A Weak Regularity Theorem for Real Analytic Optimal Control Problems

Abstract

We consider real analytic finite-dimensional control problems with a scalar input that enters linearly in the evolution equations. We prove that, whenever it is possible to steer a state $x$ to another state $y$ by means of a measurable control, then it is possible to steer $x$ to $y$ by means of a control that has an extra regularity property, namely, that of being analytic on an open dense subset of its interval of definition. Since open dense sets can have very small measure, this is a very weak property. However, it is absolutely general, depending on no assumptions other than real analyticity. This shows that real analyticity alone suffices to imply some regularity, and leaves open the question of how much more regularity can be proved in general. To show that real analyticity is essential, we prove, by constructing a class of examples, that no such theorem is true in the $C^{\infty }$ case. The regularity result implies a similar regularity theorem for time-optimal controls.