The game problem on the dolichobrachistochrone : PMM vol.40, n 6, 1976, pp. 1003–1013

The capture and evasion sets, the players' optimal strategies and the game value determined for the game problem on the dolichobrachistochrone, analysed within the framework of a position formalism similar to [1]. Singularities inherent in the game of the minimax-maximin time to contact [1, 2] become apparent; they are determined in the given problem by the specific behavior of the optimal paths close to the target set. Isaacs [4] examined the game problem on the dolichobrachistochrone, being the game analog of the classical variational problem on the brachistochrone [3]. However, as was shown in [5], the solution proposed by Isaacs contains erroneous statements.     

Optimal stabilization of linear stochastic systems : PMM vol.40, n6, 1976, pp. 1034–1039

The problem of optimal stabilization is solved for controlled linear systems with white noise. The optimal solution is obtained by the method of successive approximations each of which represents the optimal solution of the related determinate problem. Necessary and sufficient conditions of stabilizability are given.     

On a differential-difference game of escape : PMM vol. 40, n 6, 1976, pp. 995–1002

A nonlinear escape problem for conflict-controlled systems described by differential equations with a lagging argument is considered. The sufficient escape conditions which are realized in the class of piece wise-constant functions are obtained. The paper relates to the researches in [1 – 8] and is a continuation of [9, 10].     

On the problem of desalinization of soil containing readily soluble salts : PMM vol.40, n 6, 1976, pp. 1124–1126

A general solution of the problem of desalinization of soil containing rapidly soluble salts is given. The salts are initially nonuniformly distributed, and it is assumed that they pass instantaneously from the solid phase to the solution. A condition of the third kind is postulated at the soil surface, reflecting the continuity of the mass flux of the salts.A different approach was used in [1] to construct a solution for a particular case of homogeneous salinization, and the problem of uniqueness of the solution was studied. The process of diffusion of salts in the course of washing the soil was also studied in [2].     

An example of an irregular differential game : PMM vol. 40, n 6, 1976, pp. 1113–1116

The structure of the positional absorption and its relation to the programing constructions is investigated for a particular case by the method suggested in [1]. The paper is related to the investigations carried out in [2–5].     

Asymptotic stability and smooth equivalence op ordinary equations : PMM vol.40, n 6, 1976, pp. 1048–1057

Under certain specified conditions the asymptotic stability is a coarse property [1],(i.e. addition of fairly smooth functions to the right-hand sides of equations, does not disturb the asymptotic stability). It is shown below that in this cage the unperturbed system is coarse in a more general sense, namely, any smooth system acted upon by fairly small smooth perturbations, can be returned to its unperturbed state by a smooth reversible transformation. The value and order of the perturbations and the domain of existence of the transformation are all estimated explicitly. The condition required for the above assertion to hold, is that of the existence of a Liapunov function admitting, together with its derivative, specified estimates. This requirement holds, in particular, in the case when the right-hand sides of the unperturbed system are homogeneous functions, the position of equilibrium is asymptotically stable, and its neighborhood contains no solutions bounded when −∞ <t < ∞ (see [1]). If the system is analytic, the requirement will hold in at least all critical cases investigated in which the asymptotic stability with t → ∞ or t → −∞ is fixed, since in these cases the Liapunov function will be analytic, or simply polynomial. It follows therefore from the theorem which we prove, that in all the cases in question, the system is reduced by a smooth transformation, to the polynomial form. If the unperturbed system is linear, then from the theorem proved follows a theorem on linearization appearing in [2]; if the system is nonlinear but of second order, a theorem from [3] ensues. The results obtained in this paper for the nonlinear autonomous systems are extended to the case when the perturbations are continuous and bounded functions of time. This makes possible the investigation of the dynamics of the process in the neighborhood of asymptotically stable equilibria and of periodic modes, ignoring a wide range of external perturbations.     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

Vibration op two circular stamps in a layered medium : PMM vol. 40, n 6, 1976, pp. 1104–1112

The problem of the harmonic oscillations on the surface of an elastic layered medium by two circular stamps of radii a₁ and a₂ is considered. The spacing between the centers of the stamps is b > a₁ + a₂. It is assumed that there is no friction in the area of contact.By using a method developed in [1], the initial system is reduced to a system of Fredholm equations of the second kind, for whose solution approximate methods are proposed. On the basis of the results obtained, an applied theory for the vibrations of two stamps can be constructed which also takes account of the dispersion properties of the medium, in contrast to all other known applied theories.It is simple to investigate the case of vibrations of a system of n stamps by the method elucidated in this paper, however, we limit ourselves to the case of two stamps for the sake of brevity.     

Fokker-planck equation for brownian motion of rotating spherical particles : PMM vol. 40, n 6, 1976, pp. 1127–1131

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time.In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors.In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].     

On the energy flux vector for bending vibrations of a plate : PMM vol. 40, n 6, 1976, pp. 1131–1135

An expression for the energy flux vector of plate bending vibrations is obtained in invariant form. The derivation of expressions for the transverse force, bending and twisting moments in an arbitrary orthogonal coordinate system and the derivation of an orthogonality type condition for normal waves being propagated in a thin elastic strip with free edges are considered as applications.In a number of cases it turns out to be useful to consider the energy flux vector in analyzing the vibrations in systems with distributed parameters. The expressions for the Umov-Poynting vector in electrodynamics and for the energy flux vector in acoustics are well-known. An analogous vector for the bending Vibrations of a plate was mentioned only in [1 – 3], This vector is used in [1] to prove a uniqueness theorem for a two-component acoustic model consisting of an ideal compressible fluid and elastic plates in contact with it. However, the expression for the energy flux in [1] (it was later cited in [2, 3] with a reference to [1]) is erroneous. An exact expression (within the framework of the applicability of the Kirchhoff equation) is found below for the energy flux vector of the bending vibrations of a plate and some applications of the formulas obtained are mentioned.     

Contact problems of the theory of elasticity in the presence of wear : PMM vol.40, n 6, 1976, pp. 981–986

The solutions of the contact problems of the theory of elasticity in the presence of wear is given for two cases. In Sect. 1 we consider the problems in which an initially curved beam comes in contact with a half-plane. One of the initial assumptions is that the distance between certain directrices along which the body in contact is sliding and the boundary of the half-plane remains constant. In Sect. 1 the contact between the curved beam and the half-plane is discussed at the assumption that the half-plane is subject to wear. As the result of the wear, the pressure between the beam and the half-plane is gradually reduced. It is naturally assumed that the pressure at the terminal points of the contact area will, in this case, be zero. The conditions characterizing the pressure at these terminal points can be established for various types of contact problems only under certain additional assumptions; this will be discussed below.     

An alternative in the differential-difference game of approach — evasion with a functional target : PMM vol. 40, n 6, 1976, pp.987–994

An approach-evasion problem with a functional target set under constraints on the system's trajectory is studied for a conflict-con trolled system described by a differential-difference equation. The main result states: either a strategy exists for the first player resolving the approach problem or a strategy exists for the second player resolving the evasion problem. The paper is closely related to [1–6].     

On the stability of a satellite cylindrical precession in an elliptic orbit : PMM vol. 40, n 6, 1976, pp. 1040–1047

The stability of motion of a dynamically symmetric satellite with respect to its center of mass in a central Newtonian gravitational field is investigated. The satellite is a solid body whose center of mass moves on an elliptic orbit. The particular case in which the satellite axis of symmetry is normal to the orbit plane (the so-called cylindrical precession [1, 2]) and its absolute angular velocity projection on the axis of symmetry is zero, is examined. Analytical and numerical methods are used. Regions of Liapunov instability and of stability in the first approximation are. obtained in the parameter space of the problem (the inertial parameter and the orbit eccentricity). Detailed nonlinear analysis is carried out in the latter, and the formal stability of the satellite cylindrical precession is proved. The question of stability for the majority of intial conditions is also considered [4].     

Determination of the shapes of doubly-connected bar sections of maximum torsional stiffness : PMM vol. 40, n 6, 1976, pp. 1078–1084

The problem of determining the shape of the transverse section of a prismatic bar with a prismatic longitudinal cavity (hole) of given shape, subjected to torsion, from the condition that the torsional stiffness would be maximal for a given cross-sectional area, is considered. The apparatus of complex variable function theory is used to determine the outline required. Examples of computing the outlines of the sections for elliptical, square, and rectangular hole shapes are presented.     

Igniting a homogeneous reacting medium by a heat source with finite heat content : PMM vol. 40, n 6, 1976, pp. 1065–1069

An asymptotic analysis of the problem of igniting a homogeneous condensed combustible mixture by a flat, cylindrical, or spherical heated body is carried out here with allowance for temperature variation of the body during the process of igniting. An asymptotic solution defining temperature distribution in the region occupied by the condensed medium from the initial instant of time to that of ignition is obtained. The law of temperature variation of the igniting body is determined. A firing criterion that relates basic physicochemical parameters of the problem is derived on the assumption that ignition takes place when during the heat exchange between the igniter and the reacting mixture the igniter is converted from a source to a sink.     

On the self-similar solution of navier-stokes equations with volume sources and sinks of mass : PMM vol. 40, n 6, 1976, pp. 1121–1124

Unlike the investigations in [1, 2] of the motion of fluid with surface sources and sinks of mass (injection and suction), the flow is considered here in the presence of uniformly distributed mobile volume sources and sinks in flat and round channels. It is shown that far away from the inlet a self-similar solution of the system of equations of motion can be obtained. The results are applicable, for instance, to two-phase (vapor-liquid) streams with condensation or evaporation for small volume concentrations of the discrete phase and absence of phase slip.     

On the correctness of the approximate investigation of synchronous machine rotor swinging : PMM vol. 37, n6, 1973, pp. 1007–1014

We consider the complete system of equations for the dynamics of a synchronous machine with two windings on the rotor. We indicate the conditions under which the original system of equations can be reduced to the equation of motion of the rotor. The conditions for rotor selfoscillations to arise are determined as a result of investigating this equation. The complete system of equations for the dynamics of a synchronous machine containing equations describing the electrical responses and equations for the rotor's mechanical motion are obtained in [1]. Transient responses in electric circuits were investigated next, as was the expression for the electromechanical moment under a constant rotation velocity of a rotor with one circuit, e. g., field winding. However, in many of the later works the electrical equations were used only for finding the electromechanical moment under a constant spin rate of the rotor, and the problem was then reduced to the study of the equation for the rotor's mechanical motion [2, 3]. Here the conditions for which such an analysis is admissible were not mentioned. It was established that the swinging of a synchronous machine's rotor can be revealed in the form of selfoscillations. Vlasov [4] has investigated the equation of motion of a rotor and, under the assumption of a small parameter in the first derivative term, has found the conditions for the excitation of selfoscillations. Investigation in this same direction was carried out in [5]. However, in the investigation of the selfoscillations Vlasov did not examine the responses in the electrical circuits, while the expression for the electromechanical moment was obtained from power considerations. Other works have used particular expressions for the electromechanical moment, which can not explain the selfoscillation phenomenon.     

Existence of a global liapunov functional for certain classes of nonlinear distributed systems : PMM vol. 40, n 6, 1976, pp. 1135–1142

The existence of a global Liapunov functional for nonlinear evolutionary equations in a Hilbert space is investigated as a continuation of paper [1], The results obtained represent a generalization of the results of the theory of absolute stability [2, 3], for the systems with infinite dimensional phase space, and are used for investigation of the nonlocal stability and instability of nonlinear distributed systems. The conditions of existence of the global Liapunov functional obtained are illustrated by an example of a nonlinear parabolic system defined in the interval [0, 1], The concept of a Liapunov functional was first introduced and used with success in [4].