Master's Dissertation
DOI
10.11606/D.18.2016.tde-01022016-145236
Document
Author
Full name
Taís Ruoso Calliero
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2005
Supervisor
Committee
Bretas, Newton Geraldo (President)
Geromel, José Cláudio
Title in Portuguese
Um princípio de invariância para sistemas discretos
Keywords in Portuguese
Conjuntos limites
Princípio de invariância de LaSalle
Sistemas dinâmicos discretos
Abstract in Portuguese
Title in English
An invariance principle for discrete dynamic systems
Keywords in English
Discrete dynamic systems
Lasalle's invariance principle
Limit sets
Stability
Abstract in English
Many physical systems are modeled by discrete dynamic systems. With the evolution digital technology, the discrete systems became still more important, so the development of analytic tools for this type of system has high importance nowadays. ln this work, some of the main results in stability of discrete dynamic systems are studied and some new ones are proposed. lt is well known in the literature that the stability of an equilibrium point may be characterized by the Lyapunov's Direct Method, with a function known as Lyapunov auxiliary function. LaSalle, when studying the Lyapunov theory, established an important relationship between Lyapunov function and Birkhoff limit sets. Then, he created the Lasalle's lnvariance Principle. This, among other features, allows the analysis of asymptotically stability. Both the Lyapunov's Direct Method and the lnvariance Principle request the variation of the Lyapunov function to be negative semidefinite along the system trajectory. In systems with more complex behaviors, a function is hardly found with this property. This work developed a more general version of the lnvariance Principle for discrete systems, which does not require the variation of the Lyapunov function to be always negative semidefinite. This new theory enables to find these functions easily and many insoluble problems, which could not be treated with the conventional theory before, become treatable by this new result. The results of this work are useful to find estimates of discrete nonlinear systems atractors.