Fixed-time synchronization of delayed impulsive inertial neural networks with discontinuous activation functions via indefinite LKF method
Introduction
Neural networks (NNs) is an interesting topic for study in recent years with applications in pattern classification image analysis, sensor fusion, neuroscience technology, and electrical engineering, NNs have been extensively studied by many academics [1], [2], [3], [4], [5]. There are various types of NNs such as inertial NNs (INNs), cellular NNs, Hopfield NNs etc., The INNs was introduced by the authors of [6] which represented the NNs with inertial terms in the form of second order differential equations was able to contemplate the NNs more effectively. In addition, discontinuous NNs are those with discontinuous activations, have a huge advantage over standard NN models because they can perform nonlinear mapping [7], [8]. Regarding this, extensive research into the dynamical properties of NNs with irregular activation has been undertaken. For example, the authors of [9] considered INNs with discontinuity that have parameter uncertainties and performed fixed time synchronization for the same. In [10], fixed time stability (FXTST) lemmas were introduced for INNs with discontinuity which formed the basis for obtaining FXTSY for the same system. Due to the limited speed of signal transfer that will strongly impact on the stability and oscillations of the NNs, the theoretical significance and possible application of discontinuous activation, time delays, particularly time varieties, is unavoidable. The dynamical behaviors of the INNs like the stability [11], [12], [13], [14], [15], stabilization [16], [17] and synchronization [7], [18], [19], [20] are the major reasons for the effectiveness of the system. For instance, The INNs in Wang and Tian [14] have been designed utilizing the sampled data control consisting of delays which are time variant and their stability is analysed. In [15], the authors considered INNs that have discrete and distributive delays and procured stability of the considered NNs globally in Lagrange sense. The authors of [21] obtained exponential stability of INNs with memristor and time delays. The dynamic behaviors of a coupled system to achieve the same state at the same time is known as synchronization, and the synchronization problem is a subset of the stability problem (see Huang et al. [18], Chen et al. [19], Cai and Huang [20]). Synchronization is a more extensive concept. Synchronization can be classified as finite time synchronization (FNTSY) [22], [23], [24], fixed time synchronization (FXTSY) [9], [25], exponential synchronization [26], pinning synchronization [27] etc., Achieving synchronization at infinite time started losing interest among the researchers as they needed the system to be synchronized in a finite time (FNT), which henceforth introduced the concept of FNT/FXT synchronization. In [24], INNs that are neutral were taken into account which was synchronized in FNT with the presence of coefficients that vary with time and proportional delays. Hence, in this paper we try to obtain FXTSY for the INNs with discontinuous control.
A typical issue in system theory is that in different applications such as robotics, the controllers must be able to move a system to a particular location quickly. Controllers are majorly categorized into two, they are, the continuous control and the discrete control. The continuous type controllers include state feedback controller, intermittent controller etc., whereas the discrete type controllers include impulsive controller, sample data controller etc.,. The impulsive controller was concentrated by most of the researchers due to it’s property that it can effectively save the bandwidth of a network which would thereby cut the costs of control or network in the case of the systems that are controlled by networks. Impulses have been widely used to obtain stability [28], [29] and synchronization [30] in NNs. In [28], INNs with impulses were considered and the stability for distributive delayed case was obtained exponentially. Robust stability was attained in Yu et al. [31] for INNs that have uncertainty, impulses and distributive-delays. The authors of [32] attained global convergence for INNs with impulses and delays that are time variant. In addition, the major adaptive control schemes are the centralized and the decentralized adaptive controls. The centralized adaptive control concerns with the dynamic behavior of the system and uses the parameters obtained from them, whereas the decentralized adaptive control can be used to deal the systems that have many complex interconnections [33], [34]. For example, the authors of Yu et al. [35] evaluated large scale system that has time delays using impulsive controllers and attained decentralized stabilization. The decentralized controller was also used in a game theory environment [36] so that it could adapt the dynamic changes in such a framework. Thus, we use centralized and decentralized adpative controllers along with impulses in this paper.
Lyapunov’s theory is a useful technique for analyzing and designing control systems dynamical behaviors [20]. Suppose the time derivative of the considered system is taken along the trajectories and results to be negative definite but the obtained state function is positive definite, then the system attains stability. Different positive and negative definite assumptions can be imposed on the Lyapunov function and the time derivative of the considered system, respectively, so that various dynamical behaviors of the system can be obtained. To remove the strictness of negative definiteness in Lyapunov functions, the concept of indefinite derivatives was established which would bring out more prominent results. Stability and synchronization of the NNs in terms of indefinite derivatives had been widely discussed in Cai et al. [37], Li and Li [38], Zhao et al. [39], Li and Liu [40], Ning et al. [41]. For example, in Cai et al. [37], the authors considered nonautonomous systems with discontinuity and obtained FXTST through indefinite derivatives. This article’s generalized Lyapunov function is relaxed to have an indefinite derivative almost everywhere along the system’s state trajectories. On the other hand, the standard yapunov function, must have a negative definite or semi-negative definite derivative everywhere. In [38], the authors investigated the input-to-state stability and integral-input-to-state stability of nonlinear impulsive systems by using Lyapunov method involving indefinite derivative and average dwell-time method. Stability analysis of linear time-varying time-delay systems was discussed in Zhao et al. [39] by using nonquadratic Lyapunov functions and functionals, asymptotic stability criteria were established in Li and Liu [40] for nonlinear impulsive dynamic systemsvia indefinite Lyapunov functions and the input-to-state stability and integral input-to-state stability of nonlinear systems was presented in Ning et al. [41], in which the stability conditions are more relaxed than others in that they employ an indefinite Lyapunov function rather than a negative definite one. Also in Taieb [42], the author dealt with nonlinear systems and obtained stability using Lyapunov derivative. Authors of [43] established FXTXY for INNs that have discontinuous activations using the Lyapunov–Krasovskii’s indefinite functional method. Motivated by the above discussions, we consider the discontinuous INNs and obtain FXTSY with centralized impulsive controllers through the Lyapunov indefinite derivatives. The main contributions of this paper are highlighted as follows:
- 1.
In this paper the discontinuous INNs are discussed with time delays and centralized impulsive controller. In this article, a more generalised variable transformation involving two parameters is used rather than the standard reduced-order method. Moreover, by employing Filipov solutions and set valued map theory discontinuous INNs are transformed into differential inclusions with differential inclusion theory.
- 2.
In comparison to the LKF methods’ stability results on INNs with continuous /discontinuous activations [44], [45], [46], the derivative of constructed LKF is indefinite, which is novel and more generalised.
- 3.
Furthermore, centralized impulsive controller is handled separately to acquire FXTSY using the indefinite Lyapunov functional theory. Finally, two numerical examples are given to examine the feasibility of the FXTSY conditions.
This paper is organized as follows: Preliminaries and model description of the considered model are given in Section 2. In Section 3, some results about the decentralized and centralized impulsive control law with indefinite derivative are investigated. In Section 4, two numerical examples are given to illustrate the effectiveness of the derived results. The following notations are fairly utilized through the paper. and represents the set of real numbers and dimensional Euclidean space respectively. Mes means that Lebesgue measure of set . represents the ball of center and radius The convex closure of a set is represented by .
Section snippets
Preliminaries and model description
In this paper the discontinuous type INNs with time-varying delay is studied and represented as follows:where is the state of the discontinuous INNs Eq. (1), is called an inertial term of system Eq. (1), is the damping coefficient, , is the feedback template coefficient, and are delayed and without delayed activation functions respectively,
Main results
Theorem 3.1 Suppose that Assumptions (2.9)–(2.11) are satisfied. If the following conditions holds, For the dirac delta function satisfies for all t ; for all ; and where a is a constant, and f is a continuous function on . The tunable variables and satisfies the inequalities
Numerical simulations
In this section, we present two simulation examples to validate the theoretical results of this paper. Example 4.1 In this example the delayed impulsive discontinuous INNs with 4 nodes where each node has three neurons was considered. The leader node is described as:where . The coefficient matrices of Eq. (32) are given as
Conclusion
In this paper, INNs having discontinuous activation functions along with impulses and time varying delays were considered and they were synchronized at FXT. Variable transformations that were suitable to convert the impulsive discontinuous INNs into first order impulsive differential equations was carried on initially. By using the inclusion theory and set-valued map concepts, the delayed discontinuous differential equations are transformed into the differential inclusions. Furthermore, by
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (49)
Applications of neural networks to digital communications—A survey
Signal Process.
(2000)- et al.
Stability and dynamics of simple electronic neural networks with added inertia
Phys. D
(1986) - et al.
Finite-time synchronization of inertial memristive neural networks with time-varying delays via sampled-date control
Neurocomputing
(2017) - et al.
Finite-time synchronization for fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays
Fuzzy Sets Syst.
(2020) - et al.
Dynamics of simple electronic neural networks
Phys. D
(1987) - et al.
Complex networks: structure and dynamics
Phys. Rep.
(2006) - et al.
Fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks with parameter uncertainties
Neurocomputing
(2021) - et al.
Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays
Neural Netw.
(2014) - et al.
Input-to-state stability of impulsive inertial memristive neural networks with time-varying delayed
J. Frankl. Inst.
(2018) - et al.
Dengqing, distributed-delay-dependent exponential stability of impulsive neural networks with inertial term
Neurocomputing
(2018)
Global lagrange stability for neutral-type inertial neural networks with discrete and distributed time delays
Chin. J. Phys.
Global exponential stabilization and lag synchronization control of inertial neural networks with time delays
Neural Netw.
Finite-time stabilization of memristor-based inertial neural networks with discontinuous activations and distributed delays
J. Frankl. Inst.
Exponential synchronization for inertial coupled neural networks under directed topology via pinning impulsive control
J. Frankl. Inst.
Generalized Lyapunov approach for functional differential inclusions
Automatica
Finite-time synchronization for fuzzy cellular neural networks with time-varying delays
Fuzzy Sets Syst.
Decentralized finite-time adaptive fault-tolerant synchronization tracking control for multiple UAVs with prescribed performance
J. Frankl. Inst.
Finite-time synchronization for fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays
Fuzzy Sets Syst.
Fixed-time synchronization of delayed complex dynamical systems with stochastic perturbation via impulsive pinning control
J. Frankl. Inst.
Pinning synchronization of linearly coupled delayed neural networks
Math. Comput. Simul.
Distributed-delay-dependent exponential stability of impulsive neural networks with inertial term
Neurocomputing
Stability of inertial BAM neural network with time-varying delay via impulsive control
Neurocomputing
Global convergence analysis of impulsive inertial neural networks with time-varying delays
Neurocomputing
Centralized and decentralized neuro-adaptive robot controllers
Neural Netw.
Cited by (8)
Robust exponential synchronization results for uncertain infinite time varying distributed delayed neural networks with flexible delayed impulsive control
2023, Mathematics and Computers in SimulationNovel finite and fixed-time stability theorems for fractional-order impulsive discontinuous systems and their application to multi-agent systems
2022, Results in Control and OptimizationFixed-time synchronization of time-varying coupled competitive neural networks with impulsive effects
2024, Neural Computing and Applications