Global dissipativity and finite-time synchronization of mixed time-varying delayed memristor-based neural networks with discontinuous activations

Abstract

In this paper, the matters of dissipativity and finite time synchronization for memristor-based neural networks (MNNs) with mixed time-varying discontinuities are investigated. Firstly, under the framework of extending Filippov differential inclusion theory, several effective new criteria are derived. Then, the global dissipativity of Filippov solution to neural networks is proved by using generalized Halanay inequality and matrix measure method. Secondly, some novel sufficient conditions are introduced to guarantee the finite-time synchronization of the drive-response MNNs based on a simple Lyapunov function and two different feedback controllers. Finally, several numerical examples are given to verify the validity of the theoretical results.

Keywords

Memristor dissipativity finite time synchronization mixed time-varying delayed neural networks

1 Introduction

Memristor, as the fourth circuit element, was firstly proposed by Chua in 1971 [1]. In recent years, memristor has aroused widespread attention by its memory characteristics and nanoscale, which has been successfully applied in various areas throughout science and engineering [2]. The authors in [2] normally use resistors to simulate synapses among the neurons in the circuit implementation of neural network. In the other hand, a neural networks with discontinuous (or non-Lipschitz) neuron activation was employed to solve many interesting engineering tasks such as impact machineries, dry friction, power supply circuit and electrical switch etc.[3 –8]. Nevertheless, time delay in hardware implementation may lead to oscillation, divergence or instability, which may be harmful to the system. The delays, are time-varying and even vary violently with time, which affect the stability of the designed neural network. Therefore, the matters about memristor-based neural networks (MNNs) with mixed time-varying discontinuities are investigated in this paper.

In the past several years, neural network has a wide application prospect in the fields of secure communication, associative memory and image encryption. Many reports have introduced dynamic behaviors of neural networks with discontinuous neuronal activation, such as periodicity, convergence and stability. But besides convergence and stability[29 –33], the dissipativity and synchronization of neural networks are also a very important dynamic behavior[9 –28]. Epileptic seizures are characterized by abnormal synchronization of local or global neural adaptation. Seizures and synchronization occur almost at the same time. A long-range increased synchronization can terminate seizure [14]. So the synchronization and dissipativity of neural networks also have been one of the hotspots of research (see [24 –39], etc.). Up to now, there are only a few results are available on dissipativity and synchronization of neural networks with discontinuous activation functions [5 –8], etc. However, there are no reports related to the dissipativity and finite time synchronization of MNNs hybrid time-delays (with distributed time-delay) neural networks with discontinuous activation had been published. By introducing the good properties of the extended Halanay inequality and matrix measure, a less conservative method is established to ensure the dissipativity of the MNNs. In the existing literature, though there are many reports related to the synchronization control of MNNs, the main purpose of them is to study the asymptotic synchronization and exponential synchronization(see [11],[16], etc.). The asymptotic synchronization and exponential synchronization can only be realized when time tends to infinity while the finite time synchronization can be achieved in a settling time. It is worth noting that the asymptotic synchronization in [11] is not reasonable in some engineering fields. In fact, people always hope that the system can achieve synchronization quickly. Recently, due to its fast convergence speed and better interference suppression performance, the finite time synchronization of various systems has attracted more and more attention ([18, 20],etc.). Hence, the research on finite time synchronization of mixed delays MNNs with discontinuous activation is still an open problem to be solved based on our reading to the existing articles. Thus, dissipativity and finite time synchronization of hybrid time-varying delay MNNs with discontinuous activation functions are worth investigated. In the second half of this article, a discontinuous state feedback controller is designed. It is proved that the model can realize finite time synchronization under the designed controller by using inequality technique.

The main contribution of this paper:

1. We dicuss the dissipativity and finite time synchronization of mixed time-varying delays MNNs with discontinuous activation, which is less studied. It should be noted that the mixed delays in this paper include no delays, time-varying delay and distributed delays.

2. New lemmas are proved by using Cauchy inequality and some analytical technique. The slack variable ν is introduced and a less conservative condition is obtained to ensure the dissipativity of the MNNs. Furthermore, the dissipative properties of Filippov solutions are studied by using generalized Halanay inequality and matrix measure method.

3. Two discontinuous state feedback controllers are designed. Finally, finite time synchronization with discontinuous activation and mixed delay is realized.

The paper is organized as follows: the second section introduces the model of a time-delay MNNs with discontinuous activation function, and gives some necessary assumptions and definitions. In the third section, some definitions, lemmas, and main results are given. In the fourth section, we verify our results through several examples. The fifth section draws the main conclusions.

2 Model description and preliminaries

Firstly, we consider a mixed time-varying delay neural network with discontinuous activation, which can be described by the following differential equations: $\begin{matrix} \frac{d x_{i} (t)}{d t} = - h_{i} x_{i} (t) + \sum_{j = 1}^{m} l_{ij} (x_{i} (t)) g_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{m} q_{ij} (x_{i} (t)) g_{j} (x_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} r_{ij} (x_{i} (t)) \int_{- \infty}^{t} g_{j} (x_{j} (s)) p_{ij} (t - s) d s \\ + I_{i}, i = 1, 2, . . ., m, \end{matrix}$ (2.1) where x_i (t) corresponds to the voltage of the capacitor C_i ; h_i > 0 represents the neuron self-inhibitions; g_j (·) : R → R represents the neuron input-output activation of the ith neuron; the probability kernel of distributed delay is piecewise continuous function on [0, ∞) to [0, ∞), as well as $\int_{- \infty}^{0} p_{ij} (t - s) d s = 1,$ for i, j = 1, 2, . . . , m ; I_i denotes the external input to the ith neuron, l_ij (•) , q_ij (•) and r_ij (•) are the memristor-based weights given by $\begin{matrix} l_{ij} (x_{i} (t)) = \frac{M_{ij}^{a}}{C_{i}} \times {sgn}_{ij}, q_{ij} (x_{i} (t)) = \frac{M_{ij}^{b}}{C_{i}} \times {sgn}_{ij}, \\ r_{ij} (x_{i} (t)) = \frac{M_{ij}^{d}}{C_{i}} \times {sgn}_{ij}, {sgn}_{ij} = {\begin{matrix} 1, i \neq j, \\ - 1, i = j, \end{matrix} \end{matrix}$ (2.2) in which $M_{ij}^{a}, M_{ij}^{b}, M_{ij}^{d}$ denote the memductances of memristors $B_{ij}^{a}, B_{ij}^{b}, B_{ij}^{d}$ respectively. $B_{ij}^{a}$ represents the memristor between the feedback function g_j (x_j (t)) and $x_{i} (t), B_{ij}^{b}$ represents the memristor between the feedback function g_j (x_j (t - τ_ij (t))) and $x_{i} (t), B_{ij}^{d}$ represents the memristor between the feedback function g_j (x_j (s)) and x_i (t). According to the switching characteristics of memristor, we set $\begin{matrix} l_{ij} (x_{i} (t)) = {\begin{matrix} {l^{'}}_{ij}, | x_{i} (t) | \leq Δ_{i}, \\ {l^{″}}_{ij}, | x_{i} (t) | > Δ_{i}, \end{matrix} \\ q_{ij} (x_{i} (t)) = {\begin{matrix} {q^{'}}_{ij}, | x_{i} (t) | \leq Δ_{i}, \\ {q^{″}}_{ij}, | x_{i} (t) | > Δ_{i}, \end{matrix} \\ r_{ij} (x_{i} (t)) = {\begin{matrix} {r^{'}}_{ij}, | x_{i} (t) | \leq Δ_{i}, \\ {r^{″}}_{ij}, | x_{i} (t) | > Δ_{i}, \end{matrix} \end{matrix}$ (2.3) where Δ_i > 0 are switching jumps. $l_{ij}^{'}, l_{ij}^{″}, q_{ij}^{'}, q_{ij}^{″}, r_{ij}^{'}, r_{ij}^{″}, (i, j = 1, 2, . . ., m)$ are constants.

Due to (2.1) is a right-end discontinuous differential equation, the classical differential equation theory can not be used to study the dynamics of the system (2.1).

Definition 2.1. (Filippov solution)([40, 41]) Let us consider the system $\frac{d x}{d t} = g (x), x \in ℝ^{n}$ , by using discontinuous right-hand sides, the Filippov set-valued map is defined as follows: $G (x) = ⋂_{δ^{*} > 0} ⋂_{μ (N) = 0} \bar{co} [g (B (x, δ^{*}) ∖ N)],$ where $\bar{co} (Θ)$ is the closure of the convex hull of set $Θ, B (x, δ^{*}) = {y : ∥ y - x ∥ \leq δ^{*}}$ , and μ (N) denotes the Lebesgue measure of set N.

The solution in the sense of Filippov is an absolutely continuous function $x (t), t \in [0, T)$ , which satisfies the following differential inclusion: $\frac{d x}{d t} \in G (x), for a . e . t \in [0, T) .$

We apply the above theory of set-valued mapping and differential inclusion [42, 43], and the MNNs (2.1) also can be rewritten as $\begin{matrix} \frac{d x_{i} (t)}{d t} \in - h_{i} x_{i} (t) + \sum_{j = 1}^{m} \bar{co} [l_{ij} (x_{i} (t))] \bar{co} [g_{j} (x_{j} (t))] \\ + \sum_{j = 1}^{m} \bar{co} [q_{ij} (x_{i} (t))] \bar{co} [g_{j} (x_{j} (t - τ_{ij} (t)))] \\ + \sum_{j = 1}^{m} \bar{co} [r_{ij} (x_{i} (t))] \int_{- \infty}^{t} \bar{co} [g_{j} (x_{j} (s))] \\ p_{ij} (t - s) d s + I_{i}, \end{matrix}$ (2.4) for a.e. $t \in [0, T), i = 1, 2, . . ., m,$ where $\begin{matrix} \bar{co} [l_{ij} (x_{i} (t))] = {\begin{matrix} {l^{'}}_{ij}, | x_{i} (t) | \leq Δ_{i}, \\ [{\underline{l}}_{ij}, {\bar{l}}_{ij}], | x_{i} (t) | = Δ_{i}, \\ {l^{″}}_{ij}, | x_{i} (t) | > Δ_{i}, \end{matrix} \\ \bar{co} [q_{ij} (x_{i} (t))] = {\begin{matrix} {q^{'}}_{ij}, | x_{i} (t) | \leq Δ_{i}, \\ [{\underline{q}}_{ij}, {\bar{q}}_{ij}], | x_{i} (t) | = Δ_{i}, \\ {q^{″}}_{ij}, | x_{i} (t) | > Δ_{i}, \end{matrix} \\ \bar{co} [r_{ij} (x_{i} (t))] = {\begin{matrix} {r^{'}}_{ij}, | x_{i} (t) | \leq Δ_{i}, \\ [{\underline{r}}_{ij}, {\bar{r}}_{ij}], | x_{i} (t) | = Δ_{i}, \\ {r^{″}}_{ij}, | x_{i} (t) | > Δ_{i}, \end{matrix} \end{matrix}$ where $\begin{matrix} {\underline{l}}_{ij} = min {{l^{'}}_{ij}, {l^{″}}_{ij}}, {\bar{l}}_{ij} = max {{l^{'}}_{ij}, {l^{″}}_{ij}}, \\ {\underline{q}}_{ij} = min {{q^{'}}_{ij}, {q^{″}}_{ij}}, \\ {\bar{q}}_{ij} = max {{q^{'}}_{ij}, {q^{″}}_{ij}}, {\underline{r}}_{ij} = min {{r^{'}}_{ij}, {r^{″}}_{ij}}, \\ {\bar{r}}_{ij} = max {{r^{'}}_{ij}, {r^{″}}_{ij}} . \end{matrix}$

There exists measurable function $γ_{j} (t) \in \bar{co} [g_{j} (x_{j} (t))]$ , ${\frac{\overset{⌣}{d t}}{l}}_{ij} (t) \in \bar{co} [l_{ij} (x_{i} (t))],$ ${\frac{\overset{⌣}{ϱ}}{q}}_{ij} (t) \in \bar{co} [q_{ij} (x_{i} (t))],$ ${\overset{⌣}{r}}_{ij} (t) \in \bar{co} [r_{ij} (x_{i} (t))]$ such that $\begin{matrix} \frac{d x_{i} (t)}{d t} = - h_{i} x_{i} (t) + \sum_{j = 1}^{m} {\overset{⌣}{l}}_{ij} (x_{i} (t)) γ_{j} (t) \\ + \sum_{j = 1}^{m} {\overset{⌣}{q}}_{ij} (x_{i} (t)) γ_{j} (t - τ_{ij} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌣}{r}}_{ij} (x_{i} (t)) \int_{- \infty}^{t} γ_{j} (s) p_{ij} (t - s) d s + I_{i}, \end{matrix}$ (2.5) for a.e. $t \in [0, T), i = 1, 2, . . ., m,$ where the function γ_j = (γ₁, γ₂, . . . , γ_m) ^T satisfying (2.5) is called an output solution corresponding to the solution x_j (t).

In summary, system (2.1) can rewrite as follows $\begin{matrix} \frac{d x_{i} (t)}{d t} = - h_{i} x_{i} (t) + \sum_{j = 1}^{m} {\overset{⌣}{l}}_{ij} (t) g_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌣}{q}}_{ij} (t) g_{j} (x_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} {\overset{⌣}{r}}_{ij} (t) \int_{- \infty}^{t} g_{j} (x_{j} (s)) p_{ij} (t - s) d s + I_{i} . \end{matrix}$ (2.6)

Filippov set-valued map provides the convex hull of discontinuity points in order to produce continuous points (Fig. 2). The following examples of the initial value problem (IVP) are considered: $\dot{x} (t) = \tilde{g} (x (t)), x (0) = x_{0}, t \in [0, T),$ where $\tilde{g} (x) = {\begin{matrix} 2 x^{2} + 20 x < 1, \\ x^{2} + 1 x > 1 . \end{matrix}$

By differential inclusion of it is as follows: $\dot{x} (t) \in G (x) = {\begin{matrix} 2 x^{2} + 20 x < 1, \\ [2, 22] x = 1, \\ x^{2} + 1 x > 1 . \end{matrix}$

Definition 2.2. (IVP)([44]) For any continuous function $φ : [- θ, 0] \to ℝ^{m}$ and every measurable selection $Ψ : [- θ, 0] \to ℝ^{m}$ such that $\overset{´}{ψ} (s) \in \bar{co} [g (\overset{´}{φ} (s))]$ , for a . e, s ∈ [- θ, 0] by an initial value problem associated to (2.1) with initial condition, we refer to the following questions: find a couple of functions $[x, γ] : [- θ, T] \to ℝ^{m} \times ℝ^{m}$ , such that x is a solution of (2.1) on $[- θ, T]$ for some $T > 0, γ$ is said to be an output associated to x, and the following results are valid ${\begin{matrix} \dot{x} (t) = - h_{i} x_{i} (t) + \sum_{j = 1}^{m} {\overset{⌣}{l}}_{ij} (x_{i} (t)) γ_{j} (t) \\ + \sum_{j = 1}^{m} {\overset{⌣}{q}}_{ij} (x_{i} (t)) γ_{j} (t - τ_{ij} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌣}{r}}_{ij} (x_{i} (t)) \int_{- \infty}^{t} γ_{j} (s) p_{ij} (t - s) d s \\ + I_{i}, fora . e . t \in [0, T), \\ γ (t) \in \bar{co} [g (x (t))], fora . e . t \in (- \infty, T), \\ x (s) = φ (s), \forall s \in (- \infty, 0], \\ γ (s) = ψ (s), fora . e . s \in (- \infty, 0] . \end{matrix}$

Consider the neural network model (2.1) as the driver system, the controlled response system is $\begin{matrix} \frac{d y_{i} (t)}{d t} = - h_{i} y_{i} (t) + \sum_{j = 1}^{m} l_{ij} (y_{i} (t)) g_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{m} q_{ij} (y_{i} (t)) g_{j} (y_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} r_{ij} (y_{i} (t)) \int_{- \infty}^{t} g_{j} (y_{j} (s)) p_{ij} (t - s) d s \\ + I_{i} + u_{i} (t), \end{matrix}$ (2.7) for a.e. $t \in [0, T), i = 1, 2, . . ., m,$ where y_i (t) is the state of the response system, u_i (t) = (u₁ (t) , u₂ (t) , . . . u_m (t)) ^T is the controller to be designed, the other parameters are the same as those defined in system (2.1). Similarly, we have $\begin{matrix} l_{ij} (y_{i} (t)) = {\begin{matrix} {l^{'}}_{ij}, | y_{i} (t) | \leq Δ_{i}, \\ {l^{″}}_{ij}, | y_{i} (t) | > Δ_{i}, \end{matrix} \\ q_{ij} (x_{i} (t)) = {\begin{matrix} {q^{'}}_{ij}, | y_{i} (t) | \leq Δ_{i}, \\ {q^{″}}_{ij}, | y_{i} (t) | > Δ_{i}, \end{matrix} \\ r_{ij} (y_{i} (t)) = {\begin{matrix} {r^{'}}_{ij}, | y_{i} (t) | \leq Δ_{i}, \\ {r^{″}}_{ij}, | y_{i} (t) | > Δ_{i} . \end{matrix} \end{matrix}$

It follows from (2.5) that: $\begin{matrix} \dot{y} (t) \in - h_{i} y_{i} (t) + \sum_{j = 1}^{m} \bar{co} [l_{ij} (y_{i} (t))] ϑ_{j} (t) \\ + \sum_{j = 1}^{m} \bar{co} [q_{ij} (y_{i} (t))] ϑ_{j} (t - τ_{ij} (t)) \\ + \sum_{j = 1}^{m} \bar{co} [r_{ij} (y_{i} (t))] \int_{- \infty}^{t} ϑ_{j} (s) p_{ij} (t - s) d s \\ + I_{i} + u_{i} (t), \end{matrix}$ (2.8) for a.e. $t \in [0, T), i = 1, 2, . . ., m,$ $\begin{matrix} \bar{co} [l_{ij} (y_{i} (t))] = {\begin{matrix} {l^{'}}_{ij}, | y_{i} (t) | \leq Δ_{i}, \\ [{\underline{l}}_{ij}, {\bar{l}}_{ij}], | y_{i} (t) | = Δ_{i}, \\ {l^{″}}_{ij}, | y_{i} (t) | > Δ_{i}, \end{matrix} \\ \bar{co} [q_{ij} (x_{i} (t))] = {\begin{matrix} {q^{'}}_{ij}, | y_{i} (t) | \leq Δ_{i}, \\ [{\underline{q}}_{ij}, {\bar{q}}_{ij}], | y_{i} (t) | = Δ_{i}, \\ {q^{″}}_{ij}, | y_{i} (t) | > Δ_{i}, \end{matrix} \\ \bar{co} [r_{ij} (y_{i} (t))] = {\begin{matrix} {r^{'}}_{ij}, | y_{i} (t) | \leq Δ_{i}, \\ [{\underline{r}}_{ij}, {\bar{r}}_{ij}], | y_{i} (t) | = Δ_{i}, \\ {r^{″}}_{ij}, | y_{i} (t) | > Δ_{i}, \end{matrix} \end{matrix}$ for a.e, i, j = 1, 2, . . . , m, similarly, $ϑ_{j} (t) \in \bar{co} [g_{j} (y_{j} (t))],$ ${\overset{⌢}{l}}_{ij} (t) \in \bar{co} [l_{ij} (y_{i} (t))],$ ${\overset{⌢}{q}}_{ij} (t) \in \bar{co} [q_{ij} (y_{i} (t))],$ ${\overset{⌢}{r}}_{ij} (t) \in \bar{co} [r_{ij} (y_{i} (t))]$ , so (2.8) can be rewritten as $\begin{matrix} \dot{y} (t) = - h_{i} y_{i} (t) + \sum_{j = 1}^{m} {\overset{⌢}{l}}_{ij} (x_{i} (t)) ϑ_{j} (t) \\ + \sum_{j = 1}^{m} {\overset{⌢}{q}}_{ij} (x_{i} (t)) ϑ_{j} (t - τ_{ij} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌢}{r}}_{ij} (x_{i} (t)) \int_{- \infty}^{t} ϑ_{j} (s) p_{ij} (t - s) d s \\ + I_{i} + u_{i} (t), \end{matrix}$ (2.9) for a.e, i, j = 1, 2, . . . , m . Because of Definition (2.2) and Lemma 2.1, the initial value problem of system (2.7) is ${\begin{matrix} \dot{y} (t) = - h_{i} y_{i} (t) + \sum_{j = 1}^{m} {\overset{⌢}{l}}_{ij} (y_{i} (t)) ϑ_{j} (t) \\ + \sum_{j = 1}^{m} {\overset{⌢}{q}}_{ij} (y_{i} (t)) ϑ_{j} (t - τ_{ij} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌢}{r}}_{ij} (y_{i} (t)) \int_{- \infty}^{t} ϑ_{j} (s) p_{ij} (t - s) d s \\ + I_{i} + u_{i} (t), fora . e . t \in [0, T), \\ ϑ (t) \in \bar{co} [g (y (t))], fora . e . t \in [0, T), \\ y (s) = \bar{φ} (s), \forall s \in (- \infty, 0], \\ \bar{γ} (s) = \bar{ψ} (s), fora . e . s \in (- \infty, 0] . \end{matrix}$

The system (2.7) can be written as $\begin{matrix} \dot{y} (t) = - h_{i} y_{i} (t) + \sum_{j = 1}^{m} {\overset{⌢}{l}}_{ij} (t) g_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌢}{q}}_{ij} (t) g_{j} (x_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} {\overset{⌢}{r}}_{ij} (t) \int_{- \infty}^{t} g_{j} (x_{j} (s)) p_{ij} (t - s) d s \\ + I_{i} + u_{i} (t) . \end{matrix}$ (2.10)

Throughout this paper, matrix norms and vector norms are defined as follows:

Norm of vector $A = (a_{1}, a_{2}, . . ., a_{n})^{T} \in ℝ^{n}$ are denoted by

${∥ A ∥}_{1} = \sum_{i = 1}^{n} | a_{i} |, {∥ A ∥}_{2} = (\sum_{i = 1}^{n} a_{i}^{2})^{\frac{1}{2}}, {∥ A ∥}_{\infty} = max_{1 \leq i \leq n} | a_{i} | .$

Norm of matrix $H = {(h_{ij})}_{n \times n} \in ℝ^{n \times n}$ are denoted by $\begin{matrix} {∥ H ∥}_{1} = max_{j} \sum_{i = 1}^{n} | h_{ij} |, ∥ H ∥ = {∥ H ∥}_{2} \\ = {(λ_{max} (H^{T} H))}^{\frac{1}{2}}, and {∥ H ∥}_{\infty} = max_{i} \sum_{j = 1}^{n} | h_{ij} |, \end{matrix}$ where λ_max (H) represents the maximal eigenvalue of matrix H.

In this paper, we hypothesize (2.1) that the activation of discontinuous neurons has the following characteristics:

A1 For each j = 1, 2, . . . m, g_j (·) : R → R is continuous except on a countable set of isolate points ${ρ_{k}^{j}}$ , where there exist finite right and left limits, $g_{j}^{+} (ρ_{k}^{j})$ and $g_{j}^{-} (ρ_{k}^{j})$ , respectively. Moreover, g_j has at most a finite number of discontinuities on any compact interval of R.

In order to describe the system (2.1) more clearly, we cite the literature [7], the system block diagram in the neurodynamic system (2.1) described below.

Fig. 1

Circuit of neural networks. L_ij is the connection strength between the neuron activation function g_j (.) and x_i (t), Q_ij is the connection strength between the neuron activation function $g_{j}^{*} (.)$ and x_i (t), R_ij is the connection strength between the neuron activation function $g_{j}^{* *} (.)$ and x_i (t), I_i (t) is the external input, L_i is inductor, x_i (t) is current of the inductor, R_i and C_i are the resistor and capacitor, u_i, $u_{i}^{*}$ , $u_{i}^{* *}$ are the outputs, i, j = 1, 2, . . . , m.

Fig. 2

Function $\tilde{g} (x)$ and its convex hull G (x) are described above.

Fig. 3

[7] Here x denotes the neuron state. Time-varying delays represented by τ and distributed delays denoted by p are both shown in this diagram.

A2 For every j = 1, 2, . . . m, there exist nonnegative constants λ_j and ξ_j such that $\begin{matrix} sup_{ς_{j} \in \bar{co} [g_{j} (x_{j})], {\bar{ς}}_{j} \in \bar{co} [g_{j} ({\bar{x}}_{j})]} | ς_{j} - {\bar{ς}}_{j} | \leq λ_{j} | x_{j} - {\bar{x}}_{j} | \\ + ξ_{j}, \forall x_{j}, {\bar{x}}_{j} \in R, \end{matrix}$ similarly,

$\begin{matrix} sup_{ς_{j} \in \bar{co} [g_{j} (x_{j})], {\bar{ς}}_{j} \in \bar{co} [g_{j} ({\bar{x}}_{j})]} ∥ ς_{j} - {\bar{ς}}_{j} ∥ \leq λ_{j} ∥ x_{j} - {\bar{x}}_{j} ∥ + ξ_{j}, \end{matrix}$

and $\begin{matrix} sup_{ς_{j} \in \bar{co} [g_{j} (x_{j})]} ∥ ς_{j} ∥ \leq λ_{j} ∥ x_{j} ∥ + ξ_{j}, \end{matrix}$ where $\begin{matrix} \bar{co} [g_{j} (θ)] = & [min {g_{j}^{-} (θ), g_{j}^{+} (θ)}, \\ max {g_{j}^{-} (θ), g_{j}^{+} (θ)}], θ \in R . \end{matrix}$

A3 Let the time-varying delays τ_ij (t) be continuously differentiable functions, and satisfy dτ_ij (t)/dt ≤ μ < 1; here μ are positive constants.

A4 The probability kernel of distributed delay is piecewise continuous function on [0, ∞) to [0, ∞); with $\int_{0}^{\infty} p_{ij} (s) d s = 1$ for i, j = 1, 2, . . . m.

A5 There exist nonnegative constants M_j, ζ_j such that $| y_{i} (t) | \leq M_{i}, | g_{i} (.) | \leq ζ_{i}, i = 1, 2, . . ., m .$

A6 There have nonnegative function ϒ (t) and positive constant ϒ such that $\int_{0}^{ρ (t)} P (s) d s = ϒ (t) \leq ϒ;$ where 0 ≤ ρ (t) ≤ ρ, ρ is positive constant.

Lemma 2.1. (Chain rule)([45, 46]) If $ℝ^{n} \times ℝ$ is C-regular, and $x (t) : [0, + \infty) \to ℝ^{n}$ is absolutely continuous on every compact subinterval of [0, + ∞). Then, x (t) and $ℤ (x (t)) : [0, + \infty) \to ℝ$ are differentiable for almost all t ∈ [0, + ∞) and $d ℤ (x (t)) = 〈 κ (t), \frac{d x (t)}{d t} 〉, \forall κ (t) \in \partial ℤ (x (t)) .$

Lemma 2.2. (A generalized Halanay inequality) Let x (t) be a nonnegative function satisfying $\begin{matrix} D^{+} x (t) \leq (χ (t) - \tilde{α} x (t) + \tilde{β} sup_{t - τ (t) \leq s \leq t} x (t) \\ + \tilde{φ} \int_{0}^{\infty} P (s) x (t - s) d s) e^{t}, t \geq t_{0} \geq 0, \\ x (s) = ω (s), s \in (- \infty, t_{0}], \end{matrix}$ where τ (t) is a continuous, bounded function in $t \in ℝ$ , and $τ^{★} = sup_{0 \leq t < + \infty} τ (t);$ ω (s), χ (t) denote continuous and bounded functions for t ≥ 0; $\tilde{α}, \tilde{β}, \tilde{φ}$ are nonnegative constants; and the delay kernel P (s) is the same as (2.1). If there exists ϱ > 0 such that $\tilde{α} - \tilde{β} - \tilde{φ} \geq ρ, 0 \leq t < + \infty,$ where $ρ = inf_{0 \leq t < + \infty} (e^{t} (\tilde{α} - \tilde{β} - \tilde{φ}) > 0),$ then there exist positive numbers $\overset{\leftrightarrow}{\overset{\leftrightarrow}{limits}} mu$ and $\overset{\leftrightarrow}{ϱ} < ϱ$ such that $x (t) \leq χ^{★} + ({sup}_{s \in (- \infty, t_{0}]} | x (s) |) e^{- \frac{\overset{\leftrightarrow}{\overset{\leftrightarrow}{ϱ}}}{μ} (t - t_{0})}, t > t_{0} \geq 0,$ where $\begin{matrix} \overset{\leftrightarrow}{ρ} = - sup_{0 \leq t < + \infty} \overset{\leftrightarrow}{μ} + e^{t} (- \tilde{α} + \tilde{β} e^{\overset{\leftrightarrow}{μ} τ^{*}} \\ + \tilde{φ} \int_{0}^{\infty} P (s) e^{\overset{\leftrightarrow}{μ} s} d s), χ^{*} = sup_{0 \leq t < + \infty} χ (t) . \end{matrix}$

In order to facilitate the description in the proof, (2.6) is transformed into a matrix form:

$\begin{matrix} \dot{x} (t) = - H (t) x (t) + L (t) g (x (t)) + Q (t) g (x (t - τ (t))) \\ + R (t) \int_{- \infty}^{t} g (x (s)) P (t - s) d s + I (t), \end{matrix}$ (2.11)

where x (t) = col { x_i (t) } , H (t) = (h_ij (t)) _m×m, L (t) = (l_ij (t)) _m×m, Q (t) = (q_ij (t)) _m×m, R (t) = (r_ij (t)) _m×m, P (s) = (p_ij (s)) _m×m, I (t) = col { I_i (t) } , τ (t) = τ_ij (t), where g (x (t)) = (g₁ (x₁ (t)) , g₂ (x₂ (t)) , . . . , g_m (x_m (t))) ^T, i = 1, 2, . . . m.

Definition 2.3. ([47]) The matrix measure of a real square matrix $H = {(h_{ij})}_{n \times n} \in ℝ^{n \times n}$ is as follows: $μ_{p} (H) = lim_{ɛ \to 0^{+}} \frac{{∥ E + ɛ H ∥}_{p} - 1}{ɛ},$ in which ∥ .∥ is a matrix norm on $ℝ^{n \times n}$ , E is the identity matrix, and p = 1, 2, . . . , ∞.

Note that the measure of a matrix H can be regarded as the directional derivative of the norm function ∥ · ∥ _p as evaluated at the identity matrix H in the direction. Obviously, we can obtain the matrix measures as follows: $\begin{matrix} μ_{1} (H) = max_{j} {h_{jj} + \sum_{i = 1, i \neq j}^{n} | h_{ij} |}, \\ μ_{2} (H) = λ_{max} (\frac{H + H^{T}}{2}), \\ μ_{\infty} (H) = max_{i} {h_{ii} + \sum_{i = 1, j \neq i}^{n} | h_{ij} |} . \end{matrix}$

Thus, we can apply the important definition below to prove our results.

Definition 2.4. ([47]) System (2.1) is a dissipative system, if there exists a compact set $Ω^{*} \subseteq ℝ^{m},$ such that for every $x_{0} \in ℝ^{m}$ , then there exists a $T > 0$ , when for $t \geq t_{0} + T, x (t; t_{0}, x_{0}) \subseteq Ω^{*} \subseteq ℝ^{m},$ where $x (t; t_{0}, x_{0}) \subseteq Ω^{*} \subseteq ℝ^{m}$ denotes the solution of system (2.1) from initial value x₀ and initial time t₀. under these circumstances, Ω^∗ is called a globally attractive set. A set Ω^∗ is called positive invariant, if for every $x_{0} \in Ω^{*} \subseteq ℝ^{m}$ , implies $x (t; t_{0}, x_{0}) \subseteq Ω^{*} \subseteq ℝ^{m},$ for t > t₀ . Apparently, the globally attractive set is also a positive invariant.

Definition 2.5. ([14]) The MNN (2.7) with discontinuous activations is said to be finite time synchronized with system (2.1) if there exist positive constants t_*, such that $lim_{t \to t_{*}} e (t) = {(0, 0, . . ., 0)}^{T}$ and e (t) ≡ (0, 0, . . . , 0) ^T for t ≥ t_* where t_* is called the settling time.

Lemma 2.3. ([14]) For every κ ∈ R⁺, B, C ∈ R, the following results are obtained. $2 | B | | C | \leq κ B^{2} + \frac{1}{κ} C^{2} .$

Lemma 2.4. ([48]) For any constant symmetric matrix Δ ∈ R^m×m, Δ = Δ^T > 0, suppose Q (t) is a non-negative bounded scalar function defined on $[0, \infty), \int_{0}^{+ \infty} Q (v^{*}) d v^{*} = ϒ$ and vector function Φ : (- ∞ , t] → R^m for t ⩾ 0, then $\begin{matrix} {(\int_{- \infty}^{t} q (t - s) Φ (s) d s)}^{T} Δ \int_{- \infty}^{t} q (t - s) Φ (s) d s \\ \leq ϒ \int_{- \infty}^{t} q (t - s) Φ^{T} (s) Δ Φ (s) d s . \end{matrix}$

Lemma 2.5. (Jensens inequality)([49]) If ω₁, ω₂, . . . , ω_m ≥ 0 and 0 < u < v, then ${(\sum_{j = 1}^{m} ω_{j}^{v})}^{\frac{1}{v}} \leq {(\sum_{j = 1}^{m} ω_{j}^{u})}^{\frac{1}{u}} .$

Lemma 2.6. ([50]) Assume that there exists a continuous positive-definite function V (t), we have $\dot{V} (t) \leq - η^{*} V^{γ^{*}} (t), V (t) \geq 0,$ where η^* > 0 and 0 < γ^* < 1 are two constants. Then V (t) satisfies the following differential inequality: $V^{1 - γ^{*}} (t) \leq V^{1 - γ^{*}} (t_{0}^{*}) - η^{*} (1 - γ^{*}) (t - t_{0}^{*}), t_{0}^{*} \leq t \leq T^{•}$ and V (t) ≡0, ∀ t ≥ T^•, we give the settling time T^• as follows: $T^{•} = t_{0}^{*} + \frac{{(V (t_{0}^{*}))}^{1 - γ^{*}}}{η^{*} (1 - γ^{*})} .$

3 Main results

In this section, we will discuss the dissipativity and synchronization of the system (2.1). First, some lemmas are given and some coclusions are obtained.

In order to make our neural network more conservative, the slack variable ν is introduced.

Lemma 3.1. Consider the assumption (A4), then we can obtain the following inequalities hold: $\begin{matrix} sup_{ς \in \bar{co} [g (x)]} {∥ ν ς ∥}_{p} \leq {∥ λ_{f} ∥}_{p} {∥ ν x ∥}_{p} + {∥ ν ξ_{f} ∥}_{p}, \\ p = 1, 2, \infty, \forall x \in ℝ^{m}, \end{matrix}$ where ν = diag (ν₁, ν₂, . . . , ν_m) , ν_i ≠ 0, i = 1, 2, . . . , m, λ_f = diag (λ₁, λ₂, . . . , λ_m) , ξ_f = (ξ₁, ξ₂, . . . , ξ_m) ^T .

Proof. When p = 1, according to (A2), we can get $\begin{matrix} sup_{ς \in \bar{co} [g (x)]} {∥ ν ς ∥}_{1} \leq \sum_{i = 1}^{m} | ν_{i} ς_{i} | \leq \sum_{i = 1}^{m} (λ_{i} | ν_{i} x_{i} | + | ν_{i} ξ_{i} |) \\ \leq max_{i} λ_{i} \sum_{i = 1}^{m} | ν_{i} x_{i} | + \sum_{i = 1}^{m} | ν_{i} ξ_{i} | \\ \leq {∥ λ_{f} ∥}_{1} {∥ ν x ∥}_{1} + {∥ ν ξ_{f} ∥}_{1} . \end{matrix}$

When p = 2, according to (A2) and Cauchy inequality, we can get $\begin{matrix} sup_{ς \in \bar{co} [g (x)]} {∥ ν ς ∥}_{2}^{2} \\ \leq \sum_{i = 1}^{m} {| ν_{i} ς_{i} |}^{2} \leq \sum_{i = 1}^{m} {(λ_{i} | ν_{i} x_{i} | + | ν_{i} ξ_{i} |)}^{2} \\ \leq \sum_{i = 1}^{m} λ_{2}^{2} {| ν_{i} x_{i} |}^{2} + \sum_{i = 1}^{m} {| ν_{i} ξ_{i} |}^{2} \\ + 2 \sum_{i = 1}^{m} λ_{i} | ν_{i} x_{i} | | ν_{i} ξ_{i} | + \sum_{i = 1}^{m} {| ν_{i} ξ_{i} |}^{2} \\ \leq \sum_{i = 1}^{m} λ_{2}^{2} {| ν_{i} x_{i} |}^{2} + \sum_{i = 1}^{m} {| ν_{i} ξ_{i} |}^{2} \\ + 2 [\sqrt{\sum_{i = 1}^{m} λ_{2}^{2} {| ν_{i} x_{i} |}^{2}} \cdot \sqrt{\sum_{i = 1}^{m} {| ν_{i} ξ_{i} |}^{2}}] \\ = {[\sqrt{\sum_{i = 1}^{m} λ_{2}^{2} {| ν_{i} x_{i} |}^{2}} + \sqrt{\sum_{i = 1}^{m} {| ν_{i} ξ_{i} |}^{2}}]}^{2} . \end{matrix}$

Thus $\begin{matrix} sup_{ς \in \bar{co} [g (x)]} {∥ ν ς ∥}_{2} \leq max_{i} λ_{i} \sqrt{\sum_{i = 1}^{m} {| ν_{i} x_{i} |}^{2}} + \sqrt{\sum_{i = 1}^{m} {| ν_{i} ξ_{i} |}^{2}} \\ \leq {∥ λ_{f} ∥}_{2} {∥ ν x ∥}_{2} + {∥ ν ξ_{f} ∥}_{2} . \end{matrix}$ When p = ∞ , similarly, according to (A2), we can get $\begin{matrix} sup_{ς \in \bar{co} [g (x)]} {∥ ν ς ∥}_{\infty} \leq max_{i = 1} {| ν_{i} ς_{i} |} \\ \leq max_{i = 1} {(λ_{i} | ν_{i} x_{i} | + | ν_{i} ξ_{i} |)} \\ \leq max_{i} max_{i = 1} | ν_{i} x_{i} | + max_{i = 1} | ν_{i} ξ_{i} | \\ = {∥ λ_{f} ∥}_{\infty} {∥ ν x ∥}_{\infty} + {∥ ν ξ_{f} ∥}_{\infty} . \end{matrix}$ Lemma 3.1 is completed.

I. DISSIPATIVITY

In the following theorem, we discuss the global exponential dissipativity of (2.1).

Theorem 3.2. We order delay functions τ (t) are bounded functions and satisfy $0 \leq τ (t) \leq \bar{τ}$ for i, j = 1, 2, . . . , m. Then the neural networks model (2.1) will be globally dissipativity, if the assumptions (A1) and (A2) hold, and there exist one matrix measure μ_p (·) , p = 1, 2, ∞ and ϱ > 0 such that $- \tilde{α} = max {μ_{p} (- H (t)) + {∥ ν L (t) ν^{- 1} ∥}_{p} {∥ λ_{f} ∥}_{p}} < 0,$ where $- \tilde{α} + \tilde{β} + \tilde{φ} \leq - ρ$ hold, and $\begin{matrix} \tilde{β} = {∥ ν Q (t) ν^{- 1} ∥}_{p} {∥ λ_{f} ∥}_{p}, \tilde{φ} = {∥ ν R (t) ν^{- 1} ∥}_{p} {∥ λ_{f} ∥}_{p}, \end{matrix}$ where $\begin{matrix} λ_{f} = (λ_{1}, λ_{2}, . . ., λ_{m}), ξ_{f} = (ξ_{1}, ξ_{2}, . . ., ξ_{m}), \end{matrix}$ and also exist $\overset{\leftrightarrow}{ϱ} < ϱ$ and ℓ>0, we have $Ω_{1}^{*} = {x \in ℝ | {∥ ν x^{T} (t) ∥}_{p} \leq χ^{★} + ɛ}, t > ℓ,$ is a globally attractive set and positive invariant, we can derive $\begin{matrix} χ^{★} = & {∥ ν ξ_{f} ∥}_{p} ({∥ ν Q (t) ν^{- 1} ∥}_{p} + {∥ ν L (t) ν^{- 1} ∥}_{p} \\ + {∥ ν R (t) ν^{- 1} ∥}_{p}) + {∥ ν I (t) ∥}_{p}, \end{matrix}$ where $\frac{\overset{\leftrightarrow}{\overset{\leftrightarrow}{ϱ}}}{ϱ}$ is the same as Lemma 2.2 and ɛ is an arbitrarily small positive number.

Proof. Consider the following Lyapunov function for system (2.1) as $V^{*} (t) = {∥ ν x (t) ∥}_{p} .$

By calculating the time derivative of function V^* (t) along the trajectories of system (2.1), we can obtain $\begin{matrix} D^{+} V (t) \\ \leq \underset{c \to 0^{+}}{lim^{¯}} \frac{{∥ ν x (t + h) ∥}_{p} - {∥ ν x (t) ∥}_{p}}{c} \\ \leq \underset{c \to 0^{+}}{lim^{¯}} \frac{{∥ E + c e^{t} (- H (t)) ∥}_{p} - 1}{c} {∥ ν x (t) ∥}_{p} \\ + ({∥ ν L (t) ν^{- 1} ∥}_{p} \cdot {∥ ν g (x (t)) ∥}_{p} \\ + {∥ ν Q (t) ν^{- 1} ∥}_{p} \cdot {∥ ν g (x (t - τ (t))) ∥}_{p} + {∥ ν I (t) ∥}_{p} \\ + {∥ ν R (t) ν^{- 1} ∥}_{p} + {∥ \int_{- \infty}^{t} P (t - s) ν g (x (s)) d s ∥}_{p}) e^{t} . \end{matrix}$

From hypothesis (A2), it is easy to get that $\begin{matrix} {∥ g (x (t)) ∥}_{p} \leq λ {∥ x (t) ∥}_{p} + ξ, \\ {∥ g (x (t - τ)) ∥}_{p} \leq λ {∥ x (t - τ) ∥}_{p} + ξ, \\ {∥ \int_{- \infty}^{t} p (t - s) g (s) d s ∥}_{p} \\ \leq λ \int_{- \infty}^{t} p (t - s) {∥ x (s) ∥}_{p} d s + ξ, \end{matrix}$ where λ = max(λ₁, λ₂, . . . , λ_m) , ξ = max(ξ₁, ξ₂, . . . , ξ_m) .

It follows from the Definition 2.3, one has $\begin{matrix} D^{+} V (t) \\ \leq [{μ (- H (t)) + {∥ ν L (t) ν^{- 1} ∥}_{p} {∥ λ_{f} ∥}_{p}} {∥ ν x (t) ∥}_{p} \\ + {∥ ν Q (t) ν^{- 1} ∥}_{p} {∥ λ_{f} ∥}_{p} \cdot {∥ ν x (t - τ (t)) ∥}_{p} \\ + {∥ ν R (t) ν^{- 1} ∥}_{p} {∥ λ_{f} ∥}_{p} \int_{- \infty}^{t} P (t - s) {∥ ν x (s) ∥}_{p} d s \\ + {∥ ν ξ_{f} ∥}_{p} ({∥ ν Q (t) ν^{- 1} ∥}_{p} + {∥ ν L (t) ν^{- 1} ∥}_{p} \\ + {∥ ν R (t) ν^{- 1} ∥}_{p}) + {∥ ν I (t) ∥}_{p}] e^{t} \\ \leq (\tilde{α} V (t) + \tilde{β} sup_{t - \bar{τ} \leq s \leq t} (V (s)) + χ^{*} \\ + \tilde{φ} \int_{- \infty}^{t} P (t - s) V (s) d s) e^{t}, \end{matrix}$ where $\tilde{α}, \tilde{β}, \tilde{φ}$ and χ^* are given by above. According to Lemma 2.2, we can get that there exist $\overset{\leftrightarrow}{μ}$ and $\overset{\leftrightarrow}{ϱ} < ϱ$ such that $\begin{matrix} {∥ ν x (t) ∥}_{p} = & V^{*} (t) \leq χ^{★} + (sup_{s \in (- \infty, t_{0}]} V^{*} (0)) \\ e^{- \frac{\overset{\leftrightarrow}{\overset{\leftrightarrow}{ϱ}}}{μ} (t - t_{0})}, t > t_{0} \geq 0 . \end{matrix}$ And also exist $\overset{\leftrightarrow}{ϱ}$ and $\overset{\leftrightarrow}{μ}$ satisfies $\begin{matrix} \overset{\leftrightarrow}{ϱ} = & - sup_{0 \leq t < + \infty} \overset{\leftrightarrow}{μ} + (- | \tilde{α} | + | \tilde{β} | e^{\overset{\leftrightarrow}{μ} τ^{*}} \\ + | \tilde{φ} | \int_{0}^{\infty} P (s) e^{\overset{\leftrightarrow}{μ} s} d s) e^{t} . \end{matrix}$ Then, for the given sufficient small ɛ > 0, there exists ℓ>0 such that ${∥ ν x (t) ∥}_{p} \leq χ^{*} + ɛ, \forall t \geq ℓ .$ The proof of theorem 3.2 is completed.

Remark 3.1. In theorem 3.4, compared with the results in Theorem 3.2 of [45], the slack variable ν and e^t are introduced to improve the conservation of neural networks.

Remark 3.2. When the activation function becomes continuous and the term of distributed delay are zero matrices in neural networks model (2.1), we can obtain the similar results which are accordant with Theorem 3.1 in [2].

II. FINITE-TIME SYNCHRONIZATION

In the following theorem, we discuss (2.1) finite-time Synchronization.

3.1 First feedback controller.

In this subsection, we will derive some criteria to guarantee the finite-time synchronization of systems (2.1) and (2.7). Let us consider the following discontinuous state feedback controllers of the form: $u_{i} (t) = - k_{i} e_{i} (t) - v_{i} sign (e_{i} (t)),$ (3.1) where k_i, v_i are the controller gains and i = 1, 2, . . . , m . Subtracting (2.1) from (2.7), synchronization error system can be derived as follows: $\begin{matrix} {\dot{e}}_{i} (t) = - h_{i} e_{i} (t) + \sum_{j = 1}^{m} {\frac{\overset{⌣}{1 - μ}}{l}}_{ij} (t) β_{j} (e_{j} (t)) \\ + \sum_{j = 1}^{m} {\frac{\overset{⌣}{ρ}}{q}}_{ij} (t) β_{j} (e_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} {\overset{⌣}{r}}_{ij} (t) \int_{- \infty}^{t} β_{j} (e_{j} (s)) p_{ij} (t - s) d s \\ + \sum_{j = 1}^{m} ({\overset{⌢}{l}}_{ij} (t) - {\overset{⌣}{l}}_{ij} (t)) g_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{m} ({\overset{⌢}{q}}_{ij} (t) - {\overset{⌣}{q}}_{ij} (t)) g_{j} (y_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} ({\overset{⌢}{r}}_{ij} (t) - {\overset{⌣}{r}}_{ij} (t)) \int_{- \infty}^{t} g_{j} (y_{j} (s)) \\ p_{ij} (t - s) d s + u_{i} (t), \end{matrix}$ (3.2) where β_j (e_j (•)) = g_j (y_j (•)) - g_j (x_j (•)) , t ≥ 0, i, j = 1, 2, . . . , m .

Theorem 3.3. Suppose the assumptions (A1)-(A6)satisfies and the following conditions are established. Then, neural network (2.1) and (2.7) can achieve finite-time synchronization under discontinuous controller (3.1). $\begin{matrix} k_{i} \geq - h_{i} + \sum_{j = 1}^{m} [λ_{j} \vec{L} + 1 λ_{j} \vec{Q} + λ_{j} \vec{R}], \\ v_{i} \geq \sum_{j = 1}^{m} \vec{L} ξ_{j} + \sum_{j = 1}^{m} \vec{Q} ξ_{j} + \sum_{j = 1}^{m} \vec{R} ξ_{j} \\ + \sum_{j = 1}^{m} (\vec{L} - \frac{\overset{\leftarrow}{1 - μ}}{L}) (λ_{j} {\bar{M}}_{j} + ξ_{j}) \\ + \sum_{j = 1}^{m} (\vec{Q} - \overset{\leftarrow}{Q}) (λ_{j} {\bar{M}}_{j} + ξ_{j}) \\ + \sum_{j = 1}^{m} (\vec{R} - \overset{\leftarrow}{R}) (λ_{j} {\bar{M}}_{j} + ξ_{j}) = ϖ_{i} . \end{matrix}$ We set $\vec{L} = max {| {l^{'}}_{ij} |, | {l^{″}}_{ij} |}, \overset{\leftarrow}{L} = min {| {l^{'}}_{ij} |, | {l^{″}}_{ij} |}, \vec{Q} = max {| {q^{'}}_{ij} |, | {q^{″}}_{ij} |},$ $\overset{\leftarrow}{Q} = min {| {q^{'}}_{ij} |, | {q^{″}}_{ij} |},$ $\vec{R} = max {| {r^{'}}_{ij} |, | {r^{″}}_{ij} |}, \overset{\leftarrow}{R} = min {| {r^{'}}_{ij} |, | r^{″} |},$ for i, j = 1, 2, . . . , m, where $\vec{L}, \overset{\leftarrow}{L}, \vec{Q}, \overset{\leftarrow}{Q}, \vec{R}, \overset{\leftarrow}{R}$ are constants.

Proof. We define the following Lyapunov functional candidate: $V (t) = V_{a} (t) + V_{b} (t) + V_{c} (t),$ where

$\begin{matrix} V_{a} (t) = \bar{a} \sum_{i = 1}^{m} | e_{i} (t) | = \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) e_{i} (t), \\ V_{b} (t) = \bar{b} \sum_{i = 1}^{m} \sum_{j = 1}^{m} 1 λ_{j} \vec{Q} \int_{t - τ (t)}^{t} | e_{j} (s) | ds, \\ V_{c} (t) = \bar{c} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{R} \int_{- \infty}^{0} \int_{t + s}^{t} p_{ij} (- s) | e_{j} (u) | duds, \end{matrix}$

where $\bar{a}, \bar{b}, \bar{c}$ are positive numbers, then according to Lemma 2.1, we have

$\begin{matrix} {\dot{V}}_{a} (t) = \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) {- h_{i} e_{i} (t) \\ + \sum_{j = 1}^{m} {\frac{\overset{⌣}{1 - μ}}{l}}_{ij} (t) β_{j} (e_{j} (t)) \\ + \sum_{j = 1}^{m} {\overset{⌣}{q}}_{ij} (t) β_{j} (e_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} \overset{⌣}{r} (t) \int_{- \infty}^{t} β_{j} (e_{j} (s)) p_{ij} (t - s) d s \\ + \sum_{j = 1}^{m} ({\overset{⌢}{l}}_{ij} (t) - {\overset{⌣}{l}}_{ij} (t)) g_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{m} ({\overset{⌢}{q}}_{ij} (t) - {\overset{⌣}{q}}_{ij} (t)) g_{j} (y_{j} (t - τ_{ij} (t))) \\ + \sum_{j = 1}^{m} ({\overset{⌢}{r}}_{ij} (t) - {\overset{⌣}{r}}_{ij} (t)) \int_{- \infty}^{t} g_{j} (y_{j} (s)) \\ p_{ij} (t - s) d s + u_{i} (t)} . \end{matrix}$ (3.3) Obviously, $\bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) (- h_{i} e_{i} (t)) = - \bar{a} \sum_{i = 1}^{m} h_{i} | e_{i} (t) | .$ (3.4) From (A2), the following formula is obtained $\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) \sum_{j = 1}^{m} {\overset{⌣}{l}}_{ij} (t) β_{j} (e_{j} (t)) \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} \vec{L} | β_{j} (e_{j} (t)) | \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{L} | e_{j} (t) | + \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} ξ_{j} ς_{i} \vec{L}, \\ i, j = 1, 2, . . ., m, \end{matrix}$ (3.5) where ς_i = 1 if e_i (t) ≠ 0, and otherwise ς_i = 0. From hypothesis (A2), we can also get: $\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) \sum_{j = 1}^{m} {\overset{⌣}{q}}_{ij} (t) β_{j} (e_{j} (t - τ_{ij} (t))) \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{Q} | e_{j} (t - τ_{ij} (t)) | + \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} ξ_{j} ς_{i} \vec{Q} \\ i, j = 1, 2, . . ., m, \end{matrix}$ (3.6) and

$\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) \sum_{j = 1}^{m} {\overset{⌣}{r}}_{ij} (t) \int_{- \infty}^{t} β_{j} (e_{j} (s)) p_{ij} (t - s) ds \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{R} \int_{- \infty}^{t} | e_{j} (s) | p_{ij} (t - s) ds + \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} ξ_{j} ς_{i} \vec{R} \\ i, j = 1, 2, . . ., m . \end{matrix}$ (3.7)

One can obtain from (A2) and (A5) that $\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) \sum_{j = 1}^{m} ({\overset{⌢}{l}}_{ij} (t) - {\overset{⌣}{l}}_{ij} (t)) g_{j} (y_{j} (t)) \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} (\vec{L} - \overset{\leftarrow}{L}) (λ_{j} {\bar{M}}_{j} + ξ_{j}) ς_{i} . \end{matrix}$ (3.8) Similarly, one can have

$\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) \sum_{j = 1}^{m} ({\overset{⌢}{q}}_{ij} (t) - {\overset{⌣}{q}}_{ij} (t)) g_{j} (y_{j} (t - τ_{ij} (t))) \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} (\vec{Q} - \overset{\leftarrow}{Q}) (λ_{j} {\bar{M}}_{j} + ξ_{j}) ς_{i}, \end{matrix}$ (3.9)

and

$\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) \sum_{j = 1}^{m} ({\overset{⌢}{r}}_{ij} (t) - {\overset{⌣}{r}}_{ij} (t)) \\ \int_{- \infty}^{t} g_{j} (y_{j} (s)) p_{ij} (t - s) d s \\ \leq \bar{a} \sum_{i = 1}^{m} \sum_{j = 1}^{m} (\vec{R} - \overset{\leftarrow}{R}) (λ_{j} {\bar{M}}_{j} + ξ_{j}) ς_{i} . \end{matrix}$ (3.10)

It is obvious that $\begin{matrix} \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) u_{i} (t) \\ = \bar{a} \sum_{i = 1}^{m} sig n^{T} (e_{i} (t)) [- k_{i} e_{i} (t) - v_{i} sign (e_{i} (t))] \\ \leq - \bar{a} \sum_{i = 1}^{m} k_{i} | e_{i} (t) | - \bar{a} \sum_{i = 1}^{m} v_{i} ς_{i} . \end{matrix}$ (3.11) Here ς_i is the same as (3.5). Substituting (3.4)-(3.11) into (3.3), we can derive $\begin{matrix} {\dot{V}}_{a} (t) \\ \leq \bar{a} \sum_{i = 1}^{m} {- (h_{i} + k_{i}) | e_{i} (t) | \\ + \sum_{j = 1}^{m} [λ_{j} | \vec{L} | | e_{j} (t) | + λ_{j} | \vec{Q} | | e_{j} (t - τ_{ij} (t)) | \\ + λ_{j} | \vec{R} | \int_{- \infty}^{t} | e_{j} (s) | p_{ij} (t - s) ds] - (v_{i} - ϖ_{i}) ς_{i}} . \end{matrix}$ (3.12)

According to (A3) and (A4), calculating the derivative of V_b (t) and V_c (t), we get ${\dot{V}}_{b} (t) \leq \bar{b} \sum_{i = 1}^{m} \sum_{j = 1}^{m} 1 λ_{j} \vec{Q} | e_{j} (t) |$ $- \bar{b} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{Q} | e_{j} (t - τ_{ij} (t)) |,$ (3.13) and $\begin{matrix} {\dot{V}}_{c} (t) \\ \leq \bar{c} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{R} \int_{- \infty}^{0} | e_{j} (t) | p_{ij} (- s) ds \\ - \bar{c} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{R} \int_{- \infty}^{0} | e_{j} (t + s) | p_{ij} (- s) ds \\ \leq \bar{c} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{R} | e_{j} (t) | \\ - \bar{c} \sum_{i = 1}^{m} \sum_{j = 1}^{m} λ_{j} \vec{R} \int_{- \infty}^{t} | e_{j} (s) | p_{ij} (t - s) d s . \end{matrix}$ (3.14)

It is derived from (3.12)-(3.14) that $\begin{matrix} \dot{V} (t) \leq \bar{a} \sum_{i = 1}^{m} {- (h_{i} + k_{i}) + \sum_{j = 1}^{m} [λ_{j} \vec{L} + \frac{1}{1 - μ} λ_{j} \vec{Q} \\ + λ_{j} \vec{R}]} \times | e_{j} (t) | - \bar{a} \sum_{i = 1}^{m} \partial_{i} ς_{i}, \end{matrix}$ (3.15) where $\bar{a} \leq \bar{b}, \bar{a} \leq \bar{c}, \partial_{i} = min {ν_{i} - ϖ_{i}} > 0, i = 1, 2, . . . m .$ If |e_j (t) | ≠ 0, we can get that:

$\dot{V} (t) \leq - \bar{a} \sum_{i = 1}^{m} \partial_{i} ς_{i} \leq - \bar{a} \partial,$ where ∂ = min{ ∂_i, i = 1, 2, . . . m }. Therefore, there exist nonnegative constants $\hat{V}$ and t_* > 0 such that $lim_{t \to t^{*}} V (t) = \hat{V}$ and $V (t) \equiv \hat{V}, \forall t \geq t_{*}$ and ∥e (t_*) ∥ ₁ = 0, ∥ e (t) ∥ ₁ ≡ 0, ∀ t ≥ t_*.

According to Definition 2.5, system (2.1) and (2.7) achieve finite-time synchronization. $\dot{V} (t) \leq - \bar{a} \partial, \forall t \in (t_{0}, t_{*}),$ integrating this inequality, we get that $t_{*} \leq \frac{V (t_{0})}{\bar{a} \partial} + t_{0}$ . The proof is completed.

3.2. Second feedback controller

In this section, based on Lyapunov stability theorem, we will address several sufficient conditions for a finite time synchronization system (2.1) and (2.7) by designing a suitable feedback controller:

$\begin{matrix} u (t) = - sign (e (t)) M | e (t) | - sign (e (t)) N | e ((t - τ (t)) | \\ - sign (e (t)) S \int_{- \infty}^{t} P (t - s) | e (s) | d s - sign (e (t)) W ζ \\ - \tilde{ϖ} sign (e (t)) {| e (t) |}^{ς}, \end{matrix}$ (3.16)

where $\tilde{ϖ} > 0, - 1 < ς < 1$ are constants, sign (e (t)) = diag (sign (e₁ (t)) , sign (e₂ (t)) , . . . , sign (e_m (t))) , |e (t) |^ς = (|e₁ (t) |^ς, |e₂ (t) |^ς, . . . , |e_m (t) |^ς) ^T, ζ = (ζ₁, ζ₂, . . . , ζ_m) ^T, and M, N, S, W are matrices to be determined.

The (3.2) is transformed into a matrix form:

$\begin{matrix} \dot{e} (t) = - He (t) + \overset{⌣}{\overset{\leftrightarrow}{L}} (t) β (e (t)) + \frac{\overset{⌣}{χ}}{Q} (t) β (e (t - τ (t))) \\ + \overset{⌣}{R} (t) \int_{- \infty}^{t} β (e (s)) P (t - s) d s \\ + (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) g (y (t)) \\ + (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) g (y (t - τ (t))) \\ + (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) \int_{- \infty}^{t} g (y (s)) P (t - s) d s + u (t), \end{matrix}$

where y (t) = (y₁ (t) , y₂ (t) , . . . , y_m (t)) ^T, j = 1, 2, . . . m, $g (y (.)) = {(g_{1} (y_{1} (.)), g_{2} (y_{2} (.)), . . ., g_{m} (y_{m} (.)))}^{T}, H = diag (h_{1}, h_{2}, . . ., h_{m}), \overset{⌣}{L} (t) = {({\overset{⌣}{l}}_{ij} (t))}_{m \times m}, \overset{⌢}{L} (t) = {({\overset{⌢}{l}}_{ij} (t))}_{m \times m}, \overset{⌣}{Q} (t) = {({\overset{⌣}{q}}_{ij} (t))}_{m \times m}, \overset{⌢}{Q} (t) = {({\overset{⌢}{q}}_{ij} (t))}_{m \times m},$ $\overset{⌣}{R} (t) = {({\overset{⌣}{r}}_{ij} (t))}_{m \times m}, \overset{⌢}{R} (t) = {({\overset{⌢}{r}}_{ij} (t))}_{m \times m}, u (t) = {(u_{1} (t), u_{2} (t), . . ., u_{m} (t))}^{T} .$

Theorem 3.4. Suppose the assumptions (A1)-(A6) satisfies and $\begin{matrix} ∥ W ∥ \geq \frac{ξ}{ζ} (∥ \overset{⌣}{L} (t) ∥ + ∥ \overset{⌣}{Q} (t) ∥ + ϒ ∥ \overset{⌣}{R} (t) ∥) \\ + (∥ (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) ∥ + ∥ (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) ∥ \\ + ∥ (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) ∥), ∥ N ∥ \geq ∥ \overset{⌣}{Q} (t) ∥ λ, ∥ S ∥ \\ \geq ∥ \overset{⌣}{R} (t) ∥ λ ϒ, ∥ M ∥ \geq ∥ \overset{⌣}{L} (t) ∥ λ - ∥ H ∥, \end{matrix}$ establish, where ξ = max (ξ_i) , ζ = max (ζ_i) , λ = max (λ_i) , i = 1, 2, . . . , m. Then, neural network (2.1) and (2.7) can achieve finite time synchronization under discontinuous controller (3.16), and the settling time $t_{*} = \frac{2 {(V (0))}^{\frac{ς + 1}{2}}}{\tilde{ϖ} (1 - ς)}$ .

Proof. Consider the following Lyapunov-Krasovskii functional $V (t) = \frac{1}{2} e^{T} (t) e (t) .$ Calculating the upper right-hand derivative of V (t) along the positive half trajectory of system (2.1), we have

$\begin{matrix} \dot{V} (t) = e^{T} (t) \dot{e} (t) \\ = e^{T} (t) [- He (t) + \overset{⌣}{L} (t) β (e (t)) + \overset{⌣}{Q} (t) β (e (t - τ (t))) \\ + \overset{⌣}{R} (t) \int_{- \infty}^{t} β (e (s)) P (t - s) d s + (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) g (y (t)) \\ + (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) g (y (t - τ (t))) \\ + (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) \int_{- \infty}^{t} g (y (s)) P (t - s) d s + u (t)] . \end{matrix}$ (3.17)

According to (A2), we can derive $\begin{matrix} e^{T} (t) \overset{⌣}{L} (t) β (e (t)) \\ \leq {∥ e (t) ∥}^{T} ∥ \overset{⌣}{L} (t) ∥ ∥ β (e (t)) ∥ \\ \leq {∥ e (t) ∥}^{T} ∥ \overset{⌣}{L} (t) ∥ (λ ∥ \overset{⌣}{L} (t) ∥ + ξ) \\ \leq λ ∥ \overset{⌣}{L} (t) ∥ {∥ e (t) ∥}^{2} + ξ ∥ \overset{⌣}{L} (t) ∥ ∥ e (t) ∥ . \end{matrix}$ (3.18)

Similarly, we can obtain $e^{T} (t) \overset{⌣}{Q} (t) β (e (t - τ (t)))$ $\leq λ {∥ e (t) ∥}^{T} ∥ \overset{⌣}{Q} (t) ∥ ∥ e (t - τ (t)) ∥ + ξ ∥ \overset{⌣}{Q} (t) ∥ ∥ e (t) ∥ .$ (3.19)

From Lemma 2.3, for constants $\vec{χ}, \overset{\leftarrow}{Q} > 0,$

$∥ e (t) ∥ ∥ e (t - τ (t)) ∥ \leq \frac{\vec{χ}}{2} {∥ e (t) ∥}^{2} + \frac{1}{2 \vec{χ}} {∥ e (t - τ (t)) ∥}^{2} .$ (3.20)

Similarly, $∥ e (t) ∥ \int_{- \infty}^{t} P (t - s) ∥ e (s) ∥ d s$ $\leq \frac{\overset{\leftarrow}{χ}}{2} {∥ e (t) ∥}^{2} + \frac{1}{2 \overset{\leftarrow}{χ}} {(\int_{- \infty}^{t} P (t - s) ∥ e (s) ∥ d s)}^{2} .$ (3.21)

Then, by Lemma 2.4 and Assumption (A6),

${(\int_{- \infty}^{t} P (t - s) ∥ e (s) ∥ d s)}^{2} \leq ϒ \int_{- \infty}^{t} P (t - s) {∥ e (s) ∥}^{2} d s .$ (3.22)

So, (3.19) can be changed into $e^{T} (t) \overset{⌣}{χ} (t) β (e (t - τ (t))) \leq \frac{\vec{χ}}{2} ∥ \overset{⌣}{Q} (t) ∥ λ {∥ e (t) ∥}^{2}$ $+ \frac{1}{2 \vec{χ}} ∥ \overset{⌣}{Q} (t) ∥ λ {∥ e (t - τ (t)) ∥}^{2} + ξ ∥ \overset{⌣}{Q} (t) ∥ ∥ e (t) ∥ .$ (3.23)

Similarly, by Lemma 2.3 and Assumption (A6), $\begin{matrix} e^{T} (t) \overset{⌣}{R} (t) \int_{- \infty}^{t} P (t - s) f (e (s)) d s \\ \leq \frac{\overset{\leftarrow}{R}}{2} ∥ \overset{⌣}{χ} (t) ∥ λ {∥ e (t) ∥}^{2} + \frac{1}{2 \overset{\leftarrow}{R}} ∥ \overset{⌣}{χ} (t) ∥ λ ϒ \\ \int_{- \infty}^{t} P (t - s) {∥ e (s) ∥}^{2} d s + ξ ϒ ∥ \overset{⌣}{R} (t) ∥ ∥ e (t) ∥ . \end{matrix}$ (3.24)

According to assumption (A5),

$e^{T} (t) (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) g (y (t)) \leq ∥ e^{T} (t) ∥ ∥ (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) ∥ ζ,$ (3.25)

$e^{T} (t) (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) g (y (t - τ (t)))$ $\leq ∥ e^{T} (t) ∥ ∥ (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) ∥ ζ,$ (3.26) $e^{T} (t) (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) \int_{- \infty}^{t} g (y (s)) P (t - s) d s$ $\leq ∥ e^{T} (t) ∥ ∥ (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) ∥ ζ .$ (3.27)

On the other hand, $\begin{matrix} e^{T} (t) u (t) \\ \leq - ∥ e^{T} (t) ∥ ∥ M ∥ ∥ e (t) ∥ \\ - ∥ e^{T} (t) ∥ ∥ N ∥ ∥ e (t - τ (t)) ∥ \\ - ∥ e^{T} (t) ∥ ∥ S ∥ \int_{- \infty}^{t} P (t - s) ∥ e (s) ∥ d s \\ - ∥ e^{T} (t) ∥ ∥ W ∥ ζ - ∥ e^{T} (t) ∥ \tilde{ϖ} {∥ e (t) ∥}^{ς} \\ \leq - ∥ M ∥ {∥ e (t) ∥}^{2} - \frac{\vec{χ}}{2} ∥ N ∥ {∥ e (t) ∥}^{2} \\ - \frac{1}{2 \vec{χ}} ∥ N ∥ {∥ e (t - τ (t)) ∥}^{2} - \frac{\overset{\leftarrow}{L}}{2} ∥ S ∥ {∥ e (t) ∥}^{2} \\ - \frac{1}{2 \overset{\leftarrow}{χ}} ∥ Q ∥ \int_{- \infty}^{t} P (t - s) {∥ e (s) ∥}^{2} d s \\ - ∥ W ∥ ∥ e (t) ∥ ζ - \tilde{ϖ} {∥ e (t) ∥}^{ς + 1} . \end{matrix}$ (3.28)

Therefore, it is obvious that

$\begin{matrix} \dot{V} (t) \\ \leq λ ∥ \overset{⌣}{χ} (t) ∥ {∥ e (t) ∥}^{2} + ξ ∥ \overset{⌣}{L} (t) ∥ ∥ e (t) ∥ - ∥ H ∥ {∥ e (t) ∥}^{2} \\ + \frac{\vec{χ}}{2} ∥ \overset{⌣}{Q} (t) ∥ λ {∥ e (t) ∥}^{2} + \frac{1}{2 \vec{χ}} ∥ \overset{⌣}{Q} (t) ∥ λ {∥ e (t - τ (t)) ∥}^{2} \\ + ξ ∥ \overset{⌣}{Q} (t) ∥ ∥ e (t) ∥ + \frac{\overset{\leftarrow}{R}}{2} ∥ \overset{⌣}{χ} (t) ∥ λ {∥ e (t) ∥}^{2} \\ + \frac{1}{2 \overset{\leftarrow}{R}} ∥ \overset{⌣}{χ} (t) ∥ λ ϒ \int_{- \infty}^{t} P (t - s) {∥ e (s) ∥}^{2} d s \\ + ξ ϒ ∥ \overset{⌣}{R} (t) ∥ ∥ e (t) ∥ + ∥ e^{T} (t) ∥ ∥ (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) ∥ ζ \\ + ∥ e^{T} (t) ∥ ∥ (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) ∥ ζ + ∥ e^{T} (t) ∥ ∥ (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) ∥ ζ \\ - ∥ M ∥ {∥ e (t) ∥}^{2} - \frac{\vec{χ}}{2} ∥ N ∥ {∥ e (t) ∥}^{2} - \frac{1}{2 \vec{χ}} ∥ N ∥ {∥ e (t - τ (t)) ∥}^{2} \\ - \frac{\overset{\leftarrow}{L}}{2} ∥ S ∥ {∥ e (t) ∥}^{2} - \frac{1}{2 \overset{\leftarrow}{χ}} ∥ S ∥ \int_{- \infty}^{t} P (t - s) {∥ e (s) ∥}^{2} d s \\ - ∥ W ∥ ∥ e (t) ∥ ζ - \tilde{ϖ} {∥ e (t) ∥}^{ς + 1} . \end{matrix}$ (3.29)

We can derive that $\begin{matrix} \dot{V} (t) \\ \leq {- ∥ W ∥ ζ + ξ ∥ \overset{⌣}{χ} (t) ∥ + ξ ∥ \overset{⌣}{Q} (t) ∥ + ξ ϒ ∥ \overset{⌣}{R} (t) ∥ \\ + ∥ (\overset{⌢}{L} (t) - \overset{⌣}{L} (t)) ∥ ζ + ∥ (\overset{⌢}{Q} (t) - \overset{⌣}{Q} (t)) ∥ ζ \\ + ∥ (\overset{⌢}{R} (t) - \overset{⌣}{R} (t)) ∥ ζ} ∥ e (t) ∥ + {- ∥ H ∥ + λ ∥ \overset{⌣}{L} (t) ∥ \\ + \frac{\vec{χ}}{2} ∥ \overset{⌣}{Q} (t) ∥ λ + \frac{\overset{\leftarrow}{R}}{2} ∥ \overset{⌣}{χ} (t) ∥ λ - ∥ M ∥ - \frac{\vec{χ}}{2} ∥ N ∥ \\ - \frac{\overset{\leftarrow}{Q}}{2} ∥ S ∥} {∥ e (t) ∥}^{2} + {\frac{1}{2 \vec{χ}} ∥ \overset{⌣}{χ} (t) ∥ λ - \frac{1}{2 \vec{χ}} ∥ N ∥} \\ {∥ e (t - τ (t)) ∥}^{2} + {\frac{1}{2 \overset{\leftarrow}{R}} ∥ \overset{⌣}{χ} (t) ∥ λ ϒ - \frac{1}{2 \overset{\leftarrow}{2}} ∥ S ∥} \\ \int_{- \infty}^{t} P (t - s) {∥ e (s) ∥}^{2} d s - \tilde{ϖ} {∥ e (t) ∥}^{ς + 1} . \end{matrix}$ (3.30)

Suppose the conditions given in Theorem 3.8 are all established. According to Lemma 2.5; $\dot{V} (t) \leq - \tilde{ϖ} {∥ e (t) ∥}^{ς + 1} \leq - \tilde{ϖ} 2^{ς + 1} {(V (t))}^{\frac{ς + 1}{2}} .$ (3.31)

Therefore, according to Lemma 2.6, the master (2.1) and slave (2.7) system under control law (3.1) are synchronized, with the settling time $t_{*} = \frac{2 {(V (0))}^{\frac{ς + 1}{2}}}{\tilde{ϖ} (1 - ς)}$ , this proof has been completed.

Remark 3.3. For the finite time synchronization of MNNs with mixed delays and discontinuous activations, no related results can be found in the literature. Throughout this paper, Theorem 3.3 utilizes a delay-independent feedback controller and Theorem 3.8 utilizes a delay-dependent feedback controller.

Remark 3.4. Compared with previous results on synchronization of delayed neural networks [2], [14], [18], the improvement of this paper is as follows;

1. When the activation function becomes continuous and the parameter ξ_j is reduced in hypothesis (A2), we can obtain the similar results which are accordant with Theorem 2 in [2].

2. Literature [14] can be regarded as a special case in this paper, when the memristor term equals 0.

3. When the term of distributed delay in mixed delay equals 0 matrix and the activation function changes from discontinuous to continuous in neural networks model (2.1), we can obtain the similar results which are accordant with Theorem 3.1 obtained by [18].

4 Numerical examples

In this section, several examples are given to show the applicability and effectiveness of our main results.

Let’s consider the following memristor-based neural networks£º $\begin{matrix} \frac{d x_{i} (t)}{dt} = - h_{i} x_{i} (t) + \sum_{j = 1}^{2} l_{ij} (x_{i} (t)) g_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{2} q_{ij} (x_{i} (t)) g_{j} (x_{j} (t - τ_{ij} (t))) + I_{i}, \\ t \geq 0, i = 1, 2, \end{matrix}$ (4.1) where $\begin{matrix} l_{11} (x_{1} (t)) = {\begin{matrix} 0.2, | x_{1} (t) | \leq 1, \\ - 0.15, | x_{1} (t) | > 1, \end{matrix} \\ l_{12} (x_{1} (t)) = {\begin{matrix} - 1.0, | x_{1} (t) | \leq 1, \\ 0.9, | x_{1} (t) | > 1, \end{matrix} \\ l_{21} (x_{2} (t)) = {\begin{matrix} - 0.8, | x_{2} (t) | \leq 1, \\ 0.2, | x_{2} (t) | > 1, \end{matrix} \\ l_{22} (x_{2} (t)) = {\begin{matrix} 0.40, | x_{2} (t) | \leq 1, \\ - 0.10, | x_{2} (t) | > 1, \end{matrix} \\ q_{11} (x_{1} (t)) = {\begin{matrix} 0.65, | x_{1} (t) | \leq 1, \\ - 0.85, | x_{1} (t) | > 1, \end{matrix} \\ q_{12} (x_{1} (t)) = {\begin{matrix} - 0.35, | x_{1} (t) | \leq 1, \\ - 0.15, | x_{1} (t) | > 1, \end{matrix} \\ q_{21} (x_{2} (t)) = {\begin{matrix} 0.6, | x_{2} (t) | \leq 1, \\ - 0.75, | x_{2} (t) | > 1, \end{matrix} \\ q_{22} (x_{2} (t)) = {\begin{matrix} 0.9, | x_{2} (t) | \leq 1, \\ 0.55, | x_{2} (t) | > 1 . \end{matrix} \end{matrix}$

The activation function is: $\begin{matrix} g (x_{1}) = x_{1} + sign (| x_{1} | - 1), \\ g (x_{2}) = x_{2} + sign (| x_{2} | - 1) . \end{matrix}$

Now consider following slave system (2.7): $\begin{matrix} \frac{d y_{i} (t)}{dt} = - h_{i} y_{i} (t) + \sum_{j = 1}^{2} l_{ij} (y_{i} (t)) g_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{2} q_{ij} (y_{i} (t)) g_{j} (y_{j} (t - τ_{ij} (t))) + I_{i} + u_{i} (t), \\ t \geq 0, i = 1, 2 . \end{matrix}$ (4.2)

Let h₁ = 3.2, h₂ = 2.2, τ₁ = 0.1, τ₂ = 0.15, I₁ = I₂ = 0.1, λ₁ = λ₂ = 5, ξ₁ = ξ₂ = 0.5, μ = 0.2 . ξ_f = diag (1, 2) , λ_f = diag (0.5, 0.5) , ν = diag (0.5, 0.2), by simple calculation, there are ∥ξ_f ∥ ₁ = 2, ∥ λ_f ∥ ₁ = 1, μ₁ (- H (t)) = -3.2, ∥ νL (t) ν^-1 ∥ ₁ ∥ λ_f ∥ ₁ = 0.8, -h (t) = -2.4 < 0, l (t) = ∥ νQ (t) ν^-1 ∥ ₁ ∥ λ_f ∥ ₁ = 1.8, $\begin{matrix} χ^{*} = {∥ ν ξ_{f} ∥}_{1} ({∥ ν Q (t) ν^{- 1} ∥}_{1} + {∥ ν L (t) ν^{- 1} ∥}_{1} \\ + {∥ ν R (t) ν^{- 1} ∥}_{1}) + {∥ ν I (t) ∥}_{1} = 0.7, \end{matrix}$ $\overset{\leftrightarrow}{ρ} = 0.5$ , Ω₁ = χ^* = 1.4, ∥νx (t) ∥ ₁ = |0.5x₁ (t) | + |0.2x₂ (t) | ≤ 2.02. In theorem 3.2, all the conditions are satisfied, and the neural network (4.1) is globally dissipative. The results can be further validated in Figure 4. In this example, we choose ten sets of initial values, when i = 1, 2, for t ≥ 0. Especially, in Figs. 5, we also give the results in different situations for comparison. Compared with result in [7](ν = diag (1, 1)), less conservation result could be established when adding the relaxation variable ν (ν = diag (0.5, 0.2)).

Fig. 4

Globally dissipativity of the neural networks in Example with 10 groups of random initial conditions.

Fig. 5

The globally attractive set and positive invariant Ω₁ obtained by different ν.

We choose k₁ = 12.9, k₂ = 22.4, ν₁ = 8.1, ν₂ = 15.2, Based on the above data and Theorem 3.3, the system (4.2) and (4.1) are finite-time synchronization. The results can be further validated in Figure 6-8.

Fig. 6

When i= 1, the trajectories of systems (4.1) and (4.2) under controller (3.1).

Fig. 7

When i= 2, the trajectories of systems (4.1) and (4.2) under controller (3.1).

Fig. 8

Time response of synchronization error between systems (4.1) and (4.2) under the state feedback controller (3.1).

Consider the Theorem 3.8. The activation function are: $\begin{matrix} g (x_{1}) = x_{1} + sign (x_{1}), \\ g (x_{2}) = x_{2} + sign (x_{2}) . \end{matrix}$ We order ζ₁ = ζ₂ = 10, λ₁ = λ₂ = 1, ξ₁ = ξ₂ = 0.5, ϒ = 1, ς = 0 . $M = (\begin{matrix} - 2.55 & 1.12 \\ 0.85 & - 2.14 \end{matrix}), N = (\begin{matrix} 3 & 0.24 \\ 0.68 & 0.85 \end{matrix}),$ $S = (\begin{matrix} 3.56 & 0.18 \\ 0.76 & 4.85 \end{matrix}), W = (\begin{matrix} 0.84 & 2.57 \\ 2.54 & 1.44 \end{matrix}) .$

By simple calculation, we have $∥ W ∥ \geq \frac{1}{20} (∥ L ∥ + ∥ Q ∥ + ∥ R ∥) = 0.21, ∥ N ∥ \geq ∥ Q ∥ \times 1 = 1.41$ , ∥S ∥ ≥ ∥ R ∥ × 1 ×1 = 1.32, ∥ M ∥ ≥ ∥ L ∥ × 1 - ∥ H ∥ = 1.3, all the conditions are satisfied, the system (4.2) and (4.1) are finite time synchronized under the controller (3.16). The results can be further validated in Figure 9.

Fig. 9

Time response of synchronization error between systems (4.1) and (4.2) under the state feedback controller (3.16).

Remark 3.5. We define the norm of variable L,Q and take the largest of its 16 sets of values, for example ∥L ∥ = max(∥ L₁ ∥ , ∥ L₂ ∥ , . . . ∥ L₁₆ ∥).

5 Conclusions

Throughout this paper, we study the dissipativity and synchronization of memristor-based neural networks with discontinuous neuron activation and mixed delays. By using the generalized Halanay inequality and matrix measure method, the global dissipativity of Filippov solutions for MNNs with Mixed Time-Varying has been studied. Compared with other result, less conservative result could be established when adding the relaxation variable ν. In addition, under the generalized Lyapunov functional method, functional differential inclusion theory and non-smooth analysis theory, delay-independent and delay-dependent feedback controller have been designed to achieve fintite time synchronization. Finally, several numerical examples are given to verify the effectiveness of the main results.

Footnotes

Acknowledgments

The authors would like to appreciate the editor and the anonymous reviewers for their valuable comments and insightful advice, which has helped improve the quality of this paper. Supported by National Natural Science Foundation of China (Grant No. 61374028 and No. 61304162).

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