Application of duality theory to a class of composite cost control problems

Let a control system be described by a continuous linear map * from the input spaceU* (some dual Banach space) into the output spaceX* (some finite-dimensional normed space). Within the class of control problems where the constraints and cost are expressed in terms of the norms on the input and output spaces, the following two have had extensive coverage: (i)minimum effort problem: find, from amongst all inputs which have corresponding outputs lying in some closed sphere inX* centered on some desired outputx _d *, an output of minimum norm; and (ii)minimum deviation problem: find, from amongst all inputs lying in some closed sphere inU*, an input having corresponding output at a minimum distance fromx _d *. However, thecomposite cost problem, where we seek to minimizeF(u*, x _d * –x*) over elements satisfyingx* = *u* (F a certain kind of convex functional), has not received the same attention. This paper presents results for the composite cost problem paralleling known results for the minimum effort and deviation problems. It is hoped that a gap in the literature is thereby filled. We show that (a) a solution exists, (b) the solution can be characterized in terms of some closed hyperplaneH inX, and (c)H can be computed as being an element on which some concave functional over closed hyperplanes inX achieves its maximum. The treatment allows of infinite-dimensional output spaces. We make extensive use of recently developed duality theory.This research was supported by the Science Research Council of Great Britain and the Commonwealth Fund (Harkness Fellowship).