Prescribed performance synchronization of neural networks with impulsive effects

In this paper, the prescribed performance synchronization problem is addressed for a class of neural networks with impulsive effects. According to the prescribed performance control principle and the Lyapunov’s second stability theorem, a preset performance control protocol is designed. For neural networks with impulsive effects, the proposed control scheme can not only guarantee the steady-state performance of synchronization errors, but also ensure the transient performance of the synchronization process. This improves the performance of the neural networks effectively. Finally, a numerical simulation is given to illustrate the effectiveness and feasibility of the proposed control scheme.


Introduction
Neural networks have been a research hot spot in the field of artificial intelligence since 1980s. It abstracts human brain neurons from the perspective of information processing and builds a simple model. In the past few decades, it has revealed powerful information ability and marvelous intelligence characteristics, and has successfully solved many practical problems in the fields of pattern classification, automatic control, prediction and estimation, biology, medicine (Cao et al. 2004;Gopalsamy 2004;Lisena 2011;Milletari et al. 2016;Krizhevsky et al. 2012;Punam and Prakash 2018). Thus, it can be seen that the research on neural networks is significant.
However, it is worth noting that there are many inevitable emergencies, such as fluctuations in population growth, the spread of epidemics and so on, which may break the continuity and stability of the neural networks. Therefore, the impulsive effects can be introduced into the model, which can more realistically reflect the characteristics of the states of neural networks change over time, and has more practical application value. The research of impulsive differential equations began with the work of Mil'man V.D. and Myshkis A.D. in the 1960s (Mil'man and Myshkis 1960). They pioneered the theory of impulsive systems, which was greatly developed in the 1980s and made a lot of achievements (Lakshmikantham and Simeonov 1989;Shouchuan et al. 1989;Pandit 1982).
In fact, synchronization as a significant nonlinear phenomenon is widespread in our lives, such as the flight of unmanned aerial vehicles in formation (Antonio et al. 2021), the synchronized glow of fireflies, or the firing of neurons when the human brain is working. Therefore, it has drawn a lot of attention from academics (Zhang et al. 2010;Wenlian and Chen 2004;Pratap et al. 2020). In recent years, the synchronization problems of neural networks with impulse effects have aroused great interest and widespread concern of scholars. Simultaneously, a range of achievements has been obtained in the research of the synchronization problem of impulsive neural networks (Subramanian et al. 2018;Abdurahman et al. 2014;Tang et al. 2021;Li and Bohner 2010;Qian 2011;Hao et al. 2015;Luo 2008;Alzahrani et al. 2016;Karthick et al. 2019). For instance, the synchronization problem of impulsive neural networks with mixed time-varying delays and linear fractional uncertainties was discussed in Subramanian et al. (2018). In Abdurahman et al. (2014), the authors investigated the exponential function projective synchronization of impulsive neural networks with mixed time-varying delays. Tang et al. (2021) was devoted to investigating the exponential synchronization problem for a class of coupled heterogeneous neural networks. Considering the controller would suffer from impulsive effects and the disturbances, the distributed pinning control strategy was introduced. In addition, some other achievements about synchronization could be found in reference (Li and Bohner 2010;Qian 2011;Hao et al. 2015;Luo 2008;Alzahrani et al. 2016;Karthick et al. 2019).
Generally, in the synchronization process, we need to focus on steady-state performance and transient performance. The steady-state performance reflects the performance of the tracking error after the steady state, such as the control accuracy, while the transient performance reflects the process of the tracking error entering the steady state, including the convergence rate and the overshoot value. Actually, many engineering systems often require that proposed projects should satisfy the specified steadystate and transient performance in advance. Unfortunately, the synchronization control methods mentioned above only considered the steady-state performance but ignored the transient performance. To solve this problem, the prescribed performance control method has been proposed by Rovithakis and Bechlioulis, which means that the tracking error converges to an arbitrarily small bounded closed set, with convergence rate no less than a preset constant, the maximum overshoot of synchronization error does not exceed a pre-specified constant. In recent years, scholars have used prescribed performance control to solve the synchronization problems of neural networks. In (Kostarigka and Rovithakis 2012), an adaptive dynamic output feedback neural network controller for a class of mutiinput/muti-output affine in the control uncertain nonlinear systems was designed, capable of guaranteeing prescribed performance bounds on the system's output as well as boundedness of all other closed loop signals. Ni et al. (2019) was devoted to investigating fixed-time prescribed performance control problem for uncertain strict-feedback nonlinear systems with unknown dead zone. Zhou et al. (2021) aimed at the problem of finite-time prescribed performance adaptive fuzzy control for a class of unknown nonlinear systems. In addition, Aili Fan also solved many complex network synchronization problems by using prescribed performance control Li 2021, 2020). There are also abundant achievements concerned the pre-  Liu et al. (2015), Ling and Sun (2018), Shi (2021). However, to our best knowledge, there are no relevant results on the problem of the prescribed performance synchronization of neural networks with impulsive effects.
Inspired by the discussions above, we are concerned with the prescribed performance synchronization problem for a class of neural networks with impulsive effects. Based on the Lyapunov's second stability theorem, a prescribed performance control scheme is presented. The designed control protocol can not only guarantee the steady-state performance in the synchronization process, but also ensure that the convergence rate of the tracking errors is not less than the preset constant, and the overshoot is not more than the preset constant. This improves the performance of the neural networks effectively.
The innovations of this paper are as follows: 1. The proposed control scheme can not only guarantee the steady-state performance of the system, but also has better performance in the overshoot value and the convergence rate of synchronization error compared with control methods mentioned in Abdurahman et al.
(2014) that did not consider transient performance. Wang et al. (2014) considered the prescribed performance during the synchronization, the impulsive effects was not considered in the model, which is not enough to actually reflect the characteristics of the system with time. 3. It is the first time to apply the prescribed performance control method to the neural networks with impulsive effects, which can enrich the control theory and provide a new idea for solving the control problems in the future.

Although
The rest of the paper is organized as follows. In Sect. 2, the dynamic equations of Drive-Response networks are given. In addition, the basic theory of prescribed performance control and some assumptions needed are introduced. The prescribed performance control scheme is derived in Sect. 3. To illustrate the effectiveness of the results, presenting a numerical example in Sect. 4. Finally, the results are summarized in Sect. 5.

Preliminaries and problem formulation
Notations In this paper, R n represents the n-dimensional Euclidean space; R nÂn is defined as the n Â n real matrices. Z denotes the set of positive integers; f i ðÁÞ represents the activation function of each neuron; A 2 R nÂn is defined as the connection weight matrix; N denotes the number of neurons.p represents the number of impulse moments.
In this section, some assumptions and the theory of prescribed performance control which will be applied in the following discussion are supplied.

Problem formulation
In this paper, we consider a class of neural networks with impulsive effects described by: where x i t ð Þ corresponds to the state variable of the i-th unit at time t. c i [ 0 represents the rate with which the i-th neuron will reset its potential to the resting state when disconnected from the network and external inputs; I i represents the input current on the i-th unit. a ij represents the connection strength from the j-th unit to the i-th unit. ft k g k¼1;2;:::;p is a strictly monotonically increasing pulse sequence of positive numbers such that t kþ1 À t k ! # for all k ¼ 1; 2; ::: denotes the abrupt change of x i t ð Þ at impulse moment t k . We consider system (1) as the Drive neural networks, the Response neural networks is given as follows: where u i ðtÞ is the control input which will be designed later. Next, we define the following synchronization error between the Drive neural networks and the Response neural networks: k Þ; i ¼ 1; 2; :::; N; k ¼ 1; 2; :::; p:

Preliminaries
A smooth function g i t ð Þ is called a performance function, if it has the following properties: 1. g i t ð Þ is positive and strictly decreasing; 2. lim t!1 ; where i ¼ 1; 2; :::; N: Based on the prescribed performance control theory, the transient performance and steady-state performance can be achieved by ensuring that the synchronization error e i t ð Þ ¼ y i t ð Þ À x i t ð Þ evolves strictly within predefined decaying bounds as follows: where q i is a positive constant. Here, we select the exponential type performance function as follows: where l i ; g i0 ; g i1 are positive constants, respectively, g i0 [ 0 is chosen to satisfy Àq i g i ð0Þ\e i ð0Þ\q i g i ð0Þ.
It is noted that l i is the lower bound of the convergence rate of the synchronization error e i ðtÞ. Moreover, the maximum overshoot value of e i t ð Þ is no more than q i g i 0 ð Þ: Therefore, by selecting the performance function g i t ð Þ and the constant q i appropriately, then, the synchronization error e i t ð Þ can achieve prescribed performance. Throughout this paper, the following assumptions need to be given.
Assumption 1 For each i 2 1; 2; :::; N, there exist constants l i [ 0 such that for all x; y 2 R; x 6 ¼ y. Denote L ¼ max 1 i n l i ð Þ.
Assumption 2 The performance function g i t ð Þ is specified as follows: 1. g i t ð Þ is continuously differentiable on the interval ½0; 1Þ: The main purpose of this paper is to design an appropriate control input u i t ð Þ to achieve the prescribed performance synchronization Eq. (4) between the Drive-Response neural networks with impulsive effects.

Convergence analysis
To ensure the preset boundedness in Eq. (4), we can employ an error transformation to the e i t ð Þ as follows: where U i ðÁÞ : ðÀq i ; q i Þ ! ðÀ1; 1Þ is strictly smooth increasing function and selected as Prescribed performance synchronization of neural networks with impulsive effects… 12589 Then, taking the derivative of Eq. (7), we can get the system equation of transformation error as follows where C ik t k ð Þ is a continuous function of c ik , Remark 1 It can be proved that when the transformation error n i ðtÞ is bounded, then, for all t ! 0, the synchronization error e i ðtÞ can achieve the prescribed performance, namely, Eq. (4) is satisfied (See Appendix). Therefore, we only need to prove that the transformation error n i ðtÞ is bounded in the interval ½0; 1Þ.
To achieve the control objective Eq. (4), we design the following control protocol: In this paper, the convergence property of the designed control scheme is summarized in the following theorem.
Theorem 1 Based on the Lyapunov's second stability theorem. When a\c. Then the prescribed performance synchronization between Drive and Response neural networks can be achieved under the controller Eq. (10).
Proof Firstly, it is easy to get p i [ 0; and e i t ð Þ are both greater than or less than zero.
Considering the following Lyapunov function: For t 6 ¼ t k ; Consider the case at the time of the pulses. We have denote Q ¼ max 1 i n;1 k p ðjC ik ðt k ÞjÞ: According to the Eq. (13), D þ V t ð Þ\0 is true for t 2 ðt kÀ1 ; t k Þ; k ¼ 1; 2; :::; p: So, for arbitrary small positive number h, we can obtain When h ! 0 þ , we have It is not difficult to know that Then, taking the sum from i equals 1 to n, we can obtain: Hence, for t 2 t kÀ1 ; t k ð ; k ¼ 1; 2; :::; p, we get Finally, we consider the condition where t belongs to the interval ðt p ; 1Þ. By recursion, we can obtain Therefore, from what has been discussed above, we can get that the transformation error n i t ð Þ is bounded over the entire time interval, which means that the prescribed performance Eq. (4) of synchronization error e i t ð Þ can be achieved.
Remark 2 In this way, the prescribed performance synchronization of the Drive-Response neural networks with impulsive effects can be realized. Compared with Tang et al. (2021), on the basis of ensuring the steady-state performance in the synchronization process, the designed control protocol also makes the convergence rate not less than l i and the overshoot not more than q i g i ð0Þ during synchronization.

Conclusion
This paper addresses the problem of prescribed performance synchronization for neural networks with impulsive effects. Based on Lyapunov stability theorem and prescribed performance control theory, a prescribed performance control scheme is proposed, which ensures the rate of convergence of synchronization error v ! l i ,i ¼ 1; 2; :::; N, and its overshoot value does not exceed q i g i ð0Þ. Meanwhile, the transient performance is considered in the synchronization problem of impulsive neural networks for the first time, which enriches the impulsive neural networks control theory. Finally, a numerical simulation is given to demonstrate the effectiveness of the control scheme. In future studies, we will consider the delay factor in the dynamics equation of the system, which will be more consistent with the actual situation.
Then, we consider the right-hand side inequality and simplify it, we get In the same way, we also have To sum up, we can conclude that as long as n i t ð Þ is bounded, e i t ð Þ can satisfy the prescribed performance (4). Data availability Enquiries about data availability should be directed to the authors.

Declarations
Conflict of interest The authors have not disclosed any competing interests.