Preliminaries from set-valued analysis
Preliminaries from set-valued analysis
This chapter includes the necessary background on further developments in the theory of hybrid systems. It first presents the notion of convergence for a sequence of sets and how it generalizes the notion of convergence of a sequence of points. The chapter then deals with set-valued mappings and their continuity properties. Given a set-valued mapping M : ℝᵐ ⇉ ℝⁿ, the chapter defines the range of M as the set rgeM = {y ∈ ℝⁿ : Ǝx ∈ ℝᵐ such that y ∈ M(x)}; and the graph of M as the set gphM = {(x,y) ∈ ℝᵐ × ℝⁿ : y ∈ M(x)}. The chapter also specializes some of the concepts, such as graphical convergence, to hybrid arcs and provides further details in such a setting. Finally, the chapter discusses differential inclusions.
Keywords: set-valued analysis, set-valued mappings, continuity properties, graphical convergence, differential inclusions, set convergence
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