Finite-Time Stability of Fractional-Order BAM Neural Networks with Distributed Delay

Abstract

Based on the theory of fractional calculus, the generalized Gronwall inequality and estimates of mittag-Leffer functions, the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay is investigated in this paper. An illustrative example is also given to demonstrate the effectiveness of the obtained result.

1. Introduction

Fractional calculus (integral and differential operations of noninteger order) was firstly introduced 300 years ago. Due to lack of application background and the complexity, it did not attract much attention for a long time. In recent decades fractional calculus is applied to physics, applied mathematics, and engineering [1–6]. Since the fractional-order derivative is nonlocal and has weakly singular kernels, it provides an excellent instrument for the description of memory and hereditary properties of dynamical processes. Nowadays, study on the complex dynamical behaviors of fractional-order systems has become a very hot research topic.

We know that the next state of a system not only depends upon its current state but also upon its history information. Since a model described by fractional-order equations possesses memory, it is precise to describe the states of neurons. Moreover, the superiority of the Caputo’s fractional derivative is that the initial conditions for fractional differential equations with Caputo derivatives take on the similar form as those for integer-order differentiation. Therefore, it is necessary and interesting to study fractional-order neural networks both in theory and in applications.

Recently, fractional-order neural networks have been presented and designed to distinguish the classical integer-order models [7–10]. Currently, some excellent results about fractional-order neural networks have been investigated, such as Kaslik and Sivasundaram [11, 12], Zhang et al. [13], Delavari et al. [14], and Li et al. [15, 16]. On the other hand, time delay is one of the inevitable problems on the stability of dynamical systems in the real word [17–20]. But till now, there are few results on the problems for fractional-order delayed neural networks; Chen et al. [21] studied the uniform stability for a class of fractional-order neural networks with constant delay by the analytical approach; Wu et al. [22] investigated the finite-time stability of fractional-order neural networks with delay by the generalized Gronwall inequality and estimates of Mittag-Leffler functions; Alofi et al. [23] discussed the finite-time stability of Caputo fractional-order neural networks with distributed delay.

The integer-order bidirectional associative memory (BAM) model known as an extension of the unidirectional autoassociator of Hopfield [24] was first introduced by Kosko [25]. This neural network has been widely studied due to its promising potential for applications in pattern recognition and automatic control. In recent years, integer-order BAM neural networks have been extensively studied [26–29]. However, to the best of our knowledge, there is no effort being made in the literature to study the finite-time stability of fractional-order BAM neural networks so far.

Motivated by the above-mentioned works, we were devoted to establishing the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay. In this paper, we will apply Laplace transform, generalized Gronwall inequality, and estimates of Mittag-Leffler functions to establish the finite-time stability criterion of fractional-order distributed delayed BAM neural networks.

This paper is organized as follows. In Section 2, some definitions and lemmas of fractional differential and integral calculus are given and the fractional-order BAM neural networks are presented. A criterion for finite-time stability of fractional-order BAM neural networks with distributed delay is obtained in Section 3. Finally, the effectiveness and feasibility of the theoretical result is shown by an example in Section 4.

2. Preliminaries

For the convenience of the reader, we first briefly recall some definitions of fractional calculus; for more details, see [1, 2, 5], for example.

Definition 1. The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑢∶(0,∞)→𝑅 is given by 𝐼^𝛼₀⁺𝑢(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑢(𝑠)𝑑𝑠 (1) provided that the right side is pointwise defined on (0,∞), where Γ(⋅) is the Gamma function.

Definition 2. The Caputo fractional derivative of order 𝛾>0 of a function 𝑢∶(0,∞)→𝑅 can be written as ^𝐶₀𝐷^𝛾_𝑡𝑢(𝑡)=1Γ𝑛−𝛾^𝑡₀𝑢^(𝑛)(𝑠)(𝑡−𝑠)^{𝛾+1−𝑛}𝑑𝑠,𝑛−1<𝛾<𝑛. (2)

Definition 3. The Mittag-Leffler function in two parameters is defined as 𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ,𝑘𝛼+𝛽 (3) where 𝛼>0, 𝛽>0, and 𝑧∈ℂ, where ℂ denotes the complex plane. In particular, for 𝛽=1, one has 𝐸_𝛼,1(𝑧)=𝐸_𝛼(𝑧)=^∞_𝑘=0𝑧^𝑘Γ().𝑘𝛼+1 (4)
The Laplace transform of Mittag-Leffler function is ℒ𝑡^𝛽−1𝐸_𝛼,𝛽−𝜆𝑡^𝛼=𝑠^𝛼−𝛽𝑠^𝛼,ℛ(𝑠)>|𝜆|+𝜆_1/𝛼, (5) where 𝑡 and 𝑠 are, respectively, the variables in the time domain and Laplace domain and ℒ{⋅} stands for the Laplace transform.
In this paper, we are interested in the finite-time stability of fractional-order BAM neural networks with distributed delay by the following state equations: ^𝐶₀𝐷^𝛼_𝑡𝑥_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡+_{𝑛𝑗=1𝜏0})𝑟_𝑖𝑗(𝑠)𝑓_𝑗𝑦_𝑗(𝑡−𝑠)𝑑𝑠+𝐼_𝑖,𝑡≥0,_𝐶0𝐷^𝛽_𝑡𝑦_𝑗(𝑡)=−𝑐_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑗𝑖𝑔_i𝑥_𝑖(𝑡)+^𝑛_𝑖=1^𝜏₀𝑝_𝑗𝑖(𝑠)𝑔_𝑖𝑥_𝑖(𝑡−𝑠)𝑑𝑠+𝐼_𝑗,𝑖,𝑗=1,…,𝑛, (6) or in the matrix-vector notation ^𝐶₀𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)𝑦(𝑡)+=−𝐶𝑥+𝐵𝑓_𝜏0𝑅(𝑠)𝑓𝑦(𝑡−𝑠)𝑑𝑠+𝐼,_𝐶0𝐷^𝛽_𝑡𝑦(𝑡)(𝑡)(𝑥(𝑡+=−𝐶𝑦+𝐷𝑔))_𝜏0𝑃(𝑠)𝑔(𝑥(𝑡−𝑠))𝑑𝑠+𝐼,𝑡≥0, (7) where 1<𝛼, 𝛽<2. The model (6) is made up of two neural fields 𝐹_𝑥 and 𝐹_𝑦, where 𝑥_𝑖(𝑡) and 𝑦_𝑗(𝑡) are the activations of the 𝑖th neuron in 𝐹_𝑥 and the 𝑗th neuron in 𝐹_𝑦, respectively; 𝑥(𝑡),𝑦(𝑡)=𝑥₁(𝑡),…,𝑥_𝑛(𝑡),𝑦₁(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈ℝ^2𝑛 (8) is the state vector of the network at time 𝑡; the functions 𝑓𝑦(𝑡)=𝑓₁𝑦₁(𝑡),𝑓₂𝑦₂(𝑡),…,𝑓_𝑛𝑦_𝑛(𝑡)^𝑇,𝑔(𝑥(𝑡))=𝑔₁𝑥₁(𝑡),𝑔₂𝑥₂(𝑡),…,𝑔_𝑛𝑥_𝑛(𝑡)^𝑇 (9) are the activation functions of the neurons at time 𝑡; 𝐶=diag(𝑐_𝑖) is a diagonal matrix; 𝑐_𝑖>0 represents the rate with which the 𝑖th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛 and 𝐷=(𝑑_𝑗𝑖)_𝑛×𝑛 are the feedback matrix; 𝜏>0 denotes the maximum possible transmission delay from neuron to another; 𝑅=(𝑟_𝑖𝑗)_𝑛×𝑛 and 𝑃=(𝑝_𝑗𝑖)_𝑛×𝑛 are the delayed feedback matrix; 𝐼=(𝐼₁,…,𝐼_𝑛)^𝑇 and 𝐼=(𝐼₁,…,𝐼_𝑛)^𝑇 are two external bias vectors.
Let ℂ¹([−𝜏,0],ℝ^𝑛) be the Banach space of all continuously differential functions over a time interval of length 𝜏, mapping the interval [−𝜏,0] into ℝ^𝑛 with the norm defined as follows: for every 𝜑∈ℂ¹([−𝜏,0],ℝ^𝑛), 𝜑₁=max𝜑,𝜑^′=maxsup_{[]𝜃∈−𝜏,0}𝜑(𝜃),sup_{𝜃∈[−𝜏,0]}𝜑^′(𝜃). (10)
The initial conditions associated with (6) are given by 𝑥_𝑖(𝜃)=𝜑_𝑖(𝜃),𝑥^′_𝑖(𝜃)=𝜑^′_𝑖(𝜃),𝑦_𝑗(𝜃)=𝜓_𝑗(𝜃),𝑦^′_𝑗(𝜃)=𝜓^′_𝑗(𝜃),𝜃∈[−𝜏,0], (11) where 𝜑_𝑖,𝜓_𝑗∈𝐶¹([−𝜏,0],ℝ).
In order to obtain main result, we make the following assumptions.(H1)For 𝑖,𝑗=1,…,𝑛, the functions 𝑟_𝑖𝑗(⋅) and 𝑝_𝑗𝑖(⋅) are continuous on [0,𝜏].(H2)The neurons activation functions 𝑓_𝑖 and 𝑔_𝑗(𝑖,𝑗=1,…,𝑛) are bounded.(H3)The neurons activation functions 𝑓_𝑖 and 𝑔_𝑗 are Lipschitz continuous; that is, there exist positive constants ℎ_𝑖,𝑙_𝑗(𝑖,𝑗=1,…,𝑛) such that 𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)≤ℎ_𝑖|𝑢−𝑣|,𝑔_𝑗(𝑢)−𝑔_𝑗(𝑣)≤𝑙_𝑗|𝑢−𝑣|,∀𝑢,𝑣∈ℝ. (12)
Since the Caputo’s fractional derivative of a constant is equal to zero, the equilibrium point of system (6) is a constant vector (𝑥^∗,𝑦^∗)=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛,𝑦^∗₁,𝑦^∗₂,…,𝑦^∗_𝑛)^𝑇∈ℝ^2𝑛 which satisfies the system 𝑐_𝑖𝑥^∗_𝑖−^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦^∗_𝑗−^𝑛_𝑗=1^𝜏₀𝑟_𝑖𝑗(𝑠)𝑓_𝑗𝑦^∗_𝑗𝑑𝑠−𝐼_𝑖𝑐=0,𝑖=1,…,𝑛,_𝑗𝑦^∗_𝑗−^𝑛_𝑖=1𝑑_𝑗𝑖𝑔_𝑖𝑥^∗_𝑖−^𝑛_𝑖=1^𝜏₀𝑝_𝑗𝑖(𝑠)𝑔_𝑖𝑥^∗_𝑖𝑑𝑠−𝐼_𝑗=0,𝑗=1,…,𝑛. (13) By using the Schauder fixed point theorem and assumptions (H1)–(H3), it is easy to prove that the equilibrium points of system (6) exist. We can shift the equilibrium point of system (6) to the origin. Denoting (𝑢(𝑡),𝑣(𝑡))=𝑢₁(𝑡),…,𝑢_𝑛(𝑡),𝑣₁(𝑡),…,𝑣_𝑛(𝑡)^𝑇=𝑥₁(𝑡)−𝑥^∗₁,…,𝑥_𝑛(𝑡)−𝑥^∗_𝑛,𝑦₁(𝑡)−𝑦^∗₁,…,𝑦_𝑛(𝑡)−𝑦^∗_𝑛^𝑇, (14) then system (6) can be written as ^𝐶₀𝐷^𝛼_𝑡𝑢_𝑖(𝑡)=−𝑐_𝑖𝑢_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝐹_𝑗𝑣_𝑗(𝑡+_{𝑛𝑗=1𝜏0})𝑟_𝑖𝑗(𝑠)𝐹_𝑗𝑣_𝑗(𝑡−𝑠)𝑑𝑠,𝑡≥0,_𝐶0𝐷^𝛽_𝑡𝑣_𝑗(𝑡)=−𝑐_𝑗𝑣_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑗𝑖𝐺_𝑖𝑢_𝑖(𝑡)+^𝑛_𝑖=1^𝜏₀𝑝_𝑗𝑖(𝑠)𝐺_𝑖𝑢_𝑖(𝑡−𝑠)𝑑𝑠,𝑖,𝑗=1,…,𝑛, (15) with the initial conditions 𝑢_𝑖(𝜃)=𝜑_𝑖(𝜃),𝑢^′_𝑖(𝜃)=𝜑^′_𝑖(𝜃),𝑣_𝑗(𝜃)=𝜓_𝑗(𝜃),𝑣^′_𝑗(𝜃)=𝜓^′_𝑗(𝜃),𝜃∈[−𝜏,0], (16) where 𝐹_𝑗𝑣_𝑗(𝑡)=𝑓_𝑗𝑣_𝑗(𝑡)+𝑦^∗_𝑗−𝑓_𝑗𝑦^∗_𝑗,𝐺_𝑖𝑢_𝑖(𝑡)=𝑔_𝑖𝑢_𝑖(𝑡)+𝑥^∗_𝑖−𝑔_𝑖𝑥^∗_𝑖,𝜑_𝑖(𝜃)=𝜑_𝑖(𝜃)−𝑥^∗_𝑖,𝜓_𝑗(𝜃)=𝜓_𝑗(𝜃)−𝑦^∗_𝑗,𝜃∈[𝜏,0]. (17) Similarly, by using the matrix-vector notation, system (15) can be expressed as ^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡)(𝑡)(𝑣(𝑡+=−𝐶𝑢+𝐵𝐹))_𝜏0𝑅(𝑠)𝐹(𝑣(𝑡−𝑠))𝑑𝑠,𝑡≥0,_𝐶0𝐷^𝛽_𝑡𝑣(𝑡)(𝑡)(𝑢(𝑡+=−𝐶𝑣+𝐷𝐺))_𝜏0𝑃(𝑠)𝐺(𝑢(𝑡−𝑠))𝑑𝑠,𝑡≥0, (18) with the initial condition 𝑢(𝜃)=𝜑(𝜃),𝑢^′(𝜃)=𝜑^′(𝜃),𝑣(𝜃)=𝜓(𝜃),𝑣^′(𝜃)=𝜓^′(𝜃),𝜃∈[−𝜏,0], (19) where 𝐹(𝑣(𝑡))=𝐹₁𝑣₁(𝑡),𝐹₂𝑣₂(𝑡),…,𝐹_𝑛𝑣_𝑛(𝑡)^𝑇,𝐺(𝑢(𝑡))=𝐺₁𝑢₁(𝑡),𝐺₂𝑢₂(𝑡),…,𝐺_𝑛𝑢_𝑛(𝑡)^𝑇. (20)
Define the functions as follows: ℎ_𝑖(𝑡)=𝐹_𝑖𝑣_𝑖(𝑡)𝑣_𝑖(𝑡),𝑣_𝑖(𝑡)≠0,0,𝑣_𝑖(𝑡)𝑙=0,_𝑗(𝑡)=𝐺_𝑗𝑢_𝑗(𝑡)𝑢_𝑗(𝑡),𝑢_𝑗(𝑡)≠0,0,𝑢_𝑗(𝑡)=0, (21) where 𝑖,𝑗=1,…,𝑛. From assumption (H3), we can obtain |ℎ_𝑖(𝑡)|≤ℎ_𝑖, |𝑙_𝑗(𝑡)|≤𝑙_𝑗. By (21), we have 𝐹_𝑖𝑣_𝑖(𝑡)=ℎ_𝑖(𝑡)𝑣_𝑖(𝑡),𝐺_𝑗𝑢_𝑗(𝑡)=𝑙_𝑗(𝑡)𝑢_𝑗(𝑡),𝑖,𝑗=1,…,𝑛. (22) Thus, system (18) can be further written as the following form: ^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡)(𝑡)(𝑡)𝑣(𝑡)+=−𝐶𝑢+𝐵𝐻_𝜏0𝑅(𝑠)𝐻(𝑡−𝑠)𝑣(𝑡−𝑠)𝑑𝑠,𝑡≥0,_𝐶0𝐷^𝛽_𝑡𝑣(𝑡)(𝑡)(𝑡)𝑢(𝑡)+=−𝐶𝑣+𝐷𝐿_𝜏0𝑃(𝑠)𝐿(𝑡−𝑠)𝑢(𝑡−𝑠)𝑑𝑠,𝑡≥0, (23) where 𝐻(𝑡)=diag{ℎ_𝑖(𝑡)}, 𝐿(𝑡)=diag{𝑙_𝑗(𝑡)}.

Definition 4. System (23) with the initial condition (19) is finite-time stable with respect to {𝛿,𝜀,𝑡₀,𝐽}, 𝛿<𝜀, if and only if 𝜑,𝜓₁∶=𝜑₁+𝜓₁<𝛿 (24) implies ‖(𝑢(𝑡),𝑣(𝑡))‖=‖𝑢(𝑡)‖+‖𝑣(𝑡)‖<𝜀,∀𝑡∈𝐽, (25) where 𝛿 is a positive real number and 𝜀>0, 𝛿<𝜀, 𝑡₀ denotes the initial time of observation of the system, and 𝐽 denotes time interval 𝐽=[𝑡₀,𝑡₀+𝑇).
A technical result about norm upper-bounding function of the matrix function 𝐸_𝛼,𝛽 is given in [30] as follows.

Lemma 5. If 𝛼≥1, then, for 𝛽=1,2,𝛼, one has 𝐸_𝛼,𝛽𝐴𝑡^𝛼≤𝑒^𝐴𝑡𝛼,𝑡≥0. (26) Moreover, if 𝐴 is a diagonal stability matrix, then 𝐸_𝛼,𝛽𝐴𝑡^𝛼≤𝑒^−𝜔𝑡,𝑡≥0, (27) where −𝜔 (𝜔>0) is the largest eigenvalue of the diagonal stability matrix 𝐴.

Lemma 6 (see [31]). Let 𝑢(𝑡),𝑎(𝑡) be nonnegative and local integrable on [0,𝑇) (𝑇≤+∞), and let 𝑔 be a nonnegative, nondecreasing continuous function defined on [0,𝑇), 𝑔(𝑡)≤𝑀, and let 𝑀 be a real constant, 𝛼>0, with 𝑢(𝑡)≤𝑎(𝑡)+𝑔(𝑡)^t₀(𝑡−𝑠)^𝛼−1𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝑇). (28) Then 𝑢(𝑡)≤𝑎(𝑡)+^𝑡₀^∞_𝑛=1𝑔(𝑡)Γ(𝛼)^𝑛Γ()()𝑛𝛼𝑡−𝑠^𝑛𝛼−1𝑎(𝑠[).)𝑑𝑠,𝑡∈0,𝑇 (29) Moreover, if 𝑎(𝑡) is a nondecreasing function on [0,𝑇), then 𝑢(𝑡)≤𝑎(𝑡)𝐸_𝛼,1𝑔(𝑡)Γ(𝛼)𝑡^𝛼,𝑡∈[0,𝑇). (30)

3. Main Result

We first give a key lemma in the proof of our main result as follows.

Lemma 7. Let 𝑢(𝑡),𝑣(𝑡) be nonnegative and local integrable on [0,𝑇) (𝑇≤+∞), and let 𝑎₁(𝑡),𝑎₂(𝑡) be nonnegative, nondecreasing and local integrable on [0,𝑇), and let 𝑏₁, 𝑏₂ be two positive constants, 𝛼,𝛽>1, with 𝑢(𝑡)≤𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑣(𝑠)𝑑𝑠,𝑡∈[0,𝑇),𝑣(𝑡)≤𝑎₂(𝑡)+𝑏₂^𝑡₀(𝑡−𝑠)^𝛽−1𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝑇). (31)(32) Then 𝑢(𝑡)≤𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)𝑑𝑠×𝐸_𝛼+𝛽𝑏₁𝑏₂Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽[),𝑣(𝑡)≤𝑎,𝑡∈0,𝑇₂(𝑡)+𝑏₂^𝑡₀(𝑡−𝑠)^𝛽−1𝑎₁(𝑠)𝑑𝑠×𝐸_𝛼+𝛽𝑏₁𝑏₂Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽,𝑡∈[0,𝑇). (33)

Proof. Substituting (32) into (31), we obtain 𝑢(𝑡)≤𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑣(𝑠)𝑑𝑠≤𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)𝑑𝑠+𝑏₁𝑏₂^𝑡₀(𝑡−𝑠)^𝛼−1𝑠₀(𝑠−𝜉)^𝛽−1𝑢(𝜉)𝑑𝜉. (34) Changing the order of integration in the above double integral, we obtain 𝑢(𝑡)≤𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)𝑑𝑠+𝑏₁𝑏₂^𝑡₀𝑢(𝜉)𝑑𝜉^𝑡_𝜉(𝑡−𝑠)^𝛼−1(𝑠−𝜉)^𝛽−1𝑑𝑠=𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)𝑑𝑠+𝑏₁𝑏₂^𝑡₀Γ(𝛼)Γ𝛽Γ()𝛼+𝛽𝑡−𝜉^{𝛼+𝛽−1}𝑢(𝜉)𝑑𝜉. (35) Let 𝑎(𝑡)=𝑎₁(𝑡)+𝑏₁∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)𝑑𝑠, 𝑔(𝑡)=𝑏₁𝑏₂((Γ(𝛼)Γ(𝛽))/Γ(𝛼+𝛽)); then 𝑎(𝑡) is a nonnegative, nondecreasing, and local integrable function and 𝑔(𝑡) is a nonnegative, nondecreasing continuous function. Thus, by Lemma 6 (30), one has 𝑢(𝑡)≤𝑎₁(𝑡)+𝑏₁^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)𝑑𝑠×𝐸_𝛼+𝛽𝑏₁𝑏₂Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽. (36) Similarly, we get 𝑣(𝑡)≤𝑎₂(𝑡)+𝑏₂^𝑡₀(𝑡−𝑠)^𝛽−1𝑎₁(𝑠)𝑑𝑠×𝐸_𝛼+𝛽𝑏₁𝑏₂Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽. (37)
For convenience, let 𝑅=sup_{0≤𝑠≤𝜏}{‖𝑅(𝑠)‖},𝑃=sup_{0≤𝑠≤𝜏}{‖𝑃(𝑠)‖},ℎ=max_{1≤𝑖≤𝑛}ℎ_𝑖,𝑙=max_{1≤𝑗≤𝑛}𝑙_𝑗,Θ(𝑡)ℎ∶=max𝛼𝑡^𝛼𝑡1+𝜇(𝐵)𝛼+1+𝑅𝜏𝑒^𝛾𝜏,𝑙𝛽𝑡^𝛽𝑡1+𝜇(𝐷)𝛽+1+𝑃𝜏𝑒^𝛾𝜏, (38) where −𝛾 is the largest eigenvalue of the diagonal stability matrix −𝐶 and 𝜇(⋅) denotes the largest singular value of matrix (⋅).

In the following, sufficient conditions for finite-time stability of fractional-order BAM neural networks with distributed delay are derived.

Theorem 8. Let 1<𝛼,𝛽<2. If system (23) satisfies (H1)–(H3) with the initial condition (19), and 𝑒^−𝛾𝑡((1+𝑡)+Θ(𝑡))𝐸_𝛼+𝛽×ℎ𝑙𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽<𝜀𝛿, (39) where 𝑡∈𝐽=[0,𝑇), then system (23) is finite-time stable with respect to {𝛿,𝜀,0,𝐽}, 𝛿<𝜀.

Proof. By Laplace transform and inverse Laplace transform, system (23) is equivalent to 𝑢(𝑡)=𝐸_𝛼−𝐶𝑡^𝛼𝜑(0)+𝑡𝐸_𝛼,2−𝐶𝑡^𝛼𝜑^′(0)+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼−𝐶𝑡^𝛼×(𝑠)𝑣(𝑠)+𝐵𝐻_𝜏0𝑅𝜂𝐻𝑠−𝜂𝑣𝑠−𝜂𝑑𝜂𝑑𝑠,𝑣(𝑡)=𝐸_𝛽−𝐶𝑡^𝛽𝜓(0)+𝑡𝐸_𝛽,2−𝐶𝑡^𝛽𝜓^′(0)+^𝑡₀(𝑡−𝑠)^𝛽−1𝐸_𝛽,𝛽−𝐶𝑡^𝛽×(𝑠)𝑢(𝑠)+𝐷𝐿_𝜏0𝑃𝜂𝐿𝑠−𝜂𝑢𝑠−𝜂𝑑𝜂𝑑𝑠. (40)(41) From (40), (41), and Lemma 5, we obtain ‖𝑢(𝑡)‖≤𝜑+𝜑^′𝑡𝑒^−𝛾𝑡+^𝑡₀(𝑡−𝑠)^𝛼−1𝑒^{−𝛾(𝑡−𝑠)}×(𝑠)𝑣(𝑠)+𝐵𝐻_𝜏0𝑅𝜂𝐻𝑠−𝜂𝑣𝑠−𝜂𝑑𝜂𝑑𝑠,‖𝑣(𝑡)‖≤𝜓+𝜓^′𝑡𝑒^−𝛾𝑡+^𝑡₀(𝑡−𝑠)^𝛽−1𝑒^{−𝛾(𝑡−𝑠)}×𝐷𝐿(𝑠)𝑢(𝑠)+^𝜏₀𝑃𝜂𝐿𝑠−𝜂𝑢𝑠−𝜂𝑑𝜂𝑑𝑠. (42)(43) Let 𝑈(𝑡)=sup_{𝜃∈[𝑡−𝜏,𝑡]}‖𝑢(𝜃)‖𝑒^𝛾𝜃, and 𝑉(𝑡)=sup_{𝜃∈[𝑡−𝜏,𝑡]}‖𝑣(𝜃)‖𝑒^𝛾𝜃; then ‖𝑢(𝑠)‖𝑒^𝛾𝑠≤𝑈(𝑠),‖𝑢(𝑠−𝜏)‖𝑒^{𝛾(𝑠−𝜏)}≤𝑈(𝑠),‖𝑣(𝑠)‖𝑒^𝛾𝑠≤𝑉(𝑠),‖𝑣(𝑠−𝜏)‖𝑒^{𝛾(𝑠−𝜏)}≤𝑉(𝑠). (44)(45) Thus, we have by (42) and (44) that ‖𝑢(𝑡)‖𝑒^𝛾𝑡≤𝜑+𝜑^′𝑡+^𝑡₀(𝑡−𝑠)^𝛼−1×ℎ𝜇(𝐵)‖𝑣(𝑠)‖𝑒^𝛾𝑠+^𝜏₀ℎ𝑅𝑣𝑠−𝜂𝑒^{𝛾(𝑠−𝜂)}𝑒^𝛾𝜂≤𝑑𝜂𝑑𝑠𝜑+𝜑^′𝑡+^𝑡₀(𝑡−𝑠)^𝛼−1ℎ𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏𝑉(𝑠)𝑑𝑠, (46) where 𝜇(𝐵) denotes the largest singular value of matrix 𝐵. Similarly, by (43) and (45), we get ‖𝑣(𝑡)‖𝑒^𝛾𝑡≤𝜓+𝜓^′𝑡+^𝑡₀(𝑡−𝑠)^𝛽−1×𝑙𝜇(𝐷)‖𝑥(𝑠)‖𝑒^𝛾𝑠+^𝜏₀𝑙𝑃𝑥𝑠−𝜂𝑒^{𝛾(𝑠−𝜂)}𝑒^𝛾𝜂≤𝑑𝜂𝑑𝑠𝜓+𝜓^′𝑡+^𝑡₀(𝑡−𝑠)^𝛽−1𝑙𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏𝑈(𝑠)𝑑𝑠. (47) Hence, by (46) and (47), we have 𝑈(𝑡)≤𝜑₁()+1+𝑡_𝑡0(𝑡−𝑠)^𝛼−1ℎ𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏𝑉(𝑠)𝑉(𝑡)≤𝑑𝑠,𝜓₁()+1+𝑡_𝑡0(𝑡−𝑠)^𝛽−1𝑙𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏𝑈(𝑠)𝑑𝑠. (48) Set 𝑎₁(𝑡)=𝜑₁(1+𝑡),𝑎₂(𝑡)=𝜓₁(),𝑏1+𝑡₁=ℎ𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏,𝑏₂=𝑙𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏. (49) By simple computation, we have ^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)≤𝑑𝑠𝜓₁^𝑡₀(𝑡−𝑠)^𝛼−1()=1+𝑠𝑑𝑠𝜓₁𝛼𝑡^𝛼𝑡1+,𝛼+1_𝑡0(𝑡−𝑠)^𝛽−1𝑎₁(𝑠)≤𝑑𝑠𝜑₁^𝑡₀(𝑡−𝑠)^𝛽−1()=1+𝑠𝑑𝑠𝜑₁𝛽𝑡^𝛽𝑡1+.𝛽+1 (50) It follows from (48)–(50) and Lemma 7 that 𝑈(𝑡)≤(1+𝑡)𝜑₁+𝜓₁𝛼𝑡^𝛼×𝑡1+ℎ𝜇(𝐵)𝛼+1+𝑅𝜏𝑒^𝛾𝜏⋅𝐸_𝛼+𝛽ℎl𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏×𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽,𝑉(𝑡)≤(1+𝑡)𝜓₁+𝜑₁𝛽𝑡^𝛽×𝑡1+𝑙𝜇(𝐷)𝛽+1+𝑃𝜏𝑒^𝛾𝜏⋅𝐸_𝛼+𝛽ℎ𝑙𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏×𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽. (51) By (51), we obtain 𝑢(𝑡)(𝑡=‖𝑢(𝑡+‖𝑣(𝑡‖(,𝑣))‖)‖)‖≤𝑒^−𝛾𝑡𝜑,𝜓₁((1+𝑡)+Θ(𝑡))×𝐸_𝛼+𝛽ℎ𝑙𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏×𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽. (52) Thus, if condition (39) is satisfied and ‖(𝜑,𝜓)‖₁<𝛿, then ‖(𝑢(𝑡),𝑣(𝑡))‖<𝜀, 𝑡∈𝐽; that is, system (23) is finite-time stable. This completes the proof.

4. An Illustrative Example

In this section, we give an example to illustrate the effectiveness of our main result.

Consider the following two-state Caputo fractional BAM type neural networks model with distributed delay ^𝐶₀𝐷^𝛼_𝑡𝑥₁(𝑡)=−0.7𝑥₁(𝑡)−0.2𝑓₁𝑦₁(𝑡)+0.1𝑓₂𝑦₂(𝑡)+^𝜏₀𝑠^3/2𝑓₁𝑦₁()+𝑡−𝑠𝑑𝑠_𝜏0𝑠𝑓₂𝑦₂(𝑡−𝑠)𝑑𝑠,_𝐶0𝐷^𝛼_𝑡𝑥₂(𝑡)=−0.6𝑥₂(𝑡)+0.3𝑓₁𝑦₁(𝑡)+0.2𝑓₂𝑦₂(𝑡)+^𝜏₀𝑠𝑓₁𝑦₁()−𝑡−𝑠𝑑𝑠_𝜏0𝑠^3/2𝑓₂𝑦₂(𝑡−𝑠)𝑑𝑠,_𝐶0𝐷^𝛽_𝑡𝑦₁(𝑡)=−0.7𝑦₁(𝑡)+0.4𝑔₁𝑥₁(𝑡)+0.2𝑔₂𝑥₂(𝑡)−^𝜏₀𝑠𝑔₁𝑥₁()+𝑡−𝑠𝑑𝑠_𝜏0𝑠²𝑔₂𝑥₂(𝑡−𝑠)𝑑𝑠,_𝐶0𝐷^𝛽_𝑡𝑦₂(𝑡)=−0.6𝑦₂(𝑡)+0.1𝑔₁𝑥₁(𝑡)−0.3𝑔₂𝑥₂(𝑡)+^𝜏₀𝑠²𝑔₁𝑥₁()+𝑡−𝑠𝑑𝑠_𝜏0𝑠𝑔₂𝑥₂(𝑡−𝑠)𝑑𝑠 (53) with the initial condition 𝑥(𝑡)(𝑡)=1=𝜑15sin𝑡,𝑥^′(𝑡)=𝜑^′(𝑡)=1[],𝑦(𝑡)(𝑡)=115cos𝑡,𝑡∈−𝜏,0=𝜓15cos𝑡,𝑦^′(𝑡)=𝜓^′(𝑡)1=−[],15sin𝑡,𝑡∈−𝜏,0 (54) where 𝛼=1.2,𝛽=1.3, 𝜏=0.2, and 𝑓_𝑗(𝑥_𝑗)=𝑔_𝑗(𝑥_𝑗)=(1/2)(|𝑥_𝑗+1|−|𝑥_𝑗−1|), 𝑗=1,2. It is easy to know that (𝑥^∗₁,𝑥^∗₂,𝑦^∗₁,𝑦^∗₂)^𝑇=(0,0,0,0)^𝑇 is an equilibrium point of system (53). Since ‖(𝜑,𝜓)‖₁=1/15<0.07, we may let 𝛿=0.07. Take 𝑡₀[),(𝑠)=𝑠=0,𝐽=0,30,𝜀=1,𝐶=−0.700−0.6,𝐵=−0.20.10.30.2𝐷=0.40.20.1−0.3,𝑅^3/2𝑠𝑠−𝑠^3/2,𝑃(𝑠)=−𝑠𝑠²𝑠²𝑠. (55) It is easy to check that (𝐵)𝜇(𝐷)Θ(𝑡)ℎ=𝑙=1,𝛾=0.6,𝜇=0.3828,=0.4515,𝑅=0.2894,𝑃=0.24,=max0.1697𝑡^1.2(𝑡+2.2),0.1691𝑡^1.3(,𝐸𝑡+2.3)_𝛼+𝛽ℎ𝑙𝜇(𝐵)+𝑅𝜏𝑒^𝛾𝜏𝜇(𝐷)+𝑃𝜏𝑒^𝛾𝜏Γ(𝛼)Γ𝛽𝑡^𝛼+𝛽=𝐸_2.50.1867𝑡^2.5. (56) From condition (41) of Theorem 8, we can get 𝑒^−0.6𝑡(1+𝑡)+max0.1697𝑡^1.2(𝑡+2.2),0.1691𝑡^1.3(𝑡+2.3)×𝐸_2.50.1867𝑡^2.5<1.0.07 (57) We can obtain that the estimated time of finite-time stability is 𝑇≈23.78. Hence, system (53) is finite-time stable with respect to {0.07,1,0,[0,30)}.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).