Mayer control problem with probabilistic uncertainty on initial positions

In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on Rn endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, wh...

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Bibliographic Details
Published inJournal of Differential Equations Vol. 264; no. 5; pp. 3212 - 3252
Main Authors Marigonda, Antonio, Quincampoix, Marc
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.03.2018
Subjects
Differential games
Hamilton–Jacobi–Bellman equation
Optimal transport
Hamilton–Jacobi–Bellman equation
49L15
49N70
Optimal transport
Differential games
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Summary:In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on Rn endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, which are modeled by a probability measure on Rd and by a vector-valued measure on Rd, respectively. We provide a characterization of the value function of such a problem as unique solution of an Hamilton–Jacobi–Bellman equation in the space of measures in a suitable viscosity sense. Some applications to a pursuit-evasion game with uncertainty in the state space is also discussed, proving the existence of a value for the game.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2017.11.014