Strong D-stability

Abstract

A square matrix F is said to be D-stable if the eigenvalues of DF have negative real parts for any diagonal matrix D with positive diagonal elements. The perturbational properties of D-stability and block D-stability, a generalization due to khalil and Kokotovic, are examined. Notions of ‘strong D-stability’ and ‘strong block D-stability’ are introduced, and a general class of strongly block D-stable matrices is identified. Briefly, a matrix is strongly D-stable if is is D-stable and if every sufficiently small perturbation of the matrix is also D-stable. The utility of this new concept is illustrated by proving a stability theorem for time-invariant multiparameter singular perturbation problems. The novelty of this theorem is that is applies to two time scale as well as multiple time scale systems, indeed regardless of the relative magnitudes of the singular perturbation parameters. The crucial assumption of the theorem is the strong block D-stability of an associated boundary layer system.