Equations Of Neutral Type Research Articles

Existence of nonoscillatory solutions for second-order nonlinear differential equations of neutral type

In this work, we consider the existence of nonoscillatory solutions of second order nonlinear neutral differential equations. Our results include as special cases some well-known results for linear and nonlinear equations. We use the Lebesgue’s dominated convergence theorem and Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

Stable Functionals of Neutral-Type Dynamical Systems

We consider a controlled dynamical system under conditions of disturbances. Its motion is described by functional differential equations of neutral type in the form of J. Hale. A functional of the motion history is said to be stable with respect to this system if there exists a control strategy that guarantees the monotonicity of this functional for any disturbances. We study various nonlocal and infinitesimal conditions for the stability of functionals.

The continuous Galerkin finite element methods for linear neutral delay differential equations

In the paper, the superconvergence of continuous Galerkin finite element methods (CGFEMs) for linear delay differential equations of neutral type is presented. By the orthogonal analysis method and under the suitable condition, it is proven that the finite element solution is superconvergent at the nodal points and Lobatto points. Numerical experiments further confirm the effectiveness and the superconvergence of the CGFEMs.

Measure Neutral Functional Differential Equations as Generalized ODEs

In this paper, we introduce a class of measure neutral functional differential equations of type $$\begin{aligned} \mathrm {D}[N(x_t,t)]=f(x_t,t)\mathrm {D}g(t) \end{aligned}$$ through the relation with a certain class of generalized ordinary differential equations introduced in Federson and Schwabik (Differ Integral Equ 19(11):1201–1234, 2006) (we write generalized ODEs), using similar ideas to those of Federson et al. (J Differ Equ 252(6):3816–3847, 2012). By means of the correspondence with generalized ODEs, we state results on the existence, uniqueness and continuous dependences of solutions for our equation of neutral type.

On the Numerical Solution of Differential Games for Neutral-Type Linear Systems

The paper deals with a zero-sum differential game, in which the dynamics of a conflict-controlled system is described by linear functional differential equations of neutral type and the quality index is the sum of two terms: the first term evaluates the history of motion of the system realized up to the terminal time, and the second term is an integral–quadratic evaluation of the corresponding control realizations of the players. To calculate the value and construct optimal control laws in this differential game, we propose an approach based on solving a suitable auxiliary differential game, in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index evaluates the motion at the terminal time only. To find the value and the saddle point in the auxiliary differential game, we apply the so-called method of upper convex hulls, which leads to an effective solution in the case under consideration due to the specific structure of the quality index and the geometric constraints on the control actions of the players. The efficiency of the approach is illustrated by an example, and the results of numerical simulations are presented. The constructed optimal control laws are compared with the optimal control procedures with finitedimensional approximating guides, which were developed by the authors earlier.

ON A LINEAR NEUTRAL-TYPE EQUATION IN A DEGENERATE CASE

For the linear neutral-type equation, m(t)x′(t)+p(t)x′(h(t))+(Kx)(t)=f(t), t ∈ [a, b], solvability conditions for the Cauchy problem are obtained. It is allowed that the coefficient, m(·), can be zero on a positive-measure set of [a, b] (the degenerate case).

Exponential Stability of Coupled Linear Delay Time-Varying Differential–Difference Equations

Coupled linear delay time-varying differential–difference equations are considered. Explicit criteria for exponential stability of such equations are presented. The obtained results are used to derive sufficient conditions for exponential stability in linear time-varying differential equations of neutral type and in singular linear delay time-varying differential equations. Some illustrative examples are given.

Convergence Analysis of Spectral Method for Neutral Multi-pantograph Equations

The main purpose of this work is to provide a spectral method based on the operational matrices of the Legendre polynomials for solving neutral multi-pantograph equations. We analyze the convergence properties of the proposed method. It is shown that for the multi-pantograph equations of neutral type, the spectral method yields the exponential order of convergence. Some examples are given to demonstrate the high precision, fast computation and good performance of the new scheme.

Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type

We investigate oscillatory behavior of solutions to a class of second-order nonlinear neutral delay dynamic equations with nonpositive neutral coefficients. In particular, we study the corresponding noncanonical neutral differential equations. New oscillation criteria are established that complement and improve related contributions to the subject. An example is given to illustrate the main results.

Oscillation theorems for second-order nonlinear delay differential equations of neutral type

In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form \[\frac{d}{dt}\bigg[r(t)\frac{d}{dt}[x(t)+p(t)x(\tau(t))]\bigg]+q(t)G\bigl(x(\sigma(t))\bigr)=0\: for\: t\geq t_{0},\] under the assumptions $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ and $\int^{\infty}\frac{1}{r(\eta)}d\eta<\infty$ for various ranges of the bounded neutral coefficient $p$. Unlike most of the previous results, $\tau^{\prime}$ is allowed to be oscillatory. Further, some illustrative examples showing applicability of the new results are included.

Existence of a Value and a Saddle Point in Positional Differential Games for Neutral-Type Systems

For a conflict-controlled dynamical system described by functional differential equations of neutral type in Hale’s form, we consider a differential game with a performance index that estimates the motion history realized up to the terminal time and includes an integral estimation of realizations of the players’ controls. The game is formalized in the class of pure positional strategies. The main result is the proof of the existence of a value and a saddle point in this game.

Functional differential equations of neutral type with integrable weak singularity: hydrogen thermal desorption model

One of the technological challenges for hydrogen materials science (including the ITER project) is the currently active search for structural materials with various potential applications that will have predetermined limits of hydrogen permeability. One of the experimental methods is thermal desorption spectrometry (TDS). A hydrogen-saturated sample is degassed under vacuum and monotone heating. The desorption flux is measured by mass spectrometer to determine the character of interactions of hydrogen isotopes with the solid. We are interested in such transfer parameters as the coefficients of diffusion, dissolution, desorption. The paper presents a thermal desorption functional differential equations of neutral type with integrable weak singularity and a numerical method for TDS spectrum simulation, where only integration of a nonlinear system of low order ordinary differential equations (ODE) is required. This work is supported by the Russian Foundation for Basic Research (project 15-01-00744).

On stability of nonlinear neutral functional differential equations

We address the challenging problem of the exponential stability of nonlinear time-varying functional differential equations of neutral type. By a novel approach, we present explicit sufficient conditions for the exponential stability of nonlinear time-varying neutral functional differential equations. A discussion of the obtained results and illustrative examples are given.

A Descriptor Approach to Robust Leader-Following Output Consensus of Uncertain Multi-Agent Systems With Delay

In this technical note, a descriptor approach to leader-following output consensus of multi-agent systems with both stationary and dynamic leaders is given in the presence of transmission delay and model uncertainty. The proposed method can deal with stable and unstable agents described by general linear models. To this end, a new proportional-derivative-integral (PID) consensus protocol for the closed-loop multi-agent system is proposed under a directed graph. Applying this consensus protocol to the multi-agent system leads to a time-delay closed-loop equation of neutral type. To deal with the resulting neutral system, a descriptor model transformation is used to derive delay-dependent sufficient conditions for the existence of the consensus protocol in terms of certain linear matrix inequalities (LMI). The application of the proposed method is illustrated in a teleoperation system. Simulation results are given to show the effectiveness of the proposed approach.

Existence and Uniqueness of the Solution to the Cauchy Problem for the Stochastic Reaction-Diffusion Differential Equation of Neutral Type

We prove a theorem on the existence and uniqueness of a mild solution to the Cauchy problem for a stochastic differential equation of neutral type in the weighted Hilbert space.

On existence of solution of the Cauchy problem for nonlinear diffusion stochastic partial differential-difference equations of neutral type with random external perturbations

The question related to the existence of the Cauchy problem solution in the class of nonlinear diffusion stochastic partial differential-difference equations of a neutral type with random external disturbances which are independent from the Wiener process is considered. Sufficient conditions are obtained for the diffusion coefficients of nonlinear stochastic differential-difference equations of a neutral type (NDSDRRNT) that guarantee the existence of the solution with the probability of 1. The method of the proof is based on the results of O.M. Stanzhitsky and A.O. Tsukanova on the existence and uniqueness of the Cauchy problem solution for the stochastic differential reaction-diffusion equation of a neutral type. In this paper, we prove the existence of a "mild solution" of NDSDRRNT. In some cases, such equations are mathematical models of real processes, the consideration of which is planned in the future work.

Razumikhin method for impulsive functional differential equations of neutral type

Although the well-known Razumikhin method has been well developed for the stability of functional differential equations with or without impulses and it is very useful in applications, so far there is almost no result of Razumikhin type on stability of impulsive functional differential equations of neutral type. The purpose of this paper is to close this gap and establish some Razumikhin-based stability results for impulsive functional differential equations of neutral type. A kind of auxiliary function N(t) that has great randomicity is introduced to Razumikhin condition. Some examples are given to show the effectiveness and advantages of the developed method.

Periodic Solutions for Non-Autonomous Neutral Functional Differential Equations with Finite Delay

In this work, we study the existence of periodic solutions for some non-autonomous nonlinear partial functional differential equation of neutral type. We assume that the linear part is non-densely defined and generates an evolution family under the conditions introduced by N. Tanaka. The delayed part is assumed to be ω-periodic with respect to the first argument. Using a fixed-point theorem for multivalued mapping, some sufficient conditions are given to prove the existence of periodic solutions. An example is shown to illustrate our results.

Norm continuity of solution semigroups of a class of neutral functional differential equations with distributed delay

In this note, we shall consider the norm continuity of a class of solution semigroups associated with linear functional differential equations of neutral type with time lag r > 0 in Hilbert spaces. The norm continuity plays an important role in the analysis of asymptotic stability of the system under consideration by means of spectrum approaches. We shall show that for a square integrable neutral delay term and an unbounded infinitesimal generator A multiplied by a square integrable weight function in the distributed delay term, the associated solution semigroup of the system is norm continuous at every t > r .

Control by time delayed feedback near a Hopf bifurcation point

In this paper we study the stabilization of rotating waves using time delayed feedback control. It is our aim to put some recent results in a broader context by discussing two different methods to determine the stability of the target periodic orbit in the controlled system: 1) by directly studying the Floquet multipliers and 2) by use of the Hopf bifurcation theorem. We also propose an extension of the Pyragas control scheme for which the controlled system becomes a functional differential equation of neutral type. Using the observation that we are able to determine the direction of bifurcation by a relatively simple calculation of the root tendency, we find stability conditions for the periodic orbit as a solution of the neutral type equation.