BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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
Prepared in cooperation with Boeing Scientific Research Labs., Seattle, Wash. Mathematics Research Lab. Rept. nos. Mathematical note-633, D1-82-0936.