An Infinite Dimensional Variational Problem Arising in Estimation Theory

Abstract

In this paper we derive the existence of and a parameterization for the local and global minima for the (total) least squares estimation of linear models describing an infinite sequence X of data in a (separable) Hilbert space. By definition, this problem is an infinite-dimensional nonlinear variational problem; e.g. for line fitting this is the problem of finding the minima of the (least squares) distance function

$$ {f_X}:\mathbb{C}{\mathbb{P}^{\infty }} \to \mathbb{R} $$

(*)

on infinite projective space.

While the importance of such variational problems in estimation theory and in approximation theory (cf. [14],[16] ) is clear, we want to draw attention to the impact of such problems on the emerging theory of nonlinear infinite-dimensional control. The application to such diverse problems of estimation and control as nonlinear filtering and the attitude control of flexible spacecraft of an infinite-dimensional realization theory, replete with controllability and observability criteria, is widely appreciated. Of course, both the nonlinear theory in finite dimensions and the linear theory in infinite dimensions are highly developed. Yet there is presently no infinite-dimensional analogue of some of the best understood, relatively simple, nonlinear situations, e.g. controllability of left-invariant systems on homogeneous spaces(Brockett [6]) or observability of Morse-Smale systems (Aeyels [1]).

Returning to (*), we note that the solution obtained in finite dimensions by Byrnes and Willems [7] used a blend of the two techniques mentioned above; viz. by viewing ℂ ℙⁿ (or Grass (d,n)) as a homogeneous space, they studied the Morse theory of f_X using the Lie theory of coadjoint orbits. In infinite dimensions this is far more delicate since the Banach-Lie theory of general adjoint orbits is much more technical and since f_X does not satisfy Smale’s condition (C) for his infinite-dimensional Morse Theory. Among the technical contributions contained in this paper, we show that certain workable analogues of these finite-dimensional situations exist by analyzing (*) in terms of a generalization of Smale’s Morse Theory, ideally suited to the critical point analysis of linear functionals restricted to “finite rank orbits” of the infinite unitary group.

Research partially supported by NSF Grant ECS-81–21428; partially supported by National Swedish Board of Tech. Development Grant 83–3272.