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A dual to Lyapunov's stability theorem

Rantzer, Anders LU (2001) In Systems & Control Letters 42(3). p.161-168
Abstract
Lyapunov's second theorem is a standard tool for stability analysis of ordinary differential equations. Here we introduce a theorem which can be viewed as a dual to Lyapunov's result. From existence of a scalar function satisfying certain inequalities it follows that “almost all trajectories” of the system tend to zero. The scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is bounded everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin. The weaker notion of stability allows for applications also in situations where Lyapunov's theorem... (More)
Lyapunov's second theorem is a standard tool for stability analysis of ordinary differential equations. Here we introduce a theorem which can be viewed as a dual to Lyapunov's result. From existence of a scalar function satisfying certain inequalities it follows that “almost all trajectories” of the system tend to zero. The scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is bounded everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin. The weaker notion of stability allows for applications also in situations where Lyapunov's theorem cannot be used. Moreover, the new criterion has a striking convexity property related to control synthesis. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Stabilization, Nonlinear systems, Convex duality
in
Systems & Control Letters
volume
42
issue
3
pages
161 - 168
publisher
Elsevier
external identifiers
  • scopus:0034916151
ISSN
0167-6911
DOI
10.1016/S0167-6911(00)00087-6
language
English
LU publication?
yes
id
0dd7201e-4285-4db6-812d-fa46317d2114 (old id 162855)
date added to LUP
2007-07-10 08:59:14
date last changed
2018-04-08 03:33:37
@article{0dd7201e-4285-4db6-812d-fa46317d2114,
  abstract     = {Lyapunov's second theorem is a standard tool for stability analysis of ordinary differential equations. Here we introduce a theorem which can be viewed as a dual to Lyapunov's result. From existence of a scalar function satisfying certain inequalities it follows that “almost all trajectories” of the system tend to zero. The scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is bounded everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin. The weaker notion of stability allows for applications also in situations where Lyapunov's theorem cannot be used. Moreover, the new criterion has a striking convexity property related to control synthesis.},
  author       = {Rantzer, Anders},
  issn         = {0167-6911},
  keyword      = {Stabilization,Nonlinear systems,Convex duality},
  language     = {eng},
  number       = {3},
  pages        = {161--168},
  publisher    = {Elsevier},
  series       = {Systems & Control Letters},
  title        = {A dual to Lyapunov's stability theorem},
  url          = {http://dx.doi.org/10.1016/S0167-6911(00)00087-6},
  volume       = {42},
  year         = {2001},
}