Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics 2021-12-23T16:59:10+00:00 Alexander Rezounenko Open Journal Systems <p><span style="color: #009900;"><strong>Indexed/Abstracted</strong></span> in <a href="" target="_blank" rel="noopener"><strong>Zentralblatt MATH</strong></a> ( since <strong>1999</strong>; indexed more than <strong>3</strong><strong>8</strong><strong>0</strong> documents). <br>Zentralblatt MATH (<a href="" target="_blank" rel="noopener"><strong>zbMATH</strong></a>) is the world’s most comprehensive and longest-running abstracting and reviewing service in pure and applied mathematics.</p> On exact controllability and complete stabilizability for linear systems 2021-12-22T14:20:23+00:00 Rabah Rabah <div class="textLayer"> <div>In this paper we consider linear systems with control described by the equation $\dot x = \mathcal A x +\mathcal B u$ where functions&nbsp;u&nbsp;and&nbsp;x&nbsp;take values in&nbsp;U&nbsp;and&nbsp;X&nbsp;respectively. For such&nbsp;object, a short review of results concerning relations between exact controllability and&nbsp;complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis&nbsp;is done in various situations: bounded or unbounded state and control operators $\mathcal A$&nbsp;and $\mathcal B$, Banach or Hilbert spaces&nbsp;U and&nbsp;X.</div> <div class="endOfContent">&nbsp;</div> </div> <div class="annotationLayer"> <div>The well known equivalence between complete controllability and pole assignment in&nbsp;the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if&nbsp;U&nbsp;and&nbsp;X&nbsp;are Banach spaces. In Hilbert space setting this implication holds true. The converse also&nbsp;is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true.</div> <div>The corresponding results are given with some ideas for the proofs. Complete technical&nbsp;development are indicated in the cited literature. Several examples are given. Special&nbsp;attention is paid to the case of infinite dimensional systems generated by delay systems&nbsp;of neutral type in some general form (distributed delays). The question of the relation&nbsp;between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some&nbsp;classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive&nbsp;review of some research on control theory for infinite dimensional spaces. Our works in&nbsp;this area were initiated by V. I. Korobov during the 70th of the past century in Kharkov&nbsp;State University.</div> </div> 2021-11-29T00:00:00+00:00 Copyright (c) 2021 On homogeneous controllability functions 2021-12-22T15:01:28+00:00 Andrey Polyakov <div>The controllability function method, introduced by V. I. Korobov in late 1970s, is known to be an efficient tool for control systems design. It is developed for both linear/nonlinear and finite/infinite dimensional systems. This paper bridges the method&nbsp;with the homogeneity theory popular today. The standard homogeneity known since&nbsp;18th century is a symmetry of function with respect to uniform scaling of its argument.</div> <div>Some generalizations of the standard homogeneity were introduced in 20th century. This paper shows that the so-called homogeneous norm is a controllability function&nbsp;of the linear autonomous control system and the corresponding closed-loop system is&nbsp;homogeneous in the generalized sense. This immediately yields many useful properties known for homogeneous systems such as robustness (Input-to-State Stability) with</div> <div>respect to a rather large class of perturbations, in particular, with respect to bounded additive measurement noises and bounded additive exogenous disturbances. The main&nbsp;theorem presented in this paper slightly refines the design of the controllability function&nbsp;for a multiply-input linear autonomous control systems. The design procedure consists in&nbsp;solving subsequently a linear algebraic equation and a system of linear matrix inequalities. The homogeneity itself and the use of the canonical homogeneous norm essentially</div> <div>simplify the design of a controllability function and the analysis of the closed-loop system. Theoretical results are supported with examples. The further study of homogeneity-based&nbsp;design of controllability functions seems to be a promising direction for future research.&nbsp;</div> 2021-11-29T00:00:00+00:00 Copyright (c) 2021 A small gain theorem for finite-time input-to-state stability of infinite networks and its applications 2021-12-23T10:20:57+00:00 Svyatoslav Pavlichkov <div>We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a&nbsp;countable set of finite-dimensional subsystems of ordinary differential equations, each of&nbsp;which is interconnected with a finite number of its “neighbors” only and is affected by some&nbsp;external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced&nbsp;by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized&nbsp;finite-time input-to-state stabilization with respect to external disturbances for infinite networks composed&nbsp;of a countable set of strict-feedback form systems of ordinary differential equations. For&nbsp;this we combine our small-gain theorem proposed in the current work with the controllers</div> <div>design developed by S. Pavlichkov and C. K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy&nbsp;and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their&nbsp;application for stabilization, Automatica. – 2020. – 112. – 108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time ISS stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e.&nbsp;are abscent), we just obtain finite-time stability and finite-time decentralized stabilization&nbsp;of infinite networks accordingly.</div> 2021-11-29T00:00:00+00:00 Copyright (c) 2021 Linear differential-algebraic boundary value problem with singular pulse influence 2021-12-13T14:40:40+00:00 Sergey Mikhailovich Chuiko Olena Viktorivna Chuiko Kateryna Sergeyevna Shevtsova <div>The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value&nbsp;problems is associated with numerous applications of such problems in the theory of&nbsp;nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory&nbsp;of motion stability. At the same time, the study of differential algebraic boundary value&nbsp;problems is closely related to the study of pulse boundary value problems for differential&nbsp;equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk</div> <div>and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in&nbsp;particular finding the necessary and sufficient conditions for the existence of the desired&nbsp;solutions, and also the construction of the Green’s operator of the Cauchy problem and&nbsp;the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation.</div> <div> <div class="page" data-page-number="16" data-loaded="true"> <div class="textLayer">In this article we found the conditions of the existence and constructive scheme&nbsp;for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the&nbsp; analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.</div> </div> <div class="page" data-page-number="17" data-loaded="true"> <div class="canvasWrapper">&nbsp;</div> <div class="textLayer">&nbsp;</div> </div> </div> 2021-11-29T00:00:00+00:00 Copyright (c) 2021 The shape of wave-packets in a three-layer hydrodynamic system 2021-12-23T10:42:53+00:00 Diana Sergeevna Kharchenko <div>The article is devoted to the problem of wave-packet propagation in a three - layer&nbsp;hydrodynamic system "layer with a hard bottom - layer - layer with a cover stratified by&nbsp;density. The current research on selected topics is reviewed. The mathematical formulation of the problem is given in dimensionless form and contains the equations of fluid motion, kinematic and dynamic conditions on the contact surfaces, as well as the boundary&nbsp;conditions on the lid and on the bottom. Using the method of multiscale developments, the first three approximations of the studied problem are obtained, of which the first two&nbsp;are given in the article, because the third approximation has a cumbersome analytical&nbsp;form. The solutions of the first approximation and the variance relation are presented.&nbsp;The evolution equations of the circumferential wave-packets on the contact surfaces are&nbsp;derived in the form of the nonlinear Schrodinger equation on the basis of the variance</div> <div>relation and the conditions for the solvability of the second and third approximations.&nbsp;A partial solution of the nonlinear Schrodinger equation is obtained after the transition&nbsp;to a system moving with group velocity. For the first and second approximations, the&nbsp;formulas for the deviations of the contact surfaces are derived, taking into account the&nbsp;solution of the nonlinear Schrodinger equation. The conditions under which the shape&nbsp;of wave-packets on the upper and lower contact surfaces changes are derived. The regions of familiarity of the coefficients for the second harmonics on the upper and lower&nbsp;contact surfaces for both frequency pairs, which are the roots of the variance relation,&nbsp;are presented and analyzed. Also, for both frequency pairs, different cases of superimposition of maxima and minima of the first and second harmonics, in which there is&nbsp;an asymmetry in the shape of wave packets, are graphically illustrated and analyzed. All</div> <div>results are illustrated graphically. Analytical transformations, calculations and graphical representation of results were performed using a package of symbolic calculations and&nbsp;computer algebra Maple.</div> 2021-11-29T00:00:00+00:00 Copyright (c) 2021 V. I. Korobov. To the 80th anniversary 2021-12-23T16:59:10+00:00 Tetiana Smortsova Alexander Rezounenko Grigory Sklyar Svetlana Ignatovich <p>On September 27, 2021, the Editor-in-Chief of our journal, Doctor of Physical and Mathematical Sciences, Professor Valery Ivanovich Korobov turned the 80th anniversary.</p> <p>Valery Ivanovych established a Kharkiv scientific school of the mathematical control theory, which is well known in Ukraine and far beyond. Some of his results originated new scientific areas.</p> 2021-11-29T00:00:00+00:00 Copyright (c) 2021