### 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 language | English (US) |
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Article number | 7039864 |

Pages (from-to) | 3083-3088 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 2015-February |

Issue number | February |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States Duration: Dec 15 2014 → Dec 17 2014 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*2015-February*(February), 3083-3088. [7039864]. https://doi.org/10.1109/CDC.2014.7039864

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

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 2015-February, no. February, 7039864, pp. 3083-3088. https://doi.org/10.1109/CDC.2014.7039864

}

TY - JOUR

T1 - On almost Lyapunov functions

AU - Liberzon, Daniel

AU - Ying, Charles

AU - Zharnitsky, Vadim

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84988239794&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988239794&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7039864

DO - 10.1109/CDC.2014.7039864

M3 - Conference article

AN - SCOPUS:84988239794

VL - 2015-February

SP - 3083

EP - 3088

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

IS - February

M1 - 7039864

ER -