A dual to Lyapunov's stability theorem
(2001) In Systems & Control Letters 42(3). p.161168 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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/162855
 author
 Rantzer, Anders ^{LU}
 organization
 publishing date
 2001
 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
 01676911
 DOI
 10.1016/S01676911(00)000876
 language
 English
 LU publication?
 yes
 id
 0dd7201e42854db6812dfa46317d2114 (old id 162855)
 date added to LUP
 20160401 11:48:34
 date last changed
 20220428 20:22:43
@article{0dd7201e42854db6812dfa46317d2114, 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 = {{01676911}}, keywords = {{Stabilization; Nonlinear systems; Convex duality}}, language = {{eng}}, number = {{3}}, pages = {{161168}}, publisher = {{Elsevier}}, series = {{Systems & Control Letters}}, title = {{A dual to Lyapunov's stability theorem}}, url = {{http://dx.doi.org/10.1016/S01676911(00)000876}}, doi = {{10.1016/S01676911(00)000876}}, volume = {{42}}, year = {{2001}}, }