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Vol 61, No 12 (2025)
ORDINARY DIFFERENTIAL EQUATIONS
GREEN’S FUNCTION TO A STURM TYPE BOUNDARY VALUE PROBLEM FOR A FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH DELAY
Abstract
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.
Differential Equations. 2025;61(12):1587-1602
1587-1602
PARTIAL DERIVATIVE EQUATIONS
SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR STATIONARY HEAT AND MASS TRANSFER EQUATIONS WITH VARIABLE COEFFICIENTS
Abstract
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.
Differential Equations. 2025;61(12):1603-1619
1603-1619
ON MULTIDIMENSIONAL EXACT SOLUTIONS OF HYPERBOLIC EQUATION WITH MONGE–AMP`ERE OPERATOR
Abstract
A reduction method using additive, multiplicative, and functional separation of variables is applied to a hyperbolic equation with the Monge–Ampère operator. Multidimensional exact solutions are obtained, explicitly expressed in terms of elementary and special functions and/or solutions of ordinary differential equations. Examples of exact solutions anisotropic with respect to spatial variables are given.
Differential Equations. 2025;61(12):1620-1632
1620-1632
FUNDAMENTAL SOLUTION OF B-HYPERBOLIC EQUATION WITH WEAKLY NEGATIVE PARAMETERS
Abstract
The B-hyperbolic operator □γ = ∂2/∂t2 − a2ΔBγ
is considered, with the operator ΔBγ
= ∑i=1
n Bγi
, where Bγi
are Bessel operators with parameters γi > −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.
Differential Equations. 2025;61(12):1633-1647
1633-1647
УРАВНЕНИЯ В КОНЕЧНЫХ РАЗНОСТЯХ
RELATION BETWEEN SELECTION CONCEPTS FOR SYSTEMS OF DIFFERENTIAL AND DIFFERENCE EQUATIONS ON THE STANDARD SIMPLEX
Abstract
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.
Differential Equations. 2025;61(12):1648-1664
1648-1664
ASYMPTOTICS OF THE SOLUTION OF DISCRETE LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS WITH A SMALL STEP AND WEAK CONTROL IN THE CRITICAL CASE
Abstract
An algorithm is proposed for constructing an asymptotic approximation to the solution of a discrete time weakly controlled linear-quadratic optimal control problem with a small step size in the critical case. The asymptotic expansion consists of a sum of a regular series and two boundary-layer series containing boundary functions in neighborhoods of two fixed end points. The construction of the asymptotic expansion is based on the decomposition of the state space into orthogonal sums of subspaces and the use of the corresponding orthogonal projectors. Explicit relations for determining the terms of the asymptotic expansion of any order are provided. An example illustrating the proposed method is presented.
Differential Equations. 2025;61(12):1665-1685
1665-1685
CONTROL THEORY
STABILIZATION OF STATE FEEDBACK LINEARIZABLE DYNAMICAL SYSTEMS UNDER STATE CONSTRAINTS
Abstract
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.
Differential Equations. 2025;61(12):1686–1698
1686–1698
SOLUTION OF THE CONTROL PROBLEM FOR A SINGULARLY PERTURBED DYNAMICAL SYSTEM WITH PARTIAL DERIVATIVES
Abstract
A controllability criterion for a partial differential system with a small parameter at the second-order derivative was obtained. The equivalence of the obtained criterion to the Kalman criterion was proven. Control and state functions were constructed explicitly, and the solution to the limit problem was analytically determined. The problem of constructing a control that generates a boundary layer phenomenon near two boundaries of a rectangular domain of variable values was solved.
Differential Equations. 2025;61(12):1699–1718
1699–1718
Articles
AVTORSKIY UKAZATEL' TOMA 61, 2025 g.
Differential Equations. 2025;61(12):1719–1728
1719–1728