Time and norm optimality of weakly singular controls

Abstract

Let ū(t) be a control that satisfies the infinite-dimensional versionof Pontryagin’s maximum principle for a linear control system, and let z(t) be the costate associated with ū(t). It is known that integrability of ∥z(t)∥ in the control interval [0, T] guarantees that ū(t) is time and norm optimal. However, there are examples where optimality holds (or does not hold) when ∥z(t)∥ is not integrable. This paper presents examples of both cases for a particular semigroup (the right translation semigroup in L2(0, ∞)).