Research Article

A discussion on the Hölder and robust finite-time partial stabilizability of Brockett’s integrator ^∗

Chaker Jammazi

Facultédes Sciences de Bizerte, Département de Mathématiques and Laboratoire d’Ingénierie Mathématique, École Polytechnique de Tunisie, Université de Carthage, Avenue de la République, BP 77, 1054 Amilcar, Tunisia. Chaker.Jammazi@ept.rnu.tn

Abstract

We consider chained systems that model various systems of mechanical or biological origin. It is known according to Brockett that this class of systems, which are controllable, is not stabilizable by continuous stationary feedback (i.e. independent of time). Various approaches have been proposed to remedy this problem, especially instationary or discontinuous feedbacks. Here, we look at another stabilization strategy (by continuous stationary or discontinuous feedbacks) to ensure the asymptotic stability even in finite time for some variables, while other variables do converge, and not necessarily toward equilibrium. Furthermore, we build feedbacks that permit to vanish the two first components of the Brockett integrator in finite time, while ensuring the convergence of the last one. The considering feedbacks are continuous and discontinuous and regular outside zero.

(Received January 21 2010)

(Revised November 22 2010)

(Online publication April 13 2011)

Key Words:

• Brockett’s integrator;
• discontinuous feedback law;
• finite-time partial stability;
• rational partial stability;
• robust control

Mathematics Subject Classification:

• 93D15;
• 93C10;
• 93D09

Footnotes

^∗  The paper is dedicated to Professor Jean-Michel Coron for his 54th birthday.