Three-dimensional contact problem for a prestressed elastic body : PMM vol. 42, no. 6, 1978, pp 1080–1084

A half-space of an incompressible neo — Hookean [1,2] material subjected to a homogeneous bi-axial tension or compression along its boundary, is considered. A small deformation caused by the action of a smooth rigid stamp on the boundary of the half-space is superimposed on the initial finite deformation. An integral equation is obtained for the contact pressure. A solution of this equation is obtained for an inclined elliptic stamp with a flat base, and for an elliptic stamp with a curved base, for the cases when the extension coefficients in two directions are either identical, or differ little from each other. The influence of the inital loading on the distribution of the contact pressure, the displacement of the stamp and the form of the contact zone, is analysed.     

On sufficient optimality conditions : PMM vol. 42, no. 6, 1978, pp. 1131–1135

Sufficient conditions of optimality of the control in a nonlinear system are given. This involves a demand for existence of a function with specified properties. If this function is defined in a special manner, then the theorem derived in the paper yields the known theorem of Krotov [1]. A certain relaxation of the sufficient conditions given in [1] is obtained for the problems of the time optimal response in autonomous systems.     

On a problem of correction of the controlled process : PMM vol. 42, no. 6, 1978, pp. 1127–1131

The results obtained in [1] are developed further, by considering a case in which the processs is described by a nonlinear, differential vector equation of the p-th order, the correction vector appears in the right hand side of the equation of motion as one of the terms, and the noise vector is a measurable vector function. A constructive method of correcting the process in question is given, with the correction equation obtained in the explicit form.     

On theasymptotic stability of resonance systems : PMM vol. 42, no. 6, 1978 pp 1033–1038

Sufficient conditions of asymptotic stability as well as of instability derived directly from coefficients of normal form are presented.     

Solution of one of Birkhoff''s problems : PMM vol. 42, no. 6, 1978, pp. 1136–1141

The criterion of existence of an integral linear with respect to generalized velocities over the set of trajectories with the same total energy is proved for conservative dynamic systems with two degrees of freedom.     

On stable position control in differential games : PMM vol. 42, no. 6, 1978, pp 963–968

A procedure for control with a guide (1–3], yielding a solution, stable with respect to small data disturbances, of game problems of dynamics is proposed for a class, more general in comparison with [1–3], of conflict-controlled systems, i.e., systems requiring uniqueness of the program motions and their uniform boundedness. The terminology and notation follow [2].     

On the theory of dynamic thermoelasticity : PMM, vol. 42, no. 6, 1978, pp. 1093–1098

Use of the causality principle as radiation condition in dynamical problems of thermoelasticity is proposed. It follows from an analysis of the fundamental mathematical models describing the thermoelastic behavior of a continuous medium and used in the solution of specific problems, that some will yield physically unrealizable solutions. To eliminate the ambiguity in the solution which occurs, an approach is possible which has an explicit physical meaning and is based on the causality principle [1, 2]; it is required that the time source not yield a response earlier than the time of starting up of the source. Different kinds of radiation conditions of the Sommerfeld type are known in thermoelasticity problems [3 – 6].

To extract the unique solution in dynamical thermoelasticity problems, it is proposed in this paper to use the causality principle, which is equivalent to the requirement of analyticity of the solution in the upper half of the complex frequency plane; there are studied the analytic properties of the solutions of the fundamental boundary value problems for the models used most often for thermoelastic media, and there are made deductions about their physical realizability.     

The problem of minimax mean square filtration in parabolic systems : PMM vol. 42, no. 6, 1978, pp, 1016–1025

The problem of estimating parameters of state of a distributed parabolic system by observation results is considered. The system is assumed to function under conditions of undefined perturbations in the measurement channel and specified initial distribution. The problem is considered in minimax formulation [1] in conformity with the scheme accepted for ordinary differential equations [2].(^*), Analytic definition of sets X (/gJ, y (·)) (/gJ > 0) of states of a parabolic system compatible at instant /gJ with the realizable signal y (t) (t ε [0, /gJ]) is obtained. An element of region X (/gJ, y (·)) which satisfies the specified minimax criterion is chosen as the optimal estimate of the true state at instant /gJ. Integradifferential equations in partial derivatives are derived for parameters that define the evolution of regions X (/gJ, y (·)) in time. One of the methods of approximating the input problem of observation by similar problems for systems of ordinary differential equations is discussed on a specific example. Problems of observation for distributed systems in different formulations appear in [3 – 6].     

Optimal synthesis in certain terminal control problems : PMM vol. 42, no. 6, 1978, pp. 989–999

A class of problems of terminal control of multi-dimensional systems reducible to one-dimensional is investigated. Similar problems of time-optimal response for autonomous and nonautonomous systems were considered, for instance, in [1]. The solution derived here is based on sufficient conditions of the dynamic programming method [2]. Ways are developed for the derivation of an analytic solution of the problem of synthesis and, also, for the determination of optimal-phase trajectory and of programmed control. Problems of energy-optimal consumption for stabilizing the rotations of a dynamically symmetric solid body using a limited power motive system are solved [3].     

On stability of constant laplace solutions of the unrestricted three-body problem : PMM vol. 42, no. 6, 1978, pp. 1026–1032

It is shown that in the absence of third order resonances [1] Laplace solutions retain stability in the second order within the limits of the Routh —Joukowski necessary conditions of stability. When third order resonances and their interaction take place in a system, the question of stability in the second order and that of Liapunov instability is completely solved by the present investigation in conjunction with that in [2].     

Theorem on the alternative for a nonstationary differential encounter-evasion game : PMM vol. 42, no. 6, 1978, pp. 1123–1126

A position encounter-evasion differential game with non-stationary geometric constraints on the players' controls is analyzed. It is proved that the alternative is valid for this game, stating that either the position encounter game or the position evasion game is always solvable. The proof uses constructions analogous to the corresponding ones from [1].     

On the weak control of slowly damped systems : PMM vol. 42, no. 6, 1978. pp. 1000–1005

Asymptotic formulas are obtained which make it possible to derive the first approximation solution of the Riccati matrix algebraic equation of special form. Method is based on Bass' formulas [1] and the theory of perturbations [2], The problem of control of a slowly damped oscillator is investigated in detail. Formulation of the problem in this paper differs substantially from that in [3] (no assumption is made about single-frequency oscillations, and only a stationary system is considered over an infinite time interval).     

On the theory of differential games in systems with aftereffect : PMM vol. 42, no. 6, 1978 pp.969–977

An encounter-evasion differential game is studied for control systems with aftereffect [1–4]. A feature of the system being analyzed is that it has a time-lag effect with respect to the controls which provides the system with important new peculiarities. Using the investigations in [1–4], conditions for the solvability of the problem are indicated and the required control procedures are constructed.     

On the necessity of a sufficient optimality condition for pursuit time : PMM vol. 42, no. 6. 1978, pp. 1006–1015

A necessary and sufficient condition for the optimality of the upper layer time is derived for one class of linear pursuit problems satisfying local convexity conditions.     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

Stability of steady flows of perfect incompressible fluid with a free boundary : PMM vol. 42, no. 6, 1978, pp. 1049–1055

The criterion of stability of steady flow of a perfect incompressible fluid bounded by solid walls, indicated by Amol'd [1, 2], is extended to the case when a part of the flow region boundary is free and subjected to surface tension.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

On numerical realization of the function of the hereditary operator : PMM vol. 42, no. 6, 1978, pp. 1115–1122

The problem of numerical realization of a function of the hereditary operator acting on some function of time is considered. Laplace transformations are used for the operators with kernels of the Rabotnov and Rzhanitsyn type to obtain formulas which reduce the problem in question to that of computing a quadrature. When the variable assumes large values, the formulas become asymptotic equations with an estimable error of approximation.     

Contact problem for two microinhomogeneous half-spaces : Pmm vol. 43, no. 6. 1979. pp. 1082–1088

The interaction between two microinhomogeneous half-spaces in a macrohomogeneous stress-strain state is considered. The perturbation method is used to solve the problem. Finite expressions for the correlation functions and dispersion of deformations are obtained in the first order approximation. Numerical values of the coefficients characterizing the stress concentration at the boundary of interaction between the half-spaces are derived. The dependence of these coefficients on the ratio of elastic moduli of the media filling the half-spaces is studied. A particular solution of this problem for the case when the elastic moduli depend on two coordinates only, was given in [1].