Application of Duality Theory to a Class of Composite Cost Control Problems

作     者：Vinter, R. B. 

作者机构：Harkness Fellow Decision and Control Sciences Group Electronics Systems Laboratory Department of Electrical Engineering Massachusetts Institute of Technology Cambridge Massachusetts 

出 版 物：《JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS》 (优选法理论与应用杂志)

年 卷 期：1974年第13卷第4期

页      面：436-460页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Science Research Council of Great Britain and the Commonwealth Fund 

主　　题：Singular problems linear systems convex programming suboptimal control 

摘      要：Let a control system be described by a continuous linear map L* from the input space U* (some dual Banach space) into the output space X* (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 in X* centered on some desired output x(d)*, an output of minimum norm;and (ii) minimum deviation problem: find, from amongst all inputs lying in some closed sphere in U*, an input having corresponding output at a minimum distance from x(d)*. However, the composite cost problem, where we seek to minimize F(parallel to u*parallel to, parallel to x(d)* - x*parallel to) over elements satisfying x* = L*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 hyperplane H in X, and (c) H can be computed as being an element on which some concave functional over closed hyperplanes in X achieves its maximum. The treatment allows of infinite-dimensional output spaces. We make extensive use of recently developed duality theory.