# Hamilton-Jacobi-Bellman Equation for a Time-Optimal Control Problem in the Space of Probability Measures

Abstract : In this paper we formulate a time-optimal control problem in the space of probability measures endowed with the Wasserstein metric as a natural generalization of the correspondent classical problem in ${\mathbb {R}}^d$ where the controlled dynamics is given by a differential inclusion. The main motivation is to model situations in which we have only a probabilistic knowledge of the initial state. In particular we prove first a Dynamic Programming Principle and then we give an Hamilton-Jacobi-Bellman equation in the space of probability measures which is solved by a generalization of the minimum time function in a suitable viscosity sense.
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Lorena Bociu; Jean-Antoine Désidéri; Abderrahmane Habbal. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. Springer International Publishing, IFIP Advances in Information and Communication Technology, AICT-494, pp.200-208, 2016, System Modeling and Optimization. 〈10.1007/978-3-319-55795-3_18〉
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Giulia Cavagnari, Antonio Marigonda, Giandomenico Orlandi. Hamilton-Jacobi-Bellman Equation for a Time-Optimal Control Problem in the Space of Probability Measures. Lorena Bociu; Jean-Antoine Désidéri; Abderrahmane Habbal. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. Springer International Publishing, IFIP Advances in Information and Communication Technology, AICT-494, pp.200-208, 2016, System Modeling and Optimization. 〈10.1007/978-3-319-55795-3_18〉. 〈hal-01626919〉

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