Delay-range-dependent Stability Criteria of Neural Networks with Time-varying Discrete and Distributed Delays

Abstract

This article deals with the global asymptotic stability problem for a class of neural networks with time-varying discrete and distributed delays. The activation functions are assumed to be neither monotonic nor differentiable, and two types of time-varying discrete delays are considered: one is differentiable and has bounded derivatives, and the other is continuous and may vary very fast. By constructing an appropriate Lyapunov-Krasovskii functional and employing a tighter inequality, new stability criteria dependent on both the lower bound and upper bound of the time-varying time delays are established to guarantee asymptotic stability for the addressed neural networks. It is shown that the new criteria improve some results from previous studies. Two simulation examples are given to show the effectiveness and the reduced conservatism of the proposed criteria.

Keywords

Delay-Dependent Global Asymptotic Stability Neural Networks (NNs)Time-Varying Delays Linear Matrix Inequality (LMI)

1. Introduction

Neural networks (NNs) have received considerable attention due to their extensive applications in various signal-processing problems such as optimization, fixed-point computations, and other areas in recent decades. Since integration and communication delays are unavoidably encountered both in biological and artificial neural systems, and may result in oscillation and instability, increasing interest has been focused on stability analysis of NNs with time delays [1 –7, 9 –20, 34 –42].

As is well known, the use of constant fixed delays in models of delayed feedback may provide a good approximation in simple circuits consisting of a small number of cells. However, neural networks usually have a spatial limit due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there is a distribution of conduction velocities along these pathways, as well as a distribution of propagation delays. In these circumstances the signal propagation is not instantaneous and cannot be modelled with discrete delays. A more appropriate approach is to incorporate continuously distributed delays in neural network models. In recent years there has been a growing interest in neural networks with discrete and distributed delays (see for example [21 –24, 27]). In [21, 22], several sufficient conditions to ensure the existence and global asymptotic stability of an equilibrium point for the neural networks with discrete and distributed constant delays are obtained. In [23], the global exponential stability problem is investigated for the neural networks with discrete and distributed constant delays, and two sufficient conditions are given. In [24, 27], robust stability is discussed for neural networks with discrete and distributed constant delays, and several sufficient conditions are derived to guarantee global robust stability for the neural networks. However, it should be noted that most results in [21 –24, 27] are only concerned with constant time delays. Very recently, in [25, 28], the global asymptotic stability problem is investigated for neural networks with discrete and distributed time-varying delays, where time-varying delays are continuously differentiable and the derivatives of time delays are bounded. In [26], the authors consider cellular neural networks with discrete and distributed time-varying delays, where the time-varying delays are continuous and may vary very fast. It is worth pointing out that, in [25, 26, 28], the low-bound time-varying delays are assumed to 0. In fact, they are not equal to 0. To our knowledge, only one study [14] has touched on delay NNs with nonzero low-bound time-varying delays.

Motivated by the above discussions, this article considers two types of time-varying delays: one is differentiable and has a bounded derivative; the other is continuous and may vary very fast. By constructing an appropriate Lyapunov-Krasovskii functional and employing a new inequality, new stability criteria are established, without ignoring certain terms, to guarantee asymptotic stability for the addressed neural network. Two simulation examples are given to show the effectiveness and lower conservatism of the proposed criteria.

Notation. Throughout this paper, in the real symmetric matrix P > 0(≥ 0) P is a positive definite (positive semi-definite) matrix, and A > (≥)B means A – B > (≥)0. I is used to denote an identity matrix with a proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetrical terms in a symmetrical matrix are denoted by ^* [43, 44, 45].

2. Problem formulation and preliminaries

Consider the following delayed neural networks with n neurons [43, 44, 45]:

\begin{array}{l} \dot{y} (t) = - A y (t) + W_{0} g (y (t)) + W_{1} g (y (t - d (t))) \\ + W_{2} \int_{t - τ (t)}^{t} g (y (s)) d s + J \end{array}

(1)

where y(·) = [y₁ (·),y₂ (·),⋯, y_n, (·)]^τ ε Rⁿ is the neuron state vector,

g (y (\cdot)) = {[g_{1} (y_{1} (\cdot)), g_{2} (y_{2} (\cdot)), \cdot \cdot \cdot, g_{n} (y_{n} (\cdot))]}^{τ} \in R^{n}

denotes the bounded neuron activation function with g_i(0) = 0, J = [J₁,J₂,⋯,J_n]^τ ε Rⁿ is a constant input vector, A = diag {a₁, a₂,⋯,a_n} is a positive diagonal matrix, $W_{0} = {(w_{i j}^{0})}_{n \times n}, W_{1} = {(w_{i j}^{1})}_{n \times n}$ and $W_{2} = {(w_{i j}^{2})}_{n \times n}$ are the interconnection matrices representing the weight coefficients of the neurons. d(t) and τ(t) are discrete time delay and distributed time delay, respectively. τ(t) is assumed to satisfy 0 < τ(t) ≤ τ, where τ is a constant.

In the following, each neuron activation function in (1), g_i(·),i = 1,2,…,n, satisfies the following condition (see [29, 30] for details):

\begin{array}{c} l_{i}^{-} \leq \frac{g_{i} (ζ_{1}) - g_{i} (ζ_{2})}{ζ_{1} - ζ_{2}} \leq l_{i}^{+}, \forall ζ_{1}, ζ_{2} \in R, \\ ζ_{1} \neq ζ_{2}, i = 1, 2, \dots, n \end{array}

(2)

where l⁻_i and l⁺_i are constants and l_i $l_{i} = \max {| l_{i}^{-} |, | l_{i}^{+} |}$ (i = 1,2,…,n).

As the activation functions are bounded, according to [23], there exists at least one equilibrium point for the NNs (1). Assume $y^{*} = {[y_{1}^{*}, y_{2}^{*}, …, y_{n}^{*}]}^{τ}$ is an equilibrium of system (1); one can then derive from (1) that the transformation z(·) = y(·) – y^* transforms system (1) into the following system:

\begin{array}{l} \dot{z} (t) = - A z (t) + W_{0} f (z (t)) + W_{1} f (z (t - d (t))) \\ + W_{2} \int_{t - τ (t)}^{t} f (z (s)) d s \end{array}

(3)

where z(t) = [z₁ (·),z₂ (·),⋯,z_n (t)]^τ is the state vector of the transformed system, $f (z (\cdot)) = {[f_{1} (z_{1} (\cdot)), f_{2} (z_{2} (\cdot)), \cdot \cdot \cdot, f_{n} (z_{n} (\cdot))]}^{τ}$ and $f_{i} (z_{i} (\cdot)) = g_{i} (z_{i} (t) + x_{i}^{*}) - g_{i} (x_{i}^{*}), i = 1, 2, …, n$ . Note that the functions f_i (·),i = 1,2,…,n satisfy the following condition:

\begin{array}{c} l_{i}^{-} \leq \frac{f_{i} (ζ_{1}) - f_{2} (ζ_{2})}{ζ_{1} - ζ_{2}} \leq l_{i}^{+}, \forall ζ_{1}, ζ_{2} \in R, \\ ζ_{1} \neq ζ_{2}, i = 1, 2, \dots, n \end{array}

(4)

Lemma 1. (Ku [8]) For any positive symmetrical constant matrix M ε R^n×n, with a scalar γ > 0, w: [0,r] → Rⁿ such that the integrations concerned are well defined:

{(\int_{0}^{γ} ω (s) d s)}^{τ} M (\int_{0}^{γ} ω (s) d s) \leq γ \int_{0}^{γ} ω^{τ} (s) M ω (s) d s

Lemma 2. (Li [33]) For any scalars where W₁ > 0, W₂ > 0 if 0 < d_m ≤d(t) ≤ d_M; the following inequality is then true:

\frac{W_{1}}{d (t) - d_{m}} + \frac{W_{2}}{d_{M} - d (t)} \geq \min {\frac{3 W_{1} + W_{2}}{d_{M} - d_{m}}, \frac{W_{1} + 3 W_{2}}{d_{M} - d_{m}}}

From the above analysis, we can see that the stability problem of system (1) on equilibrium y^* is changed into the zero stability problem of system (3). We will now consider two cases of time-varying discrete delays [43, 44, 45]:

Case 1. d(t) is a differential function, for all t ≥ 0 and for given scalars 0 ≤ μ < 1 and 0 ≤ h₁ < h₂ satisfying the condition that h₁ ≤ d(t) ≤ h₂, ḋ(t) ≤ μ

Case 2. d (t) is a continuous function, for all t ≥ 0 and given scalars 0 ≤ h₁ < h₂ satisfying the condition that h₁ ≤ d(t) ≤ h₂. In this case, very fast continuous changes in the time delay are allowed.

3. Main results

In this section, we shall establish our main criterion based on an appropriate Lyapunov-Krasovskii functional and a tighter inequality. First, we consider the case that the time-varying discrete delays satisfy Case 1.

Theorem 1. For given scalars $L_{1} = d i a g {l_{1}^{+} l_{1}^{-}, …, l_{n}^{+} l_{n}^{-}}, L_{2} = d i a g {(l_{1}^{+} + l_{1}^{-}), …, (l_{n}^{+} + l_{n}^{-})}, L_{3} = d i a g {| l_{1}^{-} |, …, | l_{n}^{-} |}, 0 \leq h_{1} \leq h_{2}, 0 < τ$ 0 ≤ μ for delay d(t) satisfying h₁ ≤ d(t) ≤ h₂, ḋ(t) ≤ μ and τ(t) satisfying 0 < τ(t) ≤ τ, the origin of system (3) is asymptotically stable if there exist matrices P₁ > 0, Q_i > 0 (i = 1,2,3,4), R₁ > 0, R₂ > 0, Z > 0, diagonal matrices Λ = diag{λ₁,λ₂,…,λ_n} > 0, D₁ = diag{d₁₁,d₁₂,…d_1n} > 0, D₂ = diag{d₂₁,d₂₂,…,d_2n} > 0 and any matrices P_i(i = 2,…,7) such that the following LMIs hold:

\begin{array}{l} [\begin{matrix} Ξ_{11} & Ξ_{12} & P_{4}^{τ} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{12} P_{4}^{τ} & - h_{12} P_{3}^{τ} & - A^{τ} U \\ * & Ξ_{22} & P_{7}^{τ} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{12} P_{7}^{τ} & - h_{12} P_{6}^{τ} & 0 \\ * & * & - Q_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & Ξ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & 0 & W_{0}^{τ} U \\ * & * & * & * & * & Ξ_{66} & 0 & 0 & 0 & 0 & W_{1}^{τ} U \\ * & * & * & * & * & * & - Z & 0 & 0 & 0 & W_{2}^{τ} U \\ * & * & * & * & * & * & * & - h_{2} R_{1} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & - h_{12} R_{2} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & - h_{12} (R_{1} + 3 R_{2}) & 0 \\ * & * & * & * & * & * & * & * & * & * & - U \end{matrix}] < 0 \end{array}

(5a)

\begin{array}{l} [\begin{matrix} Ξ_{11} & Ξ_{12} & P_{4}^{τ} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - H_{2} P_{2}^{τ} & - H_{12} P_{4}^{τ} & - H_{12} P_{3}^{τ} & - A^{τ} U \\ * & Ξ_{22} & P_{7}^{τ} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - H_{2} P_{5}^{τ} & - H_{12} P_{7}^{τ} & - H_{12} P_{6}^{τ} & 0 \\ * & * & - Q_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & Ξ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & 0 & W_{0}^{τ} U \\ * & * & * & * & * & Ξ_{66} & 0 & 0 & 0 & 0 & W_{1}^{τ} U \\ * & * & * & * & * & * & - Z & 0 & 0 & 0 & W_{2}^{τ} U \\ * & * & * & * & * & * & * & - H_{2} R_{1} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & - 3 H_{12} R_{2} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & - H_{12} (R_{1} + R_{2}) & 0 \\ * & * & * & * & * & * & * & * & * & * & - U \end{matrix}] < 0 \end{array}

(5b)

where

Ξ_{11} = - P_{1} A - A^{τ} P_{1} + P_{2} + P_{2}^{τ} + Q_{1} + Q_{3} + Q_{4} - 2 L_{3} Λ A - 2 D_{1} L_{1}

(6)

Ξ_{12} = - P_{2}^{τ} + P_{3}^{τ} + P_{5} - P_{4}^{τ}

(7)

Ξ_{15} = P_{1} W_{0} - A^{τ} Λ + L_{3} Λ W_{0} + D_{1} L_{2}

(8)

Ξ_{22} = - P_{5} - P_{5}^{τ} + P_{6} + P_{6}^{τ} - P_{7} - P_{7}^{τ} - (1 - μ) Q_{1} - 2 D_{2} L_{1}

(9)

Ξ_{55} = τ^{2} Z + Λ W_{0} + W_{0}^{τ} Λ + Q_{2} - 2 D_{1}

(10)

Ξ_{66} = - (1 - μ) Q_{2} - 2 D_{2}

(11)

U = h_{2} R_{1} + h_{12} R_{2}

(12)

and^* denotes the symmetrical terms in a symmetrical matrix.

Proof. Consider the following Lyapunov-Krasovskii functional candidate for NNs (3):

\begin{array}{l} V (z (t)) = V_{1} (z (t)) + V_{2} (z (t)) + V_{3} (z (t)) \\ + V_{4} (z (t)) + V_{5} (z (t)) + V_{6} (z (t)) \end{array}

Where

\begin{array}{l} V_{1} (z (t)) = ξ_{0}^{τ} (t) E P ξ_{0} (t) \\ V_{2} (z (t)) = 2 \sum_{i = 1}^{n} λ_{i} \int_{0}^{z_{j}} (f_{j} (s) + | l_{i}^{-} | s) d s \\ V_{3} (z (t)) = \int_{- h_{2}}^{0} \int_{t + θ}^{t} {\dot{z}}^{τ} (s) R_{1} \dot{z} (s) d s d θ \\ V_{4} (z (t)) = \int_{- h_{2}}^{- h_{1}} \int_{t + θ}^{t} {\dot{z}}^{τ} (s) R_{2} \dot{z} (s) d s d θ \\ V_{5} (z (t)) = \int_{t - d (t)}^{t} [z^{τ} (s) Q_{1} z (s) d s + f^{τ} (z (s)) Q_{2} f (z (s))] d s \\ + \int_{t - h_{1}}^{t} z^{τ} (s) Q_{3} z (s) d s + \int_{t - h_{2}}^{t} z^{τ} (s) Q_{4} z (s) d s \\ V_{6} (z (t)) = τ \int_{- τ}^{0} \int_{t + 0}^{t} f^{τ} (z (s)) Z f (z (s)) d s d θ \end{array}

with

E = [\begin{array}{l} I & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], P = [\begin{array}{l} P_{1} & 0 \\ P_{2} & P_{5} \\ P_{3} & P_{6} \\ P_{4} & P_{7} \end{array}], ξ_{0} (t) = [\begin{matrix} z (t) \\ z (t - d (t)) \end{matrix}]

and

\begin{array}{l} E P = P^{τ} E^{τ} \geq 0, P_{1} > 0, R_{1} > 0, \\ R_{2} > 0, Q_{i} > 0, (i = 1, 2, 3, 4), Z > 0 \end{array}

Note that $ξ_{0}^{τ} (t) E P ξ_{0} (t)$ (t)EPξ₀(t) is actually z^τ (t)P₁z (t). On the other hand, from the Leibniz-Newton formula, the following equations are true:

α_{1} : = z (t) - z (t - d (t)) - \int_{t - d (t)}^{t} \dot{z} (s) d s = 0

(13)

α_{2} : = z (t - d (t)) - z (t - h_{2}) - \int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s = 0

(14)

α_{3} : = z (t - h_{1}) - z (t - d (t)) - \int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s = 0

(15)

The time derivative of V₁ (z(t)) along the trajectories of delay NNs (3) with (13)–(15), is obtained as:

\begin{array}{c} \dot{V} (z (t)) = 2 z^{τ} P_{1} \dot{z} (t) = 2 ξ_{0}^{τ} (t) P^{τ} [\begin{matrix} \dot{z} (t) \\ 0 \\ 0 \\ 0 \end{matrix}] = 2 ξ_{0}^{τ} (t) P^{τ} [\begin{matrix} \dot{z} (t) \\ α_{1} \\ α_{2} \\ α_{3} \end{matrix}] \\ = - 2 z^{τ} (t) P_{1} A z (t) + 2 z^{τ} (t) P_{1} W_{0} f (z (t)) + \\ + 2 z^{τ} (t) P_{1} W_{1} f (z (t - d (t))) \\ + 2 z^{τ} (t) P_{1} W_{2} \int_{t - τ}^{t} f (z (s)) d s + 2 z^{τ} (t) P_{2}^{τ} z (t) \\ - 2 z^{τ} (t) P_{2}^{τ} z (t - d (t)) - 2 z^{τ} (t) \\ P_{2}^{τ} \int_{t - d (t)}^{t} \dot{z} (s) d s + 2 z^{τ} (t - d (t)) P_{5}^{τ} z (t) \\ - 2 z^{τ} (t - d (t)) P_{5}^{τ} z (t - d (t)) - 2 z^{τ} (t - d (t)) \\ P_{5}^{τ} \int_{t - d (t)}^{t} \dot{z} (s) d s + 2 z^{τ} (t) P_{3}^{τ} z (t - d (t)) \\ - 2 z^{τ} (t) P_{3}^{τ} z (t - h_{2}) - 2 z^{τ} (t) \\ P_{3}^{τ} \int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s + 2 z^{τ} (t - d (t)) P_{6}^{τ} z (t - d (t)) \\ - 2 z^{τ} (t - d (t)) P_{6}^{τ} z (t - h_{2}) - 2 z^{τ} (t - d (t)) P_{6}^{τ} \int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s \\ + 2 z^{τ} (t) P_{4}^{τ} z (t - h_{1}) - 2 z^{τ} (t) P_{4}^{τ} z (t - d (t)) \\ - 2 z^{τ} (t) P_{4}^{τ} \int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s \\ + 2 z^{τ} (t - d (t)) P_{7}^{τ} z (t - h_{1}) - 2 z^{τ} (t - d (t)) P_{7}^{τ} \\ (t - d (t)) - 2 z^{τ} (t - d (t)) P_{7}^{τ} \int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s \\ = ξ_{1}^{τ} (t) Π_{1} ξ_{1} (t) . \end{array}

where

Π_{1} = [\begin{matrix} Π_{11} & Π_{12} & P_{4}^{τ} & - P_{3}^{τ} & P_{1} W_{0} & P_{1} W_{1} & P_{1} W_{2} & - P_{2}^{τ} & - P_{4}^{τ} & - P_{3}^{τ} \\ * & Π_{22} & P_{7}^{τ} & - P_{6}^{τ} & 0 & 0 & 0 & - P_{5}^{τ} & - P_{7}^{τ} & - P_{6}^{τ} \\ * & * & 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 & 0 \\ * & * & * & * & * & * & * & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & 0 & 0 \\ * & * & * & * & * & * & * & * & * & 0 \end{matrix}], ξ_{1} (t) = [\begin{matrix} z (t) \\ z (t - d (t)) \\ z (t - h_{1}) \\ z (t - h_{2}) \\ f (z (t)) \\ f (z (t - d (t))) \\ \int_{t - τ (t)}^{t} f (z (s)) d s \\ \int_{t - d (t)}^{t} \dot{z} (s) d s \\ \int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s \\ \int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s \end{matrix}]

and

\begin{array}{l} Π_{11} = - P_{1} A - A^{τ} P_{1} + P_{2} + P_{2}^{τ} \\ Π_{12} = - P_{2}^{τ} + P_{5} + P_{3}^{τ} - P_{4}^{τ} \\ Π_{22} = - P_{5} - P_{5}^{τ} + P_{6} + P_{6}^{τ} - P_{7} - P_{7}^{τ} \\ {\dot{V}}_{2} (z (t)) = 2 (f^{τ} (z (t)) + z^{τ} (t) L_{3}) Λ \dot{z} (t) \\ = 2 (f^{τ} (z (t)) + z^{τ} (t) L_{3}) \\ Λ [- A z (t) + W_{0} f (z (t)) + W_{1} f (z (t - d (t))) + W_{2} \int_{t - τ (t)}^{t} f (z (s)) d s] \end{array}

(17)

where Λ = diag {λ₁, λ₂,…, λ_n}, L₃ = diag $Λ = d i a g {λ_{1}, λ_{2}, …, λ_{n}}, L_{3} = d i a g {| l_{1}^{-} |, | l_{2}^{-} |, …, | l_{n}^{-} |}$ Similarly, computing the time derivatives V₃ (z(t)) and V₄ (z(t)) with Jensen integral inequality [8], we have:

\begin{array}{l} {\dot{V}}_{3} (z (t)) = h_{2} {\dot{z}}^{τ} (t) R_{1} \dot{z} (t) - \int_{t - h_{2}}^{t} {\dot{z}}^{τ} (s) R_{1} \dot{z} (s) d s \\ = h_{2} {\dot{z}}^{τ} (t) R_{1} \dot{z} (t) - \int_{t - h_{2}}^{t - d (t)} {\dot{z}}^{τ} (s) \\ R_{1} \dot{x} (s) d s - \int_{t - d (t)}^{t} {\dot{z}}^{τ} (s) R_{1} \dot{z} (s) d s \\ \leq h_{2} {\dot{z}}^{τ} (t) R_{1} \dot{z} (t) - \frac{1}{h_{12}} {(\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s)}^{τ} \\ R_{1} (\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s) - \frac{1}{h_{2}} (\int_{t - d (t)}^{t} \dot{z} (s) d s) R_{1} ((\int_{t - d (t)}^{t} \dot{z} (s) d s)) \end{array}

(18)

\begin{array}{l} {\dot{V}}_{4} (z (t)) = h_{12} {\dot{z}}^{τ} (t) R_{2} \dot{z} (t) - \int_{t - h_{2}}^{t - h_{1}} {\dot{z}}^{τ} (s) R_{2} \dot{z} (s) d s \\ = h_{12} {\dot{z}}^{τ} (t) R_{2} \dot{z} (t) - \int_{t - h_{2}}^{t - d (t)} {\dot{z}}^{τ} (s) R_{2} \dot{x} (s) d s \\ - \int_{t - d (t)}^{t - h_{1}} {\dot{z}}^{τ} (s) R_{2} \dot{z} (s) d s \\ \leq h_{12} {\dot{z}}^{τ} (t) R_{2} \dot{z} (t) - \frac{1}{h_{2} - d (t)} \\ {(\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s)}^{τ} R_{2} (\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s) \\ - \frac{1}{d (t) - h_{1}} {(\int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s)}^{τ} R_{2} (\int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s) \\ \leq h_{12} {\dot{z}}^{τ} (t) R_{2} \dot{z} (t) + \max {- \frac{3 W_{1} + W_{2}}{h_{12}}, - \frac{W_{1} + 3 W_{2}}{h_{12}}} \end{array}

(19)

where $W_{1} = {(\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s)}^{τ} R_{2} (\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s), W_{2} = {(\int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s)}^{τ} R_{2} (\int_{t - d (t)}^{t - h_{1}} \dot{z} (s) d s)$ . It can be verified that

{\dot{V}}_{5} (z (t)) \leq ξ_{1}^{τ} (t) Π_{2} ξ_{1} (t)

(20)

with $Π_{2} = d i a g {(Q_{1} + Q_{3} + Q_{4}), - (1 - μ) Q_{1}, - Q_{3}, - Q_{4}, Q_{2},$

$- (1 - μ) Q_{2}, 0, 0, 0, 0}$ . It is easy to compute that

\begin{array}{l} {\dot{V}}_{6} (z (t)) \leq τ^{2} f^{τ} (z (t)) Z f (z (t)) \\ - τ \int_{t - τ}^{t} f^{τ} (z (s)) Z f (z (s)) d s \\ \leq τ^{2} f^{τ} (z (t)) Z f (z (t)) - τ (t) \\ \int_{t - τ (t)}^{t} f^{τ} (z (s)) Z f (z (s)) d s \\ \leq τ^{2} f^{τ} (z (t)) Z f (z (t)) \\ - {(\int_{t - τ (t)}^{t} f (z (s)) d s)}^{τ} Z (\int_{t - τ (t)}^{t} f (z (s)) d s) \end{array}

(21)

Moreover, one can infer from (4) that

{[\begin{matrix} z (t) \\ f (z (t)) \end{matrix}]}^{τ} [\begin{matrix} l_{i}^{+} l_{i}^{-} e_{i} e_{i}^{τ} & - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} \\ - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} & e_{i} e_{i}^{τ} \end{matrix}] [\begin{matrix} z (t) \\ f (z (t)) \end{matrix}] \leq 0, i = 1, \dots, n

(22)

{[\begin{matrix} z (t - d (t)) \\ f (z (t - d (t))) \end{matrix}]}^{τ} [\begin{matrix} l_{i}^{+} l_{i}^{-} e_{i} e_{i}^{τ} & - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} \\ - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} & e_{i} e_{i}^{τ} \end{matrix}] [\begin{matrix} z (t - d (t)) \\ f (z (t - d (t))) \end{matrix}] \leq 0, i = 1, \dots, n

(23)

where e_i denotes the unit column vector with a “1” element on its ith row and zeros elsewhere. Letting $D_{1} =diag {d_{11}, d_{12}, \dots d_{1 n}}, D_{2} = diag {d_{21}, d_{22}, \dots d_{2 n}},$ $L_{1} = {l_{1}^{+} l_{1}^{-}, l_{2}^{+} l_{2}^{-}, \dots l_{n}^{+} l_{n}^{-}},$ $L_{2} = {(l_{1}^{+} + l_{1}^{-}), (l_{2}^{+} + l_{2}^{-}), \dots (l_{n}^{+} + l_{n}^{-})}$ we have from (16)–(21), (24), (25) that

\begin{array}{l} \dot{V} (z (t)) \leq ξ_{1}^{τ} (t) (Π_{1} + Π_{2}) ξ_{1} (t) \\ + 2 (f^{τ} (z (t)) + z^{τ} (t) L_{3}) Λ [- A z (t) + W_{0} f (z (t)) + W_{1} f (z (t - d (t))) \\ + W_{2} \int_{t - τ (t)}^{t} f (z (s)) d s] + h_{2} {\dot{z}}^{τ} (t) R_{1} \dot{z} (t) - \frac{1}{h_{12}} {(\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s)}^{τ} R_{1} (\int_{t - h_{2}}^{t - d (t)} \dot{z} (s) d s) \\ - \frac{1}{h_{2}} {(\int_{t - d (t)}^{t} \dot{z} (s) d s)}^{τ} R_{1} (\int_{t - d (t)}^{t} \dot{z} (s) d s) \\ + h_{12} {\dot{z}}^{τ} (t) R_{2} \dot{z} (t) + \max {- \frac{3 W_{1} + W_{2}}{h_{12}}, - \frac{W_{1} + 3 W_{2}}{h_{12}}} \\ + τ^{2} f^{τ} (z (t)) Z f (z (t)) - {(\int_{t - τ (t)}^{t} f (z (s)) d s)}^{τ} Z (\int_{t - τ (t)}^{t} f (z (s)) d s) \\ - 2 {\sum_{i = 1}^{n} d_{1 i} [\begin{matrix} z (t) \\ f (z (t)) \end{matrix}]}^{τ} [\begin{matrix} l_{i}^{+} l_{i}^{-} e_{i} e_{i}^{τ} & - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} \\ - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} & e_{i} e_{i}^{τ} \end{matrix}] [\begin{matrix} z (t) \\ f (z (t)) \end{matrix}] \\ - 2 \sum_{i = 1}^{n} d_{2 i} {[\begin{matrix} z (t - d (t)) \\ f (z (t - d (t))) \end{matrix}]}^{τ} [\begin{matrix} l_{i}^{+} l_{i}^{-} e_{i} e_{i}^{τ} & - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} \\ - \frac{l_{i}^{+} + l_{i}^{-}}{2} e_{i} e_{i}^{τ} & e_{i} e_{i}^{τ} \end{matrix}] [\begin{matrix} z (t - d (t)) \\ f (z (t - d (t))) \end{matrix}] \\ = ξ_{1}^{τ} (t) Π ξ_{1} (t) + \max {- \frac{3 W_{1} + W_{2}}{h_{12}}, - \frac{W_{1} + 3 W_{2}}{h_{12}}} \end{array}

(24)

where

\begin{array}{l} Π = \\ [\begin{matrix} Ξ_{11} + A^{τ} U A & Ξ_{12} & P_{4}^{τ} & - P_{3}^{τ} & Ξ_{15} - A^{τ} U W_{0} & Ξ_{16} & Ξ_{17} & - P_{2}^{τ} & - P_{4}^{τ} & - P_{3}^{τ} \\ * & Ξ_{22} & P_{7}^{τ} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - P_{5}^{τ} & - P_{7}^{τ} & - P_{6}^{τ} \\ * & * & - Q_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & Ξ_{55} + W_{0}^{τ} U W_{0} & Λ W_{1} + W_{0}^{τ} U W_{1} & Λ W_{2} + W_{0}^{τ} U W_{2} & 0 & 0 & 0 \\ * & * & * & * & * & Ξ_{66} + W_{1}^{τ} U W_{1} & W_{1}^{τ} U W_{2} & 0 & 0 & 0 \\ * & * & * & * & * & * & - Z + W_{2}^{τ} U W_{2} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & - \frac{1}{h_{2}} R_{1} & 0 & 0 \\ * & * & * & * & * & * & * & * & 0 & 0 \\ * & * & * & * & * & * & * & * & * & - \frac{1}{h_{12}} R_{1} \end{matrix}] \end{array}

and $Ξ_{16} = P_{1} W_{1} + L_{3} Λ W_{1} - A^{τ} U W_{1}, Ξ_{17} = P_{1} W_{2} + L_{3} Λ W_{2}$ $- A^{τ} U W_{2}, Ξ_{11}, Ξ_{12}, Ξ_{15}, Ξ_{22}, Ξ_{55}, Ξ_{66}$ are defined in (6)–(11) in order to guarantee V̇(z(t))<0, one needs to ensure that

ξ_{1}^{τ} (t) Π ξ_{1} (t) + \max {- \frac{3 W_{1} + W_{2}}{h}, - \frac{W_{1} + 3 W_{2}}{h}} < 0

(25)

Note that (25) holds if and only if

ξ_{1}^{τ} (t) Π ξ_{1} (t) - \frac{W_{1} + 3 W_{2}}{h} < 0

(26)

and

ξ_{1}^{τ} (t) Π ξ_{1} (t) - \frac{3 W_{1} + W_{2}}{h} < 0

(27)

where $- \frac{3 W_{1} + W_{2}}{h}$ and $- \frac{W_{1} + 3 W_{2}}{h}$ are defined in (19).

It is clear that inequality (26) is equivalent to (5a), and inequality (27) is equivalent to (5b), so V̇(z(t)) < 0 holds when (5a) and (5b) hold. Then, the delay NNs (5) are guaranteed to be asymptotically stable.

Theorem 1 is dependent on both the lower-bound h₁ and upper-bound h₂ of the time-varying time delays, where h₁ is not 0. When h₁ =0, the following delay-dependent stability criterion can be obtained by setting R₂ = 0,Q₃ =0,P₄ = P₇ =0.

Corollary 1. For given scalars $L_{1} = d i a g {l_{1}^{+} l_{1}^{-}, \dots l_{n}^{+} l_{n}^{-}},$ $L_{2} = d i a g {(l_{1}^{+} + l_{1}^{-}), \dots (l_{n}^{+} + l_{n}^{-})}$ $, L_{3} = d i a g {| l_{1}^{-} |, \dots, | l_{n}^{-} |}, h_{1} = 0, 0 \leq h_{2}, 0 \leq τ$ and $0 \leq μ$ . Then, for delay d(t) satisfying h₁ ≤ d(t) ≤ h₂,ḋ(t) ≤ μ and τ(t) satisfying 0 <0 < τ(t) ≤ τ, the origin of system (3) is asymptotically stable if there exist P₁ > 0, Q_i > 0(i = 1,2,4), R₁ > 0, Z > 0, diagonal matrices $Λ = d i a g {λ_{1}, λ_{2}, …, λ_{n}} > 0, D_{1} = d i a g {d_{11}, d_{12}, … d_{1 n}} > 0, D_{2} = d i a g {d_{21}, d_{22}, …, d_{2 n}} > 0$ and any matrices P_i (i = 2,3,, 5,6) such that the following LMIs hold:

\begin{array}{l} [\begin{matrix} {\bar{Ξ}}_{11} & {\bar{Ξ}}_{12} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{2} P_{3}^{τ} & - h_{2} A^{τ} R_{1} \\ * & {\bar{Ξ}}_{22} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{2} P_{6}^{τ} & 0 \\ * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\bar{Ξ}}_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & h_{2} W_{0}^{τ} R_{1} \\ * & * & * & * & Ξ_{66} & 0 & 0 & 0 & h_{2} W_{1}^{τ} R_{1} \\ * & * & * & * & * & - Z & 0 & 0 & h_{2} W_{2}^{τ} R_{1} \\ * & * & * & * & * & * & - 3 h_{2} R_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - h_{2} R_{1} & 0 \\ * & * & * & * & * & * & * & * & - h_{2} R_{1} \end{matrix}] < 0 \end{array}

(28a)

\begin{array}{l} [\begin{matrix} {\bar{Ξ}}_{11} & {\bar{Ξ}}_{12} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{2} P_{3}^{τ} & - h_{2} A^{τ} R_{1} \\ * & {\bar{Ξ}}_{22} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{2} P_{6}^{τ} & 0 \\ * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & Ξ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & h_{2} W_{0}^{τ} R_{1} \\ * & * & * & * & Ξ_{66} & 0 & 0 & 0 & h_{2} W_{1}^{τ} R_{1} \\ * & * & * & * & * & - Z & 0 & 0 & h_{2} W_{2}^{τ} R_{1} \\ * & * & * & * & * & * & - h_{2} R_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - 3 h_{2} R_{1} & 0 \\ * & * & * & * & * & * & * & * & - h_{2} R_{1} \end{matrix}] < 0 \end{array}

(28b)

Where

{\bar{Ξ}}_{11} = - P_{1} A - A^{τ} P_{1} + P_{2} + P_{2}^{τ} + Q_{1} + Q_{4} - 2 L_{3} Λ A - 2 D_{1} L_{1}

(29)

{\bar{Ξ}}_{12} = - P_{2}^{τ} + P_{3}^{τ} + P_{5}

(30)

{\bar{Ξ}}_{22} = - P_{5} - P_{5}^{τ} + P_{6} + P_{6}^{τ} - (1 - μ) Q_{1} - 2 D_{2} L_{1}

(31)

Ξ₁₅,Ξ₅₅,Ξ₆₆ are defined in (8), (10), and (11) and ^* denotes the symmetrical terms in a symmetrical matrix.

Remark 2. In [21], only constant-distributed time delay is studied. However, in this paper, we considered the stability problem of a class of NNs with time-varying delay. Meanwhile, compared with the result in [28], $l_{1}^{-}, (i = 1, 2, \dots n)$ , (i = 1,2,…n) in Theorem 1 and Corollary 1 are allowed to be positive, negative or zero, which are less restrictive bounds than those given in [28].

Now we examine Case 2. Similar to the proof of Theorem 1, we can obtain the following delay-dependent stability criterion.

Theorem 2. For a given scalar,

\begin{array}{l} L_{1} = d i a g {l_{1}^{+} l_{1}^{-}, \dots l_{n}^{+} l_{n}^{-}}, L_{2} = d i a g {(l_{1}^{+} + l_{1}^{-}), \dots (l_{n}^{+} + l_{n}^{-})}, \\ L_{3} = d i a g {| l_{1}^{-} |, \dots | l_{n}^{-} |}, 0 \leq h_{1} \leq h_{2}, 0 < τ a n d 0 < μ . \end{array}

Then, for delay ḋ (t) satisfying h₁ ≤ d (t) ≤ h₂ and τ (t) satisfying 0 < τ(t)≤ τ, the origin of system (3) is asymptotically stable if there exist matrices P₁ > 0,Q_i(i = 3,4) > 0,R₁ > 0,R₂ > 0,Z > 0, diagonal matrices $Λ = d i a g {λ_{1}, λ_{2}, \dots λ_{n}} > 0$ , $D_{1} = d i a g {d_{11}, d_{12}, \dots d_{1 n}} > 0$ , $D_{2} = d i a g {d_{21}, d_{22}, \dots d_{2 n}} > 0$ and any matrices P_i(i = 2,…,7) such that the following LMIs hold:

\begin{array}{l} [\begin{matrix} Γ_{11} & Ξ_{12} & P_{4}^{τ} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{12} P_{4}^{τ} & - h_{12} P_{3}^{τ} & - A^{τ} U \\ * & Γ_{22} & P_{7}^{τ} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{12} P_{7}^{τ} & - h_{12} P_{6}^{τ} & 0 \\ * & * & - Q_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & Γ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & 0 & W_{0}^{τ} U \\ * & * & * & * & * & - 2 D_{2} & 0 & 0 & 0 & 0 & W_{1}^{τ} U \\ * & * & * & * & * & * & - Z & 0 & 0 & 0 & W_{2}^{τ} U \\ * & * & * & * & * & * & * & - h_{2} R_{1} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & - h_{12} R_{2} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & - h_{12} (R_{1} + 3 R_{2}) & 0 \\ * & * & * & * & * & * & * & * & * & * & - U \end{matrix}] < 0 \end{array}

(32a)

\begin{array}{l} [\begin{matrix} Γ_{11} & Ξ_{12} & P_{4}^{τ} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{12} P_{4}^{τ} & - h_{12} P_{3}^{τ} & - A^{τ} U \\ * & Γ_{22} & P_{7}^{τ} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{12} P_{7}^{τ} & - h_{12} P_{6}^{τ} & 0 \\ * & * & - Q_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & Γ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & 0 & W_{0}^{τ} U \\ * & * & * & * & * & - 2 D_{2} & 0 & 0 & 0 & 0 & W_{1}^{τ} U \\ * & * & * & * & * & * & - Z & 0 & 0 & 0 & W_{2}^{τ} U \\ * & * & * & * & * & * & * & - h_{2} R_{1} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & - 3 h_{12} R_{2} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & - h_{12} (R_{1} + R_{2}) & 0 \\ * & * & * & * & * & * & * & * & * & * & - U \end{matrix}] < 0 \end{array}

(32b)

where

Γ_{11} = - P_{1} A - A^{τ} P_{1} + P_{2} + P_{2}^{τ} + Q_{3} + Q_{4} - 2 L_{3} Λ A - 2 D_{1} L_{1}

(33)

Γ_{22} = - P_{5} - P_{5}^{τ} + P_{6} + P_{6}^{τ} - P_{7} - P_{7}^{τ} - 2 D_{1} L_{1}

(34)

Γ_{55} = τ^{2} Z + Λ W_{0} + W_{0}^{τ} Λ - 2 D_{1}

(35)

Ξ₁₂, Ξ₁₅, U are defined in (7), (8) and (12) and ^* denotes the symmetrical terms in a symmetrical matrix.

In order to demonstrate the improvement over the existing criteria, Corollary 2 is established from Theorem 2 by setting h1 = 0, R2 = 0, Q3 = 0, P4 = P7 = 0.

Corollary 2. For given scalars, $L_{1} = d i a g {l_{1}^{+} l_{1}^{-}, … l_{n}^{+} l_{n}^{-}}, L_{2} = d i a g {(l_{1}^{+} + l_{1}^{-}), … (l_{n}^{+} + l_{n}^{-})}, L_{3} = d i a g {| l_{1}^{-} |, …, | l_{n}^{-} |}, h_{1} = 0, 0 \leq h_{2},$ $0 \leq τ$ and $0 \leq μ$ . Then, for any delay where d (t) and τ(t) satisfy 0 ≤ d(t)≤ h₂ and 0 ≤ τ(t)≤ τ, the origin of system (3) is asymptotically stable if there exist matrices P₁ > 0,Q₄ > 0,R₁ > 0,Z > 0, diagonal matrices $Λ = d i a g {λ_{1}, λ_{2}, \dots λ_{n}} > 0,$ $D_{1} =diag {d_{11}, d_{12}, \dots d_{1 n}} > 0, D_{2} = diag {d_{21}, d_{22}, \dots d_{2 n}} > 0$ and any matrices P_i (i = 2,3,5,6) such that the following LMIs hold:

\begin{array}{l} [\begin{matrix} {\overset{\land}{Γ}}_{11} & - P_{2}^{τ} + P_{3}^{τ} + P_{5} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{2} P_{3}^{τ} & - h_{2} A^{τ} R_{1} \\ * & {\overset{\land}{Γ}}_{22} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{2} P_{6}^{τ} & 0 \\ * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & Γ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & h_{2} W_{0}^{τ} R_{1} \\ * & * & * & * & - 2 D_{2} & 0 & 0 & 0 & h_{2} W_{1}^{τ} R_{1} \\ * & * & * & * & * & - Z & 0 & 0 & h_{2} W_{2}^{τ} R_{1} \\ * & * & * & * & * & * & - 3 h_{2} R_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - h_{2} R_{1} & 0 \\ * & * & * & * & * & * & * & * & - h_{2} R_{1} \end{matrix}] < 0 \end{array}

(36a)

\begin{array}{l} [\begin{matrix} {\overset{\land}{Γ}}_{11} & - P_{2}^{τ} + P_{3}^{τ} + P_{5} & - P_{3}^{τ} & Ξ_{15} & P_{1} W_{1} + L_{3} Λ W_{1} & P_{1} W_{2} + L_{3} Λ W_{2} & - h_{2} P_{2}^{τ} & - h_{2} P_{3}^{τ} & - h_{2} A^{τ} R_{1} \\ * & {\overset{\land}{Γ}}_{22} & - P_{6}^{τ} & 0 & D_{2} L_{2} & 0 & - h_{2} P_{5}^{τ} & - h_{2} P_{6}^{τ} & 0 \\ * & * & - Q_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & Γ_{55} & Λ W_{1} & Λ W_{2} & 0 & 0 & h_{2} W_{0}^{τ} R_{1} \\ * & * & * & * & - 2 D_{2} & 0 & 0 & 0 & h_{2} W_{1}^{τ} R_{1} \\ * & * & * & * & * & - Z & 0 & 0 & h_{2} W_{2}^{τ} R_{1} \\ * & * & * & * & * & * & - h_{2} R_{1} & 0 & 0 \\ * & * & * & * & * & * & * & - 3 h_{2} R_{1} & 0 \\ * & * & * & * & * & * & * & * & - h_{2} R_{1} \end{matrix}] < 0 \end{array}

(36b)

where

\begin{array}{l} {\overset{\land}{Γ}}_{11} = - P_{1} A - A^{τ} P_{1} + P_{2} + P_{2}^{τ} + Q_{4} - 2 L_{3} Λ A - 2 D_{1} L_{1} \\ {\overset{\land}{Γ}}_{22} = - P_{5} - P_{5}^{τ} + P_{6} + P_{6}^{τ} - 2 D_{2} L_{1} \end{array}

Ξ₁₅,Γ₅₅ are defined in (8) and (32) and ^* denotes the symmetrical terms in a symmetrical matrix.

Remark 3. For $l_{i}^{-} = - l_{i}^{+}, (i = 1, 2, … n)$ very recently, Song and Wang [26] provided a new stability condition for NNs by introducing a new type of Lyapunov functional. Compared with the Theorem 1 in [26], the advantage is that the LMI in (37) involves significantly fewer variables. Specifically, when z (t) ε Rⁿ, the number of variables to be determined in (37) as 6n² + 5n, while in [26] the number of variables is 7n² + 8n. That is, the variables in [26] are around 1.2 times those in Theorem 1. However, our result may provide less conservative results than Theorem 1 in [26], as shown in of the examples in section 4.

4. Numerical examples

In this section, we use two examples to show the benefits of our results for two types of time-varying delay.

Example 1. Consider the delayed NNs with d(t) satisfying Case 1 in [28,43,44,45].

\begin{array}{l} A = d i a g {1.2769; 0.6231; 0.9230; 0.4480}, \\ W_{0} = [\begin{matrix} - 0.0373 & 0.4852 & - 0.3351 & 0.2336 \\ - 1.6033 & 0.5988 & - 0.3224 & 1.2352 \\ 0.3394 & - 0.0860 & - 0.3824 & - 0.5785 \\ - 0.1311 & 0.3253 & - 0.9534 & - 0.5015 \end{matrix}], \\ W_{1} = [\begin{matrix} 0.8674 & - 1.2405 & - 0.5325 & 0.0220 \\ 0.0474 & - 0.9164 & 0.0360 & 0.9816 \\ 1.8495 & 2.6117 & - 0.3788 & 0.8428 \\ - 2.0413 & 0.5179 & 1.1734 & - 0.2775 \end{matrix}], \\ W_{2} = 0 \\ l_{1}^{+} = 0.1137, l_{2}^{+} = 0.1279, l_{3}^{+} = 0.7994, l_{4}^{+} = 0.2368. \end{array}

For $l_{1}^{-} = l_{2}^{-} = l_{3}^{-} = l_{4}^{-} = 0$ , it can be confirmed that Theorem 1 in [4] and Theorem 1 in [12] are not satisfied; i.e., they fail to conclude whether this system is asymptotically stable or not. However, when μ = 0, by applying the results in [12,28] to this example, the maximum allowable bounds h are 1.422 [12] and 1.932 [28], respectively, while by using Theorem 1 we have h = 3.584. When μ = 0.5 and μ = 0.9, the corresponding upper bounds are listed in Table 1. For the case of $l_{1}^{-} = 0.1, l_{2}^{-} = 0, l_{3}^{-} = - 0.2, l_{4}^{-} = 0, μ = 0.5$ and $h_{1} = 0$ , the method in [15,17] fails. In this paper, the maximum delay h, which guarantees the delayed neural networks are asymptotically stable, is calculated to be 1.973.

Table 1.

Comparisons of maximum allowed delay h2 for Example 1 with h1=0

Method	l⁻_i = 0,(i = 1…,4),μ = 0.5	= 0,(i = 1…,4),μ = 0.9	l⁻₂ = 0.2, l⁻₄ = −0.2, μ = 0.5
He et al. [17]	2.150	1.316	fail
He et al. [15]	2.224	1.584	fail
Li et al. [29]	2.224	1.584	1.868
Our paper	2.312	1.809	1.973

Example 2. Consider the delayed NNs with d(t) satisfying Case 2.

\begin{array}{l} A = d i a g {2.3; 3.4; 2.5}, \\ W_{0} = [\begin{matrix} 0.9 & - 1.5 & 0.1 \\ - 1.2 & 1 & 0.2 \\ 0.2 & 0.3 & 0.8 \end{matrix}], W_{1} = [\begin{matrix} 0.8 & 0.6 & 0.2 \\ 0.5 & 0.7 & 0.1 \\ 0.2 & 0.1 & 0.5 \end{matrix}], \\ W_{2} = [\begin{matrix} 0.3 & 0.2 & 0.1 \\ 0.1 & 0.2 & 0.1 \\ 0.1 & 0.1 & 0.2 \end{matrix}], \\ f_{1} (z) = 0.05 (| z + 1 | + | z - 1 |), f_{2} (z) = 0.1 (| z + 1 | + | z - 1 |), \\ f_{3} (z) = 0.2 (| z + 1 | + | z - 1 |) \end{array}

It is easy to see that the activation functions satisfy (4) with

l_{1}^{-} = - 0.1, l_{2}^{-} = - 0.2, l_{3}^{-} = - 0.4, l_{1}^{+} = 0.1, l_{2}^{+} = 0.2, l_{3}^{+} = 0.4

This example, with τ = 3.597 and $d (t) = 4 - 2 \sin (\frac{π}{2} t)$ , makes it obvious that 2 ≤ d(t) ≤ 6. In this case, it is found that the delay-dependent stability conditions proposed in [22,25,26] are not satisfied. However, by Theorem 2 in this paper, the neural network under consideration is globally stable. This implies that, for this example, the delay-dependent stability result proposed in Theorem 2 is less conservative than that in [22, 25, 26].

5. Conclusion

This article, by employing the newly proposed Lyapunov functional and a tighter inequality, has established new stability criteria in terms of LMIs, which are dependent on both the lower bound and upper bound of the time-varying time delays. It can be shown that the obtained criteria are less conservative than previously existing results through the numerical examples.

Footnotes

6. Acknowledgements

This paper was supported by the National Nature Science Foundation of China under grant 61105075, 61104206, the Natural Science Funds of Jiangsu Province (BK20131000), a major project to cultivate science and technology innovation from the Ministry of Education of China (708045), the Project of the National Department, Public Benefit Research Foundation (GYHY200806017), the Industry-Academic Joint Technological Innovations Fund Project of Jiangsu Province (2012t026), the Nanjing University of Information Science & Technology University Student Renovation Project (N1885012179), the Research Foundation (20070063), the College Students' Practice and Innovation Training Project (11CX017, 12CX016, 12CX050, 201310300039Z), and the Teaching Reform Project (13JY022).

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