On the NP-hardness of checking matrix polytope stability and continuous-time switching stability

Leonid Gurvits, Alexander Olshevsky

Research output: Contribution to journalArticle

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 Rn×n 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 = ⌈ nd ⌉ for any fixed real d > 0, then the problem is NP-hard, meaning that no polynomial-time algorithm in n exists provided that P ≠ N P, 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 nd matrices Ai ∈ Rn×n satisfying 0 Ai + AiT is NP-hard.

Original languageEnglish (US)
Pages (from-to)337-341
Number of pages5
JournalIEEE Transactions on Automatic Control
Volume54
Issue number2
DOIs
StatePublished - Mar 4 2009

Keywords

  • Robust control
  • Switched systems
  • Uncertain systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

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