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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2015,  vol. 18,  No 4 (64)

Contents

UDC 517.968.72
DOI 10.17377/sibjim.2015.18.401

Balakina E. Yu.
Existence and uniqueness of a soluton to the nonstationary transport equation

We consider the problem of finding the flux density of particles whose transport process is described by a nonstationary integro-differential equation. We study the case that the medium in which the process takes place is inhomogeneous; in other words, the coefficients of the equation may have jumps of the first kind.
The initial data is the density of the intput flux and the density at the initial time  moment. A solution to the problem is understood in the weak sense. It is shown that a solution exists, is unique, and can be represented as a uniformly convergent series.

Keywords: tomography, transport equation, discontinuous coefficients of the equation.
Pp. 3-8.

Balakina Ekaterina Yur'evna
Sobolev Institute of Mathematics SB RAS
4 Koptyug ave.
Novosibirsk State University
2 Pirogova st.
630090 Novosibirsk
E-mail: balakina@math.nsu.ru

UDC 519.254
DOI 10.17377/sibjim.2015.18.402

Voskoboinikova G. M. and Khairetdinov M. S.
A posteriori algorithms for soving problems of joint detection and estimation of seismic waves

We propose a new approach to finding the entry times for seismic waves in connection with solving the problem of the active geophysical monitoring of the natural environment. The efficiency of this approach is illustrated by some numerical experiments and the example of solving the model problem of the monitoring of the sounding of the borehole source location in oil and gas drilling.

Keywords: geophysical monitoring, natural and technogenic events, detection, a posteriori algorithm, numerical experiment, seismic location, borehole source.
Pp. 9-17.

Voskoboinikova Gyulnara Maratovna
Khairetdinov Marat Samatovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
6 Lavrent'ev ave.
630090 Novosibirsk
Novosibirsk State Technical University
20 Karl Marx ave.
630073 Novosibirsk
E-mail: gulya@opg.sscc.ru; marat@opg.sscc.ru

UDC 517.929.4
DOI 10.17377/sibjim.2015.18.403

Demidenko G. V. and  Matveeva I. I.
On the robust stability of solutions to linear differential equations of neutral type with periodic coefficients

We consider a class of linear systems of delay differential equations with periodic coefficients. We establish conditions on the perturbations of the coefficients under which the exponential stability of the zero solution is preserved and obtain estimates characterizing the exponential decay of solutions to the perturbed systems at infinity.

Keywords: time-delay systems of neutral type, periodic coefficients, robust stability,  Lyapunov—Krasovskii functional, Lyapunov differential equation.
Pp. 18-29.

Demidenko Gennadii Vladimirovich
Matveeva Inessa Izotovna
Sobolev Institute of Mathematics SB RAS
4 Koptyug ave.
Novosibirsk State University
2 Pirogova st.
630090 Novosibirsk
E-mail: demidenk@math.nsc.ru; matveeva@math.nsc.ru

UDC 517.98:519.677
DOI 10.17377/sibjim.2015.18.404

Derevtsov E. Yu.
Numerical solution of a problem of refractive tomography in a tube domain

A problem of refractive tomography is considered for a tube domain with a given arbitrary varying absorption and refraction of a special type modelled by means of a Riemannian metric. We propose a numerical solution of the problem based on the consecutive solution of a series of two-dimensional problems. We show that such an approach is possible if the domain has a sufficiently large family of totally geodesic submanifolds of dimension two. Riemannian metrics admitting the existence of the set are contructed. We propose an algorithm for an approximate solution of the problem based on the least squares method.

Keywords: tomography, absorption, refraction, Riemannian metric, ray transform, totally geodesic submanifold, least squares method.
Pp. 30-41.

Derevtsov Evgenii Yur'evich
Sobolev Institute of Mathematics SB RAS
4 Koptyug ave.
Novosibirsk State University
2 Pirogova st.
630090 Novosibirsk
E-mail: dert@math.nsc.ru

UDC 514.745.82
DOI 10.17377/sibjim.2015.18.405

Kazantsev M. V.
On some properties of the domain graphs of dynamical systems

We continue examining the properties of the domain graphs defining the discrete structure in the phase portraits of the dynamical systems modelling gene networks. Some necessary and sufficient conditions are described for the existence of cycles on different potential levels of the domain graph. Also the conditions are found for these graphs to beisomorphic for different dynamical systems.

Keywords: gene network, graph, dynamical system, potential level, phase portrait.
Pp. 42-48.

Kazantsev Maxim Valer'evich 
Altai State Technical University
46 Lenin ave.
656038 Barnaul
E-mail: markynaz.astu@gmail.com

UDC 519.688
DOI 10.17377/sibjim.2015.18.406

Kandryukova T. A. and Laevskii Yu. M.
On approaches to the simulation of filtration gas combustion

The first part of the article is a description of the model based on a system of first-order equations in terms of ``temperature-heat flux'', ``mass-diffusion flux'' on using the concept of  total enthalpy flow. This approach ensures strict compliance with the mesh conservation laws, which is extremely important for this class of problems. The second part of the article  introduces the concept of instantaneous velocity of the combustion front and proposes stable methods for its calculation.

Keywords: filtration combustion, heat flux, diffusion flux, mixed finite element method, explicit scheme.
Pp. 49-60.

Kandryukova Tat'yana Aleksandrovna
Laevskii Yurii Mironovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
6 Lavrent'ev ave.
630090 Novosibirsk
E-mail: kandryukova@labchem.sscc.ru; laev@labchem.sscc.ru

UDC 517.938.5
DOI 10.17377/sibjim.2015.18.407

Lakeev A. V., Linke Yu. E., and Rusanov V. A.
On the solvability of the problem of  realization of the operator functions of a nonlinear regulator of a second-order dynamical system

We study solvability of the problem of the realization of a nonlinear program-position regulator for a nonstationary second-order dynamical system which contains (as admissible solutions) a bundle of nonlinear infinite-dimensional controlled dynamical processes in a separable Hilbert space.

Keywords and phrases: inverse problem of system analysis, nonlinear differential realization.
Pp. 61-75.

Lakeev Anatolii Valentinovich
Rusanov Vyacheslav Anatol'evich
Matrosov Institute for System Dynamics and Control Theory SB RAS
134 Lermontova st.
664033 Irkutsk
Linke Yurii Ernievich
National Research Irkutsk State Technical University
83 Lermontova st.
664074 Irkutsk
E-mail: lakeyev@icc.ru; v.rusanov@mail.ru; linkeyurij@gmail.com

UDC 539.3:517.958
DOI 10.17377/sibjim.2015.18.408

Khludnev A. M.
Optimal control of inclusions in an elastic body crossing the external boundary

The paper addresses optimal control of the elastic thin inclusions located in an elastic body and crossing the external boundary. The inclusions are assumed to delaminate, thus forming a crack between the inclusions and the matrix. We impose some nonlinear boundary conditions at the crack faces that do not allow the crack faces to penetrate into each other. We prove the solvability of optimal control problems in which the quality functional characterizes the displacement of the points of the elastic inclusions located outside the elastic body, and the length of the inclusions located inside the elastic body is the control function. The case is doscussed of the zero angle between the inclusions and the external boundary.

Keywords: elastic body, elastic inclusion, crack, nonlinear boundary condition, variational inequality, optimal control.
Pp. 75-87.

Khludnev Aleksandr Mikhailovich
Lavrent'ev Institute of Hydrodynamics SB RAS
15 Lavrent'ev ave.
Novosibirsk State University
2 Pirogova st.
630090 Novosibirsk
E-mail: khlud@hydro.nsc.ru

UDC 531.36
DOI 10.17377/sibjim.2015.18.409

Chaikin S. V.
The influence of inertia and gravity forces on deformation of an elastic beam clamped in the body of an orbital gyrostat

In the restricted formulation, we consider the motion of a gyrostat with an elastic beam clamped in its body by one end along a Keplerian circular orbit in a Newtonian central force field. The rectilinear axis of the undeformed beam is placed into the symmetry plane of the principal central ellipsoid of intertia of the mechanical system. The inextensible beam is subjected to infinitesimal space deformations in the process of the motion of the system. Under certain assumptions, in the semi-inverse statement we study the special relative equilibria (the states of rest of the system except for a rotor in the orbital coordinate system). In the  equilibria, the deformed axis of the beam lies either in the plane perpendicular to the local vertical line, or in the orbital plane, or in the plane perpendicular to orbit. The choice for studying such special equilibria is determined by the purpose of the simplest separation of the influence of the inertia and gravitation on the deformation of the beam.

Keywords: orbital gyrostat, elastic beam, circular orbit, Newtonian attraction, relative equilibria, action of inertia and gravity on deformation.
Pp. 88-97.

Chaikin Sergei Vasil'evich
Matrosov Institute for System Dynamics and Control Theory SB RAS
134 Lermontov st.
664033 Irkutsk
E-mail: schaik@yandex.ru

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