Time-delay Systems Research Articles

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

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

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

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

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.

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)

Stability and Bifurcation Analysis for the Transmission Dynamics of Skin Sores with Time Delay

Impetigo is a highly contagious skin infection that primarily affects children and communities in low-income regions and has become a significant public health issue impacting both individuals and healthcare systems. A nonlinear deterministic model based on the transmission dynamics of skin sores (impetigo) is developed with a specific emphasis on the time delay effects in the infection and recovery processes. To address this complexity, we introduce a delay differential equation (DDE) to describe the dynamic process. We analyzed the existence of Hopf bifurcations associated with the two equilibrium points and examined the mechanisms underlying the occurrence of these bifurcations as delays exceeded certain critical values. To obtain more comprehensive insights into this phenomenon, we applied the center manifold theory and the normal form method to determine the direction and stability of Hopf bifurcations near bifurcation curves. This research not only offers a novel theoretical perspective on the transmission of impetigo but also lays a significant mathematical foundation for developing clinical intervention strategies. Specifically, it suggests that an increased time delay between infection and isolation could lead to more severe outbreaks, further supporting the development of more effective intervention approaches.

On the Evolution Operators of a Class of Linear Time-Delay Systems

This paper studies the properties of the evolution operators of a class of time-delay systems with linear delayed dynamics. The considered delayed dynamics may, in general, be time-varying and associated with a finite set of finite constant point delays. Three evolution operators are defined and characterized. The basic evolution operator is the so-called point delay operator, which generates the solution trajectory under point initial conditions at t0=0. Furthermore, this paper also considers the whole evolution operator and the delay strip evolution operator, which define the solution trajectory, respectively, at any time instant and along a strip of time whose size is that of the maximum delay. These operators are defined for any given bounded piecewise continuous function of initial conditions on an initialization time interval of measure being identical to the maximum delay. It is seen that the semigroup property of the time-invariant undelayed dynamics, which is generated by a C0-semigroup, becomes lost by the above evolution operators in the presence of the delayed dynamics. This fact means that the point evolution operator is not a strongly and uniformly continuous one-parameter semigroup, even if its undelayed part has a time-invariant associated dynamics. The boundedness and the stability properties of the time-delay system, as well as the strong and uniform continuity properties of the evolution operators, are also discussed.

Tube-based integral sliding mode predictive control scheme for constrained uncertain nonlinear time-delay system

For a class of constrained uncertain nonlinear time-delay systems, how to achieve their stability and robustness is a crucial issue. To this end, this paper proposes a tube-based integral sliding mode predictive control scheme (ISMPC). The model predictive control (MPC) is designed as a nominal controller to generate an optimal ‘reference trajectory’ for the actual (controlled) system. The integral sliding mode control (ISMC) acts as an auxiliary controller to combat disturbances and ensure that the actual trajectory is restricted in a robust tube centred on the reference trajectory. In addition, during the design process, the practical constraints are satisfied through the sliding mode boundary layer, without the need for iterative computation of robust invariant sets. The proposed control scheme combines the advantages of model predictive control MPC and ISMC, ensuring the optimal performance and robustness of the controlled system, and effectively reducing the computational burden required for the control scheme. Finally, the stability of the closed-loop system is proved theoretically, and the effectiveness of the proposed method is verified by the simulation results of two examples.

Replay attack detection based on smooth watermarking and encoding and decoding

The proposed detection method in this study, which involves smooth watermarking, encoding, and decoding, aims to address replay attacks in time-delay systems during communication. Implementing an event-triggered technique helps save communication resources and cut down on unnecessary information transmission. A weighted sliding average method is employed to smooth the system’s watermark influenced by time delay, decreasing unnecessary fluctuations and boosting anti-attack resilience. The encoding and decoding systems were ingeniously designed to increase the invisibility of the smooth watermarking, making it more sensitive in detecting replay attacks. Multiple attack scenarios are meticulously examined, after which rigorous simulation tests are carried out to prove the proposed scheme’s reliability, effectiveness, and substantial superiority.

A Periodic Delay Differential Equation Model for Mosquito Suppression Based on Beverton–Holt-Type of Birth

A Periodic Delay Differential Equation Model for Mosquito Suppression Based on Beverton–Holt-Type of Birth

New Oscillation Conditions Test for Neutral Differential Equations

In this work, we investigate new oscillation conditions of delay differential equations of second-order. We start by finding new relations and inequalities between the corresponding function and its solution, also with its derivatives. The new inequalities allow us to improve the asymptotic and oscillation conditions of the solutions of the this equation. By using an improved method and approach, we obtain new criteria that test the all solutions oscillation. The article provides some examples that highlight the significance of new conditions and properties.

The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited

The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited

Relaxed optimal control problem for a finite horizon G-SDE with delay and its application in economics

This paper investigates the existence of a G-relaxed optimal control of a controlled stochastic differential delay equation driven by G-Brownian motion (G-SDDE in short). First, we show that optimal control of G-SDDE exists for the finite horizon case. We present as an application of our result an economic model, which is represented by a G-SDDE, where we studied the optimisation of this model. We connected the corresponding Hamilton–Jacobi–Bellman equation of our controlled system to a decoupled G-forward backward stochastic differential delay equation (G-FBSDDE in short). Finally, we simulate this G-FBSDDE to get the optimal strategy and cost.

Sampled-data control of a class of stochastic feedforward nonlinear systems with large delays and sparse sampling

This paper investigates the problem of sampled-data control for a class of stochastic nonlinear systems with delays in the input and state. It is shown that under an upper-triangular linear growth condition, a linear memoryless sampled-data feedback controller is able to globally stabilise the stochastic nonlinear time-delay systems, no matter how large the time delay and sampling period are. The memoryless sampled-data feedback controller is designed by using the emulation method together with the feedback domination technique. With the aid of Lyapunov-Krasovskii functional method, the constructed memoryless sampled-data feedback controller achieves global exponential stability (GES) in mean square of the hybrid closed-loop systems. Application of the sampled-data control scheme is demonstrated through a practical example with simulation.

Practical stability of stochastic differential delay equations driven by G-Brownian motion with general decay rate

This article is concerned with the quasi sure practical stability of nonlinear stochastic differential delay equations driven by G-Brownian motion (G-SDDEs) with a general decay rate. Sufficient conditions are established by constructing appropriate G-Lyapunov functionals. Moreover, we provide some numerical examples to demonstrate the effectiveness of the obtained results. For more information see https://ejde.math.txstate.edu/Volumes/2024/70/abstr.html

Improvements on the Leighton‐type oscillation criteria for impulsive differential equations and extension to non‐canonical case

We investigate the oscillatory behavior of solutions of second‐order linear differential equations with impulses. These equations can be categorized into two types: canonical and non‐canonical. This study examines the canonical and non‐canonical impulsive differential equations in connection with the famous Leighton‐type oscillation theorem. The well‐known theorem states that every solution of where and are continuous with , is oscillatory if and This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition fails, that is, when the equation is of non‐canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non‐canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

Synchronization Between Kerr Cavity Solitons and Broad Laser Pulse Injection

The synchronization of a soliton frequency comb in a Kerr cavity with pulsed laser injection is studied numerically. The neutral delay differential equation is used to model the light dynamics in the cavity. This model allows for the investigation of both cases where the pulse repetition period is close to the cavity round-trip time and where the repetition period of the injection pulses is close to a rational fraction M/N of the round-trip time. It is demonstrated that solitons can exist in this latter case, provided that the injection pulses are of a higher amplitude, which is directly proportional to the number M. Furthermore, it is shown that the synchronization range of the solitons is also proportional to the number M. The solitons excited by pulses with a period slightly different from the M:N-resonance can be destabilized by the Andronov–Hopf bifurcation, which occurs when the injection level at the soliton position decreases to M times the injection amplitude corresponding to the saddle-node bifurcation in a model equation with uniform injection.