Time-delay Systems Research Articles

Numerical algorithms based on splines for singularly perturbed parabolic partial differential equations with mixed shifts

In this paper, we discuss singularly perturbed time-dependent convection–diffusion problems that arise in computational neuroscience. Specifically, we provide approaches for one-dimensional singularly perturbed parabolic partial differential difference equations (SPPPDDEs) with mixed shifts in the spatial variable using fitted operator spline in compression and adaptive spline. Temporal discretization is done by backward Euler’s method, and spline methods with exponential fitting on uniform mesh are implemented in the spatial domain. For better approximations, the Richardson extrapolation technique is used, which is demonstrated by two numerical examples. The convergence of the proposed methods is investigated and found to be uniform with respect to the perturbation parameter. Graphical representations are provided to show how the shifts affect the proposed solution to the problem.

Coupled Aero–Hydrodynamic Analysis in Floating Offshore Wind Turbines: A Review of Numerical and Experimental Methodologies

Floating offshore wind turbines (FOWTs) have received increasing attention as a crucial component in renewable energy systems in recent years. However, due to the intricate interactions between aerodynamics and hydrodynamics, accurately predicting the performance and response remains a challenging task. This study examines recent advancements in the coupled aero–hydrodynamic numerical simulations for horizontal-axis FOWTs, categorizing existing research by coupling methods: uncoupled, partially coupled, and fully coupled. The review summarizes models, methodologies, and key parameters investigated. Most partially coupled analyses rely on forced oscillation, while the interplay between aerodynamics and elasticity, as well as interactions among multiple FOWTs, remain under-explored. Additionally, this review describes relevant physical model tests, including wave basin tests, wind tunnel tests, and real-time hybrid tests (RTHT). Although RTHT faces issues related to system time delays, they have garnered significant attention for addressing scale effects. The paper compares the three coupling methods, emphasizing the importance of selecting the appropriate approach based on specific design stage requirements to balance accuracy and computational efficiency. Finally, it suggests future research directions, offering a meaningful reference for researchers engaged in studying the aero–hydrodynamic behavior of FOWTs.

Design of functional estimation-based controller for the stabilization of time-delayed systems

Design of functional estimation-based controller for the stabilization of time-delayed systems

Transcendental entire solutions for systems of trinomial difference and differential-difference equations

Transcendental entire solutions for systems of trinomial difference and differential-difference equations

Impact of multiple time delays on bifurcation of a class of fractional nearest-neighbor coupled neural networks

In this paper, the impacts of multiple time delays on bifurcation of a class of fractional nearest-neighbor coupled neural networks are considered. Firstly, the sum of time delays is selected as a parameter, and the fractional nearest-neighbor coupled neural network model is linearized to obtain the corresponding characteristic equation. Then, utilizing stability and bifurcation theory of fractional-order delay differential equations, we investigate the effect of time delays on the system’s stability and bifurcations. The results show that when the time lag exceeds the critical value, the system will lose stability and generate Hopf bifurcation. Finally, the correctness of the conclusions in this paper is verified through numerical simulation.

Adaptive Neural Indirect Inverse Control for a Class of Fractional-Order Hysteretic Nonlinear Time-Delay Systems and Its Application

Adaptive Neural Indirect Inverse Control for a Class of Fractional-Order Hysteretic Nonlinear Time-Delay Systems and Its Application

Complex Dynamics and PID Control Strategies for a Fractional Three-Population Model

In recent decades, there have been many studies on Hopf bifurcation and population stability with time delay. However, the stability and Hopf bifurcation of fractional-order population systems with time delay are lower. In this paper, we discuss the dynamic behavior of a fractional-order three-population model with pregnancy delay using Laplace transform of fractional differential equations, stability and bifurcation theory, and MATLAB software. The specific conditions of local asymptotic stability and Hopf bifurcation for fractional-order time-delay systems are determined. A fractional-order proportional–integral–derivative (PID) controller is applied to the three-population food chain system for the first time. The convergent speed and vibration amplitude of the system can be changed by PID control. For example, after fixing the values of the integral control gain ki and the differential control gain kd, the amplitude of the system decreases and the convergence speed changes as the proportional control gain kp decreases. The effectiveness of the PID control strategy in complex ecosystem is proved. The numerical simulation results are in good agreement with the theoretical analysis. The research in this paper has potential application values concerning the management of complex population systems. The bifurcation theory of fractional-order time-delay systems is also enriched.

A Complex Delay Differential Equations Model for Acute Lymphoblastic Leukemia

In this paper, we introduce an innovative mathematical model designed to capture the dynamics of Acute Lymphoblastic Leukemia (ALL) under therapeutic interventions, employing delay-differential equations to account for the time delays inherent in biological processes. The model consists of 13 delay-differential equations, incorporating six distinct delays to represent various time-dependent factors such as drug effects, immune responses, and tumor growth cycles. To facilitate the analysis, we first identified the equilibrium points, which serve as critical benchmarks for understanding the system’s behavior under steady-state conditions, followed by a detailed stability analysis to assess the robustness of these points against perturbations. Utilizing the critical case theorem, we translated the system by shifting the equilibrium point to zero, simplifying the stability examination. A series of transformations were applied to aid this process, allowing for deeper insights into the dynamics of ALL under treatment. Our findings contribute to understanding treatment efficacy and tumor progression, offering a mathematical framework that not only highlights the complex interplay between treatment, tumor dynamics, and time delays but also provides a foundation for future research aimed at optimizing therapeutic strategies for ALL management.

Phase-Change-Material-Based True Time-Delay System

This study explores the achievement of a tunable true time-delay (TTD) system for a microwave phased-array antenna (MPAA) by incorporating the reversible phase-transition property of phase-change material (PCM) with Bragg gratings (BGs) and a cascade of three phase-shifted Bragg grating resonators (CPSBGRs). The goal was to design a low-power-consuming, non-volatile highly tunable compact TTD system for beam steering. A programmable on/off reflector was designed by changing a PCM-incorporated BG/CPSBGR from one phase to another. By arranging several programmable on/off reflectors in a row, a delay line was realized, and by incorporating several delay lines, the TTD system was achieved. Numerical simulations and parametric analyses were conducted for the evaluation of the TTD system’s performance at an operating wavelength of 1550 nm and 1550.6 nm for programmable on/off reflectors with BGs and CPSBGRs. The findings point out the effectiveness of incorporating PCMs with BGs/CPSBGRs, thereby maintaining a high performance with less complexity.

Electric Circuit Analogs of First-Order Dual-Phase-Lag Diffusion

Abstract The dual-phase-lag (DPL) model of non-classical diffusion plays an expanding role for applications involving transient heat, mass, and momentum transfer. To open a new realm of applications for the model, the work reported here introduces RLC-type (resistor-inductor-capacitor) and RCC-type (resistor-capacitor-capacitor) electric circuits that behave as discrete analogs of first-order, continuum DPL diffusion. Importantly, the phase lags are two time-scales representing departures from classical diffusion that define the wave-like and over-diffusive regimes of DPL behavior. Expressions for these phase lags arise while deriving the difference-differential equations that govern the circuits. Furthermore, simulations exhibit DPL behavior for simple, one-dimensional ladder circuits subjected to step increases in boundary voltages. In particular, the subsequent voltage disturbances in the circuits propagate faster in the over-diffusive regime than in the wave-like regime. The simulations also show that DPL behavior stems from transient divisions of current between RL components in the wave-like circuit and between RC components in the over-diffusive circuit. Consequently, adopting the DPL perspective in the design of electronic systems may inspire novel applications that take advantage of DPL behavior. Finally, the circuits introduced here may lead to the study of DPL diffusion in analogous scenarios of heat, mass, and momentum transfer that are difficult to access experimentally.

Dynamic reliability-based assessment for the seismic performance of semi-actively base-isolated buildings with hydraulic dampers

Seismic base isolation faces the challenge of the large displacement demands for the isolators in case of long-period ground motions. Semi-active control has the potential to mitigate the risk against such extreme scenarios and therefore increase the seismic reliability of base-isolated structures. However, the superstructural nonlinearity and the time delay in the control system are always neglected in related studies. Such a knowledge gap may overestimate the benefit of semi-active control in enhancing the seismic reliability of base-isolated buildings. This study explores the seismic reliability of semi-actively base-isolated buildings with realistic considerations of the superstructural nonlinearity and the time delay in a practical hydraulic control system in a fully probabilistic framework. It features the use of the probability density evolution method (PDEM) and the development of a MATLAB-OpenSees interaction technique. The reliability of the semi-actively isolated structure and its passive counterpart are compared, highlighting the effects of nonlinearity and time delay. The results show the superstructural nonlinearity and the inevitable time delay in a practical hydraulic control system may significantly degrade the control performance of semi-actively base-isolated buildings, potentially leading to higher failure probabilities than passive buildings.

Delay-dependent stability conditions for delay differential equations with unbounded operators in Banach spaces

We consider the equation \(du(t)/dt=Au(t)+B u(t-h)\) where \(t>0\), \(h\) is a positive constant, and \(A\) is a linear unbounded and \(B\) is a linear bounded operators. We establish explicit delay-dependent conditions for exponential stability, and present applications to partial integro-differential equations with delay. For more information see https://ejde.math.txstate.edu/Volumes/2024/76/abstr.html

Boundary-Value Problems for Differential-Difference Equations with Finite and Infinite Orbits of Boundaries

Boundary-Value Problems for Differential-Difference Equations with Finite and Infinite Orbits of Boundaries

Fixed-time Lyapunov criteria of stochastic impulsive time-delay systems and its application to synchronization of Chua’s circuit networks

In this paper, we investigate the fixed-time stability and synchronization issues of stochastic impulsive delay complex networks (SICNs) with delayed impulses. Firstly, a new fixed-time stability criterion, which provides a direct connection between impulsive strength, impulsive instants, delays and the stability time, is established for stochastic delay systems with delayed impulsive. In this criterion, two types of delays: system delay and impulsive delay are considered, and the impulsive intensity at each impulse instant can be different, (facilitate or disrupt stability). It is the first time to consider the fixed-time stability issues of this case. Secondly, based on this criterion, a novel feedback controller and an adaptive controller are designed to study the fixed-time synchronization of SICNs with delayed impulses; Thirdly, by utilizing Lyapunov functional theory, stochastic analysis technique, matrix inequality analysis techniques and proposed stability criterion, the fixed-time synchronization conditions of SICNs, are derived in the form of linear matrix inequalities (LMIs). Finally, a numerical example and an application example, Chua’s circuit networks, are provided to illustrate the validity of the theoretical analysis.

Nonlinear Differential-Difference Equations of Elliptic and Parabolic Type and Their Applications to Nonlocal Problems

Nonlinear Differential-Difference Equations of Elliptic and Parabolic Type and Their Applications to Nonlocal Problems

Adaptive Neural Control Design for Strict-Feedback Time-Delay Nonlinear Systems Based on Fast Finite-Time Stabilization: A Case Study of Synchronous Generator Systems

This study aims to investigate the finite-time control problem for a class of strict-feedback time-delay nonlinear systems with unknown functions. The control design is based on a fast finite-time practical stability criterion. Unknown nonlinear functions can be estimated using the universal approximation performance of neural networks. Finite-time control design is performed using adaptive backstepping technology. By performing closed-loop stability analyses and choosing appropriate Lyapunov–Krasovskii functionals, all signals in a closed-loop system can be bounded within a finite time. Subsequently, the proposed control method can be applied for the excitation control of synchronous generators. The effectiveness of the proposed method is verified using a numerical model of a single-machine power system.

Adaptive fixed-time fuzzy control for delayed recycling continuous stirred tank reactor with asymmetric time-varying full-state constraints

Recently, there are some results concerning stability and finite-time stability control for continuous stirred tank reactors (CSTR). However, when considering the control of time-delay CSTR systems or fixed-time stability of CSTR systems, there are relatively few articles involved. In this paper, we first consider the CSTR system that includes both of the above factors simultaneously, that is, the adaptive fixed-time fuzzy control design problem for a CSTR with a delayed recycle stream and asymmetric time-varying full-state constraints. Firstly, based on existing works, we present an improved practical fixed-time stability criterion, which can be applied to more complex nonlinear systems. Then, we construct time-varying asymmetric integral barrier Lyapunov functions to handle asymmetric time-varying full-state constraints. Next, fuzzy logic systems are used to approximate the unknown functions of the system, and a sliding mode differentiator is introduced to avoid the “explosion of complexity” of the time derivative of the virtual controller. Furthermore, using the improved practical fixed-time stability criterion, we present the adaptive fixed-time fuzzy controller design processes that can ensure tracking performance in a fixed-time, and states of the system never violate their constraint bounds. Finally, a simulation example is provided to verify the effectiveness of the proposed control strategy.

Memory and anticipation: two main theorems for Markov regime-switching stochastic processes

We present two main theorems for stochastic processes with a Markov regime-switching model. First, we work on an existence-uniqueness theorem for a Stochastic Differential Delay Equation with Jumps and Regimes (SDDEJRs). Then we provide the duality between an SDDEJR and an Anticipated Backward Stochastic Differential Equation with Jumps and Regimes (ABSDEJRs). Our goal is to provide two technical and fundamental theorems for the future theoretical and applied developments of time-delayed and time-advanced models.

Smoothness of Generalized Solutions of a Boundary-Value Problem for a Second-Order Differential-Difference Equation with Mixed Boundary Conditions

Smoothness of Generalized Solutions of a Boundary-Value Problem for a Second-Order Differential-Difference Equation with Mixed Boundary Conditions

Solving Fuzzy Delay Differential Equations using Fuzzy Elzaki Transform and Fuzzy Elzaki Decomposition Method

Objective: In this work, the fuzzy Elzaki transform and fuzzy Elzaki decomposition method are used to solve fuzzy delay differential equations. The fuzzy Elzaki decomposition method is a combination of the fuzzy Elzaki transform and Adomian decomposition method in a fuzzy environment. The use of transforms in solving differential equations makes the solution process simpler. Methodology: The fuzzy Elzaki transform is applied in the fuzzy delay differential equation. The nonlinear terms are decomposed using Adomian polynomials in a nonlinear fuzzy delay differential equation. In the transformed domain, the resulting algebraic equation is solved. The solution in the original time domain is obtained by applying the inverse Elzaki transform. Findings: The fuzzy Elzaki transform and fuzzy Elzaki decomposition method are useful for handling the complexities associated with fuzzy functions and delays in differential equations. The technique used to solve the fuzzy delay differential equation gives results with better accuracy compared to the exact solution. The applicability of the method is illustrated with numerical examples. Novelty: The Elzaki transform is a helpful tool for solving delay differential equations with discontinuities. For the resilient solution of complicated differential equations, especially those including nonlinearity and fuzzy logic, the fuzzy Elzaki decomposition approach is useful. It gives an organized way to find the solution to fuzzy delay differential equation by utilizing the advantages of both the Elzaki transform and the Adomian decomposition method in a fuzzy situation. Keywords: Triangular Fuzzy Number, Fuzzy Elzaki Transform, Adomian decomposition method, Fuzzy Elzaki decomposition method, Fuzzy Delay Differential Equations (FDDE)