Input-to-State Stability for Dynamical Neural Networks with Time-Varying Delays

Abstract

A class of dynamical neural network models with time-varying delays is considered. By employing the Lyapunov-Krasovskii functional method and linear matrix inequalities (LMIs) technique, some new sufficient conditions ensuring the input-to-state stability (ISS) property of the nonlinear network systems are obtained. Finally, numerical examples are provided to illustrate the efficiency of the derived results.

1. Introduction

Recently, the dynamical neural networks (DNNs), which are firstly introduced by Hopfield in [1], have been extensively studied due to its wide applications in various areas such as associative memory, parallel computation, signal processing, optimization, and moving object speed detection. Since time delay is inevitably encountered in implementation of DNNs and is frequently a source of oscillation and instability, neural networks with time delays have become a topic of great theoretical and practical importance, and many interesting results have been derived (see, e.g., [2–5] and [6–9]). Furthermore, in practical evolutionary processes of the networks, absolute constant delay may be scarce and is only the poetic approximation of the time-varying delays. Delays are generally varied with time because information transmission from one neuron to another neuron may make the response of networks with time-varying delays. Accordingly, dynamical behaviors of neural networks with time-varying delays have been discussed in the last decades (see, e.g., [3, 8–11], etc.).

It is well known that neural networks are often influenced by external disturbances and input errors. Thus many dissipative properties such as robustness [12], passivity [13], and input-to-state stability [4, 10, 11, 14–19] are apparently significant to analyze its dynamical behaviors of the networks. For instance, Ahn incorporated robust training law in switched Hopfield neural networks with external disturbances to study boundedness and exponentially stability [12], and studied passivity in [13]. Especially, the ISS implies not only that the unperturbed system is asymptotically stable in the Lyapunov sense but also that its behavior remains bounded when its inputs are bounded. It is one of the useful classes of dissipative properties for nonlinear systems, which is firstly introduced in nonlinear control systems by Sontag in [20], and then extended by Praly and Jiang [21] and Angeli et al. [19 and Ahn (see [17, 19], and references therein). Due to these research background, the ISS properties of neural networks are investigated in recent years (see, e.g., [16–19] and references therein). For example, by using the Lyapunov function method, some nonlinear feedback matrix norm conditions for ISS have been developed for recurrent neural networks ([16]). Moreover, Ahn utilized Lyapunov function method to discuss robust stability problem for a class of recurrent neural networks, and also some LMI sufficient conditions have been proposed to guarantee the ISS (see [17]). In [18], by employing a suitable Lyapunov function, some results on boundedness, ISS, and convergence are established. Also, in [19] a new sufficient condition is derived to guarantee ISS of Takagi-Sugeno fuzzy Hopfield neural networks with time delay. However, there is few results to deal with the ISS of dynamical neural networks (DNNs) with time-varying delays ([11]).

Motivated by the above discussions, we discuss the ISS properties of DNNs with time-varying delays in this paper. By using Lyapunov-Krasovskii functional technique, ISS conditions for the considered dynamical neural networks are given in terms of LMIs, which can be easily calculated by certain standard numerical packages. We also provide two illustrative examples to demonstrate the effectiveness of the proposed stability results.

The organization of this paper is as follows. In Section 2, our mathematical model of dynamical neural networks is presented and some preliminaries are given. In Section 3, the main results for both ISS and asymptotically stability of dynamical neural networks with time-varying delays are proposed. In Section 4, two numerical examples are illustrated to demonstrate the effectiveness of the theoretical results. Concluding remarks are collected in Section 5. Proof of Lemma 2.4 is given in the appendix.

Notions
Let 𝑅^𝑛 denote the 𝑛-dimensional Euclidean space and |⋅| denote the usual Euclidean norm. Denote 𝐶=𝐶([−𝜏,0],𝑅^𝑛) and designate the norm of an element in 𝐶 by ‖𝜙‖_𝜏=sup_{−𝜏≤𝜗≤0}‖𝜙(𝜗)‖. 𝑅^𝑛×𝑛 is the set of all 𝑛×𝑛 real matrices. Let 𝐵^𝑇, 𝐵⁻¹, 𝜆_max(𝐵), 𝜆_min(𝐵), and ‖𝐵‖=𝜆_max(𝐵^𝑇𝐵) denote the transpose, the inverse, the largest eigenvalue, the smallest eigenvalue, and the Euclidean norm of a square matrix 𝐵, respectively. The notation 𝑃>0(≥0) means that 𝑃 is real symmetric and positive definite (positive semidefinite). The notion 𝑋>𝑌 (𝑋≥𝑌), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋−𝑌 is positive definite (positive semidefinite). 𝐼 denotes the element matrix. The set of all measurable locally essentially bounded functions 𝑢∶𝑅⁺→𝑅^𝑛, endowed with (essential) supremum norm ‖𝑢‖_∞=sup{‖𝑢(𝑡)‖,𝑡≥0}, is denoted by 𝐿^𝑚_∞. In addition, denote 𝑢_𝑡 the truncation of 𝑢 at 𝑡; that is, 𝑢_𝑡(𝑠)=𝑢(𝑠) if 𝑠≤𝑡, and 𝑢(𝑠)=0 if 𝑠>𝑡. We recall that a function 𝛾∶𝑅⁺→𝑅⁺ is a 𝐾-function if it is continuous, strictly increasing, and 𝛾(0)=0; it will be a 𝐾_∞-function if it is a 𝐾-function and also 𝛾(𝑠)→∞ as 𝑠→∞. A function 𝛽∶𝑅⁺×𝑅⁺→𝑅⁺ is a 𝐾𝐿-function if for each fixed 𝑡≥0 the function 𝛽(⋅,𝑡) is a 𝐾-function, and, for each fixed 𝑠≥0, it is decreasing to zero as 𝑡→∞.

2. Mathematical Model and Preliminaries

Consider the following nonlinear time-delay system 𝑥̇=𝑓𝑡,𝑥_𝑡,𝑢(𝑡), (2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝑢(𝑡)∈𝑅^𝑛 is the input function; 𝑥_𝑡∈𝐶 is the standard function given by 𝑥_𝑡(𝜏)=𝑥(𝑡+𝜏). Without loss of generality, we suppose that 𝑓(0,0,0)=0, which ensuring that 𝑥(𝑡)=0 is the trivial solution for the unforced system 𝑥̇(𝑡)=𝑓(𝑡,𝑥_𝑡,0). Define 𝑥(𝑡)≜𝑥(𝑡,𝑡₀,𝜙) is a solution of system with initial value 𝜙 at time 𝑡₀.

Given a continuous functional 𝑉∶𝑅⁺×𝐶→𝑅⁺, the upper right-hand derivative 𝑉̇ of the function 𝑉 is given by 𝑉̇𝑡,𝜙=lim_ℎ→0+sup1ℎ𝑉𝑡+ℎ,𝑥_𝑡+ℎ𝑡,𝑥_𝑡−𝑉𝑡,𝑥_𝑡. (2.2)

For delayed dynamical system, we first give the input-to-state stable (ISS) definition as usual case.

Definition 2.1. System (2.1) is ISS if there exist a 𝐾𝐿-function𝛽 and a 𝐾-function𝛾 such that, for each input 𝑢∈𝐿^𝑚_∞ and each 𝜉_𝑡0∈𝐶, it satisfies 𝑥(𝑡;𝜉,𝑢)≤𝛽‖𝜉_𝑡0‖_𝜏,𝑡−𝑡₀+𝛾‖𝑢_𝑡‖_∞,∀𝑡≥𝑡₀. (2.3) Note that, by causality, the same definition would result if one could replace ‖𝑢_𝑡‖_∞ by ‖𝑢‖_∞.

Definition 2.2. A continuous differentiable functional 𝑉(𝑡,𝜙)∶𝑅⁺×𝐶→𝑅⁺ is called the ISS Lyapunov-Krasovskii functional if there exist functions 𝛼₁, 𝛼₂ of class 𝐾_∞, a function 𝜒 of class 𝐾 and a continuous positive definite function 𝑊 such that 𝛼₁𝜙(0)≤𝑉𝑡,𝜙≤𝛼₂‖𝜙‖_𝜏,𝑉̇𝑡,𝜙≤−𝑊𝜙(0)if𝜙(0)≥𝜒‖𝑢‖_∞,∀𝜙∈𝐶,𝑢∈𝐿^𝑚_∞. (2.4)(2.5)

Remark 2.3. A continuous differential functional 𝑉(𝑡,𝜙)∶𝑅⁺×𝐶→𝑅⁺ is an ISS Lyapunov-Krasovskii functional if and only if there exist 𝛼₃,𝛼₄∈𝐾_∞ such that (2.4) holds and 𝑉̇𝑡,𝜙≤−𝛼₃𝜙(0)+𝛼₄‖𝑢‖_∞. (2.6) The proof is similarly to one of Remark 2.4 in [23]. We omit it here.

Similarly to the case of ordinary differential equation (ODE), we will establish a link between the ISS property and the ISS Lyapunov-Krasovskii functional for time-delay systems in the following Lemma.

Lemma 2.4. The system (2.1) is ISS if it admits an ISS Lyapunov-Krasovskii functional.
For completeness, the proof is given in appendix.

To obtain our results, we need the following two useful lemmas.

Lemma 2.5 (Schur Complement [24]). For given symmetric matrix 𝑆=^𝑆11𝑆12_𝑆21𝑆22, where 𝑆₁₁∈𝑅^𝑟×𝑟, 𝑆₂₁=𝑆^𝑇₁₂, the following three conditions are equivalent: (i)𝑆<0; (ii)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂; (iii)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂.

Lemma 2.6 (see [25]). Given any matrix 𝑋,𝑌, and Λ with appropriate dimensions such that Λ=Λ^𝑇 and any scalar 𝜀>0, then 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝜀𝑋^𝑇1Λ𝑋+𝜀𝑌^𝑇Λ⁻¹𝑌. (2.7)

In this paper, we consider the following dynamical neural networks with time-varying delays 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡−𝜏(𝑡))+^𝑛_𝑖=1𝑢_𝑖(𝑡),𝑖=1,2,…,𝑛, (2.8) or equivalently 𝑑𝑥(𝑡)(𝑡)(𝑥(𝑡𝑑𝑡=−𝐴𝑥+𝑊𝑔))+𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡), (2.9) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the neuron state, 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))^𝑇 is the input, and 𝑔(𝑥(𝑡))=(𝑔₁(𝑥₁(𝑡)),𝑔₂(𝑥₂(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡)))^𝑇 denotes the nonlinear neuron activation function. 𝐴=diag{𝑎₁,𝑎₂,…,𝑎_𝑛} is the positive diagonal matrix. 𝑊=(𝑏_𝑖𝑗)_𝑛×𝑛 and 𝑊₁=(𝑐_𝑖𝑗)_𝑛×𝑛 are the interconnection matrices representing the weighting coefficients of neurons. 𝜏(𝑡) is the time-varying delays.

Throughout this paper, we always suppose that 𝑔_𝑖𝑥_𝑖(𝑡)≤𝑙_𝑖𝑥_𝑖,∀𝑖,∀𝑥_𝑖∈𝑅,𝑖=1,2,…,𝑛,0≤𝜏(𝑡)≤𝜏,0≤𝜏̇(𝑡)≤1. (𝐴1)(𝐴2) From (𝐴1), we easily see that 𝑥(𝑡)=0 is the solution of (2.9) with 𝑢(𝑡)=0.

3. ISS Analysis

In this section, we give two theorems on ISS in form of LMIs.

Theorem 3.1. Let (𝐴1) and (𝐴2) hold. If there exist a positive definite matrix 𝑃 and a positive diagonal matrix 𝐷 such that −𝐴^𝑇𝑃−𝑃𝐴𝑀𝐼𝑃𝑊₁𝑀𝐷⁻²+𝑊^𝑇𝑊𝑃𝑊00𝐼0𝑃0_𝑇1𝑃00−𝐷⁻²<0, (3.1) where 𝑀=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}, and then the system (2.9) is ISS.

Proof. We consider the following functional: 𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑔^𝑇(𝑥(𝜁))𝐷⁻²𝑔(𝑥(𝜁))𝑑𝜁. (3.2) Its derivative along the solution 𝑥(𝑡) of (2.9) is given as 𝑉̇(𝑥(𝑡))=2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑔^𝑇(𝑥(𝑡))𝐷⁻²𝑔(𝑥(𝑡−(𝜏̇(𝑡𝑔))1−))^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐷⁻²𝑔(𝑥(𝑡−𝜏(𝑡)))=2𝑥^𝑇(𝑡)𝑃−𝐴𝑥(𝑡)+𝑊𝑔(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡)+𝑔^𝑇(𝑥(𝑡))𝐷⁻²𝑔(𝑥(𝑡))−(1−𝜏̇(𝑡))𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐷⁻²𝑔(𝑥(𝑡−𝜏(𝑡)))=2𝑥^𝑇(𝑡)−𝐴^𝑇𝑃−𝑃𝐴𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊𝑔(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑔^𝑇(𝑥(𝑡))𝐷⁻²𝑔(𝑥(𝑡−(𝜏̇(𝑡𝑔))1−))^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐷⁻²𝑔(𝑥(𝑡−𝜏(𝑡)))+2𝑥^𝑇(𝑡)𝑃𝑢(𝑡). (3.3) We have −(1−𝜏̇(𝑡))𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐷⁻²𝑔(𝑥(𝑡−𝜏(𝑡)))+2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))=−(1−𝜏̇(𝑡))𝐷⁻¹𝑔(𝑥(𝑡−𝜏(𝑡)))−𝐷𝑊^𝑇₁𝑃𝑥(𝑡)^𝑇⋅(1−𝜏̇(𝑡))𝐷⁻¹𝑔(𝑥(𝑡−𝜏(𝑡)))−𝐷𝑊^𝑇₁𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝑊₁𝐷²𝑊^𝑇₁𝑃𝑥(𝑡). (3.4) Since the first term of the right-hand side of (3.4) is negative semidefinite, we obtain −(1−𝜏̇(𝑡))𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐷⁻²𝑔(𝑥(𝑡−𝜏(𝑡)))+2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡)𝑃𝑊₁𝐷²𝑊^𝑇₁𝑃𝑥(𝑡). (3.5) From (𝐴1), we obtain 𝑔^𝑇(𝑥(𝑡))𝐷⁻²𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑀𝐷⁻²𝑀𝑥(𝑡), (3.6) where 𝑀=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}.
Then by Lemma 2.6, we have 2𝑥^𝑇(𝑡)𝑃𝑊𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑃𝑊𝑔(𝑥(𝑡))^𝑇𝑃⁻¹𝑃𝑊𝑔(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑔^𝑇(𝑥(𝑡))𝑊^𝑇𝑃𝑊𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑀𝑊^𝑇𝑃𝑊𝑀𝑥(𝑡),2𝑥^𝑇(𝑡)𝑃𝑢(𝑡)≤𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+[𝑃𝑢(𝑡)]^𝑇𝑃⁻¹𝑃𝑢(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑢^𝑇(𝑡)𝑃𝑢(𝑡). (3.7) Substituting (3.5), (3.6), and (3.7) into (3.3), we finally obtain 𝑉̇(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝐺𝑥(𝑡)+𝑢^𝑇(𝑡)𝑃𝑢(𝑡)≤𝜆_min(𝐺)|𝑥(𝑡)|²+𝜆_max(𝑃)|𝑢(𝑡)|², (3.8) where 𝐺=−𝐴^𝑇𝑃−𝑃𝐴+𝑀(𝐷⁻²+𝑊^𝑇𝑃𝑊)𝑀+𝑃𝑊₁𝐷²𝑊^𝑇₁𝑃+𝑃.
Define 𝛼₃(𝑟)=−𝜆_min(𝐺)⋅𝑟², 𝛼₄(𝑟)=𝜆_max(𝑃)⋅𝑟², then we can obtain that 𝑉̇(𝑥(𝑡))≤−𝛼₃(|𝑥(𝑡)|)+𝛼₄‖𝑢‖_∞. (3.9)
Note that 𝐺<0 is equivalent to (3.1) by Lemma 2.5. Then the defined 𝑉 is an ISS Lyapunov-Krasovskii functional. It follows from Lemma 2.4 and Remark 2.3 that the delayed neural networks (2.9) are ISS. The proof is complete.

Remark 3.2. Theorem 3.1 reduces to asymptotically stability condition for dynamical neural networks with time-varying delays when 𝑢(𝑡)=0.

Remark 3.3. Recently, some results on ISS or IOSS were obtained in [10, 17–19, 26]. However, these results were restricted to nondelay or constant delay. In contrast to the results [10, 17–19, 26], we consider dynamical neural networks with time-varying delays and propose a set of delay-independent criteria for asymptotically convergent state estimation of these neural networks in this paper.

In the following, we give a delay-dependent sufficient criterion.

Theorem 3.4. Let (𝐴1) and (𝐴2) hold. The system (2.9) is ISS if there exist a symmetric positive definite matrix 𝑃 and a positive definite matrix 𝑄 such that −𝐴^𝑇𝑃−𝑃𝐴+𝜏𝑄=−𝜇𝐼,𝜇>0,∀𝑢∈𝐿^𝑚_∞,2‖𝑀‖𝑊₁1−𝜏̇(𝑡)+‖𝑀‖²‖𝑊‖²+2<𝜇, (3.10)(3.11) where 𝑀=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}.

Proof. We consider the following functional: 𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+⁰_{(𝑡)−𝜏}^𝑡_𝑡+𝜁𝑥^𝑇𝜂𝑄𝑥𝜂𝑑𝜂𝑑𝜁+^𝑡_{(𝑡)𝑡−𝜏}𝑔^𝑇(𝑥(𝜁))𝑅𝑔(𝑥(𝜁))𝑑𝜁, (3.12) where 𝑅 is a positive definite matrix.
The derivative of (3.12) along the trajectories of the system is obtained as follows: 𝑉̇(𝑥(𝑡))=2𝑥^𝑇(𝑡)𝑃−𝐴𝑥(𝑡)+𝑊𝑔(𝑥(𝑡))+𝑊₁𝑔(𝑥((𝑡(𝑡(𝑡)𝑥𝑡−𝜏)))+𝑢)+𝜏^𝑇(𝑡)𝑄𝑥(𝑡)−(1−𝜏̇(𝑡))^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝜁)𝑄𝑥(𝜁)𝑑𝜁+𝑔^𝑇(𝑥(𝑡))𝑅𝑔(𝑥(𝑡))−(1−𝜏̇(𝑡))𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑅𝑔(𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡)−𝐴^𝑇𝑃−𝐴𝑃+𝜏𝑄𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊𝑔(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))−(1−𝜏̇(𝑡))^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝜁)𝑄𝑥(𝜁)𝑑𝜁+𝑔^𝑇(𝑥(𝑡))𝑅𝑔(𝑥(𝑡))−(1−𝜏̇(𝑡))𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑅𝑔(𝑥(𝑡−𝜏(𝑡))). (3.13) From (3.10), which reduces to 𝑉̇(𝑥(𝑡))≤−𝜇𝑥^𝑇(𝑡)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊𝑔(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))−(1−𝜏̇(𝑡))^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝜁)𝑄𝑥(𝜁)𝑑𝜁+𝑔^𝑇(𝑥(𝑡))𝑅𝑔(𝑥(𝑡))−(1−𝜏̇(𝑡))𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑅𝑔(𝑥(𝑡−𝜏(𝑡))). (3.14) From (𝐴1), we obtain that 𝑔^𝑇(𝑥(𝑡))𝑅𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑀𝑅𝑀𝑥(𝑡)≤‖𝑅‖‖𝑀‖²|𝑥(𝑡)|², (3.15) where 𝑀=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}.
From Lemma 2.6, we have 2𝑥^𝑇(𝑡)𝑃𝑊𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+[𝑃𝑊𝑔(𝑥(𝑡))]^𝑇𝑃⁻¹𝑃𝑊𝑔(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑔^𝑇(𝑥(𝑡))𝑊^𝑇𝑃𝑊𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑀𝑊^𝑇(𝑡)≤𝑃𝑊𝑀𝑥1+‖𝑀‖²‖𝑊‖²‖𝑃‖|𝑥(𝑡)|²,2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))≤(1−𝜏̇(𝑡))𝑔^𝑇(𝑥((𝑡(𝑥((𝑡+1𝑡−𝜏)))𝑅𝑔𝑡−𝜏)))𝜏̇(𝑡)𝑥1−^𝑇(𝑡)𝑃𝑊₁𝑅⁻¹𝑊^𝑇₁𝑃𝑥(𝑡). (3.16) Then 2𝑥^𝑇(𝑡)𝑃𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡)))−(1−𝜏̇(𝑡))𝑔^𝑇(𝑥((𝑡(𝑥((𝑡≤1𝑡−𝜏)))𝑅𝑔𝑡−𝜏)))𝜏̇(𝑡)𝑥1−^𝑇(𝑡)𝑃𝑊₁𝑅⁻¹𝑊^𝑇₁(𝑡)≤1𝑃𝑥1−𝜏̇(𝑡)‖𝑅‖⁻¹‖𝑊₁‖²‖𝑃‖²|𝑥(𝑡)|². (3.17) For the third term of (3.14), we have 2𝑥^𝑇(𝑡)𝑃𝑢(𝑡)≤𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+[𝑃𝑢(𝑡)]^𝑇𝑃⁻¹𝑃𝑢(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑢^𝑇(𝑡)(𝑡)≤‖𝑃‖𝑃𝑢|𝑥(𝑡)|²+‖𝑃‖|𝑢(𝑡)|². (3.18) Substituting (3.15), (3.16), (3.17), and (3.18) into (3.14), we can obtain the following inequality: 𝑉̇(𝑥(𝑡))≤𝜆|𝑥(𝑡)|²+‖𝑃‖|𝑢(𝑡)|², (3.19) where we denote that 𝜆=−𝜇+‖𝑅‖‖𝑀‖²+‖𝑃‖‖𝑀‖²‖𝑊‖²‖𝑃‖+1+21−𝜏̇(𝑡)‖𝑅‖⁻¹‖𝑊₁‖²‖𝑃‖²,‖𝑅‖=𝑊₁‖𝑃‖𝜏̇(𝑡𝑀‖.1−)‖ (3.20) From (3.11), we easily obtain that 𝜆<0.
Define 𝐾_∞-functions 𝛼₃(𝑟)=−𝜆𝑟², 𝛼₄(𝑟)=‖𝑃‖⋅𝑟². Then we can obtain that 𝑉̇(𝑥(𝑡))≤𝛼₃(|𝑥(𝑡)|)+𝛼₄‖𝑢‖_∞. (3.21) From Lemma 2.4 and Remark 2.3, the system (2.9) is ISS. The proof is complete.

4. Illustrative Examples

In this section, we will give two examples to show the efficiency of the results derived in Section 3.

Example 4.1. Consider a 3-dimension dynamical neural network (2.9) with parameters defined as 𝐴=400040004,𝑊=𝑊₁=.1−0.1−0.2−0.11−0.3−0.2−0.31 (4.1)
Letting 𝑔_𝑖(𝑥_𝑖)=1/(1+𝑒^−𝑥𝑖) and the time-varying delay is chosen as 𝜏(𝑡)=0.6|sin𝑡|. They satisfy assumptions (𝐴1) and (𝐴2), respectively. Obviously, there exist 𝑙₁=𝑙₂=𝑙₃=1 and 𝜏=0.5,0<𝜏̇(𝑡)<0.6 that satisfy the conditions. Then 𝑀=¹⁰⁰⁰¹⁰₀₀₁.
By using MATLAB to solve the LMIs (3.1), we have 01𝑃=100201003,𝐷=.100010001 (4.2) From Theorem 3.1, we can see that delayed neural network (2.9) achieves ISS.

Example 4.2. Consider a 3-dimension dynamical neural network (2.9) with parameters followed as 𝐴=7.60009.50008.8,𝑊=𝑊₁=.0.400.120.320.450.020.100.120.040.42 (4.3)
Letting 𝑔_𝑖(𝜃)=(|𝜃+1|−|𝜃−1|)/2, 𝜃∈𝑅 and the time-varying delay is chosen as 𝜏(𝑡)=1/(𝑡+2). We can check the assumptions (𝐴1) and (𝐴2) with 𝑙₁=𝑙₂=𝑙₃=1 and 0≤𝜏(𝑡)≤1, 0≤𝜏̇(𝑡)≤0.5 for any 𝑡>0. Also we have 𝑀=¹⁰⁰⁰¹⁰₀₀₁.

By solving (3.10) and (3.11), we get 𝑃=.0.70.10.30.10.70.10.30.10.7,𝑄=2.820.8552.460.8554.150.9152.460.9153.16,𝑅=0.927−0.03−0.02−0.039.28−0.08−0.01−0.029.25 (4.4)

From Theorem 3.4, we can see that delayed neural network (2.9) obtains ISS.

However, the above results cannot be obtained by using criteria on ISS in existing publications (e.g., [10, 11, 17–19, 26]).

5. Conclusions

In this paper, dynamical neural networks with time-varying delays were considered. By using Lyapunov-Krasovskii functional method and linear matrix inequalities (LMIs) techniques, several theorems with regarding to judging the ISS property of DNNs with time-varying delays have been obtained. It is shown that the ISS can be determined by solving a set of LMIs, which can be checked by using some standard numerical packages in MATLAB. At last, two numerical examples were given to illustrate the theoretical results.

Appendix

Proof of Lemma 2.4. We divided into four parts to prove this lemma.
Claim  1. (i) The solution 𝑥=0 of the system (2.9) is uniformly asymptotically stable if and only if there exists a function 𝛽 of class 𝐾𝐿 and a positive number 𝑐 independent of 𝑡₀ such that for forall  𝑡≥𝑡₀,forall ‖𝑥_𝑡0‖_𝜏≤𝑐 it satisfies that |𝑥(𝑡)|≤𝛽‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀. (A.1) Particularly, the system (2.9) is uniformly global asymptotically stable if and only if (A.1) admits for any 𝑥_𝑡0∈𝐶.
The Claim is so trivial that we omit the proof here.

Claim  2. For each (𝑡,𝜙)∈𝑅⁺×𝐶, if there exist a continuous functional 𝑉(𝑡,𝜙)∶𝑅⁺×𝐶→𝑅⁺, functions 𝛼₁,𝛼₂ of class 𝐾_∞, and a continuous positive definite function 𝑊 such that 𝛼₁𝜙(0)≤𝑉𝑡,𝜙≤𝛼₂‖𝜙‖_𝜏,𝑉̇𝑡,𝜙≤−𝑊𝜙(0), (A.2)(A.3) then the solution 𝑥(𝑡)=0 is globally uniformly asymptotically stable, and there exist a 𝛽∈𝐾𝐿 such that |𝑥(𝑡)|≤𝛽‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀,∀𝑡₀,𝑥_𝑡0∈𝑅⁺×𝐶,𝑡≥𝑡₀. (A.4)

Proof. From [27], the solution 𝑥(𝑡)=0 is globally uniformly asymptotically stable. Then by Claim 1, we obtain (A.4). The proof is complete.

Claim  3. Let (A.3) in Claim 2 replaced by 𝑉̇𝑡,𝜙≤−𝑊𝜙(0),∀𝜙(0)≥𝜇>0. (A.5) Then for any 𝑥_𝑡0∈𝐶, there exist 𝛽∈𝐾𝐿,𝑇≜𝑇(𝑥_𝑡0,𝜇), such that |𝑥(𝑡)|≤𝛽‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀,∀𝑡∈𝑡₀,𝑡₀,|𝑥(𝑡+𝑇)|≤𝛼₁⁻¹𝛼₂𝜇,∀𝑡≥𝑡₀+𝑇. (A.6)

Proof. Let 𝐵_𝜇≜{𝑥∈𝑅^𝑛∣‖𝑥_𝑡0‖_𝜏≤𝜇}, 𝐵^𝑐_𝜇≜𝑅^𝑛−𝐵_𝜇, 𝐷_𝜇≜{𝑥∈𝑅^𝑛∣‖𝑥_𝑡0‖_𝜏≤𝛼⁻¹(𝛼₂(𝜇))} (no loss generality, we assume that 𝛼₁(𝜇)≤𝛼₂(𝜇), then 𝜇≤𝛼₁⁻¹(𝛼₂(𝜇))). Then 𝐵_𝜇⊆𝐷_𝜇. In the following, we divided 𝑥_𝑡0∈𝐶 into two parts.
Case  1.  𝑥_𝑡0∈𝐵_𝜇.
We make the claim that 𝑥(𝑡) will be always remain in 𝐵_𝜇. Define 𝑡₁=inf{𝑡≥𝑡₀∶|𝑥(𝑡)|=𝜇}, if |𝑥(𝑡)|>𝜇, 𝑡>𝑡₁, then ‖𝑥(𝑡)‖≥𝜇, 𝑡≥𝑡₁ and 𝑉̇(𝑡,𝜙)≤−𝛼₃(|𝜙(0)|)<0, and we have 𝛼₁(|𝑥(𝑡)|)≤𝑉(𝑡,𝑥_𝑡)≤𝑉(𝑡,𝑥_𝑡1)≤𝛼₂(‖𝑥_𝑡1‖_𝜏), 𝑡≥𝑡₁. Then |𝑥(𝑡)|≤𝛼₁⁻¹(𝛼₂(‖𝑥_𝑡1‖_𝜏))=𝛼₁⁻¹(𝛼₂(𝜇)). If |𝑥(𝑡)|<𝜇, 𝑡>𝑡₁, let 𝑡₂=inf{𝑡≥𝑡₁∶|𝑥(𝑡)|=𝜇}, we will analyze them as the above. Then we obtain |𝑥(𝑡)|≤𝛼₁⁻¹(𝛼₂(𝜇)),𝑡≥𝑡₀.
Case  2.  𝑥_𝑡0∈𝐵^𝑐_𝜇, that is, ‖𝑥_𝑡0‖_𝜏>𝜇.
Let 𝑡₀+𝑇(𝑥_𝑡0,𝜇)=inf{𝑡≥𝑡₀∶|𝑥(𝑡)|=𝜇} and 𝑇=𝑇(𝑥_𝑡0,𝜇). We prove that 𝑡₀+𝑇 is limit. From |𝑥(𝑡)|≥𝜇, 𝑡∈[𝑡₀,𝑡₀+𝑇], (A.2), (A.5), and Case 2, we have |𝑥(𝑡)|≤𝛽(‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀), where 𝛽∈𝐾𝐿. Since 𝛽(‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀) is strictly decreasing, and 𝛽(‖𝑥_𝑡0‖_𝜏,⋅)→0 as 𝑡→∞, 𝑡₀+𝑇 is limit. Then from Case 1, 𝑥(𝑡) will be always remain in 𝐷_𝜇 if arrive the boundary of 𝐵_𝜇. Then we obtain (A.6). The proof is complete.

Claim  4. Let (A.3) in Claim 2 replaced by 𝑉̇𝑡,𝜙≤−𝑊𝜙(0),if𝜙(0)≥𝜌‖𝑢‖_∞, (A.7) where 𝜌∈𝐾. Then the system is ISS.

Proof. From Claim 3, we have |𝑥(𝑡)|≤𝛽‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀+𝛼₁⁻¹𝛼₂𝜌‖𝑢‖_∞,𝑡≥𝑡₀. (A.8) Since 𝑥(𝑡) only depends on the 𝑢(𝑠) defined on [𝑡₀,𝑡], we obtain |𝑥(𝑡)|≤𝛽‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀+𝛼₁⁻¹𝛼₂𝜌‖𝑢_𝑡‖_∞,𝑡≥𝑡₀. (A.9) Then |𝑥(𝑡)|≤𝛽‖𝑥_𝑡0‖_𝜏,𝑡−𝑡₀+𝛾‖𝑢_𝑡‖_∞,𝑡≥𝑡₀, (A.10) where 𝛾≜𝛼₁⁻¹(𝛼₂(𝜌)). This proves that the system is ISS.

Acknowledgment

The work is supported partially by National Natural Science Foundation of China under Grant no. 10971240, 61263020, and 61004042, Key Project of Chinese Education Ministry under Grant no. 212138, Natural Science Foundation of Chongqing under Grant CQ CSTC 2011BB0117, and Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ120630.