Vol 60, No 5 (2024)

A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed and conditions are determined under which the absolute stability of the investigated system can be proved with the aid of such a functional. In the case of nolinearities of the power form, estimates for the convergence rate of solution to the origin are derived. The stability analysis of the corresponding system with parameter switching is fulfiled. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained.

A stochastic differential-difference hybrid system is a system of interacting variables whose dynamics are described by stochastic differential equations for some of them and difference equations for others. Systems with two types of difference equations are examined: first, a difference equation in the form of a process involving the multiplicative Wiener process, and second, a difference equation with delay. The existence and uniqueness theorems for both systems have been proven. The basic conditions on the system’s parameters are local Lipschitz conditions and linear growth order.

Formal asymptotics are substantiated that describe typical dropping cusp singularity of quasiclassical approximations to solutions of two cases of the integrable nonlinear Schr¨odinger equation −𝑖𝜀Ψ′𝑡=𝜀2Ψ′′𝑥𝑥±2|Ψ|2Ψ, where 𝜀 is a small parameter. The substantiation uses the ideology and facts of the mathematical catastrophe theory and the part of the theorem of Yu.F. Korobeinik, concerning analytical as ℎ → 0 solutions 𝐺(ℎ, 𝑢) of the mixed type linear equation ℎ𝐺′′ℎℎ = 𝐺′′𝑢𝑢 to which the hodograph images of both cases of the systems of equations of these quasiclassical approximations are equivalent.

A two-parameter multiplicative control problem is studied for a model of electron-induced charging of an inhomogeneous polar dielectric. Exact estimates of the local stability of its optimal solutions with respect to small perturbations of both the cost functionals and the given function of the boundary value problem are derived. For one of the controls, the relay property or the bang-bang principle is established.

A nonlinear system of ordinary differential equations with control parameters is considered. Pointwise restrictions are imposed on the possible values of these parameters. It is required to solve the problem of transferring the trajectory of the system from an arbitrary initial position to the smallest possible neighborhood of a given target set at a fixed time interval by selecting the appropriate feedback control. To solve this problem, it is proposed to construct a continuous piecewise cubic function of a special kind. The level sets of this function correspond to internal estimates of the solvability sets of the system. Using this function, it is also possible to construct a feedback control function that solves the target control problem at a fixed time interval. The paper proposes formulas for calculating the values of a piecewise cubic function, examines its properties, and considers an algorithm for searching for parameters defining this function.

A new symmetric variational functional-algebraic formulation of the eigenvalue problem in the Hilbert space with linear dependence on the spectral parameter for a class of mathematical models of thin-walled structures with an attached oscillator is proposed. The existence of eigenvalues and eigenvectors is established. A new symmetric approximation of the problem in a finite-dimensional subspace with linear dependence on the spectral parameter is constructed. Error estimates of the approximate eigenvalues and eigenvectors are obtained. The theoretical results are illustrated on the example of the problem of structural mechanics.