Discretional Convexity and the Maximum Principle for Discrete Systems
J.M. Holtzman, Hubert Halkin · SIAM Journal on Control · 1966
Directional convexity is a property of sets closely related to, but weaker than, convexity. It is the existence of supporting hyperplanes at all boundary points of a convex set that makes convexity important in optimal control theory. However, there need be supporting hyperplanes on only one side of the sets for much of the development. Directionally convex sets have this property. The concept of directional convexity, a property of sets, is a generalization, in the following sense, of the concept of convexity, a property of functions. The graph of every convex function is a directionally convex set. However, not every directionally convex set is a graph of a function. Since directional convexity is more general than convexity (for both sets and functions), it may unify a method of investigation which uses convex (or concave) functions with another which uses convex sets. Properties of directional convexity and of matrices that preserve directional convexity are given. Directional convexity was introduced recently to extend the applicability of results on the optimal control of discrete-time systems. It is shown here that the results may be further generalized.