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Generalized Control Systems in the Space of Probability Measures.
- Source :
- Set-Valued & Variational Analysis; Sep2018, Vol. 26 Issue 3, p663-691, 29p
- Publication Year :
- 2018
-
Abstract
- In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on ℝd<inline-graphic></inline-graphic>. We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1-16, 2008). We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savaré in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 18770533
- Volume :
- 26
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Set-Valued & Variational Analysis
- Publication Type :
- Academic Journal
- Accession number :
- 131207312
- Full Text :
- https://doi.org/10.1007/s11228-017-0414-y