The computation of convex invariant sets via Newton's method

Titelangaben

Baier, Robert ; Dellnitz, Michael ; Hessel-von Molo, Mirko ; Kevrekidis, Yannis G. ; Sertl, Stefan:
The computation of convex invariant sets via Newton's method.
In: Journal of Computational Dynamics. Bd. 1 (2014) Heft 1 . - S. 39-69.
ISSN 2158-2491
DOI: https://doi.org/10.3934/jcd.2014.1.39

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Abstract

In this paper we present a novel approach to the computation of convex invariant sets of dynamical systems. Employing a Banach space formalism to describe differences of convex compact subsets of ℝⁿ [$\R^n$] by directed sets, we are able to formulate the property of a convex, compact set to be invariant as a zero-finding problem in this Banach space. We need either the additional restrictive assumption that the image of sets from a subclass of convex compacts under the dynamics remains convex or we have to convexify these images. In both cases we can apply the Newton's method in Banach spaces to approximate such invariant sets if an appropriate smoothness of a set-valued map holds. The theoretical foundations for realizing this approach are analyzed, and it is illustrated first by analytical and then by numerical examples.