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Given second-order Ito stochastic equations, we construct almost surely equivalent stochastic equations of the Lagrangian structure. We establish the conditions for the direct and indirect analytical representations of the Lagrangian in the presence of random perturbations. The results obtained are illustrated by examples. [ABSTRACT FROM AUTHOR]

In this paper based on a given stochastic system, an equivalent almost sure Lagrangian structure equation is constructed in direct and indirect representation, assuming that nonholonomic relations are imposed on the original system. [ABSTRACT FROM AUTHOR]

We consider inverse problems for differential systems in the presence of random perturbations. By the method of additional variables, we establish sufficient conditions for the representation of stochastic differential equations of the second order in the form of stochastic Lagrange equations, as well as for the representation of stochastic differential equations of the first order in the form of stochastic canonical equations. The obtained results are illustrated by specific examples. [ABSTRACT FROM AUTHOR]

Nanoscale coatings formed by thermal vacuum deposition of metal on an aluminum substrate have been investigated by Mössbauer spectroscopy, X-ray diffraction (XRD) analysis, and electron microscopy. The film coatings have been grown by depositing enriched 57Fe in a magnetron vacuum chamber at a residual atmospheric pressure. It is shown that carbon penetration occurs over grain boundaries and inhomogeneities of interboundary formations, and an amorphous phase is formed during deposition of fine-grained 57Fe. After annealing at a temperature of 500°C, the amorphous structure is crystallized and a monophase (cementite Fе3С) is formed. High-temperature annealing leads to the formation of ferrite α-Fе(C) and graphite due to the decrease in the Fе3С content, according to the equilibrium phase diagram of the Fе–C system. [ABSTRACT FROM AUTHOR]

Conversion electron Mössbauer spectroscopy was used to study the thin films obtained by co-precipitation of coatings from two sources: magnetron sputtering of a metal tantalum target and thermovacuum evaporation of metallic iron enriched in the 57Fe isotope. The films of required thickness were formed by alternating exposure to a magnetron-generated tantalum flow and a flow of iron vapor from a thermal evaporator. The analysis of the Mössbauer spectra showed that co-precipitation of iron atoms transferred to the vapor state by heating and tantalum atoms from the magnetron plasma forms a solid solution of iron in tantalum. The film structure is mainly a metastable β-modification of tantalum of the tetragonal crystal system. Annealing of the samples at 1100°С led to the transition of the β-tantalum structure into the body-centered cubic structure of the α-modification of tantalum. It is found that magnetron sputtering creates favorable conditions for formation of solid solutions of two metals at temperatures far from the melting temperatures of their components. [ABSTRACT FROM AUTHOR]

By the method of Lyapunov functions, we study the problem of optimal stabilization of an analytically defined integral manifold in the class of stochastic differential equations in the case where random perturbations belong to the class of processes with independent increments. [ABSTRACT FROM AUTHOR]

In the class of stochastic differential systems of equations of Itô type with indirect control by the first or second derivative, we consider the inverse dynamic problem of constructing a system controller such that a given manifold is an integral manifold of the system. In both cases, the quasi-inversion method is used to construct the whole set of equations of controllers providing the solution of this problem. [ABSTRACT FROM AUTHOR]

By the method of Lyapunov functions, we establish sufficient conditions of stability and asymptotic stability in probability for the integral manifold of the Itô differential equations in the presence of random perturbations from the class of processes with independent increments. Theorems on stochastic stability of the analytically given integral manifold of differential equations are proved in the first approximation and under the permanent action of small (in the mean) random perturbations. [ABSTRACT FROM AUTHOR]

We consider the reconstruction problem in the class of stochastic differential equations of Itô type on the basis of given motion properties that depend only on part of the variables. We determine the set of controls providing necessary and sufficient conditions for the existence of a given integral manifold. [ABSTRACT FROM AUTHOR]

The separation method is used obtain sufficient conditions for the solvability of the main (according to Galiullin’s classification) inverse problem in the class of first-order Itô stochastic differential systems with random perturbations from the class of Wiener processes and diffusion degenerate with respect to a part of variables. [ABSTRACT FROM AUTHOR]

We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces. [ABSTRACT FROM AUTHOR]

An Erratum to this paper has been published: https://doi.org/10.1134/S1063774521230012 [ABSTRACT FROM AUTHOR]

An obituary for Marat Isimgalievich Rakhimberdiev, a specialist in qualitative theories of differential equations, is presented.