On the stability of hybrid difference-differential systems

We consider hybrid difference-differential systems and analyze statements of initial-boundary value problems for such systems as compared with systems with retarded argument of the neutral type. We study the stability of solutions of linear stationary hybrid difference-differential systems and derive necessary and sufficient conditions for their asymptotic and exponential stability. In the scalar case, these conditions are refined and expressed via the original coefficients of the system in parametric form, which permits one to keep track of how the perturbations in the coefficients affect the solutions and to find the limiting value of the delay for which stability is preserved.     

A Liapunov functional for a matrix neutral difference-differential equation with one delay

For the matrix neutral difference-differential equation $\text{x}\text{?}\text{(t) + A}\text{x}\text{?}\text{(t ? τ)  Bx(t) + Cx(t ? τ)}$ we construct a quadratic Liapunov functional which gives necessary and sufficient conditions for the asymptotic stability of the solutions of that equation. We consider a difference equation approximation of the difference-differential equation, and for this difference equation we construct a Liapunov function from which we obtain the desired Liapunov functional by an appropriate limiting process. The Liapunov functional thus obtained gives the best possible estimate for the rates of growth or decay of the solutions of the matrix neutral difference-differential equation. The results obtained are natural generalizations of previous results obtained for a matrix retarded difference-differential equation with one delay.     

Gr?bner bases in difference-differential modules and difference-differential dimension polynomials

In this paper we extend the theory of Gr?bner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for constructing these Gr?bner bases counterparts. To this aim we introduce the concept of “generalized term order” on ℕ ^m ×ℤ ⁿ and on difference-differential modules. Using Gr?bner bases on difference-differential modules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations. This work was supported by the National Natural Science Foundation of China (Grant No. 60473019) and the KLMM (Grant No. 0705)     

Global solutions and invariant tori of difference-differential equations

We prove the existence of an m-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.     

Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues

Electrical networks containing lossless transmission lines are often modeled by difference-differential equations of neutral type. This paper finds sufficient conditions for asymptotic stability for linear systems of these equations. Also given is a modification of the direct method of Liapunov for difference equations. This method is applied to finding asymptotic stability criteria for the discrete analogs of the linear system of difference-differential equations.     

Some results on zeros and uniqueness of difference-differential polynomials

We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions.We also investigate the uniqueness problems of difference...     

On the classical and generalized solutions of boundary-value problems for difference-differential equations with variable coefficients

The first boundary-value problem for second-order difference-differential equations with variable coefficients on a finite interval (0, d) is considered. The following question is studied: Under what conditions will the boundary-value problem for a difference-differential equation have a classical solution for an arbitrary continuous right-hand side? It is proved that a necessary and sufficient condition for the existence of a classical solution is that certain coefficients of the difference operators on the orbits generated by the shifts be equal to zero.     

一类不连续时滞系统的一致最终有界性

主要讨论不连续的时滞自治系统,在Filippov解意义下的一致最终有界性问题．基于Lyapunov-Krasovskii泛函给出了全局强一致最终有界的Lyapunov定理,并将其应用到一类带有不连续摩擦项的时滞力学系统．     

Construction of the Lyapunov–Krasovskii Functionals for some Classes of Positive Delay Systems

A criterion is established for the diagonal stability of positive linear difference-differential systems with discrete and distributed delay. The problem is studied of existence of a common diagonal Lyapunov–Krasovskii functional for a family of these systems. The approach developed is used to obtain conditions of the absolute stability and estimates of the time of transient processes for nonlinear difference-differential systems with sector type nonlinearities.     

Modal control of hybrid differential-difference systems and associated delay systems of neutral type in scales of differential-difference controllers

We study the statements and solvability of the modal control problem (the pole assignment problem) for linear time-invariant hybrid difference-differential systems in symmetric form and for the associated delay systems of neutral type. We obtain constructive necessary and sufficient parametric conditions for the modal controllability of the systems in question in various scales of difference-differential controllers. Methods for the construction of such controllers solving the corresponding modal controllability problem are indicated. The results are illustrated by examples and counterexamples.     

Diffraction of light by superposed parallel supersonic waves,being harmonics of the same fundamental. Solution of the system of difference-differential equations for the amplitudes

Starting from the general system of difference-differential equations for the amplitudes of the diffracted beams of light, given by Mertens, and using the method of Kuliasko, Mertens and Leroy for the diffraction of light by one supersonic wave, it is possible to reduce the solution of the system of difference-differential equations, to the solution of a partial differential equation. In this way it is possible to calculate the intensities of the ordern and ?n, as a series expansion in ρ. Here we only considered terms up to ρ². It was also possible to verify the general symmetry properties for the intensities studied by Leroy and Mertens.     

Singular control of delay systems

A necessary condition, equivalent to Jacobson's condition, for optimality of a singular control is derived for systems represented by difference-differential equations.     

Stable Solutions to Homogeneous Difference-Differential Equations with Constant Coefficients: Analytical Instruments and an Application to Monetary Theory

In economic systems, reactions to external shocks often come with a delay. On the other hand, agents try to anticipate future developments. Both can lead to difference-differential equations with an advancing argument. These are more difficult to handle than either difference or differential equations, but they have the merit of added realism and increased credibility. This paper generalizes a model from monetary economics by von Kalckreuth and Schröder. Working out its stability properties, we present a general method for determining the stability of any solution to a homogeneous linear difference-differential equation with constant coefficients and advancing arguments.     

A projection-iterative method for finding periodic solutions of nonlinear systems of difference-differential equations with impulses

A method is developed for finding approximately the periodic solutions of nonlinear systems of difference-differential equations with impulses at fixed moments.     

Periodic solutions of ẋ(t) = –f(x(t), x(t – 1))

The existence of periodic solutions of general first order nonlinear difference-differential equations is shown using an appropriate cone of initial data and an asymptotic fixed point theorem.     

Stability of discontinuous differential equations with finite number of retarded terms by lyapunov functions

We consider here a class of second-order discontinuous differential equations with a finite number of nterms with retarded arguments; using lyapunov-type functions we prove, under suitable conditions, that the zero solutions of these Equations are globally uniformly asympototically stable.     

A note on linear equations modeling birth-and-death processes

Cauchy's method of characteristics is applied to derive a comprehensive solution for a class of differential, partial differential and difference-differential equations encountered in the study of branching processes. The results are then used to address an unsolved Markov's generalized birth process.     

Degenerate difference-differential equations: Algebraic theory

In this paper we consider the algebraic aspects of the theory of degenerate difference-differential equations. It will be shown that the fundamental algebraic concepts to be used are module theoretic. We have to consider similarity of polynomial matrices in one or more indeterminates. In the case of systems with commensurable lags the underlying modules have a simple structure, because the corresponding ring of scalars is the principal ideal domain of polynomials in one indeterminate. This fact makes it possible to prove a structure theorem for degenerate difference-differential equations with commensurable lags. This theorem shows that degenerate systems of this type essentially are trivial in the sense of Henry [15], i.e., the characteristic quasipolynomial is a polynomial. Further it is shown that coordinate transforms with “time lag” play an essential role for the construction of degenerate equations. The power of the method is demonstrated by some examples, some of which are equations with incommensurable lags.     

Uniqueness of Difference-Differential Polynomials of Meromorphic Functions

Ukrainian Mathematical Journal - We study the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions sharing a small function (ignoring multiplicities)...     

The Dressing Chain and One-Point Commuting Difference Operators of Rank 1

We construct solutions to the difference-differential equation that are associated with onepoint commuting difference operators of rank 1 in the case of spectral curves of genus 1.