Abstract
This paper studies the stability analysis of fractional-order bidirectional associative memory neural networks with mixed time-varying delays. The orders of these systems lie in the interval . Firstly, a sufficient condition is derived to ensure the finite-time stability of systems by resorting to some analytical techniques and some elementary inequalities. Next, a sufficient condition is obtained to guarantee the global asymptotic stability of systems based on the Laplace transform, the mean value theorem, the generalized Gronwall inequality, and some properties of Mittag–Leffler functions. In particular, these obtained conditions are expressed as some algebraic inequalities which can be easily calculated in practical applications. Finally, some numerical examples are given to verify the feasibility and effectiveness of the obtained main results.
1. Introduction
Neural networks have drawn increasing interests due to their powerful applications in physics, mechanics, biology, information science, and sociology In order to meet the requirements in practical applications, some researchers have proposed various types of neural networks, such as cellular neural networks [1], Hopfield neural networks [2], recurrent neural networks [3], bidirectional associative memory (BAM) neural networks [4], and memristor-based neural networks [5]. In [4], Kosko first proposed bidirectional associative memory neural networks to store and invoke pattern pairs. A BAM neural network consists of two layers of associative neurons, and the neurons arranged in one layer are fully interconnected with those in the other layer, but there are no interconnections in the same layer. It has been revealed that BAM neural networks can provide potential applications in pattern recognition, signal processing, and combinatorial optimization [6, 7].
In the recent decades, the fractional calculus has been paid considerable attention owing to its great development in theory and application. As is well known, fractional-order derivative is a very powerful tool to describe memory and hereditary properties of many materials and dynamical processes, and hence it can be well used to characterize a large number of systems [8–10] in many fields. In order to describe the dynamical behavior of neurons better, fractional-order neural networks have been put forward by combining the fractional calculus with neural networks. As an important type of neural network, fractional-order BAM neural network has been widely studied. In recent years, many great contributions have been made to the dynamics of systems [11–22] which is closely related to the applications in various fields. For example, the references [11, 14–20] reported some research on the global asymptotic stability, Mittag–Leffler stability, uniform stability, and finite-time stability. In [12, 21, 22], the authors investigated the adaptive synchronization, the finite-time synchronization, and the Mittag–Leffler synchronization for several kinds of fractional-order BAM neural networks.
Notice that the order of systems in most works lies in the interval . In the real world, it is also very significant to focus on fractional-order neural networks with the order due to their successful applications. For example, for the second-order multiagent dynamics, a fractional-order observer with the order can be used to capture the velocity information which is not always available [23]. In addition, the fractional-order systems with have been extensively studied in mechanics, physics, and information science [24–26]. To the best of our knowledge, most results for could not be directly extended to the case of due to its more complicated mathematical theory. Thus, it is very attractive to carry out the study on fractional-order neural networks with (see [12–14, 27–30] and the references therein). In [12, 13, 30], the authors considered the finite-time synchronization for several classes of fractional-order memristor-based neural networks with time delays. In [27], Wu et al. studied the finite-time stability for fractional-order delayed neural networks. Rakkiyappan et al. [28] reported the finite-time stability for a class of fractional-order complex-valued memristor-based neural networks with time delays. Chen et al. [29] discussed the finite-time stability for a class of fractional-order memristor-based neural networks with time delays. Recently, Xu et al. [20] considered the finite-time stability for a class of fractional-order BAM delayed neural networks. In the above works, the proofs are mainly based on the Laplace transform, the generalized Gronwall inequality and some properties of the Mittag–Leffler functions, and hence the obtained sufficient conditions are closely related to the Mittag–Leffler functions.
In the aforementioned works, it is noted that only the discrete constant delays are involved in the network models. However, as revealed in [31–33], neural networks usually have a spatial nature due to the presence of a large number of parallel pathways with various axon sizes and lengths, so that there is a distribution of propagation delays over a period of time. In this situation, discrete delays cannot well characterize the neural networks since the signal propagation is no longer instantaneous. Consequently, the distributed delays should be also taken into account in the description of neural network models. In the recent decades, lots of researchers have made great efforts to the dynamics of neural networks with both discrete and distributed delays and there have been some excellent results [33–38]. Notice that these works were mainly concerned with integer-order neural networks. The research on fractional-order neural networks with discrete and distributed time delays has received little attention. In [39, 40], the authors considered the global Mittag–Leffler stability and the global exponential stability for Caputo fractional-order neural network with discrete and infinite-time distributed delays. Recently, Zhang et al. [41] investigated the asymptotic stability for a class of Riemann–Liouville fractional-order neural networks with discrete and finite-time distributed constant delays. In [42], Wu et al. studied the uniform stability of Caputo fractional-order neural networks with discrete and finite-time distributed constant delays. It is worth pointing out that the order of systems in these works lies in the interval . To the best of our knowledge, the stability analysis for the case of has not been dealt within the existing literature.
In this paper, we focus on a class of Caputo fractional-order BAM neural networks with discrete and distributed time-varying delays for . These networks can be described aswhere and denote the state vectors in the X-layer and Y-layer, respectively and and are the discrete and distributed time-varying delays which are continuous and bounded. The remaining notations will be specifically introduced in the next section. The main contributions of this paper include the following several aspects: (i) We consider the stability analysis for a class of fractional-order BAM neural networks with discrete and distributed time-varying delays for which has not been discussed in the existing literature. (ii) For the finite-time stability problem of the system under consideration, the classical method [13, 27–30] can give a sufficient condition which is closely related to the Mittag–Leffler function. This condition cannot be easily calculated, and the settling time cannot be easily estimated. Here, we apply some analytical techniques and some elementary inequalities to investigate this problem. Under the assumption (μ is a positive constant), we derive a sufficient condition independent of the Mittag–Leffler function. It is noted that our sufficient condition can be expressed as an algebraic inequality and the estimated settling time T can be easily obtained in practical applications. (iii) To guarantee the global asymptotic stability of systems, a sufficient condition is obtained by resorting to the Laplace transform, the mean value theorem, the generalized Gronwall inequality, and some properties of Mittag–Leffler functions. Different from the above finite-time stability analysis, this does not require the assumption . In addition, the obtained condition generalizes Corollary 1 in [27].
The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries and network model. In Section 3, a sufficient condition is firstly derived to ensure the finite-time stability of systems. Next, a sufficient condition is obtained to ensure the global asymptotic stability of systems. Finally, some results related to the equilibrium point are directly given. In Section 4, some examples are provided to verify the effectiveness of the obtained main results. Finally, Section 5 gives some conclusions and presents some possible research in future.
2. Preliminaries and Network Model
In this section, we first recall some definitions and properties related to the Caputo derivative and Mittag–Leffler functions. Next we list some important inequalities and introduce the network model.
Definition 1 (see [43]). Let and let m be a positive integer such that . For , the Caputo derivative of fractional order a of is defined bywhere is the Gamma function, i.e.,
Definition 2 (see [44]). The fractional integral with noninteger order of a function is defined aswhere and is the Gamma function.
Proposition 1 (see [45]). Let and let m be a positive integer such that . For , we have
Definition 3 (see [43]). The Mittag–Leffler function with two parameters is defined aswhere , , and .
Let denote the Mittag–Leffler function with one parameter, i.e., . Obviously, we have .
Let be the Laplace transform operator. Then, we havewhere denotes the Laplace transform of .
Proposition 2 (see [43]). Let . Suppose that µ satisfies . Then, for and ,where and are two positive real constants.
Proposition 3 (see [46]). Let and . Suppose that A is a diagonal stability matrix. For any , we havewhere is the largest eigenvalue of the matrix A and denotes any vector or induced matrix norm.
Now we present some inequalities which are crucial to our main results.
Proposition 4 (generalized Bernoulli inequality [47]). Let and . For , we have . Moreover, .
Proposition 5 (see [48]). Let and let , , and be nonnegative functions on . Ifthen,where .
Proposition 6 (generalized Gronwall inequality [49]). Let . Let be a nonnegative function locally integrable on (some ). Suppose that is a nonnegative, nondecreasing continuous function defined on , then there exists a constant M such that . Let be nonnegative and locally integrable on . If satisfieson , thenMoreover, if is a nondecreasing function on , then we haveIn what follows, we consider a class of Caputo fractional-order BAM neural networks with mixed time-varying delays which can be described as equation (1) orwhere , , and . and denote the state of neurons in the X-layer and Y-layer, respectively. The constants and are the self-regulating parameters of the neurons. The discrete time-varying delay and the distributed time-varying delay are continuous functions such that and , where and are two positive constants. The constants , , and are the connection, the discretely delayed connection, and the distributively delayed connection. The constants , , and have the same meanings as , , and . , , , , , and represent the activation functions. and stand for the external inputs.
Besides, the activation functions , , , and satisfy the following Lipschitz conditions:for any , where and and , , , , , and are some positive constants.
Let and let denote the Banach space of k-dimensional continuous vector functions defined on the interval . The norm of is defined as .
Let and be any two solutions of system (15) with different initial conditions. Let and . The initial conditions of are given as follows:where and .
Definition 4. System (15) is said to be asymptotically stable ifwhere and .
Definition 5. Suppose that δ and ε are any positive constants such that . If impliesthen system (15) is said to achieve the finite-time stability with respect to .
3. Main Results
In this section, we will consider the finite-time stability and the global asymptotic stability for a class of fractional-order BAM neural networks with mixed time-varying delays for .
3.1. Finite-Time Stability Analysis
In this subsection, some sufficient conditions are derived to ensure the finite-time stability of system (15). For the time-varying delay , we need to make further assumptions.
Assumption 1. The discrete time-varying delay is a differentiable function such thatwhere µ is a constant.
In what follows, we state the main theorem. For convenience, let us first introduce some notation related to the parameters of system (15). Let and . LetLet
Theorem 1. Let δ and ε be any positive constants such that . Suppose that Assumption 1 holds. If andwhere and , and then system (15) is finite-time stable with respect to .
Proof. Let and be two solutions of system (15) with different initial conditions. Let and . Assume that satisfies the initial condition (17).
Proposition 1 yieldsAccording to the Lipschitz conditions, we obtainSimilarly, we haveMoreover, it follows thatMeanwhile, we haveTogether with the estimates of and , it follows thatMaking use of the Cauchy–Schwartz inequality, we obtainFor the integral , we deriveFor the integral , we deduceCombining this with the assumption on , we obtainMoreover, it follows thatFrom the inequalities (30)–(34), we getLet , where . Obviously, we haveIn addition, for any fixed t, we use the mean value theorem to getwhere . Moreover,From the inequalities (35)–(38), we haveMoreover,Hence,where and .
In view of Proposition 5, we obtainWith the help of Proposition 4, we getCombining this with the inequality (41), we obtainMoreover, we getAccording to the condition of Theorem 1, we get for any . This indicates that system (15) is finite-time stable with respect to . The proof is completed.
Remark 1. It is obvious that the left hand side of (23) increases with the increase of time t. Hence, the settling time T can be easily calculated in practical applications.
If some parameters of system (15) are assumed to satisfy some further conditions, then system (15) has a unique equilibrium point. Based on Theorem1, the finite-time stability of the equilibrium point can be ensured under the assumption of Theorem 1. In order to discuss the existence and uniqueness of the equilibrium point, the distributed time-varying delay is assumed to be a constant, i.e., .
Corollary 1. Assume that the distributed time-varying delay is a constant, i.e., . Let Assumption 1 hold. If the parameters of system (15) satisfy the condition in Theorem 1 and the following conditions:then the system has a unique equilibrium point which is finite-time stable with respect to .
In fact, from the proof of Theorem 1, we only need to prove the existence and uniqueness of the equilibrium point. The proof mainly depends on the contraction mapping principle. For the convenience of readers, we only show some key points.
Proof. We define a mapping as follows:whereHere, and .
In the following, we prove that is a contraction mapping on . For any two different points and , we haveThe Lipschitz conditions lead toTogether with the condition (46), we getThis indicates that is a contraction mapping on . As a result, there exists a unique point such that , i.e.,This implies that system (15) has a unique equilibrium point .
3.2. Asymptotic Stability Analysis
In this subsection, some sufficient conditions are derived to ensure the global asymptotic stability of system (15). Let , , , , , and be defined as in (21).
Theorem 2. Let . Ifwhere , , and , then system (15) is globally asymptotically stable.
Proof. Let and be any two solutions of system (15) with different initial conditions. Let and . Assume that satisfies the initial condition (17).
Making use of the Laplace transform and the inverse Laplace transform, we deriveIn the same way, it follows thatAccording to Proposition 3, we getLet , , , and . From the inequality (56), we obtainFrom the inequality (57), we havewhere , , , and .
Obviously, we have and . Together with the estimates of and , it follows thatLet , where . Obviously, we haveIn addition, for any fixed t, we use the mean value theorem to getwhere . Moreover,Together with the inequalities (60) and (61), we havewhere , , and . Thus,Owing to Proposition 6, we obtainMoreover, we haveWith the help of Proposition 2, we getwhere and are two positive constants. If , thenThis indicates that system (15) is globally asymptotically stable. The proof is completed.
Based on Corollary 1 and Theorem 2, we can easily obtain the following result related to the equilibrium point.
Corollary 2. Assume that the distributed time-varying delay is a constant, i.e., . If the parameters of system (15) satisfy the conditions (46) and (53), then the system has a unique equilibrium point which is globally asymptotically stable.
Remark 2. Proposition 3 plays a key role in the proof of Theorem 2. Notice that the letter α in Proposition 3 satisfies . For , the inequality (9) in Proposition 3 should be changed towhere M is a positive constant. It is worth pointing out that M cannot be definitely given [46]. Hence, for , if we follow the proof of Theorem 2, we will obtain a condition depending on M, which cannot be easily verified.
Remark 3. In [42], Wu et al. investigated the quasi-uniform stability of a class of fractional-order neural networks with discrete and finite-time distributed constant delays for . In this paper, we focus on a class of fractional-order BAM neural networks with discrete and finite-time distributed time-varying delays for . Two sufficient conditions (see Theorem 1 and Theorem 2) are obtained to ensure the finite-time stability and the global asymptotic stability of systems, respectively.
Remark 4. For , Tyagi et al. [39] considered the uniform stability and the Mittag–Leffler stability of fractional-order complex-valued neural networks with discrete and continuously distributed constant delays; Srivastava et al. [40] studied the exponential stability of a class of fractional-order impulsive neural networks with discrete time-varying delay and distributed delay. These two kinds of mixed time delays are completely different from that considered in this paper.
Remark 5. In the existing literature, there have been some works [14, 20, 27–30] on the finite-time stability of fractional-order neural networks with the orders satisfying . The obtained sufficient conditions in these works are some inequalities related to the Mittag–Leffler functions whose proofs mainly depend on the Laplace transform, the generalized Gronwall inequality, and some inequalities related to Mittag–Leffler functions. Following this method, we can also get a sufficient condition ensuring the finite-time stability of system (15), i.e.,where γ, , , and are defined as in Subsection 3.2. Obviously, this condition is closely related to the Mittag–Leffler function, and hence it cannot be easily calculated. It is also noted that the settling time cannot be easily estimated. In order to overcome this problem, we use some analytical techniques and some elementary inequalities to investigate the finite-time stability of system (15). We derive a sufficient condition (see Theorem 1) which can be expressed as an algebraic inequality. In particular, the estimate of the settling time can be easily obtained in practical applications. In addition, the size relationship between and cannot be obtained from the theoretical perspective under the same parameters.
Remark 6. When the distributed time-varying delay vanishes and the discrete time-varying delay is constant, i.e., and , Theorem 2 is reduced to Corollary 1 in [27].
Remark 7. In [15], Wu et al. reported the global Mittag–Leffler stabilization of fractional-order BAM neural networks via linear feedback control or partial feedback control. For a class of uncertain fractional-order nonlinear systems, the robust stabilization by proposing a kind of fractional-order sliding mode controller is considered. In [50], Yin et al. discussed the robust stability for a class of uncertain fractional-order nonlinear systems with external disturbance by using an adaptive sliding mode technique based on a fractional-order switching type control law. Notice that the order of systems in these works lie in the interval . In future, we will consider the stabilization problem of system (15) by designing a kind of fractional-order sliding mode controller. In addition, we will also investigate some applications of fractional-order systems with in electrical engineering based on a recent research.
4. Numerical Simulations
In this section, some numerical examples are provided to illustrate the effectiveness of our main results.
Consider a class of fractional-order BAM neural networks with mixed time-varying delays which can be described as follows:
Example 1. For system (72), the parameters are given as follows: , , , , , , , , , , , , , , , andA simple calculation gives the following data: , , , and . Let and be two solutions of systems with the following initial conditions:for any . The state trajectories are shown in Figure 1, and the trajectories of the error are depicted in Figure 2.
By calculating, it follows that , , , , and . Let us choose . According to the initial data, we take . Moreover, from Theorem 1, the estimated settling time can be easily obtained. The norm of error is depicted in Figure 2 which indicates the correctness of Theorem 1. On the other hand, by the sufficient condition (71) based on the existing technique, the estimated settling time . This implies the effectiveness of Theorem 1.
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Example 2. For the system (72), the parameters are given as follows: , , , , , , , , , , , , , , , andObviously, , , and . Let and be two solutions of this system with the following initial conditions:for any .
By calculating, it follows that , , , and . Moreover, . Obviously, . Thus, the condition of Theorem 2 is satisfied. The state trajectories and the norm of error are depicted in Figures 3 and 4, which indicates the feasibility of Theorem 1.
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Example 3. For system (72), the parameters are given as follows: , , , , , , , , , , , , , , , andObviously, , , , and . Let , , and be three solutions of systems with the following initial conditions:for any . The state trajectories and the trajectories of errors are depicted in Figures 5 and 6, respectively.
By calculating, we easily get the following data: , , , , and .
Let us take . For any two of the above three solutions, the initial conditions allow us to take the values of δ as follows:(1)For and , the value of δ is taken as . Moreover, the estimated settling time can easily obtained from Theorem 1.(2)For and , the value of δ is taken as . Theorem 1 gives the estimate of settling time .(3)For and , the value of δ is taken as . Based on Theorem 1, the estimated settling time can easily derived.For the above three cases, the norms of errors are depicted in Figure 7 which indicates the correctness of Theorem 1.
In addition, by the sufficient condition (71) based on the existing technique, the estimated settling time is 2.6256, 2.7558, and 2.1774 for the above three cases. This implies the effectiveness of Theorem 1.
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Example 4. For the system (72), the parameters are given as follows: , , , andAn immediate calculation gives the following parameters: , , and . Let , , and be three solutions of this system with the following initial conditions:for any .
By calculating, it follows that , , , and . Moreover, . Obviously, . Thus, the condition of Theorem 2 is satisfied. For the above three solutions, the time evolutions and the norms of errors are shown in Figures 8 and 9, respectively. These indicate the feasibility of Theorem 2.
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5. Conclusions
This paper investigated the stability of fractional-order BAM neural networks with discrete and finite-time distributed time-varying delays. The order of systems is between 1 and 2. Firstly, to ensure the finite-time stability of systems, a sufficient condition was derived based on some analytical techniques and some elementary inequalities which are completely different from those in some earlier works. Unlike those existing conditions for fractional-order neural networks with discrete constant delay, our sufficient condition is not related to the Mittag–Leffler function. It is worth noting that this condition is expressed as an algebraic inequality, and the settling time can be easily estimated in practical applications. Next, a sufficient condition was obtained to guarantee the global asymptotic stability of systems based on the Laplace transform, the mean value theorem, the generalized Gronwall inequality, and some properties of Mittag–Leffler functions. Different from the above finite-time stability analysis, this does not require the additional assumption on the distributed time-varying delay. Moreover, this obtained condition generalizes that in the earlier work. In addition, two conditions were directly obtained to ensure the stability of the equilibrium point. Finally, some numerical examples were provided to verify the effectiveness of our main results. In future, for fractional-order systems with , we will discuss the stabilization problem based on fractional-order sliding mode control and investigate some practical applications in electrical engineering. This may be a big challenge for us since the mathematical theory for is more complicated than that for .
Data Availability
All data used are provided within the article in Section 4.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research is supported by the National Natural Science Foundation of China (Grant no. 11401595). The first author is very grateful to Professor Xiaoqun Wu (Wuhan University) for her kind help. The authors also want to thank Professor Jun Mei (University of Pretoria) for his valuable suggestions on the revision of this manuscript.