Global Exponential Stability Criteria for Bidirectional Associative Memory Neural Networks with Time-Varying Delays

Abstract

The global exponential stability for bidirectional associative memory neural networks with time-varying delays is studied. In our study, the lower and upper bounds of the activation functions are allowed to be either positive, negative, or zero. By constructing new and improved Lyapunov-Krasovskii functional and introducing free-weighting matrices, a new and improved delay-dependent exponential stability for BAM neural networks with time-varying delays is derived in the form of linear matrix inequality (LMI). Numerical examples are given to demonstrate that the derived condition is less conservative than some existing results given in the literature.

1. Introduction

A class of neural networks related to bidirectional associative memory (BAM) has been introduced by Kosko [1]. This model generalized the single-layer autoassociative Hebbian correlator to a two-layer pattern-matched heteroassociative circuit. It is an important model with the ability of information memory and information association, which is crucial for various applications such as pattern recognition, solving optimization problems, and automatic control engineering [2–10]. In [1, 11], Kosko investigates the global stability of BAM models and obtains a severe constraint of having a symmetric connection weight matrix. Since it is impossible to maintain an absolutely symmetric connection weight matrix, asymmetric connection has been a focus of this field. Some of these applications require that there should be a well-defined computable solution for all possible initial states. From a mathematical point of view, this means that the equilibrium point of the designed cellular neural networks (CNNs) is globally asymptotically stable (GAS) or globally exponentially stable (GES). Moreover, in biological and artificial neural networks, time delays arise in the process of information transmission; for example, in the electronic implementation of analogue neural networks, time delays occur in the communication and response of neurons owing to the finite switching speed of amplifiers. It is known that they can create an oscillatory or an unstable phenomenon. Therefore, the study of the stability and convergent dynamics of BAM neural networks with delays has raised considerable interest in recent years; see for examples [5, 7, 9, 10, 12–23] and the references cited therein. In [14, 15, 18, 20–22, 24–27], several sufficient conditions on the global exponential stability of BAM neural networks with time-varying delays have been derived. It is worth pointing out that the given criteria in [14, 15, 18, 20–22, 24–27] required the following hypothesis: the time-varying delays are continuously differentiable, the derivative of time-varying delays is smaller than one, and activation functions are bounded and monotonically nondecreasing. The common approach for studying stability of BAM neural networks is Lyapunov stability theory. With a properly designed Lyapunov-Krasovskii functional as well as introducing free-weighting matrices, one may derive stability criteria in term of linear matrix inequality (LMI) which is easily solved by several available algorithms.

Based on the above discussion, we propose to study the problem of global exponential stability of BAM neural networks with time-varying delays and generalized activation functions. The main contributions of our works are that the system consists of both memoryless and delayed activation functions, and the lower and upper bounds of the activation functions are allowed to be either positive, negative, or zero which is more general than systems considered in [14, 15, 18, 21, 22, 24–27]. By constructing a new and improved Lyapunov-Krasovskii functional which contains some integral terms of the activation functions, less conservative results are obtained by introducing appropriate free-weighting matrices and by using some improved integral inequality. Finally, two numerical examples are presented to show that our result is less conservative than some existing ones.

Notations. Throughout the paper, ℝ denotes the set of all real numbers. ∗ denotes the elements below the main diagonal of a symmetric block matrix. diag{⋯} denotes the diagonal matrix. For symmetric matrices 𝑋 and 𝑌, the notation 𝑋>𝑌 (resp., 𝑋≥𝑌) means that the matrix 𝑋−𝑌 is positive definite (resp., nonnegative). 𝜆_𝑚(⋅) and 𝜆_𝑀(⋅) denote the smallest and largest eigenvalue of given square matrix, respectively.

2. Model Description and Preliminaries

Consider the following BAM neural network with time-varying delays of the form 𝑢̇_𝑖(𝑡)=−𝑐_𝑖𝑢_𝑖(𝑡)+^𝑚_𝑗=1𝑎⁽¹⁾_𝑖𝑗𝑔_𝑗𝑣_𝑗(𝑡+_𝑚𝑗=1)𝑎⁽²⁾_𝑖𝑗𝑔_𝑗𝑣_𝑗(𝑡−ℎ(𝑡))+𝐼_𝑖,𝑣̇_𝑗(𝑡)=−𝑑_𝑗𝑣_𝑗(𝑡)+^𝑛_𝑖=1𝑏⁽¹⁾_𝑗𝑖𝑓_𝑖𝑢_𝑖(𝑡)+^𝑛_𝑖=1𝑏⁽²⁾_𝑗𝑖𝑓_𝑖𝑢_𝑖(𝑡−𝑑(𝑡))+𝐼_𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, (1) where 𝑢_𝑖(𝑡) and 𝑣_𝑗(𝑡) are the state of the 𝑖th neurons from the neural field 𝐹_𝑢 and the 𝑗th neurons from the neural field 𝐹_𝑣, at time 𝑡, respectively; 𝑐_𝑖 and 𝑑_𝑗 denote the neuron charging time constants and passive delay rates, respectively; 𝑎⁽¹⁾_𝑖𝑗 and 𝑏⁽¹⁾_𝑗𝑖 are the synaptic connection strengths; 𝑎⁽²⁾_𝑖𝑗 and 𝑏⁽²⁾_𝑗𝑖 are delayed synaptic connection strengths; 𝑓_𝑖(⋅) and 𝑔_𝑗(⋅) denote the activation functions of the 𝑖th neurons from the neural field 𝐹_𝑢 and the 𝑗th neurons from the neural field 𝐹_𝑣, respectively; 𝐼_𝑖 and 𝐼_𝑗 denote the external inputs; and 𝑑(𝑡) and ℎ(𝑡) represent the time-varying differentiable functions which satisfy (i)(𝑡)𝑑̇(𝑡)()(𝑡)ℎ̇(𝑡)0≤𝑑≤𝑑,≤𝜏<1,ii0≤ℎ≤ℎ,≤𝜇<1, (2) where 𝑑,ℎ,𝜇, and 𝜏 are positive scalars.

The initial conditions associated with (1) are assumed to be 𝑢_𝑖(𝑠)=𝜙_𝑖(𝑠),𝑣_𝑗(𝑠)=𝜑_𝑗(𝑠),𝑠∈[−max{𝑑,ℎ},0],𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (3)

Throughout this paper, we make the following assumption on the activation function 𝑓_𝑖(⋅), 𝑔_𝑗(⋅).(A1)𝑓_𝑖(⋅) and 𝑔_𝑗(⋅) are bounded on ℝ.(A2) For any 𝛼,𝛽∈ℝ,𝛼≠𝛽, there exist four constant matrices 𝐸=diag(𝐸₁,𝐸₂,…,𝐸_𝑛), 𝐹=diag(𝐹₁,𝐹₂,…,𝐹_𝑛), 𝑀=diag(𝑀₁,𝑀₂,…,𝑀_𝑚), and 𝑁=diag(𝑁₁,𝑁₂,…,𝑁_𝑚) satisfying 𝐸_𝑖≤𝑓_𝑖(𝛼)−𝑓_𝑖𝛽𝛼−𝛽≤𝐹_𝑖𝑁,𝑖=1,2,…,𝑛,_𝑗≤𝑔_𝑗(𝛼)−𝑔_𝑗𝛽𝛼−𝛽≤𝑀_𝑗,𝑗=1,2,…,𝑚. (4)It is clear that under (A1) and (A2), the system (1) has at least one equilibrium; see [20]. In order to simplify our proof, we shift the equilibrium point 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇, 𝑣^∗=(𝑣^∗₁,𝑣^∗₂,…,𝑣^∗_𝑚)^𝑇 of system (1) to the origin. Let  𝑥_𝑖(𝑡)=𝑢_𝑖(𝑡)−𝑢^∗_𝑖, 𝑦_𝑗(𝑡)=𝑣_𝑗(𝑡)−𝑣^∗_𝑗, 𝑓_𝑖(𝑥_𝑖(𝑡))=𝑓_𝑖(𝑥_𝑖(𝑡)+𝑢^∗_𝑖)−𝑓_𝑖(𝑢^∗_𝑖), 𝑔_𝑗(𝑦_𝑗(𝑡))=𝑔_𝑗(𝑦_𝑗(𝑡)+𝑣^∗_𝑗)−𝑔_𝑗(𝑣^∗_𝑗), 𝑓_𝑖(𝑥_𝑖(𝑡−𝑑(𝑡)))=𝑓_𝑖(𝑥_𝑖(𝑡−𝑑(𝑡))+𝑢^∗_𝑖)−𝑓_𝑖(𝑢^∗_𝑖), 𝑔_𝑗(𝑦_𝑗(𝑡−ℎ(𝑡)))=𝑔_𝑗(𝑦_𝑗(𝑡−ℎ(𝑡))+𝑣^∗_𝑗)−𝑔_𝑗(𝑣^∗_𝑗), 𝑖=1,2,…,𝑛 and 𝑗=1,2,…,𝑚. Then the system (1) can be transformed to𝑥̇(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑎⁽¹⁾_𝑖𝑗𝑔_𝑗𝑦_𝑗(𝑡+_𝑚𝑗=1)𝑎⁽²⁾_𝑖𝑗𝑔_𝑗𝑦_𝑗((𝑡,𝑦̇(𝑡)𝑡−ℎ))=−𝑑_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑏⁽¹⁾_𝑗𝑖𝑓_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑖=1𝑏⁽²⁾_𝑗𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝑑(𝑡)). (5) The activation functions 𝑓_𝑖(⋅) and 𝑔_𝑗(⋅) satisfy the following properties.(H1)𝑓_𝑖 and 𝑔_𝑗 are bounded on ℝ.(H2) For any 𝛼,𝛽∈ℝ,𝛼≠𝛽, there exist constant matrices 𝐸=diag(𝐸₁,𝐸₂,…,𝐸_𝑛),𝐹=diag(𝐹₁,𝐹₂,…,𝐹_𝑛), 𝑀=diag(𝑀₁,𝑀₂,…,𝑀_𝑚), and 𝑁=diag(𝑁₁,𝑁₂,…,𝑁_𝑚) satisfying 𝐸_𝑖≤𝑓_𝑖(𝛼)−𝑓_𝑖𝛽𝛼−𝛽≤𝐹_𝑖𝑁,𝑖=1,2,…,𝑛,_𝑗≤𝑔_𝑗(𝛼)−𝑔_𝑗𝛽𝛼−𝛽≤𝑀_𝑗,𝑗=1,2,…,𝑚. (6)(H3)𝑓_𝑖(0)=0 and 𝑔_𝑗(0)=0, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚.

Rewrite the system (7) into the vector form 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦((𝑡,𝑦̇(𝑡)(𝑡)𝑡−ℎ))=−𝐷𝑦+𝐵₁𝑓(𝑥(𝑡))+𝐵₂𝑓(𝑥(𝑡−𝑑(𝑡))). (7) The initial conditions associated with (7) are assumed to be 𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠),𝑦_𝑗(𝑠)=𝜑_𝑗(𝑠),𝑠∈[−max{𝑑,ℎ},0],𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, (8) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇, 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑚(𝑡))^𝑇, 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛), 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛), 𝐴₁=(𝑎⁽¹⁾_𝑖𝑗)_𝑚×𝑛, 𝐴₂=(𝑎⁽²⁾_𝑖𝑗)_𝑚×𝑛, 𝐵₁=(𝑏⁽¹⁾_𝑖𝑗)_𝑛×𝑚, 𝐵₂=(𝑏⁽²⁾_𝑖𝑗)_𝑛×𝑚, 𝑓(⋅)=(𝑓₁(⋅),𝑓₂(⋅),…,𝑓_𝑛(⋅))^𝑇, 𝑔(⋅)=(𝑔₁(⋅), 𝑔₂(⋅),…,𝑔_𝑚(⋅))^𝑇.

Definition 1 (see [14]). The trivial solution of system (7) is said to be globally exponentially stable if there exist constants 𝑘>0 and 𝜌≥1 such that ‖𝑥(𝑡)‖²+𝑦(𝑡)²≤𝜌𝑒^−2𝑘𝑡𝜙²+𝜑²,∀𝑡≥0, (9) where one denotes 𝜙²+𝜑²=sup_{−max{𝑑,ℎ}≤𝑠≤0}𝜙(𝑠)²+sup_{−max{𝑑,ℎ}≤𝑠≤0}𝜑(𝑠)². (10)

Lemma 2 (see [28]). If there exist symmetric positive-definite matrix 𝑋₃₃>0 and arbitrary matrices 𝑋₁₁,𝑋₁₂,𝑋₁₃,𝑋₂₂, and 𝑋₂₃ such that 𝑋𝑋=¹¹𝑋₁₂𝑋₁₃𝑋^𝑇₁₂𝑋₂₂𝑋₂₃𝑋^𝑇₁₃𝑋^𝑇₂₃𝑋₃₃≥0, (11) then, −^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑋₃₃𝑥̇(𝑠)≤𝑑𝑠_{𝑡𝑡−ℎ(𝑡)}𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑥̇^𝑇(𝑠)×𝑋₁₁𝑋₁₂𝑋₁₃𝑋^𝑇₁₂𝑋₂₂𝑋₂₃𝑋^𝑇₁₃𝑋^𝑇₂₃0𝑥(𝑡)𝑥((𝑡𝑥̇(𝑠)𝑡−ℎ))𝑑𝑠. (12)

Lemma 3 (see [25]). For any real vectors 𝑎, 𝑏 and any matrix 𝑄>0 with appropriate dimensions, it follows that 2𝑎^𝑇𝑏≤𝑎^𝑇𝑄𝑎+𝑏^𝑇𝑄⁻¹𝑏. (13)

Lemma 4 (see [25]). Suppose that (H2) holds; then ^𝑢_𝑣𝑓_𝑖(𝑠)−𝑓_𝑖(𝑣)𝑑𝑠≤[𝑢−𝑣]𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣),𝑖=1,2,…,𝑛,_𝑢𝑣𝑔_𝑗(𝑠)−𝑔_𝑗(𝑣)𝑑𝑠≤[𝑢−𝑣]𝑔_𝑗(𝑢)−𝑔_𝑗(𝑣),𝑗=1,2,…,𝑚. (14)

3. Main Result

In this section, we present a theorem which states the conditions that guarantee the global exponential stability of the system (7) employing the Lyapunov stability theory and linear matrix inequality approach.

Theorem 5. Under the assumptions (H1)–(H3), for given four diagonal matrices 𝐸=diag(𝐸₁,𝐸₂,…,𝐸_𝑛), 𝐹=diag(𝐹₁,𝐹₂,…,𝐹_𝑛), 𝑀=diag(𝑀₁,𝑀₂,…,𝑀_𝑚), and 𝑁=diag(𝑁₁,𝑁₂,…,𝑁_𝑚) and positive constants 𝑑,ℎ,𝜏,𝜇,  and 𝑘, the system (7) is globally exponentially stable with the convergent rate 𝑘, if there exist positive matrices 𝑃_𝑖,𝑊_𝑖,𝑍_𝑖, 𝑖=1,2, 𝑄_𝑗,𝑗=1,2,3,4, positive diagonal matrices 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑛), 𝑅=diag(𝑟₁,𝑟₂,…,𝑟_𝑚), and positive-definite matrices 𝑆𝑆=¹¹𝑆₁₂𝑆₁₃𝑆^𝑇₁₂𝑆₂₂𝑆₂₃𝑆^𝑇₁₃𝑆^𝑇₂₃𝑍₁,𝑇𝑇=¹¹𝑇₁₂𝑇₁₃𝑇^𝑇₁₂𝑇₂₂𝑇₂₃𝑇^𝑇₁₃𝑇^𝑇₂₃𝑍₂, (15) such that the following LMI holds: ΣΞ=¹¹Σ¹₂Σ¹₃000Σ¹₇Σ¹₈∗Σ²₂000000∗∗Σ³₃Σ³₄Σ³₅0Σ³₇Σ³₈∗∗∗Σ⁴₄Σ⁴₅0Σ⁴₇0∗∗∗∗Σ⁵₅Σ⁵₆Σ⁵₇0∗∗∗∗∗Σ⁶₆00∗∗∗∗∗∗Σ⁷₇Σ⁷₈∗∗∗∗∗∗∗Σ⁸₈<0, (16) where Σ¹₁=2𝑘𝑃₁−𝑃₁𝐶−𝐶^𝑇𝑃₁−2𝑘𝐸^𝑇𝐾−2𝑘𝐾^𝑇𝐸+𝐸^𝑇𝐾𝐶+𝐶^𝑇𝐾𝐸+𝑄₁+𝑑𝐶^𝑇𝑍₁𝐶+𝑒^−2𝑘𝑑𝑑𝑆₁₁+𝑆^𝑇₁₃+𝑆₁₃+𝑊₁,Σ¹₂=𝑒^−2𝑘𝑑𝑑𝑆₁₂+𝑆^𝑇₂₃−𝑆₁₃,Σ¹₃Σ=2𝑘𝐾,₁₇=𝑃₁𝐴₁−𝐸^𝑇𝐾𝐴₁−𝑑𝐶^𝑇𝑍₁𝐴₁,Σ¹₈=𝑃₂𝐴₂−𝐸^𝑇𝐾𝐴₂−𝑑𝐶^𝑇𝑍₁𝐴₂,Σ²₂=𝑒^−2𝑘𝑑𝑑𝑆₂₂−𝑆^𝑇₂₃−𝑆₂₃−𝑒^−2𝑘𝑑𝑄₁,Σ³₃=ℎ𝐵^𝑇₁𝑍₂𝐵₁+𝑄₃−2𝐾𝐶𝐹⁻¹,Σ³₄=ℎ𝐵^𝑇₁𝑍₂𝐵₂,Σ³₅=𝑃₂𝐵₁−𝑁^𝑇𝑅𝐵₁−ℎ𝐵^𝑇₁𝑍₂Σ𝐷,₃₇=𝐾𝐴₁+𝑅𝐵₁,Σ³₈=𝐾𝐴₂,Σ⁴₄=ℎ𝐵^𝑇₂𝑍₂𝐵₂−(1−𝜏)𝑒^−2𝑘𝑑𝑄₃−(1−𝜏)𝑒^−2𝑘𝑑𝐹⁻¹𝑊₁𝐹⁻¹,Σ⁴₅=𝑃₂𝐵₂−𝑁^𝑇𝑅𝐵₂−ℎ𝐵^𝑇₂𝑍₂Σ𝐷,₄₇=𝑅𝐵₂,Σ⁵₅=2𝑘𝑃₂−𝑃₂𝐷−𝐷^𝑇𝑃₂−2𝑘𝑁^𝑇𝑅−2𝑘𝑅^𝑇𝑁+𝑁^𝑇𝑅𝐷+𝐷^𝑇𝑅𝑁+𝑄₂+ℎ𝐷^𝑇𝑍₂𝐷+𝑒^−2𝑘ℎℎ𝑇₁₁+𝑇^𝑇₁₃+𝑇₁₃+𝑊₂,Σ⁵₆=𝑒^−2𝑘ℎℎ𝑇₁₂+𝑇^𝑇₂₃−𝑇₁₃,Σ⁵₇Σ=2𝑘𝑅,₆₆=𝑒^−2𝑘ℎℎ𝑇₂₂−𝑇^𝑇₂₃−𝑇₂₃−𝑒^−2𝑘ℎ𝑄₂,Σ⁷₇=𝑑𝐴^𝑇₁𝑍₁𝐴₁+𝑄₄−2𝑅𝐷𝑀⁻¹,Σ⁷₈=𝑑𝐴^𝑇₁𝑍₁𝐴₂,Σ⁸₈=𝑑𝐴^𝑇₂𝑍₁𝐴₂−1−𝜇𝑒^−2𝑘ℎ𝑄₄−1−𝜇𝑒^−2𝑘ℎ𝑀⁻¹𝑊₂𝑀⁻¹. (17)

Proof. Choose the Lyapunov-Krasovskii function candidate for the system (7) to be 𝑉(𝑡)=⁵_𝑙=1𝑉_𝑙(𝑡), (18) where 𝑉₁(𝑡)=𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+2^𝑛_𝑖=1𝑘_𝑖𝑒^{2𝑘𝑡𝑥𝑖}₀^(𝑡)𝑓_𝑖(𝑠)−𝐸_𝑖𝑠𝑑𝑠+𝑒^2𝑘𝑡𝑦^𝑇(𝑡)𝑃₂𝑦(𝑡)+2^𝑚_𝑗=1𝑟_𝑗𝑒^{2𝑘𝑡𝑦𝑗}₀^(𝑡)𝑔_𝑗(𝑠)−𝑁_𝑗𝑠𝑉𝑑𝑠,₂(𝑡)=^𝑡_𝑡−𝑑𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ}𝑒^2𝑘𝑠𝑦^𝑇(𝑠)𝑄₂𝑦(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_{𝑡−𝑑(𝑡)}𝑒^2𝑘𝑠𝑓^𝑇(𝑥(𝑠))𝑄₃𝑓(𝑥(𝑠))𝑑𝑠+^𝑡_{𝑡−ℎ(𝑡)}𝑒^2𝑘𝑠𝑔^𝑇𝑦(𝑠)𝑄₄𝑔𝑦(𝑠)𝑉𝑑𝑠,₄(𝑡)=^𝑡_{𝑡−𝑑(𝑡)}𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑊₁𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−ℎ(𝑡)}𝑒^2𝑘𝑠𝑦^𝑇(𝑠)𝑊₂𝑦(𝑠)𝑉𝑑𝑠,₅(𝑡)=^𝑡_𝑡−𝑑^𝑡_𝑠𝑒^2𝑘𝜃𝑥̇^𝑇(𝜃)𝑍₁𝑥̇(𝜃)+𝑑𝜃𝑑𝑠_{𝑡𝑡−ℎ𝑡𝑠}𝑒^2𝑘𝜃𝑦̇^𝑇(𝜃)𝑍₂𝑦̇(𝜃)𝑑𝜃𝑑𝑠. (19) The derivative of 𝑉(𝑡) along the trajectories of system (7) is given by 𝑉̇₁(𝑡)=2𝑘𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+2𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑃₁𝑥̇(𝑡)+4𝑘^𝑛_𝑖=1𝑘_𝑖𝑒^{2𝑘𝑡𝑥𝑖}₀^(𝑡)𝑓_𝑖(𝑠)−𝐸_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑘_𝑖𝑒^2𝑘𝑡𝑥̇_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)−𝐸_𝑖𝑥_𝑖(𝑡)+2𝑘𝑒^2𝑘𝑡𝑦^𝑇(𝑡)𝑃₂𝑦(𝑡)+2𝑒^2𝑘𝑡𝑦^𝑇(𝑡)𝑃₂𝑦̇(𝑡)+4𝑘^𝑚_𝑗=1𝑟_𝑗𝑒^{2𝑘𝑡𝑦𝑗}₀^(𝑡)𝑔_𝑗(𝑠)−𝑁_𝑗𝑠𝑑𝑠+2^𝑚_𝑗=1𝑟_𝑗𝑒^2𝑘𝑡𝑦̇_𝑗(𝑡)𝑔_𝑗𝑦_𝑗(𝑡)−𝑁_𝑗𝑦_𝑗(𝑡)≤𝑒^2𝑘𝑡2𝑘𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+2𝑥^𝑇(𝑡)×𝑃₁−𝐶𝑥(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦((𝑡𝑓(𝑥(𝑡(𝑡𝑡−ℎ))+4𝑘))−𝐸𝑥)^𝑇(𝑡)𝑓(𝑥(𝑡(𝑡𝐾𝑥+2))−𝐸𝑥)^𝑇×𝐾−𝐶𝑥(𝑡)+𝐴₁𝑔𝑦(𝑡)+𝐴₂𝑔𝑦(𝑡−ℎ(𝑡))+2𝑘𝑦^𝑇(𝑡)𝑃₂𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃₂−𝐷𝑦(𝑡)+𝐵₁𝑓(𝑥(𝑡))+𝐵₂𝑓(𝑥((𝑡𝑔𝑦(𝑡)(𝑡𝑡−𝑑)))+4𝑘−𝑁𝑦)^𝑇(𝑡)𝑔𝑦(𝑡)(𝑡𝑅𝑦+2−𝑁𝑦)^𝑇𝑅×−𝐷𝑦(𝑡)+𝐵₁𝑓(𝑥(𝑡))+𝐵₂𝑓(𝑥(𝑡−𝑑(𝑡))). (20) By (H2), we have −2𝑓^𝑇(𝑥(𝑡))𝐾𝐶𝑥(𝑡)≤−2𝑓^𝑇(𝑥(𝑡))𝐾𝐶𝐹⁻¹𝑓(𝑥(𝑡)),−2𝑔^𝑇𝑦(𝑡)𝑅𝐷𝑦(𝑡)≤−2𝑔^𝑇𝑦(𝑡)𝑅𝐷𝑀⁻¹𝑔𝑦(𝑡). (21) Substituting (21) into (20), we obtain 𝑉̇₁(𝑡)≤𝑒^2𝑘𝑡×2𝑘𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)−2𝑥^𝑇(𝑡)𝑃₁𝐶𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃₁𝐴₁𝑔𝑦(𝑡)+2𝑥^𝑇(𝑡)𝑃₁𝐴₂×𝑔𝑦(𝑡−ℎ(𝑡))+4𝑘𝑓^𝑇(𝑥(𝑡))𝐾𝑥(𝑡)−4𝑘𝑥^𝑇(𝑡)𝐸^𝑇𝐾𝑥(𝑡)−2𝑓^𝑇(𝑥(𝑡))𝐾𝐶×𝐹⁻¹𝑓(𝑥(𝑡))+2𝑓^𝑇(𝑥(𝑡))𝐾𝐴₁𝑔𝑦(𝑡)+2𝑓^𝑇(𝑥(𝑡))𝐾𝐴₂𝑔𝑦(𝑡−ℎ(𝑡))+2𝑥^𝑇(𝑡)𝐸^𝑇𝐾𝐶𝑥(𝑡)−2𝑥^𝑇(𝑡)𝐸^𝑇𝐾𝐴₁𝑔𝑦(𝑡)−2𝑥^𝑇(𝑡)𝐸^𝑇𝐾𝐴₂𝑔𝑦(𝑡−ℎ(𝑡))+2𝑘𝑦^𝑇(𝑡)𝑃₂𝑦(𝑡)−2𝑦^𝑇(𝑡)𝑃₂𝐷𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃₂𝐵₁𝑓(𝑥(𝑡))+2𝑦^𝑇(𝑡)𝑃₂𝐵₂𝑓(𝑥(𝑡−𝑑(𝑡)))+4𝑘𝑔^𝑇𝑦(𝑡)𝑅𝑦(𝑡)−4𝑘𝑦^𝑇(𝑡)𝑁^𝑇𝑅𝑦(𝑡)−2𝑔^𝑇𝑦(𝑡)𝑅𝐷𝑀⁻¹𝑔𝑦(𝑡)+2𝑔^𝑇𝑦(𝑡)𝑅𝐵₁𝑓(𝑥(𝑡))+2𝑔^𝑇𝑦(𝑡)𝑅𝐵₂𝑓(𝑥(𝑡−𝑑(𝑡)))+2𝑦^𝑇(𝑡)𝑁^𝑇𝑅𝐷𝑦(𝑡)−2𝑦^𝑇(𝑡)𝑁^𝑇𝑅𝐵₁𝑓(𝑥(𝑡))−2𝑦^𝑇(𝑡)𝑁^𝑇𝑅𝐵₂𝑓(𝑥((𝑡,𝑉̇𝑡−𝑑)))₂(𝑡)=𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)−𝑒^−2𝑘𝑑𝑥^𝑇(𝑡−𝑑)𝑄₁𝑥(𝑡−𝑑)+𝑦^𝑇(𝑡)𝑄₂𝑦(𝑡)−𝑒^−2𝑘ℎ𝑦^𝑇(𝑡−ℎ)𝑄₂𝑦(,𝑉̇𝑡−ℎ)₃(𝑡)=𝑒^2𝑘𝑡×𝑓^𝑇(𝑥(𝑡))𝑄₃𝑓(𝑥(𝑡−𝑑̇(𝑡)𝑒))1−^{−2𝑘𝑑(𝑡)}𝑓^𝑇(𝑥(𝑡−𝑑(𝑡)))×𝑄₃𝑓(𝑥(𝑡−𝑑(𝑡)))+𝑔^𝑇𝑦(𝑡)𝑄₄𝑔𝑦(𝑡)−1−ℎ̇(𝑡)𝑒^{−2𝑘ℎ(𝑡)}×𝑔^𝑇𝑦(𝑡−ℎ(𝑡))𝑄₄𝑔𝑦(𝑡−ℎ(𝑡))≤𝑒^2𝑘𝑡𝑓^𝑇(𝑥(𝑡))𝑄₃𝑓(𝑥(𝑡−()𝑒))1−𝜏^−2𝑘𝑑𝑓^𝑇(𝑥(𝑡−𝑑(𝑡)))×𝑄₃𝑓(𝑥(𝑡−𝑑(𝑡)))+𝑔^𝑇𝑦(𝑡)𝑄₄𝑔𝑦(𝑡)−1−𝜇𝑒^−2𝑘ℎ𝑔^𝑇𝑦(𝑡−ℎ(𝑡))×𝑄₄𝑔𝑦((𝑡,𝑉̇𝑡−ℎ))₄(𝑡)=𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑊₁𝑥(𝑡)−1−𝑑̇(𝑡)𝑒^{−2𝑘𝑑(𝑡)}𝑥^𝑇(𝑡−𝑑(𝑡))×𝑊₁𝑥(𝑡−𝑑(𝑡))+𝑦^𝑇(𝑡)𝑊₂𝑦(𝑡)−1−ℎ̇(𝑡)𝑒^{−2𝑘ℎ(𝑡)}×𝑦^𝑇(𝑡−ℎ(𝑡))(𝑠)𝑊₂𝑦^𝑇(𝑡−ℎ(𝑡))≤𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑊₁𝑥(𝑡)−(1−𝜏)𝑒^−2𝑘𝑑×𝑥^𝑇(𝑡−𝑑(𝑡))𝑊₁𝑥(𝑡−𝑑(𝑡))+𝑦^𝑇(𝑡)𝑊₂𝑦(𝑡)−1−𝜇𝑒^−2𝑘ℎ×𝑦^𝑇(𝑡−ℎ(𝑡))(𝑠)𝑊₂𝑦^𝑇(𝑡−ℎ(𝑡)). (22)(23) By (H2), we have −𝑒^−2𝑘𝑑𝑥^𝑇(𝑡−𝑑(𝑡))𝑊₁𝑥(𝑡−𝑑(𝑡))≤−𝑒^−2𝑘𝑑𝑓^𝑇(𝑥(𝑡−𝑑(𝑡)))𝐹⁻¹𝑊₁×𝐹⁻¹𝑓(𝑥(𝑡−𝑑(𝑡))),−𝑒^−2𝑘ℎ𝑦^𝑇(𝑡−ℎ(𝑡))(𝑠)𝑊₂𝑦^𝑇(𝑡−ℎ(𝑡))≤−𝑒^−2𝑘ℎ𝑔^𝑇𝑦(𝑡−ℎ(𝑡))𝑀⁻¹𝑊₂×𝑀⁻¹𝑔𝑦(𝑡−ℎ(𝑡)). (24) By (24), we conclude that 𝑉̇₄(𝑡)≤𝑒^2𝑘𝑡𝑥^𝑇(𝑡)𝑊₁𝑥(𝑡)−(1−𝜏)𝑒^−2𝑘𝑑×𝑓^𝑇(𝑥(𝑡−𝑑(𝑡)))𝐹⁻¹𝑊₁𝐹⁻¹×𝑓(𝑥(𝑡−𝑑(𝑡)))+𝑦^𝑇(𝑡)𝑊₂𝑦(𝑡)−1−𝜇𝑒^−2𝑘ℎ×𝑔^𝑇𝑦(𝑡−ℎ(𝑡))𝑀⁻¹𝑊₂𝑀⁻¹𝑦((𝑡.𝑉̇×𝑔𝑡−ℎ))₅(𝑡)=𝑑𝑒^2𝑘𝑡𝑥̇^𝑇(𝑡)𝑍₁𝑥̇(𝑡)−^𝑡_𝑡−𝑑𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠+ℎ𝑒^2𝑘𝑡𝑦̇^𝑇(𝑡)𝑍₂𝑦̇(𝑡)−^𝑡_𝑡−ℎ𝑒^2𝑘𝑠𝑦̇^𝑇(𝑠)𝑍₂𝑦̇(𝑠)𝑑𝑠≤𝑒^2𝑘𝑡𝑑𝑥̇^𝑇(𝑡)𝑍₁𝑥̇(𝑡)−𝑒^{−2𝑘𝑑𝑡}_𝑡−𝑑𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑦̇𝑑𝑠×ℎ^𝑇(𝑡)𝑍₂𝑦̇(𝑡)−𝑒^{−2𝑘ℎ𝑡}_𝑡−ℎ𝑦̇^𝑇(𝑠)𝑍₂𝑦̇(𝑠)𝑑𝑠. (25)(26) By Lemma 2, we obtain −^𝑡_𝑡−𝑑𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)≤𝑑𝑠_{𝑡𝑡−𝑑}𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝑑)𝑥̇^𝑇(𝑠)×𝑆₁₁𝑆₁₂𝑆₁₃𝑆^𝑇₁₂𝑆₂₂𝑆₂₃𝑆^𝑇₁₃𝑆^𝑇₂₃0×𝑥(𝑡)𝑥()𝑥̇(𝑠)=𝑡−𝑑𝑑𝑠_{𝑡𝑡−𝑑}𝑥^𝑇(𝑡)𝑆₁₁+𝑥^𝑇(𝑡−𝑑)𝑆^𝑇₁₂+𝑥̇^𝑇(𝑠)𝑆^𝑇₁₃𝑥(𝑡)+𝑑𝑠_{𝑡𝑡−𝑑}𝑥^𝑇(𝑡)𝑆₁₂+𝑥^𝑇(𝑡−𝑑)𝑆₂₂+𝑥̇^𝑇(𝑠)𝑆^𝑇₂₃×𝑥(𝑡−𝑑)𝑑𝑠+^𝑡_𝑡−𝑑𝑥^𝑇(𝑡)𝑆₁₃+𝑥^𝑇(𝑡−𝑑)𝑆₂₃𝑥̇(𝑠)𝑑𝑠=𝑥^𝑇(𝑡)𝑑𝑆₁₁𝑥(𝑡)+𝑥^𝑇(𝑡−𝑑)𝑑𝑆^𝑇₁₂𝑥(𝑡)+^𝑡_𝑡−𝑑𝑥̇^𝑇(𝑠)𝑑𝑠𝑆^𝑇₁₃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑑𝑆₁₂𝑥(𝑡−𝑑)+𝑥^𝑇(𝑡−𝑑)𝑑𝑆₂₂𝑥(𝑡−𝑑)+^𝑡_𝑡−𝑑𝑥̇^𝑇(𝑠)𝑑𝑠𝑆^𝑇₂₃𝑥(𝑡−𝑑)+𝑥^𝑇(𝑡)𝑆₁₃^𝑡_𝑡−𝑑𝑥̇(𝑠)𝑑𝑠+𝑥^𝑇(𝑡−𝑑)𝑆₂₃^𝑡_𝑡−𝑑𝑥̇(𝑠)𝑑𝑠=𝑥^𝑇(𝑡)𝑑𝑆₁₁𝑥(𝑡)+𝑥^𝑇(𝑡−𝑑)𝑑𝑆^𝑇₁₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑑𝑆₁₂𝑥(𝑡−𝑑)+𝑥^𝑇(𝑡−𝑑)𝑑𝑆₂₂𝑥(𝑡−𝑑)+[𝑥(𝑡)−𝑥(𝑡−𝑑)]^𝑇×𝑆^𝑇₁₃𝑥(𝑡)+𝑆^𝑇₂₃𝑥(+𝑥𝑡−𝑑)_𝑇(𝑡)𝑆₁₃+𝑥^𝑇(𝑡−𝑑)𝑆₂₃×[𝑥(𝑡)(,−−𝑥𝑡−𝑑)]_{𝑡𝑡−ℎ}𝑦̇^𝑇(𝑠)𝑍₂𝑦̇(𝑠)≤𝑑𝑠_{𝑡𝑡−ℎ}𝑦^𝑇(𝑡)𝑦^𝑇(𝑡−ℎ)𝑦̇^𝑇(𝑠)×𝑇₁₁𝑇₁₂𝑇₁₃𝑇^𝑇₁₂𝑇₂₂𝑇₂₃𝑇^𝑇₁₃𝑇^𝑇₂₃0×𝑦(𝑡)𝑦()𝑦̇(𝑠)=𝑡−ℎ𝑑𝑠_{𝑡𝑡−ℎ}𝑦^𝑇(𝑡)𝑇₁₁+𝑦^𝑇(𝑡−ℎ)𝑇^𝑇₁₂+𝑦̇^𝑇(𝑠)𝑇^𝑇₁₃𝑦(𝑡)+𝑑𝑠_{𝑡𝑡−ℎ}𝑦^𝑇(𝑡)𝑇₁₂+𝑦^𝑇(𝑡−ℎ)𝑇₂₂+𝑦̇^𝑇(𝑠)𝑇^𝑇₂₃()+×𝑦𝑡−ℎ𝑑𝑠_{𝑡𝑡−ℎ}𝑦^𝑇(𝑡)𝑇₁₃+𝑦^𝑇(𝑡−ℎ)𝑇₂₃𝑦̇(𝑠)𝑑𝑠=𝑦^𝑇(𝑡)ℎ𝑇₁₁𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ)ℎ𝑇^𝑇₁₂𝑦(𝑡)+^𝑡_𝑡−ℎ𝑦̇^𝑇(𝑠)𝑑𝑠𝑇^𝑇₁₃𝑦(𝑡)+𝑦^𝑇(𝑡)ℎ𝑇₁₂𝑦(𝑡−ℎ)+𝑦^𝑇(𝑡−ℎ)ℎ𝑇₂₂𝑦()+𝑡−ℎ_{𝑡𝑡−ℎ}𝑦̇^𝑇(𝑠)𝑑𝑠𝑇^𝑇₂₃𝑦(𝑡−ℎ)+𝑦^𝑇(𝑡)𝑇₁₃^𝑡_𝑡−ℎ𝑦̇(𝑠)𝑑𝑠+𝑦^𝑇(𝑡−ℎ)𝑇₂₃^𝑡_𝑡−ℎ𝑦̇(𝑠)𝑑𝑠=𝑦^𝑇(𝑡)ℎ𝑇₁₁𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ)ℎ𝑇^𝑇₁₂𝑦(𝑡)+𝑦^𝑇(𝑡)ℎ𝑇₁₂𝑦(𝑡−ℎ)+𝑦^𝑇(𝑡−ℎ)ℎ𝑇₂₂𝑦(𝑡−ℎ)+𝑦(𝑡)−𝑦(𝑡−ℎ)^𝑇×𝑇^𝑇₁₃𝑦(𝑡)+𝑇^𝑇₂₃𝑦(+𝑦𝑡−ℎ)_𝑇(𝑡)𝑇₁₃+𝑦^𝑇(𝑡−ℎ)𝑇₂₃×𝑦(𝑡)−𝑦(𝑡−ℎ). (27) Substituting (27) into (26), we have 𝑉̇₅(𝑡)≤𝑒^2𝑘𝑡𝑑𝑥̇^𝑇(𝑡)𝑍₁𝑥̇(𝑡)+𝑒^−2𝑘𝑑𝑑𝑥^𝑇(𝑡)𝑆₁₁𝑥(𝑡)+2𝑑𝑥^𝑇(𝑡)𝑆^𝑇₁₂𝑥()[𝑥(𝑡)(𝑡−𝑑+2−𝑥𝑡−𝑑)]^𝑇×𝑆^𝑇₁₃𝑥(𝑡)+𝑆^𝑇₂₃𝑥(𝑡−𝑑)+𝑑𝑥^𝑇(𝑡−𝑑)𝑆₂₂𝑥(𝑦̇𝑡−𝑑)+ℎ^𝑇(𝑡)𝑍₂𝑦̇(𝑡)+𝑒^−2𝑘ℎℎ𝑦^𝑇(𝑡)𝑇₁₁𝑦(𝑡)+2ℎ𝑦^𝑇(𝑡)𝑇₁₂𝑦()𝑦(𝑡)(𝑡−ℎ+2−𝑦𝑡−ℎ)^𝑇×𝑇^𝑇₁₃𝑦(𝑡)+𝑇^𝑇₂₃𝑦(𝑡−ℎ)+ℎ𝑦^𝑇(𝑡−ℎ)𝑇₂₂𝑦(𝑡−ℎ). (28) From (22), (25), and (28) we obtain 𝑉̇(𝑡)≤𝑒^2𝑘𝑡𝜉^𝑇(𝑡)Ξ𝜉(𝑡), (29) where Ξ is defined as in (16), and 𝜉(𝑡)=[𝑥(𝑡),𝑥(𝑡−𝑑),𝑓(𝑥(𝑡)),𝑓(𝑥(𝑡−𝑑(𝑡))),𝑦(𝑡),𝑦(𝑡−ℎ),𝑔(𝑦(𝑡)),𝑔(𝑦(𝑡−ℎ(𝑡)))]. Since the matrix Ξ given in Theorem 5 is the negative definite matrix, we have 𝑉̇(𝑡)≤0, for all 𝑡≥0 which implies that 𝑉(𝑡)≤𝑉(0). From the definition of 𝑉(𝑡) in (20), we obtain 𝑉(0)=𝑥^𝑇(0)𝑃₁𝑥(0)+2^𝑛_𝑖=1𝑘_𝑖^𝑥𝑖₀⁽⁰⁾𝑓_𝑖(𝑠)−𝐸_𝑖𝑠+𝑑𝑠_0−𝑑𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑑𝑠_0−𝑑(0)𝑒^2𝑘𝑠𝑓^𝑇(𝑥(𝑠))𝑄₃𝑓(𝑥(𝑠))𝑑𝑠+⁰_−𝑑(0)𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑊₁𝑥(𝑠)𝑑𝑠+𝑦^𝑇(0)𝑃₂𝑦(0)+2^𝑚_𝑗=1𝑟_𝑗^𝑦𝑗₀⁽⁰⁾𝑔_𝑗(𝑠)−𝑁_𝑗𝑠+𝑑𝑠_0−ℎ𝑒^2𝑘𝑠𝑦^𝑇(𝑠)𝑄₂𝑦(𝑠)𝑑𝑠+⁰_−ℎ(0)𝑒^2𝑘𝑠𝑔^𝑇𝑦(𝑠)𝑄₄𝑔𝑦(𝑠)+𝑑𝑠_0−ℎ(0)𝑦^𝑇(𝑠)𝑊₂𝑦(𝑠)+𝑑𝑠_{0−𝑑0𝑠}𝑒^2𝑘𝜃𝑥̇^𝑇(𝜃)𝑍₁𝑥̇(𝜃)+𝑑𝜃𝑑𝑠_0−ℎ0𝑠𝑒^2𝑘𝜃𝑦̇^𝑇(𝜃)𝑍₂𝑦̇(𝜃)𝑑𝜃𝑑𝑠≤𝜆_𝑀𝑃₁𝜙²+2𝐾_𝑀(𝐹−𝐸)𝜙²+𝜆_𝑀𝑄₁+𝜆_𝑀𝑄₃(𝐹−𝐸)+𝜆_𝑀𝑊₁×⁰_−𝑑𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠+𝜆_𝑀𝑃₂𝜑²+2𝑅_𝑀(𝑀−𝑁)𝜑²+𝜆_𝑀𝑄₂+𝜆_𝑀𝑄₄(𝑀−𝑁)+𝜆_𝑀𝑊₂×⁰_−ℎ𝑒^2𝑘𝑠𝑦^𝑇(𝑠)𝑦(𝑠)𝑑𝑠+𝜆_𝑀𝑍₁⁰_−𝑑⁰_𝑠𝑥̇^𝑇(𝜃)𝑥̇(𝜃)𝑑𝜃𝑑𝑠+𝜆_𝑀𝑍₂⁰_−ℎ⁰_𝑠𝑦̇^𝑇(𝜃)𝑦̇(𝜃)𝑑𝜃𝑑𝑠, (30) where 𝐾_𝑀=max_{1≤𝑖≤𝑛}(𝑘_𝑖), 𝐹−𝐸=max_{1≤𝑖≤𝑛}(𝐹_𝑖−𝐸_𝑖), 𝑅_𝑀=max_{1≤𝑗≤𝑚}(𝑟_𝑗) and 𝑀−𝑁=max_{1≤𝑗≤𝑛}(𝑀_𝑗−𝑁_𝑗).
It follows from Lemma 3 that 𝑥̇^𝑇(𝜃)𝑥̇(𝜃)=−𝐶𝑥(𝜃)+𝐴₁𝑔𝑦(𝜃)+𝐴₂𝑔𝑦(𝜃−ℎ(𝜃))^𝑇×−𝐶𝑥(𝜃)+𝐴₁𝑔𝑦(𝜃)+𝐴₂𝑔𝑦(𝜃−ℎ(𝜃))=𝑥^𝑇(𝜃)𝐶^𝑇𝐶𝑥(𝜃)+𝑔^𝑇𝑦(𝜃)𝐴^𝑇₁𝐴₁𝑔𝑦(𝜃)+𝑔^𝑇𝑦(𝜃−ℎ(𝜃))×𝐴^𝑇₂𝐴₂𝑔𝑦(𝜃−ℎ(𝜃))−2𝑥^𝑇(𝜃)𝐶^𝑇𝐴₁𝑔𝑦(𝜃)−2𝑥^𝑇(𝜃)𝐶^𝑇𝐴₂×𝑔𝑦(𝜃−ℎ(𝜃))+2𝑔^𝑇𝑦(𝜃)𝐴^𝑇₁𝐴₂𝑔𝑦(𝜃−ℎ(𝜃))≤3𝑥^𝑇(𝜃)𝐶^𝑇𝐶𝑥(𝜃)+3𝑔^𝑇𝑦(𝜃)𝐴^𝑇₁𝐴₁𝑔𝑦(𝜃)+3𝑔^𝑇𝑦(𝜃−ℎ(𝜃))𝐴^𝑇₂×𝐴₂𝑔𝑦((𝜃,𝑦̇𝜃−ℎ))_𝑇(𝜃)𝑦̇(𝜃)=−𝐷𝑦(𝜃)+𝐵₁𝑓(𝑥(𝜃))+𝐵₂𝑓(𝑥(𝜃−𝑑(𝜃)))^𝑇×−𝐷𝑦(𝜃)+𝐵₁𝑓(𝑥(𝜃))+𝐵₂𝑓(𝑥(𝜃−𝑑(𝜃)))=𝑦^𝑇(𝜃)𝐷^𝑇𝐷𝑦(𝜃)+𝑓^𝑇(𝑥(𝜃))𝐵^𝑇₁𝐵₁𝑓(𝑥(𝜃))+𝑓^𝑇(𝑥(𝜃−𝑑(𝜃)))×𝐵^𝑇₂𝐵₂𝑓(𝑥(𝜃−𝑑(𝜃)))−2𝑦^𝑇(𝜃)𝐷^𝑇𝐵₁𝑓(𝑥(𝜃))−2𝑦^𝑇(𝜃)𝐷^𝑇𝐵₂×𝑓(𝑥(𝜃−𝑑(𝜃)))+2𝑓^𝑇(𝑥(𝜃))𝐵^𝑇₁𝐵₂𝑓(𝑥(𝜃−𝑑(𝜃)))≤3𝑦^𝑇(𝜃)𝐷^𝑇𝐷𝑦(𝜃)+3𝑓^𝑇(𝑥(𝜃))𝐵^𝑇₁𝐵₁𝑓(𝑥(𝜃))+3𝑓^𝑇(𝑥(𝜃−𝑑(𝜃)))𝐵^𝑇₂×𝐵₂𝑔𝑦(𝜃−ℎ(𝜃)). (31) Substituting (31) into (30), we obtain the bound of 𝑉(0) as follows: 𝑉(0)≤𝜆_𝑀𝑃₁𝜙²+2𝐾_𝑀(𝐹−𝐸)𝜙²+1−𝑒^−2𝑘𝑑×𝜆2𝑘_𝑀𝑄₁+𝜆_𝑀𝑄₃×(𝐹−𝐸)+𝜆_𝑀𝑊₁+𝜆_𝑀𝑃₂𝜑²+2𝑅_𝑀(𝑀−𝑁)𝜑²+1−𝑒^−2𝑘ℎ×𝜆2𝑘_𝑀𝑄₂+𝜆_𝑀𝑄₄(𝑀−𝑁)+𝜆_𝑀𝑊₂+𝑑²2𝜆_𝑀𝑍₁3𝜆_𝑀𝐶^𝑇𝐶𝜙²+3𝜆_𝑀𝐴^𝑇₁𝐴₁𝜑²+3𝜆_𝑀𝐴^𝑇₂𝐴₂𝜑²+ℎ²2𝜆_𝑀𝑍₂3𝜆_𝑀𝐷^𝑇𝐷𝜑²+3𝜆_𝑀𝐵^𝑇₁𝐵₁𝜙²+3𝜆_𝑀𝐵^𝑇₂𝐵₂𝜙². (32) Thus, 𝑉(0)≤𝜒₁𝜙²+𝜒₂𝜑², (33) where 𝜒₁=𝜆_𝑀𝑃₁+2𝐾_𝑀(𝐹−𝐸)+1−𝑒^−2𝑘𝑑×𝜆2𝑘_𝑀𝑄₁+𝜆_𝑀𝑄₃(𝐹−𝐸)+𝜆_𝑀𝑊₁+3𝑑²2𝜆_𝑀𝑍₁𝜆_𝑀𝐶^𝑇𝐶+ℎ²2𝜆_𝑀𝑍₂3𝜆_𝑀𝐵^𝑇₁𝐵₁+3𝜆_𝑀𝐵^𝑇₂𝐵₂,𝜒₂=𝜆_𝑀𝑃₂+2𝑅_𝑀(𝑀−𝑁)+1−𝑒^−2𝑘ℎ×𝜆2𝑘_𝑀𝑄₂+𝜆_𝑀𝑄₄(𝑀−𝑁)+𝜆_𝑀𝑊₂+3ℎ²2𝜆_𝑀𝑍₂𝜆_𝑀𝐷^𝑇𝐷+𝑑²2𝜆_𝑀𝑍₁3𝜆_𝑀𝐴^𝑇₁𝐴₁+3𝜆_𝑀𝐴^𝑇₂𝐴₂. (34) On the other hand, we have 𝑉(𝑡)≥𝑒^2𝑘𝑡𝜆_𝑚𝑃₁‖𝑥(𝑡)‖²+𝜆_𝑚𝑃₂𝑦(𝑡)². (35) Therefore, ‖𝑥(𝑡)‖²+𝑦(𝑡)²≤𝜌𝑒^−2𝑘𝑡𝜙²+𝜑², (36) where 𝜌=max(𝜒₁,𝜒₂)/min(𝜆_𝑚(𝑃₁),𝜆_𝑚(𝑃₂))≥1. Therefore, the system (7) is global exponentially stable with the convergent rate 𝑘>0. This completes the proof.

Remark 6. In hypothesis (H2), lower bounds 𝐸_𝑖,𝑁_𝑗 and upper bounds 𝐹_𝑖, 𝑀_𝑗, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, of activation functions are allowed to be either positive, negative, or zero. Clearly, hypothesis (H2) in our paper is more general than those given in [14, 15, 18, 21, 22, 24–27]. Hence, our result is less conservative than some existing results given in the literature.

4. Numerical Examples

Example 1. Consider the BAM neural networks in (7) with 𝐶=diag(1,1,1), 𝐷=diag(2,2,2),𝐴₁=𝐵₁=0, 𝐴₂=,𝐵0.050.250.050.10.050.150.150.150.05₂=.0.750.750.9500.500.150.150.150.05 (37)

In this example, the activation function and time delay are given as follows: 𝑓₁(𝑥)=𝑓₂(𝑥)=𝑓₃(𝑥)=(1/2)(|𝑥+1|−|𝑥−1|), 𝑔₁(𝑦)=𝑔₂(𝑦)=𝑔₃(𝑦)=(1/2)(|𝑦+1|−|𝑦−1|), 𝑑(𝑡)=0.5, ℎ(𝑡)=1. It follows that 𝑑=0.5, 𝜏=0.3, ℎ=1, and 𝜇=0.3. The assumption (H2) is satisfied with 𝐹=diag(1,1,1), 𝑀=diag(1,1,1), 𝐸=𝑁=0. Let 𝑘=0.1. By using the LMI Toolbox in MATLAB, the LMI (16) of Theorem 5 is feasible with 𝑘=0.1 and a set of solutions of (16) is given by 𝑃₁=,𝑃28.63820.68430.75010.684328.20611.09810.75011.098127.7326₂=,𝑄11.76310.56710.41700.567125.82960.79340.41700.793426.2860₁=,𝑄14.4157−0.0495−0.0610−0.049514.4217−0.0911−0.0610−0.091114.4462₂=,𝑄12.8030−0.9294−0.9704−0.929416.9982−0.4287−0.9704−0.428717.2419₃=,𝑄16.77250.44950.25280.449515.06690.59690.25280.596913.1708₄=,𝑊11.9146−0.2010−0.3870−0.201015.1767−0.6757−0.3870−0.675715.3739₁=,𝑊14.62181.08580.99971.085813.02601.30010.99971.300111.6739₂=,𝑍8.4065−0.0639−0.2523−0.063911.99590.1677−0.25230.167711.6957₁=,𝑍13.39930.21420.38990.214213.57460.39410.38990.394113.7729₂=,𝑆2.35110.74680.67360.746811.26270.35000.67360.350011.7724₁₁=,𝑆12.4617−0.0242−0.0266−0.024212.4732−0.0399−0.0266−0.039912.4949₁₂=,𝑆−1.1535−0.0211−0.0257−0.0211−1.1444−0.0335−0.0257−0.0335−1.1320₁₃=,𝑆−0.9075−0.0415−0.0544−0.0415−0.9093−0.0805−0.0544−0.0805−0.9019₂₂=,𝑆12.2314−0.0115−0.0101−0.011512.2463−0.0163−0.0101−0.016312.2626₂₃=,𝑇1.63060.01290.02080.01291.64300.02700.02080.02701.6490₁₁=,𝑇8.6800−0.5650−0.6089−0.565012.0158−0.3052−0.6089−0.305212.2653₁₂=,𝑇−1.8596−0.2702−0.2795−0.2702−1.2902−0.1294−0.2795−0.1294−1.2317₁₃=,𝑇−1.7089−0.3795−0.4069−0.3795−0.6642−0.2437−0.4069−0.2437−0.5078₂₂=,𝑇8.2813−0.2103−0.2276−0.210310.2975−0.1027−0.2276−0.102710.4221₂₃=,,.1.97070.18710.18700.18712.30620.07830.18700.07832.3346𝐾=14.942700014.273400013.4467𝑅=5.83150008.19630008.3562 (38) Thus, the system (7) is 0.1-exponentially stable and the value 𝜌=13.4606. The solution of the closed-loop system satisfies ‖𝑥(𝑡)‖²+𝑦(𝑡)²≤13.4606𝑒^{−2(0.1)𝑡}𝜙²+𝜑²,∀𝑡∈ℝ⁺. (39)

By applying Theorem 5 and by solving the LMI (16) using MATLAB LMI Toolbox, we obtain the convergence rate 𝑘 which guarantees that the global exponential stability is 0.998. In Table 1, we give comparison of maximum allowable convergence rate 𝑘 obtained by Theorem 5 and by other methods in some previous existing results. From Table 1, it is shown that the proposed global exponential stability criterion is less conservative than those obtained in [24, 25, 29].

Table 1 

Maximum allowable convergence rate.

Example 2. Consider the BAM neural networks in (7) with ,𝐴𝐶=500050005₁=,𝐴−0.72−0.44−0.21−0.72−0.83−0.1−0.010.01−0.04₂=,,𝐵−0.01−0.12−0.240.17−0.33−0.43−0.250.33−0.05𝐷=500050005₁=,𝐵−0.31−0.310.920.34−0.33−0.780.340.470.25₂=.−0.83−0.12−0.52−0.650.5−0.14−0.05−0.14−0.65 (40)

In this example, the activation function and time-varying delay are given as follows: 𝑓₁(𝑥)=𝑓₂(𝑥)=𝑓₃(𝑥)=(1/2)(|𝑥+1|−|𝑥−1|), 𝑔₁(𝑦)=𝑔₂(𝑦)=𝑔₃(𝑦)=(1/2)(|𝑦+1|−|𝑦−1|), 𝑑(𝑡)=sin²(0.5𝑡), ℎ(𝑡)=0.1cos²(𝑡); the assumption (H2) is satisfied with 𝐸=diag(−0.2,−0.25,−0.2), 𝐹=diag(0.3,0.4,0.5), 𝑁=diag(−0.2,−0.25,−0.2), 𝑀=diag(0.3,0.4,0.5). It follows that 𝑑=0.5,𝜏=0.3,ℎ=0.1, and 𝜇=0.3. Let 𝑘=0.1. By using the LMI Toolbox in MATLAB, the LMI (16) is feasible with 𝑘=0.1 and a set of solutions of (16) is given by 𝑃₁=,𝑃110.3704−30.1135−5.4152−30.113590.30138.8948−5.41528.8948115.0609₂=,𝑄43.9625−2.30317.3341−2.303138.16714.99697.33414.996962.3183₁=,𝑄125.4489−1.88020.9399−1.8802127.5495−1.00630.9399−1.0063130.0399₂=,𝑄100.3079−4.1254−0.7526−4.1254101.84901.0897−0.75261.0897114.9297₃=,𝑄136.0956−4.291938.6712−4.291990.480819.202238.671219.2022113.1213₄=,𝑊42.6986−34.8491−15.2241−34.849196.77282.0801−15.22412.0801145.8684₁=,𝑊140.805925.3914−0.935725.3914146.4301−4.4760−0.9357−4.476011.6128₂=,𝑍105.4681−0.14055.1590−0.1405107.441916.76375.159016.7637134.1991₁=,𝑍69.1695−24.4444−4.5155−24.444448.71777.8545−4.51557.854565.4657₂=,𝑆99.94901.335735.24701.335780.217910.668235.247010.6682111.8243₁₁=,𝑆103.3199−2.98950.4043−2.9895103.2019−0.14040.4043−0.1404106.3073₁₂=,𝑆−2.9235−0.63140.1584−0.6314−2.4889−0.16720.1584−0.1672−1.9572₁₃=,𝑆1.0569−2.46490.0237−2.46490.40900.21130.02370.21132.4210₂₂=,𝑆97.0839−2.17720.0865−2.177296.32840.19800.08650.198098.4407₂₃=,𝑇8.9358−1.7429−0.3501−1.74297.02420.7119−0.35010.71198.3850₁₁=,𝑇105.0332−0.41930.0331−0.4193105.22660.30430.03310.3043107.5064₁₂=,𝑇−1.5261−0.6096−0.1149−0.6096−1.24150.0894−0.11490.08940.3340₁₃=,𝑇−7.6562−3.2414−0.6934−3.2414−5.76500.5527−0.69340.55274.8321₂₂=,𝑇104.9648−0.5090−0.0471−0.5090105.19400.0964−0.04710.0964106.5444₂₃=,,.2.60720.14200.20770.14202.6111−0.04040.2077−0.04042.2053𝐾=106.595700063.39210006.3380𝑅=86.005900072.023100071.2919 (41) Thus, the system (7) is 0.1-exponentially stable and the value 𝜌=14.1811. The solution of the closed-loop system satisfies ‖𝑥(𝑡)‖²+𝑦(𝑡)²≤14.1811𝑒^{−2(0.1)𝑡}𝜙²+𝜑²,∀𝑡∈ℝ⁺. (42) The maximum allowable convergence rate 𝑘 for different values of 𝜏=𝜇 is given in Table 2. The trajectory of solutions of BAM neural networks with time-varying delays is shown in Figure 1, where the initial conditions are chosen as 𝜙₁=cos(𝑠), 𝜙₂=sin(𝑠),𝜙₃=sin(𝑠)−1, 𝜑₁=cos(𝑠)+1,𝜑₂=sin(𝑠)−2, 𝜑₃=cos(𝑠)+1, 𝑠∈[−0.5,0].

Table 2 

Maximum allowable convergence rate.

Figure 1 

Time responses of state variables.

5. Conclusion

This paper has proposed a new sufficient condition guaranteeing the global exponential stability criteria for bidirectional associative memory neural networks with time-varying delays and generalized activation functions. The developed stability condition is in terms of LMI, which can be easily solved by some existing software packages. Furthermore, the proposed stability conditions are less conservative than some works in the literature.

Acknowledgments

The first author is supported by the Graduate School, Chiang Mai University and the Science Achievement Scholarship of Thailand. The second author is supported by the Centre of Excellence in Mathematics, Thailand and Commission for Higher Education, Thailand.