An algorithm for constructing Lyapunov functions
DOI:
https://doi.org/10.58997/ejde.mon.08Abstract
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.
For more information see https://ejde.math.txstate.edu/Monographs/08/abstr.html
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