Stochastic Delay Research Articles

Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system’s solutions, we investigate stochasticity’s influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

Complex global dynamics of conditionally stable slopes: effect of initial conditions

In the present paper, we investigate the effect of the initial conditions on the dynamics of the spring-block landslide model. The time evolution of the studied model, which is governed by a system of stochastic delay differential equations, is analyzed in the mean-field approximation, which qualitatively exhibits the same dynamics as the initial model. The results of the numerical analysis show that changing the initial conditions has different effects in different parts of the parameter space of the model. Namely, moving away from the fixed-point initial conditions has a stabilizing effect on the dynamics when the noise, the friction parameters a (higher values) and c as well as the spring stiffness k are taken into account. The stabilization manifests itself in a complete suppression of the unstable dynamics or a partial limitation of the effect of some friction parameters. On the other hand, the destabilizing effect of changing the initial conditions occurs for the lower values of the friction parameters a and for b. The main feature of destabilization is the complete suppression of the sliding regime or a larger parameter range with a transient oscillatory regime. Our approach underlines the importance of analyzing the influence of initial conditions on landslide dynamics.

On the qualitative behaviors of stochastic delay integro-differential equations of second order

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE).To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature.

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results.

Stability for Markov switching stochastic delay systems binding event-triggered mechanism to activate multi-impulse jumps

This paper focuses on the pth moment exponentially stability for Markov switching and multi-impulse jumps stochastic time-varying delay system, where the switching behavior among subsystems of the target system is determined by Markov chains, and the occurrence of impulsive jumps is decided according to event-triggered impulsive mechanism when certain well-designed conditions are satisfied. By applying the Itô formula, Gronwall inequality and Razumikhin theorem, some novel sufficient criteria are provided to assure the system stability and get rid of Zeno phenomenon. It is worth pointing out that the multi-impulse jumps are our research aim and the range of delays considered is relatively wide, i.e., the daily bounded delay τ(t)∈[0,1) and the unupper bound delay τ(t)∈[1,∞). Subsequently, two diverse event trigger mechanism about impulsive jumps are proposed for such two types of delays, namely the defined event-triggered impulsive mechanism with delay. Finally, the validity and feasibility of the developed theoretical results are verified by two numerical simulations.

Exponential mean‐square string stability of vehicular platoon with stochastic disturbance and time delay

AbstractPlatooning is an effective way to reduce air resistance and fuel consumption in vehicular transportation. However, stochastically generated disturbances can significantly impact the string stability of platoons. This article proposes a novel control law, based on sliding mode control and leader‐predecessor communications, designed to improve the exponential mean‐square string stability of platoons with Ito‐type stochastic disturbances that consider friction force. Specifically, our approach employs a third‐order nonlinear dynamic model to capture the relevant spatiotemporal correlations present in vehicle platoons. The proposed longitudinal control strategy effectively stabilizes the platoons with time delay and was verified through simulation experimentation. In this study, the stability of sliding mode motion is thoroughly investigated by means of sub‐reachable and practical stability analysis with sliding mode control. The findings of this study provide a theoretical foundation for further research on vehicle platoons with increasingly complicated interference and time delays. Overall, our results demonstrate that our approach can achieve significant improvements in both fuel economy and exhaust emissions while maintaining high levels of string stability, further contributing to the development of efficient and sustainable transportation systems.

Dynamics of a stochastic delay predator-prey model with fear effect and diffusion for prey

We establish a stochastic delay predator-prey model with fear effect and prey diffusion, and investigate the dynamic behavior of the model. Initially, we use Itô's formula to prove the existence and uniqueness of a global positive solution and the stochastic ultimate boundedness of the system. Subsequently, we provide sufficient conditions for the extinction and persistence of prey and predator, and discuss three possible scenarios. Beside, we construct a suitable Lyapunov function and obtain the conditions for the existence of an ergodic stationary distribution. Finally, we present some numerical simulations to verify the analytical results.

Stability in Distribution of Highly Nonlinear Hybrid Stochastic Delay Systems by Delay Feedback Control

Stability in Distribution of Highly Nonlinear Hybrid Stochastic Delay Systems by Delay Feedback Control

Periodic Self-Triggered Impulsive Synchronization of Hybrid Stochastic Complex-Valued Delayed Networks

This paper studies the synchronization problem for hybrid stochastic delay complex-valued networks with time-varying multi-links (HSDCNT) via periodic self-triggered impulsive control (PSIC). Therein, PSIC combines with the benefits of impulsive control and periodic self-triggered strategy. Moreover, time delays, stochastic disturbances, and time-varying multi-links are considered simultaneously. Particularly, the synchronization for HSDCNT is studied on the complex space and no split real and imaginary parts. Actually, different from the previous PSIC-based results, this paper extends the real-valued neural networks to HSSs, delayed systems, and complex-valued networks, which is significant and challenging. Then, as a practical application, we consider the synchronization of a kind of Cohen-Grossberg neural network in the complex space. Finally, a numerical example is utilized for demonstration.

Limiting dynamics for stochastic delay p-Laplacian equation on unbounded thin domains

Limiting dynamics for stochastic delay p-Laplacian equation on unbounded thin domains

The nonlinear dynamics analysis of stochastic delay Jeffcott rotor-seal system with the elastic support

The establishment of stochastic delay Jeffcott rotor-seal system not only considers the track irregularity caused by random noise and the possible influence of stochastic parameter excitation, but also considers the time-delay characteristics of sealing force. The rotation of the rotor shaft causes the gas in the chamber to produce a dynamic effect, resulting in a rotating force, which delays the feedback on the rotating shaft. Firstly, the one-dimensional average Itoˆ differential equation is obtained by simplifying the infinite dimensional stochastic delay differential equation with the perturbation method. Secondly, the global and local stability of the rotor system are obtained by analyzing the singular boundary theory and the maximum Lyapunov exponent. Then, the conditions and types of stochastic bifurcation of rotor system are obtained by analyzing the steady-state joint probability density function. Finally, numerical simulation verifies the accuracy of the theoretical analysis, showing that the time delay affects rotor system to reach the stable critical speed for the first time, and the noise disturbance has a certain stability effect on the rotor system.

BUFFERING POLICIES FOR PREFABRICATED CONSTRUCTION SUPPLY CHAIN SUBJECT TO MATERIAL LEAD TIME AND ACTIVITY DURATION UNCERTAINTIES

Supply chain management plays a pivotal role in the smooth execution of prefabricated construction. One key aspect involves strategically placing and sizing buffers to handle uncertainties (e.g., stochastic material lead-times and activity durations) within the prefabricated construction supply chain (PCSC). This study examines three buffering policies based on varying combinations of time and inventory buffers to mitigate stochastic material delays and activity prolongs in PSCS, namely, pure inventory buffering policy, pure time buffering policy, and mixed inventory-time buffering policy. To enable this analysis, we characterize how stochastic material delays originating from off-site supply chains impact project schedules, and then develop mathematical procedures for sizing inventory and/or time buffers under the three buffering policies. Case application and numerical analysis are conducted to investigate the performance of these buffering policies and the impact of the project characteristics on them (e.g., due date and arrival rate). Finally, insights are extracted to assist managers in choosing appropriate policies for projects with different characteristics. In general, combining inventory and time buffers results in better performance, particularly under tight project deadlines and high arrival rates. The pure time buffering policy can also be a viable option in specific situations, allowing managers to have more choices.

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

A Proactive-Reactive-Based Approach for Continuous Berth Allocation and Quay Crane Assignment Problems with Hybrid Uncertainty

Port operations have been suffering from hybrid uncertainty, leading to various disruptions in efficiency and tenacity. However, these essential uncertain factors are often considered separately in literature during berth and quay crane assignments, leading to defective, even infeasible schedules. This paper addressed the integrated berth allocation and quay crane assignment problem (BACAP) with stochastic vessel delays under different conditions. A novel approach that combines both proactive and reactive strategies is proposed. First, a mixed-integer programming model is formulated for BACAP with quay crane maintenance activities under the ideal state of no delay. Then, for minor delays, buffer time is added to absorb the uncertainty of the arrival time of vessels. Thus, a robust optimization model for minimizing the total service time of vessels and maximizing the buffer time is developed. Considering that the schedule is infeasible when a vessel is seriously delayed, a reactive model is built to minimize adjustment costs. According to the characteristics of the problem, this article combined local search with the genetic algorithm and proposed an improved genetic algorithm (IGA). Numerical experiments validate the efficiency of the proposed algorithm with CPLEX and Squeaky Wheel Optimization (SWO) in different delay conditions and problem scales. An in-depth analysis presents some management insights on the coefficient setting, uncertainty, and buffer time.

Stochastic Delay Guarantees for Devices With Dual Connectivity

Stochastic Delay Guarantees for Devices With Dual Connectivity

Comments to ‘Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shift’

In Liu et al. (2023), the authors set out to prove orbital stability for traveling waves in a (stochastic) delay reaction–diffusion equation. For the deterministic proof, they follow a phase-tracking technique as developed by Zumbrun and Howard (1998) [1] and for the stochastic version, they use a stochastic phase-tracking technique as developed by Hamster and Hupkes (2019, 2020a,b) [2,3] (which in turn is based on the aforementioned technique by Howard and Zumbrun). Note that the authors do not acknowledge the origins of their techniques.In the deterministic case, stability results are known (Mei, 2009) [4], but the application of phase-tracking to this problem appears to be new. However, there is a very subtle interplay between the time-dependent phase and the delay in the delay equation which is not accounted for, nor is explained or proved if the chosen phase has the same properties in the general framework of delay semigroups as for regular semigroups.The proof of the stochastic case of course has the same issues, but let us assume that the issues can be resolved, then there are still several issues with the proof. Especially, the phase tracking introduces terms that depend on the gradient of the solution which must be bounded in the H1 norm. This introduces singularities in the bounds that do not show in these proofs.

Numerical approximations of stochastic delay differential equations with delayed impulses

This paper focuses on approximating stochastic delay differential equations with delayed impulses using Euler-Maruyama-type approximations. One key difference from previous literature is that the impulsive perturbations considered in this paper are past-dependent. Additionally, both the time delays in the stochastic delay differential equations and in the impulsive functions are functions of time. We establish the mean square convergence of the Euler-Maruyama approximations under a local Lipschitz condition and a linear growth condition. Furthermore, we determine the order of convergence under a global Lipschitz condition and provide an illustrative example.

Saturated H∞ control of networked systems under stochastic transmission delays: The random sampling periods case

AbstractIn this paper, the saturated control problems of networked systems with random sampling periods and stochastic transmission delays are investigated. The main issue is to design an saturated controller such that the controlled close‐loop system is exponentially mean‐square stable with a desired performance. A matrix exponential which is different from some existing literature is used to reconstruct the discrete‐time stochastic system to an equivalent yet tractable augmented model without any additional conditional constraint. In virtue of reduced‐order confluent Vandermonde matrix approach and Kronecker product operation, the stability condition in terms of linear matrix inequality is derived and then the desired saturated controller is designed. It should be emphasized that the improved method can also be used for the stability analysis and controller design when the system matrix contains complex roots. Finally, two numerical examples and a three‐axis milling machine system are utilized to manifest the reliability and practicality of the designed strategy.

Attack-tolerant control for Markovian jump systems with stochastic sampling: A sliding mode scheme

This work discusses the sampled-data sliding mode control problem for Markovian jump systems in the presence of Denial-of-Service (DoS) attacks, where the state signals are sampled stochastically and transmitted via vulnerable communication networks. The successful transmitted sequence with double randomness is established, and the probability distribution on both the attacked signals and stochastic delays are given. With the aid of the coordinate transformation, a linear sliding surface is constructed and the mode-dependent sliding mode controller is designed by means of only successfully transmitted state signals. Moreover, the resultant closed-loop systems composed of partial state and sliding mode variable are described as a continuous time-delay system with double randomness depending on the probability distribution characteristics of both the sampling periods and DoS attacks. Both the mean-square exponential ultimate boundedness of the closed-loop system and the reachability of the sliding surface are analyzed and the corresponding sufficient conditions are derived. Meanwhile, an optimization problem with genetic algorithm is formulated for achieving fast convergence of the state trajectory, small ultimate bound of the system state and small sliding bound. Finally, the simulation results via the numerical example are provided to illustrate the proposed sampled-data sliding mode control scheme.