Accession Number : AD0701160

Title :   SEMICONTINUITY OF THE FACE-FUNCTION OF A CONVEX SET,

Descriptive Note : Technical Rept.,

Corporate Author : BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB

Personal Author(s) : Klee,Victor ; Martin,Michael

Report Date : OCT 1969

Pagination or Media Count : 30

Abstract : The paper began as a study of the convergence properties of an algorithm of H. S. Witsenhausen. His algorithm deals with a linear differential system driven by a bounded perturbation and a bounded control, with a cost that is a convex function of the state reached at a given final time. The controller receives exact samples of the current state of the system at a finite number of sampling times and seeks to minimize the supremum (over-all possible perturbations) of the cost. Witsenhausen proposes a sequence of approximate algorithms, all related to the boundary X of a certain d-dimensional compact convex set K associated with the problem, and shows that the sequence has desirable convergence properties for all points of a certain subset X sub e of X. For the procedure to be fully applicable, X sub e should be all of X, and he shows that this is the case if K is polyhedral or strictly convex. Here we show that X sub e = X when d = 2, thus proving a conjecture of J. B. Kruskal, but that the situation is more complicated when d = or > 3. Specifically, X sub e must be a dense G sub delta subset of X but its (d-1)-dimensional measure may be zero. Thus Witsenhausen's algorithm has good convergence properties with respect to category but not necessarily with respect to measure. (Author)

Descriptors :   (*CONVEX SETS, THEOREMS), PERTURBATION THEORY, CONTROL SYSTEMS, TOPOLOGY, CONVERGENCE

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE