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