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The boundedness of all products of a pair of matrices is undecidable. (English) Zbl 0985.93042
Summary: We show that the boundedness of the set of all products of a given pair \(\Sigma\) of rational matrices is undecidable. Furthermore, we show that the joint (or generalized) spectral radius \(\rho(\Sigma)\) is not computable because testing whether \(\rho(\Sigma)<1\) is an undecidable problem. As a consequence, the robust stability of linear systems under time-varying perturbations is undecidable, and the same is true for the stability of a simple class of hybrid systems. We also discuss some connections with the so-called “finiteness conjecture”. Our results are based on a simple reduction from the emptiness problem for probabilistic finite automata, which is known to be undecidable.

MSC:
93D09 Robust stability
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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