How violent are fast controls?
type: Item
instance of: scholarly article
title: How violent are fast controls?
zbMATH Open document ID: 0663.49018
DOI: 10.1007/BF02551238
author: Thomas I. Seidman
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 1988-01-01T00:00:00Z
review text: As is clearly stated in the title, this paper investigates the question of the asymptotic growth of the minimum norm control to reach a fixed target as the horizon is decreased. Consider the linear control system: \[ \dot x=Ax+Bu,\quad x(0)=0,\quad x\in {\mathbb{R}}^ n,\quad u\in {\mathbb{R}}^ m \] where A and B are fixed matrices; the target \(\xi\) is to be reached at time T: \(x(T;u)=\xi\). One seeks the control of minimal \(L^ 2\) norm: \[ \| u\|^ 2=\int^{T}_{0}\| u(t)\|^ 2dt,\quad \| u_{opt}\| =\min \{\| u\| /x(T;u)=\xi \}. \] Let \(C_ T\) be defined by: \[ C_ T:\xi \to u_{opt}(.;T,\xi). \] The system is supposed to be controllable. Let K be the smallest integer such that \(rank[B,AB,...,A^ kB]=n\). Then the main result of the paper is that \(\| C_ T\| \sim \gamma T^{-(k+)}\) as \(T\to 0\).
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 49J15
zbMATH DE Number: 4084442
zbMATH Keywords: LQ optimal control
zbMATH Keywords: asymptotic growth
zbMATH Keywords: minimum norm control
zbMATH Keywords: linear control system
zbMATH Keywords: time-invariant
reviewed by: Jacques Henry
MaRDI profile type: Publication
cites work: A lower bound of the norm of the control operator for the heat equation
cites work: On boundary controllability of a vibrating plate
cites work: Two results on exact boundary control of parabolic equations
full work available at URL: https://doi.org/10.1007/bf02551238
OpenAlex ID: W2096463704
description: scientific article; zbMATH DE number 4084442