Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ

Infinite horizon optimal control problem with general endpoint constraints is reduced to a family of standard problems on finite time intervals containing the value of the conditional cost of the phase vector as a terminal term. New version of the Pontryagin maximum principle containing an explicit characterization of the adjoint variable is obtained for the problem with a general asymptotic endpoint constraint. In the case of the problem with free final state this approach leads to a normal form version of the maximum principle formulated completely in the terms of the conditional cost function.

We consider a competitive facility location problem with two competing parties operating in a situation of uncertain demand scenario. The problem to find the best solutions for the parties is formulated as a discrete bi-level mathematical programming problem. In the paper, we suggest a procedure to compute an upper bound for the objective function on subsets. The procedure could be employed in implicit enumeration schemes capable to compute an optimal solution for the problem under study. Within the procedure, additional constraints iteratively augment the high-point relaxation of the initial bi-level problem, what strengthens the relaxation and improves the upper bound’s quality. New procedure to generate such cuts allows to construct the strongest cuts without enumerating the parameters encoding them.

It is given a stability estimate of a solution of a source problem for the stationary radiative transfer equation. It is suppose that the source is an isotropic distribution. Earlier stability estimates for this problem were known in a partial case of the emission tomography problem only, when the scattering operator vanishes, and for the complete transfer equation under additional and difficult in checking conditions for the absorption coefficient and the scattering kernel. In the present work, we suggest a new and enough simple approach for obtaining a stability estimate for the problem under the consideration. The transfer equation is considered in a circle of the two-dimension space. In the forward problem, it is assumed that incoming radiation is absent. In the inverse problem for recovering the unknown source some data for solutions of the forward problem related to outgoing radiation are given. The obtained result can be used for an estimation of the summary density of distributed sources of the radiation.

Based on the geometric characteristics of the tetrahedron, quantitative estimates of its degeneracy are proposed and their relationship with the condition number of local bases generated by the edges emerging from the same vertex is established. The concept of the tetrahedron degeneracy index is introduced in several versions and their practical equivalence to each other is established. To assess the quality of a particular tetrahedral partition, it is proposed to calculate the empirical distribution function of the degeneracy index on its tetrahedral elements. A model irregular triangulation (tetrahedralization or tetrahedral partition) of three-dimensional space is proposed, depending on the control parameter that determines the quality of its elements. The coordinates of the tetrahedra vertices of the model triangulation tetrahedrons are the sums of the corresponding coordinates of the nodes of some given regular grid and random increments to them. For various values of the control parameter, the empirical distribution function of the tetrahedron degeneration index is calculated, which is considered as a quantitative characteristic of the quality of tetrahedra in the triangulation of a three-dimensional region.

On a two-dimensional sphere, a distributed renewable resource is considered, the dynamics of which is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource, carried out by constant or periodic impulse harvesting. It is shown that after choosing an admissible exploitation strategy, the dynamics of the resource tend to the limiting dynamics corresponding to this strategy, and that there is an admissible harvesting strategy that maximizes the time averaged harvesting of the resource.

We suggest the derivation of gravitation equations in the framework of Vlasov-Poisson relativistic equations with Lambda-term from the classical least action and use Hamilton-Jacobi consequence for cosmological solutions and investigate Lagrange points.

In this paper, we construct a strongly continuous operator group generated by a one-dimensional Dirac operator acting in the space \(\mathbb{H} = {{\left( {{{L}_{2}}[0,\pi ]} \right)}^{2}}\). The potential is assumed to be summable. It is proved that this group is well-defined in the space \(\mathbb{H}\) and in the Sobolev spaces \(\mathbb{H}_{U}^{\theta }\), \(\theta > 0\), with fractional index of smoothness \(\theta \) and under boundary conditions \(U\). Similar results are proved in the spaces \({{\left( {{{L}_{\mu }}[0,\pi ]} \right)}^{2}}\), \(\mu \in (1,\infty )\). In addition we obtain estimates for the growth of the group as \(t \to \infty \).

In this paper, the first fixed-ratio approximation algorithms are proposed for the series of asymmetric settings of the well-known combinatorial routing problems. Among them are the Steiner cycle problem, the prize-collecting traveling salesman problem, the minimum cost cycle cover problem by a bounded number of cycles, etc. Almost all the proposed algorithms rely on original reductions of the considered problems to auxiliary instances of the Asymmetric Traveling Salesman Problem and employ the breakthrough approximation results for this problem obtained recently by O. Svensson, J. Tarnawski, L. Végh, V. Traub and J. Vygen. On the other hand, approximation of the cycle cover problem was proved by more deep extension of their approach.

In many applications, multidimensional integrals over the unit hypercube arise, which are calculated using Monte Carlo methods. The convergence of the best of them turns out to be quite slow. In this paper, fundamentally new cubatures with super-power convergence based on the improved Korobov grids and special variable substitution are proposed. A posteriori error estimates are constructed, which are practically indistinguishable from the actual accuracy. Examples of calculations illustrating the advantages of the proposed methods are given.

We consider Bernstein inequality for the Riesz derivative of order \(0 < \alpha < 1\) of entire functions of exponential type in the uniform norm on the real line. The interpolation formula for this operator is obtained; this formula has non-equidistant nodes. By means of this formula, the sharp Bernstein inequality is obtained for all \(0 < \alpha < 1\), more precisely, the extremal entire function and the exact constant are written out.

The paper presents the results of mathematical modeling of plasma transfer in a spiral magnetic field using new experimental data obtained at the SMOLA trap created at the Budker Institute of Nuclear Physics SB RAS. Plasma confinement in the trap is carried out by transmitting a pulse from a magnetic field with helical symmetry to a rotating plasma. New mathematical model is based on a stationary plasma transfer equation in an axially symmetric formulation. The distribution of the concentration of the substance obtained by numerical simulation confirmed the confinement effect obtained in the experiment. The dependences of the integral characteristics of the substance on the depth of corrugation of the magnetic field, diffusion and plasma potential are obtained.