Mathematical Setting and Motivation

In this chapter we introduce the general setting in terms of differential inclusions and their solutions. Additionally, we recall stability notions for differential equations and corresponding Lyapunov-like characterizations. Since in the context of differential inclusions smooth control Lyapunov functions are not sufficient to describe weak stability properties, we use nonsmooth control Lyapunov functions in the Dini sense. Nonsmooth control Lyapunov functions of this form were introduced by Sontag in [63]. The Dini derivative as an extension of the directional derivative is discussed in the last part of this chapter.