Vol 28, No 4 (2024)

The article deals with a model of optimal dynamic measurement with multiplicative influence, considered as an optimal control problem for a nonstationary Leontief-type system. The existence of a solution to this problem in a stochastic formulation is established. The main objective is to find a recoverable signal (control action) that brings the system state as close as possible to the observed indicators, given the presence of an additional input process modeling noise. Solutions to the system must be sought in spaces of random processes. To achieve this, the optimal control problem in spaces of
differentiable "noises" is preliminarily analyzed. The linearity of the transducer model, described by a non-stationary Leontief-type system, allows the original system to be decomposed into deterministic and stochastic subsystems. Based on the results regarding the solvability of optimal control problems for each subsystem, a solution to the original problem is obtained.
The first part of the article presents the solvability conditions for a stochastic non-stationary Leontief-type system. The second part explores the optimal control problem in the stochastic case and derives estimates for the minimized functionals using results previously obtained for the deterministic counterpart. In conclusion, an algorithm for studying the problem of optimal dynamic measurement with multiplicative influence in spaces of "noises" is presented.

The work is devoted to the calculation of the geometry of an equi-strength annular disk taking into account the anisotropy and strength differential effect. The disk is under centrifugal forces and tractions on the inner and outer surfaces. The problem statement is based on the anisotropic elasticity theory and the plane stress assumption. General quadratic failure criterion is used, the only requirement for which is ellipticity. In particular cases, the used condition is reduced to many known strength criteria (Tsai–Wu, Hill, Drucker–Prager, von Mises, etc.).
The governing system of equations consists of the compatibility equation, the equilibrium equation and the condition of constant equivalent stress. This condition is satisfied by a trigonometric substitution and an introduced auxiliary function. The two remaining equations are solved sequentially in an implicit form, in which the auxiliary function is treated as independent variable. The found analytical solution allows to construct the geometry of the disk (profile and inner radius of the disk) of equal strength, and also to determine the distribution of stresses in it. It is established that the solution may not exist and be non-unique. In particular cases, the solution is reduced to solutions for many known failure criteria, as well as to the classical solution of Rabotnov. Comparison of calculations obtained for the Tsai–Wu and von Mises criteria showed that anisotropy and different strengths under tension and compression can have a significant effect on the geometry of a disk of equal strength and the stress state in it.

The paper presents an analytical solution to the problem of axisymmetric antiplane shear. The deformable material is enclosed between two cylindrical surfaces, one of which is fixed, and the other moves along the generatrix. This problem models a shear-off testing scheme. We use a geometrically nonlinear formulation of the elastic-plastic problem, taking the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts. The elastic properties of the specimen are described by the Mooney–Rivlin hyperelastic model. We consider an isotropic hardening material with the hardening law that is an arbitrary monotonic function of the accumulated plastic strain. The Tresca yield condition is utilized. The original nonlinear coupled system of partial differential equations is reduced to ordinary linear differential equations, the solution of which requires the calculation of definite integrals. The resulting solution includes deformation in the elastic range, the initiation of plastic flow, propagation of the plastic deformation region, and subsequent intensive plastic flow. The solution is illustrated with examples of materials with linear hardening, quadratic hardening, and Voce-type hardening with saturation. For these examples, ''force – displacement'' relationships, the distribution of accumulated plastic strain over the sample cross-section, and data on the distortion of material fibers, which were located in the radial direction before deformation, are
presented.

A mathematical model of gas oscillations induced by external harmonic loading has been developed, taking into account spatiotemporal nonlocality. The model is based on the equilibrium (motion) equation and a modified Hooke’s law, which incorporates relaxation terms accounting for the mean free path and time of microparticles (electrons, atoms, molecules, ions, etc.).
Numerical studies of the model have shown that resonance occurs when the natural frequency of gas oscillations coincides with the frequency of the external load. This resonance is characterized by a sharp increase in the amplitude of oscillations, which is limited by the gas friction coefficient. When the frequency of the external load is close to the natural frequency of gas oscillations, bifurcation-flutter oscillations (beats) are observed, accompanied by periodic increases and decreases in the oscillation amplitude at each point of the spatial variable. In this case, the gas oscillations exhibit an infinite
number of amplitudes and frequencies.
Periodic variations in gas displacement and pressure, ranging from zero to a certain maximum value and propagating along the length of the methane pyrolysis reactor, contribute to the cleaning of its internal surfaces from loose carbon deposits. The carbon removed from the reactor walls accumulates in the lower part between two gas-tight shut-off valves, allowing for its removal without interrupting the pyrolysis process. This model can be useful for optimizing reactor cleaning processes and improving the efficiency of methane pyrolysis.

Solutions to some boundary value problems for systems of hyperbolic partial differential equations can be constructed explicitly in terms of the Riemann matrix. In this regard, the question of explicitly constructing the Riemann matrix for high-order hyperbolic systems of equations is relevant.
We consider a system of third-order hyperbolic partial differential equations with three independent variables. For the specified system, the Riemann matrix is constructed as a solution to a special Goursat problem. Furthermore, the Riemann matrix satisfies a Volterra integral equation. The Riemann matrix is expressed explicitly in terms of a hypergeometric function of a matrix argument. Similarly, a system of fourth-order hyperbolic partial differential equations with four independent variables is considered. These results are generalized for a system of hyperbolic partial differential equations of order $n$ that does not contain derivatives of order less than $n$.