Robust stability and a criss-cross algorithm for pseudospectra

J. V. Burke, A. S. Lewis, M. L. Overton

Research output: Contribution to journalArticlepeer-review


A dynamical system ẋ = Ax is robustly stable when all eigenvalues of complex matrices within a given distance of the square matrix A lie in the left half-plane. The 'pseudospectral abscissa', which is the largest real part of such an eigenvalue, measures the robust stability of A. We present an algorithm for computing the pseudospectral abscissa, prove global and local quadratic convergence, and discuss numerical implementation. As with analogous methods for calculating H norms, our algorithm depends on computing the eigenvalues of associated Hamiltonian matrices.

Original languageEnglish (US)
Pages (from-to)359-375
Number of pages17
JournalIMA Journal of Numerical Analysis
Issue number3
StatePublished - Jul 2003


  • Eigenvalue optimization
  • H norm
  • Hamiltonian matrix
  • Pseudospectrum
  • Robust control
  • Robust optimization
  • Spectral abscissa
  • Stability

ASJC Scopus subject areas

  • General Mathematics
  • Computational Mathematics
  • Applied Mathematics


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