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This article deals with passivity of coupled reaction–diffusion neural networks (CRDNNs) with multi-proportional delays and impulses. Two aspects are mainly considered: one is that the coupling term of the system depends on the delay, and the other is that the coupling term of the system is delay-independent. In order to simplify the complex system, the Kronecker product and its properties are applied to rewrite the complex system. According to whether the transfer function is a multi-proportional delayed function or a single proportional delayed function, several seemly Lyapunov Krasovskii functionals (LKFs) are constructed. Then, through the use of Kronecker product, Green's formula and inequality analysis method, several criteria of input-strictly passivity (ISP) and output-strictly passivity (OSP) of the studied systems are attained, which all rely on proportional delayed factors. In addition, these criteria are exhibited in the form of matrix inequalities and can be checked with conventional software. Eventually, the passivity criteria are validated by two numerical examples and simulations. • Without the use of nonlinear transformations, the structure of paper is clearer. • Kronecker product and properties simplify the computational process of the paper. • The influence of impulses is considered, which is better describe the studied system. • These passive criteria are exhibited in the form of LMI and are easy to check. [ABSTRACT FROM AUTHOR]

This article is concerned with the problems of high-order moment interval stability/stabilization and observability of stochastic impulsive T–S fuzzy systems (SITSFSs). Firstly, the definition of high-order moment interval stability is proposed of SITSFSs in this paper. In consideration of the connection between pole placement and performance of systems, sufficient conditions for the distribution of the eigenvalues of systems within a defined interval are provided, serving as the criterion for the interval stability in high-order moment of SITSFSs. Different from the general concept of stability, besides determining the stability in high-order moment of the system, high-order moment interval stability can also describe the dynamic performance in high-order moment of systems. Then, the proposed approach in combination with fuzzy control theory provides a reliable and effective method for designing an interval state feedback controller that can ensure the stability of systems and regulate convergence rate of systems. Furthermore, moment observability of SITSFSs has also been addressed in this article. Finally, the numerical simulation verifies the validity and suitability of the proposed method. • A novel 2 p th moment interval stability/stabilization criterion is put forward. • Using the H -representation method, an extended system model is constructed. • A new fuzzy controller design method is proposed to reflect the dynamic and static performances of the system in high-order moment. [ABSTRACT FROM AUTHOR]

A special kind of Poisson stable trajectory, which is called unpredictable and leads to sensitivity in the quasi-minimal set, was proposed by Akhmet and Fen (2016) for semiflows. In the present paper we carry this finding one step further by defining a new kind of trajectory, called asymptotically unpredictable. We prove that such motions also lead to sensitivity in the dynamics. This feature is now achieved under a weaker hypothesis. Benefiting from the Bebutov dynamical system, continuous asymptotically unpredictable functions on the real axis are defined, and it is shown that the set of these functions properly includes the set of unpredictable ones. Moreover, results on producing new asymptotically unpredictable functions from a given one are obtained. The existence and uniqueness of bounded asymptotically unpredictable solutions of quasi-linear systems are also investigated, and an application to Hopfield neural networks is provided. • A new type of motion, called asymptotically unpredictable, is introduced on semiflows. • It is proved that asymptotically unpredictable trajectories lead to sensitivity in the dynamics. • Sensitivity is achieved under a weaker hypothesis compared to unpredictable motions. • Continuous asymptotically unpredictable functions are defined via the Bebutov dynamical system. • Asymptotically unpredictable solutions of quasi-linear systems are additionally discussed. [ABSTRACT FROM AUTHOR]

In this paper, stochastic integral input-to-state stability (SiISS) is studied for stochastic delayed networked control systems (SDNCSs). With the assistance of Lyapunov–Krasovskii functional, as well as stochastic analysis and inequality techniques, we establish a Lyapunov-type criterion that guarantees SiISS for SDNCSs. What is more, another sufficient criterion is proposed by means of coefficients in SDNCSs and it is proved to be very useful in practice. Based on the main theorems obtained, we investigate SiISS of two typical SDNCSs in details including stochastic coupled van der Pol oscillators on networks with time delay and stochastic Hopfield neural networks with time delay as applications. Last but not least, the effectiveness and applicability of our results are illustrated via corresponding numerical examples and simulation outcomes. • We provide a systematic approach to construct a Lyapunov–Krasovskii functional for stochastic delayed networked control systems. • Some novel criteria are developed to ensure stochastic integral input-to-state stability for stochastic delayed networked control systems. • Two examples and their simulation results are provided to show the effectiveness and feasibility of our method. [ABSTRACT FROM AUTHOR]

The objective of the paper is to give a unitary approach to nonconservative conditions of the equilibrium stability. The focus is on a real world system defined by a switched linear mathematical model having control delay. A control synthesis based on predictive state feedback is first applied to compensate the actuator delay. This results in a closed loop switching system free of delay. Then, a theorem giving conservative sufficient conditions of global uniform exponential stability for a switched system is proved. Next, a theorem expressing necessary and sufficient stability conditions is cited, in which the differences compared to the proved theorem are highlighted, but whose mechanism could not be applied in practice. Finally, it is shown how the problem of nonconservativeness can be solved starting from an important theorem given in the literature that uses the idea of homogeneous polynomial Lyapunov functions originating in the Hilbert's famous 17th Problem. The paper investigates the complexity of the method in terms of numerical computation volume, and offers a way to circumvent the conflict between the method and the mathematical model. In this manner older and newer results are presented in a unitary concept and in a selfcontained approach. Numerical simulations are applied to a real-world model: an active vibration control for a physical model of an intelligent airplane wing in a wind tunnel. The substantial result of the simulations is the minimum guaranteed dwell time that characterizes the equilibrium stability of the switching system with control delay. • Development of a unitary approach to the nonconservative stability of a real world switched linear system with control delay. • Use of a state-predictive feedback to convert the switched system with control delay into a delay free switched system. • Solution for the nonconservative LMI conditions by using SOS programming algorithms and algorithm given in Fig. 3. • Use for the first time of the theory of nonconservativeness based on SOS programming algorithms to a real-world object. • Achievement of the minimum guaranteed dwell time characterizing equilibrium stability of switching system with control delay. [ABSTRACT FROM AUTHOR]

This paper comes up with an novel augmented Lyapunov–Krasovskii functional (LKF) approach to process finite-time passivity-based H ∞ non-fragile state feedback control issue of Takagi–Sugeno fuzzy system (TSFS) via taking the effects of both time-varying delays and actuator faults. Designing a non-fragile controller to guarantee that TSFS is finite-time bounded (FTB) and meet finite-time mixed H ∞ and passive performance is the focus of this paper. On the basis of the new augmented LKF and improved boundary techniques, sufficient delay-dependent conditions are inferred to acquire the desired results of the TSFS. Furthermore, the derived finite-time stability conditions are derived in terms of linear matrix inequalities (LMIs) and the controller gain matrices are confirmed through solving LMIs. Subsequently, three examples are addressed to certify the applicability and advantage of the put forward methodology. • The LKF selected in this paper is composed of non-integral term, single-integral term, double-integral term and triple-integral term which are augmented by introducing additional vectors including more delay and derivative information. • Via constructing augmented LKF and employing extended Wirtinger's inequality and Jensen's inequality to process the integral term, the novel stability criteria on delay-dependent are set up for TSFS. • Via dealing with LMIs, the non-fragile controller is devised for TSFS under the premise of actuator faults to ensure that the TSFS is FTB and meet finite-time mixed H ∞ and passive performance, then the numerical examples are presented to illustrate the advantage of the developed criteria. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential stability and stabilization problems are investigated for memristive neural networks (MNNs) with stochastic disturbances, spatial diffusions and distributed delays. The spatial diffusions are not assumed to be symmetric and distributed delays are relaxed to be unbounded. Then, the presented MNNs are modeled as a class of stochastic partial systems with hybrid delays. Based on nonsmooth analysis, the Lyapunov–Krasovskii functional and inequality approach, some simple algebraic criteria are derived for the global exponential stability as well as the stabilization via two kinds of designed feedback controllers. The derived criteria are easily verified and the obtained results are available for other delayed systems with or without stochastic disturbances and spatial diffusions. Finally, the stability and stabilization results are validated by numerical simulations. • The stochastic model with spatial diffusions and distributed delays is discussed. • The derived results are generalized and they contain previous ones as special cases. • The method is available for other stochastic delayed partial differential systems. • The space set of the spatial diffusions in our model is not symmetrical. • The distributed delays are unbounded. [ABSTRACT FROM AUTHOR]

The Lyapunov direct method is a powerful tool for analyzing the stability of nonlinear dynamical systems. However, most existing research has primarily focused on asymptotic stability, neglecting the crucial aspect of boundedness. In light of this, this paper aims to address the issues concerning the generalized Lyapunov stability theory of distributed-order systems and its application to ensure uniform boundedness and global attractiveness for distributed-order Hopfield neural networks (DOHNNs) with time-varying external input. To achieve this, several essential inequalities that form the foundation of our main work are proposed. Subsequently, two stability criteria in the form of generalized distributed-order Lyapunov theorem are derived by employing Laplace transform techniques. Meanwhile, building upon the proposed inequalities and Lyapunov theorem, sufficient conditions that guarantee the uniform boundedness and global attractiveness of DOHNNs with time-varying external input are established. By the way, all our proposed conclusions are presented in both continuous-time and discrete-time cases to present the distinction and relationship between these two cases. Finally, various simulation examples are designed meticulously to validate the correctness of our proposed theorems and the effectiveness to real-world applicability of our approach. • A novel inequality on the derivative of discrete distributed-order is proposed to establish the dynamical theories of DOHNNs. • The theory of distributed-order Lyapunov stability for continuous-time systems has undergone improvements, and its applicability has been expanded to more general functions and systems, thereby broadening its range of use. Furthermore, it has been generalized to discrete-time distributed-order systems to facilitate the analysis of stability conditions for both continuous-time and discrete-time distributed-order neural networks. • Sufficient conditions for the uniform boundedness and attractiveness of the continuous-time and discrete-time DOHNNs with time-varying external inputs have been established, filling in the blank for the application of distributed-order systems. [ABSTRACT FROM AUTHOR]

Nowadays, online information is disseminated in multiple languages on different platforms, and research on the dissemination trend and control of online rumors is a topic worth discussing. In the bilingual environment, we study the S 2 I S rumor propagation model with time delay and saturated conversion function which implies the limitation of resources in homogeneous and heterogeneous networks. First and foremost, the analysis of the boundedness of the solutions and the existence of the rumor-spreading equilibrium points is given in our work, and we figure out the basic reproduction number. Next, we conduct the local stability analysis of the rumor-free equilibrium point and the rumor-spreading equilibrium points according to the characteristic equation. The global stability of the rumor-free equilibrium point is shown based on the Lyapunov stability theory. Besides, we study the forward and backward bifurcation phenomenon and optimal control problem in theory, which is seldom discussed in the bilingual environment. Finally, numerical simulations are performed to verify the correctness of our main results. • This paper establishes a S2IS rumor propagation model with saturated conversion function. • Stability analysis and bifurcation phenomenon are investigated in theory. • Optimal control strategy is theoretically given by studying optimal system. [ABSTRACT FROM AUTHOR]

In this paper, the stabilization problem of stochastic Lévy noise coupled systems with time delays and regime switching diffusions (SLCSTR) is researched via intermittent control with a time delay (ICTD). It is worth pointing out that both Lévy noise and regime switching diffusions are considered into coupled systems. Furthermore, different from previous intermittent control based on current state, ICTD combining the benefits of aperiodically intermittent control and delay feedback control is based on past state. Meanwhile, the stabilization problem of SLCSTR is investigated via delay feedback control. Then, by Lyapunov method, graph theory and some techniques of inequalities, some stabilization criteria of SLCSTR in the sense of exponential stability in mean square are obtained. In addition, we investigate the stabilization problem of stochastic delayed coupled oscillators with Lévy noise and regime switching diffusions as a practical application of our theoretical results. Finally, numerical simulations are given for illustration. • A novel ICTD is designed to solve the stabilization issue of CSs, which is more practical. • Compared with the previous delay controls, time delay of ICTD and DFC in this paper is time-varying. • we consider a new SLCSTR, which has wider scope of applications compared to the existing results. • The stabilization criterion shows that the control law depends strongly on the coupling strength, the perturbed intensity of noises, time delays and the maximum rest time ratio. [ABSTRACT FROM AUTHOR]

This paper studies the synchronization of coupled neural networks (NNs) using static pinning impulsive control, which incorporates stability delays and event-triggered coupling mechanism. The term "static pinning" refers to a fixed set of pinned nodes throughout the control process. The pinned layer and follower layer are employed to depict the hierarchical structure generated by each static pinning strategy. Based on hierarchical structure, the time delay information present in the impulses is effectively extracted through the static pinning control and integrated into the synchronization of the pinning layer, and follower layer, utilizing an event-triggered coupling mechanism. The study establishes flexible criteria that ensure synchronization for coupled NNs while excluding the Zeno behavior. In addition, the pinning layer is equipped with time-delay free impulsive control input with asynchronous actuation, and the synchronization of coupled NNs are also obtained. The results show that the delays in the pinning impulsive control input contribute significantly to the synchronization of pinning layer. Finally, the simulation results are given to illustrate the effectiveness of the control method. • A new multilayer method is proposed to implement static pinning. • Pinning layer is designed with delayed impulsive control input. • Sufficient conditions ensuring synchronization of coupled NNs are presented. [ABSTRACT FROM AUTHOR]

This paper mainly studies stability of delayed nonlinear systems (DNSs) under state-dependent delayed impulses (SDDIs). Firstly, under some appropriate hypotheses, it is proved that the pulse phenomenon can be avoided. Then, by the method of comparison principle, new theoretical frameworks are established for stability of DNSs under state-dependent delayed impulsive control and delayed impulsive interference, respectively. Furthermore, the proposed criteria are applied in the stability of delayed neural networks (DNNs) under SDDIs. Finally, examples are provided to illustrate the correctness of obtained results. • Compared with [1,2] , the SDDIs are considered. Sufficient conditions are given to eliminate the pulse phenomenon and to make the solution of system intersect with every impulsive surface once and only once. • The delayed impulsive sequence depends on the state of the DNSs. Some stability results are proposed for DNSs under state-dependent delayed impulsive control and delayed impulsive interference, respectively. • The obtained theoretical results are applied to DNNs with SDDIs. It should be noted that delays exist in both the state-dependent impulse and the system, which extend the existing models [1,2]. [ABSTRACT FROM AUTHOR]

Highlights • We discuss the decay synchronization problem of CMTDNs. • We design a novel nonlinear feedback controller and a Lyapunov function. • Some sufficient conditions are derived to ensure the decay synchronization of CMTDNs/CMDNs. • The main results can easily extended to exponential synchronization, polynomial synchronization and logarithmic synchronization. • Two simulations are shown to present the correctness of the main results. Abstract In this paper, we discuss the decay synchronization of complex multi-links time-varying dynamic networks (CMTDNs) for the first time. The exponential synchronization, polynomial synchronization and logarithmic synchronization can be seen as the special cases of the decay synchronization. CMTDNs are more general and realistic than the complex network with the single-link. By virtue of the definitions of the ψ -type function and ψ -type stability, we give the definition of the decay synchronization of the drive-response CMTDNs. Some novel and easily validated sufficient conditions are deduced to ensure the decay synchronization of drive-response CMTDNs based on the Lyapunov stability theory with the help of a novel nonlinear feedback controller and a new Lyapunov–Krasovskii function. Finally, two numerical simulations have been given to verify the correctness of main results. [ABSTRACT FROM AUTHOR]

In this paper we study a three-dimensional tumor–immune system interaction model consisted of tumor cells, activated T cells, and immune checkpoint inhibitor anti-PD-1. Based on the uncontrollable character of tumor cells in the absence of immune response and treatment, the growth of tumor cells is assumed to be exponential. We discuss the distribution of equilibria qualitatively and study the stability of all possible equilibria with and without anti-PD-1 drug. When no drug is applied, the model has a tumor-free equilibrium and at most one tumorous equilibrium. Biologically, there exists a threshold d T 1 for the death rate d T of T cells: when d T ≥ d T 1 tumor cells will keep growing; when d T < d T 1 tumor cells may be eradicated for some positive initial values and keep growing for some other positive initial values. For the case with anti-PD-1 treatment, the model has at most five tumor-free equilibria and two interior equilibria. Our analysis indicates that there exists a threshold γ A 1 for the intravenous continuous injection γ A : when γ A ≤ γ A 1 the fate of tumor cells is the same as the case with no drug applied; when γ A > γ A 1 the model may exhibit bistable phenomena and periodic orbits. Furthermore, we establish the existence of local Hopf bifurcation around the interior equilibrium and determine the stability of the bifurcating periodic orbits. Our simulations show that the model exhibits a stable periodic orbit which implies the long term coexistence and balance of the tumor and immune system. • Study a tumor–immune system interaction model with an immune checkpoint inhibitor. • Discuss the number and stability of all possible equilibria. • Establish the existence of Hopf bifurcation and the stability of the periodic orbit. • Carry out numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

• In this paper, we investigate the chaotic dynamics of a Hopfield neural network under different types of external stimulus. • The research results show that the neural network can exhibit different dynamical attractors under different external stimulus. • Particularly, the neural network simultaneously stimulated by electromagnetic radiation and multi-level-logic pulse can generate multi-scroll attractors previously unobserved in Hopfield type neural networks. • Furthermore, the Hamilton energy of the neural network in different cases is calculated and analyzed, and it is found that lower average Hamilton energy can be detected in the neural network when complexity of external stimulus is enhanced. • Finally, mathematical models of the neural network under with different stimulus are physically implemented by using simple analog circuit. Biological nervous system is very sensitive to external disturbances, and appropriate stimulus is beneficial for improving neural function in the neural system. In this paper, the effect of different external stimuli on chaotic dynamics in a Hopfield neural network with three neurons is explored. Mathematical model of the neural network is respectively established under three different cases, namely without external stimulus, with only electromagnetic radiation stimulus, and with both electromagnetic radiation stimulus and multi-level-logic pulse stimulus. Under the three cases, equilibrium points, stabilities, and attractors of the neural network are investigated carefully. The research results demonstrate that the neural network with periodic attractors can induce abundant chaotic attractors by imposing electromagnetic radiation on its one neuron. And when this neuron is simultaneously stimulated via electromagnetic radiation and multi-level-logic pulse, the neural network can produce complex multi-scroll attractors previously unobserved in Hopfield-type neural networks. Numerical results are verified by hardware experiments, effectively. Furthermore, based on the Helmholtzâs theorem, the Hamilton energy of the neural network is calculated and analyzed. It is found that lower average Hamilton energy can be detected in the neural network when complexity of external stimuli is enhanced. These new findings could offer a new insight into the occurrence mechanism of some neurological diseases. [ABSTRACT FROM AUTHOR]