The paper considers the problem of bringing vibrations of a flat membrane to rest, controlled by forces applied to the entire area of the membrane and limited in absolute value. Sufficient conditions are given for the initial data of the deviation and velocity of the membrane, under which a complete stop of motion in a finite time is possible. The resting time is also evaluated. The theorem on estimating the eigenfunctions of the Dirichlet problem for the Laplace equation used in the work makes it possible to refine the mentioned sufficient condition in comparison with the work of F.L. Chernousko. In this work, a similar problem is considered, and the method of expanding the unknown control and the corresponding solution in terms of eigenfunctions of the Dirichlet problem for the Laplace equation is also applied. In this paper, the problem of bringing various elastic oscillatory systems with distributed parameters (membrane, rod, plate, etc.) to a state of rest in a finite time is reduced by means of expansion into a Fourier series in the corresponding system of eigenfunctions to the study of the problem of stopping a counting system of pendulums, connected with each other only through the values of external control actions, the sum of the values of which should not exceed in absolute value some given constant. In order to fulfill this limitation, it is necessary to use estimates for the absolute values of the eigenfunctions, normalized in the mean square. In this paper, we use some estimates for the absolute values of eigenfunctions, previously obtained by Eidus D.M. This allows us to refine the results of F.L. Chernousko for sufficient conditions on the initial functions of the oscillatory system, under which we must dampen the oscillations. These conditions consist in the fact that the initial functions belong to Sobolev spaces with certain indices and in the fulfillment of some additional boundary conditions on the boundary of the domain in which the system is defined.

Keywords: control, wave equation, limited distributed force, Fourier method, counting system of harmonic oscillators

The paper presents analytical estimates of the proximity of solutions to boundary value problems for elastic-creeping layered composite materials widely used in technology under long-term loading, and also gives the corresponding averaged model for such materials. The estimates show the possibility of using the averaged model over a long time interval for the problem of loading by a constantly acting force. Previously, this statement was confirmed by numerical experiments comparing solutions of boundary value problems for an effective (averaged) model and direct numerical calculation using the original model for a highly inhomogeneous layered material. Analytical estimates are based on previously obtained estimates of the proximity of solutions to stationary problems of elasticity theory. For the one-dimensional model considered in this work, the following property is established: if the constitutive relations for various phases of the composite material are written as dependences of deformations on stresses, then the coefficients for the same form of writing the constitutive relations of the averaged model are obtained as simple weighted average values ​​of similar coefficients for individual phases.

Keywords: layered composite materials, creep theory, averaging method, evaluating the efficiency of the averaging method, asymptotically long time interval

The principles of constructing a mathematical model of water purification based on the use of a biologically active layer, the bacteria of which absorb harmful impurities contained in the water, are considered. A system of equations is given, on the basis of which a water treatment model is constructed in the simplest element, which is a rod covered with a biofilm. The system of equations is a system that includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on a part of the surface of the domain, connected to each other through the boundary condition and the potential in an equation of hyperbolic type. Next, an asymptotic analysis of this system is carried out, which makes it possible to reduce the model of an individual element to the solution of a simple ordinary differential equation. On this basis, a model is proposed for the operation of the entire water treatment device containing a large number of such elements.

Keywords: water treatment, biologically active layer, asymptotic analysis of solutions in a thin area, biofilm, mathematical model of pollution treatment, system of differential equations of mixed type, optimization of biofilter designs, urban wastewater