The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples.