Time and Norm Optimality of Weakly Singular Controls

Abstract

Let $\bar{u}(t)$be a control that satisfies the infinite-dimensional version of Pontryagin’s maximum principle for a linear control system, and let ${z}(t)$be the costate associated with $\bar{u}(t)$. It is known that integrability of ${z}(t)$in the control interval [0, T] guarantees that $\bar{u}(t)$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 ${L^2}(0,\infty )$).

Mathematics Subject Classification (2000). 93E20, 93E25.