Minimum-Time Optimal Control Problems

Abstract

In Chap. 8 we derived a weaker version of Pontryagin’s minimum principle using the dynamic programming procedure. A rigorous proof of the general statement of the principle is tedious. Even in the minimum-time optimal control problem where the cost functional is simply (t₁ — t₀), an easy proof of the principle is not available without using functional analysis. In this chapter we will study the minimum-time optimal control problem for a continuous-time linear system in some detail and derive the minimum principle for this setting. In order to give a rigorous and yet somewhat elegant treatment, it is necessary to use some terminology and results from measure theory and functional analysis. Our original intention of presenting an elementary treatment of the subject matter is maintained if the reader is willing to accept two existence results (namely: Lemma 9.1 and the last portion of the proof of Theorem 9.2), consider “measurable functions” as “piecewise continuous functions”, regard the “almost everywhere” notion as the weaker notion “everywhere with an exception of a finite number of points”, and assume a set E with positive measure to be a nonempty interval.