Important Moments in Systems and Control

The moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory, and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space $C[a,b]$ and a “positive” sequence $c$, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive “rational” measures in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. The analysis of these functions, and related maps, yields interesting corollaries for the moment problem and its applications, which we compare to those in the recent literature and which play a crucial role in part of our proof.