State Feedback Stabilization for Neutral-Type Neural Networks with Time-Varying Discrete and Unbounded Distributed Delays

Abstract

The problem of stabilization for a class of neutral-type neural networks with discrete and unbounded distributed delays is investigated. By introducing an appropriate Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative, delay-range-dependent and rate-dependent stabilization criteria are presented in the form of LMIs with nonlinear constraints. In order to solve the nonlinear problem, a cone complementarity linearization (CCL) algorithm is offered. In addition, several numerical examples are provided to illustrate the applicability of the proposed approach.

1. Introduction

In 1943, McCulloch and Pitts [1] proposed the concept of artificial neurons and proved that a single neuron can perform logic functions. This is usually viewed as the beginning of era of artificial neural networks, which is called neural networks for short. Recently, neural networks have received considerable attention due to their wide applications in solving some optimization problems, pattern recognition, image processing, and signal processing. Since time delay is frequently a source of instability and/or oscillation of many practical systems, several approaches have been proposed for analysis and synthesis of delayed neural networks (see [2–8] and the references therein).

A neutral-type time-delay system contains delays in the state and its derivatives. Such system can be found in population ecology, lossless transmission lines, heat exchangers, and so forth. Because of its wider applications, the stability for the class of neutral-type systems has received considerable attention in the last several decades (see, e.g., [9–13]). Corresponding to the class of neutral-type time-delay systems, we will also get a class of neutral-type neural networks. Stability criteria for neutral-type neural networks have been proposed in [3, 14–21] by constructing appropriate Lyapunov-Krasovskii functional and applying the linear matrix inequality (LMI) approach. Recently, the stability conditions for the neutral-type neural networks with discrete and unbounded distributed delays were provided in [17–21]. Specifically, Rakkiyappan and Balasubramaniam [21] studied the global asymptotic stability for a class of neutral-type neural networks with two unbounded distributed delays by the so-called LMI technique. Lu [22] designed a state feedback controller stabilizing the neutral-type neural networks with discrete and bounded distributed delays. However, there are no results about the stabilization for neutral-type neural networks with unbounded distributed delay(s), this motivates our research.

This paper deals with the stabilization problem of a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays. New sufficient conditions for the existence of the state feedback controller are proposed by introducing a Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative. Since these sufficient conditions are presented in the form of LMIs with nonlinear constraints, the so-called CCL technique is employed to deal with the nonlinear constraints. This allows us to obtain a state feedback gain by LMI Control Toolbox of MATLAB. Several numerical examples are given to illustrate the effectiveness of the proposed approach.

2. Problem Formulation

Consider the class of neutral-type neural networks with time-varying discrete and unbounded distributed delays described by the integro differential equation: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1+𝑐_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡−𝜏(𝑡))_𝑛_𝑗=1𝑒_𝑖𝑗^𝑡−∞𝑧_𝑗(𝑡−𝑠)𝑓_𝑗𝑥_𝑗(𝑠)d𝑠+^𝑛_𝑗=1+𝑑_𝑖𝑗𝑥̇_𝑗(𝑡−ℎ(𝑡))_𝑛_𝑗=1𝑓_𝑖𝑗^𝑡−∞𝑥𝑧_𝑗(𝑡−𝑠)𝑥̇_𝑗(𝑠)d𝑠+ℓ_𝑖+𝐼_𝑖(𝑡),(1)_𝑖(𝜃)=𝜙_𝑖(𝜃),∀𝜃∈(−∞,0],(2) where 𝑖=1,2,…,𝑛,𝑥_𝑖(𝑡) is the state variable of the 𝑖th neuron, 𝑎_𝑖>0 is a constant, 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, 𝑑_𝑖𝑗, 𝑒_𝑖𝑗, and 𝑓_𝑖𝑗 are connection weight coefficients of the neurons, 𝑓_𝑗 is the neuron activation function, ℓ_𝑖 is the external bias vector element, 𝐼_𝑖(𝑡) is the external control input vector, 𝜙_𝑖(𝜃) denotes the initial condition, 𝜏(𝑡) and ℎ(𝑡) are time-varying delays satisfying 00≤≤ℎ𝜏₁₁≤𝜏(𝑡)≤𝜏₂,𝜏̇(𝑡)≤𝜏₃,≤ℎ(𝑡)≤ℎ₂,ℎ̇(𝑡)≤ℎ₃,(3) and 𝑧_𝑗(𝑡) is a real value nonnegative continuous function on [0,+∞) and satisfies ^∞0𝑧_𝑗(𝑠)d𝑠=1.(4) When 𝐼_𝑖(𝑡)≡0 for all 𝑖, the global asymptotic stability of the system (1) has been considered in [21]. Here, the control input item 𝐼_𝑖(𝑡) in (1) is borrowed from [22].

Throughout this paper, we assume that the neuron activation functions 𝑓_𝑗 (𝑗=1,2,…,𝑛) satisfy the following condition: ||𝑓_𝑗𝜁₁−𝑓_𝑗𝜁₂||≤𝐿_𝑗||𝜁₁−𝜁₂||∀𝜁₁,𝜁₂∈ℝ,𝜁₁≠𝜁₂,(5) where 𝐿_𝑗 (𝑗=1,2,…,𝑛) are known positive constants.

Let 𝑥^∗=(𝑥^∗1,𝑥^∗2,…,𝑥^∗𝑛) be an equilibrium point of the system (1). Set 𝑦_𝑖=𝑥_𝑖−𝑥^∗𝑖,   𝑖=1,…,𝑛. Then the system (1) can be written as 𝑦̇(𝑡)=−++𝐶𝐸𝐴𝑔𝑦((𝑦𝑡_𝑡−∞)(+𝑡−𝐵𝜏𝑔((𝑡𝑦))()𝑡)+)𝐷𝑦̇(𝑡−ℎ(𝑡))+𝐹_𝑡−∞𝑍(𝑡−𝑠)𝑔(𝑦(𝑠))d𝑠𝑦(𝜃)=𝜙(𝜃𝑍),(𝑡−∀𝜃𝑠)∈𝑦̇((𝑠−)∞d𝑠,0+]𝐼,(𝑡),(6) where 𝑦(𝑡)=[𝑦₁(𝑡)𝑦₂(𝑡)⋯𝑦_𝑛(𝑡)]^𝑇, 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛), 𝐵=[𝑏_𝑖𝑗], 𝐶=[𝑐_𝑖𝑗], 𝐷=[𝑑_𝑖𝑗], 𝐸=[𝑒_𝑖𝑗], 𝐹=[𝑓_𝑖𝑗], 𝑍(𝑡)=diag(𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡)), 𝐼(𝑡)=[𝐼₁(𝑡)⋯𝐼_𝑛(𝑡)]^𝑇, 𝑔(𝑦(𝑡))=[𝑔₁(𝑦₁(𝑡))g₂(𝑦₂(𝑡))⋯𝑔_𝑛(𝑦_𝑛(𝑡))]^𝑇 with 𝑔_𝑗(𝑦_𝑗(𝑡))=𝑓_𝑗(𝑦_𝑗(𝑡)+𝑥^∗𝑗)−𝑓_𝑗(𝑥^∗𝑗), and 𝜙(𝜃)=[𝜙₁(𝜃)−𝑥^∗1⋯𝜙_𝑛(𝜃)−𝑥^∗𝑛]. From (5), we have 𝑔^2𝑗(𝜁)≤𝐿^2𝑗𝜁², 𝑗=1,2,…,𝑛, for all 𝜁∈ℝ.

The main purpose of this paper is to develop a delay-range-dependent and rate-dependent condition for the existence of a state feedback controller 𝐼(𝑡)=𝐾𝑦(𝑡),(7) which stabilizes the neutral-type neural network (6) with time-varying discrete and unbounded distributed delays.

A delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained based on a Lyapunov-Krasovskii functional approach accompanied with a CCL technique, a delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained. Set 𝑦_𝑡(𝜃)=𝑦(𝑡+𝜃), 𝜃∈(−∞,0]. Assume that there exists a scalar 𝑉>0 such that ∫^0−∞‖𝑦_𝑡(𝑠)‖²d𝑠≤𝑉 and ∫^0−∞‖𝑦̇_𝑡(𝑠)‖²d𝑠≤𝑉. This guarantees that the so-called Lyapunov-Krasovskii functional theory can be applied to the considered stabilization problem.

3. Main Results

In this section, we will investigate a delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6). This requires the following several lemmas.

Lemma 1 (Jensen inequality [23]). Given a real symmetry positive-definite matrix 𝑃∈ℝ^𝑛×𝑛, a pair of scalars 𝑎 and 𝑏 satisfying 𝑏≥𝑎≥0. If a vector-valued function 𝑥∶[𝑎,𝑏]→ℝ^𝑛 is derivable on [𝑎,𝑏], then (𝑏−𝑎)^𝑏𝑎𝑥̇^𝑇(𝑤)𝑃𝑥̇(𝑤)d𝑤≥[𝑥(𝑏)−𝑥(𝑎)]^𝑇𝑃[𝑥(𝑏)−𝑥(𝑎)].(8)

Lemma 2 (Schur complementary lemma [24]). For a given matrix 𝑆∶=^𝑆11𝑆12_{𝑆𝑇12𝑆22} with 𝑆₁₁=𝑆^𝑇11 and 𝑆₂₂=𝑆^𝑇22, the following conditions are equivalent.(i)𝑆<0. (ii)𝑆₂₂<0 and 𝑆₁₁−𝑆₁₂𝑆⁻¹²²𝑆^𝑇12<0.(iii)𝑆₁₁<0 and 𝑆₂₂−𝑆^𝑇12𝑆⁻¹¹¹𝑆₁₂<0.

Lemma 3 (Cauchy-Schwarz inequality [25]). If the functions 𝑓 and 𝑔 are integral on [𝑎,𝑏], then ^𝑏𝑎𝑓(𝑠)𝑔(𝑠)d𝑠²≤^𝑏𝑎𝑓²(𝑠)d𝑠^𝑏𝑎𝑔²(𝑠)d𝑠.(9)

Based on the above three lemmas, the following delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6) can be investigated by constructing an appropriate Lyapunov-Krasovskii functional and applying the CCL technique.

Theorem 4. For given scalars ℎ₁, ℎ₂, ℎ₃, 𝜏₁, 𝜏₂, and 𝜏₃ such that ℎ₂>ℎ₁>0 and 𝜏₂>𝜏₁>0, the system (6) subject to (3) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices 𝑃,𝑃,𝑇_𝑖,𝑇_𝑖,𝑄_𝑖(𝑖=1,…,6),𝑄₄,𝑄₅,𝑄₆, 𝑅_𝑗(𝑗=1,…,5),𝑅₄,𝑅₅, 𝑀∶=diag(𝑚₁,…,𝑚_𝑛), 𝑁∶=diag(𝑛₁,…,𝑛_𝑛), 𝑁∶=diag(𝑛₁̃,…,𝑛_𝑛̃), a matrix 𝐾,  and positive scalars 𝜀₁ and 𝜀₂,  such that ⎡⎢⎢⎢⎢⎢⎢⎢⎣Π𝐴^𝑇𝐴^𝑇𝐴^𝑇𝐴^𝑇𝐴^𝑇𝐴^𝑇𝐴𝐴−0𝑇₁−0𝑇₂0000𝐴00−0𝑇₃000𝐴000−0𝑇₄00𝐴0000−0𝑇₅0𝐴00000−0𝑇₆−𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎦<0,(10)₁𝑃−𝑃𝑄₄≤0,−𝑇²𝑃−𝜏𝑃₋₁₁₂−𝑇𝑅₄≤0,₃𝑃−𝜏𝑃₂₋₁−𝑇𝑅₅≤0,(11)₄𝑃−ℎ𝑃₋₁₁₂𝑄₅≤0,−𝑇⁵𝑃−ℎ𝑃₂₋₁−𝑇𝑄₆≤0,₆𝑃𝑃𝑃=𝐼,−𝑃𝑁𝑇_𝑖≤0,(12)𝑅𝑇_𝑖=𝐼(𝑖=1,…,6),₄𝑄𝑅₄=𝐼,𝑅₅𝑅₅=𝐼,₄𝑄𝑄₄=𝐼,𝑄₅𝑄₅=𝐼,₆𝑄₆=𝐼,𝑁𝑁=𝐼,(13)

where Π∶=𝑒^𝑇1³_𝑖=1𝑅_𝑖+³_𝑖=1𝑄_𝑖+𝜀₁𝐿^𝑇𝐿𝑒₁−−𝑒^𝑇21−1ℎ₃−𝜏₃𝑅₃−𝜀₂𝐿^𝑇𝐿𝑒₂+𝐴^𝑇𝑒₁+𝑒^𝑇1𝐴−𝜏₋₁₁₂𝑒^𝑇5𝑄₃𝑒₅−ℎ⁻¹¹²𝑒₅−𝑒₆^𝑇𝑄₆𝑒₅−𝑒₆+𝑒_𝑇9𝑒₂−𝑒₄^𝑇𝑅₅𝑒₂−𝑒₄𝑀−𝜀₁𝐼𝑒₉−𝑒^𝑇11𝑀𝑒₁₁−𝑒^𝑇12𝑁𝑒₁₂−𝜀₂𝑒^𝑇10𝑒₁₀−1−ℎ₃𝑒^𝑇8𝑄₄𝑒₈−−𝜏𝜏⁻¹¹²₋₁₁₂𝑒₂−𝑒₄^𝑇𝑅₄𝑒₂−𝑒₄−𝑒_𝑇3𝑒₂−𝑒₃^𝑇𝑅₄𝑒₂−𝑒₃𝑅₁𝑒₃−𝑒^𝑇4𝑅₂𝑒₄−−𝜏𝜏⁻¹¹²₁₋₁𝑒₂−𝑒₃^𝑇𝑅₅𝑒₂−𝑒₃−𝑒_𝑇6𝑒₁−𝑒₃^𝑇𝑅₅𝑒₁−𝑒₃𝑄₁𝑒₆−𝑒^𝑇7𝑄₂𝑒₇−−ℎℎ⁻¹¹²₋₁₁₂𝑒₅−𝑒₆^𝑇𝑄₅𝑒₅−𝑒₆−ℎ₋₁₁₂𝑒₅−𝑒₇^𝑇𝑄₆𝑒₅−𝑒₇−ℎ₁₋₁𝑒₅−𝑒₇^𝑇𝑄₅𝑒₅−𝑒₇𝐴=−𝑃𝐴+𝑒₁𝐿𝐾−=0𝑒₆d0i^𝑇a0g𝑄0₆0𝐿𝑒₁0₁−𝑃𝐷𝑒₆𝑃,𝐵𝑃𝐶𝑃𝐸𝑃𝐹,𝑒_𝑖,…,𝐿_𝑛,=ℎ0_{𝑛×(𝑖−1)}𝐼_𝑛0_{𝑛×(12−𝑖)}(𝑖=1,…,12),₁₂=ℎ₂−ℎ₁,𝜏₁₂=𝜏₂−𝜏₁.(14)

Furthermore, when the LMIs (10)–(12) with the constraint (13) are feasible, a desired state feedback gain is given by 𝐾=𝑃⁻¹𝐾.

Proof. Choose a Lyapunov-Krasovskii functional as follows: 𝑉𝑦_𝑡=⁶_𝑖=1𝑉_𝑖𝑦_𝑡,(15) with 𝑉𝑉₁𝑦_𝑡=𝑦^𝑇(𝑡)𝑃𝑦(𝑡),(16)₂𝑦_𝑡=+^{𝑡𝑡−ℎ(𝑡)}𝑦^𝑇(𝑠)𝑄₃𝑦(𝑠)+𝑦̇^𝑇(𝑠)𝑄₄𝑦̇(𝑠)d𝑠₂_𝑖=1^{𝑡𝑡−ℎ}_𝑖+𝑦^𝑇(𝑠)𝑄_𝑖𝑦(𝑠)d𝑠_{𝑡𝑡−𝜏(𝑡)}𝑦^𝑇(𝑠)𝑅₃𝑦(𝑠)d𝑠+²_𝑖=1^{𝑡𝑡−𝜏}_𝑖𝑉𝑦^𝑇(𝑠)𝑅_𝑖𝑦(𝑠)d𝑠,₃𝑦_𝑡=+^−𝜏1_−𝜏2^{𝑡𝑡+𝜃}𝑦̇^𝑇(𝑠)𝑅₄𝑦̇(𝑠)d𝑠d𝜃_0−𝜏2^{𝑡𝑡+𝜃}𝑉𝑦̇^𝑇(𝑠)𝑅₅𝑦̇(𝑠)d𝑠d𝜃,₄𝑦_𝑡=+^−ℎ1_−ℎ2^{𝑡𝑡+𝜃}𝑦̇^𝑇(𝑠)𝑄₅𝑦̇(𝑠)d𝑠d𝜃_0−ℎ2^{𝑡𝑡+𝜃}𝑉𝑦̇^𝑇(𝑠)𝑄₆𝑦̇(𝑠)d𝑠d𝜃,₅𝑉𝑦_𝑡=^𝑛_𝑗=1𝑚_𝑗^∞0𝑧_𝑗(𝛿)^{𝑡𝑡−𝛿}𝑔^2𝑗𝑦_𝑗(𝜉)d𝜉d𝛿,₆𝑦_𝑡=^𝑛_𝑗=1𝑛_𝑗^∞0𝑧_𝑗(𝛿)^{𝑡𝑡−𝛿}𝑦̇^2𝑗(𝜉)d𝜉d𝛿.(17)
Set ‖𝑦_𝑡‖^2𝑤=‖𝑦(𝑡)‖²+∫^0−∞‖𝑦_𝑡(𝜃)‖²𝑑𝜃+∫^0−∞‖𝑦̇_𝑡(𝜃)‖²d𝜃. By some derivation, we have 𝜆_min(𝑃)‖𝑦(𝑡)‖²≤𝑉𝑦_𝑡≤𝑎₁+𝑎₂+𝑎₃‖‖𝑦_𝑡‖‖^2𝑤,(18) where 𝑎₁=𝜆_max(𝑃𝑎)₂+𝜆_max𝑄₁+𝜆_max𝑄₂+𝜆_max𝑄₃+𝜆_max𝑄₄,𝑎₃=𝜆_max𝑅₁+𝜆_max𝑅₂+𝜆_max𝑅₃,=𝜏+₁₂ℎ𝜆_max2𝑅₄+𝜏₂𝜆_max𝑅₅+ℎ₁₂𝜆_max𝑄₅𝜆_max𝑄₆+𝜆_max𝑀𝐿^𝑇𝐿+𝜆_max(𝑁),(19) which implies that the functional 𝑉(𝑦_𝑡) is a legitimate due to [26]. Next we deal with the derivatives of 𝑉_𝑖(𝑦_𝑡)(𝑖=1,…,6).
Set 𝜉𝐴(𝑡₁)==[−𝑦_𝑇𝐴+𝐾000000𝐷𝐵𝐶𝐸𝐹],𝑦_𝑇(𝑡),𝑦^𝑇(𝑡−𝜏(𝑡)),𝑦^𝑇𝑡−𝜏₁,𝑦𝑡−𝜏₂,𝑦^𝑇(𝑡−ℎ(𝑡)),_𝑇𝑔𝑡−ℎ₁,𝑦^𝑇𝑡−ℎ₂,𝑦̇^𝑇(𝑡−ℎ(𝑡)),_𝑇(𝑦(𝑡)),𝑔^𝑇(𝑦(𝑡−𝜏(𝑡))),_𝑡−∞_𝑡−∞𝑍(𝑡−𝑠)𝑔(𝑦(𝑠))d𝑠^𝑇,𝑍(𝑡−𝑠)𝑦̇(𝑠)d𝑠^𝑇^𝑇.(20)
When the controller (7) is applied to the system (6), the resultant closed-loop system is obtained as follows: 𝑦̇(𝑡)=𝐴₁𝜉(𝑡).(21) Due to Lemma 1, (3), the derivatives of 𝑉_𝑖(𝑦_𝑡) (𝑖=1,…,4) along the trajectory of (21) are 𝑉₁̇𝑦_𝑡==2𝜉_𝑇𝑦^𝑇(𝑡)𝑃𝑦̇(𝑡)=2𝑦^𝑇(𝑡)𝑃𝐴₁𝜉(𝑡)𝑉̇(𝑡)𝑒^𝑇1𝑃𝐴₁+𝐴^𝑇1𝑃𝑒₁𝜉(𝑡),(22)₂𝑦_𝑡=𝑦^𝑇(𝑡)³_𝑖=1𝑄_𝑖+³_𝑖=1𝑅_𝑖−1−ℎ̇(𝑡)𝑦_𝑇𝑦(𝑡)−1−ℎ̇(𝑡)𝑦̇_𝑇(𝑡−ℎ(𝑡))𝑄₃𝑦(𝑡−ℎ(𝑡))−𝑦_𝑇(𝑡−ℎ(𝑡))𝑄₄𝑦̇(𝑡−ℎ(𝑡))−𝑦_𝑇𝑡−ℎ₁𝑄₁𝑦𝑡−ℎ₁−(1−𝑡𝜏−̇(𝑡ℎ))₂𝑦_𝑇𝑄₂𝑦𝑡−ℎ₂−𝑦_𝑇(𝑡−𝜏(𝑡))𝑅₃𝑦(𝑡−𝜏(𝑡))−𝑦_𝑇𝑡−𝜏₁𝑅₁𝑦𝑡−𝜏₁≤𝜉_𝑇𝑡−𝜏₂𝑅₂𝑦𝑡−𝜏₂+𝑦̇^𝑇(𝑡)𝑄₄𝑦̇(𝑡)(𝑡)𝑒^𝑇1³_𝑖=1𝑄_𝑖+³_𝑖=1𝑅_𝑖𝑒¹−𝜉_𝑇𝜉(𝑡)+𝜉^𝑇(𝑡)𝐴^𝑇1𝑄₄𝐴₁𝜉(𝑡)−1(−𝑡)ℎ𝑒^𝑇7₃𝑄₂𝑒₇𝜉(𝑡)−1−ℎ₃𝜉^𝑇(𝑡)𝑒^𝑇8𝑄₄𝑒₈𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒^𝑇6𝑄₁𝑒₆𝜉(𝑡)−1−𝜏₃𝜉^𝑇(𝑡)𝑒^𝑇5𝑄₃𝑒₅𝜉(𝑡)−𝜉_𝑇𝜉^𝑇(𝑡)𝑒^𝑇2𝑅₃𝑒₂𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒^𝑇3𝑅₁𝑒₃𝜉(𝑡)𝑉̇(𝑡)𝑒^𝑇4𝑅₂𝑒₄𝜉(𝑡),(23)₃𝑦_𝑡=𝑦−̇^𝑇(𝑡)𝜏₁₂𝑅₄+𝜏₂𝑅₅𝑦̇(𝑡)_{𝑡−𝜏1𝑡−𝜏2}=𝑦̇_𝑇𝑦̇^𝑇(𝑠)𝑅₄𝑦̇(𝑠)d𝑠−^{𝑡𝑡−𝜏}₂𝑦̇^𝑇(𝑠)𝑅₅𝑦̇(𝑠)d𝑠−(𝑡)𝜏₁₂𝑅₄+𝜏₂𝑅₅𝑦̇(𝑡)−^{𝑡−𝜏(𝑡)𝑡−𝜏}₂𝑦̇^𝑇(𝑠)𝑅₄𝑦̇(𝑠)d𝑠_{𝑡−𝜏1𝑡−𝜏(𝑡)}−𝑦̇^𝑇(𝑠)𝑅₄𝑦̇(𝑠)d𝑠−^{𝑡−𝜏(𝑡)𝑡−𝜏}₂𝑦̇^𝑇(𝑠)𝑅₅𝑦̇(𝑠)d𝑠_{𝑡−𝜏1𝑡−𝜏(𝑡)}≤𝜉_𝑇𝑦̇^𝑇(𝑠)𝑅₅𝑦̇(𝑠)d𝑠−^{𝑡𝑡−𝜏}₁𝑦̇^𝑇(𝑠)𝑅₅𝑦̇(𝑠)d𝑠−𝜏(𝑡)₋₁₁₂𝐴^𝑇1𝜏₁₂𝑅₄+𝜏₂𝑅₅𝐴₁𝜉(𝑡)−𝜏₋₁₁₂𝜉^𝑇(𝑡)𝑒₂−𝑒₄^𝑇𝑅₄𝑒₂−𝑒₄𝜉(𝑡)−𝜏₋₁₁₂𝜉^𝑇(𝑡)𝑒₂−𝑒₃^𝑇𝑅₄𝑒₂−𝑒₃𝜉(𝑡)−𝜏₋₁₁₂𝜉^𝑇(𝑡)𝑒₂−𝑒₄^𝑇𝑅₅𝑒₂−𝑒₄𝜉(𝑡)−𝜏₁₋₁𝜉^𝑇(𝑡)𝑒₂−𝑒₃^𝑇𝑅₅𝑒₂−𝑒₃𝜉(𝑡)𝑉̇𝜉^𝑇(𝑡)𝑒₁−𝑒₃^𝑇𝑅₅𝑒₁−𝑒₃𝜉(𝑡),(24)₄𝑦_𝑡=𝑦−̇^𝑇(𝑡)ℎ₁₂𝑄₅+ℎ₂𝑄₆𝑦̇(𝑡)_{𝑡−ℎ1𝑡−ℎ2}=𝑦̇_𝑇𝑦̇^𝑇(𝑠)𝑄₅𝑦̇(𝑠)d𝑠−^{𝑡𝑡−ℎ}₂𝑦̇^𝑇(𝑠)𝑄₆𝑦̇(𝑠)d𝑠−(𝑡)ℎ₁₂𝑄₅+ℎ₂𝑄₆𝑦̇(𝑡)−^{𝑡−ℎ(𝑡)𝑡−ℎ}₂𝑦̇^𝑇(𝑠)𝑄₅𝑦̇(𝑠)d𝑠_{𝑡−ℎ1𝑡−ℎ(𝑡)}−𝑦̇^𝑇(𝑠)𝑄₅𝑦̇(𝑠)d𝑠−^{𝑡−ℎ(𝑡)𝑡−ℎ}₂𝑦̇^𝑇(𝑠)𝑄₆𝑦̇(𝑠)d𝑠_{𝑡−ℎ1𝑡−ℎ(𝑡)}≤𝜉_𝑇𝑦̇^𝑇(𝑠)𝑄₆𝑦̇(𝑠)d𝑠−^{𝑡𝑡−ℎ}₁𝑦̇^𝑇(𝑠)𝑄₆𝑦̇(𝑠)d𝑠−ℎ(𝑡)₋₁₁₂𝐴^𝑇1ℎ₁₂𝑄₅+ℎ₂𝑄₆𝐴₁𝜉(𝑡)−ℎ₋₁₁₂𝜉^𝑇(𝑡)𝑒₅−𝑒₇^𝑇𝑄₅𝑒₅−𝑒₇𝜉(𝑡)−ℎ₋₁₁₂𝜉^𝑇(𝑡)𝑒₅−𝑒₆^𝑇𝑄₅𝑒₅−𝑒₆𝜉(𝑡)−ℎ₋₁₁₂𝜉^𝑇(𝑡)𝑒₅−𝑒₇^𝑇𝑄₆𝑒₅−𝑒₇𝜉(𝑡)−ℎ₁₋₁𝜉^𝑇(𝑡)𝑒₅−𝑒₆^𝑇𝑄₆𝑒₅−𝑒₆𝜉(𝑡)𝜉^𝑇(𝑡)𝑒₁−𝑒₆^𝑇𝑄₆𝑒₁−𝑒₆𝜉(𝑡).(25)
By Lemma 3 and (4), we can obtain the derivatives of 𝑉₅(𝑦_𝑡) and 𝑉₆(𝑦_𝑡) as follows: 𝑉₅̇𝑦_𝑡=^𝑛_𝑗=1𝑚_𝑗^∞0−𝑧_𝑗(𝛿)𝑔^2𝑗𝑦_𝑗(𝑡)d𝛿_𝑛_𝑗=1𝑚_𝑗^∞0=𝑧_𝑗(𝛿)𝑔^2𝑗𝑦_𝑗(𝑡−𝛿)d𝛿_𝑛_𝑗=1𝑚_𝑗𝑔^2𝑗𝑦_𝑗(𝑡)^∞0𝑧_𝑗(𝛿)d𝛿−^𝑛_𝑗=1𝑚_𝑗^∞0×𝑧_𝑗(𝛿)d𝛿_∞0≤𝑔_𝑇𝑧_𝑗(𝛿)𝑔^2𝑗𝑦_𝑗(𝑡−𝛿)d𝛿−(𝑦(𝑡))𝑀𝑔(𝑦(𝑡))_𝑛_𝑗=1𝑚_𝑗^∞0𝑧_𝑗(𝛿)𝑔_𝑗𝑦_𝑗(𝑡−𝛿)d𝛿²=𝑔^𝑇(𝑦(𝑡))𝑀𝑔(𝑦(𝑡))−^0−∞𝑍(𝑡−𝑠)𝑔(𝑦(𝑠))d𝑠^𝑇=𝜉×_𝑇𝑀^0−∞𝑍(𝑡−𝑠)𝑔(𝑦(𝑠))d𝑠𝑉̇(𝑡)𝑒^𝑇9𝑀𝑒₉𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒^𝑇11𝑀𝑒₁₁𝜉(𝑡),(26)₆𝑦_𝑡≤𝑦̇^𝑇(𝑡)𝑁𝑦̇(𝑡)−^𝑛_𝑗=1𝑛_𝑗^∞0𝑧_𝑗(𝛿)𝑦̇_𝑗(𝑡−𝛿)d𝛿²=𝜉^𝑇(𝑡)𝐴^𝑇1𝑁𝐴₁𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒^𝑇12𝑁𝑒₁₂𝜉(𝑡).(27) For the functions 𝑔_𝑗, 𝑗=1,2,…,𝑛, using 𝑔^2𝑗(𝜁_𝑗)≤𝐿^2𝑗𝜁^2𝑗, we get that 00≤≤−−𝜀𝜀₁₂𝜉^𝑇(𝑡)𝑒^𝑇9𝑒₉𝜉(𝑡)+𝜀₁𝜉^𝑇(𝑡)𝑒^𝑇1𝐿^𝑇𝐿𝑒₁𝜉(𝑡),𝜉^𝑇(𝑡)𝑒^𝑇10𝑒₁₀𝜉(𝑡)+𝜀₂𝜉^𝑇(𝑡)𝑒^𝑇2𝐿^𝑇𝐿𝑒₂𝜉(𝑡),(28) where 𝐿=diag(𝐿₁,…,𝐿_𝑛). Noting that 𝐾=𝑃⁻¹𝐾, we have 𝐴=𝑃𝐴₁. The combination of (22)–(28) gives 𝑉̇𝑦_𝑡=⁶_𝑖=1𝑉_𝑖̇𝑦_𝑡≤𝜉^𝑇(𝑡)Π𝜉(𝑡),(29) where Π=Π+𝐴^𝑇1𝑄₄+𝜏₁₂𝑅₄+𝜏₂𝑅₅+ℎ₁₂𝑄₅+ℎ₂𝑄₆+𝑁𝐴₁.(30)
By Lemma 2 and (10)–(13), we can get that ⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑃Π𝐴₁𝐴^𝑇1𝑃𝐴^𝑇1𝑃𝐴^𝑇1𝑃𝐴^𝑇1𝑃𝐴^𝑇1𝑃𝐴^𝑇1𝑃𝑃𝐴Π₂₂00000₁𝑃𝐴0Π₃₃0000₁𝑃𝐴00Π₄₄000₁𝑃𝐴000Π₅₅00₁𝑃𝐴₁0000Π₆₆000000Π₇₇⎤⎥⎥⎥⎥⎥⎥⎥⎦<0,(31) where ΠΠ₂₂=−𝑃𝑄⁴⁻¹𝑃,Π₃₃=−𝜏⁻¹¹²𝑃𝑅⁴⁻¹𝑃,₄₄Π=−𝜏²⁻¹𝑃𝑅⁵⁻¹𝑃,Π₅₅=−ℎ⁻¹¹²𝑃𝑄⁵⁻¹𝑃,₆₆=−ℎ²⁻¹𝑃𝑄⁶⁻¹𝑃,Π₇₇=−𝑃𝑁⁻¹𝑃.(32)
Thus it is easy to see that 𝑉̇(𝑦_𝑡)<0, which, together with (7) and [26, Theorem  3.1.6], implies that the system (6) is asymptotically stabilizable. The proof is completed.

Remark 5. A delay-range-dependent and rate-dependent condition under which the system (6) is asymptotically stabilizable is investigated in Theorem 4. In [7, 8], the stability of a class of the neural networks with unbounded distributed delays has been studied; however, the neutral term was not considered in their models. In [18], only a unbounded distributed delay is taken into account. But in this paper the neutral-type neural networks with two unbounded distributed delays are studied. Therefore, Theorem 4 in this paper can be viewed as an extension of the corresponding results in [7, 8, 18].

Remark 6. In order to apply [26, Theorem 3.1.6] to conclude that the system (6) is asymptotically stabilizable, we denote a new norm ‖⋅‖^2𝑤 on a space of functions.
Noting that the stabilization criterion proposed in Theorem 4 includes the nonlinear constraint (13), we cannot obtain the desired gain 𝐾 by using the LMI Controller Toolbox of MATLAB. In order to deal with the nonlinear constraint (13), we offer the following CCL algorithm. The algorithm is available to determine the maximum value of ℎ₂ for given ℎ₁, ℎ₃, 𝜏₁, 𝜏₂, and 𝜏₃ and the corresponding gain 𝐾.

Algorithm 7 (CCL algorithm). Step 1. Choose a sufficiently small initial ℎ₂>ℎ₁, such that there exists a feasible solution to (10)–(12) and 𝑃𝐼𝑃𝐼≥0,𝑁𝐼𝑁𝐼≥0,𝑅⁴𝐼𝑅𝐼₄𝑅≥0,₅𝐼𝑅𝐼₅≥0,𝑇^𝑖𝐼𝑇𝐼_𝑖𝑄≥0(𝑖=1,…,6),_𝑗𝐼𝑄𝐼_𝑗≥0(𝑗=4,5,6).(33) Set ℎ_max=ℎ₂.

Step 2. Find a feasible set of 𝑃₀, 𝑃₀, 𝑁₀, 𝑁₀, 𝑇_𝑖0, 𝑇_𝑖0(𝑖=1,…,6), 𝑅_𝑙0, 𝑅_𝑙0(𝑙=4,5), 𝑄_𝑗0, and 𝑄_𝑗0(𝑗=4,5,6) satisfying (10)–(12), (33). Set 𝑘=0.

Step 3. Solve the following LMI problem for the variables 𝐾, 𝑃, 𝑃, 𝑁, 𝑁, 𝑇_𝑖, 𝑇_𝑖(𝑖=1,…,6), 𝑅_𝑙, 𝑅_𝑙(𝑙=4,5), 𝑄_𝑗, and 𝑄_𝑗(𝑗=4,5,6): min_{subjectto(8)−(10),(29)}trΘ_𝑘,(34) where Θ_𝑘=𝑃_𝑘𝑃+𝑃𝑃_𝑘+𝑁_𝑘𝑁+𝑁𝑁_𝑘+⁶_𝑖=1𝑇^𝑖𝑇_𝑖𝑘+𝑇_𝑖𝑘𝑇_𝑖+₅_𝑙=4𝑅^𝑙𝑅_𝑙𝑘+𝑅_𝑙𝑘𝑅_𝑙+⁶_𝑗=4𝑄^𝑗𝑄_𝑗𝑘+𝑄_𝑗𝑘𝑄_𝑗.(35)

Set 𝑃_𝑘+1=𝑃, 𝑃_𝑘+1=𝑃, 𝑁_𝑘+1=𝑁, 𝑁_𝑘+1=𝑁, 𝑇_𝑖𝑘+1=𝑇_𝑖, 𝑇_𝑖𝑘+1=𝑇_𝑖(𝑖=1,…,6), 𝑅_𝑙𝑘+1=𝑅_𝑙, 𝑅_𝑙𝑘+1=𝑅_𝑙(𝑙=4,5), 𝑄_𝑗𝑘+1=𝑄_𝑗 and 𝑄_𝑗𝑘+1=𝑄_𝑗(𝑗=4,5,6).

Step 4. If the LMI. Π∶=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Π𝐴^𝑇1𝑄₄𝑄₄𝐴^𝑇1𝑅₄𝐴^𝑇1𝑅₅𝐴^𝑇1𝑄₅𝐴^𝑇1𝑄₆𝐴^𝑇1𝑁𝑅𝐴₁−𝑄₄00000₄𝑅𝐴₁0−𝜏⁻¹¹²𝑅₄0000₅𝑄𝐴₁00−𝜏²⁻¹𝑅₅000₅𝑄𝐴₁000−ℎ⁻¹¹²𝑄₅00₆𝑁𝐴𝐴₁₁0000−ℎ²⁻¹𝑄₆000000−𝑁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0(36) is feasible for the variables 𝑁, 𝑇_𝑖(𝑖=1,…,6), 𝑅_𝑙(𝑙=4,5), and 𝑄_𝑗(𝑗=4,5,6) and the matrices 𝐾 and 𝑃 obtained in Step 3, then set 𝐾=𝑃⁻¹𝐾 and stop. If the LMI (36) is infeasible within a specified number of iterations, then stop; otherwise, set 𝑘=𝑘+1 and go to Step 3.

Remark 8. The idea of the above algorithm is taken from [27]. Compared with the CCL algorithms proposed in [28, 29], the merit of the above algorithm is that more freedoms are provided to test the iteration stop conditions (see [27, Remark  4] for details).

Next, we consider the following neutral-type neural network (37), which is a special case of (6). A stability criterion for the neural network (37) has been presented in [21]. The following corollary can be immediately obtained from Theorem 4, which gives a stabilization criterion for the neutral-type neural network (37): 𝑦̇(𝑡)=−++𝐶𝐸𝐴𝑔𝑦((𝑦𝑡_𝑡−∞)(+𝑡−𝐵ℎ𝑔((𝑦𝑡))()𝑡)+)𝐷𝑦̇(𝑡−ℎ(𝑡))+𝐹_𝑡−∞𝑍(𝑡−𝑠)𝑔(𝑦(𝑠))d𝑠𝑍(𝑡−𝑠)𝑦̇(𝑠)d𝑠+𝐼(𝑡),(37) where 0≤ℎ(𝑡)≤ℎ₂,ℎ̇(𝑡)≤ℎ₃.(38)

Corollary 9. For given scalars ℎ₂ and ℎ₃ such that ℎ₂>0, the neutral-type neural network (37) subject to (38) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices 𝑃,𝑃,𝑄_𝑖(𝑖=1,…,4),𝑄₂,𝑄₄,𝑇_𝑗,𝑇_𝑗(𝑗=1,2,3),𝑀∶=diag(𝑚₁,…,𝑚_𝑛),𝑁∶=diag(𝑛₁,…,𝑛_𝑛), and 𝑁∶=diag(𝑛₁̃,…,𝑛_𝑛̃), a matrix 𝐾 and positive scalars 𝜀₁ and 𝜀₂, such that Σ∶=⎡⎢⎢⎣Σ𝒜^𝑇𝒜^𝑇𝒜^𝑇𝒜𝒜−0𝑇₁−0𝑇₂0𝒜00−0𝑇₃−𝑇⎤⎥⎥⎦<0,₁𝑃−𝑃𝑄₂≤0,−𝑇²𝑃−ℎ𝑃₂₋₁−𝑇𝑄₄≤0,₃𝑃𝑃=𝐼,𝑃𝑇_𝑗−𝑃𝑁≤0,(39)𝑄𝑇_𝑗=𝐼(𝑗=1,2,3),₂𝑄₂=𝐼,𝑄₄𝑄₄=𝐼,𝑁𝑁=𝐼,(40) where Σ∶=𝑒^𝑇1̂𝑄₁+𝑄₃+𝜀₁𝐿^𝑇𝐿𝑒₁̂−𝑒₂̂1−ℎ₃𝑄₁−𝜀₂𝐿^𝑇𝐿𝑒₂̂+𝒜^𝑇𝑒₁̂+𝑒^𝑇1̂𝒜−1−ℎ₃𝑒^𝑇4̂𝑄₂𝑒₄̂−𝑒^𝑇3̂𝑄₃𝑒₃̂+𝑒^𝑇5̂𝑀−𝜀₁𝐼𝑒₅̂−𝑒^𝑇7̂𝑀𝑒₇̂−−ℎℎ²⁻¹₂₋₁𝑒₂̂−𝑒₃̂^𝑇𝑄₄𝑒₂̂−𝑒₃̂−𝑒_𝑇8̂𝑒₁̂−𝑒₂̂^𝑇𝑄₄𝑒₁̂−𝑒₂̂𝒜𝑒_𝑖̂=−𝑃𝐴𝑁+𝑒₈̂𝐾−𝜀0₂𝑒^𝑇6̂0𝑒₆̂,𝑃𝐷𝑃𝐵𝑃𝐶𝑃𝐸𝑃𝐹,=0_{𝑛×(𝑖−1)}𝐼_𝑛0_{𝑛×(8−𝑖)},𝑖=1,…,8.(41)
Furthermore, when the LMIs (39) with the constraint (40) are feasible, a desired state feedback gain is given by 𝐾=𝑃⁻¹𝐾.

Proof. If we use the following 𝜉̂(𝑡) and 𝑉(𝑦_𝑡) instead of 𝜉(𝑡) and 𝑉(𝑦_𝑡) in the proof of Theorem 4, respectively, then the proof can be easily completed: 𝑉𝑦_𝑡=⁵_𝑖=1𝑉_𝑖𝑦_𝑡,(42) with 𝑉𝑉₁𝑦_𝑡=𝑦^𝑇(𝑡)𝑃𝑦(𝑡),₂𝑦_𝑡=+^{𝑡𝑡−ℎ(𝑡)}𝑦^𝑇(𝑠)𝑄₁𝑦(𝑠)+𝑦̇^𝑇(𝑠)𝑄₂𝑦̇(𝑠)d𝑠_{𝑡𝑡−ℎ2}𝑉𝑦^𝑇(𝑠)𝑄₃𝑦(𝑠)d𝑠,₃𝑉𝑦_𝑡=^0−ℎ₂^{𝑡𝑡+𝜃}𝑦̇^𝑇(𝑠)𝑄₄𝑦̇(𝑠)d𝑠d𝜃,₄𝑉𝑦_𝑡=^𝑛_𝑗=1𝑚_𝑗^∞0𝑍_𝑗(𝛿)^{𝑡𝑡−𝛿}𝑔^2𝑗𝑦_𝑗(𝜉)d𝜉d𝛿,₅𝜉̂(𝑡)=𝑦𝑦_𝑡_𝑇=^𝑛_𝑗=1𝑛_𝑗^∞0𝑍_𝑗(𝛿)^{𝑡𝑡−𝛿}𝑦̇^2𝑗(𝜉)d𝜉d𝛿.𝑦̇_𝑇(𝑡),𝑦^𝑇(𝑡−ℎ(𝑡)),𝑦^𝑇𝑡−ℎ₂,(𝑡−ℎ(𝑡)),𝑔^𝑇(𝑦(𝑡)),𝑔^𝑇(𝑦(𝑡−ℎ(𝑡))),_𝑡−∞_𝑡−∞𝑍(𝑡−𝑠)𝑔(𝑦(𝑠))d𝑠^𝑇,𝑍(𝑡−𝑠)𝑦̇(𝑠)d𝑠^𝑇^𝑇.(43)

The following system model is a practical partial element equivalent circuit (PEEC) that is described in [30, Figure 1] by Bellen et al.: 𝑦̇(𝑡)=𝐴𝑦𝑦((𝜃𝑡))=+𝐵𝜙𝑦(𝜃()t,−𝜏∀)𝜃+∈𝐶𝑦[−̇(𝑡𝜏−,0𝜏]),+𝐼(𝑡),((4445)) where 𝜏 is a positive scalar representing the system delay.

It should be emphasized that the matrix 𝐴 in (44) is not necessarily diagonal, which is different from the one in (6). However, one can easily find that the proof of Theorem 4 is always available whether the matrix 𝐴 is diagonal or not. For this reason, we can derive the following Corollaries 10 and 12 by a method similar to obtaining Theorem 4.

Corollary 10. For a given scalar 𝜏>0, the neutral-type system (44) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices 𝑃, 𝑃,  𝑄_𝑖(𝑖=1,…,3), 𝑄₂, 𝑄₃, 𝑇_𝑗, and 𝑇_𝑗(𝑗=1,2) and a matrix 𝐾, such that Ξ∶=⎡⎢⎢⎢⎣Ξ𝐴^𝑇𝐴^𝑇𝐴𝐴−0𝑇₁−0𝑇₂−𝑇⎤⎥⎥⎥⎦<0,(46)₁𝑃−𝑃𝑄₂≤0,−𝑇²𝑃−𝜏𝑃₋₁𝑃𝑃=𝐼,𝑇_𝑗𝑄₃≤0,(47)𝑄𝑇_𝑗=𝐼(𝑗=1,2),₂𝑄₂=𝐼,𝑄₃𝑄₃=𝐼,(48) where Ξ∶=𝑒−^𝑇1𝑄𝑒_1𝑇3𝑒₁−𝑒^𝑇2𝑄₁𝑒₂+𝐴^𝑇𝑒₁+𝑒^𝑇1𝐴𝐴𝑒_𝑖=𝑃𝐴+𝑄₂𝐾𝑒₃𝑃−𝐵𝜏𝑃⁻¹𝐶𝑒₁,−𝑒₂^𝑇𝑄₃𝑒₁−𝑒₂,=0_{𝑛×(𝑖−1)}𝐼_𝑛0_{𝑛×(3−𝑖)},𝑖=1,2,3.(49)
Furthermore, when the LMIs (46) and (47) with the nonlinear constraint (48) are feasible, a desired state feedback gain is given by 𝐾=𝑃⁻¹𝐾.

Remark 11. Similar to Algorithm 7, one can solve the stabilization criteria proposed in Corollaries 9 and 10.

Corollary 12. For a given scalar 𝜏>0, the unforced system of (44) is asymptotically stable if there exist real symmetry positive-definite matrices 𝑃 and 𝑄_𝑖(𝑖=1,2,3) such that Θ∶=𝑒^𝑇1𝑄₁𝑒₁−𝑒^𝑇2𝑄₁𝑒₂+𝐴^𝑇𝑃𝑒₁+𝑒^𝑇1𝑃𝐴−𝑒^𝑇3𝑄₂𝑒₃−𝜏⁻¹𝑒₁−𝑒₂^𝑇𝑄₃𝑒₁−𝑒₂+𝐴^𝑇𝑄₂+𝜏𝑄₃𝐴<0,(50) where 𝐴=𝐴𝐵𝐶,(51) and 𝑒_𝑖 is defined as in Corollary 10.

4. Numerical Examples

In this section, we will illustrate our approach by several numerical examples.

Example 13. Consider the neutral-type neural network (6), where 𝐶𝐸==𝐴=−000..046..119201𝐿5−030−.030,.015..218,𝐵,=𝐹=−𝐷−00=.003.1..435000−.−01.100.01.02_𝑗.,51,,𝜏=1,𝑗=1,2,₃ℎ=0.5,𝜏₂=4.8,𝜏₁=4.1,₃=0.5,ℎ₁=4.5,ℎ₂=7.5,𝑔(𝑦(𝑠))=tanh𝑦(𝑠)2.(52) Solving the LMIs (10)–(12) with nonlinear constraint (13) by Algorithm 7 and the LMI Control Toolbox of MATLAB, we can get a desired state feedback gain stabilizing the system as follows: 𝐾=−00.7.1706730−10.7.1204720.(53)
The state response curves of the closed-loop system are given in Figures 1 and 2, where the initial functions in Figures 1 and 2 are, respectively, chosen as 𝜙(𝜃)=^−0.1−20, 𝜃∈(−∞,0] and 𝜙(𝜃)=^e𝜃_e2𝜃, 𝜃∈(−∞,0].

(a)

(b)

Figure 1 

State responses of the closed-loop system for 𝑧1(𝑠)=𝑧2(𝑠)=1−|𝑠−1|, 0≤𝑠≤2.

(a)

(b)

Figure 2 

State responses of the closed-loop system for 𝑧1(𝑠)=𝑧2(𝑠)=(√2/𝜋)𝑒−(1/2)𝑠2, 𝑠∈(−∞,0].

Example 14. Consider the system (37) with the following parameter matrices: 𝐿𝐴𝐶𝐸===030000..0...614353−00.0060..1.2.126,,,𝐵𝐹𝐷===10.100.019..0.3582−10−0.00.102..911.2_𝑗85,,,𝜃∈=(1−,∞𝑗,0=],1𝑔,2(𝑦,(𝑠))𝜙=(𝜃ta)n=h−−𝑦02(.𝑠01),𝑧(𝑠)=1−||𝑠−1||(0≤𝑠≤2),2ℎ₂,=8.(54) When ℎ₃=0.15 by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.1] is not feasible, so it cannot guarantee that the unforced system is stable. Now we check the stabilization condition proposed in Corollary 9 of this paper and find that it is feasible. So, by Corollary 9 the system is asymptotically stabilizable by state feedback controller, and a desired state feedback gain is given by 𝐾=−10.6.3615658−10.8.3016698,(55) and, further, the state response curves of the closed-loop system are given in Figure 3.
When ℎ₃=0, by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.2] is not feasible, which cannot guarantee the unforced system is stable. However, the stabilization condition proposed in Corollary 9 can be satisfied. Hence, by Corollary 9, the neutral-type neural network is asymptotically stabilizable via state feedback controller, and a desired state feedback controller gain is given by 𝐾=−10.6.3814170−10.8.3119180,(56) and, further, the state response curves of the closed-loop system are given in Figure 4.

(a)

(b)

Figure 3 

State responses of the closed-loop system when ℎ3=0.15.

(a)

(b)

Figure 4 

State responses of the closed-loop system when ℎ3=0.

Example 15. Consider the PEEC system (44) with the following parameter matrices: 𝐴=100×𝛽31−219𝐶−02=61,𝐵=100×−−001..55−−010..55−−031,72−−412540213.(57)
Applying Corollaries 12 and 10, the maximum delay bounds 𝜏_𝑀 for the various 𝛽 are presented in Table 1, which clearly shows the effectiveness of our approach. Through Table 1, we can get that the stability criterion proposed in this paper is less conservative than the ones in [31–33].
The following example is provided for applying the approach proposed in this paper to a realistic problem which is motivated by the small PEEC model in [30, Figure 1].

Table 1 

Allowable upper bound of 𝜏_𝑀 for different 𝛽.

Example 16. Consider the PEEC system (44) with 𝐴=100×−317𝐶−=1291−206,𝐵=100×−−001..55−−010..55−−031,72−−412540213,𝜏=1.(58) When 𝐼(𝑡)≡0, Bellen et al. [30] gave sufficient conditions for the asymptotic stability of the zero solution to (44) by utilizing a suitable reformulation of the system.
By applying Algorithm 7 and the LMI Control Toolbox of MATLAB to solve LMIs (46) and (47) with the nonlinear constraint (48), a required state feedback gain is obtained as follows: 𝐾=−−647684.71..66726112835−−421584.366..09418564080−−130.17.7.7111754175.(59) The simulation results for the PEEC system are shown by Figures 5 and 6 with the initial value 𝜙(𝜃)=[sin(𝑡)sin(2𝑡)sin(3𝑡)]^𝑇. Figure 5 represents the state responses of the PEEC system when 𝐼(𝑡)≡0, and Figure 6 represents the state responses of the resultant closed-loop system. From Figures 5 and 6, we can see that both the open-loop and closed-loop systems are stable, however, the speed of convergence towards the null point in the closed-loop system is faster than one in the open-loop system.

Figure 5 

State responses of the open-loop system.

Figure 6 

State responses of the closed-loop system.

5. Conclusions

In this paper, the delay-range-dependent and rate-dependent stabilization criteria for a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays are established. The criteria are derived by constructing an appropriate Lyapunov-Krasovskii functional and using certain matrix technique. A CCL algorithm is developed to obtain the state feedback gain 𝐾. Numerical examples are provided to show that our results are more suitable than some existing ones, which illustrate the merits of the proposed approach.

Acknowledgments

This work is supported by the fund of Heilongjiang Education Committee under Grant no. 12521429 and the fund of Heilongjiang University Innovation Team Support Plan under Grant no. Hdtd2010-03. The authors thank the anonymous referees for their helpful comments and suggestions that improved greatly this paper.