On the NP-hardness of checking matrix polytope stability and continuous-time switching stability
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Gurvits_On-the-NP-Hardness.pdf
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Author(s)
Gurvits, Leonid
Olshevsky, Alexander
Date Issued
February 2009
Journal
IEEE Transactions on Automatic Control
Publisher
Institute of Electrical and Electronics Engineers
Citation
Gurvits, L., and A. Olshevsky. “On the NP-Hardness of Checking Matrix Polytope Stability and Continuous-Time Switching Stability.” Automatic Control, IEEE Transactions on 54.2 (2009): 337-341. © 2009 IEEE
Version
Final published version
Abstract
Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R[superscript ntimesn] are stable. In particular, we are interested whether there exist algorithms which can solve this problem in time polynomial in n and k. We show that if k=n[superscript d] for any fixed real d > 0, then the problem is NP-hard, meaning that no polynomial-time algorithm in n exists provided that P ne NP, a widely believed conjecture in computer science. On the other hand, when k is a constant independent of n, then it is known that the problem may be solved in polynomial time in n. Using these results and the method of measurable switching rules, we prove our main statement: verifying the absolute asymptotic stability of a continuous-time switched linear system with more than n[superscript d] matrices A[subscript i] isin R[superscript ntimesn] satisfying 0 ges A[subscript i] + A[subscript i] [superscript T] is NP-hard.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
http://dx.doi.org/10.1109/TAC.2008.2007177
