The Bang-Bang, Purification and Convexity Principles in Infinite Dimensions: Additional Characterizations of the Saturation Property∗

The authors apply their recent work on the Lyapunov theorem in locally convex Hausdorff spaces to the bang-bang principle for control systems in infinite dimensions. They show that the bang-bang principle holds for every integrably bounded, measurable, weakly compact convex-valued multifunction if and only if the underlying measure space is saturated. They also demonstrate the equivalence of the bangbang principle to what is termed the purification and convexity principles. Applications to variational problems with integral constraints are indicated.