |b i j ||k j |M j |x j | -λ n ∑ i=1 δ i |J i | + n ∑ j=1 n ∑ k=1 |T i jk |ω 2 ≥ λ n ∑ i=1 δ i c i |x i | -δ i n ∑ j=1 M j |a i j ||x j | -δ i n ∑ j=1 M j |b i j ||k j ||x j | -λ n ∑ i=1 δ i |J i | + n ∑ j=1 n ∑ k=1 |T i jk |ω 2 = λ n ∑ i=1 δ i c i - n ∑ j=1 δ j |a ji |M i - n ∑ j=1 δ j |b ji |M i |k i | |x i | -λ n ∑ i=1 δ i |J i | + n ∑ j=1 n ∑ k=1 |T i jk |ω 2 Define δ 0 = min 1≤i≤n δ i c i - n ∑ j=1 δ j |a ji |M i - n ∑ j=1 δ j |b ji |M i |k i | ∆ 0 = max 1≤i≤n δ i |J i | + n ∑ j=1 n ∑ k=1 |T i jk |ω 2 Let Ω = x : |x i | < β = n(∆ 0 + 1) δ 0 ( 5
)
then Ω is non-empty from (4). It follows from ( 5) that for any

x ∈ ∂ Ω there exists 1 ≤ i 0 ≤ n such that |x i 0 | = β . So we have n ∑ i=1 δ i F i (x, λ ) ≥ λ n ∑ i=1 δ i 0 c i 0 - n ∑ j=1 δ j M i 0 |a ji 0 | - n ∑ j=1 δ j M i 0 |b ji 0 ||k i 0 | |x i 0 | -λ n ∑ i=1 ∆ 0 ≥ λ δ 0 |x i 0 | -λ n∆ 0 > 0. For all λ ∈ (0, 1], it means that F(x, λ ) = 0 for any x ∈ ∂ Ω and λ ∈ (0, 1]. If λ = 0, we have F(x, λ ) = x = 0 for any x ∈ ∂ Ω. Therefore F(x, λ ) = 0, for any x ∈ ∂ Ω, λ ∈ [0, 1]. We have deg(I d , Ω, 0) = 1 where I d (x) = x and deg (I d , Ω, 0)
is the topological degree. Thus from the homotopy invariance theorem given for instance in [61, page 13], we obtain

deg(I d , Ω, 0) = deg (l(.), Ω, 0) = 1.
By using the topological degree theory [START_REF] Cho | Topological degree theory and applications[END_REF], we can conclude that the equation l(x) = 0 has at least one solution in Ω. It implies that System (1) has at least one equilibrium point.

Finite Time boundedness analysis

Finite time Boundedness

Compared with some existing results in [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF][START_REF] Feng | Delay-dependent stability of neutral type neural networks with distributed delays[END_REF][START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF], System ( 1) is more general to some extent. Moreover, the usual Lyapunov stability is studied in [START_REF] Feng | Delay-dependent stability of neutral type neural networks with distributed delays[END_REF][START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF] whereas the concept of FTB is discussed in this section. Assume that x * = (x * 1 , . . . , x * n ) T is an equilibrium point of System [START_REF] Bishop | Neural networks for pattern recognition[END_REF]. By a simple transformation

z(t) = x(t) -x *
we can shift the equilibrium point x * to the origin. By using (2), System (1) can be rewritten as (see [START_REF] Lou | Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays[END_REF]):

     ż(t) = -Cz(t -σ ) + A + Γ T T * g (z(t -τ(t))) +B t -∞ K(t -s)g (z(s)) ds + Dż (t -h(t)) z(s) = φ (s) -x * , s ∈ (-∞, 0] where g (z(.)) = f (z(.) + x * ) -f (x * ), T i = [T i jk ] n×n , T * = T 1 + T T 1 , ..., T n + T T n T , Γ = diag[ξ , ..., ξ ], ξ = [ξ 1 , ..., ξ n ] T , ξ i = T i jk T i jk + T ik j f k (x k (t -τ(t)) + T ik j T i jk + T ik j f k (x * k ).
We will use this z-form of System (1) for the proof of the results of our article. We introduce the following notations:

Γ + = diag[ξ + , ..., ξ + ], ξ + = [ω 1 , ..., ω n ] T , c + = max 1≤i≤n c i .
The methods used in [START_REF] Wang | Extended finite-time H ∞ control for uncertain switched linear neutral systems with time-varying delays[END_REF][START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF][START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF][START_REF] Ali | Delay-dependent stability criteria of uncertain Markovian jump neural networks with discrete interval and distributed time-varying delays[END_REF][START_REF] Wang | New results on robust finite-time boundedness of uncertain switched neural networks with time-varying delays[END_REF] for ensuring the FTB of NNs require the differentiability and the boundedness of the derivative of the time-varying delays. For improving these results, we remove this restriction by establishing the following theorem where the time-varying delays are not necessary differentiable.

Theorem 4.1. Under assumptions (H1) -(H2) -(H3), Sys- tem (1) is FTB with respect to (c 1 , c 2 , R, T ) if there exist a posi- tive scalar α, two n × n matrices Q 1 , Q 2 , three n × n symmetric positive definite matrices P, Q 3 , Q 4 , four n × n positive diago- nal matrices U 1 , U 2 , Q 5 , Q 6 ,
and a 2n × 2n matrix

Q 7 = T 11 T 12 * T 22 > 0
such that the following conditions hold:

Ξ =              Π 11 0 Π 13 Π
             < 0 (6)
and

c 2 e -αT √ c 1 √ c 1 ω -1 1 > 0 ( 7
)
where

Π 11 = -PC -CP + Q 3 + σ 2 Q 4 -U 1 Σ 1 -αP, Π 13 = PD, Π 14 = T T 12 , Π 16 = CPC + αCP, Π 17 = U 1 Σ 2 , Π 18 = P Ã, Π 19 = PB, Π 22 = τT 22 + Q 6 -Q 1 -Q T 1 , Π 23 = Q 1 D + Q T 2 D, Π 25 = -Q 1 C, Π 28 = Q 1 Ã, Π 29 = Q 1 B, Π 33 = -Q 6 (1 -h * ) -D T Q 2 D -D T Q T 2 D, Π 35 = D T Q 2 C, Π 36 = -D T PC, Π 38 = -D T Q 2 Ã, Π 39 = -D T Q 2 B, Π 44 = τT 11 -T 12 -T T 12 -U 2 Σ 1 , Π 48 = U 2 Σ 2 , Π 55 = -Q 3 , Π 66 = -Q 4 -αCPC, Π 68 = -CP Ã, Π 69 = -CPB, Π 77 = Q 5 κ -U 1 , Π 88 = -U 2 , Π 99 = -Q 5 ,
and

ω 1 = 2λ max (P)(1 + σ 2 c + 2 ) + σ λ max (Q 3 ) + hλ max (Q 6 ) + σ 3 λ max (Q 4 ) + τ 2 λ max (T 22 ) + n ∑ j=1 q j k j max j M 2 j ∞ 0 uk j (u)du × nc + σ λ min (Q 3 ) 1 2

+ (λ min (P))

-1 22 cond(P)cond( P)

with

Σ 1 = diag(M - 1 M + 1 , . . . , M - n M + n ), Σ 2 = diag M - 1 + M + 1 2 , . . . , M - n + M + n 2 , Q 5 = diag(q 1 , . . . , q n ), κ = diag(k 2 1 , . . . , k 2 n ), Ã = A + Γ + T T * , P = R 1 2 PR 1 2 ,
cond(P) = λ max (P) λ min (P) the condition number of P.

The proof of Theorem 4.1 is inspired by the proof of Theorem 2 in [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF].

PROOF. Let us consider the following LKF

V (t, z(t)) = 6 ∑ i=1 V i (t, z(t)) (8)
where

V 1 (t, z(t)) = z(t) -C t t-σ z(s)ds T P z(t) -C t t-σ z(s)ds V 2 (t, z(t)) = t t-σ z T (s)Q 3 z(s)ds + t t-h(t) żT (s)Q 6 ż(s)ds V 3 (t, z(t)) = σ t t-σ t s z T (u)Q 4 z(u)duds V 4 (t, z(t)) = t 0 u u-τ(u) z(u -τ(u)) ż(s) T Q 7 z(u -τ(u)) ż(s) ds du V 5 (t, z(t)) = 0 -τ t t+u żT (s)T 22 ż(s)ds du V 6 (t, z(t)) = n ∑ j=1 q j k j ∞ 0 k j (u) t t-u g 2 j (z j (s))ds du By calculating V (t, z(t)) (see AppendixA), we obtain V (t, z(t)) ≤ ζ T (t, z(t)) Ξζ (t, z(t))
where

ζ (t, z(t)) = z T (t), żT (t), żT (t -h(t)) , z T (t -τ(t)) , z T (t -σ ), t t-σ z(s)ds T , g T (z(t)) , g T (z(t -τ(t))) , t -∞ K(t -s)g(z(s))ds T T and Ξ =              Π11 0 Π 13 Π
             < 0 (9) with Π11 = -PC -CP + Q 3 + σ 2 Q 4 -U 1 Σ 1 , Π16 = CPC, Π66 = -Q 4 . Therefore V (t, z(t)) ≤ ζ T (t, z(t)) Ξζ (t, z(t)) ≤ ζ T (t, z(t))Ξζ (t, z(t)) + αV 1 (t, z(t)) ≤ ζ T (t, z(t))Ξζ (t, z(t)) + αV (t, z(t)) with Ξ = Ξ +               αP 0 0 0 0 -αCP 0 0 0 * 0 0 0 0 0 0 0 0 * * 0 0 0 0 0 0 0 * * * 0 0 0 0 0 0 * * * * 0 0 0 0 0 * * * * * αCPC 0 0 0 * * * * * * 0 0 0 * * * * * * * 0 0 * * * * * * * * 0               Since Ξ < 0, it leads to V (t, z(t)) ≤ αV (t, z(t)). (10)
Integrating ( 10) from 0 to t ∈ [0, T ], we obtain

V (t, z(t)) ≤ e αt V (z(0)) (11)
where

V (z(0)) ≤ 2λ max (P)(1 + σ 2 c + 2 ) + σ λ max (Q 3 ) + hλ max (Q 6 ) + σ 3 λ max (Q 4 ) + τ 2 λ max (T 22 ) + n ∑ j=1 q j k j max j M 2 j ∞ 0 uk j (u)du φ 2 h . (12)
On one hand, by using the Cauchy-Schwartz inequality and Lemma 2.6 we have

z(t) ≤ C t t-σ z(s)ds + V 1 (t, z(t))
λ min (P)

≤ C t t-σ z(s)ds + V (t, z(t)) λ min (P) (13)
and since

t t-σ z(s)ds 2 = t t-σ z(s)ds T t t-σ z(s)ds ≤ σ t t-σ z T (s)z(s)ds ≤ σ λ min (Q 3 ) t t-σ z T (s)Q 3 z(s)ds ≤ σ λ min (Q 3 ) V (t, z(t))
we obtain

z(t) ≤ nc + σ λ min (Q 3 ) 1 2 + λ min (P) -1 2 V (t). (14)
On the other hand, Lemma 2.6, implies that

λ max ( P) z(t) 2 ≥ z T (t) Pz(t) ≥ λ min (P)z T (t)Rz(t), λ min ( P) z(0) 2 ≤ z T (0) Pz(0) ≤ λ max (P)z T (0)Rz(0). (15)
So ( 11)-( 12)-( 14) and [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF] prove that

z(t) ≤ nc + σ λ min (Q 3 ) 1 2 + λ min (P) -1 2 e α 2 T V (z((0)) ≤ e α 2 T 2λ max (P)(1 + σ 2 c + 2 ) + σ λ max (Q 3 ) + hλ max (Q 6 ) + σ 3 λ max (Q 4 ) + τ 2 λ max (T 22 ) + n ∑ j=1 q j k j max j M 2 j ∞ 0 uk j (u)du 1 2 × nc + σ λ min (Q 3 ) 1 2
+ λ min (P)

-1 2
λ max (P) λ min ( P) c 1 [START_REF] Li | Asymptotic stability analysis on nonlinear systems with leakage delay[END_REF] and

z T (t)Rz(t) ≤ e αT 2λ max (P) 1 + σ 2 c + 2 + σ λ max (Q 3 ) + hλ max (Q 6 ) + σ 3 λ max (Q 4 ) + τ 2 λ max (T 22 ) + n ∑ j=1 q j k j max j M 2 j ∞ 0 uk j (u)du × nc + σ λ min (Q 3 ) 1 2 + λ min (P) -1 2 2
× cond(P)cond( P)c 1 [START_REF] Peng | Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms[END_REF] The inequality [START_REF] Peng | Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms[END_REF] and the Schur complement lemma, given for instance in [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF], applied to condition [START_REF] Chen | Exponential stability criterion for interval neural networks with discrete and distributed delays[END_REF] prove that

z(t) T Rz(t) < c 2 for all t ∈ [0, T ].
So, the proof is completed.

Remark 4.1. If conditions ( 6) and ( 7) of Theorem 1 are satisfied with α = 0, then System (1) is asymptotically stable in the sense of Lyapunov. The LKF inequality ( 10) is used to deal with the problem of FTB of the NHOHNNs with infinite distributed delays and leakage delay. Compared with [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF], some new sufficient conditions in terms of LMIs are established to make our results less conservative with respect to the delays. When the number of neurons n increases, the complexity increases strongly because 8.5n 2 + 4n + 1 variables are involved in the LMIs that must be resolved. Such a complexity is caused by the use of the LKF inequality [START_REF] Peng | LMI-based global exponential stability of equilibrium point for neutral delayed BAM neural networks with delays in leakage terms via new inequality technique[END_REF].

Remark 4.2. Condition ( 7) is not standard LMIs, however it can be guaranteed by the following conditions

λ 1 I < P < λ 2 I, λ 3 I < Q 3 < λ 4 I, Q 4 < λ 5 I, Q 6 < λ 6 I, T 22 < λ 7 I; (18)
-c 2 r 2 1 λ 2 1 e -αT + c 1 2λ 2 (1 + σ 2 c + 2 ) + σ λ 4 + hλ 6 + σ 3 λ 5 + τ 2 λ 7 + n ∑ j=1 q j k j max j M 2 j ∞ 0 uk j (u)du nc + σ λ 3 + 1 λ 1 2 r 2 2 λ 2 2 < 0; (19)
where r 1 = λ min (R), r 2 = λ max (R) and λ i , i = 1, . . . 7 are unknown positive variables. It should be pointed out that ( 19) is not a LMI w.r.t. λ i , i = 1, 2, 3 because λ i appears in a nonlinear fashion. Therefore, we first find the scalars λ i from ( 6) and then we solve [START_REF] Balasubramaniam | Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays[END_REF] which then becomes a LMI.

When τ(t) = 0 and K = 0, System (1) turns into

                   ẋi (t) = -c i x i (t -σ ) + n ∑ j=1 a i j f j (x j (t)) + n ∑ j=1 n ∑ k=1 T i jk f k (x k (t)) f j (x j (t)) + n ∑ j=1 d i j ẋ j (t -h(t)) x(s) = φ (s), s ∈ (-∞, 0] (20)
Let us give the definition of finite time boundedness for System [START_REF] Lakshmanan | Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays[END_REF].

Definition 4.2. ( [START_REF] Jiang | Finite time stability of Cohen-Grossberg neural network with time-varying delays[END_REF]) System (20) is said to be Finite Time Bounded (FTB) with respect to

(c 1 , c 2 , R, T ), 0 < c 1 ≤ c 2 , T > 0, R > 0 if for all t ∈ [0, T ] max sup -σ <θ ≤0 {x T (θ )Rx(θ )}, sup -h≤θ ≤0 { ẋT (θ )R ẋ(θ )} ≤ c 1 implies that x T (t)Rx(t) < c 2 .
According to Definition 4.2, we deduce the following corollary.

Ξ =         Π11 0 Π 13 0 Π 16 Π17 * Π22 Π 23 Π 25 0 0 * * Π 33 Π 35 Π 36 0 * * * Π 55 0 0 * * * * Π 66 0 * * * * * -U 1         < 0 ( 21
)
and

c 2 e -αT √ c 1 √ c 1 ω -1 2 > 0 ( 22
)
where

Π11 = -PC -CP + Q 3 + σ 2 Q 4 -U 1 Σ 1 -U 2 Σ 1 -αP, Π17 = (U 1 +U 2 ) Σ 2 + P Ã, Π22 = Q 6 -Q 1 -Q T 1 .
and

ω 2 = e αT 2λ max (P)(1 + σ 2 c + 2 ) + σ λ max (Q 3 ) + hλ max (Q 6 ) + σ 3 λ max (Q 4 ) nc + σ λ min (Q 3 ) 1 2 + λ min (P) -1 2 2
× cond(P)cond( P).

PROOF. Consider the following LKF

V (t, z(t)) = 3 ∑ i=1 V i (t, z(t)) (23)
By using the LKF [START_REF] Wang | Reliable H ∞ control for discrete-time piecewise linear systems with infinite distributed delays[END_REF] and similar arguments to the ones of Theorem 4.1 we obtain easily the result.

Remark 4.3. The lower order class of Hopfield NNs is investigated by many authors [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF][START_REF] Li | Asymptotic stability analysis on nonlinear systems with leakage delay[END_REF][START_REF] Peng | Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms[END_REF] and can be considered as a theoretical basis for solving optimization problems. As reported in [START_REF] Cooper | Stability analysis of higher-order neural networks for combinatorial optimization[END_REF], this class of NNs is expected to produce the poorest quality of solution with a great complexity as measured by the order of the network. Therefore, our work offers a theoretical basis for the design of the second-order class of NNs with mixed time delays more effective in the resolution of optimization problems thanks to the second order synaptic terms T i jk .

Now, when the high-order terms T i jk = 0, System (1) turns into

           ẋi (t) = -c i x i (t -σ ) + n ∑ j=1 a i j f j (x j (t -τ(t))) + n ∑ j=1 b i j t -∞ k j (t -s) f j (x(s)) ds + n ∑ j=1 d i j ẋ j (t -h(t)) x(s) = φ (s), s ∈ (-∞, 0] (24)
Then, we have the following result.

Ξ =               Π 11 0 Π 13 Π
              < 0 ( 25
)
and

c 2 e -αT √ c 1 √ c 1 ω -1 1 > 0 ( 26
)
where Πi8 = Π i8 -PΓ + T T * for i = 1, 2, 3, 6.

The proof of Corollary 4.4 is similar to the one that of Theorem 4.1, so it is omitted.

The computation can be simplified by reducing the parameters involved in the LKF. It is also possible to replace the LKF by a Lyapunov function with the Razumukhin techniques. However, this will have a great influence on the conservatism with respect to the delays, in particular on the upper bounds of the delays [START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF][START_REF] Zhou | Razumikhin and Krasovskii stability theorems for time-varying time-delay systems[END_REF].

In order to simplify the calculations without losing the conservatism with respect to the delays, we establish the following Corollary 4.5 which provides a simplified criterion for NHOHNNs without delay in the leakage term. This criterion is based on the Schur complement and has been presented for the Lyapunov stability of a special class of NHOHNNs in [START_REF] Singh | Simplified LMI condition for global asymptotic stability of delayed neural networks[END_REF]. The impact of this criterion on the conservatism with respect to the delays will be illustrated in Example 5.3.

Λ =         -Π 22 -Π 23 0 0 -Π 28 -Π 29 * -Π 33 0 0 -Π 38 -Π 39 * * -Π 44 0 -Π 48 0 * * * -Π 77 0 0 * * * * -Π 88 0 * * * * * -Π 99         > 0 (27) and c 2 e -αT √ c 1 √ c 1 ω1 > 0 ( 28
)
where

P = R -1 2 PR -1 2 , ω1 = λ min (P) cond( P) 2λ max (P) + hλ max (Q 6 ) + τ 2 λ max (T 22 ) + n ∑ j=1 q j k j max j M 2 j ∞ 0 u k j (u)du -1
.

PROOF. Let us consider the following LKF

V 0 (t, z(t)) = 2 ∑ i=1 Vi (t, z(t)) + 6 ∑ i=4 V i (t, z(t)) (29)
where

V1 (t, z(t)) = z(t) T Pz(t), V2 (t, z(t)) = t t-h(t) żT (s)Q 6 ż(s)ds, V 4 (t, z(t)) = t 0 u u-τ(u) z(u -τ(u)) ż(s) T Q 7 z(u -τ(u)) ż(s) dsdu; V 5 (t, z(t)) = 0 -τ t t+u żT (s)T 22 ż(s)dsdu, V 6 (t, z(t)) = n ∑ j=1 q j k j ∞ 0 k j (u) t t-u g 2 j (z j (s))ds du.
By using the LKF [START_REF] Li | A new model transformation of discrete-time systems with time-varying delay and its application to stability analysis[END_REF] and similar arguments to the ones of Theorem 4.1, we obtain

V0 (t, z(t)) ≤ -ζ T 0 (t, z(t)) Λζ 0 (t, z(t)) + αV 0 (t, z(t))
where

ζ 0 (t, z(t)) = z T (t), żT (t), żT (t -h(t)), z T (t -τ(t)), g T (z(t)), g T (z(t -τ(t))), t -∞ K(t -s)g(z(s))ds T T and Λ =           Λ 11 Λ 12 Λ 13 -Π 14 -Π 17 -Π 18 -Π 19 * -Π 22 -Π 23 0 0 -Π 28 -Π 29 * * -Π 33 0 0 -Π 38 -Π 39 * * * -Π 44 0 -Π 48 0 * * * * -Π 77 0 0 * * * * * -Π 88 0 * * * * * * -Π 99           with Λ 11 = PC +CP +U 1 Σ 1 + αP, Λ 12 = Q 1 C Λ 13 = -PD -CQ T 2 D Clearly, if Λ > 0 ( 30
)
then we have V0 (t, z(t)) ≤ αV 0 (t, z(t)).

By using Schur complements, (30) can be expressed as

cR -Γ > 0 ( 32
)
where

R =         Λ22 Λ23 0 0 Λ28 Λ29 * Λ33 0 0 Λ38 Λ39 * * Λ44 0 Λ48 0 * * * Λ77 0 0 * * * * Λ88 0 * * * * * Λ99         with c > 0, Qi , > 0, i = 1, 2, 5, 7, Ǔ1 > 0, Ǔ2 > 0, Q 1 = c Q1 , Q 2 = c Q2 , Q 5 = c Q5 , Q 6 = c Q6 Q 7 = c Q7 , U 1 = c Ǔ1 , U 2 = c Ǔ2 , T 11 = c Ť11 , T 12 = c Ť12 , T 22 = c Ť22 , Λ22 = -τ Ť22 -Q6 + Q1 + QT 1 , Λ23 = -Q1 D -QT 2 D, Λ28 = -Q1 Ã, Λ29 = -Q1 B, Λ33 = Q6 (1 -h * ) + D T Q2 D + D T QT 2 D, Λ38 = -D T Q2 Ã, Λ39 = D T Q2 B, Λ44 = -τ Ť11 + Ť12 + Ť T 12 + Ǔ2 Σ 1 , Λ48 = -Ǔ2 Σ 2 , Λ77 = -Q5 κ + Ǔ1 , Λ88 = Ǔ2 , Λ99 = Q5 . and Γ =         Λ T 12 -Π T 13 -Π T 14 -Π T 17 -Π T 18 -Π T 19         (Λ 11 ) -1 Λ 12 -Π 13 -Π 14 -Π 17 -Π 18 -Π 19
Since Λ 11 > 0 then it leads to Γ > 0 and consequently R > 1 c Λ > 0 from condition [START_REF] Huang | State estimation of recurrent neural networks with time-varying delay: A novel delay partition approach[END_REF]. Therefore ( 31) is satisfied by taking c > λ max (T ) λ max (R) . Thus, condition [START_REF] Huang | State estimation of recurrent neural networks with time-varying delay: A novel delay partition approach[END_REF] implies that V0 (t, z(t)) ≤ αV 0 (t, z(t)) and consequently V (t, z(t)) ≤ e αT V (z(0)).

Furthermore, we have

x(0) 2 ≤ 1 λ min (P) x T (0)Px(0) ≤ 1 λ min (P) x T (0)R 1 2 R -1 2 PR -1 2 R 1 2 x(0) ≤ λ max ( P) λ min (P) x T (0)Rx(0) Therefore V (0) ≤ γc 1 ( 33
)
where

γ = λ max ( P) λ min (P) 2λ max (P) + hλ max (Q 6 ) + τ 2 λ max (T 22 ) + n ∑ j=1 q j k j max j M 2 j ∞ 0 uk j (u)du . ( 34
)
and

x T (t)Rx(t) ≤ 1 λ min ( P) x T (t)Px(t) ≤ 1 λ min ( P) V (t, x(t)) .
So conditions ( 28) and [START_REF] Yan | Finite-time stability and stabilization of itô stochastic systems with Markovian switching: mode-dependent parameter approach[END_REF] imply that x T (t)Rx(t) < c 2 which achieves the proof.

Remark 4.4. In [START_REF] Cheng | Stochastic finite-time boundedness for Markovian jumping neural networks with time-varying delays[END_REF], the problem of FTB for Markovian jumping NNs with time-varying delays is studied by using the reciprocally convex combination technique where a double integral term appears in the LKF. Very recently, the authors of [START_REF] Rajavel | Finite-time nonfragile passivity control for neural networks with time-varying delay[END_REF] have used the same approach where a triple integral term is added in the LKF. However, these novel LKFs also lead to a greater complexity in terms of inequalities and variables to be calculated and this complexity can cause numerical problems with a large number of neurons ( [START_REF] Yan | Finite-time stability and stabilization of itô stochastic systems with Markovian switching: mode-dependent parameter approach[END_REF]). In our article we use a well know LKF [START_REF] Huang | Global exponential stability of neutral highorder stochastic Hopfield neural networks with Markovian jump parameters and mixed time delays[END_REF], but based on the Jensen's integral inequality the upper bound of the LKF is estimated more tightly and then the conservatism with respect to the delays is reduced (see Example 5.1).

Finite time Boundedness stabilization

In this section, sufficient conditions are given for solving the FTB-stabilization problem of a general class of NHOHNNs with time delay in the leakage term and mixed time delays of the form

                           ẋi (t) = -c i x i (t -σ ) + n ∑ j=1 a i j f j (x j (t -τ(t))) + n ∑ j=1 n ∑ k=1 T i jk f k (x k (t -τ(t))) f j (x j (t -τ(t))) + n ∑ j=1 b i j t -∞ k j (t -s) f j (x j (s)) ds + n ∑ j=1 d i j ẋ j (t -h(t)) + u, t > 0, i = 1, . . . , n x(s) = φ (s), s ∈ (-∞, 0] ( 35
)
where u is the control variable. The following state feedback control is considered

u (x(t)) = K 2 x(t) ( 36
)
where K 2 is to be determined. The corresponding closed-loop z-system is given by

     ż(t) = -Cz(t -σ ) + Ag(z(t -τ(t)) + Γ T T * g (z(t -τ(t))) +B t -∞ K(t -s)g (z(s)) ds + Dż (t -h(t)) K 2 z(t) z(s) = φ (s) -x * , s ∈ (-∞, 0] (37)
We now introduce the definition of finite time bounded stabilization. Definition 4.6. ( [START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF]) The NNs described by System (35) is said to be FTB-stabilizable w.r.t. (c 1 , c 2 , R, T ) if there exists a controller u (z(t)) of the form [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF] such that the corresponding closed-loop z-System (37) is FTB with respect to (c 1 , c 2 , R, T ).

We can now state the main result of this section.

Ψ =              Ψ 11 Ψ 12 Ψ 13 0 Π 15 Ψ 16 Π 17 Π 18 Π 19 * Π 22 Π 23 0 Π 25 0 0 Π 28 Π 29 * * Π 33 0 Π 35 Π 36 0 Π 38 Π 39 * * * Π 44 0 0 0 Π 48 Π 49 * * * * Π 55 0 0 0 0 * * * * * Π 66 0 Π 68 Π 69 * * * * * * Π 77 0 0 * * * * * * * Π 88 0 * * * * * * * * Π 99              < 0 (38)
and

c 2 e -αT √ c 1 √ c 1 ω -1 1 > 0 ( 39
)
where

C = C -K 2 , Ψ 11 = -PC -CP + PK 2 + K T 2 P + Q 3 + σ 2 Q 4 -U 1 Σ 1 -αP, Ψ 12 = -Q 1 K 2 , Ψ 13 = PD + D T Q 2 K 2 , Ψ 16 = CP C + αCP,
PROOF. According to Definition 4.6, if we apply Theorem 4.1 to the closed-loop System (37) then we can easily obtain the result. The details of the proof is left to the reader.

When σ = 0, Theorem 4.7 cannot be directly used to design a feedback control of the form of [START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF]. To overcome this obstacle, the following proposition is established. such that the following conditions hold:

Ξ 1 =           Ξ 11 Ξ 12 Ξ 13 Ξ 14 Ξ 15 Ξ 16 Ξ 17 * Ξ 22 Ξ 23 0 0 Ξ 26 Ξ 27 * * Ξ 33 0 0 Ξ 36 Ξ 37 * * * Ξ 44 0 Ξ 46 0 * * * * Ξ 55 0 0 * * * * * Ξ 66 0 * * * * * * Ξ 77           < 0 (40) and c 2 e -αT √ c 1 √ c 1 ω-1 2 > 0 (41)
where

Ξ 11 = -CX -XC +Y +Y T -Ũ1 Σ 1 -αX, Ξ 12 = XC +Y, Ξ 13 = D + D T CX + D T Y, Ξ 14 = T T 12 , Ξ 15 = Ũ1 Σ 2 , Ξ 16 = ÃX, Ξ 17 = BX, Ξ 22 = τT 22 + Q 6 -2I, Ξ 23 = XD + D T X, , Ξ 26 = X Ã, Ξ 27 = XB, Ξ 33 = -Q 6 (1 -h * ) -2D T D, Ξ 36 = -XD T Ã, Ξ 37 = -XD T B, Ξ 44 = τ T11 -T12 -T T 12 -Ũ2 Σ 1 , Ξ 46 = Ũ2 Σ 2 , Ξ 55 = Q5 κ -Ũ1 , Ξ 66 = -Ũ2 , Ξ 77 = -Q5 . and ω2 = 1 λ min ( X) 2λ max (X -1 ) + hλ max (Q 6 ) + τ 2 λ max (T 22 ) + n ∑ j=1 ρ -1 j q j ρ -1 j k j max j M 2 j ∞ 0 u k j (u)du with Q5 = diag( q1 , . . . , qn ), X = diag(ρ 1 , . . . , ρ n ), X = R -1 2 X -1 R -1 2 .
PROOF. Let Q 1 = Q 2 = I. Now, we make some transformations for the above inequalities ( 40)- [START_REF] Zhou | Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations[END_REF]. Pre and post-multiplying (40) by diag(X -1 , I, I, X -1 , . . . , X -1 ) and letting

X = P -1 , Y = KX, Q5 = P -1 Q 5 P -1 ; Ũ1 = P -1 U 1 P -1 , Ũ2 = P -1 U 2 P -1 , T12 = P -1 T 12 P -1 . ( 42
)
we obtain Λ > 0. That is, [START_REF] Zhao | Finite-time boundedness analysis of memristive neural network with time-varying delay[END_REF] implies Condition [START_REF] Huang | State estimation of recurrent neural networks with time-varying delay: A novel delay partition approach[END_REF]. Then, by replacing [START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF] in Condition [START_REF] Zhou | Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations[END_REF] we obtain easily [START_REF] Lakshmanan | Delay-dependent state estimation of neural networks with mixed timevarying delays[END_REF]. Therefore, the conditions of Corollary 4.5 are obtained which achieves the proof.

Remark 4.5. Based on the inequality S -1 ≥ 2ξ Iξ 2 S (S being a non-singular matrix), the following LMI conditions

-λ 1 I I * -X i < 0, X i < λ 2 I, Q5 < 2ξ 1 I -ξ 2 1 λ 3 I -(2ξ 3 I -ξ 2 3 λ 2 )π 1 c 2 e -αT + c 1 2λ 1 + hλ 6 + τ 2 λ 7 √ c 1 r 3 λ 1 * -λ 3 . < 0
where

π 1 = λ min (R -1 ), r 3 = n ∑ j=1 k j max j M 2 j ∞
0 uk j (u)du, ξ 1 , ξ 3 are adjustable parameters and λ i , i = 1, 2, 3, 6, 7 are unknown positive scalars, ensure Condition (41) [START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF].

When σ = 0, the following proposition can be applied directly to design a feedback control which is able to FTB-stabilize the NHOHNNs. Proposition 4.9. Under the assumptions (H1) -(H2) -(H3), if there exist a positive scalar α, three positive symmetric definite matrices P, Q 3 , Q 4 , five n × n diagonal matrices X, U 1 > 0, U 2 > 0, Q 5 > 0, Q 6 > 0, and a 2n × 2n matrix

Q 7 = T 11 T 12 * T 22 > 0
such that the following conditions hold:

Θ =              Θ 11 X Π
             < 0 (43)
and

c 2 e -αT √ c 1 √ c 1 ω -1 1 > 0 ( 44
)
where

Θ 11 = -PC -CP + Q 3 + σ 2 Q 4 -U 1 Σ 1 -αP + 2X, Θ 16 = CPC + αCP -CX, Θ 22 = τT 22 + Q 6 -2P, Θ 23 = PD, Θ 25 = -PC, Θ 28 = P Ã, Θ 29 = PB, Θ 33 = -Q 6 (1 -h * )
and other parameters are the same as in Theorem 4.1, then the feedback control u 2 (x(t)) = P -1 Xx(t) FTB-stabilize System (35) w.r.t. (c 1 , c 2 , R, T ).

PROOF. Let Q 1 = P and Q 2 = 0. The proof of Proposition 4.9 is similar to the one of Theorem 4.7 so it is omitted here.

Remark 4.6. In the results presented in [START_REF] Wang | Extended finite-time H ∞ control for uncertain switched linear neutral systems with time-varying delays[END_REF][START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF][START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF][START_REF] Ali | Delay-dependent stability criteria of uncertain Markovian jump neural networks with discrete interval and distributed time-varying delays[END_REF][START_REF] Wang | New results on robust finite-time boundedness of uncertain switched neural networks with time-varying delays[END_REF], the feedback control given for ensuring the FTB-stabilization of the NNs cannot be designed with a leakage delay. When σ = 0, the conditions established in these works are not LMI conditions which renders the control algorithm more complicated.

To overcome these difficulties, the matrix gain should be of the following form K = P -1 X. In this case, it is possible to find LMI conditions ensuring the FTB-stabilization of NNs of the above mentioned works, even if there is a leakage delay.

Numerical examples

In this section, three numerical examples are presented to show the effectiveness of the results.

Example 1

Consider System (1) with n = 2 and

f 1 (s) = f 2 (s) = tanh(s), τ(t) = 0.2 -0.1 cost, h(t) = σ = 0.1, k 1 (s) = k 2 (s) = e -s , J = (1, 2)

It leads to

τ = 0.3, h * = 0, M - j = 0, M + j = 1, k j = 1 for j = 1, 2. Note that C -A + M -B + K + M = 3.2 -0.38 -1 2.57
is a M-matrix. By using Theorem 3.1, we know that System (1) has an equilibrium point.

For c 1 = 0.35, T = 5 and R = I, solving ( 6)-( 7) with the Matlab LMI toolbox [START_REF] Lofberg | Yalmip : a toolbox for modeling and optimization in matlab[END_REF] The minimum value of c 2 satisfies min c 2 > 1.8157 for α = 0.02. Moreover, we see on Figure 2 that the considered system is not FTB w.r.t. (0.35, 0.9, I, 100). Finally for α = 0.01, Figure 4 shows the minimum of c 2 with respect to (T, c 1 ) for having the FTB. h * = 0 and σ = 0 Methods τ max [START_REF] Arbi | Stability analysis of delayed Hopfield neural networks with impulses via inequality tech-niques[END_REF] 0.87 [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF] 0.3527 [START_REF] Wang | Global exponential stability of high-order bidirectional associative memory (BAM) neural networks with time delays in leakage terms[END_REF] 0.4 Corollary 4.5 1.123 When σ = 0, the upper bound of the delay τ for which LMIs (6) of Theorem 4.1 remain feasible is τ max = 1.088. However, we obtain τ max = 1.123 if we use the simplified criterion given in Corollary 4.5.

Remark 5.1. The stability criterion given in [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF][START_REF] Arbi | Stability analysis of delayed Hopfield neural networks with impulses via inequality tech-niques[END_REF][START_REF] Sakthivel | Design of state estimator for bidirectional associative memory neural networks with leakage delays[END_REF][START_REF] Wang | Global exponential stability of high-order bidirectional associative memory (BAM) neural networks with time delays in leakage terms[END_REF] fails for small delays (τ max ≤ 0.87) and becomes infeasible when a leakage delay σ extends beyond 0.15. The method proposed in our work overcomes these difficulties by proving sufficient conditions that are able to ensure the FTB and even the asymptotic stability for a delay larger than the one given in [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF][START_REF] Arbi | Stability analysis of delayed Hopfield neural networks with impulses via inequality tech-niques[END_REF][START_REF] Wang | Global exponential stability of high-order bidirectional associative memory (BAM) neural networks with time delays in leakage terms[END_REF]. Table 1 Remark 5.2. The method used in our article improves and extends the results given in [START_REF] Li | Delay-dependent stability of neural networks of neutral type with time delay in the leakage term[END_REF][START_REF] Sakthivel | Design of state estimator for bidirectional associative memory neural networks with leakage delays[END_REF] by reducing the conservatism with respect to the delays and by simplifying the calculus simultaneously. This improvement is illustrated in Tables 3-4 and 5 by presenting a comparison of the computational load. Noted that FWM and DV stand respectively for the number of h * = 0 Methods τ max for σ = 0.15 τ max for σ = 0.2 [START_REF] Sakthivel | Design of state estimator for bidirectional associative memory neural networks with leakage delays[END_REF] Infeasible Infeasible [Theorem 4.1] 0.245 0.095 The state feedback control u 2 (x(t)) = K 2 x(t) with

K 2 = -0.1 0 0 -0.9
FTB-stabilize the system but it also asymptotically stabilize the system if we consider an infinite time evolution. The history of x T (t)x(t) is then illustrated in Figure 7. In order to FTB stabilize System [START_REF] Onori | Finite time stability design via feedback linearization[END_REF] with the above numerical values, we build the following control U (x(t)) = P -1 Xx(t) according to the strategy given in Subsection 4.2. Proposition 4.9 implies that the closed-loop System [START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF] Remark 5.3. It should be pointed out that the results in [START_REF] Wang | Extended finite-time H ∞ control for uncertain switched linear neutral systems with time-varying delays[END_REF][START_REF] Wu | Finite-time boundedness and stabilization of uncertain switched neural networks with timevarying delay[END_REF][START_REF] Ali | Robust finite-time H ∞ control for a class of uncertain switched neural networks of neutral-type with distributed time varying delays[END_REF][START_REF] Ali | Delay-dependent stability criteria of uncertain Markovian jump neural networks with discrete interval and distributed time-varying delays[END_REF][START_REF] Wang | New results on robust finite-time boundedness of uncertain switched neural networks with time-varying delays[END_REF] fail for System (1) with the time-varying delay (45) even without leakage delay because [START_REF] Lin | Finite-time boundedness and L 2 -gain analysis for switched delay systems with norm-bounded disturbance[END_REF] is not differentiable. This proves the advantage of the proposed approach in our article.

Conclusion

This article deals with the problem of finite time boundedness and finite time boundedness stabilization of a general class of neutral high-order Hopfield neural networks with time delay in the leakage term and mixed time delays. By using the topological degree theory, sufficient conditions are given to prove the existence of equilibrium points. The Lyapunov-Krasovskii functional method and the LMIs technique are used to establish some sufficient conditions which ensure the finite time boundedness and finite time boundedness stabilization of the class of systems considered in our article. Finally, numerical examples are presented to show the effectiveness and the interest of our proposed results.

AppendixA. The calculus of V (t) System (1) has an equivalent form given by

Lemma 2 . 5 .

 25 ([15]) Given any real matrix M = M T > 0 of appropriate dimension and a vector field ω : [a, b] → R n such that the integrations concerned are well defined, then we have

Theorem 3 . 1 .

 31 Under assumptions (H1) -(H2) -(H3), System (1) has at least one equilibrium point. PROOF. If x * = (x * 1 , . . . , x * n ) T denotes an equilibrium point of System (1), then x * satisfies for all i = 1, . . . , n

Corollary 4 . 3 .

 43 Under the assumptions and notations of Theorem 4.1, System (20) is FTB w.r.t. (c 1 , c 2 , R, T ) if the following conditions hold:

Corollary 4 . 4 .

 44 Under the assumptions and notations of Theorem 4.1 , System (24) is FTB w.r.t. (c 1 , c 2 , R, T ) if the following conditions hold:

Corollary 4 . 5 .

 45 Under the assumptions and notations of Theorem 4.1, System (1) with σ = 0 is FTB w.r.t. (c 1 , c 2 , T, R) if the following conditions hold:

Theorem 4 . 7 .

 47 Under the assumptions and notations of Theorem 4.1, System (35) is FTB-stabilizable w.r.t. (c 1 , c 2 , R, T ) if the following conditions hold:

Proposition 4 . 8 .

 48 Under assumptions (H 1 ) -(H 3 ), System (35) without leakage delay is FTB-stabilizable w.r.t. (c 1 , c 2 , T, R) if there exist a positive scalar α, three n × n matrices Q1 , Q2 , Y , five n × n positive diagonal matrices X, Ũ1 , Ũ2 , Q5 , Q 6 and a 2n × 2n matrix Q7 = T11 T12 * T 22 > 0

  Theorem 4.1 leads to the FTB of the considered system w.r.t. (0.35, 3, I, 5). Time history of x T (t)x(t) is illustrated on Fig 1.

Figure 1 :Figure 2 :

 12 Figure 1: Time history of x T (t)x(t) with initial condition (0.305, 0.5) T

Figure 3 :

 3 Figure 3: max c 1 with respect to (T, c 2 )

Figure 4 :

 4 Figure 4: min c 2 with respect to (T, c 1 )

Figure 7 : 2 5

 72 Figure7: The history of x T (t)x(t) for System[START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF] with initial condition (0.1, 0.1) T under controller[START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF] with K = K 2

Figure 8 :

 8 Figure8: The history of x T (t)x(t) for System[START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF] without controller[START_REF] Kamenkov | On stability of motion over a finite interval of time[END_REF]

Figure 10 :

 10 Figure 10: The history of x T (t)x(t) for System (37) under time-varying delay (45).

  T , and parameters C, A, T 1 , T 2 , B and D given as follows:

	C =	4 0 0 4	, A =	0.1 -0.8 -0.93 0.18	, T 1 =	0.2 0.07 0.03 0.01	,
	T 2 =	0.1 0.05 0.04 0.02	, B =	0.7 -0.2 -0.2 0.5	, D =	0.1 0 0 0.1	.

  with α = 0.02 leads to the following solutions

	P =	128.0037 19.7975 19.7975 62.5951	, Q 1 =	10.08 3.3704 3.3704 6.3560
	Q 2 =	16.7695 1.7013 1.7013 7.4374	, Q 3 =	220.9824 57.9710 57.9710 131.3136
	Q 4 =	7.3255 1.2523 1.2523 5.1806	, Q 5 =	110.8764 0	0 65.8265
	Q 6 =	2.0464 0	0 0.8882	, U 1 =	232.5544 0	0 156.5080
	U 2 =	65.8755 0	0 58.9563	, T 11 =	112.1715 54.9582 54.9582 85.9251
	T 12 =	45.4573 20.3443 20.3443 31.9253	, T 22 =	19.7736 8.8448 8.8448 13.8120

Table 1 :

 1 The maximum allowable bounds τ max for h

* = 0 and σ = 0

  and Table 2 compare the maximum allowable bounds τ max of τ(t) derived from Theorem 4.1, Corollary 4.5 and [15, 73, 75, 74].

Table 2 :

 2 The maximum allowable bounds of τ max for h * = 0

		σ = 0
	Methods	FWM	DV	τ max
	[15]	2	4n 2 + 4n	0.3527
	[Corollary 4.5]	2	3n

2 + 4n + 1 1.123

Table 3 :

 3 A comparison of computational load when σ = 0

		σ = 0.15
	Methods	FWM	DV	τ max
	[74]	2	28n 2 + 16n	Infeasible
	[Theorem 4.1]	2	8.5n 2 + 4n + 1	0.245

Table 4 :

 4 A comparison of computational load when σ = 0.15 free-weight matrices and the number of decision variables used.

	C =	1 0 0 0.2

5.2. Example 2

Consider System (1) with n = 2, and other parameters similar to Subsection 5.1. By setting c 1 = 6.5, T = 5 and R = I, we have that System (1) is FTB w.r.t. (6.5, 1500, I, 5) but is not FTB w.r.t. (6.5, 1500, I, 10). Moreover, we have |x i | → +∞, i = 1, 2 when t → +∞ as shown if Figure

5

which proves that this system is not asymptotically stable.

Table 5 :

 5 The maximum allowable bounds of τ(t) when σ = 0.2 So, we have shown that FTB and Lyapunov asymptotic stability are two independent concepts. Now we consider the controlled System[START_REF] Onori | Finite time stability design via feedback linearization[END_REF] with the above numerical values. In order to FTB-stabilize System[START_REF] Onori | Finite time stability design via feedback linearization[END_REF], we consider the feedback control according to the method proposed in Theorem 4.7. The feedback control is given by u (x(t)) =

	K 1 x(t), with K 1 =	-0.1 0	0 0.042	. From Theorem 4.7, we de-
	duce that System (35) is FTB stabilizable w.r.t. (6.5, 1500, I, 10).
	Figure 6 shows the history of x T (t)x(t).
	1500
					x T (t)x(t)
	1000
	x T (t)x(t)
	500
	0
	0	2	4	6	8	10
			Time (s)

Figure 6: The history of x T (t)x(t) for System (35) with initial condition (0.1, 0.1) T under controller (36) with K = K 1

  is FTB-stabilizable w.r.t. (c 1 , c 2 , T, I) for α = 0.02, with the following solutions The history of x T (t)Rx(t) with the previous controller u (x(t)) = P -1 Xx(t) is illustrated on Figure9. Figure 9: The history of x T (t)x(t) for System (37)under controller P -1 Xx(t).Now, if the time-varying delay τ(t) defined by the non-differentiable function is as follows:τ(t) = 0.3 -0.1 sint i f t ∈ I = ∪ k≥0 [2kπ, (2k + 1)π] ; 0 i f t ∈ R + \I.(45)System[START_REF] Onori | Finite time stability design via feedback linearization[END_REF] stays FTB-stabilizable w.r.t. (c 1 , c 2 , T, I) which is illustrated in Figure10.

		2						x T (t)x(t)
		1.8
		1.6
		1.4
	x T (t)x(t)	1 1.2
		0.8
		0.6
		0.4
		0.2
		0	2		4	6		8		10
						Times (s)
		2
								x
		1.8
		1.6
		1.4
	x T (t)x(t)	1 1.2
		0.6 0.8									P =	6.5871 0	0 4.3602	, Q 3 =	-0.1380 0.2511 0.3573 -0.1380
		0.2 0.4									Q 4 =	4.8974 -0.5024 -0.5024 4.5153	, Q 5 =	0 6.1356	3.5997 0
		0	1	2	3	4 Time (s) 5	6	7	8	9	Q 6 =	0.5134 0	0 0.3434	,U 1 =	0 12.0818	7.2642 0
											U 2 =	11.2816 0	0 8.7200	, T 11 =	4.6319 0.5660 0.5660 3.5918
											T 12 =	4.0306 1.0331 1.0331 3.1048	, T 22 =	3.7924 1.3756 1.3756 2.8913
											X =	-14.7082 0	0 -5.5260
											and the controller u (x(t)) = P -1 Xx(t) with
												P -1 X =	-2.2329 0	0 -1.2674	.

T (t)x(t)

  -C t t-σ z(u)du = -Cz(t) + (A + Γ T T * )g(z(tτ(t))) +B t -∞ K(ts)g(z(s))ds + Dż(th(t)) z(s) = φ (s)x * -Cz(t) + (A + Γ T T * )g(z(t -(τ(t))) s)g(z(s))ds = -2z T (t)PCz(t) + 2z T (t)P(A + Γ T T * )g(z(tτ(t))) s)g(z(s))ds + 2z T (t)PDż(th(t))

	         dt z(t) Moreover, we have d
	V1 (t, z(t)) = 2 z(t) -C	t	z(u)du
			t-σ
				t
	+ Dż(t -h(t)) + B t K(t + 2z T (t)PB -∞ -∞ t K(t + 2z T (t)CPC z(u)du -2	t	z(u)du	T	CPDż(t -h(t))
			t-σ			t-σ
		t
	-2	z(u)du
		t-σ
		t	T	t
	-2	z(u)du	CPB	K(t -s)g(z(s))ds	(A.1)
		t-σ		-∞
						t t-σ	z T (u)Q 4 z(u)du
	≤ σ 2 z T (t)Q 4 z(t) -		t t-σ	z(u)du	T	Q 4	t t-σ	z(u)du
								(A.3)
	and
	V4 (t, z(t)) =	t t-τ(t)	g(z(t -τ(t))) ż(s)	T T 11 T 12 * T 22	g(z(t -τ(t))) ż(s)	ds
	= τ(t)z T (t -τ(t))T 11 z(t -τ(t)) + 2z T (t)T T 12 z(t -τ(t))
	-2z T (t -τ(t))T T 12 z(t -τ(t)) +	t t-τ(t)	żT (s)T 22 ż(s)ds
	≤ z

T P T CP(A + Γ T T * )g(z(t -(τ(t))) and V2 (t, z(t)) = z T (t)Q 3 z(t)z T (tσ )Q 3 z(tσ ) + żT (t)Q 6 ż(t) -żT (th(t))Q 6 ż(th(t))(1 -ḣ(t)) ≤ z T (t)Q 3 z(t)z T (tσ )Q 3 z(tσ ) + żT (t)Q 6 ż(t) -żT (th(t))Q 6 ż(th(t))(1h * ) (A.2)

It follows from Lemma 2.5 that

V3 (t, z(t)) = σ 2 z T (t)Q 4 z(t)σ T (tτ(t))[τT 11 -2T T 12 ]z(tτ(t))] + 2z T (t)T T 12 z(tτ(t)) + t t-τ

żT (s)T 22 ż(s)ds (A.4)

and

In addition, we note that

By using (H1), the following inequality holds

for any n × n diagonal matrices U 1 > 0, U 2 > 0. By using (A.1)-(A.9), we finally get V (t, z(t)) ≤ ζ T (t, z(t)) Ξζ (t, z(t)).


Dat-Duong Phan

Emmanuel Moulay

Patrick Coirault

Frédéric Launay

Pierre Combeau

Potential feedback control for the power control in LTE

Keywords: Potential feedback control, Lyapunov function, discrete-time systems, input delay, LTE

This article deals with the study of uplink (from the mobile phone to the base station) transmitted power control in Long Term Evolution (LTE). It provides a nonlinear Potential Feedback Control (PFC) for SISO discrete-time systems with input delays and disturbances by using the Lyapunov theory and the Artstein transform. It is an original approach in which a stabilization problem of a linear SISO system with a constraint on the state space is transformed into a problem of nonlinear stabilization. We use this strategy of stabilization for the uplink transmission power control in LTE.

I. INTRODUCTION

The uplink power control is an important task for mobile phone communication systems. It improves the Quality of Service (QoS) in mobile telephony, and more specifically for the new communication standard LTE. On one hand, it sets up a power controller on mobile phones in order to ensure that the Signal to Interference plus Noise Ratio (SINR) stays above a limit below which the communication is interrupted. On the other hand, it consists of minimizing the overall transmitted power in a cell in order to minimize the interferences between users and to maximize the battery life of mobile phones.

Since several years, the interest in wireless cellular networks has stimulated the investigation of several strategies to solve the uplink power control problem [START_REF] Charalambous | Power control in wireless ad hoc networks: Stability and convergence under uncertainties[END_REF], [START_REF] Frantti | Adaptive fuzzy power control for WCDMA mobile radio systems[END_REF], [START_REF] Gunnarsson | Control theory aspects of power control in UMTS[END_REF], [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF], [START_REF] Shoarinejad | A distributed scheme for integrated predictive dynamic channel and power allocation in cellular radio networks[END_REF], [START_REF] Subramanian | A robust power and rate control method for state-delayed wireless networks[END_REF], [START_REF] Xiao | A utility-based powercontrol scheme in wireless cellular systems[END_REF]. The Transmission Power Control (TPC) strategy, which is an algorithm, has been developed for the new LTE standard [START_REF] Muhammad | Uplink closed loop power control for LTE system[END_REF], [START_REF] Mullner | Contrasting open-loop and closed-loop power control performance in UTRAN LTE uplink by UE trace analysis[END_REF], [START_REF] Song | Evolved cellular network planning and optimization for UMTS and LTE[END_REF], [START_REF]Physical layer procedures, 3GPP (Third Generation Parternship Program)[END_REF]. Moreover, the uplink power control in LTE can be modeled as a problem of stabilization of an input delay discrete-time system with a constraint on the state space. So, a suitable strategy for the power control in LTE is the Model Predictive Control (MPC) for a discrete-time system taking into account a constraint on the state space variable [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], [START_REF] Gutman | A linear programming regulator applied to hydroelectric reservoir level control[END_REF]. In this article, we provide a new simple feedback control strategy for a SISO system as an alternative solution to the TPC and MPC strategies. It is an original approach in which the stabilization problem of a linear SISO system with a constraint on the state space is transformed into a problem of nonlinear stabilization involving a potential controller. The potential control has essentially been used in robotics to D. Phan and P. Coirault and F. Launay are with LIAS (EA 6315), École Nationale Supérieure d'Ingénieurs de Poitiers, France (email: dat.duong.phan@univ-poitiers.fr; patrick.coirault@univ-poitiers.fr; frederic.launay@univ-poitiers.fr) E. Moulay and P. Combeau are with XLIM-SIC (UMR CNRS 7252), Université de Poitiers, France (email: emmanuel.moulay@univ-poitiers.fr, pierre.combeau@univ-poitiers.fr) avoid collisions with obstacles [START_REF] Ogren | Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment[END_REF], [START_REF] Rimon | Exact robot navigation using artificial potential functions[END_REF]. To the best of our knowledge, this is the first time that a PFC is used for a discrete-time system. The term "potential" is used to refer to the Coulomb potential which is an impassable barrier. A PFC is a nonlinear feedback control ensuring that the system output is always higher than a lower limit. Applying this approach in LTE, it leads to the easy adjustment of two parameters for each mobile phone. The main difficulty of this strategy of stabilization is due to the nonlinearity of the potential term which implies the use of the Lyapunov theory. In order to take into account the mobility of mobile phones, an input delay which denotes the propagation time and computation time is added. So, we use the Artstein transform for discretetime systems to deal with this input delay [START_REF] Ionete | Controllability and observability of input/output delayed discrete systems[END_REF], [START_REF] Nilsson | Real-time control systems with delays[END_REF].

The paper is organized as follows. The uplink power control in LTE is described in section II. Then, the PFC for discretetime systems is developed in section III. It is applied and simulated for a cellular network in LTE in section IV and a comparison with the TPC and MPC strategies is given. Finally, a conclusion is addressed in section V.

II. UPLINK POWER CONTROL IN LTE

A. Modeling of a transmission channel

In this article, we are interested in the uplink channel between a mobile phone denoted by k and a base station of the cell denoted by i. In urban environments, the channel gain of a radio link can be modeled by three components: path-loss, log-normal shadowing, and multi-path fading. The gain of the channel is defined as

g ik (n) = γ t γ r λ 2 (4π) 2 d 2 ik • 10 0.1 δ(n) • |r(n)| 2 (1)
where n is the sample time. The term γtγrλ 2

(4π) 2 d 2 ik
is the far field gain with γ t the transmitting antenna gain, γ r the receiving antenna gain, d ik the distance between the transmitter of the cell i and the receiver k in meters, the system loss factor not related to propagation ( ≥ 1) and λ the wavelength in meters [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF]. The term 10 0.1 δ(n) is used to model large-scale lognormal shadowing where δ(n) is a normal Gaussian random variable [START_REF] Yang | OFDMA system analysis and design[END_REF]. The term |r(n)| 2 is used to model Rayleigh fading where r(n) is the random variable that has a Rayleigh distribution [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF].

The received SINR is the ratio of the received signal power to the power of the interference plus noise within the bandwidth of the transmitted signal. The SINR is commonly used in wireless communication as a way to measure the quality of transmission. In LTE, the use of Orthogonal Frequency Division Multiple Access (OFDMA) and Single Carrier Frequency Division Multiple Access (SC-FDMA) avoids the intra-cell interferences [START_REF] Khan | LTE for 4G mobile broadband: air interface technologies and performance[END_REF], [START_REF] Song | Evolved cellular network planning and optimization for UMTS and LTE[END_REF]. Therefore, after frequency synchronization, only the inter-cell interferences affect the SINR in the model.

Notation 1: The lower case is used for the linear domain and the upper case for the logarithm domain in dB of the parameters. For instance, x ik stands for the SINR in linear domain and X ik stands for the SINR in the logarithm domain and we have X ik = 10 log 10 x ik .

The SINR of the mobile phone k at the base station i is given by

x ik (n) = g ik (n) • p ik (n) =k g i (n) • p i (n) + σ 2 ik (n) (2)
where p i and g i are respectively the transmitted power and channel gain from the mobile phone belonging to the neighboring base stations of i, σ 2 ik the thermal noise affecting the channel between the mobile phone k and the base station of the cell i. The term =k g i (n) • p i (n) corresponds to the inter-cell interferences. As it was explained above, the intracell interferences in LTE are negligible. So, we can rewrite (2) as follows

x ik (n) = g ik (n) • p ik (n) i ik (n) + σ 2 ik (n) (3)
where i ik = =k g i (n) • p i (n) are the inter-cell interferences.

B. Uplink PUSCH power control

A presentation of the LTE standard and the LTE physical layer is given in [START_REF] Stefania Sesia | LTE -The UMTS Long Term Evolution From Theory to Practice[END_REF]. There are several kinds of uplink physical channels such as: -the Physical Uplink Control Channel (PUCCH) which is used to support the transmission of acknowledgment, scheduling requests, the Channel Quality Indicator (CQI) and the Sounding Reference Symbol (SRS) [START_REF]Physical layer procedures, 3GPP (Third Generation Parternship Program)[END_REF]Section 5.1.3] which is used at the base station for the channel estimation; -the Physical Uplink Shared Channel (PUSCH) which is the main physical channel used for the data transmission and paging information.

We are interested in the power control of the PUSCH channel which is the main physical channel of the physical layer. In LTE, the power control is computed at the base station [START_REF]Physical layer procedures, 3GPP (Third Generation Parternship Program)[END_REF]Page 11].

The setting of the User Equipment (UE) (which can be for instance a mobile phone) transmission power for a PUSCH transmission is defined as follows [START_REF]Physical layer procedures, 3GPP (Third Generation Parternship Program)[END_REF] P P U SCH,c (j) = min(P CM AX,c , 10

log 10 M P U SCH,c (j) +P O,P U SCH,c (β) + α c (β) • P L c (j) +Θ T F,c (j) + F c (j)) (4
) where j is the number of the subframe, β = 0, 1, 2 a specific parameter in LTE, P CM AX,c the maximum transmit power for serving cell c, M P U SCH (j) the measured bandwidth in number of resource blocks valid for subframe j and serving cell c, P O,P U SCH,c (β) the parameter provided by higher layers which has a cell-specific and nominal part expressing the power to be contained in one physical resource block, α c (β) ∈ [0, 1] a cell-specific parameter allowing the compensation of partial pathloss and provided by higher layers, P L c the downlink pathloss estimate calculated at the UE for serving cell c, Θ T F,c the UE specific parameter depending on the chosen Modulation and Coding Scheme (MCS), F c (j) a current P U SCH power control adjustment state for serving cell c defined by

F c (j) = F c (j -1) + ∆ P U SCH (j -K P U SCH (j)) (5)
where ∆ P U SCH (j -K P U SCH (j)) is a correction value, also referred to as a Transmit Power Command (TPC) which was signaled on subframe (j -K P U SCH (j)), K P U SCH (j) is the downlink delay corresponding to the propagation time and the calculation time of the processor,

K P U SCH (j) ∈ {4, • • • , 7}.
P CM AX,c depends on the maximum allowed power configured by higher layers and the maximum UE power class.

In the remainder of this paper, we will replace the number of the subframe j by the time variable n because frames and subframes are sent successively in time. The indexes c, P U SCH and T F which are specific to the LTE standard will be omitted. For instance,

F c (i) = F c (i -1) + ∆ P U SCH (i -K P U SCH (i)) becomes F (n) = F (n -1) + ∆(n -K) (6)
In the linear domain, the SINR is defined by equation ( 3). Let us define the pathloss pl by pl ik

(n) = 1 g ik (n) and i o t ik (n) = i ik (n)+σ 2 ik (n) σ 2 ik (n)
. We can rewrite equation (3) as follows

x ik (n) = p ik (n) pl ik (n) • σ 2 ik (n) • i o t ik (n)
.

We obtain the equation of the SINR in the logarithm domain

X ik (n) = P ik (n) -P L ik (n) -Σ 2 ik (n) -I o T ik (n). (7)
In the linear domain, the interference term i ik and the thermal noise σ 2 ik are measured at the base station [START_REF] Kreher | LTE Signaling: Troubleshooting and Optimization[END_REF]. So, we have the terms Σ 2 ik (n) and I o T ik (n). The term P ik is known at the base station and the term P L ik is estimated with the SRS. Then, we have the SINR X ik at the base station by using [START_REF] Xiao | A utility-based powercontrol scheme in wireless cellular systems[END_REF]. Now, we develop a state space model for the uplink power control in LTE. In the following, and for simplicity, we will omit the indexes i and k in the equations and X(n) and P (n) stands for X ik (n) and P ik (n). The U stands for ∆ P U SCH defined in the previous section. Therefore, equation [START_REF] Subramanian | A robust power and rate control method for state-delayed wireless networks[END_REF] becomes

F (n) = F (n -1) + U (n -K) (8)
We use the z-transform from equation ( 8) and we obtain

F (z) = z -K 1 -z -1 U (z). (9)
In the sequel, and for clarity, we will omit the saturation term P CM AX given in ( 4) but it will be taken into account in the simulations of section IV with the use of a saturation bloc (see the figure 1). However, no elaborate strategy is provided to treat this saturation. From equation ( 4), we have

P (z) = W 1 (z) + F (z) = W 1 (z) + z -K 1 -z -1 U (z) (10)
where W 1 (z) = 10 log 10 M (z) + P 0 (z) + α(j) • P L(z) + Θ(z). As the round-trip delay M, which is the sum of the downlink and the uplink delays, can be considered constant (see [START_REF] Dahlman | 4G: LTE/LTE-Advanced for Mobile Broadband[END_REF], [START_REF] Motorola | Long term evolution (LTE): Overview of LTE air-interface[END_REF]), the uplink delay is equal to M -K with K ∈ {4, • • • , 7}. From equation ( 7) and taking into account the propagation delay, we approximate

X(z) = (W 2 (z) + P (z))z -(M -K) ( 11
)
where

W 2 (z) = -P L(z) -Σ 2 (z) -I o T (z).
We substitute equation [START_REF] Song | Evolved cellular network planning and optimization for UMTS and LTE[END_REF] in equation ( 11) and we obtain

X(z) = z -M 1 -z -1 U (z) + (W 1 (z) + W 2 (z))z -(M -K) . (12)
Multiplying both sides of the equation ( 12) by (1 -z -1 ), we get

(1-z -1 )X(z) = z -M U (z)+(W1(z)+W2(z))(1-z -1 )z -(M -K) .
Now, using the z-inverse transform we obtain the following discrete-time system

X(n) = X(n -1) + U (n -M ) + H(n -1) (13)
with the known input

H(n-1) = W 2 (n-(M -K))-W 2 (n- (M -K) -1) + W 1 (n -(M -K)) -W 1 (n -(M -K) -1)
and a constant delay M -1 on the control. It is worth noting that the values of W 1 and W 2 must be archived in a register to be used in equation ( 13), the values of K ∈ {4, • • • , 7} are known.

The round-trip delay is a constant in LTE M = 10ms including the calculation time of the processor [START_REF] Dahlman | 4G: LTE/LTE-Advanced for Mobile Broadband[END_REF], [START_REF] Motorola | Long term evolution (LTE): Overview of LTE air-interface[END_REF]. The delays on the exogenous known inputs of the system W 1 and W 2 are not critical because it only means that data are stored. Let Ĥ(n) be the estimation of the exogenous input such that H(n) -Ĥ(n) < . We rewrite (13) as follows

X(n) = X(n -1) + V (n -M ) -Ĥ(n -1) + H(n -1) where V (n -M ) = U (n -M ) + Ĥ(n -1) is a new virtual control variable. It leads to X(n) = X(n -1) + V (n -M ) + D(n -1) (14)
where D(n -1) = H(n -1) -Ĥ(n -1) is the disturbance on the exogenous input H(n -1).

Let us give the block diagram of the system in figure 1, including a receiver and a transmitter corresponding to a base station and a mobile phone.

III. PFC FOR DISCRETE-TIME SYSTEMS

The basic definitions of asymptotic stability and Lyapunov functions for discrete-time systems with disturbances are given in [25, Definitions 2.1 and 2.6]. The Lyapunov theorems for discrete-time systems used in this article are given in [START_REF] Astrom | Computer-controlled systems: theory and design[END_REF], [START_REF] Elaydi | An Introduction to Difference Equations[END_REF]. In this section, we develop our PFC for SISO systems with disturbances in order to take into account the constraint on the state variable.

Theorem 1: Consider the system given by

x(n + 1) = x(n) + u(n) + d(n), n ∈ N ( 15
)
where x(n) ∈ R is the state, u(n) ∈ R the input and |d(n)| ≤ a bounded disturbance. Let c ≤ 0, under the initial condition x(0) > c the system (15) with the feedback control

u(n) = k 1 (x(n) -c) + k2 x(n)-c (16)
with

-1 2 < k 1 < 0 and k 2 ≥ 2 2(2k1+1)
is asymptotically stable with respect to the attractive and invariant set

S = -+ √ 2 -4k1k2 -2k1 + c; -(k1-1)+ √ 2 (k1-1) 2 -4k 3 1 k2 2 + c . Moreover, we have x(n) ≥ -+ √ 2 -4k1k2 -2k1 + c for all n ∈ N * = N \ {0}.
Proof. We have two possible cases which depend on the sign of the initial condition x(0) of the system (15). First case: x(0) > 0. We take c = 0. Firstly, we seek the equilibrium points of the closed-loop system. With the following input u(n) = k 1 x(n) + k2

x(n) , the system (15) becomes

x(n + 1) = (k 1 + 1)x(n) + k 2 x(n) + d(n). (17)
The positive equilibrium interval is given by

S + eq = x > 0 : x = (k 1 + 1)x + k 2 x + d(n); |d(n)| ≤ . If x = 0 then we have k 1 x2 + d(n)x + k 2 = 0. If k 1 < 0 and k 2 > 0 then we have a positive root x + (n) = d(n)+ √ ∆ -2k1 where ∆ = d(n) 2 -4k 1 k 2 . So, we obtain S + eq = -+ √ 2 -4k1k2 -2k1 ; + + √ 2 -4k1k2 -2k1
.

We seek the minimum value of x(n + 1). From (17), we have

∂x(n + 1) ∂x(n) = (k 1 + 1)x(n) 2 -k 2 x(n) 2 (18)
for n ∈ N. For k 1 > -1, the roots of (18) are x(n) = ± k2 k1+1 . As x(0) > 0, we have x(n+1) ≥ 4 (k 1 + 1) k 2d for all n ∈ N. So, the interval of possible minimum values for x(n + 1) is given by

S min = [ 4(k 1 + 1)k 2 -; 4(k 1 + 1)k 2 + ].
The smallest value of S min is min S min = 4(k 1 + 1)k 2 -. We want that min S min > 0, therefore k 2 > 2 4(k1+1) . We impose min S min ≥ min S + eq under the conditions -1 < k 1 < 0, k 2 > 2 4(k1+1) and > 0. It leads to the following algebraic inequality

-(1 + 2k 1 ) + 2 -4k 1 k 2 + 2k 1 4k 2 (k 1 + 1) ≤ 0. (19)
A symbolic calculator gives the following sufficient conditions > 0, -1 2 < k 1 < 0, k 2 ≥ 2 2(2k1+1) on , k 1 and k 2 in order to satisfy the inequality [START_REF] Yang | OFDMA system analysis and design[END_REF]. Under these conditions, we have

x(n) ≥ -+ √ 2 -4k1k2 -2k1
for all n ∈ N * and a relevant minimum value for the attractive and invariant set. Now we have to prove the stability of the closed-loop system (15), [START_REF] Ionete | Controllability and observability of input/output delayed discrete systems[END_REF]. Let us consider the following Lyapunov function

V (n, x(n)) = x(n) -x + (n) for x(n) > max S + eq = + √ 2 -4k1k2 -2k1
and n ∈ N * . We have

∆V (n, x(n)) = V (n + 1, x(n + 1)) -V (n, x(n)) = k1x(n) 2 +(d(n)-x + (n+1)+x + (n))x(n)+k2 x(n)
.

It is easy to see that ∆V (n,

x(n)) < 0 if x(n) > x * (n) where x * (n) = (d(n) -x + (n + 1) + x + (n)) + √ δ with δ = (d(n) -x + (n + 1) + x + (n)) 2 -4k 1 k 2 .
We have

max n∈N * x * (n) = -(k 1 -1) + 2 (k 1 -1) 2 -4k 3 1 k 2 2
and this maximum value is reached for d

(n) = , x + (n) = + √ 2 -4k1k2 -2k1
, and

x + (n + 1) = -+ √ 2 -4k1k2 -2k1
. As the maximum value max n∈N * x * (n) is greater than max S + eq for - 1 2 < k 1 < 0, we define the attractive and invariant set by

S0 = -+ √ 2 -4k 1 k 2 -2k 1 ; -(k1 -1) + 2 (k1 -1) 2 -4k 3 1 k2 2
and we have ∆V (n, x(n)) < 0 for all x(n) > max S 0 .

To conclude, we have to prove that S 0 is invariant. Due to the previous calculus of S min , we have min x(n)∈S0

x(n + 1) ≥ min S 0 . Now, let us calculate max x(n)∈S0 x(n + 1) and check that max x(n)∈S0 x(n + 1) ≤ max S 0 . Due to the variations of the continuous function x(n + 1) with respect to the variable x(n) and the intermediate value theorem, we only have to prove that x(n + 1)| min S0 ≤ max S 0 and x(n + 1)| max S0 ≤ max S 0 . It is quite easy but lengthy to check both conditions and this is left to the reader.

By using the Lyapunov theorem for discrete systems given in [25, Theorem 1], we conclude that if > 0, -1 2 < k 1 < 0 and k 2 ≥ 2 2(2k1+1) then the system ( 15) is asymptotically stable with respect to the interval S 0 .

Second case: x(0) ≤ 0. From equations ( 15) and ( 16) we have

x(n + 1) = x(n) + k 1 (x(n) -c) + k 2 x(n) -c + d(n). ( 20
)
Subtracting c in both sides of the equation ( 20), we get

x(n+1)-c = x(n)-c+k 1 (x(n)-c)+ k 2 x(n) -c +d(n). (21)
Consider the following change of variable

ψ(n) = x(n) -c (22)
that leads to the new system

ψ(n + 1) = (k 1 + 1)ψ(n) + k 2 ψ(n) + d(n). ( 23
)
We take c < x(0), it leads to ψ(0) > 0. Therefore, we can apply the proof of the first case. We deduce that if c < x(0), > 0, -1 2 < k 1 < 0 and k 2 ≥ 2 2(2k1+1) then the system ( 15) is asymptotically stable with respect to the interval

S S = -+ √ 2 -4k1k2 -2k1 + c; -(k1-1)+ √ 2 (k1-1) 2 -4k 3 1 k2 2 + c (24) Remark 1: If the value of k 1 is close to -1
2 then max S is rather close to min S and we can ensure a good interval of convergence. If the value of k 1 is close to zero then max S is distant from min S and we cannot ensure a good interval of convergence. Moreover, a small value of k 2 satisfying the condition

x(n) ≥ -+ √ 2 -4k1k2 -2k1
+ c for all n ∈ N * ensures a small interval of convergence. The stability of the system (15) remains even if there are bounded disturbances which implies a certain robustness.

In order to take into account a constant delay on the input during the transmission in LTE, we develop a new result on the stabilization for SISO systems with disturbances and constant delay on the input involving a potential controller. The proof of this result is based on the discrete-time Artstein transform.

Theorem 2: Consider the system given by

x(n + 1) = x(n) + u(n -T d ) + d(n), n ∈ N (25)
where

x(n) ∈ R is the state, u(n) ∈ R the input, T d ∈ N *
the delay and |d(n)| ≤ a bounded disturbance. Let c ≤ 0, under the initial condition x(0) > c the system (25) with the feedback control

u(n) = k 1 (ξ(n) -c) + k 2 ξ(n) -c , n ∈ N (26)
with

u(n) = 0 for n < 0, -1 2 < k 1 < 0, k 2 ≥ 2 2(2k1+1) and ξ(n) = x(n)+ T d i=1 u(n-i)
, is asymptotically stable with respect to the attractive and invariant set

S d = [min S d ; max S d ] where min S d = -+ √ 2 -4k1k2 -2k1 + c -T d and max S d = -(k1-1)+ √ 2 (k1-1) 2 -4k 3 1 k2 2 + c + T d . Moreover, we have x(n) ≥ -+ √ 2 -4k1k2 -2k1 + c -T d for all n ≥ T d .
Proof. The discrete-time Artstein transform allows us to transform a linear discrete-time controlled system having a constant input delay into a system without delay. This transformation is presented in [START_REF] Ionete | Controllability and observability of input/output delayed discrete systems[END_REF], [START_REF] Nilsson | Real-time control systems with delays[END_REF] for MIMO discrete-time systems. Applying the Artstein transform to the system (25), we obtain the following system without delay

ξ(n + 1) = ξ(n) + u(n) + d(n) (27)
where

ξ(n) = x(n) + T d i=1 u(n -i). (28)
We have that the state ξ(.) of the system ( 27) is asymptotically stable with respect to the interval S with the feedback control ( 16) due to theorem 1. By using the Artstein transform, we deduce that x(.) is asymptotically stable with respect to a set S d we have to determine. From equation ( 27) we deduce [START_REF] Castellanos | Performance of uplink fractional power control in UTRAN LTE[END_REF] and from equations ( 28), ( 29) we obtain

T d i=1 u(n -i) = ξ(n) -ξ(n -T d ) - T d i=1 d(n -i + 1),
x(n) = ξ(n -T d ) + T d i=1 d(n -i + 1). ( 30
)
As d(.) is bounded by assumption and ξ(.) is bounded due to theorem 1, then the state x(.) is also bounded due to (30). Therefore, there exists a minimum value x min such that the state x(.) is always higher than x min . From equation (30), we deduce

x min = ξ min -T d (31)
where

ξ min = -+ √ 2 -4k1k2 -2k1
due to theorem 1. This minimum value x min is also the minimum value min S d of the attractive and invariant set of system [START_REF] Jiang | A converse Lyapunov theorem for discrete-time systems with disturbances[END_REF]. Moreover, it is easy to check that the maximum value max S d of the attractive and invariant set of system [START_REF] Jiang | A converse Lyapunov theorem for discrete-time systems with disturbances[END_REF] satisfies

max S d = max S + T d . (32)
So, we have found the attractive and invariant set S d . As we have ξ(n) ≥ ξ min for all n > 0, then we have x(n) ≥ x min for all n > T d . This concludes the proof.

IV. SIMULATION RESULTS

We now provide the simulations of the SINR in LTE for the modeling presented in section II. For the simulations, we suppose that there is an estimated error of 1, 1dB on the exogenous input H(n). The evaluation of this value depends on the quality of the channel gain estimator. In the figure 1, x min (z) is a minimal SINR corresponding to a highest acceptable Block Error Rate (BLER) for communication between transmitter and receiver. The figure 2 allows us to find the minimum value of the SINR denoted by SINR min . The maximum BLER given by the LTE standard is 10 -1 . For a given modulation rate r, the relation between the BLER and the SINR is illustrated by the figure 2. We suppose for the simulations that the SINR is equal to the SNR, as in [START_REF] Lei | Wireless link SNR mapping onto an indoor testbed[END_REF]. We choose the 64QAM modulation and r = 2/3 for our simulation. We find the value of SINR min = 14, 37dB. For the simulations, we suppose that the distance between the mobile phone and the base station varies between 18, 80m and 342, 67m, so the P L varies between 63, 93dB and 89, 17dB. We assume the values of P 0 (n) = -78dBm, M (n) = 50, α(j) = 0, 8 given in [START_REF] Castellanos | Performance of uplink fractional power control in UTRAN LTE[END_REF]. Moreover, we suppose that I o T (n) varies between 3dB and 7dB and Σ 2 (n) varies between -110dBm and -120dBm. The coherence time of the channel is given by

T c = 9 16πf 2 m = 0.423 f m (33)
where f m = v λ is the maximum Doppler shift with v the velocity [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF]. In LTE, the carrier frequency f c = c λ = 1.8GHz. The control sample time T e at which we refresh the feedback control cannot exceed the coherence time of the channel. We apply our feedback control ( 16) with a sample time of 50ms for a mobile phone which moves slowly with a velocity of 5km.h -1 in a cell having a radius of 500 meters. So, the input delay which is much lower than the sample time is neglected. Due to the theorem 1, we choose the value of k 1 = -0.49, c = 0 and we impose SINR min = min S. Then, the minimum value of k 2 , being an integer such as x(n) ≥ 14, 37dB for all n N * , is given by k 2 = 117. The figure 3a presents the evolution of the SINR of a mobile phone which moves with a velocity of 5km.h -1 with T e = 50ms. We can see in the figure 3a that the SINR is always higher than the SINR min and lower than the upper bound of the attractive and invariant set S in the steady-state with the PFC. So, the system is asymptotically stable with respect to S. The term k2

x(n-1) allows the SINR to be higher than the SINR min in one sample time. Now, suppose that the velocity of the mobile phone is 260km.h -1 then T c = 1ms. So, we chose the sample time T e = 1ms which is a classical sample time in LTE for the power control [START_REF] Khan | LTE for 4G mobile broadband: air interface technologies and performance[END_REF]. In order to apply the power control, we have to take into account the input delay which is then higher than the sample time. Due to the theorem 2, we have ξ min = x min + M -1 Te = 24.27dB. We chose the values of k 1 = -0.49, c = 0 and we impose SINR min = min S d . We deduce the value of k 2 = 316. The figures 3b, 3c correspond to the evolution of the SINR of a mobile phone which moves with a velocity of 260km.h -1 with T e = 1ms. We can see in the figure 3b that, by using the PFC (26), the system with input delay is above the SINR min after 9 steps which corresponds to M -1

Te and is asymptotically stable. A comparison with the TPC and the MPC (having a minimal prediction horizon of 1) and taking into account the same perturbation of 1, 1dB is also highlighted in the figures 3a, 3b. We can see that the efficiency of both strategies is quite similar. We have to take into account the input delay in the PFC strategy when we have a high mobility. Indeed, we see in the figure 3c that the SINR is not always higher than the SINR min if we do not apply the PFC (26) of the theorem 2 but the PFC [START_REF] Ionete | Controllability and observability of input/output delayed discrete systems[END_REF] of the theorem 1 to the system (14). Moreover, we see in the figure 4 that the execution time, for a simulated time period of the system of 10s, of the PFC is always shorter than the one of the TPC and the MPC. This is an advantage for the real-time computing due to the fact that we have no prediction to do with the PFC. The figure 4 highlights that the real-time computing can become critical because the number of mobile phones can be very high in a cellular network. V. CONCLUSION

In this article, we have developed a new feedback control strategy for discrete-time systems having a constraint on the state space variable and an input delay. This strategy is based on a nonlinear PFC. It is well suited for the power control of a wireless cellular network due to the constraints on the minimum value of the SINR imposed by the telecommunication standards and on the real-time computing. In our application in LTE, it ensures that the SINR is always higher than a given minimum value. As future research lines, it will be interesting to extend the PFC to MIMO systems.

Fig. 1 .

 1 Fig. 1. The closed-loop power control system for each user

Fig. 2 .

 2 Fig.2. The BLER versus the SINR (dB) for a different modulation types and rates r[START_REF] Stefania Sesia | LTE -The UMTS Long Term Evolution From Theory to Practice[END_REF]

Fig. 3 .

 3 Fig. 3. (a) SINR with a velocity of 5km.h -1 ; (b) (c) SINR with a velocity of 260km.h -1 with and without Artstein transform

Fig. 4 .

 4 Fig. 4. Execution time for a simulated time period of the system of 10s