Measure Driven Differential Inclusions

Measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data. We introduce the concept of a robust solution to a measure driven inclusion, which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by Dal Maso and Rampazzo and others. Closure properties of sets of robust solutions are established, and notions of relaxation investigated. Implications for optimality conditions for impulsive control problems are pursued in a companion paper.