Set-Valued and Variational Analysis

, Volume 26, Issue 3, pp 663–691 | Cite as

Generalized Control Systems in the Space of Probability Measures

  • Giulia Cavagnari
  • Antonio Marigonda
  • Khai T. Nguyen
  • Fabio S. Priuli


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 \(\mathbb {R}^{d}\). 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.


Optimal transport Differential inclusions Time-optimal control Set-valued analysis 

Mathematics Subject Classification (2010)

34A60 49J15 


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TrentoTrentoItaly
  2. 2.Department of Computer ScienceUniversity of VeronaVeronaItaly
  3. 3.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  4. 4.Istituto per le Applicazioni del Calcolo “M.Picone”Consiglio Nazionale delle RicercheRomaItaly

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