卷 23, 编号 1 (2023)

We consider the Lezanski – Polyak – Lojasiewicz inequality for a real-analytic function on a real-analytic compact manifold without boundary in finite-dimensional Euclidean space. This inequality emerged in 1963 independently in works of three authors: Lezanski and Lojasiewicz from Poland and Polyak from the USSR. The inequality is appeared to be a very useful tool in the convergence analysis of the gradient methods, firstly in unconstrained optimization and during the past few decades in problems of constrained optimization. Basically, it is applied for a smooth in a certain sense function on a smooth in a certain sense manifold. We propose the derivation of the inequality from the error bound condition of the power type on a compact real-analytic manifold. As an application, we prove the convergence of the gradient projection algorithm of a real analytic function on a real analytic compact manifold without boundary. Unlike known results, our proof gives explicit dependence of the error via parameters of the problem: the power in the error bound condition and the constant of proximal smoothness first of all. Here we significantly use a technical fact that a smooth compact manifold without boundary is a proximally smooth set.

It is well-known that the Lagrange interpolation based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere), like the Fourier series of a summable function. On the other hand, any measurable almost everywhere finite function can be “adjusted” in a set of an arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises whether the class of continuous functions has a similar property with respect to any interpolation process. In the present paper, we prove that there exists the matrix of nodes $\mathfrak{M}_\gamma$ arbitrarily close to the Jacoby matrix $\mathfrak{M}^{(\alpha,\beta)}$, $\alpha,\beta>-1$ with the following property: any function $f\in{C[-1,1]}$ can be adjusted in a set of an arbitrarily small measure such that interpolation process of adjusted continuous function $g$ based on the nodes $\mathfrak{M}_\gamma$ will be uniformly convergent to $g$ on $[a,b]\subset(-1,1)$.

The possibilities of applying the qualitative theory of differential equations to one problem of heat and mass transfer in multilayer planar semiconducting structures are studied. The consideration is carried out on the example of a mathematical model of a stationary process of diffusion of nonequilibrium minority  charge carriers generated by a wide excitation source. The use of a wide source of external influence makes it possible to reduce modeling problems to one-dimensional ones and describe these mathematical models by ordinary differential equations. These are the processes in various nanosystems exposed to wide beams of charged particles or electromagnetic radiation. The paper reviews the results of recent studies of such models. The main object of study was the questions of the correctness of the considered mathematical models, special attention is paid to the mathematical assessment of the influence of external factors on the state of the object under study. Previously, the methods of the qualitative theory of differential equations, in our case, the assessment of the influence of external influence on the distribution of nonequilibrium minority charge carriers as a result of their diffusion in a semiconductor, in combination with the consideration of the uniqueness of the solution of differential equations of heat and mass transfer and the correctness of the mathematical models used, were considered very rarely, and for wide electron beams, a quantitative analysis of such problems has not previously been carried out at all. In the present work, the main attention is paid to the influence of the right side of the differential equation, the excitation function of minority charge carriers, on the solution of the differential diffusion equation, which describes the distribution of nonequilibrium charge carriers that have diffused in each layer of such a structure. The uniqueness of the solution of the problem under consideration and the continuous dependence of the solution on the right side of the differential equation are proved. Estimates are obtained for the influence of external factors on the diffusion of generated carriers in each layer of a multilayer planar semiconductor structure.

To obtain the deformation matrix of the prismatic finite element at the loading step, taking into account the physical nonlinearity, three variants of physical equations were used. In the first variant, the defining equations of the theory of plastic flow are implemented, according to which the increment of deformations is divided into elastic and plastic parts. The increment of elastic deformations is related to the increments of stresses by Hooke's law. The relationship of plastic strain increments with stress increments is determined based on the hypothesis of the proportionality of the components of the plastic strain increment tensor to the components of the stress deviator. In the second variant, the components of the plastic strain increment tensor are obtained on the basis of the proposed hypothesis about the proportionality of these components to the components of the stress increment deviator at the loading step. In this variant, as well as in the first variant, the hypothesis of incompressibility of the material during plastic deformation is accepted. In the third variant, the defining equations at the loading step were obtained on the basis of the proposed hypothesis about the proportionality of the components of the deformation increment deviator to the components of the stress increment deviator without dividing the deformation increments into elastic and plastic parts. The proportionality coefficient turned out to be a function of the chord modulus of the deformation diagram. The hypothesis of incompressibility of the material during plastic deformation was not accepted, but the dependence between the first invariants of strain tensors and stress tensors obtained from the experiment was realized. For comparison with the first and second variants of the defining equations, this dependence between the first invariants of strain and stress tensors is determined by the elastic deformation formula. A prismatic element with triangular bases is adopted as the finite element. Displacement increments and stress increments are taken as nodal unknowns. Approximation of the desired values of the finite element method, in a mixed formulation through nodal values, was carried out using linear functions. The stress-strain state matrix is presented on the basis of a mixed functional obtained from the physical expression of the equality of the possible and actual work of external and internal forces at the loading step with the replacement of the actual work of internal forces by the difference of the full and additional work of internal forces. The calculation example shows an adequate correspondence in the calculation results based on the considered variants of the physical relations and the preference of the third variant of the defining equations of the theory of plasticity is noted.

The classical problem of optimal control of the attitude maneuver of a spacecraft as a rigid body of arbitrary dynamic configuration under arbitrary boundary conditions for the angular position and angular velocity of a spacecraft without restriction on the control vector function and with a fixed transition time is considered. As a criterion of optimality, the functional of the energy spent on the rotation of a spacecraft is used. Within the bounds of the Poinsot concept describing arbitrary angular motion of a rigid body in terms of generalized conical motion, a modification of the problem of optimal control of the angular motion of a spacecraft is carried out and its trajectory is given in this class of motions. At the same time, the generality of the original problem is practically not violated, since the known exact solutions to the classical problem of optimal angular motion of a dynamically symmetric spacecraft in cases of plane rotation or regular precession and similar solutions of the modified problem completely coincide; in other cases, in numerical calculations of the classical and modified problems, the discrepancy between the values of the optimization functional is no more than a few percent, including spacecraft rotations at large angles. Therefore, the proposed solution of the modified problem can be used as quasi-optimal with respect to the classical problem. Explicit expressions for the quaternion of the orientation and the vector of angular velocity of a spacecraft are given, a formula for the vector of the control moment of a spacecraft is obtained based on the solution of the inverse problem of the dynamics of a rigid body. The quasi-optimal algorithm for optimal rotation of a spacecraft is given. Numerical examples showing the proximity of solutions to the classical and modified problems of optimal reorientation of a spacecraft are given.

The purpose of our study was to formulate provisions for obtaining an integral assessment characterizing the rating of forest areas in terms of fire hazard. We obtained this estimate based on the aggregation of many parameters characterizing climatic conditions and factors that take into account anthropogenic influence in a given area of the forest. Considering the heterogeneity of such parameters, we used the methods of fuzzy inference and the theory of fuzzy sets to aggregate them. The complex for determining the assessment of the forest area is implemented in the form of a hierarchical fuzzy inference system. We investigated the process of functioning of the formed complex and found that its output pattern is predominantly stepwise. This result makes it possible to classify the analyzed forest areas into states groups. Further studies of the classes of states formed by us by the methods of cluster analysis make it possible to identify areas with similar characteristics. The use of the classification results makes it possible to rank forest areas according to the order of preventive or preparatory measures to reduce  fire hazard or increase  responsiveness in case of a fire. The results obtained by us are aimed at using in decision support systems for the management of forests and other types of adjacent territories.