Vol 25, No 132 (2020)

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].

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.

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.

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.