Vol 241 (2025)
Articles
Stability criteria of systems of nonlinear ordinary differential equations based on additive transformations of the formula of finite increments
Abstract
Based on additive transformations of difference schemes and the formula of finite increments, we obtain criteria for the Lyapunov stability of systems of ordinary differential equations. The mathematical structure of the criteria admits the possibility of software implementation. Approximate solutions of the system can be computed by the piecewise interpolation method with iterative refinement. The search for Lagrange points is based on calculating the minimum of the modulus function using the sorting by stable address merging. Practical applications of the criteria allow one to perform the stability analysis in real-time mode.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:3-12
3-12
On the maximal number of spanning trees in cacti with given order
Abstract
The number of spanning trees of a graph is an important characteristic of its reliability as a data transmission network. We found the maximal number of spanning trees in a cactus with a given number of vertices and also in a bipartite cactus with a given number of vertices. In particular, friendship graphs and Koch networks are extremal graphs.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:13-17
13-17
Feedback minimum principle for optimal control problems with terminal conditions and its extensions
Abstract
The nonlocal necessary optimality condition, the so-called feedback minimum principle (F-PM) obtained in previous publications of the author for free endpoint problems, is generalized for problems with terminal constraints. The proof of the new necessary condition is based on abstract methods of support majorants and modified Lagrange functions (MLF) with a quadratic penalty. But the corresponding unconstrained problem does not necessarily have to be solved. If the reference process is optimal, then there is no descent for the MLF from it using F-PM. If this necessary optimality is violated, then we obtain an improved admissible process. The constructive basis of the feedback minimum principle is the descent method with feedback strategies. However, it is natural to use this descent method for minimizing the modified Lagrangian in the well-known Krotov and Pontryagin optimality conditions. As a result of such an extension of the F-PM descent method, we obtain feedback versions of the Krotov and Pontryagin methods, which are significantly more efficient than the traditional methods.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:18-29
18-29
Probability distributions defined by Lah numbers and generalized Lah numbers
Abstract
Probability distributions described by means of Lah numbers and generalized Lah numbers are considered. Roots of generating functions are examined and numerical characteristics are found. The asymptotic behavior of distributions is studied.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:30-39
30-39
Application of the parametrization method for solving optimal control problems with discontinuous phase trajectories
Abstract
In this paper, we consider an optimal control problem involving delays in phase variables and controlled discontinuities in the phase trajectory. To solve this type of problem, we propose the parametrization method based on representing control functions as generalized splines with movable nodes and subsequently reducing the original problem to a finite-dimensional nonlinear programming problem for control parameters. In this work, the moments of discontinuity of the phase trajectory are considered as variables of the resulting finite-dimensional problem. Also, we propose an algorithm for calculating the derivatives of the objective function based on the use of the adjoint variables of the original optimal control problem. The algorithm proposed for solving the considered problems was tested by examples presented in the work.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:40-54
40-54
Energy loss of a high-speed color particle in a non-Abelian plasma within the framework of the Hamiltonian formalism
Abstract
A general formula for the energy loss of a fast color-charged particle in the process of its scattering on soft boson excitations of quark-gluon plasma is derived within the framework of the classical Hamiltonian formalism. The concept of the effective current of the scattering process is introduced, its relationship with the classical scattering matrix is determined, and a formula for the energy loss of a high-energy color particle moving in a high-temperature plasma with a non-Abelian interaction type is found.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:55-63
55-63
Parametric transformation of nonconvex optimal control problems
Abstract
A nonconvex linear-quadratic optimal control problem with indefinite matrices of quadratic forms is considered. The transformation and parametrization of the target functional are performed; this leads to a family of identical problems in the sense of a global solution. Conditions on the parameters that distinguish convex problems are obtained. A parametric optimization procedure is implemented according to the criterion of proximity to the original problem. As a result, convex linear-quadratic problems are constructed, which provide the possibility of improving extremal controls in a nonconvex problem.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:64-70
64-70
About one $SI^{\ast}$-interval of rank $2$ multioperations
Abstract
We consider rank-$2$ multi-operations. The superposition is defined on the basis of a set-theoretic intersection operation with selection of a single element. The interval between the clone of self-transitive operations and the multiclone of all multi-operations is described. The results obtained can be applied to the description of other intervals in the lattice of rank-$2$multiclones.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:71-82
71-82
The first boundary-value problem for some mixed equations of thermal conductivity of the second and fourth order
Abstract
In this paper, we present a computational model for solving the first boundary-value problem for a mixed differential equation in a spatially two-dimensional case using an implicit finite-difference scheme. For a fourth-order equation, we use two different second-order operators that are similar to a one-dimensional mixed thermal conductivity operator in the spatial variable, generalizing the purely hyperbolic case and used in mathematical modeling of the shutdown process for an electric arc. The well-posedness of the boundary-value problem is proved.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:83-89
83-89
Generalized Navier–Stokes equations associated with the Dolbeault complex
Abstract
We consider the Cauchy problem for a system of nonlinear differential equations structurally similar to the classical evolutional Navier–Stokes equations for an incompressible liquid. The main difference of this system is that it is generated not by the standard gradient, divergence, and curl operators, but by the multidimensional Cauchy–Riemann operator, its the compatibility complex (which is usually called the Dolbeault complex) and its formally adjoint operator. The similarity of the structure makes it possible to prove the theorem of the existence of weak solutions for this problem and the open mapping theorem on the scale of specially constructed Bochner–Sobolev spaces. In addition, a criterion for the existence of a “strong” solution in these spaces is obtained.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;241:90-100
90-100