Averaged time-optimal control problem in the space of positive Borel measures^★,★★

¹ Department of Mathematical Sciences, Rutgers University – Camden, 311 N. 5th Street Camden, NJ 08102, USA
piccoli@camden.rutgers.edu
² Department of Computer Science, University of Verona, Strada Le Grazie 15, 37134 Verona, Italy
mrgntn80@univr.it

^a Corresponding author: giulia.cavagnari@rutgers.edu

Received: 8 February 2017
Revised: 3 August 2017
Accepted: 3 August 2017

Abstract

We introduce a time-optimal control theory in the space ℳ⁺ (ℝ^d) of positive and finite Borel measures. We prove some natural results, such as a dynamic programming principle, the existence of optimal trajectories, regularity results and an HJB equation for the value function in this infinite-dimensional setting. The main tool used is the superposition principle (by Ambrosio–Gigli–Savaré) which allows to represent the trajectory in the space of measures as weighted superposition of classical characteristic curves in ℝ^d.