On Almost Lyapunov Functions for Systems with Inputs

Shenyu Liu, Daniel Liberzon

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In this work an almost Lyapunov function theorem from our recent work is generalized to systems with inputs. It is shown that if for any inputs and initial conditions, the time that solutions of the system can stay in a bad region where the Lyapunov function does not decrease fast enough has a sufficiently small upper bound, then the system is globally exponentially stable uniformly with respect to the inputs. In our analysis, the almost Lyapunov function is directly expressed as a function of time along arbitrary solution and the upper bound of the ratio of this function at the time the solution trajectory leaves and enters the bad region is found to be less than 1. Consequently all solutions are shown to converge to the origin asymptotically with some careful justification. It is also concluded that a system with inputs is exponentially input-to-state stable if its auxiliary system satisfies all the hypotheses in our main theorem. The result is then applied on an example adopted and modified from our previous work and it shows an improvement in the sense that stability can still be verified even when there is stronger perturbation to the example's stable dynamics.

Original languageEnglish (US)
Title of host publication2019 IEEE 58th Conference on Decision and Control, CDC 2019
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781728113982
StatePublished - Dec 2019
Event58th IEEE Conference on Decision and Control, CDC 2019 - Nice, France
Duration: Dec 11 2019Dec 13 2019

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Conference58th IEEE Conference on Decision and Control, CDC 2019

ASJC Scopus subject areas

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


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