Differential-Algebraic Decision Methods and some Applications to System Theory

Abstract This paper provides a general view of differential-algebraic decision methods and their applications to system theory. It includes the basic properties of differential polynomials, reduction procedures and culminates in the concept of characteristic set and its computation. Such topics are well known from the works by Ritt (1950). A characteristic set of a differential ideal is a finite subset from which many properties of the differential ideal are often readily obtainable merely by inspecting its elements. This is the main point of decision methods in differential algebra. We show through some theorems that basic tests in system theory are thus performable by means of a characteristic set of the differential ideal defining a system. Such tests are, say, invertibility, observability, universal external trajectories computation, etc. As far as computation of characteristic sets is constructive, these tests are now available for algebraic systems. Computation of characteristic sets is actually constructive in principle, but a general algorithm which is fit for use is wanting. Interesting partial results are proposed. Reduce programs of the algorithms described in this paper are written.