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 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.     

Oscillations of mechanical systems that do not become linear when the parameter vanishes : PMM vol. 42, no. 6, 1978, pp. 1039–1048

The results of investigations in [1] are extended to multidimensional systems that become nonlinear at μ = 0. Two-dimensional mechanical systems were investigated in [2,3]. The characteristic equations of systems considered here contain in the critical system either a pair of pure imaginary roots or two zero roots with one or two groups of solutions and n roots with negative real parts in the adjoint system. It is shown that the investigation of such systems necessitates the imposition on the system of some constraints that supplement those specified in [1], The auxilliary function u⁽¹⁾_k (θ) used in the determination of Liapunov's function is derived by a different method than in [1 – 3], In two of the three investigated cases the problem is reduced to the determination of roots of some integral real irrational function. An example is presented.     

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 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 membrane state of multiply-connected convex shells : PMM vol. 37, n6, 1973, pp. 1141–1145

Conditions for the realization of the membrane state of equilibrium of multiplyconnected shells of positive Gaussian curvature subjected to surface and edge forces, are investigated; the concepts of correctness and stability of the membrane states are introduced. The terminology and notation correspond to that used in [1, 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.     

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 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].     

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].     

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.     

Contact problem for a layer with two stamps : PMM vol. 42, no. 6, 1978, pp. 1068–1073

A problem of impressing coaxial stamps of circular cross section into the upper and lower surface of a homogeneous elastic layer is studied. The bases of the stamps have axial symmetry. The parts of the layer surfaces lying oustide the contact zone are stress-free, there is no friction or coupling between the layer and the stamps. A system of two integral equations with two unknown functions is obtained, and provides a solution of the problem. The method of separating the singularities provides the way of reducing this system to the Fredholm equations of second kind. An approximate solution of the equations is obtained for the case of flat stamps under the assumptions that the two parameters entering the system are sufficiently small.

Problems of a layer with various boundary conditions were formulated and solved in many papers and books, e.g. [1, 2]. However, to the best of the author's knowledge, in all these problems the conditions at the boundary were assumed different only on one side of the layer; in the present problem the boundary conditions are mixed at both sides of the layer, and this results in a system of two integral equations.     

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 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.     

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].     

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 the approximate synthesis of the optimal control of stochastic quasilinear systems with aftereffect : PMM vol. 42, no. 6, 1978, pp. 978–988

Control problems for quasilinear deterministic systems without time lag were analyzed in [1, 2]. In the present paper the control of quasilinear stochastic systems, whose theory has been presented in [3–6], is studied. The approximate synthesis of the control of stochastic systems with aftereffect is of importance since the construction of their exact optimal control is successful only in exceptional cases [7, 8]. In the paper an approximate optimal control synthesis algorithm is proposed and a method for obtaining error bounds, different from ones previously obtained [9, 10], is developed.     

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].