The geometry of the minimum cost problem

Abstract

In recent years considerable attention has been given to a class of linear control problems known as minimum effort problems. There are various approaches to the problem and many generalizations. In 1962, Neustadt [l] formulated the minimum effort problem, and in 1964 he extended this problem to a formulation in Lebesque spaces. Also in 1962, Reid [2] con- sidered a problem of this type and again the setting was in Lebesque spaces. Reid and Neustadt both attacked the problem by transforming the mini- mum norm problem to an equivalent finite moment problem. In 1966, Porter and Williams [3] formulated the minimum effort control problem in an abstract Banach space and applied classical techniques of functional analysis to analyze the problem. In [4], Minamide and Nakamura considered a problem which includes as special cases the minimum effort problems and a class of approximation problems. The method of Minamide and Nakamura is very geometric and informative. In this paper, we present a problem which is a direct generalization of Problem (P) in [4], and an existence result. The problem is formulated without any references to topological structure, and we prove the existence of an optimal solution by applying an affine separation theorem. We take this abstract approach for two reasons. The abstract approach is geometric and provides insight into the structure of the problem. Also, this geometric approach illustrates that affine separation is sufficient in many existence proofs while, in general, continuous supporting functionals are needed to discuss necessary conditions. Moreover, there is no requirement that the operators be continuous, and hence the results obtained may be applied without “solving” the “state equations.”