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In the article the mutual coupling between radiators influence on electrical parameters of convex antenna array with doubly curved radiators placement surface is investigated. Yagi-Uda antennas are used as radiators. The integral equations method is incorporated.

The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time interval for a linear steady–state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time–optimal problems. © 2020 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved.

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space $\mathbb R^n$ are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin’s integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in $\mathbb R^n$ and the sets approximating them. The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in $\mathbb R^2$ of sufficiently small radius.

Пусть Q- выпуклое (не обязательно ограниченное) множество в C с непустой внутренностью, обладающее счетным базисом окрестностей из выпуклых областей; A(Q) - пространство ростков всех функций, аналитических на Q, с естественной топологией индуктивного предела. В статье доказан критерий того, что фиксированный ненулевой дифференциальный оператор бесконечного порядка с постоянными коэффициентами, действующий в A(Q), имеет линейный непрерывный правый обратный. Этот критерий получен в терминах существования специального семейства субгармонических функций., №4 (2018)