Alam, R.; Bora, S.; Karow, M.; Mehrmann, V.; Moro, J. Perturbation theory for Hamiltonian matrices and the distance to bounded-realness. (English) Zbl 1227.93081 SIAM J. Matrix Anal. Appl. 32, No. 2, 484-514 (2011). Summary: Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis. Cited in 28 Documents MSC: 93C73 Perturbations in control/observation systems 93B36 \(H^\infty\)-control 93B40 Computational methods in systems theory (MSC2010) 49N35 Optimal feedback synthesis 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 93B52 Feedback control 93C05 Linear systems in control theory Keywords:Hamiltonian matrix; Hamiltonian eigenvalue problem; perturbation theory; passive system; bounded-realness; purely imaginary eigenvalues; sign characteristic; Hamiltonian pseudospectra; structured mapping problem; distance to bounded-realness Software:Eigtool; Seigtool PDFBibTeX XMLCite \textit{R. Alam} et al., SIAM J. Matrix Anal. Appl. 32, No. 2, 484--514 (2011; Zbl 1227.93081) Full Text: DOI