Abstract
Aiming at the interference of the delay term in continuous dynamics to the impulsive systems, we study the potential effects of time delay on the stability of a class of impulsive neural networks (INNs) in this paper. Two cases of delay are considered. For the case of small delay, a sufficient condition for the stability of delayed INNs is obtained by virtue of the average impulsive interval (AII) method. The derived results illustrate that within limits, the convergence rate of the system becomes larger with the increase of time delay. For another case, a strict comparison principle is proposed to prove that the impulsive system still maintains the original stability for any large but bounded delay under certain conditions. In particular, as an extension, the stability of delayed INNs for hybrid impulses containing both stabilizing and destabilizing impulses is also discussed. Finally, three examples are simulated to demonstrate the validity of the theoretical results.
1. Introduction
As a mathematical model of information processing, neural network (NN) is one of the most active branches of computational intelligence and machine learning. There are many kinds of NNs, and as a special kind of NNs, impulsive neural networks (INNs) have unique research value. INN was first proposed by Alan Lloyd Hodgkin and Andrew Huxley in 1952. The simulation of its neurons is closer to reality because it characterizes the transient state mutation of neurons in neural networks at a certain moment. The impulsive system is a mixture of continuous dynamic system and discrete-time system, which is different from the pure continuous-time dynamic system and pure discrete-time system. It is suitable for studying a class of dynamic systems affected by sudden change or instantaneous disturbance [1]. Furthermore, impulsive phenomena exist in various fields such as secure communication, automatic control, and mechanical system. With the help of impulsive control, we can reduce a lot of application costs. So far, many interesting results have been obtained on INNs (see [2, 3] and their references).
Time delay is known to exist in many complex networks and control systems due to the influence of some practical situations. Over the past decades, time-delay systems have been vigorously studied because of their wide applications in NNs, sampling data control, biological modeling, and other fields. Meanwhile, various types of delays are discussed in NNs, such as distributed delay [4], time-varying delay [5], and state-dependent delay [6]. However, in previous studies, time delay is generally considered to be an important source of poor system performance and system instability. Few researchers have noticed that time delay may be beneficial to system stability. This is because our impression of time delay is so rigid that we ignore the stabilizing effects of time delay. Actually, we can also extract the stabilizing information of time delay through some analysis methods. For instance, in [7], the authors make a point that the increase of time delay has a dual effect on the stability of the system, that is, it may stabilize a previously unstable system or destabilize a previously stable system.
Combining the two points of time delay and impulsive effects, many scholars have done a lot of work on INNs with delay [8–12]. For example, in [8], Chen et al. utilized an auxiliary state variable to transform the impulsive delayed system into an equivalent augmented model. On the basis of this model, the stability criterion of the system was derived. In [9], Zhang et al. firstly designed an impulsive controller for the time-delay discrete system to guarantee that the system can achieve stability. In [11], Jiang et al. investigated the impacts of time delay in impulses on system stability through average impulsive delay and average impulsive interval (AII) methods. In [12], Li and Song focused on the stabilization of time-delay systems under impulsive control, and the results show that the delay term in impulses may be conducive to the stabilization of the system. Obviously, studying a system with both time delay and impulsive effects is challenging because we need to consider the interaction of the two on the system.
Furthermore, it can be observed that both references [11, 12] have investigated the impacts of the delay term in impulses on stability of the system, but few papers have studied the latent effects of the delay term in continuous dynamics on stability of the impulsive system. Knowing that an impulsive system is a combination of continuous and discrete subsystems, it is interesting to think about the overall effects of the delay term in the continuous subsystem and the impulsive effect in the discrete subsystem on the system stability. In addition, looking back at the fact that time delays may facilitate the stability of systems, a natural problem emerges: under what conditions does the delay term in continuous dynamics play a positive role in the stability of systems?
In view of the above discussion, this paper mainly studies the potential effects of delay term in continuous dynamics on the stability of a class of INNs. Compared with some existing results, this paper fully captures the information that time delay can enhance stability. With regard to small delay and large delay, the stability of INNs with delay is investigated by using AII condition, and the hidden role that delay plays in the stability of system is revealed. With regard to hybrid impulses, the AII condition is replaced by the dwell-time condition so as to deal with the impulsive parameters as a whole, and the stability criterion of INNs is also derived. On the whole, the main features of this paper can be generalized as follows:(1)The time delay in two cases is considered. When the delay is small, we capture the stabilizing information of time delay by means of the impulsive delay inequality and then integrate it into the Lyapunov-based function. Finally, with the help of the AII condition, the stability criterion of a kind of general INNs is derived. The results show that in a certain range, the system converges more quickly when the delay value is larger.(2)In order to handle the case of large delay, we adopt a strict comparison principle which is different from the comparison-like principle, and it is proved that these kinds of INNs are robust to any large but bounded delay.(3)Considering the dual effects of impulses, we extend the ideas of the first two points to the hybrid INNs containing stabilizing and destabilizing impulses.
This paper is organized as follows. In Section 2, a kind of general INNs with delay is introduced, and some requisite definitions and assumptions are presented. In Section 3, the main theorem results of this paper are derived, which fully illustrate the latent effects of delay term in continuous dynamics on stability of a kind of INNs. In Section 4, three numerical examples are simulated to indicate the validity of the derived results. Finally, Sections 5 gives a brief conclusion and prospects of the feasibility of the future research.
2. Preliminaries
2.1. Notations
Let , , and stand for the set of real numbers, the set of nonnegative real numbers, and the set of dimensional real-valued vectors, respectively. Denote and as the set of positive integer numbers and nonnegative integer numbers, respectively. For vector , let . Denote as the set of piecewise right continuous function , where . Denote the upper right-hand Dini derivative of function as .
2.2. Model
In this paper, based on relevant work in reference [13], we consider a class of INNs, the main form of which is as follows:where are constants, is the number of neurons, represents the state variable of the neuron at time , represents the derivative of , is the transmission delay, and are the neuron activation functions at time and , respectively, and are real constants representing the connection weight, is the impulse sequence satisfying and , and , where and . Generally, we suppose that the solution of network (1) is right continuous, that is, , and the sequence is called the impulsive control rule. As a matter of convenience, we let , which means . Define the solution of network (1) through as , where represents the initial state.
For subsequent needs, we give some requisite definitions and assumptions as follows.
Definition 1 (see [14]). Suppose that there exist positive constants and such thatwhere represents the number of impulses in the interval . Then, is called the elasticity number, and denotes the AII constants.
Remark 1. The concept of AII is proposed to handle various types of impulses. In fact, AII condition (2) allows an upper bound, that is, . Particularly, at least one impulse is required for each interval of length in the case of . For AII constant , it can be observed that it contains more impulsive instant sequences when the elasticity number is larger.
Definition 2. For any given initial value , if there exist positive numbers and such thatholds for every sequence , then we can say that the network (1) is globally uniformly exponentially stable (GUES) over the class .
Remark 2. mentioned in the above definition represents a collection of impulsive instant sequences that satisfy AII condition (2).
Assumption 1. The functions satisfy , , and function satisfies .
Remark 3. Clearly, Assumption 1 guarantees that is an equilibrium point to network (1).
Assumption 2. The functions are all Lipschitz continuous and meetfor all , where are constants.
Assumption 3. The impulsive operator meetsfor all , where is Lipschitz constant.
In order to facilitate the subsequent expression, we make
3. Main Results
We will discuss the stability of INNs from the following three aspects in current section. Firstly, we consider the stability of a kind of INNs with small delay (the delay does not exceed any two consecutive impulsive time intervals, i.e., ). Besides, the latent effects of time delay are also explored. Secondly, the stability of INNs with arbitrarily finite delay is considered. Compared with small delay, we use large delay (which implies that the delay may be greater than a certain impulsive time interval, namely, may not be true) to represent relatively larger delay (which is collectively referred to as arbitrarily finite delay here). For arbitrarily finite delay, we analyze the robustness of the stability of INNs with delay and verify that the system can remain stable for any large but bounded delay under certain conditions. Finally, in view of the fact that the impulsive effects may promote or suppress system stability, we extend the ideas of the first two points to delayed INNs with hybrid impulses.
3.1. INNs with Small Delay
In what follows, we will discuss the case where the delay is small. We capture the stabilizing information of time delay with the help of the impulsive delay inequality and then integrate it into the Lyapunov-based function. Finally, through the AII method, we can derive the stability criterion of INNs.
Theorem 1. If there exist constants , and the following conditions hold:wherethen under Assumptions 1–3, network (1) is GUES over the class .
Proof. Construct a function and make . For any , let , and design an auxiliary functionTo start with, let , and then we will confirm thatTogether with (7) and (8),Firstly, we demonstrate that (9) is true for , namely, . Note that . If the above statement is incorrect, then there is an instant such thatWhen , it is apparent that , and in combination with (10), we derive . For , we can calculate thatWhen , combining the definition of and conditions (6), (10), and (11), one hasObviously, it could be observed that it contradicts , namely, (9) holds for .
Next, through mathematical induction method, we assume that (9) is true for , i.e.,which meansSubsequently, we demonstrate thatRecall (7) and (14), and we obtainThus, (15) holds for . On the contrary, it is assumed that there is an instant which makesIf , referring to (12), we derive thatSimilarly, what calls for special attention is that when , it follows from (14) thatAt this time, if , on account of , and together with (6), (16), and (17), we could compute thatwhich contradicts , namely, (15) holds.
Hence, we can getwhich means (8) is true, namely, .
Let , and then we obtainSince , the AII method further yields thatwhere represents the number of impulses in the interval .
According to (18) and the definition of , we havewhere . Until now, we have done the proof.
Remark 4. From (6) and (8) in Theorem 1, we can notice the latent impacts of time delay on the decay rate of Lyapunov function . In particular, if , then we have . Meanwhile, the implicit function is determined by , which decreases as increases. Obviously, the result derived from the above theorem shows that the convergence rate is related to parameter , and in view of the relationship between and delay , it can be concluded that the convergence rate of the system will become larger with the increase of delay, which means that we have captured the stabilizing effects of time delay. In addition, what needs special attention is that in the majority of the available literature about the stability of delayed INNs, we can see that the stability of the system tends to be destroyed as delay increases, but different results are obtained in this paper.
Remark 5. It should be noted that the conclusion of the relationship between time delay and system stability derived from Remark 4 is based on , so the results may be conservative to some extent. Furthermore, the conclusion of Theorem 1 is a sufficient condition rather than a necessary condition; then, it is possible for the system to be stable when is small and does not meet the conditions of Theorem 1.
3.2. INNs with Arbitrarily Finite Delay
For the case of arbitrarily finite delay, based on strict comparison principle and the concept of AII, a stability criterion of INNs is also derived.
Lemma 1 (see [13]). Let and . Suppose that meetfor all . Then, implies that .
Theorem 2. If there exists constant such thatthen under Assumptions 1–3, network (1) is GUES over the class for arbitrarily finite delay .
Proof. Construct a function , and let .
Next, similar to (11), we haveWhen , according to Assumptions 1–3, we could getIntroduce an impulsive delayed system with as its unique solution:Apparently, when . In accordance with Lemma 1, we haveFrom the variable parameter formula, we obtainwhere denotes the Cauchy matrix of the following system:According to the properties of the Cauchy matrix, combining and , we can obtainwhere , and it is evident that by using condition (21).
Reviewing (26) and (28), we haveSince , and , this implieswhere . In fact, by using (21), we can obtain . Then, we shall confirm thatOn the contrary, suppose (31) is untenable; then, there exists an instant which makesSubsequently, from (29) and (33), one haswhich is in contradiction with (32). Thus, we can derive that (31) holds. Finally, combining (25), we havei.e.,So far, we have done the proof.
Remark 6. It can be observed from Theorems 1 and 2 that the AII constant should be small enough to meet conditions (5) and (21) in the case of . The AII constant means the frequency of impulsive control. The smaller is, the higher the impulsive frequency will be. In addition, it should be noted that the result derived from Theorem 2 involves elasticity number , and equation (21) may not hold when is sufficiently large. However, for delay-free system (see [14]), the derived result does not involve , so we cannot obtain such a conclusion. Moreover, we can obverse that the elasticity number is not involved in the result of Theorem 1, which is the difference between Theorems 1 and 2. Therefore, under the condition of AII, Theorem 2 further illustrates the internal relationship between large delay and system stability.
Remark 7. Although Theorems 1 and 2 are proposed for small and large delays, respectively, this does not mean that Theorem 2 and Theorem 1 are mutually exclusive. Actually, Theorem 2 is a supplement to Theorem 1 because the so-called large delay in Theorem 2 just means that may not be true, in which case it covers the case of small delay. Therefore, Theorem 2 is also applicable to the case of small delay.
3.3. Extension to INNs with Hybrid Impulses
In recent years, hybrid impulse as an important topic has attracted wide attention, and numerous meaningful results have emerged. Particularly, in [15], considering the influence of hybrid impulses on the synchronization process, the authors designed an effective hybrid impulsive controller so as to achieve the quasi-synchronization of NNs. In this case, a sufficient delay-dependent criterion for quasi-synchronization is obtained. Meanwhile, in [15, 16], the authors adopted AII and average impulsive gain methods to deal with the hybrid impulses. In this paper, an improved dwell-time condition is introduced to treat the hybrid impulses. In view of the above discussion, we extend the results of the first two theorems in this section.
First of all, in order to extend Theorem 1, we put forward Theorem 3 by referring to the processing procedure of the small delay case.
Theorem 3. If Assumption 3 is replaced by the following condition:where , then under Assumptions 1and 2 and (35), for given constants , the following conditions are satisfied:where , and satisfies with ; then, network (1) is GES.
Proof. The same analysis method as Theorem 1 is used here, except that the impulsive parameters are changed, so we shall take advantage of and to acquire the following statement:Then, by using and , one hasThat is,The proof is completed.
Next, referring to the analysis method of arbitrarily finite delay case, we obtain Theorem 4 as an extension of Theorem 2.
Theorem 4. Suppose that the parameter in Assumption 3 is replaced by , then under Assumptions 1 and 2 and modified Assumption 3, for given constants , the following conditions are fulfilled:Then, network (1) is GES.
Proof. It is easy to prove this theorem by combining the analytical methods of Theorems 2 and 3, so we leave out it for brevity.
Remark 8. The parameter in Theorems 3 and 4 is used to describe the variable of hybrid impulses in impulsive control systems. As you can see, in Theorems 1 and 2 is required to be , but in Theorem 3 and 4 satisfies , that is, if and if . It implies that stabilizing impulses and destabilizing impulses may exist at the same time. Furthermore, in order to handle these parameters overall, we propose conditions and , and in a sense, the parameter may be approximately regarded as the “average value” of . In fact, condition combined with or combined with can be considered as an improvement of dwell-time condition.
Remark 9. Compared with reference [11], this paper studies the effects of time delay in continuous dynamics on system stability. Bear in mind that the relationship between time delay in continuous dynamics and the stability of impulsive systems is not easy to find, and the derived results based on AII method contain both time delay and the AII constant , which is not obtained in previous results. Furthermore, we extend the results to systems with hybrid impulses. In addition, when discussing the delay effects in reference [12], the authors limit the time delay to be less than any two consecutive impulsive time intervals. However, we loosen the condition of time delay in this paper, that is, the time delay can be smaller than any two consecutive impulsive intervals, or it can be greater than any two consecutive impulsive time intervals.
Remark 10. On the basis of this paper, we could also investigate the case of neural network models with time-varying delays. Actually, the results of this paper are still valid after the constant delay is replaced by time-varying delay in the model. We will continue to explore this question in depth in future studies.
4. Illustrative Examples
Finally, for the purpose of verifying the above achievements, we put forward the following three examples in current section.
Example 1. Consider a 2-dimensional INN with small delay:under impulsive controlwhere .
Obviously, . By calculating,Here, we set the initial value and choose impulsive instants , which means . When , we can figure out and , and it can be tested that all conditions in Theorem 1 hold. Therefore, we can derive that systems (39) and (40) are GUES when . In addition, according to Remark 4, it can be seen that the system may converge faster with the increase of delay, which corresponds to the simulation results in Figures 1–3. Moreover, we calculate its corresponding parameter and estimate its convergence rate for different time delays , which are shown in Table 1. More importantly, it also reveals the potential stabilizing effect of time delay.
Example 2. Consider another 2-dimensional INN with large delay:under impulsive controlwhere and . Here, we set the initial value . As shown in Figure 4, the impulse-free system is unstable.
In addition, it is apparent that . By calculating,Therefore, we can conclude that . That is to say, the conditions in Theorem 2 are fulfilled, and it is deduced that systems (43) and (44) are GUES, which is well reflected in Figure 5. By the way, the time delay could be much larger and the system would still be stable. In what follows, we calculate the corresponding convergence rate for different time delays , which are shown in Table 2. Actually, the delay is not limited to 100, and it can even be greater than 100. As long as the delay is bounded under certain conditions, the initial stability of impulsive system can be guaranteed. Here we only calculate the convergence rate of the system when the delay increases to 100.
Example 3. Consider a 3-dimensional INN:under hybrid impulsive controlwhereSuppose that and . We give the initial value and take time delay . As shown in Figure 6, the impulse-free system is unstable.
Additionally, it is evident that . By calculating,Furthermore, we can figure out by Matlab. That is, and in conditions of Theorem 3 are fulfilled. At the same time, a series of conditions in Theorem 3 are completely true. Therefore, systems (43) and (44) are GES, which is well illustrated in Figure 7. From the simulation results, the impulsive effects indeed have both stabilizing and destabilizing effects.
5. Conclusion
In this paper, we have discussed the stability of a kind of INNs with delay. Particularly, the internal relation between time delay and system stability has been revealed. Firstly, we have investigated the case where the delay is small. By constructing Lyapunov function, combining the impulsive delay inequality and AII condition, we have obtained a sufficient condition to assure the exponential stability of INNs. The results have shown that within limits, the system converges more quickly with the increase of time delay. Secondly, we have explored the case where the delay is arbitrarily large but bounded and derived a Lyapunov-based stability criterion by virtue of the strict comparison principle. Finally, as an extension, we have considered the case where INN is a system with hybrid impulses. In future studies, we may discuss the delay effects of a kind of INN with state-dependent delay.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the Natural Science Foundation of China under grant no. 61976084, the Natural Science Foundation of Hubei Province of China under grant no. 2021CFA080, and the Young Top-Notch Talent Cultivation Program of Hubei Province of China.