Neutral Delay Research Articles

On stability of numerical solutions of neutral stochastic delay differential equations with time‐dependent delay

This paper focuses on exponential stability of numerical solutions of neutral stochastic delay differential equations. A novel approach is introduced to study numerical approximation methods of neutral stochastic differential equations with time‐dependent delay. In contrast to the advances in the literature, this work provides a new method and new criteria for which the Euler‐Maruyama approximation method and the backward Euler‐Maruyama approximation method can reproduce exponential stability in mean square and almost sure exponential stability for sufficiently small step sizes. Two examples are provided to demonstrate the effectiveness of our criteria.

Oscillation of Nonlinear Neutral Delay Difference Equations of Fourth Order

This paper focuses on the study of the oscillatory behavior of fourth-order nonlinear neutral delay difference equations. The authors use mathematical techniques, such as the Riccati substitution and comparison technique, to explore the regularity and existence properties of the solutions to these equations. The authors present a new form of the equation: Δ(a(m)(Δ3z(m))p1−1)+p(m)wp2−1(σ(m))=0, where z(m)=w(m)+q(m)w(m−τ) with the following conditions: ∑s=m0∞1a(1p1−1(s))=∞. The equation represents a system where the state of the system at any given time depends on its current time and past values. The authors demonstrate new insights into the oscillatory behavior of these equations and the conditions required for the solutions to be well-behaved. They also provide a numerical example to support their findings.

Exponential input-to-state stability for neutral stochastic delay differential equations with Lévy noise and Markovian switching

This paper mainly focuses on the pth ()-moment input-to-state stability (ISS) of neutral stochastic delay differential equations (NSDDEs) with Lévy noise and Markovian switching. By using the generalized integral inequality and the Lyapunov function methodology, the ISS, integral input-to-state stability (iISS), and stochastic input-to-state stability (SISS) of such equations are obtained. When the input signal is a constant signal and a zero signal, the pth ()-moment ISS reduces to the pth ()-moment practical exponential stability and the pth ()-moment exponential stability, respectively. Finally, an example of the mass–spring–damping (MSD) model under the stochastic perturbation is given to verify the validity of the results.

Nonoscillatory Properties of Fourth Order Nonlinear Neutral Differential equation

In this paper, the oscillatory and nonoscillatory qualities for every solution of fourth-order neutral delay equation are discussed. Some conditions are established to ensure that all solutions are either oscillatory or approach to zero as . Two examples are provided to demonstrate the obtained findings.

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2^{-n}, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

H∞and Asymptotic Stability via delay feedback for hybrid neutral stochastic delay differential equations with Lévy noise

AbstractThis work addresses existence and stabilization problem for a hybrid neutral stochastic delay differential equations with Lévy noise (HNSDDELN). The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. We first prove the existence and uniqueness of the solution. We then design a delay feedback controller to make an unstable HNSDDELN $H_{\infty }$ and asymptotically stable in ${\mathbb{L}}^{p}$. We end up with a numerical example that corroborates our theoretical findings.

Asymptotic stability of nonlinear neutral delay integro-differential equations

Abstract In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.

Improved Properties of Positive Solutions of Higher Order Differential Equations and Their Applications in Oscillation Theory

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we improve these properties by giving them an iterative character, allowing us to apply the criteria more than once. The results obtained in this paper are characterized by the fact that they do not require the existence of unknown functions and do not need the commutation condition to composition of the delay functions, which are necessary conditions for the previous related results.

Fault diagnosis and fault-tolerant control design for neutral time delay system

This paper presents a new approach of fault-tolerant control (FTC) for the transmission line as a neutral variable time-delay system. The main goal of this work guarantees faulty neutral variable time delay system stabilization using the state feedback control design based on Lyapunov function and the Linear Matrix Inequality resolution. The use of the FTC method is to achieve actuator and sensor fault compensation. This method is based on two steps. The first one is the synthesis of a nominal control, which remains to maintain the closed-loop system stability. The second step is based on adding a new control law to the nominal one to compensate the fault effect on system behaviour and maintain the desired performance in the closed loop system. Then, a conception of an adaptive observer is used to detect and estimate the fault. Finally, the developed approach is applied for the transmission line. The given results are presented to prove the effectiveness of this approach.

Mixed-Delay-Dependent Augmented Functional for Synchronization of Uncertain Neutral-Type Neural Networks with Sampled-Data Control

In this paper, the synchronization problem of uncertain neutral-type neural networks (NTNNs) with sampled-data control is investigated. First, a mixed-delay-dependent augmented Lyapunov–Krasovskii functional (LKF) is proposed, which not only considers the interaction between transmission delay and communication delay, but also takes the interconnected relationship between neutral delay and transmission delay into consideration. Then, a two-sided looped functional is also involved in the LKF, which effectively utilizes the information on the intervals [tk,t], [tk−τ,t−τ],[t,tk+1),[t−τ,tk+1−τ). Furthermore, based on the suitable LKF and a free-matrix-based integral inequality, two synchronization criteria via a sampled-data controller considering communication delay are derived in forms of linear matrix inequalities (LMIs). Finally, three numerical examples are carried out to confirm the validity of the proposed criteria.

On the approximate controllability of non-densely defined Sobolev-type nonlocal Hilfer fractional neutral Volterra-Fredholm delay integro-differential system

The main focus of our discussion is the approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral Volterra–Fredholm integro-differential systems. The important outcomes are obtained by applying concepts and ideas from fractional calculus, Dhage’s fixed point theorem, and multivalued maps. Finally, an example is given for deriving theoretical results.

Exponential Stability for Neutral Stochastic Differential Delay Equations with Markovian Switching and Nonlinear Impulsive Effects

Exponential Stability for Neutral Stochastic Differential Delay Equations with Markovian Switching and Nonlinear Impulsive Effects

Oscillation of higher-order canonical delay differential equations: comparison theorems

In this work, we study the oscillatory properties of a higher-order neutral delay differential equation. By using the principle of comparison with equations of the first order we establish a set of new oscillation criteria for this equation under the canonical condition. Furthermore, the new criteria extend and complement some previous results in the literature. To that end, we compare these criteria by applying them to special cases of the equations under consideration in order to determine which one is the most efficient and least restrictive.

Arbitrary decay for a nonlinear Euler-Bernoulli beam with neutral delay

In this paper, the free transverse vibration of a nonlinear Euler- Bernoulli beam under a neutral type delay is considered. In order to suppress the beam transverse vibrations, a boundary control based on the Lyapunov method is designed. The novelty of this paper is the ability to get a wide variety of energy decay rates under free vibration conditions.

Explicit numerical approximations for McKean-Vlasov neutral stochastic differential delay equations

This paper studies the numerical methods to approximate the solutions for a sort of McKean-Vlasov neutral stochastic differential delay equations (MV-NSDDEs) that the growth of the drift coefficients is super-linear. First, we obtain that the solution of MV-NSDDE exists and is unique. Then, we use a stochastic particle method, which is on the basis of the results about the propagation of chaos for a stochastic particle system, to deal with the approximation of the law. Furthermore, we construct the tamed Euler-Maruyama numerical scheme with respect to the corresponding particle system and obtain the rate of convergence. Combining propagation of chaos and the convergence rate of the numerical solution to the particle system, we get the convergence error between the numerical solution and exact solution of the original MV-NSDDE in the stepsize and number of particles.

Exponential stability and numerical simulation of a Bresse-Timoshenko system subject to a neutral delay

<abstract><p>In the present work, we consider a one-dimensional Bresse-Timoshenko system with neutral delay term and a viscous damping acting on vertical displacement of the beam. Under appropriate assumptions on the kernel of this kind of delay and based on the multipliers method, we construct a suitable Lyapunov functional that allows us to establish an exponential decay of the energy in spite of the existence of the delay. Moreover, our result does not depend on any condition on the coefficients of the system. Finally, we present some numerical results to illustrate the theoretical result obtained.</p></abstract>

Sharp results for oscillation of second-order neutral delay differential equations

The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which essentially improve a number of related ones from the literature. A couple of examples illustrate the value of the results obtained.

Stability test for neutral type delay systems: a piecewise linear approximation scheme*

A necessary and sufficient test for the stability of neutral type linear time-delay systems is presented. Our approach combines the Lyapunov functional expressed in terms of the delay Lyapunov matrix for this class of systems with a piecewise linear approximation of the functional argument. As a result, a stability criterion is given in terms of the non-negative definiteness test of a matrix. An example is presented to validate the obtained result.

Study on the oscillation of solution to second-order impulsive systems

<abstract><p>In the present article, we set the if and only if conditions for the solutions of the class of neutral impulsive delay second-order differential equations. We consider two cases when it is non-increasing and non-decreasing for quotient of two positive odd integers. Our main tool is the Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.</p></abstract>

Linearized instability for differential equations with dependence on the past derivative

We provide a criterion for instability of equilibria of equations in the form x ˙ ( t ) = g ( x t ′ , x t ) , which includes neutral delay equations with state-dependent delay. The criterion is based on a lower bound Δ > 0 for the delay in the neutral terms, on regularity assumptions of the functions in the equation, and on spectral assumptions on a semigroup used for approximation. The spectral conditions can be verified studying the associated characteristic equation. Estimates in the C 1 -norm, a manifold containing the state space X 2 of the equation and another manifold contained in X 2 , and an invariant cone method are used for the proof. We also give mostly self-contained proofs for the necessary prerequisites from the constant delay case, and conclude with an application to a mechanical example.