Weakest Conditions for Existence of Lipschitz Continuous Krotov Functions in Optimal Control Theory

Abstract

Let Y be a metric space; let $(y_ * ^ - ,y_ * ^ + )$ be a point in the product space $Y \times Y$, and let v be an “admissible value function” on $Y \times Y$ taking values in the extended real line. We give a necessary and sufficient condition for existence of some Lipschitz continuous function $\phi :Y \to \mathbb{R}$ such that \[\begin{gathered} \phi \left( {y^ + } \right) - \phi \left( {y^ - } \right) \leqq v\left( {y^ - ,y^ + } \right),\quad {\text{all }}\left( {y^ - ,y^ + } \right) \in Y \times Y, \hfill \\ \phi \left( {y_ * ^ + } \right) - \phi \left( {y_ * ^ - } \right) = v\left( {y_ * ^ - ,y_ * ^ + } \right) \hfill \\ \end{gathered} \] (such functions are called Krotov functions). These results provide conditions, which are in a certain sense weakest, for the applicability of methods of Carathéodory in the calculus of variations and optimal control theory concerning validation of extremals.