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An algorithm for constructing Lyapunov functions

Author(s): Sigurdur Freyr Hafstein

Journal: Electronic Journal of Differential Equations
ISSN 1072-6691

Volume: Monograph 08 ;
Start page: 1;
Date: 2007;
Original page

Keywords: Lyapunov functions | switched systems | converse theorem | piecewise affine functions

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 $dot{mathbf{x}}=mathbf{f}_p(t,mathbf{x})$, $pinmathcal{P}$, 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 $dot{mathbf{x}}=mathbf{f}_p(t,mathbf{x})$, $pinmathcal{P}$, 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.

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