Measurable Multifunctions and Differential Inclusions

Abstract

Differential inclusions $$\begin{array}{*{20}c}{\begin{array}{*{20}c}{\dot x\left( t \right) \in F\left( {t,x\left( t \right)} \right)} & {{\rm{a}}.{\rm{e}}.} & {t \in I} \\\end{array}}\\\end{array}$$ (2.1) It feature prominently in modern treatments of Optimal Control. This has come about for several reasons. One is that Condition (2.1), summarizing constraints on allowable velocities, provides a convenient framework for stating hypotheses under which optima; control problems have solutions and optimality conditions may be derived. Another is that, even when we choose not to formulate an optimal control problem in terms of a differential inclusion, in cases when the data are nonsmooth, often the very statement of optimality conditions makes reference to differential inclusions. It is convenient then at this stage to highlight important properties of multifunctions and differential inclusions of particular relevance in Optimal Cotrol.