On Quasi Differential Quotients

We discuss basic properties and some applications of a generalized notion of differential for set-valued maps, called Quasi Differential Quotient. The latter was proved to be extremely useful to deal with several open questions in Optimal Control, such as the so called “Infimum Gap” problem. Furthermore, it recently revealed to be a valuable tool to prove second order necessary conditions expressed in terms of non-smooth Lie-brackets when the vector field is merely Lipschitz continuous. In this paper, we present a first step towards a more systematic theory of Quasi Differential Quotients and we study the relation between Quasi Differential Quotients and other generalized differentials such as the Sussmann’s Generalized Differential Quotient and Sussman’s Approximate Generalized Differential Quotient, the Clarke’s Generalized Jacobian and the Warga’s Derivative Container.