Stabilization of controllable and observable systems
type: Item
instance of: scholarly article
title: Stabilization of controllable and observable systems
zbMATH Open document ID: 0839.93061
author: Jean-Michel Coron
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1996-06-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73343
review text: The author presents a nice survey about the stabilization of continuous-time smooth systems, from the definitions to his recent major contributions. Let \(\dot x= f(x, u)\) with \(f(0, 0)= 0\) and \(x\in \mathbb{R}^n\) be such that the origin is locally small time continuously reachable and that the strong accessibility rank condition holds at the origin. If \(n\geq 4\), then the origin is locally small time stabilizable with periodic continuous static state feedback. The idea of the proof consists in showing that one can drive (with a feedback) any initial condition in a neighbourhood of the origin to a vicinity of a smooth closed one-dimensional manifold \(\gamma\). Then, a new open-loop control can be chosen so that the image of a vicinity \(V\) of \(\gamma\) under the action of the flow is embedded in \(\mathbb{R}^n\backslash \{0\}\) (so that the open-loop control can be represented as a feedback) for all \(t< T\), and so that all points of \(V\) are driven to the origin at \(t= T\), provided that \(n\geq 4\) and that the variational system is controllable. For \(n= 1\), if \(f\) is analytic, using that time has the same dimension as the state, the author shows how to stabilize locally the origin with a continuous autonomous feedback. If \(f\) is affine without drift, one can stabilize the origin globally with a smooth periodic feedback. Under additional observability assumptions, the origin is locally small time stabilizable by periodic continuous dynamic output feedback. The idea of the proof involves a first step where the state is first observed using the dynamic extension, and another step where stabilization is performed using the available state. Concerning the first result, let us remark that in [the author, SIAM J. Control Optimization 33, 804-833 (1995; Zbl 0828.93054)], the feedback belongs to \(C^0(\mathbb{R}^n\times [0, T]; \mathbb{R}^m)\) of class \(C^\infty\) on \((\mathbb{R}^n)\backslash \{0\}\times [0, T]\) and vanishing on \(\{0\}\times [0, T]\…
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93B52
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 824189
zbMATH Keywords: stabilization
zbMATH Keywords: periodic
zbMATH Keywords: continuous
zbMATH Keywords: dynamic output feedback
reviewed by: A. Akutowicz
MaRDI profile type: Publication
OpenAlex ID: W2017433728
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DOI: 10.5802/AFST.791
description: scientific article; zbMATH DE number 824189