Applied Mathematics and Control Sciences

Applied Mathematics and Control Sciences is an open-access periodical scientific peer-reviewed journal.

Full Journal Title: Applied Mathematics and Control Sciences

Abbreviation: Appl. Math. Control Sci.

Publisher: Perm National Research Polytechnic University, Perm, Russian Federation

DOI: 10.15593/2499-9873

Languages: Russian, English

Editor-in-Chief: Professor, Dr. Sci. Valerii Yu. Stolbov

Executive Editor: C.Sci. Aleksandr O. Alekseev

Editorial Contact: 

Address: Editorial Board "Applied Mathematics and Control Sciences", Russian Federation, Perm, 614990, Komsomolsky ave., 29
Phone: +7 (342) 219-85-87; + 7 (909) 1000-150
E-mail: aoalekseev@pstu.ru  

Frequency: Quarterly

Applied Mathematics and Control Sciences has no article processing and/or article submission charges.

All Journal's Content, including articles,  is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). Editorial of the Journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles and allows readers to use them for any other lawful purpose in accordance with Budapest Open Access Initiative's definition of Open Access.

Journal intended for researchers specializing in the field of applied mathematics, mathematical modeling, differential equations, dynamical systems, optimal control, organizational behavior control, automation, senior students of natural areas.

Since 2019 the journal included in the List of peer-reviewed scientific journals, formed by the Higher Attestation Commission under the Ministry of Science and Higher Education of the Russian Federation.

In 2022 the journal is categorized as category K1 (included in 25% leading Russian peer-reviewed scientific journals).

The years 2022-2031 have been declared the Decade of Science and Technology by decree of the president of the Russian Federation

Current Issue

No 3 (2024)

Investigation of the stability of a linear autonomous difference equation with complex coefficients
Aksenenko I.A.

Abstract

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.
Applied Mathematics and Control Sciences. 2024;(3):13-23
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On the estimates of the fundamental solution and the Cauchy function for a class of linear autonomous differential equations of neutral type
Balandin A.S.

Abstract

In the paper we consider a class of linear autonomous differential equations of neutral type. The equation under study arises in applications such that dynamics of cell population, motion of 2 dimensional elastic plates with friction, and ultrasonic flaw detection. On the other hand, this equation has a large variety of asymptotic properties of solutions and is therefore also interesting from a theoretical point of view, which is confirmed by a significant number of purely theoretical studies. The equation in question is a good example of an object that is simple enough to allow effective stability conditions to be obtained, and at the same time complex enough to exhibit all the variety of asymptotic properties of solutions of autonomous equations of neutral type.The study of the stability of the considered equation is reduced to the study of asymptotic properties of its fundamental solution and Cauchy function. For the equation under study, the exponential stability criterion is known, and the domain of stability is constructed in the space of the coefficients.In this paper, we study the positivity of the fundamental solution and the Cauchy function of the given equation, and establish two-sided exponential estimates for these functions. To do this, a well-known lemma on differential inequality is generalized for the linear autonomous differential equation of neutral type. Further, we obtain that if the equation in question is exponentially stable and its characteristic function has at least one real root, then the fundamental solution and the Cauchy function are positive on the positive semi-axis. We give a geometric form to this condition, describing a domain in the parameter space of the equation. Based on the positiveness of the fundamental solution and the Cauchy function, we construct their two-sided exponential estimates. The exponents and coefficients in the estimates obtained for the fundamental solution and the Cauchy function are exact. The effectiveness of the results established in the article is illustrated by an example.
Applied Mathematics and Control Sciences. 2024;(3):24-37
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Conditions of everywhere solvability for a linear functional differential equation of the first order
Bravyi E.I.

Abstract

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.
Applied Mathematics and Control Sciences. 2024;(3):38–52
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The Cauchy matrix of a system with fractional derivative and aftereffect
Maksimov V.P.

Abstract

A system of linear functional differential equations with fractional derivative and aftereffect is considered. The question of the representation of solutions is investigated, the existence f the Cauchy matrix as the kernel of an integral representation is proved, and the main defining relationships for the Cauchy matrix are derived.The well-known definition of the fractional Caputo derivative of the order is used. The system under study includes, in addition to the Caputo derivative, a linear Volterra operator of a general form. Using the Riemann–Liouville fractional integration operator the initial system is reduced to a linear integral Volterra equation, for which the convergence of the Neumann series and the integral representation of the solution using a resolvent integral operator are established. It is shown that the Cauchy matrix is expressed explicitly through the resolvent kernel of this operator. In the case of a transition to the integer order of the derivative, the obtained defining relationship for the Cauchy matrix coincides with the known one. The use of the Cauchy matrix opens up wide possibilities for the study of fractional systems in terms of obtaining effective conditions of solvability of boundary value problems, control problems and the description of asymptotic behavior of solutions similar to how it is done for wide classes of systems with integer derivatives.All the constructions are based on the use of the basic statements of the theory of abstract functional differential equations developed by the heads of the Perm seminar, Professors N.V. Azbelev and L.F. Rakhmatullina.
Applied Mathematics and Control Sciences. 2024;(3):53–63
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On the Cauchy matrix of a class of hybrid systems
Mulyukov M.V.

Abstract

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.
Applied Mathematics and Control Sciences. 2024;(3):64–72
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On invertibility of the operator at a derivative in the neutral type differential equation with two incommensurable delays
Postanogova I.Y.

Abstract

This article considers a neutral type functional differential equation with two incommensurable delays at the derivative, with focusing on stability issues. The invertibility of the operator at the derivative in Lebesgue spaces Lp is investigated, and the location of the roots of its characteristic equation on the complex plane is analyzed.To study the invertibility of the operator at the derivative, the spectrum of the internal superposition operator S is defined, and its description is provided in terms of coefficients of the original equation. The resulting description of the spectrum allows formulating the conditions under which the operator at the derivative is invertible. Further, the invertibility of this operator facilitates the identification of criteria for exponential stability and instability.A connection between the coefficients of the operator S, the type of stability of the original equation, and the invertibility of the operator in arbitrary Lebesgue functional space, as well as the location of roots of the characteristic equation, is established.It has been demonstrated that the presence of roots of the characteristic equation to the right of the imaginary axis is equivalent to the instability of the original neutral type equation and the non-invertibility of the operator at the derivative. Conversely, if all the roots of the characteristic equation are located to the left of the imaginary axis and are separated from it, then the operator at the derivative is invertible, and the neutral type equation is exponentially stable. These conditions have been shown to be effectively verifiable in terms of the coefficients of the original equation. A “critical” case has also been described, when the roots of the characteristic equation lie to the left of the imaginary axis but are not separated from it; specifically, a vertical chain of roots approaches arbitrarily close to the imaginary axis. In this scenario, it is established that the operator at the derivative is non-invertible, and the neutral type equation cannot be exponentially stable.
Applied Mathematics and Control Sciences. 2024;(3):73-90
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To constructive study of asymptotic properties of the delay differential equation with periodic parameters
Rumyantsev A.N.

Abstract

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
Applied Mathematics and Control Sciences. 2024;(3):91-98
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On estimates of solutions to systems of linear autonomous functional differential equations of delayed type
Sabatulina T.L.

Abstract

In the paper we consider systems of linear autonomous functional differential equations (FDEs) of delayed type, the coefficients in a system can be of any sign. These FDE systems include equations with various types of aftereffects, including concentrated and distributed delays.The purpose of this paper is to obtain new effective conditions of stability for systems of linear autonomous FDEs. The study is based on an idea of constructing a so-called “comparison system”, which, on the one hand, has a simpler structure, and on the other hand, the same asymptotic properties as the original system. The comparison system may contain a delay, and not only concentrated, but also distributed. The comparison system is constructed in such a way that all components of its fundamental matrix are non-negative. Since the coefficient matrices in the comparison system are diagonal, it can be considered as a set of independent scalar equations. For the fundamental solutions of such equations in the papers of V.V. Malygina and K.M. Chudinov, exact two-sided exponential estimates were obtained, which also give an exponential estimate for the fundamental matrix of the comparison system.For autonomous FDEs of a delayed type, as is known, approaching zero always occurs exponentially, which means the existence of such positive constants N and α that . However, without specifying estimates for the coefficient N and exponent α or an algorithm for their effective calculation, the exponential stability problem cannot be considered completely solved. In the article, along with new conditions of stability, estimates of the rate at which solutions approach zero are found. The effectiveness of the results obtained is illustrated by several examples in which FDEs with different types of aftereffects are selected as comparison systems.
Applied Mathematics and Control Sciences. 2024;(3):99-112
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