Linear Neutral Equations Research Articles

Nondegeneracy and Uniqueness of Periodic Solution for a Neutral Differential Equation

We analyze the nondegeneracy of second-order linear neutral differential equation (x(t)-cx(t-τ))′′=a(t)x(t),where c is a constant. By applications of the nondegeneracy of this linear neutral equation and an extension of Mawhin’s continuation theorem, we obtain existence and uniqueness of periodic solution for the prescribed second-order neutral differential equations. At last, we give two examples to show the applications of the theorems.

Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure

The stability theory for linear neutral equations subjected to delay perturbations is addressed. It is assumed that the delays cannot necessarily vary independently of each other, but depend on a possibly smaller number of independent parameters. As a main result, necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum, which take into account the precise dependency structure of the delays. In the derivation of the stability theory, results from realization theory and determinantal representations of multivariable polynomials play an important role. The observations and results obtained in the paper are first illustrated and validated with a numerical example. Next, the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed.

Stability impact of small delays in proportional–derivative state feedback

This paper points out specific stability problems of the state derivative and proportional–derivative state feedback which may result from the presence of otherwise negligible delays in the control loops. It is shown that the stability problems arise from the neutral character of the closed-loop system with the state derivative action under small delay perturbations. As the main result, a stability criterion is derived for a safe implementation of the state derivative feedback. Furthermore, a scheme including filtered state derivative feedback is proposed to overcome the destabilizing effects of small feedback delays. Two application examples are presented. From a theoretical point of view, the main contribution of the paper lies in the stability theory of linear neutral equations with dependent delays, subjected to delay perturbations.

2T-periodic solution form order neutral type differential equations with time delays

Periodic solution of m order linear neutral equations with constant coefficient and time delays was studied. Existence and uniqueness of 2 T-periodic solutions for the equation were discussed by using the method of Fourier series. Some new necessary and sufficient conditions of existence and uniqueness of 2T-periodic solutions for the equation are obtained. The main result is used widely. It contains results in some correlation paper for its special case, improves and extends the main results in them. Existence of periodic solution for the equation in larger number of particular case can be checked by using the result, but cannot be checked in another paper. In other words, the main result in this paper is most generalized for (1), the better result cannot be found by using the same method.

Oscillation and Small Solutions of Autonomous Linear Neutral Equations

In this paper we deal with the linear neutral functional differential equation d[ f( x t )]/ dt + g( x t ) = 0. We conclude that all small components of solutions vanish after a finite time, as do all small solutions, and that the equation is oscillatory if and only if its characteristic equation has no real roots.

The time control theory of linear differential equations of neutral type

The paper tackles the time optimal control theory of linear neutral equations. Controllability questions in function space are answered. The existence and forms of an optimal control are established both in function space and in Euclidean space.

Coincidence degree and periodic solutions of neutral equations

The problem of existence of periodic solutions for some nonautonomous neutral functional differential equations is examined. It is an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations and of a generalized Leray-Schauder theory. Although proofs are simple, the results are nontrivial extensions to the neutral case of existence theorems for periodic solutions of functional differential equations.