GDQ criteria of viability for differential inclusions

The viability problem for differential inclusions is studied. It is assumed that the right-hand side of the differential inclusion is given by a multifunction (an orientor field) defined by the graph of another multifunction (called a constraint multifunction), which depends on time. We use Generalized Differential Quotients as a differentiation tool in tangential condition. We assume that the constraint multifunction has a GDQ-regular multiselection and that the orientor field is upper semi-continuous with respect to the state variable. We also impose some weak measurability conditions. In order to formulate the main viability theorem we present some auxiliary results on Cellina continuously approximable multifunctions and Generalized Differential Quotients. The main result states that the differential inclusion has a global solution.