Robust stability and a criss-cross algorithm for pseudospectra. (English) Zbl 1042.65060

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

IMA J. Numer. Anal. 23, No. 3, 359-375 (2003).

A dynamical system \(\dot{x}=Ax\) 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 \(A\). A criss-cross algorithm is presented for computing the pseudospectral abscissa; global and local quadratic convergence is proved, and a numerical implementation is discussed. As with analogous methods for calculating \(H_\infty\) norms, the algorithm depends on computing the eigenvalues of associated Hamiltoniam matrices. The criss-cross method is implemented in MATLAB and extensively tested.

Reviewer: Guido Vanden Berghe (Gent)

MSC:

65L15	Numerical solution of eigenvalue problems involving ordinary differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations

Keywords:

pseudospectrum; robust stability; spectral abscissa; numerical examples; Hamiltonian system; dynamical system; convergence; criss-cross algorithm

Software:

Algorithm 800; Matlab