Vol 233, No 3 (2024): СОВРЕМЕННАЯ МАТЕМАТИКА И ЕЕ ПРИЛОЖЕНИЯ. Тематические обзоры

For the Sturm–Liouville equation of the standard form, we examine properties of the transfer matrix $\hat{C}$ along a closed path starting at a point $z_0$ and going counterclockwise around the boundary of a convex domain containing exactly one singular point $z_s$ of the potential (the boundary of the domain does not contain singular points). The main attention is paid to the study of singular points that are not branching points; we prove that in this case, if the trace of the matrix $\hat{C}$ is not equal to two, then all its elements are entire functions of the spectral parameter of order $1/2$ and type $2\left|z_0-z_s\right|$ with a trigonometric indicator.

In this paper, we examine a one-dimensional boundary-value problem for the heat equation with a loaded term in the form of the Caputo fractional derivative with respect to a spatial variable. The problem is reduced to the Volterra integral equation with a kernel containing a Wright-type function, for which solvability conditions are obtained. 

In this paper, we present a step-by-step method for the approximate analytical calculation of the matrix of covariance functions for a system of linear stochastic ordinary integro-differential equations with finite concentrated and distributed delays perturbed by additive fluctuations in the form of a vector standard Wiener process with independent components. The method proposed is a combination of the classical method of steps and the expansion of the state space and consists of several stages that make it possible to pass from a non-Markov system of stochastic equations to a chain of Markov systems without delay. Based on these systems, we construct sequences of systems of auxiliary linear ordinary differential equations for elements of vectors of mathematical expectations and covariance matrices of extended state vectors, and then obtaib the required equations for covariance functions.

We study a neighborhood of singular second-order extremals in optimal control problems that are affine in a two-dimensional control in a disk. We study the stabilization problem for a linear system of second-order differential equations for which the origin is a singular second-order extremal. This problem can be considered as a perturbation of an analog of the Fuller problem with two-dimensional control in a disk. We prove that for this class of problems, optimal solutions keep their form of logarithmic spirals that arrive at a singular point in a finite time, while optimal controls make an infinite number of revolutions along the circle. Finally, we present a brief review of problems whose solutions have the form of such logarithmic spirals.

This paper is devoted to the study of the behavior as  of solutions of the Cauchy problem for a first-order differential equation in a Banach space with quadratic operator pencils with the derivative of the unknown function. The branching equation is obtained and analyzed by using the Newton diagram. The conditions of the appearing of a boundary layer near the initial point are identified and the structure of boundary-layer functions is determined.

In this work, continuous stochastic processes with fuzzy states are studied. The main attention is paid to the class of stationary fuzzy stochastic processes. The properties of their numerical characteristics are established: fuzzy expectations, expectations, and correlation functions. Their spectral representation and the generalized Wiener–Khinchin theorem are substantiated. The results obtained are based on the properties of fuzzy stochastic variables and numerical stochastic processes. Triangular fuzzy stochastic processes are considered as examples.