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Vol 211, No 6 (2020)
Local infimum and a family of maximum principles in optimal control
Abstract
The notion of a local infimum for the optimal control problem, which generalizes the notion of an optimal trajectory, is introduced. For a local infimum the existence theorem is proved and necessary conditions in the form of a family of ‘maximum principles’ are derived. The meaningfulness of the necessary conditions, which generalize and strengthen Pontryagin's maximum principle, is illustrated by examples. Bibliography: 9 titles.
Matematicheskii Sbornik. 2020;211(6):3-39
3-39
Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$
Abstract
Let $S\subset\mathbb R^n$ be a nonempty closed set such that for some $d\in[0,n]$ and $\varepsilon>0$ the $d$-Hausdorff content $\mathscr H^d_\infty(S\cap Q(x,r))\geq\varepsilon r^d$ for all cubes $Q(x,r)$ with centre $x\in S$ and edge length $2r\in(0,2]$. For each $p>\max\{1,n-d\}$ we give an intrinsic characterization of the trace space $W_p^1(\mathbb R^n)|_S$ of the Sobolev space $W_p^1(\mathbb R^n)$ to the set $S$. Furthermore, we prove the existence of a bounded linear operator $\operatorname{Ext}\colon W_p^1(\mathbb R^n)|_S\to W_p^1(\mathbb R^n)$ such that $\operatorname{Ext}$ is the right inverse to the standard trace operator. Our results extend those available in the case $p\in(1,n]$ for Ahlfors-regular sets $S$. Bibliography: 36 titles.
Matematicheskii Sbornik. 2020;211(6):40-94
40-94
A canonical basis of a pair of compatible Poisson brackets on a matrix algebra
Abstract
Given an arbitrary complex matrix $A$ and a generic matrix $X$ we find a canonical basis for the Kronecker part of the bi-Lagrangian subspace with respect to the corresponding Poisson brackets on the Lie algebra $\mathfrak{gl}_n(\mathbb C)$, and also find a system of functions in bi-involution corresponding to this basis. In particular, for nilpotent matrices $A$ we prove that all nonzero functions obtained by applying the Mishchenko-Fomenko argument shift method to the coefficients of the characteristic polynomial form the Kronecker part of the complete system of functions in bi-involution. Bibliography: 9 titles.
Matematicheskii Sbornik. 2020;211(6):95-106
95-106
Functions with universal Fourier-Walsh series
Abstract
We prove results on the existence of functions whose Fourier series in the Walsh system are universal in some sense or other in the function classes $L^p[0,1]$, $0< p< 1$, and $M[0,1]$. We also give a description of the structure of these functions.
Bibliography: 30 titles.
Matematicheskii Sbornik. 2020;211(6):107-131
107-131
Three-webs $W(r,r,2)$
Abstract
Local differential-geometric properties of three-webs $W(r,r,2)$ formed on a $2r$-dimensional manifold by foliations of codimension $r,r$ and $2$, respectively, are considered. In particular, three-webs defined by complex analytic functions of $r$ complex arguments belong to this class of webs. The structure equations of a three-web $W(r,r,2)$ in an adapted co-frame (in particular, in a natural co-frame) are deduced; the canonical connection $\Gamma$ on the manifold of a web $W(r,r,2)$ is introduced; formulae are obtained to calculate (in a natural co-basis) the components of the first structure tensor of a three-web $W(r,r,2)$ in terms of the derivatives of the function of this web. Three special classes of three-webs $W(r,r,2)$ are considered in detail: regular and group three-webs and also three-webs $W(r,r,2)$ generated by holomorphic functions. Bibliography: 17 titles.
Matematicheskii Sbornik. 2020;211(6):132-156
132-156