Vol 26, No 133 (2021)

A structured population the individuals of which are divided into $n$ age or typical groups $x_1,\ldots,x_n$ is considered.
We assume that at any time moment $k,$ $k=0,1,2\ldots$ the size of the population $x(k)$  is determined by
the normal autonomous system of difference equations $x(k+1)=F\bigl(x(k)\bigr)$,
where $F(x)={\rm col}\bigl(f_1(x),\ldots,f_n(x)\bigr)$ are given vector functions with real non-negative components $f_i(x),$ $i=1,\ldots,n.$
We investigate the case when it is possible to influence the population size by means of harvesting.
The model of the exploited population under discussion has the form
$x(k+1)=F\left(\right(1-u\left(k\right) \left)x\right(k) ),$
where $u(k)=\bigl(u_1(k),\dots,u_n(k)\bigr)\in[0,1]^n$ is a control vector, which can be varied to achieve the best result of harvesting the resource.
We assume that the cost of a conventional unit
of each of $n$ classes is constant and equals to $C_i\geqslant 0,$ $i=1,\ldots,n.$
To determine the cost of the resource obtained as the result of harvesting, the discounted income function is introduced into consideration. It has the form

$H_{\alpha }(u¯,x(0\left)\right)=\sum _{j=0}^{\infty } {\sum _{i=1}^n}_{(}{C}_i{x}_i\left(j\right)u_i\left(j\right)e^{(}-\alpha j),$

where $\alpha>0$ is the discount coefficient.
The problem of constructing controls on finite and infinite time intervals at which the discounted income from the extraction of a renewable resource reaches the maximal value is
solved. As a corollary, the results on the construction of the optimal harvesting mode for a homogeneous population are obtained (that is, for $n =1$).

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

Linked and maximal linked systems (MLS) on $\pi$-systems of measurable (in the wide sense) rectangles are considered ($\pi$-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with $\pi$-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or $\sigma$-algebra of sets. In the general case, the family of measurable rectangles itself forms a  $\pi$-system of set-product (the measurability is identified with belonging to a  $\pi$- system) which allows to consider MLS on a given $\pi$-system (of measurable rectangles). The following principal property is established: for all considered variants of $\pi$-system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.