Visnyk of V.N.Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

Open access journal                   ISSN 2523-4641 (Online), ISSN 2221-5646 (Print)

Volume 89, 2019

Ця сторінка Українською / This page in Ukrainian

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A.L. Vyshnevetskiy, Convergence of random walks on double transitive group generated by its permutational character. P. 4-9.

Short abstract: Let $P$ be a probability on a finite group $G$, $U(g)=\textstyle\frac{1}{|G|}$ the uniform probability on $G$, $P^{(n)}$ an $n$-fold convolution of $P$. Under well-known mild conditions, $P^{(n)}\rightarrow U$ if $n\rightarrow\infty$. A lot of estimates of the rate of the convergence are found in different norms. We consider the groups that have a double transitive presentation, and the probability $P$ that naturally arises in this presentation. An exact formula for rate of convergency for these groups for the norm $\|F\|=\sum\limits_{g\in G} |F(g)|$, where $F(g)$ is a function on group $G$, is given.

Extended abstract: Let $P$ be a probability on a finite group $G$, $U(g)=\textstyle\frac{1}{\rule{0pt}{6pt}|G|}$ the uniform (trivial) probability on the group $G$, $P^{(n)}=P *\ldots*P$ an $n$-fold convolution of $P$. A lot of estimates of the rate of the convergence $P^{(n)}\rightarrow U$ are found in different norms. It is well known conditions under which $P^{(n)}\rightarrow U$ if $n\rightarrow\infty$. Many papers are devoted to estimating the rate of this convergence for different norms. We consider finite groups that have a double transitive representation by substitutions and the probability that naturally arises in this image. This probability on each element of the group is proportional to the number of fixed (or stationary) points of this element, which is considered as a substitution. In other words, this probability is a character of the substitution representation of the group. A probability is called class if it takes the same values on each class of conjugate elements of a group, that is, it is a function of the class. The considered probability is class because any character of a group takes on the same values on conjugate elements. Any probability (and, in general, functions with values in an arbitrary ring) on a group can be associated with #an element of the group algebra of this group over this ring. The class probability corresponds to an element of the center of this group algebra; that is why the class probability is also called central. On an abelian group, any probability is class (central). In the paper convergence with respect to the norm $\|F\|=\sum\limits_{g\in G} |F(g)|$, where $F(g)$ is a function on group $G$, is considered. For the norm an exact formula not estimate only, as usual for rate of convergence of convolution $P^{(n)}\rightarrow U$ is given. It turns out that the norm of the difference $\|P^{(n)}-U\|$ is determined by the order of the group, degree the group as a substitution group, and the number of regular substitutions in the group. A substitution is called regular if it has no fixed points. Special cases are considered the symmetric group, the alternating group, the Zassenhaus group, and the Frobenius group of order $p(p-1)$ with the Frobenius core of order $p$ ($p$ is a prime number). A Zassenhaus group is a double transitive substitution group of a finite set in which only a trivial substitution leaves more than two elements of this set fixed.

Keywords: probability; finite group; convergency; convolution.

2010 Mathematics Subject Classification: 20D99; 60B15; 60B10. [ Full-text available (PDF) ]   Top of the page.

O.V. Nesmelova, Seminonlinear boundary value problems for nondegenerate differential-algebraic system. P. 10-20.

Short abstract: We obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations. We studied the case of the nondegenerate system of differential-algebraic equations, namely: the differential-algebraic system reduced to the system of ordinary differential equations with the arbitrary continuous function.

Extended abstract: In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of A.I. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

Keywords: nonlinear Noetherian boundary value problems; differential-algebraic equations; pseudoinverse matrices.

2010 Mathematics Subject Classification: 34B15.   [ Full-text available (PDF) ]   Top of the page.

A.A. Makarov, I.G. Nikolenko, Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer, P. 21-32.

Short abstract: A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated.

Extended abstract: A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa --- an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Keywords: boundary-value problem; pseudodifferential equations; Fourier transform; parabolicity; hypoellipticity.

2010 Mathematics Subject Classification: 35S15. [ Full-text available (PDF) ]   Top of the page.

M.S. Filipkovska, A block form of a singular pencil of operators and a method of obtaining it, P. 33-58.

Short abstract: A block form of a singular operator pencil, which consists of singular and regular blocks, where invertible blocks and zero blocks are separated out, is described. A method of obtaining the block form of a singular pencil and the corresponding direct decompositions of spaces, and also methods for the construction of projectors onto subspaces from the direct decompositions, are shown. The projectors enable one to obtain the form of the blocks. Examples of the block representations of singular pencils are given for various cases.

Extended abstract: A block form of a singular operator pencil $\lambda A+B$, where $\lambda$ is a complex parameter, and the linear operators $A$, $B$ act in finite-dimensional spaces, is described. An operator pencil $\lambda A+B$ is called regular if $n = m = rk(\lambda A+B)$, where $rk(\lambda A+B)$ is the rank of the pencil and $m$, $n$ are the dimensions of spaces (the operators map an $n$-dimensional space into an $m$-dimensional one); otherwise, if $n \ne m$ or $n = m$ and $rk(\lambda A+B) < n$, the pencil is called singular (irregular). The block form (structure) consists of a singular block, which is a purely singular pencil, i.e., it is impossible to separate out a regular block in this pencil, and a regular block. In these blocks, zero blocks and blocks, which are invertible operators, are separated out. A method of obtaining the block form of a singular operator pencil is described in detail for two special cases, when $rk(\lambda A+B) = m < n$ and $rk(\lambda A+B) = n < m$, and for the general case, when $rk(\lambda A+B) < n, m$. Methods for the construction of projectors onto subspaces from the direct decompositions, relative to which the pencil has the required block form, are given. Using these projectors, we can find the form of the blocks and, accordingly, the block form of the pencil. Examples of finding the block form for the various types of singular pencils are presented. To obtain the block form, in particular, the results regarding the reduction of a singular pencil of matrices to the canonical quasidiagonal form, which is called the Weierstrass-Kronecker canonical form, are used. Also, methods of linear algebra are used. The obtained block form of the pencil and the corresponding projectors can be used to solve various problems. In particular, it can be used to reduce a singular semilinear differential-operator equation to the equivalent system of purely differential and purely algebraic equations. This greatly simplifies the analysis and solution of differential-operator equations.

Keywords: operator pencil; matrix pencil; singular; regular block; block form; structure.

2010 Mathematics Subject Classification: 47A05; 15A22; 47N20.   [ Full-text available (PDF) ]   Top of the page.

P.V. Lukianov, BVI-noise simulation of two-blade helicopter’s rotor sinshape, P. 59-75.

Short abstract: In this paper a problem of BVI-noise generation by two-blade rotor sin-shape is set and solved. A sound density and a pressure level for far and near-field have been calculated. A comparative analysis o f the data with ones for two-blade rotor with rectangular blades has been carried out. Sin-shape rotor noise for most case of calculations has 3-5\,Db less then the noise of rotor with rectangular blade. Here essential reapportionment of energy of longitudinal sound waves to s-waves is observed. Interference figure says about complex non-liner character of generated sound. Its specter activates more high frequencies. Blade shape variation along the blade sweep allows controlling character and level of BVI-noise.

Keywords: sound generation; helicopter; BVI-noise.

2010 Mathematics Subject Classification: 76Q05; 76G25.   [ Full-text available (PDF) ]   Top of the page.

S.Yu. Ignatovich, Yu.V. Sukhinina, Time-optimal control problem with two final points for a kinematic model of an UAV, P. 76-92.

Short abstract: We study the time-optimal control problem for an unmanned aerial vehicle (drone) moving in the plane of a constant altitude; a kinematic model is considered where the angular velocity is a control. The drone must reach a given unit circle in the minimal possible time and stay on this circle rotating clockwise or counterclockwise. We obtain a complete solution of this time-optimal control problem and give a solution of the optimal synthesis problem.

Extended abstract: We study the time-optimal control problem for an unmanned aerial vehicle (drone) moving in the plane of a constant altitude. A kinematic model is considered where the angular velocity is a control. Such a system is described by Markov-Dubins equations; a large number of works are devoted to solving different optimal and admissible control and stabilization problems for such models. In the papers [T. Maillot, U. Boscain, J.-P. Gauthier, U. Serres, Lyapunov and minimum-time path planning for drones, J. Dyn. Control Syst., V. 21 (2015)] and [M.A. Lagache, U. Serres, V. Andrieu, Minimal time synthesis for a kinematic drone model, Mathematical Control and Related Fields, V. 7 (2017)] the time optimal control problem is solved where the drone must reach a given unit circle in the minimal possible time and stay on this circle rotating counterclockwise. In particular, in the mentioned works it is shown that is this case the problem is simplified; namely, the problem becomes two-dimensional. In the present paper we consider a natural generalization of the formulation mentioned above: in our problem, the drone must reach a given unit circle in the minimal possible time and stay on this circle, however, both rotating directions are admissible. That is, the drone can rotate clockwise or counterclockwise; the direction is chosen for reasons of minimizing the time of movement. Such a reformulation leads to the time-optimal control problem with two final points. In the paper, we obtain a complete solution of this time-optimal control problem. In particular, we show that the optimal control takes the values $\pm1$ or $0$ and has no more than two switchings. If the optimal control is singular, i.e., contains a piece $u=0$, then this piece is unique and the duration of the last piece equals $\pi/3$; moreover, in this case the optimal control ins non-unique and the final point can be $(0,1)$ as well as $(0,-1)$. If the optimal control is non-singular, i.e., takes the values $\pm1$, then it is unique (except the case when the duration of the last piece equals $\pi/3$) and the optimal trajectory entirely lies in the upper or lower semi-plane. Also, we give a solution of the optimal synthesis problem.

Keywords: a kinematic model; time-optimal control problem; optimal synthesis.

2010 Mathematics Subject Classification: 49N35; 93C10.   [ Full-text available (PDF) ]   Top of the page.

V.I. Korobov, A.I. Derevianko, Controllability of the linear switched dynamical systems of the special type, P. 93-101.

Short abstract: In this paper, the controllability of a special type of linear switched systems is studied. Switch is carried out between two 2 x 2 matrices with purely imaginary eigenvalues. Such a system describes oscillations of a spring pendulum with a switchable stiffness coefficient. The main result of the work is an algorithm that allows finding a set of switching signals for switching from point to point, and a theorem for switching systems with a block-diagonal matrix.

Extended abstract: Switched systems is a special case of hybrid dynamical systems with discrete and continuous dynamics. They are widely applied when a real system cannot be described by one single model. In theoretical works on switched systems, switching signals and times can be random or given by some law. Stability depends both on vector fields and on the switching law. In the present paper, a different formulation of the problem is considered, that is the case, when switching signal is under our control. Namely, a switched system is called controllable if for any two points there exists a switching signal that allows to get from the first point to the second one. In the paper the controllability of linear switched systems of a special type is studied. More specifically, we consider a switch, that is carried out between two 2 x 2 matrices with purely imaginary eigenvalues of both matrices. In the first section we discuss the physical meaning of switched systems of this type. Namely, the problem of oscillation of a spring pendulum with a switchable stiffness coefficient is considered with the series and parallel connection of an additional spring to the system with one given spring. We prove that such a system is controllable, and propose the method of finding the controlling switching signal. In the second section we present the main result of the work. We formulate an algorithm that allows finding a set of switching signals for getting from any given initial point to any given end point. We present an example of such controlling switching signals, simulated in MATLAB. In the last section we propose a generalization of the obtained result and formulate the theorem that states the controllability of the special type switched system with a block-diagonal matrix of high dimension. The method presented in the paper can be generalized to study of controllability of linear switched systems of more general form.

Keywords: linear switched systems; controllability; switching way; getting to the given point; spring pendulum.

2010 Mathematics Subject Classification: 93C15; 93B05.   [ Full-text available (PDF) ]   Top of the page.

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Visnyk Kharkivs'koho natsional'noho universytetu imeni V. N. Karazina, Seriya «Matematyka, prykladna matematyka i mekhanika»

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