# American Institute of Mathematical Sciences

June  2017, 7(2): 213-233. doi: 10.3934/mcrf.2017007

## Regularity results for a time-optimal control problem in the space of probability measures

 Department of Mathematics, University of Trento, Via Sommarive 14, Ⅰ-38123 Povo (Trento), Italy

Received  April 2016 Revised  September 2016 Published  April 2017

Fund Project: The author has been supported by INdAM-GNAMPA Project 2015: Set-valued Analysis and Optimal Transportation Theory Methods in Deterministic and Stochastics Models of Financial Markets with Transaction Costs

This paper investigates some regularity properties of the minimum time function for a time-optimal control problem in the space of probability measures endowed with the topology induced by the Wasserstein metric. The main motivation leading us to the generalization of the classical theory to this framework is to model situations in which we have only a probabilistic knowledge of the initial state, as it happens in real settings where noises and measurement errors may occur. We consider a deterministic evolution for a system ruled by a controlled continuity equation and, pursuing the goal of studying a generalization of the classical results for this setting, we prove an attainability result and a locally Lipschitz continuity property for the generalized minimum time function.

Citation: Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007
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