Guardian Maps and the Generalized Stability of Parametrized Families of Matrices and Polynomials
MARYLAND UNIV COLLEGE PARK SYSTEMS RESEARCH CENTER
Pagination or Media Count:
The generalized stability of families of real matrices and polynomials is considered. Generalized stability is meant in the usual sense of confinement of 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 study of this problem. Basically these are scalar maps which vanish when their matrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to 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 are derived. For the general multiparameter case, the problem is transformed into one of checking that a given map is nonzero for the allowed parameter values.
- Theoretical Mathematics