Electronic Journal of Differential Equations,
Monograph 08, 2007, (101 pages).
Title: An algorithm for constructing Lyapunov functions
Author: Sigurdur Freyr Hafstein (Reykjavik Univ., Iceland)
Abstract:
In this monograph we develop an algorithm for constructing Lyapunov functions
for arbitrary switched dynamical systems
$\dot{\mathbf{x}} = \mathbf{f}_\sigma(t,\mathbf{x})$,
possessing a uniformly asymptotically stable equilibrium.
Let $\dot{\mathbf{x}}=\mathbf{f}_p(t,\mathbf{x})$, $p\in\mathcal{P}$,
be the collection of the ODEs, to which the switched system corresponds.
The number of the vector fields $\mathbf{f}_p$ on the right-hand side
of the differential equation is assumed to be finite and we assume that
their components $f_{p,i}$ are $\mathcal{C}^2$ functions and that we
can give some bounds, not necessarily close, on their second-order
partial derivatives.
The inputs of the algorithm are solely a finite number of the
function values of the vector fields $\mathbf{f}_p$ 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 $\dot{\mathbf{x}} = \mathbf{f}_\sigma(t,\mathbf{x})$ is
equivalent to the concept of a common Lyapunov function for the systems
$\dot{\mathbf{x}}=\mathbf{f}_p(t,\mathbf{x})$, $p\in\mathcal{P}$,
and that if $\mathcal{P}$ contains exactly one element, then the
switched system is just a usual ODE
$\dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x})$.
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