Jurdjevic-Quinn Conditions and Discontinuous Bounded Damping Control

Alex Bombrun — Jean-Baptiste Pomet

INRIA Sophia Antipolis, B.P. 93, 06902 Sophia Antipolis cedex, France.

Alex.Bombrun@sophia.inria.fr, Jean-Baptiste.Pomet@sophia.inria.fr.

ABSTRACT. This note presents a practical stabilization result for discontinuous damping control (or Jurdjevic-Quinn control), in the case where the bound on controls is small. The motivation is to estimate, as that bound goes to zero, how the time taken to reach a neighborhood of the target tends to infinity.

KEYWORDS: damping control, small control, low-thrust transfer

1. Introduction

For smooth control systems whose drift possesses a first integral V which is minimum at some desired configuration (this already makes that configuration Lyapunovstable for the system with zero control), a well known strategy to obtain asymptotic stability, called damping control, or Jurdjevic-Quinn control, consists in using the control to make V decrease; this strengthens stability of the desired configuration, and under some non-degeneracy assumptions, yields convergence, i.e. asymptotic stability. This is recalled in Section 2.

In general, there is a subset W of the state space where images = 0 for any choice of the control, and at each point outside W, there is a choice of the control that renders negative (more precisely, at any such point, the control space ...

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