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Accession Number:
AD0701160
Title:
SEMICONTINUITY OF THE FACE-FUNCTION OF A CONVEX SET,
Corporate Author:
BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
Report Date:
1969-10-01
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 Witsenhausens algorithm has good convergence properties with respect to category but not necessarily with respect to measure. Author
Descriptive Note:
Technical Rept.,
Supplementary Note:
Prepared in cooperation with Boeing Scientific Research Labs., Seattle, Wash. Mathematics Research Lab. Rept. nos. Mathematical note-633, D1-82-0936.
Pages:
0030
Contract Number:
N00014-67-A-0103-0003
File Size:
0.00MB
