Abstract This paper focuses mainly on the Euler scheme of stochastic delay differential equations with locally Lipschitz coefficients. The convergence in probability of the Euler scheme and the corresponding weak limit process of the normalized error process are derived. Furthermore, this paper also considers a class of specific degenerate stochastic delay equations and obtains the associated weak limit process for the stronger error process. The error parameter of this stronger error process for such a degenerate system is n instead of $\sqrt{n}$ in the normalized error process. This causes substantial challenges in the analysis and proofs and the weak limit process also becomes more complicated and involves some additional terms. This result is new and interesting even for the non-delay case.

In this paper, we study the tail distribution, smoothness and density estimates of the solution of a fundamental class stochastic differential delay equations. Base on the techniques of the Malliavin calculus we obtain an explicit estimate for tail distributions and upper and lower Gaussian estimates for density.

The exact response theory, also known as Transient Time Correlation Function formalism, is a powerful method concerning how observables respond to a given perturbation of the dynamics of the systems of interest, and it extends linear response theory to generic (autonomous) dynamical systems. Its main ingredient is the so-called dissipation function. In this paper, we adapt this theory for time-lagged systems, and we illustrate its applicability considering simple examples of delay equations, with different memory terms. Adopting the technique already used for time deterministic as well as stochastic time-dependent perturbations, the dynamics is described in a higher dimensional phase space, in which the delay-dependent dynamics is mapped into an augmented phase space: the new dynamics is proven to be autonomous and suitable for the exact responses to be computed. In addition, we explore the comparison between linear and exact approaches for a specific kernel choice.

This paper investigates qualitative properties of fractional delay differential equations formulated in terms of the Atangana–Baleanu–Caputo (ABC) fractional derivative of order 1<ϱ<2. Three related problem settings are examined: equations with variable delay, the constant-delay case, and a multi-delay extension involving several discrete delay terms. For each formulation, sufficient conditions ensuring existence and uniqueness of solutions are established in both the supremum norm and an exponentially weighted Maksoud norm. The analysis is carried out using Banach’s fixed point theorem in conjunction with progressive contractions and suitable Lipschitz-type conditions. In addition, Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) stability results are derived, providing quantitative estimates on the sensitivity of solutions with respect to perturbations. To complement the theoretical findings, numerical examples are presented, one of which illustrates the behavior of approximate solutions for various fractional orders.

To overcome the constraint of communication delay on tracking accuracy, a time-delay model predictive control method is proposed. Integrating trajectory coordinate dynamics and time-delay state equations to construct an enhanced composite model with time-varying delay characteristics. Designed the optimal weighting method to predict communication delay, transformed the uncertain delay equation into a time series model with external inputs, and constructed an augmented prediction equation based on the Diophantine equation. The controller solves the objective function online through rolling optimization and introduces feedback correction to compensate for time delay errors. The key content is to abandon the traditional ideal time series assumption and achieve effective modeling of time-delay uncertainty, as well as collaborative optimization of time-delay prediction and feedback correction. The results show that TDMPC significantly reduces the lateral tracking error compared to standard MPC under communication delay, and the amplitude of lateral angular velocity fluctuation is significantly reduced, significantly improving the tracking stability of complex time-delay scenarios.

Populations (in ecology, epidemics, biotechnology, economics, social processes) do not only interact over time but also age over time. It is therefore common to model them as “age-structured” partial differential equations (PDEs), where age is the ‘space variable.' Since the models also involve integrals over age, both in the birth process and in the interaction among species, they are in fact integro-partial differential equations (IPDEs) with positive states. To regulate the population densities to desired profiles, harvesting is used as input. But non-discriminating harvesting, where wanting to repress one (overpopulated) species will inevitably repress the other (near-extinct) species as well, the positivity restriction on the input (no insertion of population, only removal), and the multiplicative (nonlinear) nature of harvesting, makes control challenging even for ordinary differential equation (ODE) versions of such dynamics, let alone for their IPDE versions, on an infinite-dimensional nonnegative state space. With this paper, we introduce a design for a benchmark version of such a problem: a two-population predator-prey setup. The model is equivalent to two coupled ODEs, actuated by harvesting which must not drop below zero, and strongly (“exponentially”) disturbed by two autonomous but exponentially stable integral delay equations (IDEs). We develop two control designs. With a modified Volterra-like control Lyapunov function, we design a simple feedback which employs possibly negative harvesting for global stabilization of the ODE model, while guaranteeing regional regulation with positive harvesting. With a more sophisticated, restrained controller we achieve regulation for the ODE model globally, with positive harvesting. For the full IPDE model, with the IDE dynamics acting as large disturbances, for both the simple and saturated feedback laws we provide explicit estimates of the regions of attraction. Simulations illustrate the nonlinear infinite-dimensional solutions under the two feedbacks. The paper charts a new pathway for control designs for infinite-dimensional multi-species dynamics and for nonlinear positive systems with positive controls.

Well-posedness and stability of boundary delay equations

Early warning of financial crises through critical field dynamics: A nonlocal trend-inhibition delay equation framework

Large deviations for numerical approximation of stochastic differential delay equations

We considered a model for an infectious disease outbreak, when the depletion of susceptible individuals is negligible, and assumed that individuals adapt their behavior according to the information they receive about new cases. In line with the information index approach, we supposed that individuals react to past information according to a memory kernel that is continuously distributed in the past. We analyzed equilibria and their stability, with analytical results for selected cases. Thanks to the recently developed pseudospectral approximation of delay equations, we studied numerically the long-term dynamics of the model for memory kernels defined by gamma distributions with a general non-integer shape parameter, extending the analysis beyond what is allowed by the linear chain trick. In agreement with previous studies, we showed that behavior adaptation alone can cause sustained waves of infections even in an outbreak scenario, and notably in the absence of other processes like demographic turnover, seasonality, or waning immunity. Our analysis gives a more general insight into how the period and peak of epidemic waves depend on the shape of the memory kernel and how the level of minimal contact impacts the stability of the behavior-induced positive equilibrium.

The aim of this paper is to investigate the delay-dependent stability of highly nonlinear hybrid neutral stochastic differential delay equations (NSDDEs). Departing from most existing studies, the system under consideration incorporates a time-varying delay that is not required to be differentiable. A novel decomposition scheme for the drift coefficient is introduced, relaxing the conventional restrictive Lipschitz condition on the delay component. By constructing appropriate Lyapunov functionals and employing an M-matrix approach, delay-dependent conditions are derived to ensure moment stability for the considered highly nonlinear NSDDEs. Finally, an example is given to demonstrate the effectiveness of our new theory.

This paper addresses the control problem for linear difference delay equations (LDDEs). It is shown that the L 1 exact controllability criterion derived in Chitour et al. [(2023). Hautus-Yamamoto criteria for approximate and exact controllability of linear difference delay equations. Discrete and Continuous Dynamical Systems, 43(9), 3306–3337] can be reduced to a condition that requires verification only on a compact subset of the complex plane, thereby simplifying the analysis of controllability for LDDEs. This result can be seen as an analogue of the Hale–Silkowski criterion for exponential stability. Furthermore, a continuity property of exact controllability with respect to the delays is established. As an application, an exact controllability criterion is derived for one-dimensional hyperbolic partial differential equations.

Abstract In this paper, we discuss the existence and uniqueness of solution of Atangana-Baleanu-Caputo impulsive fractional delay differential equations with caratheo-dory function. We further introduce modified Ulam-Hyers-Rassias stability criteria by considering a real-valued function that is Lebesque integrable. This new concept makes the theory more realistic, flexible, and mathematically consistent with modern analysis (fractional calculus, impulsive system and delay equations). It covers unbounded but integrable disturbances, accommodates Caratheordory conditions, and extends applicability to a much larger class of dynamical systems. Extending Ulam-Hyers-Rassias stability to Lebesque integrable perturbations makes it compatible with stronger existence and uniqueness theorems using Banach and Schauder fixed point theorems which often require mappings to be continuous and bounded in $L^{\frac{1}{\theta}}([t_{0}, T])$ L 1 θ ( [ t 0 , T ] ) type norms. The stability of the solution of Atangana-Baleanu-Caputo impulsive fractional delay differential equations with caratheo-dory function is also investigated by using the modified Ulam-Hyers-Rassias stability concept. The outcome will aid in the theoretical development of fractional differential equations with memory effects, impulse perturbations, and delay factors

The increased number of controller area network bus nodes in the four-wheel independent steer-by-wire (4WISBW) system introduces uncertain network communication delays in the steering mechanism’s control inputs, reducing tracking accuracy and synchronization performance. To address this issue, we propose a multi-agent adaptive formation control strategy, comprising a nonlinear time-delay estimator (NTDE) and a multi-agent formation controller (MAFC). The NTDE reduces high-frequency oscillations and steady-state errors in delay estimation using a nonlinear integral sliding surface and a chatter-free supertwisting delay equation, while deriving the implicit time-delay system of the steering mechanism through nonsingular transformations. The MAFC constructs a leader-follower formation topology for the 4WISBW implicit time-delay system and designs adaptive coupling coefficients to dynamically adjust the time-varying formation of steering mechanisms, compensating for tracking and synchronization errors caused by network delays. Hardware-in-the-loop testing validates the proposed strategy’s effectiveness.

Delay differential equations (DDEs) are recognized as a significant class of functional differential equations, arising naturally in various scientific and engineering disciplines such as biology, physics, and control systems. In this study, a novel collocation framework is proposed for solving generalized pantograph-type differential equations (PTDEs). Furthermore, the methodology is extended to handle systems of second-order DDEs effectively. The approach is constructed upon a set of carefully chosen basis functions derived from shifted generalized Chebyshev polynomials (SGCPs), whose mathematical properties underpin the theoretical foundation of the proposed algorithms. A rigorous error analysis is provided to assess the accuracy of the SGCP-based expansion. To demonstrate the robustness and practicality of the method, a set of illustrative examples is included, highlighting the effectiveness and reliability of the developed numerical schemes.

ABSTRACT This paper investigates the well‐posedness and positivity of solutions to a class of delayed transport equations on a network. The material flow is delayed at the vertices and along the edges. The problem is reformulated as an abstract boundary delay equation, and well‐posedness is proved by using the Staffans–Weiss theory. We also establish spectral theory for the associated delay operators and provide conditions for the positivity of the semigroup.

By virtue of the novel technique, this paper focuses on the mean-square convergence of the backward Euler method (BEM) for stochastic differential delay equations (SDDEs) without using the moment boundedness of numerical solutions. The convergence rate for SDDE whose drift and diffusion coefficients can both grow polynomially is shown. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, some numerical experiments are implemented to illustrate the reliability of the theories.

A meshless finite point method for a class of parabolic differential equation of neutral delay

Abstract We present an exact analytical equation for the Shapiro time delay (STD) due to a spherical non-rotating body. As a result, accurate values of the STD in comparison with first and second-order expressions for Schwarzschild spacetime (1Sch and 2Sch) and first-order post-Newtonian formalism (1PN) are achieved. Accordingly, the lowest STD discrepancies between our exact equation and these approximations lie within the picosecond and sub-picosecond level for light beams affected by the Sun’s gravity. Our results might be useful for time delay measurements in the solar system or extragalactic binary pulsar systems, where a high accuracy level is required.

This paper investigates a stochastic differential investment and reinsurance game with delay between two competitive constant absolute risk aversion (CARA) insurers in a defaultable market. Each insurer purchases a proportional reinsurance contract to transfer the claim risk and invests its wealth in three assets: a risk-free asset, a risky asset whose price process is described by the Heston local-stochastic volatility model, and a defaultable bond. Introducing the delay feature, we obtain the wealth process modeled by the stochastic differential delay equation. The competitive relationship between the two insurers is characterized by the non-zero-sum stochastic differential game in which two insurers consider the relative performance measured by the difference in their terminal wealth. The objective of each insurer is to maximize the expected CARA utility of the combination of its terminal wealth and the relative performance with delay. Applying the dynamic programming approach, we obtain the Hamilton-Jacobi-Bellman equation. Then, we employ the perturbation method to solve complex nonlinear partial differential equations and obtain the asymptotic solutions for the Nash equilibrium strategies as well as the corresponding value functions in the post-default case and pre-default case. Furthermore, sensitivity analysis is conducted to explain the effects of model parameters on the equilibrium strategy.