Linear Neutral Functional Differential Equations Research Articles

Almost automorphy for abstract neutral differential equations via control theory

In this paper we study the existence of almost automorphic solutions for a class of linear neutral functional differential equations with finite delay and values in a Banach space. We show that the existence of an almost automorphic mild solution is related to the approximate controllability of a distributed control system. We applied our results to establish the existence of an almost automorphic solution for a neutral wave equation with delay.

Hyperbolicity of solution semigroups for linear neutral differential equations

Consider the linear neutral functional differential equation of the form $$\left\{\begin{array}{l@{\quad }l}\frac{\partial}{\partial t}Fu_t=BFu_t+\Phi u_t, & t\ge 0,\\[3pt]u_0(s)=\varphi(s),& s\in [-r,0],\end{array}\right.$$ where the function u(⋅) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ we prove that the solution semigroup for this equation is hyperbolic provided that the differential operator B generates a hyperbolic semigroup on X.

Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in Lp Spaces

AbstractWe present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) in Lp spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the NFDE as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems.

Spectral Properties and Finite Pole Assignment of Linear Neutral Systems in Banach Spaces

We will consider a pole assignment problem for a class of linear neutral functional differential equations in Banach spaces. We will think of the neutral system studied as that of involving no time delays and reduce the study of adjoint semigroups and spectral properties of neutral equations to those of Cauchy problems. Under the assumption that both the control and eigenspace of pole are finite dimensional, we establish the rank conditions for finite pole assignability.

Characterizations of Positive Linear Functional Differential Equations

In this paper, we first prove that if a linear neutral functional differential equation is positive then it must degrade into a linear functional differential equation of retarded type. Then, we give some explicit criteria for positive linear functional differential equations. Consequently, we obtain a novel criterion for exponential stability of positive linear functional differential equations.

An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type

An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented.

Some sufficient conditions for exponential stability of linear neutral functional differential equations

Abstract In this paper, we give some sufficient conditions for exponential stability of linear neutral functional differential equations. The obtained results are extensions of the recent results in [D.Q. Cao, P. He, Stability criteria of linear neutral systems with a single delay, Appl. Math. Comput. 148 (2004) 135–143, D.Q. Cao, P. He, Sufficient conditions for stability of linear neutral systems with a single delay, Appl. Math. Lett. 17 (2004) 139–144], [G.-D. Hu, G.-D. Hu, X.F. Zou, Stability of linear neutral systems with multiple delays: boundary criteria, Appl. Math. Comput. 148 (2004) 707–715; J.H. Park, S. Won, A note on stability of neutral delay-differential systems, J. Franklin Inst. 336 (1999) 1–9] to linear neutral functional differential equations.

Problem of periodic solutions for neutral functional differential equation

The problem of periodic solutions for a kind of kth-order linear neutral functional differential equation is studied. By using the theory of Fourier expansions, a sufficient and necessary condition to guarantee the existence and uniqueness of periodic solution is obtained. Further, by applying this result and Schauder's fixed point principle, a kind of kth-order nonlinear neutral functional differential equation is investigated, and some new results on existence of the periodic solutions are given as well. These results improve and extend some known results in recent literature.

Strong stabilization of neutral functional differential equations

A linear neutral functional differential equation is called strongly exponentially stable if it is exponentially stable when subjected to small variations in the delays. In this paper we present necessary and sufficient conditions for the existence of a feedback control so that the resulting closed-loop neutral functional differential equation is strongly exponentially stable.

A functional model approach to linear neutral functional differential equations

A functional model approach to linear neutral functional differential equations

Density of images of semigroup operators for linear neutral functional differential equations

In this paper we consider properties of a system described by a linear functional differential equation of a neutral type. By T(t) we mean the operator: an initial state --+ the state at t. The state space is the Sobolev space WI’, hence T(t) maps W,P into itself, but other spaces may also be used. We mention, for example, that Hale [7] has considered the space of continuous functions C and Manitius [l] the space LTP x L2 (retarded case only). The main problem we consider here is to find conditions for density of the image of the operator T(t) (in the considered space). This problem was solved by Jakubczyk [2] for a simple neutral system described by a difference-differential equation. His paper was an inspiration for the author to solve this problem in a general case. The method used in our paper is based on properties of convolution operators and a ring structure of such operators. These ideas may be found in the papers of Jakubczyk [2] and that of Kamen [3]. In Section 3 we prove that a convolution operator in the space LP([O, T]; Iwn), T < +co, has a dense image if and only if a determinant of a kernel matrix of this operator over the ring K (defined in Section 2) is not a zero-divisor in this ring. This fact is then used to derive a criterion for density of im T(t). In the proof of the main result we use some representation of the operator T(h). A similar representation was used by Manitius in [l]. Density of the image of T(t) means that the system described by the functional differential equation is, in some sense, nondegenerate in the state space. In a neutral case the image of T(t) may cover the whole state space. The problem considered in this paper is closely related to the problem of completness of eigenfunctions associated with the system. Completness means that the space spanned by generalized eigenfunctions is dense in the state space. From papers of Henry [8] and Manitius [l], it follows that the above problems are equivalent (in the spaces C and [w” x L”). Thus our main result on density of im T(t) gives

Representation for solutions of linear neutral functional differential equations

Representation for solutions of linear neutral functional differential equations