Search Results (66)

This paper investigates the quasi-synchronization (QS) problem for drive-response discrete-time delayed inertial neural networks (DTDINNS). Unlike existing studies that mainly rely on classical stability theorems, linear matrix inequality (LMI) methods, and matrix measure approaches (MMA), this work establishes three innovative quasi-synchronization criteria for DTDINNS by adopting the solution formula of second-order linear difference equations (SOLDES), infinite series summation techniques, and the solution formula of first-order linear difference equation group (Lemma 6). To the best of our knowledge, this study is the first attempt to introduce the general solution theory of second-order linear difference equations and infinite series summation methods to analyze the synchronization behavior of neural networks (NNS). The proposed framework offers a novel theoretical tool for the synchronization analysis of discrete-time delayed neural networks (DTDNNS), which bears important theoretical significance for relevant research fields. Full article

This paper studies mild solutions and Bolza optimal control for Riemann–Liouville fractional stochastic integro-differential systems incorporating fourth-order diffusion, time-varying control, non-instantaneous impulses, and infinite delay. Based on our self-developed $(\mu ,\nu ,\xi ,e,k)$-resolvent family, we derive the mild solution formulation and prove its existence via the Krasnoselskii–Schaefer fixed-point theorem. Using the Arzel$\acute{a}$-Ascoli theorem, Mazur’s lemma, and Balder’s lower semicontinuity principle, we further establish the existence of optimal control pairs. A numerical example from Euler–Bernoulli beam dynamics illustrates the theoretical results. Full article

Cooperative adaptive cruise control (CACC), as an extension of adaptive cruise control (ACC), is an intelligent transportation approach for connected and automated vehicles. By using vehicle-to-vehicle information, CACC improves longitudinal tracking performance, traffic throughput, and string-stable platoon behavior. However, controller tuning remains sensitive to vehicle-dynamics parameters, spacing-policy selection, fractional-order dynamics, and communication delay. This paper presents an analytical parameter-space-based fractional-order PD (FOPD) controller synthesis framework for string-stable CACC systems. For the constant-time headway spacing policy, the controller parameters are investigated in the $(k_p,k_d,\mu )$ parameter space, where the fractional differentiation order $\mu$ is considered as an additional design variable. To obtain the feasible stabilizing regions, the fractional-order characteristic equation is evaluated on the imaginary axis, and the delay-dependent stability boundaries are derived through a frequency-domain boundary-locus formulation. The stabilizing gain regions are constructed through the complex-root boundary (CRB), real-root boundary (RRB), and infinite-root boundary (IRB), which provide an interpretable graphical basis for controller-gain and fractional-order selection. In addition, the effect of the headway time on the admissible stability region is examined jointly with the fractional order. The proposed structure is implemented with a feedforward controller that uses the acceleration information of the preceding vehicle under a predecessor-vehicle-following communication topology. The selected fractional-order CACC (FO-CACC) controller is validated in an eight-vehicle platoon simulation environment and compared with integer-order ACC (IO-ACC), fractional-order ACC (FO-ACC), and integer-order CACC (IO-CACC) configurations. The results show that the proposed parameter-space approach enables systematic FOPD tuning and that the selected FO-CACC controller satisfies the frequency-domain string-stability requirement while maintaining smooth time-domain responses in position, velocity, acceleration, headway time, spacing error, and control input. Additional simulations under the New European Driving Cycle (NEDC) and the FTP-75 (Federal Test Procedure 1975) driving cycles further indicate that the proposed FO-CACC structure maintains accurate spacing regulation and bounded acceleration behavior under standard drive-cycle conditions. Overall, the results indicate that the fractional-order parameter provides an effective design freedom for improving string-stable cooperative platoon performance. Full article

The Hilfer fractional derivative effectively captures non-locality, historical dependence, and memory effects, making it valuable for modeling real-world systems, and exponential growth can describe explosive growth phenomena in real-world problems. This paper focuses on the existence of mild solutions for infinite-delay differential equations involving Hilfer fractional derivatives, fractional Laplacian operator ${(-\Delta )}^{\delta }$, and exponentially growing functions in Orlicz spaces. First, by utilizing standard $L^p$-$L^q$ estimates for strongly continuous semigroups generated by fractional Laplacian operator, the existence of global solutions in the Orlicz space $\exp L^p\left({\mathbb{R}}^d\right)$ and the time-weighted $L^z\left({\mathbb{R}}^d\right)$ space is established. Then, by leveraging Hölder’s interpolation inequality, the existence of local solutions in $L^1\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)$ is established. Full article

This paper presents a generalized epidemic modeling framework based on g-tempered Caputo fractional derivatives with discrete time delays. The proposed approach incorporates nonlocal memory effects, nonlinear temporal scaling, and delayed epidemiological responses within a unified mathematical structure. The introduction of the nonlinear time transformation $g\left(t\right)$ and the tempering parameter $\lambda$ eliminates the unrealistic infinite-memory behavior associated with classical power-law kernels while simultaneously introducing new challenges related to parameter identifiability and inverse problems. We investigate the structural properties of the resulting dynamical systems and show that the associated inverse problem is inherently ill-posed. To illustrate the practical implications of these results, the framework is applied to a delayed SIQR epidemiological model. Numerical simulations are performed using a generalized L1-type scheme adapted to delayed fractional histories, and a multi-phase parameter estimation procedure is proposed to address the ill-posedness of the reconstruction problem. The results demonstrate the ability of the model to capture both short- and long-term memory effects in epidemic evolution while highlighting the challenges of statistical identifiability in generalized fractional systems. Full article

We provide sufficient conditions for the existence of $(\omega ,c)$-periodic solutions of a general class of nonlinear functional integral equations. This study extends and generalizes previous contributions in the literature. As an application of the developed theory, we establish the existence of $(\omega ,c)$-periodic solutions for recurrent neural networks with time-varying coefficients and mixed delays, as well as for a class of nonlinear Volterra–Stieltjes integral equations with infinite delay. Full article

We analyze the population dynamics of a microbial cross-protection model and derive the exact conditions under which a Fold–Hopf bifurcation emerges. By applying center-manifold reduction and normal-form theory, we reduce the infinite-dimensional delay differential system to a finite-dimensional ordinary differential system, enabling rigorous bifurcation analysis. Numerical simulations reveal a rich repertoire of dynamical behaviors, including stable equilibria, sustained oscillations, and noise-induced irregularities. Our findings identify time-delay-induced Fold–Hopf bifurcation as a fundamental mechanism driving oscillatory coexistence in cross-protection mutualisms, for previously reported experimental observations. Full article

Fog computing is one of the growing distributed computing platforms incorporated by Industries today as it performs real-time data analysis closer to the edge of the IoT network. It offers cloud capabilities at the edge of the fog networks through improved efficiency and flexibility. As the demands of Internet of Things (IoT) devices keep varying, it is important to rapidly modify the resource allocation policies to satisfy them. Constant fluctuation of the demands leads to over or under provisioning of resources. The computing capability of the fog nodes is small, and hence there is a necessity to develop resource provisioning policies that reduce the delay and bandwidth consumption. In this paper, a novel large language model (LLM)-guided Q-learning framework is designed and developed. The uncertainty in the fog environment in terms of delay incurred, bandwidth usage, and heterogeneity of fog nodes is represented using the LLM model. The reward shaping of a Q-learning agent is enriched by considering the heuristic value of the LLM model. The experimental results ensure that the proposed framework is good with respect to processing delay, energy consumption, load balancing, and service level agreement violation under a finite and infinite fog computing environment. The results are further validated through the expected value analysis statistical methodology. Full article

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler–Karhunen–Loève method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler–Karhunen–Loève method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics. Full article

The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor $u(t+\theta )$ in the delay term may lead to oscillations, hysteresis effects, and other phenomena in the solution, which increases the difficulty of studying the well-posedness and asymptotic behavior of the solution. This study investigates the global well-posedness and asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations incorporating infinite delays. To establish global well-posedness, we first construct several suitable function spaces and then prove them using the Galekin approximation method. Then, by accurately estimating the number of determining nodes, we reveal the asymptotic behavior of the solution. The results indicate that the long-term behavior of a strong solution can be determined by its values at a finite number of nodes. Full article

Uncertainty, time delays, and jumps often coexist in dynamic game problems due to the complexity of the environment. To address such issues, we can utilize uncertain delay differential equations with jumps to depict the dynamic changes in differential game problems that involve uncertain noise, delays, and jumps. In this paper, we first examine a linear quadratic differential game optimistic value problem within an uncertain environment characterized by jumps and delays. By applying the $Z(x,y)$ transform, we convert the infinite-dimensional problem into a finite-dimensional one. We then demonstrate that the condition for the existence of a Nash equilibrium strategy is equivalent to the existence of solutions to two cross-coupled matrix Riccati equations. Furthermore, we explore the saddle point equilibrium strategy of the linear quadratic differential game optimistic value model and derive the corresponding saddle point equilibrium solution. Finally, we apply our results to solve a carbon emission reduction game problem. Full article

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (${\mathbb{E}}_{\alpha ,\beta }$), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models. Full article

This research relies on several kinds of Volterra-type integral differential systems and their associated stability concerns under the impulsive effects of the Volterra integral terms at certain time instants. The dynamics are defined as delay-free dynamics contriobution together with the contributions of a finite set of constant point delay dynamics, plus a Volterra integral term of either a finite length or an infinite one with intrinsic memory. The global asymptotic stability is characterized via Krasovskii–Lyapuvov functionals by incorporating the impulsive effects of the Volterra-type terms together with the effects of the point delay dynamics. Full article

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases. Full article

The susceptible–infected–recovered/removed–vaccinated (SIRV) epidemic model is an important generalization of the SIR epidemic model, as it accounts quantitatively for the effects of vaccination campaigns on the temporal evolution of epidemic outbreaks. Additional to the time-dependent infection ($a\left(t\right)$) and recovery ($\mathit{\mu }\left(t\right)$) rates, regulating the transitions between the compartments $S\to I$ and $I\to R$, respectively, the time-dependent vaccination rate $v\left(t\right)$ accounts for the transition between the compartments $S\to V$ of susceptible to vaccinated fractions. An accurate analytical approximation is derived for arbitrary and different temporal dependencies of the rates, which is valid for all times after the start of the epidemics for which the cumulative fraction of new infections $J\left(t\right)\ll 1$. As vaccination campaigns automatically reduce the rate of new infections by transferring persons from susceptible to vaccinated, the limit $J\left(t\right)\ll 1$ is even better fulfilled than in the SIR-epidemic model. The comparison of the analytical approximation for the temporal dependence of the rate of new infections $\mathring{J}\left(t\right)=a\left(t\right)S\left(t\right)I\left(t\right)$, the corresponding cumulative fraction $J\left(t\right)$, and $V\left(t\right)$, respectively, with the exact numerical solution of the SIRV-equations for different illustrative examples proves the accuracy of our approach. The considered illustrative examples include the cases of stationary ratios with a delayed start of vaccinations, and an oscillating ratio of recovery to infection rate with a delayed vaccination at constant rate. The proposed analytical approximation is self-regulating as the final analytical expression for the cumulative fraction $J_{\infty }$ after infinite time allows us to check the validity of the original assumption $J\left(t\right)\le J_{\infty }\ll 1$. Full article