Electron. J. Diff. Eqns.,
Monograph 08, 2007, (101 pages).
An algorithm for constructing Lyapunov functions
Sigurdur Freyr Hafstein
Abstract:
In this monograph we develop an algorithm for constructing Lyapunov functions
for arbitrary switched dynamical systems
,
possessing a uniformly asymptotically stable equilibrium.
Let
,
,
be the collection of the ODEs, to which the switched system corresponds.
The number of the vector fields
on the righthand side
of the differential equation is assumed to be finite and we assume that
their components
are
functions and that we
can give some bounds, not necessarily close, on their secondorder
partial derivatives.
The inputs of the algorithm are solely a finite number of the
function values of the vector fields
and these bounds.
The domain of the Lyapunov function constructed by the algorithm is
only limited by the size of the equilibrium's region of attraction.
Note, that the concept of a Lyapunov function for the arbitrary switched
system
is
equivalent to the concept of a common Lyapunov function for the systems
,
,
and that if
contains exactly one element, then the
switched system is just a usual ODE
.
We give numerous examples of Lyapunov functions constructed by our
method at the end of this monograph.
Submitted August 29, 2006. Published August 15, 2007.
Math Subject Classifications: 35J20, 35J25.
Key Words: Lyapunov functions; switched systems; converse theorem;
piecewise affine functions.
Show me the
PDF file (6694 KB),
TEX file, and other files for this article.

Sigurdur Freyr Hafstein
School of Science and Engineering
Reykjavik University
Reykjavik, Iceland
email: sigurdurh@ru.is

Return to the EJDE web page