Measure Driven Differential Inclusions

Abstract

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. ᮊ

Invariance for impulsive control systems

“…The proof is similar to that of a similar result (see Theorem 4.1 of [1]) in which the control measure is scalar-valued and, therefore, we omit it.…”

Examples of nonclassical feedback control problems

“…Therefore, optimal trajectories may well have impulsive character. For optimization problems with impulsive controls we refer to [7,10,11]. Aim of the present paper is to understand what happens in the case where the initial data is not assigned in advance, and one seeks a feedback u = u(t, x) that is optimal in connection with a whole collection of possible initial data.…”

State-Constrained Optimal Control Problems of Impulsive Differential Equations

“…It is known that each graph completion may lead to a different solution [6]. Further properties of the Graph completion concept and generalization to measure driven differential inclusions can be found in [25,26,31,30].…”