APPLIED MATHEMATICS
AND CONTROL SCIENCES

The stability of a linear autonomous difference equation with two complex coefficients and different delays is investigated. The starting point of the study is the Schur-Cohn theorem on the location of the roots of the characteristic equation in the complex plane with respect to the unit disk. To construct the domain of the exponential stability of the considered equation in the parameter space, the method of D-decomposition is used, which consists in constructing surfaces in the phase space such that when these surfaces are crossed by a point moving in the phase space, then the number of roots of the corresponding to the point characteristic equation, located outside the unit disk in the complex plane, changes. The region to which the zero number of roots corresponds is the domain of stability. This scheme is implemented for the above-mentioned difference equation: geometric stability criteria are found and the domains of exponential stability in the four-dimensional space of coefficients are described. Uniform stability is studied separately. Its domain is the domain of exponential stability supplemented by a part of its boundary. For the exact description of the domain of uniform stability, the description of a “multiplicity curve” was required, that is the line, all points of which correspond to multiple roots of the characteristic equation. The obtained results can be applied to the study of processes in physics, engineering, economics, and biology, for the modeling of which discrete models in the form of difference equations are used.

This article discusses linear functional differential equations that can serve as the basis for modern modeling in various fields of science, technology, and economics, including the study of neural networks and machine learning. These equations describe a wide class of processes where the rate of change of a certain quantity depends not only on the values at the current time, but also on the values in the past and future.The aim of the work is to obtain precise conditions on the parameters of the equation, under which the equation has a solution for any integrable right-hand side, which reflects the existence of the modeled object for a reasonably large class of external influences.It is shown that to establish the fact of everywhere solvability of a first-order functional differential equation, it is sufficient to study only three boundary value problems: a periodic boundary value problem, a Cauchy problem, and a problem with a boundary condition at the right end.In terms of the values of the norms of the positive and negative parts of the functional operator, necessary and sufficient conditions are obtained for a linear functional differential equation of the first order to be solvable everywhere. If these conditions on the norms are not satisfied, then there exists an operator with the given norms of the positive and negative parts such that the equation will have no solutions for some integrable right-hand sides.The developed research methods are based on the apparatus of the theory of functional differential equations and can be applied to study other classes of functional equations, in particular, higher-order equations.The results obtained can be used to analyze and model various dynamic systems with delays and (or) advances. These delays and advances can be described by the most general functional operators, including both positive and negative parts, which corresponds to the consideration of systems with both positive and negative feedback. This allows for a more accurate description and prediction of the behavior of such systems.

We consider the issue of asymptotic stability of a linear continuous-discrete system of functional differential equations with constant coefficients. Such systems consist of two subsystems, continuous and discrete, and are often called hybrid. The continuous subsystem is a system of differential equations. The feature of the hybrid system under consideration is that its continuous part is a system of differential equations with concentrated delay, while the overwhelming majority of papers consider hybrid systems whose continuous part is a system of ordinary differential equations. The standard approach to studying the stability of the latter systems is integration on each finite interval and the construction of the monodromy matrix. However, this approach is generally inapplicable to the problem of studying the stability of hybrid systems such that the continuous part is a system of differential equations with deviating argument. In the present paper, the method of generating functions and the analysis of the spectrum of shift operator along the trajectory of the solution of a hybrid system are applied to study the stability of hybrid systems. Construction of the generating function for the Cauchy matrix and for the fundamental solution allows us to reduce the problem of asymptotic stability of a hybrid system to the problem of studying the location of the roots of a certain function in the complex plane. For this function, it is natural to introduce the term «the characteristic function of the hybrid system», which was done. In addition, it is proved that for these hybrid systems, the asymptotic stability is equivalent to the uniform exponential stability. This approach is compatible with the D-partition method, which allows us to use it to obtain new effective coefficient conditions of asymptotic stability for hybrid systems: in particular, to construct the stability region. In this article, a new simple necessary criterion for the asymptotic stability of a hybrid system is constructed, which is reduced to checking two elementary numerical inequalities.

The article is devoted to the constructive study of the stabilizability of the solution of the Cauchy problem to a periodic function for system of differential equations with delay and periodic parameters. The results obtained in this paper are natural continuation of research in this area (see, for example, the works of J.S.P. Munembe). The proposed research method is based on the use of the Cauchy matrix of the system under consideration. Knowledge of the Cauchy matrix allows us to construct an auxiliary numerical matrix and reduce the problem to estimating the spectral radius of this matrix. Fulfillment of the condition that spectral radius of the specified matrix is less than one guarantees the presence of the stabilizability property.The source of the effective implementation of the proposed method for studying the problem under consideration is the possibility of accurately constructing the Cauchy matrix of a differential equation with piecewise linear delay based on the approach proposed by the author. As an illustration, the article considers an example of the Cauchy problem for a differential equation with piecewise linear delay and periodic parameters. For a given equation, the Cauchy function is constructed using precision computing software, and it is also proved that the solution of the Cauchy problem has the property of stabilizability to a periodic function