Том 58, № 5 (2018)

We consider an optimal guaranteed control problem for a linear time-varying system that is subject to unknown bounded disturbances. A control strategy is defined that guarantees steering the system to a given terminal set for any realization of disturbances and takes into account that at one future time instant the control loop will be closed. An efficient method for constructing the optimal control strategy and an algorithm for optimal feedback control based on this type of strategies are proposed.

A singularly perturbed parabolic system of nonlinear reaction–diffusion equations is studied. Systems of this class are used to simulate autowave processes in chemical kinetics, biophysics, and ecology. A detailed algorithm for constructing an asymptotic approximation of a travelling front solution is proposed. In addition, methods for constructing an upper and a lower solution based on the asymptotics are described. According to the method of differential inequalities, the existence of an upper and a lower solution guarantees the existence of a solution to the problem under consideration. These methods can be used for asymptotic analysis of model systems in applications. The results can also be used to develop and justify difference schemes for solving problems with moving fronts.

A technique for analyzing the spatial stability of viscous incompressible shear flows in ducts of constant cross section, i.e., a technique for the numerical analysis of the stability of such flows with respect to small time-harmonic disturbances propagating downstream is described and justified. According to this technique, the linearized equations for the disturbance amplitudes are approximated in space in the plane of the duct cross section and are reduced to a system of first-order ordinary differential equations in the streamwise variable in a way independent of the approximation method. This system is further reduced to a lower dimension one satisfied only by physically significant solutions of the original system. Most of the computations are based on standard matrix algorithms. This technique makes it possible to efficiently compute various characteristics of spatial stability, including finding optimal disturbances that play a crucial role in the subcritical laminar–turbulent transition scenario. The performance of the technique is illustrated as applied to the Poiseuille flow in a duct of square cross section.

A Cauchy problem for the time-dependent radiative transfer equation in a three-dimensional multicomponent medium with generalized matching conditions describing Fresnel reflection and refraction at the interface of the media is considered. The unique solvability of the problem is proven, a Monte Carlo method for solving the initial-boundary value problem is developed, and computational experiments for different implementations of the algorithm are conducted.

A boundary value problem describing the equilibrium of a two-dimensional linear elastic body with a thin rectilinear elastic inclusion and possible delamination is considered. The stress and strain state of the inclusion is described using the equations of the Euler–Bernoulli beam theory. Delamination means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions preventing crack face interpenetration are imposed on the crack faces. As a result, problem with an unknown contact domain is obtained. The problem is solved numerically by applying an iterative algorithm based on the domain decomposition method and an Uzawa-type algorithm for solving variational inequalities. Numerical results illustrating the efficiency of the proposed algorithm are presented.

Numerical methods for computing plausibility and belief distributions of consequences of a subjective model are considered. More precisely, related constrained optimization problems are studied. Error estimates of the proposed algorithms are obtained. Techniques for taking into account the information about the consequence available to the researcher for improving the accuracy of computations are discussed.

In computational methods and mathematical modeling, it is often required to find vectors of a linear manifold or a polyhedron that are closest to a given point. The “closeness” can be understood in different ways. In particular, the distances generated by octahedral, Euclidean, and Hölder norms can be used. In these norms, weight coefficients can also be introduced and varied. This paper presents the results on the properties of a set of octahedral projections of the origin of coordinates onto a polyhedron. In particular, it is established that any Euclidean and Hölder projection can be obtained as an octahedral projection due to the choice of weights in the octahedral norm. It is proven that the set of octahedral projections of the origin of coordinates onto a polyhedron coincides with the set of Pareto-optimal solutions of the multicriterion problem of minimizing the absolute values of all components.