Vol 60, No 11 (2024)

For a nonlinear second order ordinary differential equation, arising while mathematical modeling cryochemical synthesis of medicinal nanoforms, the behavior of its positive monotonic solutions is studied as well as the existence, uniqueness and properties of solutions of various boundary value problems with fixed and free boundaries.

For an ordinary differential equation with a fractional discretely distributed differentiation operator, the Naimark problem is studied, where the boundary conditions are specified in the form of linear functionals. This allows us to cover a fairly wide class of linear local and nonlocal conditions. A necessary and sufficient condition for the unique solvability of the problem is obtained. A representation of the solution to the problem under study is found in terms of special functions. The theorem of existence and uniqueness of the solution is proven.

The Cauchy problem for Petrovskii second-order parabolic systems in a strip on the plane is considered. The coefficients of the system satisfy the double Dini condition. The unique solvability of the problem in the space of functions that are continuous and bounded together with their spatial derivatives of the first order in the closure of the strip is established and corresponding estimates are obtained. An integral representation of the solution is given.

For a switched affine system closed by stabilizing static feedback, a method is presented for constructing a parametric family of partitions of the state space, relative to which this closed system remains stable.

The problem of optimal choice of system parameters with respect to a given control quality criterion is studied. To compare systems with different parameter values, a quantitative estimation of controllability is introduced. This estimation is based on the average value of the function that defines the quality criterion. As an example, a very simplified model of an underwater vehicle is considered. The problem of finding an arrangement of its control propellers, in which either the movement time or the energy consumption of the vehicle is minimal is investigated. To test the used approach, the trajectories of the underwater vehicle are randomly generated. The energy consumption and the movement time along these trajectories of systems with different parameter values and different indexes are compared.

The work is devoted to the problem of verifying that the state of a linear controlled system of differential equations will hit the target set over a finite time interval, despite the uncertainties (noise). Some geometric, pointwise convex constraints on uncertainties are imposed. In the case of a two-dimensional state space a method is proposed for constructing a solvability set without the calculation of the convex hulls of the functions necessary to construct a support function of the geometric difference of the sets. A Hamilton–Jacobi–Bellman type equation is obtained, which is satisfied by the distance function to the solvability set.

When solving localization problem numerically, the main problem is to construct a universal cross section corresponding to a given localizing function. The paper proposes two methods for solving this problem, which use estimates of the first and second order derivatives. A comparative analysis of these methods with a method based on the use of all nodes of a regular grid was carried out. A comparative analysis shows that the proposed methods are superior both in terms of computational complexity and in the quality of the resulting approximation of the universal section.

A solution of the wave equation with three spatial variables is given, which has a finite energy integral, but does not tend to a spherical wave at infinity.

Ниже публикуются аннотации докладов, состоявшихся в осеннем семестре 2024 г. (предыдущее сообщение о работе семинара дано в журнале “Дифференциальные уравнения”. 2024. Т. 60. № 6).∗∗