Vol 234 (2024)
Статьи
On a discrete two-parameter fractional control problem
Abstract
In this paper, we examine a fractional difference analog of an optimal control problem occupying an intermediate position between problems with lumped and distributed parameters and obtain various first-order optimality necessary conditions.
3-10
The optimal control problem of the thermal effect of a laser beam on a two-layer biomaterial
Abstract
In this paper, we propose a constructive approach to constructing a function for optimal control of the thermal effect of a laser beam on a two-layer biomaterial. Under the thermal influence constructed, the distribution of the temperature state of a two-layer biomaterial transfers from a given initial state at a certain time interval into a given final state and minimizes the value of the quality criterion. The proposed approach is based on the method of variable separation and methods of the theory of optimal control of dynamic systems.
11-20
On asymptotics of solution of nonlinear difference equation of convolution type
Abstract
Nonlinear difference equations appear in many problems of probability theory, computer science, and combinatorics. In this paper, a nonlinear difference equation of the convolution type with parameters is considered. Asymptotics of solutions of such equations are used for the enumeration of labeled connected graphs. To obtain the asymptotics, we apply Bender’s theorem for the coefficients of formal power series.
21-26
The influence of lower derivatives on the solvability of the Dirichlet problem for multidimensional elliptic systems
Abstract
By means of the Fourier transform, we examine the Dirichlet problem for a multidimensional elliptic system containing lower derivatives. We prove that the lower terms significantly influence the solvability of the first boundary value problem for elliptic systems of equations in contrast to the case of a single elliptic equation. The problem is reduced to a single second-order equation; the nature of the solvability of the original problem depends on the type of this equation.
27-34
Cauchy problem for a special case of fluid motion in a pressure pipeline
Abstract
In this paper, we obtain an exact solution of the Cauchy problem for a system of inhomogeneous first-order partial differential equations of hyperbolic type, which describes a motion of a fluid in a pressure pipeline.
35-42
Support majorants and feedback minimum principles for discrete optimal control problems
Abstract
Support conditions for two classes of problems are found: problems for which the discrete maximum principle is valid and for generalized solutions that are optimal in a convex problem with trajectories realized in the original formulation.
43-49
On some systems of partial differential equations with a small parameter in the principal part
Abstract
In this paper, we examine systems of partial differential equations containing a small positive parameter in the principal part. We establish a relationship between solutions of the system with a small parameter and solutions of the limit system obtained if the parameter is equal to zero. We present classes of systems that preserve the properties of regularly perturbed problems under singular perturbations are admit constructing asymptotic solutions by methods of regular perturbation theory.
50-58
On one class of exact solutions of the multidimensional nonlinear heat equation with a zero front
Abstract
We consider a class of exact solutions of a multidimensional nonlinear heat equation with a source. The construction of these solutions leads to the solution of a family of second-order ordinary differential equations. If appropriate Cauchy conditions are specified, exact solutions can be interpreted as nontrivial solutions with zero front. An existence theorem is proved and a solution is constructed in the form of a converging power series. An approximate algorithm based on the collocation method of radial basis functions is proposed. Test calculations and numerical analysis of the solutions obtained are performed.
59-66
Composition of numbers with constraints and the hierarchical structure of planar sections of Pascal’s pyramid
Abstract
In this paper, we examine compositions of natural numbers with constraints on natural parts and their relationship with hierarchical combinatorial objects. We derive a formula for calculating the number of such compositions with three constraints based on the sums of elements of planar sections of Pascal’s pyramid. Also, we obtain recurrence relations and generating functions for the numbers of compositions and examine some important special cases for well-known combinatorial numbers.
67-74
Quantum search with entanglement-breaking channel of queries to the oracle
Abstract
This paper is devoted to the study of quantum search in the case of entanglement-breaking distortions in queries to the oracle. We examine an influence of entanglement-breaking distortions on evolution of the success probability and the register coherence with respect to the computational basis.
75-82
The problem of identifying the input signal of dynamic systems modeled by Volterra polynomials
Abstract
In this paper, we consider one class of Volterra equations of the first kind that appear in the problem of identifying the input signal of a dynamic system. We discuss an approach to the approximate solution of Volterra polynomial equations that model nonlinear dynamics by integro-power Volterra series. A method for constructing a numerical solution using the Newton–Kantorovich iterative process is proposed. Based on standard quadrature methods and the product integration method, we obtain calculation formulas.
83-90
On some SI*-precomplete sets of multifunctions of rank 2
Abstract
In this paper, we consider multifunctions defined on a two-element set and returning subsets of a given set as values. We discuss the question of describing all sets that are precomplete with respect to the superposition operation. We give examples of two precomplete sets described in the language of predicate preservation by function; their closedness and precompleteness with respect to the closure under consideration are proved, and an example of a complete set is given.
91-98
99-107
Lyapunov stability criteria for systems of ordinary differential equations in multiplicative and additive forms
Abstract
Various Lyapunov stability criteria for systems of ordinary differential equations are presented in the form of necessary and sufficient conditions. The criteria are obtained under the conditions of existence and continuity of the solution on the semi-axis, continuity of the right part of the system and its continuous differentiability on the semi-axis. The criteria are constructed on the basis of recurrent transformations of difference schemes of numerical integration with a residual term at each step. The multiplicative and additive form of the criteria entails the possibility to computerize the stability analysis and perform it in real time.
108-117
An approach to calculating degenerate extremal controls based on fixed point problems
Abstract
In the class of optimal control problems linear with respect to the control, an approach to calculating degenerate controls that satisfy the maximum principle, is proposed. This approach is based on new optimality conditions in the form of fixed point problems that are equivalent to the well-known conditions of the maximum principle. New forms of conditions for the maximum principle make it possible to construct effective methods for searching for degenerate controls. The efficiency of the approach proposed is illustrated using model examples.
118-132
Methods of research of some systems with linear delay
Abstract
In this paper, we discuss methods for obtaining sufficient conditions for systems of linear differential equations with linear delay of neutral type and methods for analyzing asymptotic properties of certain classes of systems of neutral type.
133-142
Hamiltonian formalism for hard and soft excitations in a plasma with a non-Abelian interaction
Abstract
Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy color-charged test particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie–Poisson bracket to the case of a continuous medium involving bosonic normal field variable and a non-Abelian color charge is performed and the corresponding Hamilton equations are derived. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations in the hot gluon plasma and the degree of freedom of a hard test particle associated with its color charge are presented. A complete system of the canonicity conditions for these transformations is obtained. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of this particle is obtained.
143-158
On the exact solution of the evolution equations for two interacting narrow wave packets propagating in a non-Abelian plasma
Abstract
In this paper, we present and discuss a self-consistent system of kinetic equations of the Boltzmann type, which takes into account the time evolution of soft non-Abelian plasma excitations and the mean value of the color charge in the interaction of a high-energy color-charged particle with a plasma. Based on these equations, we examine a model problem of interaction of two infinitely narrow wave packets and obtain a system of first-order nonlinear ordinary differential equations, which governs the dynamics of interacting the colorless and color components of the density of the number collective bosonic excitations. Due to the autonomy of the right-hand sides, we reduce this system to a single nonlinear Abel differential equation of the second kind. Finally, we show that at a certain ratio between the constants involved in this nonlinear equation, one can obtain an exact solution in the parametric form.
159-169