This article addresses the issue of output synchronization via intermittent dynamic event-triggered sampled-data (IDETSD) security control of reaction-diffusion neural networks (RDNNs) under spatially local averaged measurements (SLAMs) subject to both delays and random deception attacks, where a Bernoulli distribution is utilized to describe whether channels suffer from the cyberattacks. An IDETSD security control method under SLAMs and random deception attacks is presented to achieve the output synchronization of delayed RDNNs. Compared with time-triggered intermittent sampled-data (SD) control strategies, a dynamic event-triggered (ET) mechanism to more effectively mitigate the impact induced by random deception attacks that intentionally tamper with the state transmission signals from sensors to controllers is introduced in this article. Moreover, new output synchronization criteria are established by applying an ET-dependent switched Lyapunov functional (LF) and inequality techniques. Then, the desired IDETSD controller is obtained by solving linear matrix inequalities (LMIs). To validate the efficacy of the proposed approach, simulation outcomes from two numerical studies are presented.

In this article, novel results on the maximality of discrete fractional Green's functions are established and corresponding explicit Lyapunov inequalities for delta fractional systems, with applications to chaos analysis and robust control design, are derived. For the proposed Riemann-Liouville fractional difference system with the delta boundary conditions, explicit expressions for the maximum values of the associated Green's function over its domain are obtained. These results lead to a refined Lyapunov delta-type inequality establishing a necessary condition for the existence of nontrivial solutions, where the lower bound explicitly depends on the maximum values of the fractional order and the Green's function. Furthermore, it is demonstrated that violation of this inequality implies the existence of nontrivial solutions and can induce chaotic behavior in fractional difference systems. For control applications, robust stability conditions for uncertain fractional systems are established and stabilizing state feedback controllers is designed. Finally, the numerical examples validate the emergence of chaos under inequality violation and confirm the control design's efficacy for robust stability.

Fixed-Time Stability Lemmas with Generalized Lyapunov Inequality Conditions: Applications to Leakage-Delayed Competitive Networks

In this paper, the fixed-time bipartite consensus of linear multi-agent systems (MASs) under event-triggered intermittent control (ETIC) is studied. In existing bipartite consensus, both cooperative and competitive relationships among agents are necessary. However, we focus on the scenario where all agents exhibit cooperative behaviour. Firstly, a new control protocol is devised by integrating the event-triggered strategy with the centralised sampled-data control (CSDC), and auxiliary functions are introduced to determine the protocol interval. Secondly, because the CSDC requires global information, to overcome this limitation, an improved ETIC embedded with distributed sampled-data control (DSDC) is proposed. Additionally, using Lyapunov stability theory and inequality methods, some sufficient conditions for fixed-time bipartite consensus of MASs are provided, while the Zeno phenomenon can be avoided. Ultimately, two numerical examples are employed to verify the theoretical conclusions.

In this article, the practical fixed-time (FxT) control of switched neutral Filippov systems (SNFSs) on networks is considered. The perturbation functions that can be discontinuous, and the neutral logics expressed by the difference operators, are addressed by new approaches. Several novel Lyapunov inequalities with indefinite functions are proposed, where detailed estimations of the settling-time (ST) are obtained by discussing the different values of the exponents of the Lyapunov functions, which can include the existing results. Considering that the states generally cannot converge to the origin accurately under finite-time control in real applications, practical FxT stability lemmas with indefinite functions are established for the first time, in which the bounded condition imposed on the indefinite function is more practical than the previously unbounded ones. By designing the adaptive control strategies, the FxT and practical FxT synchronization control are investigated based on the Lyapunov-Krasovskii functionals (LKFs), which show the delay characteristic via the adaptive update law containing delay values. Notably, the theoretical deficiency arising from the Lyapunov function when studying the FxT stability of real systems with delays by using the FxT stability lemmas with indefinite function is solved in a successful way. Finally, the validity of the main results is verified by numerical simulations on an electrical device containing an LC transmission line.

This article focuses on the issue of finite-time stability (FTS) of nonlinear systems with stabilizing and destabi-lizing impulses exhibiting proportional delays. Some Lyapunov-based local/global FTS criteria with settling-time estimation are established under a relaxed Lyapunov inequality condition. Specifically, two local FTS criteria are obtained by constructing an appropriate set of impulsive instant sequences that can be used to estimate the upper bound of the number of impulses experienced by the state before reaching zero. Furthermore, by utilizing the average impulsive interval (AII) approach, we expand the domain of attraction to a global scope in the case of stabilizing delayed impulses, while only local FTS can be achieved in the case of destabilizing delayed impulses. Our results indicate that under the relaxed Lyapunov condition, the time delay in stabilizing impulses can either promote or compromise the system stability, depending on the continuous dynamics of the system. In either case, sufficient conditions are established to guarantee the FTS of the system, regardless of the impact of the delayed impulses. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

This article considers a Lyapunov delta-type inequality with Green’s functions including fractional falling functions. We define a fractional difference problem of Riemann-Liouville type with a fractional boundary condition and, using the Green’s function, obtain the ordering property in a discrete domain. Moreover, we apply the properties of this function to find the existence of a delta Lyapunov inequality.

We study the synchronization of delayed reaction-diffusion neural networks (RDNNs) with Neumann boundary conditions, considering both distributed and discrete delays. Particularly, boundary sampled-data (SD) control is proposed to synchronize delayed RDNNs. In the proposed synchronization strategy, boundary SD control is based on boundary and distributed SD measurements. Based on the Lyapunov stability theory and inequality techniques, some synchronization criteria via the boundary SD control are proposed for delayed RDNNs. The boundary SD control gains are obtained by solving the conditions with linear matrix inequalities. Finally, a numerical example is presented to demonstrate the feasibility and effectiveness of the proposed method.

Stability and stabilization of discrete-time linear time-varying systems via Lyapunov difference equations and inequalities

In this article, new q-analogues of Lyapunov-type inequalities are presented for two-point fractional boundary value problems involving the Riemann–Liouville fractional q-derivative with well-posed q-boundary conditions. The study relies on the properties of the q-Green’s function, which is constructed to solve such problems and allows for the analytical derivation of the inequalities. These inequalities find application in two directions: establishing precise lower bounds for the eigenvalues of corresponding q-fractional spectral problems and formulating criteria for the absence of real zeros in q-analogues of Mittag-Leffler functions. The obtained results generalize classical and fractional Lyapunov inequalities, offering new perspectives for the analysis of stability and spectral properties of q-fractional differential systems. The relevance of the work is driven by the growing interest in q-calculus in discrete models, such as viscoelastic systems or quantum circuits, where discrete dynamics play a key role. The convenience of closed-form analytical expressions makes the results practically applicable. The research lays the foundation for further generalizations, including Caputo derivatives or multidimensional q-systems, which may stimulate new discoveries in discrete fractional analysis.

This paper addresses the synchronisation of multiple fuzzy neural networks with time delays. Firstly, a quantised iterative learning control strategy is constructed by the quantised state information among the neighbouring nodes to iteratively update the control and achieve the synchronisation of multiple fuzzy neural networks with time delays. Quantized state information is delivered to the communication channel as control inputs to reduce the bandwidth stress caused by continuous communication. Second, sufficient criteria for achieving the synchronisation of multiple fuzzy neural networks with time delays are derived in light of the Lyapunov functions and inequality techniques. The obtained results can be considered as an extension of relative works on the synchronisation of nonlinear systems. Finally, the validity of theoretical results are verified by means of a numerical example.

In this paper, the problem of global non-fragile and fixed-time output-feedback stabilisation is considered for a class of nonlinear systems with a lower-triangular form and uncertain measured noise. First, two Lyapunov matrix inequalities with uncertainties are presented. Then, a positive and a negative homogeneous system with unknown perturbations are given and proven to be globally asymptotically stable and finite-time stable, respectively, based on these two matrix inequalities. Besides, two homogeneous Lyapunov functions are constructed, and two important Lyapunov inequalities are derived. Second, by applying an auxiliary variable, the considered nonlinear system is augmented. Then, a non-fragile augmented output-feedback controller, incorporating positive and negative homogeneous terms, is presented to ensure the fixed-time stability of the closed-loop system. Finally, two simulation examples are used to verify the effectiveness of the proposed control scheme.

Decomposition method-based global Mittag-Leffler synchronization for fractional-order Clifford-valued neural networks with transmission delays and impulses.

Fixed-time synchronization of proportional delay memristive complex-valued competitive neural networks.

Novel Generalizations of Lyapunov and Wintner-Hartman Inequalities via the $$\digamma$$-Caputo Fractional Operator

This paper investigates the anti-synchronization problem of delay-coupled fractional memristor-based discrete-time neural networks within the T-S fuzzy framework via an event-triggered mechanism. First, fractional-order, coupling topology, and T-S fuzzy rules are incorporated into the discrete-time network model to enhance its applicability. Subsequently, a T-S fuzzy-based event-triggered mechanism is designed, which determines control updates by evaluating whether the system state satisfies predefined triggering conditions, thereby significantly reducing the communication load. Moreover, using diverse fuzzy rules enhances controller flexibility and accuracy. Finally, Zeno behavior is proven to be absent. Using the Lyapunov direct method and inequality techniques, we derive sufficient conditions to ensure anti-synchronization of the proposed system.Numerical simulations confirm the effectiveness of the proposed control scheme and support the theoretical results.

Under the influence of reaction-diffusion, fuzzy logic (fuzzy MIN and MAX feedback templates), and time-varying delays, the synchronisation problem for fractional-order gene regulatory networks is addressed in this article. First, novel adaptive boundary controllers are developed under Neumann boundary conditions, which can effectively reduce control costs using dynamic control gains and actuators installed at the boundary of the spatial domain. Then, through the designed adaptive boundary controllers, new sufficient criteria based on linear matrix inequality (LMI) are obtained to guarantee both asymptotic and finite-time stability of the error dynamic system by using the direct Lyapunov approach and inequality techniques, which ensure that the drive system is synchronised with the response system. Lastly, a simulation example illustrates the effectiveness of designed adaptive boundary control protocols.

ABSTRACTIn this article, the authors investigate the global and exponential dissipativity of quaternion‐valued inertial neural networks (QVINNs) with mixed time‐varying delays, without utilizing order reduction of inertial neural networks (INNs) and quaternion separation methods. Using innovative Lyapunov functional and inequality techniques, several fruitful sufficient criteria with multi‐parameters are derived for QVINNs to ensure global dissipativity (GD), which generalizes and refines existing results. This article estimates the attractive sets and exponentially attractive sets globally. Unlike previous studies in which quaternion‐valued neural networks (QVNNs) are separated into real‐valued neural networks (RVNNs) and INNs are reduced into first‐order systems, the foundation of this article rests upon approaches that diverge from the traditional methods of separation and order reduction. Unlike existing results on the GD of traditional neural networks (NNs) with bounded discrete time delays, this article focuses on INNs with unbounded discrete time‐varying delays, which is more realistic because neurons consider their entire past rather than partial history within bounded time delays. In general, in exponential stability, synchronization, and dissipativity results, researchers typically impose an upper bound on the rate of convergence , but in the present article, the authors investigate dissipativity criteria without such a restriction on the convergence rate in global exponential dissipativity (GED). Finally, to demonstrate the efficiency of our theoretical work, three consecutive examples are proposed to illustrate the effectiveness of the obtained results. The first two examples verify the proposed results, and the third one, related to QVNNs, redemonstrates the efficiency of storing true color image patterns.

This paper investigates delayed event-triggered impulsive control for simplicial complexes under stochastic hybrid cyber attacks, such as denial-of-service (DoS) and deception attacks. Firstly, leveraging the properties of d-simplices and link-dependent topology, an equivalent adjacency matrix is introduced to model D-simplicial complexes as a linear system. Secondly, a delayed event-triggered mechanism is developed to ensure efficient resource utilisation while preventing the Zeno phenomenon through a delayed waiting time. The proposed control strategy framework combines an event-triggered mechanism with topology-dependent impulsive control, where impulsive instants are dynamically determined. Additionally, the impact of hybrid attacks is modelled by incorporating DoS and deception attacks that follow a Bernoulli distribution. Thirdly, sufficient conditions for mean-square exponential synchronisation of simplicial complexes under attacks are derived using Lyapunov functions and inequality recursion techniques. Finally, numerical simulations and examples are presented to validate the effectiveness and robustness of the proposed approach.

In this paper, we address the challenge of achieving H ∞ synchronisation within a specific class of complex networks with multi-weights (CNMWs) operating under both fixed and switching topologies. To tackle this challenge, several key steps are taken. Initially, two distinct CNMWs with different topological structures are formulated. Subsequently, employing Lyapunov functionals and inequality techniques, a set of sufficient conditions is derived to ensure robust H ∞ synchronisation. The study then proceeds to the design of two distinct Proportional-Derivative (PD) controllers, which are employed to facilitate the analysis of H ∞ synchronisation in this network context. Finally, the effectiveness of the proposed methodology is confirmed through the presentation of two illustrative numerical examples.