Neutral Delay Research Articles

Asymptotic-numerical solvers for linear neutral delay differential equations with high-frequency extrinsic oscillations

We present a method to compute efficiently and easily solutions of systems of linear neutral delay differential equations with highly oscillatory forcing terms. This method is based on asymptotic expansions in inverse powers of a perturbed oscillatory parameter. Each term of the asymptotic expansion is derived by recursion. The cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided and show that with few terms of the asymptotic expansion, the solutions are approximated with high accuracy.

Stability Analysis of Linear Neutral Delay Systems With Two Delays via Augmented Lyapunov–Krasovskii Functionals

Stability analysis is considered in this paper for linear neutral delay systems subject to two different delays in both the state variables and the retarded derivatives of state variables. By choosing a suitable state vector indexed by an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , a new augmented Lyapunov-Krasovskii functional (LKF) is constructed, and a stability criterion based on linear matrix inequalities is developed accordingly. It is shown that the proposed condition is less conservative than the existing methods due to the introduction of the delay-product-type integral terms in the LKF. The resulting stability criterion is then applied to the robust stability analysis of neutral delay systems with norm-bounded uncertainty. Moreover, a delay-independent stability criterion is developed based on the proposed LKF, and its frequency-domain interpretation is also given. These developed stability criteria indexed by an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> exhibit a hierarchical character: the larger the integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , the less conservatism of the resulting stability criterion. Finally, two numerical examples are carried out to illustrate the effectiveness of the proposed method.

Stability analysis of some neutral delay-differential equations with a frequency-domain approach

&lt;p style='text-indent:20px;'&gt;Two general schemes of neutral delay differential equations are considered. They represent non delayed systems with delayed, nonlinear feedback control. The stability conditions in terms of system parameters are derived using an approach based on feedback systems, namely, the Nyquist stability criterion. It is also shown that those results coincide with some already found in the literature, and others derived here, which are based on classical tools, that investigate directly the roots of characteristic equation. Finally, some examples are given to illustrate the usefulness of this approach.&lt;/p&gt;

An implicit system of delay differential algebraic equations from hydrodynamics

Direct spring operated pressure relief valves connected to a constantly charged vessel and a downstream pipe have a complex dynamics. The vessel-valve subsystem is described with an autonomous system of ordinary differential equations, while the presence of the pipe adds two partial differential equations to the mathematical model. The partial differential equations are transformed to a delay algebraic equation coupled to the ordinary differential equations. Due to a square root nonlinearity, the system is implicit. The linearized system can be transformed to a standard system of neutral delay differential equations (NDDEs) having more elaborated literature than the delay algebraic equations. First, the different forms of the mathematical model are presented, then the transformation of the linearized system is conducted. The paper aims at introducing this unusual mathematical model of an engineering system and inducing research focusing on the methodology to carry out bifurcation analysis in implicit NDDEs.

Differential equations of the neutral delay type: More efficient conditions for oscillation

&lt;abstract&gt;&lt;p&gt;In this article, we derive an optimized relationship between the solution and its corresponding function for second- and fourth-order neutral differential equations (NDE) in the canonical case. Using this relationship, we obtain new monotonic properties of the second-order equation. The significance of this paper stems from the fact that the asymptotic behavior and oscillation of solutions to NDEs are substantially affected by monotonic features. Based on the new relationships and properties, we obtain oscillation criteria for the studied equations. Finally, we present examples and review some previous theorems in the literature to compare our results with them.&lt;/p&gt;&lt;/abstract&gt;

Global exponential bipartite synchronization for neutral memristive inertial coupling mixed time-varying delays neural networks with antagonistic interactions

In this paper, the global exponential bipartite synchronization (GEBS) for neutral-type memristive inertial coupling mixed time-varying delays neural networks (NTMI-CMTVDNNs) with synergistic and antagonistic interactions (SAI) is investigated. Firstly, the NTMI-CMTVDNNs with SAI between neurons are modeled by a signed graph. Inertial network is equivalently transformed into the first-order differential equations via variable substitution. Secondly, a novel lemma is proposed to deal with parameter mismatch of the above modeled network. By designing delay-dependent and delay-independent discontinuous control laws, the delay-dependent sufficient conditions under the bounded and unbounded activation functions are obtained to ensure the GEBS of the NTMI-CMTVDNNs with SAI with or without a leader neurons, respectively. Finally, two numerical simulation examples illustrate the validity of our theoretical results.

On exponential stabilization of a nonlinear neutral wave equation

This work aims to study a nonlinear wave equation subject to a delay of neutral type. The nonlinearity and the delay appear in the second time derivative. In spite of the fact that delays by nature, have an instability effect on the structures, the strong damping is sufficient to allow the system to reach its equilibrium state with an exponential manner. The difficulties arising from the nonlinearity have been overcome by using an inequality due to a Sobolev embedding theorem. The main result has been established without any condition on the coefficient of the neutral delay.

Amended Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of Higher-Order Functional Differential Equations

Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them.

Stability of quaternion-valued neutral-type neural networks with leakage delay and proportional delays

This paper is concerned with the stability issue of quaternion-valued neural networks with neutral delay, proportional delay and leakage delay. Taking use of the principle of homeomorphism, techniques of matrix inequality and Lyapunov stability theory, a main stability criterion is derived in the form of quaternion-valued linear matrix inequality for ensuring the unique existence and global stability of the equilibrium point for the considered quaternion-valued neural networks. An illustrative example and its simulations are given to show the effectiveness of the theoretical result.

Pinning synchronization of fractional memristor-based neural networks with neutral delays and reaction–diffusion terms

This article explores the pinning synchronization of fractional memristor-based neural networks under the effect of neutral delays and reaction–diffusion. First, we introduce neutral delays into the Riemann–Liouville fractional reaction–diffusion memristor-based neural networks model. Subsequently, two innovative pinning controllers incorporating the current and past states are designed. Next, we provide a new modified Lyapunov functional for obtaining two less conservative criteria to enable asymptotic synchronization of fractional reaction–diffusion memristor-based neural networks with neutral delays with the aid of Green’s theorem, inequality techniques, and pinning control techniques. In addition, the results based on algebraic inequalities correct and improve some existing ones. Finally, simulations are employed to validate the obtained conclusions.

Existence of mild solution to the fractional order impulsive nonlinear control system

In this paper we study the existence of mild solution to the fractional impulsive nonlinear mixedtype integro-differential partial equation with neutral infinite delay and nonlocal conditions in a Banach space. As a prove of this investigation we address some important concepts of work such as fractional calculus, semigroup of operators and Krasnoselskii's fixed point theorem.

A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations

A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations

New Delay-Partitioning LK-Functional for Stability Analysis with Neutral Type Systems

This paper investigates the stability issues associated with neutral-type delay systems. Firstly, the delay-partitioning method is employed to construct a brand-new LK-functional candidate. The discrete delay and a neutral delay are divided into several piecewise points through a relaxable sequence of constant numbers, are increasing at a steady rate and are not larger than 1. Secondly, to fully use the interconnection information among the delayed state vectors, a new LK-functional is constructed. Thirdly, the recently published single/multiple integral inequalities are employed to bound the derivative of the newly developed LK function. Finally, a novel stability criterion for neutral systems is developed based on the above treatment. Furthermore, a new corollary is also proposed for the condition of τ=h. The benefits and productivities of our method are demonstrated by numerical examples.

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.

Functional Integro-Differential Equations with State-Dependent Delay and Non-Instantaneous Impulsions: Existence and Qualitative Results

This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with measures of noncompactness. The obtained result is illustrated by an example at the end.

Characterization of First Order 2-dim Neutral Delay Difference Systems

Characterization of First Order 2-dim Neutral Delay Difference Systems

An Investigation on the Optimal Control for Hilfer Fractional Neutral Stochastic Integrodifferential Systems with Infinite Delay

The main concern of this manuscript is to study the optimal control problem for Hilfer fractional neutral stochastic integrodifferential systems with infinite delay. Initially, we establish the existence of mild solutions for the Hilfer fractional stochastic integrodifferential system with infinite delay via applying fractional calculus, semigroups, stochastic analysis techniques, and the Banach fixed point theorem. In addition, we establish the existence of mild solutions of the Hilfer fractional neutral stochastic delay integrodifferential system. Further, we investigate the existence of optimal pairs for the Hilfer fractional neutral stochastic delay integrodifferential systems. We provide an illustration to clarify our results.

Pinning synchronization of stochastic neutral memristive neural networks with reaction–diffusion terms

This paper investigates the pinning synchronization of stochastic neutral memristive neural networks with reaction–diffusion terms. Firstly, two novel pinning controllers, which contain both current state and past state, are designed. Subsequently, in terms of Green’s theorem, inequality technology, stochastic analysis theory and pinning control technology, two easy-to-test sufficient conditions based on algebraic inequalities are obtained to ensure the mean-square asymptotic synchronization of stochastic memristive neural networks with neutral delays and reaction–diffusion terms by providing a new Lyapunov–Krasovskii functional. In addition, some existing results can be regarded as special cases of our work. Finally, illustrative examples further verify the correctness and validity of the derived results.

Exploiting delay differential equations solved by Eta functions as suitable mathematical tools for the investigation of thickness controlling in rolling mill

This work is dedicated to introducing the properties and application of Eta functions. We derive the properties of the Eta function, such as the generating function, integral representation, and the Laplace transform. Also, some properties of the Eta-based functions are introduced. To show the advantages of the Eta-based functions in the computational method, we develop a new numerical method to solve the state-dependent and time-dependent neutral delay differential equation based on the Eta-based function. We introduce the operational matrix of derivative for the Eta-base functions to develop the new numerical method. This method uses the operational matrix of derivative and collocation method to convert the delay differential equation to a system of nonlinear algebraic equations. We derive the technique’s error bound and establish the method’s accuracy by solving some examples, which are state-dependent and time-dependent delay differential equations. In the end, we study the model of the metal forming process by rolling the mill using the new numerical method to show the advantages of using the Eta-based function for solving a more practical problem.

Hopf bifurcation calculation in neutral delay differential equations: Nonlinear robotic arms subject to delayed acceleration feedback control

This study gives the derivation of the Hopf bifurcation calculation for neutral delay differential equations using the centre manifold reduction theorem and normal form calculation. The whole concept was inspired by a case study where a robotic arm subjected to nonlinear stiffness with delayed acceleration feedback controller is modelled. Two different configurations are distinguished depending on the location of the acceleration sensor, a collocated and a non-collocated one. After a brief investigation of the linear stability, the bifurcation occurring at the loss of stability is calculated with the presented analytic equations for neutral delay differential equations. Then, a nonlinear term is introduced in the control law that can modify the occurring subcritical behaviour and improve the robustness of the system. The analytic results are carefully analysed and validated via a numerical continuation software, which also provided useful information about the global behaviour of the bifurcations in addition to the locally valid analytic results.