Control of linear dynamical systems by time transformations
type: Item
instance of: scholarly article
title: Control of linear dynamical systems by time transformations
zbMATH Open document ID: 1268.34109
author: S. V. Bodnarchuk
published in: Ukrainian Mathematical Journal
publication date: 2013-01-23T00:00:00Z
review text: Consider a linear integral system \(S\) described by the equation \[ x(t)=x(0)+\int_0^tAx(s)\,ds+B\gamma (t),\quad t\geq 0,\tag{1} \] where \(x(0)\in{\mathbb{R}}^m\), \(A\) and \(B\) are \(m\times m\) and \(m\times d\) constant matrices, resp., \(\gamma :{\mathbb{R}}_+\rightarrow{\mathbb{R}}^d\) is a measurable bounded function called a configuration, and \(x :{\mathbb{R}}_+\rightarrow{\mathbb{R}}^m\) is an unknown function which describes the system \(S\) at time \(t>0\). Denote a solution of (1) as \(x(t,x,\gamma )\). Define a time transformation on the interval \([0, T]\) as a continuous strictly incremental function \(\lambda:\,[0, T]\rightarrow [0, T]\) such that \(\lambda (0)=0\) and \(\lambda (T)=T\). Denote by \(\Lambda_T\) the set of all time transformations on the interval \([0, T]\). \(\Lambda_T\)-control of the system \(S\) at zero on the interval \([0,T]\) with given \(\gamma\) is defined by the existence of a neighborhood \(U\) of \(0\in{\mathbb{R}}^m\) such that, for any \(x\in U\), there is a \(\lambda\in\Lambda_T\) satisfying the relation \(x(T,x,\gamma (\lambda (t))) =x(T,0,\gamma )\). Denote further by \(\sum_m^d\) the set of piecewise constant functions from \({\mathbb{R}}_+\) into \({\mathbb{R}}^d\) with \(m\) jumps. The main result is the proven equivalence among three controllability conditions (i)--(iii): (i) \(\operatorname{Rank}[AB,\dotsc,A^mB]=m\); (ii) there exist \(T>0\) and a bounded measurable configuration \(\gamma\) which admits the \(\Lambda_T\)-control of the system \(S\); (iii) there exists a configuration \(\gamma\in\sum_m^d\) for every \(T>0\) which admits a \(\Lambda_T\)-control of system \(S\). An illustrative example is supplied.
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6128306
zbMATH Keywords: linear inhomogeneous integral equations
zbMATH Keywords: control
zbMATH Keywords: controllability
zbMATH Keywords: time transformation
reviewed by: Lubomír Bakule
MaRDI profile type: Publication
full work available at URL: s11253-012-0646-9
OpenAlex ID: W2011949263
cites work: Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion
cites work: Q5341279
cites work: Absolute continuity and convergence in variation for distributions of functionals of Poisson point measure
cites work: Densities for Ornstein-Uhlenbeck processes with jumps
cites work: Conditions for the existence and smoothness of the distribution density of the Ornstein–Uhlenbeck process with Lévy noise
cites work: Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with Lévy noise
cites work: On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes
cites work: Q5818478
DOI: 10.1007/S11253-012-0646-9
description: scientific article; zbMATH DE number 6128306