On almost Lyapunov functions

Research output: Contribution to journalConference article

Abstract

We study asymptotic stability properties of nonlinear systems in the presence of 'almost Lyapunov' functions which decrease along solutions in a given region not everywhere but rather on the complement of a set of small volume. Nothing specific about the structure of this set is assumed besides an upper bound on its volume. We show that solutions starting inside the region approach a small set around the origin whose volume depends on the volume of the set where the Lyapunov function does not decrease, as well as on other system parameters. The result is established by a perturbation argument which compares a given system trajectory with nearby trajectories that lie entirely in the set where the Lyapunov function is known to decrease, and trades off convergence speed of these trajectories against the expansion rate of the distance to them from the given trajectory.

Original languageEnglish (US)
Article number7039864
Pages (from-to)3083-3088
Number of pages6
JournalProceedings of the IEEE Conference on Decision and Control
Volume2015-February
Issue numberFebruary
DOIs
StatePublished - Jan 1 2014
Event2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States
Duration: Dec 15 2014Dec 17 2014

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Lyapunov functions
Lyapunov Function
Trajectories
Trajectory
Decrease
Complement of a set
Speed of Convergence
Asymptotic stability
Asymptotic Stability
Nonlinear systems
Nonlinear Systems
Trade-offs
Upper bound
Perturbation

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

On almost Lyapunov functions. / Liberzon, Daniel; Ying, Charles; Zharnitsky, Vadim.

In: Proceedings of the IEEE Conference on Decision and Control, Vol. 2015-February, No. February, 7039864, 01.01.2014, p. 3083-3088.

Research output: Contribution to journalConference article

Liberzon, Daniel ; Ying, Charles ; Zharnitsky, Vadim. / On almost Lyapunov functions. In: Proceedings of the IEEE Conference on Decision and Control. 2014 ; Vol. 2015-February, No. February. pp. 3083-3088.
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