# 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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https://hal.inria.fr/hal-01626919
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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. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. pp.200-208, ⟨10.1007/978-3-319-55795-3_18⟩. ⟨hal-01626919⟩

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