Stabilization via symmetry switching in hybrid dynamical systems

Sebastian Hage-Packhäuser; Michael Dellnitz

doi:10.3934/dcdsb.2011.16.239

Article Contents

2011, Volume 16, Issue 1: 239-263. Doi: 10.3934/dcdsb.2011.16.239

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Stabilization via symmetry switching in hybrid dynamical systems

Sebastian Hage-Packhäuser^1, and
Michael Dellnitz^1,

1.
Chair of Applied Mathematics, University of Paderborn, D-33098 Paderborn

Received: May 31, 2010

Revised: January 31, 2011

Published: June 2011

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Abstract

With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of symmetry switching and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of hybrid symmetries for switched systems is established whereupon orbital switching is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.

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Mathematics Subject Classification: Primary: 34C14, 34D05, 34A30; Secondary: 34A36.

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