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-Savare) which allows to represent the trajectory in the space of measures as weighted superposition of classical characteristic curves in Rd.
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