Taming Heterogeneity and Complexity of Embedded Control

A Rigorous Numerical Algorithm for Controllability

Fritz Colonius ¹ — Tomasz Kapela ²

¹ Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany [e-mail]

² Institute of Computer Science, Jagiellonian University, Kraków, Poland [e-mail]

ABSTRACT. The paper presents a rigorous numerical algorithm for controllability, based on the Krawczyk operator. For given two sets X and Y we check whether each point from the set X is connected by some controlled trajectory with each point in the set Y . Two examples are included.

KEYWORDS: controllability, rigorous numerics, Krawczyk operator, Takens-Bogdanov oscillator, escape equation

1. Introduction

The purpose of this paper is to provide a rigorous numerical method that allows us to confirm controllability results. The notion of ‘rigorous’ in this context may need some explanation. It is very different from standard notions of convergence and, maybe, error estimates. It refers to a by now well established line of research which aims at computer assisted proofs for mathematical results. In particular, methods from interval arithmetics are used, see e.g. Neumair [Neu 90] or Jaulin et al. [JKDW 01], in order to take errors of floating point numerics into account. Thus the numerical results have the status of mathematically proven theorems. To the best of our knowledge, such algorithms for controllability have not yet been provided in the literature, with the exception of Marquardt [Mar 05]. ...