Guardian Maps and the Generalized Stability of Parametrized Families of Matrices and Polynomials

Abstract

The generalized stability of families of real matrices and polynomials is considered. (Generalized stability is meant in the usual sense of confinementof matrix eigenvalues or polynomial zeros to a prespecified domain in the complex plane, and includes Hurwitz and Schur stability as special cases.)"Guardian maps" and "semiguardian maps" are introduced as a unifying tool for the studyof this problem. Basically these are scalar maps that vanish when theirmatrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to a generalized stability with respect to domains of the complex plane. In the case of one- and two-parameter families of matrices or polynomials, concise necessary and sufficient conditions for generalized stability arederived. For the general multiparameter case, the problem is transformedinto one of checking that a given map is nonzero for the allowedparameter values.Note: This is TR 88-69-r1. A previous version of this report, TR 88-69, had a different title.