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# An algorithm for constructing Lyapunov functions

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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

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})\$, \$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.

# Tangokurs Rapperswil-Jona 