Vol 61, No 12 (2025)

In this paper, a boundary value problem with generalized boundary conditions of the Sturm type is studied for a linear ordinary delay differential equation with the Dzhrbashyan–Nersesyan derivative of arbitrary order. The solution to the problem is written out in the terminology of the Green function. The existence and uniqueness theorem of the solution to the problem is formulated and proved.

A new boundary value problem for stationary heat and mass transfer equations with variable coefficients is considered. It is assumed that the leading coefficients of viscosity, thermal conductivity, and diffusion, as well as the buoyancy force included in the original equations, depend on the temperature and the concentration of the substance dissolved in the base medium. A mathematical framework is developed to study the mentioned boundary value problem based on a variational approach. Using the developed framework, the global existence of a weak solution to the boundary value problem under study is proved, and sufficient conditions on the problem data are established that ensure the local uniqueness of the weak solution with the additional smoothness property of the temperature and concentration.

The B-hyperbolic operator □_γ = ∂²/∂t² − a²Δ_Bγ is considered, with the operator Δ_Bγ = ∑_i=1 ⁿ B_γi , where B_γi are Bessel operators with parameters γ_i > −1. The definition of the δ_−γ-Dirac distribution is introduced and the formula for the Bessel transform of the δ_−γ-Dirac distribution is obtained. Three types of fundamental solutions of the B-hyperbolic operator are given. A solution to the inhomogeneous B-hyperbolic equation is given.

The conditions linking the strict selection property in continuous and discrete dynamic systems on a standard simplex are investigated. These conditions allow for the correct selection of the integration step without losing the strict selection property in the system. An attempt is made to link the concepts of selection for difference systems with the corresponding differential analog. It is shown that, when the solution of a difference system converges uniformly to the vertex of the simplex, the original differential systems also possess the selection property and are widely used in constructing mathematical models of various real-world processes.

The problem of stabilizing the origin is solved for dynamical systems written in a form that admits state feedback linearization, taking into account the magnitude constraints on the state variable values. Based on known results on the possibility of obtaining identical control laws when using the integrator backstepping and the state feedback linearization methods to design stabilizing feedbacks, sufficient conditions are proposed for the gain coefficients and roots of the characteristic polynomial of the closed-loop system that ensure fulfillment of the specified constraints. The derived conditions guaranteeing that the constraints hold are based on the results obtained using the integrator backstepping method combined with logarithmic barrier Lyapunov functions. As an example, a solution to the problem of a generalized coordinate regulation is considered for a mechanical system, whose dynamics with respect to the selected generalized variable can be represented as a chain of fourth-order integrators, taking into account the constraints on the values of the generalized coordinate, velocity, acceleration, and jerk.