Pseudospectra of Linear Operators
SIAM Review
Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL
ACM Transactions on Mathematical Software (TOMS)
Optimization and Pseudospectra, with Applications to Robust Stability
SIAM Journal on Matrix Analysis and Applications
Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
SIAM Journal on Control and Optimization
Journal of Computational and Applied Mathematics
Krylov-Based Model Order Reduction of Time-delay Systems
SIAM Journal on Matrix Analysis and Applications
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The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours.