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When is a switched linear system stable?

We survey several aspects of an apparently simple matrix question : given two square matrices A and B, how to verify that all possible infinite products of the type ABBABAAAB... converge to zero? This question arises in a number of applications which we briefly describe (including hybrid systems, wavelets and the design of codes). We prove that the related problem of determining if all infinite products remain bounded, is algorithmically undecidable. We then exhibit the occurence of Sturmian sequences and of a devil's staircase in the analysis of possible optimal periodic products. We conclude with a recent efficient approximation algorithm based on semi-definite liftings. All these results allow a better understanding of the problem, which we will nevertheless leave unsolved. The results reported have been obtained in part with J. Tsitsiklis, A. Vladimirov and Y. Nesterov.
Type of Seminar:
Public Seminar
Prof. Vincent Blondel
Dept of Engineering Mathematics, Université Catholique de Louvain, Belgium.
Jun 09, 2004   17:15

ETH Zentrum, Gloriastrasse 35, Building ETZ, Room E6
Contact Person:

Prof. P. Parrilo
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Biographical Sketch:
Vincent D. Blondel is professor of Applied Mathematics at the Université catholique de Louvain (Belgium) and is head of the Department of Mathematical Engineering. He obtained a MSc in Pure Mathematics from Imperial College (London, UK) and a PhD in Applied Mathematics from the Université catholique de Louvain. In 1993, he was a visiting scientist at Oxford University. During the academic year 1993-1994, he was the Göran Gustafsson Fellow at the Royal Institute of Technology (Stockholm, Sweden). In 1993-1994 he was a research fellow a the French National Research Center in Control and Computer Science (INRIA, Rocquencourt-Paris). From 1994 to 1999 he was an associate professor at the Institute of Mathematics of the Université de Liège in Belgium. Since October 1999 he is with the Université catholique de Louvain. Vincent Blondel has been a visitor with the Australian National University (1991), the University of California at Berkeley (1998), the Santa Fe Institute (2000), Harvard University (2001) and has been visiting the Massachusetts Institute of Technology every year since 1994. He has also been an invited professor at the Ecole Normale Supérieure in Lyon (1998) and at the University of Paris VII (1999 and 2000). Dr Blondel's major current research interests lie in several area of mathematical control theory and theoretical computer science. He has been a recipient of a Grant from the Trustees of the Mathematics Institute of Oxford University (1992), the Prize Agathon De Potter of the Belgian Royal Academy of Science (1993) and the Prize Paul Dubois of the Montefiore Institute (1993). He is a former associate editor of the European Journal of Control and is an associate editor of Systems and Control Letters and of the Journal on Mathematics of Control, Signals, and Systems. He was awarded the triennal SIAM prize on control and systems theory in 2001.