Vol 27, No 137 (2022)
Articles
New properties of recurrent motions and limit sets of dynamical systems
Abstract
In the earlier article by the authors [A.P. Afanas’ev, S. M. Dzyuba “About new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5-14] a connection between general motions and recurrent motions in a compact metric space is established, and a very simple behavior of recurrent motions is proved. Based on these results, we introduce here a new definition of recurrent motion which, in contrast to the one widely used in modern literature, provides fairly complete information about the structure of a recurrent motion as a function of time and, therefore, is more illustrative. At the same time, we show that in an abstract metric space, the proposed definition is equivalent to Birkhoff’s definition and is equivalent to the generally accepted modern definition in a complete metric space. Necessary and sufficient conditions for recurrence (in the sense of the definition proposed in the article) of a motion in a compact metric space are obtained. It is proved that α - and ω -limit sets of any motion are minimal in a compact metric space (this assertion was announced in an earlier paper by the authors). From the minimality of α - and ω -limit sets, it is deduced that in a compact metric space, each positively (negatively) Poisson-stable point lies on the trajectory of a recurrent motion, i.e. is a point of a minimal set, and thus, in a compact metric space with a finite positive invariant measure almost all points are points of minimal sets.
Russian Universities Reports. Mathematics. 2022;27(137):5-15
5-15
About the methods of renewable resourse extraction from the structured population
Abstract
The problem of optimal extraction of a resource from the structured population consisting of individual species or divided into age groups, is considered. Population dynamics, in the absence of exploitation, is given by a system of ordinary differential equations and at certain time moments, part of the population, is extracted. In particular, it can be assumed that we extract various types of fish, each of which has a certain value. Moreover, there exist predatorprey interactions or competition relationships for food and habitat between these species. We study the properties of the average time benefit which is equal to the limit of the average cost of the resource with an unlimited increase in times of withdrawals. Conditions are obtained under which the average time benefit goes to infinity and a method for constructing a control system to achieve this value is indicated. We show that for some models of interaction between two species, this method of extracting a resource can lead to the complete extinction of one of the species and unlimited growth to the other. Therefore, it seems appropriate to study the task of constructing a control to achieve a fixed final value of the average time benefit. The results obtained here are illustrated with examples of predator-prey models and models of competition of two species and can be applied to other various models of population dynamics.
Russian Universities Reports. Mathematics. 2022;27(137):16-26
16-26
Inclusions with mappings acting from a metric space to a space with distance
Abstract
The article deals with an inclusion in which a multivalued mapping acts from a metric space ( X, ρ) into a set Y with distance d . This distance satisfies only the first axiom of the metric: d y 1 , y 2 is equal to zero if and only if y1 = y2 . The distance does not have to be symmetric or to satisfy the triangle inequality. For the space ( Y, d ) , the simplest concepts (of a ball, convergence, the distance from a point to a set) are defined, and for a multivalued map G : X⇉Y , the sets of covering, Lipschitz and closedness are introduced. In these terms (allowing us to adapt the classical conditions of covering, Lipschitz property and closedness of mappings of metric spaces to the maps with values in ( Y, d ) and to weaken such conditions), a theorem on solvability of the inclusion F(x , x)∋ y is formulated, and an estimate for the deviation in the space (X , ρ) of the set of solutions from a given element x0 ∈ X is given. The main conditions of the obtained statement are the following: for any x from some ball, the pair ( x, y) belongs to the α -covering set of the mapping F (·, x) and to the β -Lipschitz set of the mapping Fx , ∙ , where α>β . The proof of the corresponding statement is based on the construction of the sequences { xn }⊂ X and { yn }⊂ Y satisfying the relations y n ∈Fx n ,x n , y ∈Fx n+1 ,x n , αρ (x n+1 , x n )≤d(y , y n )≤ βρ(x n , x n-1 ) . Also, in the paper, we obtain sufficient conditions for the stability of solutions of the considered inclusion to changes in the multivalued mapping F and in the element y .
Russian Universities Reports. Mathematics. 2022;27(137):27-36
27-36
Spectral properties of an even-order differential operator with a discontinuous weight function
Abstract
This article proposes a new method for studying differential operators with a discontinuous weight function. It is assumed that the potential of the operator is a piecewise smooth function on the segment of the operator definition. The conditions of «conjugation» at the point of discontinuity of the weight function are required. The spectral properties of a differential operator defined on a finite segment with separated boundary conditions are studied. The asymptotics of the fundamental system of solutions of the corresponding differential equation for large values of the spectral parameter is obtained. With the help of this asymptotics, the «conjugation» conditions of the differential operator in question are studied. The boundary conditions of the operator under study are investigated. As a result, we obtain an equation for the eigenvalues of the operator, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular polygon, is studied. In various sectors of the indicator diagram, the asymptotics of the eigenvalues of the investigated differential operator is found. The formula for the first regularized trace of this operator by using the found asymptotics of the eigenvalues by the Lidsky-Sadovnichy method is obtained. In the case of the passage to the limit, the resulting formula leads to the trace formula for the classical operator with a smooth potential and constant weight function.
Russian Universities Reports. Mathematics. 2022;27(137):37-57
37-57
On ill-posed problems, extremals of the Tikhonov functional and the regularized Lagrange principles
Abstract
The problem of finding a normal solution to an operator equation of the first kind on a pair of Hilbert spaces is classical in the theory of ill-posed problems. In accordance with the theory of regularization, its solutions are approximated by the extremals of the Tikhonov functional. From the point of view of the theory of problems for constrained extremum, the problem of minimizing a functional, equal to the square of the norm of an element, with an operator equality constraint (that is, given by an operator with an infinite-dimensional image) is equivalent to the classical ill-posed problem. The paper discusses the possibility of regularizing the Lagrange principle (LP) in the specified constrained extremum problem. This regularization is a transformation of the LP that turns it into a universal tool of stable solving illposed problems in terms of generalized minimizing sequences (GMS) and preserves its “general structural arrangement” based on the constructions of the classical Lagrange function. The transformed LP “contains” the classical analogue as its limiting variant when the numbers of the GMS elements tend to infinity. Both non-iterative and iterative variants of the regularization of the LP are discussed. Each of them leads to stable generation of the GMS in the original constrained extremum problem from the extremals of the regular Lagrange functional taken at the values of the dual variable generated by the corresponding procedure for the regularization of the dual problem. In conclusion, the article discusses the relationship between the extremals of the Tikhonov and Lagrange functionals in the considered classical ill-posed problem.
Russian Universities Reports. Mathematics. 2022;27(137):58-79
58-79
Stability of a three-layer symmetric differential-difference scheme in the class of functions summable on a network-like domain
Abstract
In the paper, the stability conditions of a three-layer symmetric differential-difference scheme with a weight parameter in the class of functions summable on a network-like domain are obtained. To analyze the stability of the differential-difference system in the space of feasible solutions H , a composite norm is introduced that has the structure of a norm in the space H2 = H⊕H . Namely, for Y={Y 1 , Y 2 }∈ H2 , Yl ∈ H (l=1,2) , ∥ Y∥ H 2 = ∥ Y 1 ∥ 1, H 2 + ∥ Y 2 ∥ 2, H 2 , where ∥·∥ 1, H 2 ∥·∥ 2, H 2 are some norms in H . The use of such a norm in the description of the energy identity opens the way for constructing a priori estimates for weak solutions of the differential-difference system, convenient for practical testing in the case of specific differentialdifference schemes. The results obtained can be used to analyze optimization problems that arise when modeling network-like transfer processes with the help of formalisms of differentialdifference systems.
Russian Universities Reports. Mathematics. 2022;27(137):80-94
80-94
Dynamic programming in the routing problem: decomposition variant
Abstract
The questions of applying the dynamic programming (DP) apparatus to the routing problem with constraints and cost functions with the tasks list dependence are investigated. It is supposed that binary partition of the task set is given; tasks of the first task group must be fulfilled before the fulfillment of the task of the second group begins. In each of the groups, precedence conditions may be present. This setting can be applied in the case of sheet cutting on CNC machines, where two above-mentioned groups form zones planned at the cutting stage. In general case, for the optimal solution construction, the two-stage variant of DP is used. Linking two versions of DP is realized by identification of the criterion terminal component for service problem of the first group with extremum function connected with the second group. The connection of optimal solutions for above-mentioned two problems allows to construct an optimal solution for the initial joint problem. Based on the theoretical constructions algorithm realized on personal computer is constructed; computing experiment is realized.
Russian Universities Reports. Mathematics. 2022;27(137):95-124
95-124