Linear Retarded Functional Differential Equations Research Articles

Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.

On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Some regularity results on stochastic convolutions in point delay differential equations perturbed by noise

In this work, we obtain some regularity results on stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded coefficient operators driven by white noise. We first establish some estimates on fundamental solutions which play an important role in the subsequent theory. Then we apply these estimates to some stochastic convolutions arising in stochastic delay differential equations to obtain the desired regularity property.

On regularity of stochastic convolutions of functional linear differential equations with memory

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $ C_0 $-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

On stationarity of stochastic retarded linear equations with unbounded drift operators

ABSTRACTThis article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.

Converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations

In this article we give a collection of converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly-degenerate Lyapunov–Krasovskii functional is a necessary and sufficient condition for the global exponential stability of linear retarded functional differential equations. This is carried out using a switched system representation approach.

Almost periodic solutions of partial differential equations with delay

In this paper we study the existence of almost periodic solutions for linear retarded functional differential equations with finite delay and values in a Banach space. We relate the existence of almost periodic solutions with the stabilization of distributed control systems. We apply our results to transport models and to the wave equation.

One method for the investigation of linear functional-differential equations

We consider a scalar linear retarded functional-differential equation $$ \dot{x}(t)=ax\left( {t-1} \right)+bx\left( {\frac{t}{q}} \right)+f(t),\quad q>1. $$ The study of linear retarded functional-differential equations mainly deals with two types of initial-value problems: initial-value problem with initial functions and initial-value problems with initial point (when it is necessary to find a classical solution whose substitution in the original equation reduces it to the identity). In the paper, the initial-value problem with initial point is studied by the method of polynomial quasisolutions. We prove theorems on the existence of polynomial quasisolutions and exact polynomial solutions of the analyzed linear retarded functional-differential equation. The results of numerical experiments are presented.

Stability property and essential spectrum of linear retarded functional differential equations

In this work, we shall study stability and stabilization of a class of retarded functional differential equations in Banach spaces. We present sufficient (and necessary, sometimes) conditions for weak, asymptotic and exponential stability properties. It is shown that retarded growth bounds are totally determined by retarded non-isolated spectra, isolated eigenvalues with infinite algebraic multiplicity and spectral bounds. Stabilization problem via compact perturbations, a topic which is practically useful in the context of control theory of dynamical systems, is considered as well.

Approximation of Eigenvalues of Evolution Operators for Linear Retarded Functional Differential Equations

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.

Equivariant versal unfoldings for linear retarded functional differential equations

We continue our investigation of versality for parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds. In this paper, we consider RFDEs equivariant with respect to the action of a compact Lie group. In a previous paper (Buono and LeBlanc, <em>J. Diff. Eqs.</em>, <strong>193 </strong>, 307-342 (2003)), we have studied this question in the general case (i.e. no <em>a priori</em> restrictions on the RFDE). When studying the question of versality in the equivariant context, it is natural to want to restrict the range of possible unfoldings to include only those which share the same symmetries as the original RFDE, and so our previous results do not immediately apply. In this paper, we show that with appropriate projections, our previous results on versal unfoldings of linear RFDEs can be adapted to the case of linear equivariant RFDEs. We illustrate our theory by studying the linear equivariant unfoldings at double Hopf bifurcation points in a $\mathbb D_3$-equivariant network of coupled identical neurons modeled by delay-differential equations due to delays in the internal dynamics and coupling.

Nonoscillations for odd-dimensional systems of linear retarded functional differential equations

This paper is concerned with the nonoscillatory problems of odd-dimensional systems of linear retarded functional differential equations. Based upon the corresponding characteristic equations, we get some criteria for nonoscillations by utilizing the matrix measures.

An identification problem for linear retarded functional differential equations

An identification problem is studied for linear retarded functional differential equations (FDEs) in Banach spaces. The unique identification of solution states and scalar forcing functions from the given data is proved.

An identification problem for linear retarded functional differential equations

An identification problem is studied for linear retarded functional differential equations (FDEs) in Banach spaces. The unique identification of solution states and scalar forcing functions from the given data is proved.

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the theory with several examples, and comment on the applicability of these results to bifurcation analyses of nonlinear RFDEs.

Differentiability of solutions of second-order functional differential equations with unbounded delay

In this paper we study the differentiability of solutions of the second-order semilinear abstract retarded functional differential equation with unbounded delay, specially when the underlying space is reflexive or at least has the Radon–Nikodym property. We apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically.

DIFFERENTIABILITY OF SOLUTIONS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this note we establish a criterion to obtain classical solutions for a quasi-linear abstract retarded functional differential equation with infinite delay and we apply this result to characterize the infinitesimal generators of several strongly continuous semigroup of linear operators that arise in the theory of linear abstract retarded functional differential equations with infinite delay on a phase space defined axiomatically.

Structural Operators and Eigenmanifold Decomposition for Functional Differential Equations in Hilbert Spaces

This paper studies spectral properties of linear retarded functional differential equations in Hilbert spaces with the emphasis on their relations to structural operators. The equations involve unbounded operators acting on the discrete and distributed delayed terms, and the operators acting on the instantaneous term are defined through sesquilinear forms. The main concern of this paper is studying the spectral properties of the infinitesimal generators associated with the solution semigroups by means of structural operators. The characterizations of eigenmanifolds are derived and the relations between the manifolds and structural operators are shown by using the properties of structural operators.

Periodic Solutions of Linear and Quasilinear Neutral Functional Differential Equations

In this paper we will use the Fourier expansions to generalize Hale′s alternative results for the existence of periodic solutions of linear retarded functional differential equations to neutral equations. Using some a priori estimation and the Leray-Schauder degree theory, we will obtain some existence theorems of periodic solutions for a class of quasilinear neutral equations. The results also generalize the works of Hatvani and Krisztin on retarded equations.

Numerical Approximations for Hereditary Systems with Input and Output Delays: Convergence Results and Convergence Rates

In this paper, the averaging approximation scheme for linear retarded functional differential equations with delays in control and observation is considered in the context of the state space theory developed by Pritchard and Salamon [SIAM J. Control Optim., 25 (1987), pp. 121–144]. Using known results from linear semigroup theory, convergence and estimate of convergence rate of the approximating semigroups are established. These extend results due to Banks and Burns [SIAM J. Control Optim., 16 (1978), pp. 169–208] and Lasiecka and Manitius [SIAM J. Numer. Anal., 25 (1988), pp. 883–907] on hereditary systems with delays in state, to the case when delays in control and observation are included. The main difference from the case when delays in input and output are excluded is that unbounded input and output operators must be dealt with in the abstract formulation. Moreover, in the presence of the unboundedness of the input and output operators, new convergence results of the state solutions and the output are also obtained.