Fixed-time synchronization of multilayered complex dynamic networks via quantized variable-gain saturated control

This article is concerned about fixed-time (FT) synchronization of spatiotemporal networks (STNs) with the Robin boundary condition. Above all, a switching-type FT stability theorem and an integral inequality are established, which provide a novel theoretical tool for the rigorous analysis of FT control in STNs. Subsequently, three kinds of nontrivial power-law controllers are developed which are separately acted on the interior, the boundary, and the whole of the spatial domain. Based on these control schemes and Lyapunov-like method, several flexible criteria are obtained to achieve FT synchronization of STNs, and the upper bound of the synchronization time is explicitly estimated. Note that, the derived results here are also perfectly applicable to STNs with Neumann or Dirichlet boundary condition. Several illustrate examples are presented at final to confirm the developed controllers and criteria.

This paper investigates fixed-time (FXT) synchronization in switched coupled neural networks (SCNNs) building on an improved FXT stability theorem. First, the FXT stability criterion and the settling time (ST) are enhanced by employing the method of reduction to absurdity. As a key application, FXT synchronization is developed for SCNNs with both discontinuous and continuous activation functions. By designing a discontinuous controller incorporating switched signals and employing the theory of differential inclusions, several criteria are established to guarantee FXT synchronization in the considered networks. Ultimately, two numerical illustrations are provided to showcase the efficacy and correctness of the theoretical findings.

This paper investigates modified fixed-time synchronization (FxTS) of complex networks (CNs) with time-varying delays based on continuous and discontinuous controllers. First, for the sake of making the settling time (ST) of FxTS is independent of the initial values and parameters of the CNs, a modified fixed-time (FxT) stability theorem is proposed, where the ST is determined by an arbitrary positive number given in advance. Then, continuous controller and discontinuous controller are designed to realize the modified FxTS target of CNs. In addition, based on the designed controllers, CNs can achieve synchronization at any given time, or even earlier. And control strategies effectively solve the problem of ST related to the parameters of CNs. Finally, an appropriate simulation example is conducted to examine the effectiveness of the designed control strategies.

This article is concerned with the problem of fixed-time (FXT) and preassigned-time (PAT) synchronization for discontinuous dynamic networks by improving FXT stability and developing simple control schemes. First, some more relaxed conditions for FXT stability are established and several more accurate estimates for the settling time (ST) are obtained by means of some special functions. Based on the improved FXT stability, FXT synchronization for discontinuous networks is discussed by designing a simple controller without a linear feedback term. Besides, the PAT synchronization is also explored by developing several nontrivial control protocols with finite control gains, where the synchronized time can be prespecified according to actual needs and is irrelevant with any initial value and any parameter. Finally, the improved FXT stability and the synchronization for complex networks are confirmed by two numerical examples.

Finite-time and fixed-time bipartite synchronization of complex networks with signed graphs

Fixed-time output synchronization of multi-layer complex networks under dynamic event-triggering control

This article considers the fixed-time stability (FXTS) issue for discontinuous system described by differential equation (DE) with time-varying parameters. Using the tool of differential inclusion (DI), several improved FXTS criteria and estimation formulas of settling time (SET) are derived by employing a relaxed Lyapunov method. The special novelty of the developed Lyapunov method is that the derivative of Lyapunov function possesses indefiniteness. As an important application, the established FXTS theorems are utilized to deal with fixed-time (FXT) synchronization issue for fuzzy neural networks (FNNs) possessing state discontinuity, where the fuzzy operation is used in the synaptic law computing, and one typical time-varying switching control protocol is designed. Moreover, a series of estimations of SET for FXT synchronization are given out. Finally, the simulation examples are present to substantiate the validity of the obtained results.

Synchronizability of multi-layer small-world dynamical networks

Exploring synchronizability of complex dynamical networks from edges perspective

Finite-time synchronization of complex dynamical networks under delayed impulsive effects

SummaryThe issues of cluster synchronization and state estimation for a class of nonlinear complex dynamical networks are concurrently focused in this study. In precise, led by the design of complex dynamical networks, a proportional integral‐based observer is implemented for achieving robustness, by the virtue of which the states of examined networks are estimated. Thereupon, a proportional integral observer‐based hybrid triggered control scheme is proposed, wherein, the switch‐over between time‐ and event‐ triggered scheme is governed through a Bernoulli distributed random variable. Further, with the application of mixed and passivity performance technique, the external disturbances for the addressed complex dynamical networks are reduced. Based on the Lyapunov stability theory, the requisite sufficient conditions in respect of linear matrix inequalities are provided which ensures the discussed complex dynamical network's synchronization phenomenon. In a further, the presented criteria are exploited for the attainment of required controller, proportional and integral gain matrices. In conclusion, the applicability of proposed results and efficacy of the undertaken strategy are evidently proven with the illustration of two numerical examples with Chua's circuit and Goodwin oscillator model.

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

Complex dynamical networks consist of a large set of interconnected nodes with each node being a fundamental unit with detailed contents. A great number of natural and man-made complex networks such as social networks, food networks, neural networks, the World Wide Web, computer networks, electrical power grid and so on, can be effectively modelled by complex dynamical networks. Synchronization of a group of dynamical nodes in a complex network topology is one of the most interesting and significant phenomenon in complex dynamical networks. In this section, we shall discuss impulsive synchronization of the discrete delay networks. By employing the impulsive control method and the Razumikhin-type results obtained in previous chapter, two criteria for exponential synchronization will be derived.

Function projective synchronization in complex dynamical networks with time delay via hybrid feedback control

Finite-time synchronization of complex dynamic networks with impulsive control and actuator saturation: An LMI approach