Vol 25, No 132 (2020)
Articles
Some questions of the analysis of mappings of metricand partially ordered spaces
Abstract
The questions of existence of solutions of equations and attainability of minimum values of functions are considered. All the obtained statements are united by the idea of existence for any approximation to the desired solution or to the minimum point of the improved approximation. The relationship between the considered problems in metric and partially ordered spaces is established. It is also shown how some well-known results on fixed points and coincidence points of mappings of metric and partially ordered spaces are derived from the obtained statements. Further, on the basis of analogies in the proofs of all the obtained statements, we propose a method for obtaining similar results from the theorem being proved on the satisfiability of a predicate of the following form. Let X, ≤ - be a partially ordered space, the mapping Φ:X ×X →0, 1 satisfies the following condition: for any x ∈X there exists x ' ∈X such that x ' ≤x and Φx ' ,x =1. The predicate F (x )= Φ(x ,x ) is considered, sufficient conditions for its satisfiability, that is, the existence of a solution to the equation F (x )=1 . This result was announced in [Zhukovskaya T.V., Zhukovsky E.S. Satisfaction of predicates given on partially ordered spaces // Kolmogorov Readings. General Control Problems and their Applications (GCP-2020). Tambov, 2020, 34-36].
Russian Universities Reports. Mathematics. 2020;25(132):345-358
345-358
Minimax differential game with delay
Abstract
The paper considers a minimax positional differential game with aftereffect based on the i -smooth analysis methodology. In the finite-dimensional (ODE) case for a minimax differential game, resolving mixed strategies can be constructed using the dynamic programming method. The report shows that the i -smooth analysis methodology allows one to construct counterstrategies in a completely similar way to the finite-dimensional case. Moreover as it is typical for the use of i -smooth analysis, in the absence of an aftereffect, all the results of the article pass to the corresponding results of the finite-dimensional theory of positional differential games.
Russian Universities Reports. Mathematics. 2020;25(132):359-369
359-369
Elements of analytical solutions constructor in a class of time-optimal control problems with the break of curvature of a target set
Abstract
A planar velocity control problem with a disc indicatrix and a target set with a smooth boundary having finite discontinuities of second-order derivatives of coordinate functions is considered. We have studied pseudo-vertices-special points of the goal boundary that generate a singularity for the optimal control function. For non-stationary pseudo-vertices with discontinuous curvature, one-way markers are found, the values of which are necessary for analytical and numerical construction of branches of a singular set. It is proved that the markers lie on the border of the spectrum-the region of possible values. One of them is equal to zero, the other takes an invalid value -∞ . In their calculation, asymptotic expansions of a nonlinear equation expressing the transversality condition are applied. Exact formulas for the extreme points of branches of a singular set are also obtained based on markers. An example of a control problem is presented, in which the constructive elements are obtained using the developed methods (pseudo-vertex, its markers, and the extreme point of a singular set), are sufficient to construct a singular set and an optimal result function in an explicit analytical form over the entire area of consideration.
Russian Universities Reports. Mathematics. 2020;25(132):370-386
370-386
New method for the numerical solution of the Fredholm linear integral equation on a large interval
Abstract
The traditional numerical process to tackle a linear Fredholm integral equation on a large interval is divided into two parts, the first is discretization, and the second is the use of the iterative scheme to approach the solutions of the huge algebraic system. In this paper we propose a new method based on constructing a generalization of the iterative scheme, which is adapted to the system of linear bounded operators. Then we don’t discretize the whole system, but only the diagonal part of the system. This system is built by transforming our integral equation. As discretization we consider the product integration method and the Gauss-Seidel iterative method as iterative scheme. We also prove the convergence of this new method. The numerical tests developed show its effectiveness.
Russian Universities Reports. Mathematics. 2020;25(132):387-400
387-400
Fractional order differential pursuit games with nonlinear controls
Abstract
The article is devoted to the problems of extending the results and methods of the theory of differential games and optimal control to systems of fractional order. The research is motivated by numerous applications of fractional calculus in control problems of industrial facilities, chemical and biochemical plants, etc. The article considers the problem of pursuit in games represented by nonlinear differential equations of arbitrary fractional order in the sense of Caputo. To study this pursuit problem, we use an approach similar to the method of L. S. Pontryagin, developed for linear differential games of integer orders. In this paper, new sufficient conditions are obtained for solving the pursuit problem in the class of games under study. It has been proven that if these conditions are met, the game can be completed within a certain limited period of time. When solving the pursuit problem, we also used the representation of the solution to a differential equation in terms of generalized matrix functions.
Russian Universities Reports. Mathematics. 2020;25(132):401-409
401-409
On differential equations in Banach algebras
Abstract
We consider higher-order linear differential equations with constant coefficients in Banach algebras (this is a direct generalization of higher-order matrix differential equations). The presentation is based on higher algebra, differential equations and functional analysis. The results obtained can be used in the study of matrix equations, in the theory of small oscillations in physics, and in the theory of perturbations in quantum mechanics. The presentation is based on the original research of the authors.
Russian Universities Reports. Mathematics. 2020;25(132):410-421
410-421
Volterra funktional equations in the stability problem for the existence of global solutions of distributed controlled systems
Abstract
Earlier the author proposed a rather general form of describing controlled initial-boundary value problems (CIBVPs) by means of Volterra functional equations (VFE) z t =f t, A z t , vt , t≡ t 1 ,⋯, t n ∈ Π⊂ Rn , z∈ L p m ≡ L p Πm , where f .,.,. : Π× Rl ×Rs →Rm ; v (.)∈D⊂ L k s - control function; A : L p m Π→ L q l ( Π )- linear operator; the operator A is a Volterra operator for some system T of subsets of the set Π in the following sense: for any H ∈T , the restriction A[ z ] H does not depend on the values of z Π\\H ; (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets H ∈T ). This continuation is realized with the help of the chain H 1 ⊂ H 2 ⊂…⊂ H k-1 ⊂ H k≡ Π, where H i ∈T , i=1, k. A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case p =q =k =∞ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator F vz . (t )≡f (t ,Az t , v (t )) satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space L ∞ m (H ) such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case 1≤p , q , k ≤∞, (this case covers a much wider class of CIBVP), the operator F v ; as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space L p m (H ) , for which the operator F v is a contraction operator. The corresponding basic theorem ( equivalent norm theorem ) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.
Russian Universities Reports. Mathematics. 2020;25(132):422-440
422-440
Controlled differential equations with a parameter and with multivalued impulses
Abstract
We study the Cauchy problem for a controlled differential system with a parameter which is an element of some metric space Ξ , containing phase constraints on the control. It is assumed that at the given time instants t k , k = 1, 2,…, p , the solution x is continuous from the left and suffers a discontinuity, the value of which is x t k +0 -x t k , belongs to some non-empty compact set of the space Rn . The notions of an admissible pair of this controlled impulsive system are introduced. The questions of continuity of admissible pairs are considered. Definitions of a priori boundedness and a priori collective boundedness on a given set S ×K (where S ⊂ Rn is a set of initial values, K ⊂ Ξ is a set of parameter values) of the set of phase trajectories are considered. It is proved that if at some point x 0 ,ξ ∈ Rn × Ξ the set of phase trajectories is a priori bounded, then it will be a priori bounded in some neighborhood of this point.
Russian Universities Reports. Mathematics. 2020;25(132):441-447
441-447