This research concerns a system-theoretic approach to determine the stability (assessed in terms of the strict Hurwitz property) of interval polynomials (i.e. those having coefficient uncertainties which vary independently over specified ranges) having real or complex coefficients. The underlying procedure not only proves the recent results of Kharitonov but, more importantly, offer potentialities for generalization to the bivariate and, subsequently, the multivariate cases for both continuous-time/space and discrete-time/space dynamical systems. This research is important because multidimensional feedback systems have been proposed for various purposes such as iterative image processing and image restoration. Such image processing systems that contain feedback loops are sometimes known to oscillate in space and time and these undesirable oscillations can only be avoided if proper stability conditions are imposed on the feedback systems, especially subject to the practical requirement that the coefficients of the characteristic polynomial are each bounded from above and below instead of being known with certainty. The first part of this research will be concerned with establishing conditions to guarantee that the zero-set of multivariate interval polynomials belong to commonly specified polydomains of interest. Second, the scopes for adapting the system-theoretic approach developed to handle the interval or "boxed" type of coefficient uncertainties to the case when dependencies in the variation of the polynomial coefficient set are allowed, will be investigated. Finally, the counterpart of the results derived in the first and, possibly, the second phase of research which exist in the case of more general polydomains will be investigated and research into the stability of robust matrix polynomials will be initiated. In summary, the aim of this research is to develop a system-theoretic approach allowing the determination of the stability of interval polynomials (i.e. those having coefficient uncertainties which vary over specified ranges). This research has potential applications to image processing and restoration techniques.
