Electron. J. Diff. Eqns., Monograph 08, 2007, (101 pages).

An algorithm for constructing Lyapunov functions

Sigurdur Freyr Hafstein

In this monograph we develop an algorithm for constructing Lyapunov functions for arbitrary switched dynamical systems $\dot{\hbox{\bf x}} = \hbox{\bf f}_\sigma(t,\hbox{\bf x})$, possessing a uniformly asymptotically stable equilibrium. Let $\dot{\hbox{\bf x}}=\hbox{\bf f}_p(t,\hbox{\bf x})$, $p\in\mathcal{P}$, be the collection of the ODEs, to which the switched system corresponds. The number of the vector fields $\hbox{\bf f}_p$ 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 $\hbox{\bf f}_p$ 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 $\dot{\hbox{\bf x}} = \hbox{\bf f}_\sigma(t,\hbox{\bf x})$ is equivalent to the concept of a common Lyapunov function for the systems $\dot{\hbox{\bf x}}=\hbox{\bf f}_p(t,\hbox{\bf x})$, $p\in\mathcal{P}$, and that if $\mathcal{P}$ contains exactly one element, then the switched system is just a usual ODE $\dot{\hbox{\bf x}}=\hbox{\bf f}(t,\hbox{\bf x})$. 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.

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

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