Ky-Fan and Sion Theorems for the Lexicographic Order and Applications to Vectorial Games and Min-Max Control Problems

In many situations it is necessary to consider problems in which decisions have to be taken in a given sequence. For this purpose the lexicographic order of $\mathbb{R}^n$ appears to be an appropriate concept to employ. Therefore, we will study several of its properties in this paper. In a first step we introduce some definitions and regularity concepts with respect to this type of order. Secondly, we prove several classical results redesigned for the vectorial setting as well as several new specific results which are interesting consequences of our approach. Finally, as an application, we study min-max control problems with order. We investigate the existence of the value and we provide results concerning the existence of saddle points for a vectorial cost.