Impulsive Delay Research Articles

Asymptotical stability of stochastic delayed time-varying systems with stochastic delayed impulses

This paper investigates the asymptotical stability of stochastic delayed time-varying systems (SDTVS) with stochastic delayed impulses, which is a challenging and important problem in impulsive control theory. We utilise the impulsive density function (IDF) without linear constraints to estimate the number of impulses and the size of impulsive delay, which can better capture the impulsive effects than the conventional average impulsive interval (AII). We also relax the constraints between impulsive delay and the delay in the continuous dynamics by comparing their sizes, rather than requiring them to satisfy certain inequalities. We employ the comparison principle and the Razumikhin condition to derive Razumikhin-type stability results for our systems. We illustrate our results with two numerical examples that demonstrate their effectiveness and applicability.

Exponential Synchronization of Coupled Inertial Neural Networks With Hybrid Delays and Stochastic Impulses.

The synchronization problem of the coupled delayed inertial neural networks (DINNs) with stochastic delayed impulses is studied. Based on the properties of stochastic impulses and the definition of average impulsive interval (AII), some synchronization criteria of the considered DINNs are obtained in this article. In addition, compared with previous related works, the requirement on the relationship among the impulsive time intervals, system delays, and impulsive delays is removed. Furthermore, the potential effect of impulsive delay is studied by rigorous mathematical proof. It is shown that within a certain range, the larger the impulsive delay, the faster the system converges. Numerical examples are provided to show the correctness of the theoretical results.

Stability analysis of aperiodic impulsive delay systems using sum of squares approach

AbstractThe stability problem of aperiodic impulsive delay systems with unstable continuous and discrete dynamics is addressed in this article. New delay‐dependent stability conditions are proposed by employing a clock‐dependent Lyapunov–Krasovskii‐like functional composed of a Lyapunov–Krasovskii functional and a high‐order polynomial matrix function. The stability conditions are then solved using sum of squares programming techniques, yielding less conservative results than the looped‐functionals approach. Finally, the simulation results demonstrate the effectiveness of the results.

Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

Research in impulsive delay differential equations has been undergoing some exciting growth in recent times. This to a large extent can be attributed to the quest by mathematicians in particular and the science community as a whole to unveil nature the way it truly is. The realization that differential equations, in general, and indeed impulsive delay differential equations are very important models for describing the true state of several real-life processes/phenomena may have been the tunic. One can attest that in most human processes or natural phenomena, the present state is most often affected significantly by their past state and those that were thought of as continuous may indeed undergo abrupt change at several points or even be stochastic. In this study, a special strictly ascending continuous delay is constructed for a class of system of impulsive differential equations. It is demonstrated that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. The integral equivalence of the formulated system of equations is also obtained via a scheme similar to that of Perron by making use of certain assumptions.

Fractional exponential stability of nonlinear conformable fractional-order delayed systems with delayed impulses and its application

This article studies the fractional exponential stability (FES) of nonlinear conformable fractional-order delayed systems (CFODSs) and the fractional exponential synchronization of conformable fractional-order delayed inertial neural networks (CFODINNs), both with delayed impulses (DIs). Under the conformable fractional-order derivative framework, a novel Halanay inequality is established by generalizing the average impulsive interval (AII), which is a key basis for further investigations. Additionally, based on the proposed inequality, the requirement of the magnitude relationship among the system delay, the impulsive intervals, and the impulsive delays is removed. Moreover, using the improved average impulsive delay (AID), a relaxed FES criterion for nonlinear CFODSs with DIs is derived. Furthermore, as an application of the obtained theoretical results, the fractional exponential synchronization of CFODINNs with DIs is studied. Finally, simulations are given to illustrate the validity of our results, where the positive effect of impulsive delays is verified.

Synchronization of nonlinear neural networks with hybrid couplings and uncertain time-varying perturbations: A novel distributed-delay impulsive comparison principle

This paper investigates the synchronization of nonlinear drive-response neural networks subject to uncertain time-varying perturbations, non-delayed coupling, and distributed delay coupling. To address the influence of distributed and discrete delays on the system, we establish a novel impulsive comparison principle, extending the Halanay inequality. By leveraging Lyapunov stability theory, we derive sufficient conditions for the exponential synchronization of the neural networks using a delayed impulsive controller with historical status information. This approach relaxes the conventional constraint that impulsive delays must be smaller than impulsive intervals, thereby generalizing existing synchronization results for distributed delay networks. Numerical simulations for chaotic neural networks validate the theoretical results and demonstrate the sensitivity of the control gain matrix.

On Stability for Non-Instantaneous Impulsive Delay Differential Equations

On Stability for Non-Instantaneous Impulsive Delay Differential Equations

Delay-compensatory synchronization on uncertain chaotic neural networks via average delayed impulsive gains

This paper mainly studies the exponential synchronization issue of chaotic neural networks with parametric uncertainties and time-varying delays via hybrid impulsive control and delay compensatory strategy. Considering the hybrid impulses and flexible delays in the controller, the idea of average delayed impulsive gain (ADIG) is introduced to quantify the effects of hybrid impulses from a comprehensive viewpoint, i.e., not only the desynchronizing/synchronizing delay-free impulses but also the desynchronizing/synchronizing delayed impulses are discussed. Meanwhile, the impulsive delays are fetched and integrated to compensate the desynchronizing impulses dynamically, which removes certain strict restrictions on the impulsive gains and impulsive delays compared with existed works. Correspondingly, some relaxed sufficient criteria for exponential synchronization are derived by jointly utilizing the concept of average delayed impulsive gain, the delay compensatory scheme and the mathematical induction methodology. Ultimately, two examples with some comparisons are given to verify the applicability and superior performance of the obtained results.

Input-to-state hybrid impulsive formation stabilization for multi-agent systems with impulse delays

This paper addresses the input-to-state formation stabilization problem of nonlinear multi-agent systems within a hybrid impulsive framework, considering delay-dependent impulses, strong nonlinearity, and deception attack signals. By leveraging Lyapunov functionals, impulsive comparison theory, average impulsive interval methods, and graph theory, we develop novel criteria for possessing locally input-to-state and integral input-to-state formation stabilization across different impulse sequence classes. These criteria are expressed in terms of continuous/impulsive feedback gains, time delay size, nonlinearity strength, uniform upper bound of impulsive interval, and length of average impulsive interval. Notably, the design of control impulses benefit the destabilizing continuous dynamics in the formation stabilization process. To demonstrate the effectiveness and validity of the analytical results, we provide numerical simulation examples involving various types of external attack signals.

Exponential stability of switched systems with state-dependent delayed impulses via B-equivalent method

This paper studies the stability problem of switched systems with state-dependent delayed impulses (SDDIs), where switching times satisfy the definition of average dwell-time. Firstly, the pulse phenomenon of the systems with SDDIs is avoided under some necessary assumptions. Subsequently, the state-dependent delayed impulsive switched systems can be transformed into the corresponding time-dependent ones by using the B-equivalent method. Furthermore, the stability of stable and unstable systems with time-dependent delayed impulses (TDDIs) are analyzed, where the limitation that the impulsive delays are less than the impulsive intervals is relaxed. Based on the equivalence of the original system and the corresponding switched system with TDDIs, some new stability conditions for the switched systems with SDDIs are derived. Finally, the theoretical results are applied to a switched neural network with SDDIs to demonstrate the validity of the results.

Exponential synchronization of stochastic coupled neural networks with stochastic delayed impulsive effect

This paper studies the synchronization of stochastic coupled neural networks (SCNNs) with stochastic delayed impulses, which are more realistic and complex than the deterministic ones. Unlike previous works, we incorporate two distinct types of stochastic factors into the model: random noise affecting the systems continuous dynamics, and stochastic impulsive intensity applied at discrete impulsive instances. We extend the differential inequality with stochastic delayed impulses to stochastic systems without requiring that impulsive delay to be smaller than impulsive interval. We also model the impulsive intensity as a discrete-time Markov chain. By using the generalized inequality, graph theory, and stochastic analysis techniques, we derive a synchronization criterion for SCNNs with stochastic delayed impulses. We present numerical examples to demonstrate the effectiveness of our results.

Robust Stability Analysis of Switched Neural Networks with Application in Psychological Counseling Evaluation System

In this work, the effectiveness and stability of psychological counseling are evaluated using the switched complex-valued neural networks (SCVNN) model, which includes parameter disturbances, impulsive perturbations, variable and continuously distributed delays in the system state, and impulsive delay. How to analyze and judge the stability of the network simply and effectively is the primary prerequisite for its successful application. Therefore, we explore the dynamic behavior of SCVNN with both variable and distributed delays along with impulsive effect. Initially, the proposed conditions for the existence and uniqueness of equilibrium in SCVNN are presented. Subsequently, employing the inequality technique and impulsive average dwell time approach, sufficient conditions for the robust exponential stability of SCVNN under both arbitrary and restricted switching are obtained. Lastly, the psychological counseling evaluation system (PCES) is established, and a simulation example is used to verify the correctness and effectiveness of the presented findings.

Relative controllability for conformable impulsive delay differential equations

Abstract In this paper, we mainly study a class of conformable impulsive delay differential equations (CIDDEs). We first define a conformable impulsive delayed matrix function, and construct an explicit solution for linear CIDDEs by virtue of variation of constants method. Subsequently, based on impulsive delayed Grammian matrix, we study the relative controllability for the addressed linear equations. Moreover, with the help of Krasnoselskii’s fixed point theorem, relative controllability for the considered semilinear equations is proposed. Finally, two examples with numerical simulations are given to illustrate the main results.

Exponential stability of impulsive random delayed nonlinear systems with average-delay impulses

In this paper, we investigate the pth moment exponential stability of impulsive random delayed nonlinear systems (RDNS) with average-delay impulses. Utilizing the average impulsive interval, average impulsive delay, and the Lyapunov method, we present several criteria for pth exponential stability in these systems under a novel condition, diverging from previous literature. Additionally, we explore the effects of delayed impulses, analyzing both destabilizing and stabilizing impulses. Our findings reveal the dual effects of time delay in impulses, which can potentially destabilize a stable system or stabilize an unstable one.

Exponential stability of delayed discrete‐time systems with averaged delayed impulses

AbstractThis article investigates the exponential stability of nonlinear discrete‐time systems with time‐varying state delay and delayed impulses, where the delays in impulses are not fixed. Specifically, the study is separated into two cases: (1) stability of delayed discrete‐time systems with destabilizing delayed impulses, where the time delays in impulses can be flexible and even larger than the length of the impulsive interval; (2) stability of delayed discrete‐time systems with mixed delayed impulses (means stabilizing and destabilizing delayed impulses exist simultaneously), where the time delays in impulses are unfixed between two adjacent impulsive instants. First, the concept of average impulsive delay (AID) is extended to discrete‐time systems with delayed impulses. Then, some Lyapunov‐based exponential stability criteria are provided for delayed discrete‐time systems with delayed impulses, where the impulses satisfy the average impulsive interval (AII) condition and the delays in impulses satisfy the AID condition. For the delayed discrete‐time systems with mixed delayed impulses, the exponential stability criterion is provided by limiting the geometric average of the impulses' strengths. The obtained results can be used to study the delayed discrete‐time systems with some large impulses which may even be unbounded. It's also shown that the delays in impulses can have a stabilizing effect on the stability of delayed discrete‐time systems. Some numerical examples are also provided to illustrate the effectiveness and advantage of the theoretical results.

Stability analysis of nonlinear systems with delayed impulses: a generalised average dwell-time scheme

This paper studies the input-to-state stability (ISS) and uniform stability for impulsive nonlinear systems with delayed impulses, where delays are flexible between two consecutive impulsive instants. Illustrating effects caused by delays, concepts of generalised average impulsive delay and generalised reverse average impulsive delay are introduced, and more applicable in practice. Moreover, sufficient conditions for stability properties are formulated with destabilising and stabilising impulses through a general candidate ISS-Lyapunov function and the generalised conditions. It is shown that the double effects of delays in impulses (i.e. stabilising an initial unstable system, or contrarily, destabilising an initial stable system) coincide with the existing results using the exponential candidate ISS-Lyapunov function. Finally, two examples are provided to demonstrate our developed techniques.

Exponential Stability of Stochastic Time-Delay Neural Networks with Random Delayed Impulses

The mean square exponential stability of stochastic time-delay neural networks (STDNNs) with random delayed impulses (RDIs) is addressed in this paper. Focusing on the variable delays in impulses, the notion of average random delay is adopted to consider these delays as a whole, and the stability criterion of STDNNs with RDIs is developed by using stochastic analysis idea and the Lyapunov method. Taking into account the impulsive effect, interference function and stabilization function of delayed impulses are explored independently. The results demonstrate that delayed impulses with random properties take a crucial role in dynamics of STDNNs, not only making stable STDNNs unstable, but also stabilizing unstable STDNNs. Our conclusions, specifically, allow for delays in both impulsive dynamics and continuous subsystems that surpass length of impulsive interval, which alleviates certain severe limitations, such as presence of upper bound for impulsive delays or requirement that impulsive delays can only exist between two impulsive events. Finally, feasibility of the theoretical results is verified through three simulation examples.

Thermal stress coupling mechanism during nanosecond impulse delay for the synergistic study of continuous-nanosecond combined laser removal of the rubber layer on a concrete surface.

Based on the principle of laser ablation and elastic vibration effect, a model of continuous nanosecond combined laser removal of rubber marks on a concrete surface was established. The model can explain the evolution of temperature, stress, and removal depth on time and laser energy density during laser cleaning. The results show that the theoretical adsorption force between the rubber layer and the concrete base is approximately 3.88×10-9 N. The continuous laser cleaning threshold is 561.31J/c m 2. In the combined laser, the continuous laser is 534.41J/c m 2, and the nanosecond laser is 0.35J/c m 2. As the delay time between the 2ns laser beams increases, the maximum peak in the temperature curve gradually decreases. The optimal cleaning delay was obtained as Δ t=0.65S. The peak temperature at the characteristic position (0µm, 0µm) is 592.13K, which is lower than the vaporization temperature of the rubber layer. The thermal stress values generated at this characteristic position exceed the adsorption stress values, indicating that the elastic removal mechanism is the main removal mechanism at the junction between the rubber layer and the concrete substrate.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

Synchronization of Time-Delay Coupled Neural Networks With Stabilizing Delayed Impulsive Control.

This brief studies the distributed synchronization of time-delay coupled neural networks (NNs) with impulsive pinning control involving stabilizing delays. A novel differential inequality is proposed, where the state's past information at impulsive time is effectively extracted and used to handle the synchronization of coupled NNs. Based on this inequality, the restriction that the size of impulsive delay is always limited by the system delay is removed, and the upper bound on the impulsive delay is relaxed, which is improved the existing related results. By using the methods of average impulsive interval (AII) and impulsive delay, some relaxed criteria for distributed synchronization of time-delay coupled NNs are obtained. The proposed synchronization conditions do not impose on the upper bound of two consecutive impulsive signals, and the lower bound is more flexible. Moreover, our results reveal that the impulsive delays may contribute to the synchronization of time-delay systems. Finally, typical networks are presented to illustrate the advantage of our delayed impulsive control method.