Top

Published in:

2018 | OriginalPaper | Chapter

Superposition Principle for Differential Inclusions

Authors : Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli

Published in: Large-Scale Scientific Computing

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We prove an extension of the Superposition Principle by Ambrosio-Gigli-Savaré in the context of a control problem. In particular, we link the solutions of a finite-dimensional control system, with dynamics given by a differential inclusion, to a solution of a continuity equation in the space of probability measures with admissible vector field. We prove also a compactness and an approximation result for admissible trajectories in the space of probability measures.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Spreading Rumors and External Actions

next chapter Estimation of Star-Shaped Reachable Sets of Nonlinear Control Systems

Ambrosio, L.: Transport equation and Cauchy problem for non-smooth vector fields. In: Dacorogna, B., Marcellini, P. (eds.) Contribute in Calculus of Variations and Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol. 1927, pp. 1–41. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-75914-0_1 CrossRef

Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)MATH

Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318(3), 527–555 (2000)MathSciNetCrossRefMATH

Bernard, P.: Young measures, superpositions and transport. Indiana Univ. Math. J. 57(1), 247–276 (2008)MathSciNetCrossRefMATH

Cavagnari, G.: Regularity results for a time-optimal control problem in the space of probability measures. Math. Control Relat. Fields 7(2), 213–233 (2017)MathSciNetCrossRefMATH

Cavagnari, G., Marigonda, A.: Time-optimal control problem in the space of probability measures. In: Lirkov, I., Margenov, S.D., Waśniewski, J. (eds.) LSSC 2015. LNCS, vol. 9374, pp. 109–116. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26520-9_11 CrossRef

Cavagnari, G., Marigonda, A.: Measure-theoretic Lie brackets for nonsmooth vector fields (Submitted)

Cavagnari, G., Marigonda, A., Nguyen, K.T., Priuli, F.S.: Generalized control systems in the space of probability measures. Set-Valued Var. Anal. (2017). https://doi.org/10.1007/s11228-017-0414-y

Cavagnari, G., Marigonda, A., Orlandi, G.: Hamilton-Jacobi-Bellman equation for a time-optimal control problem in the space of probability measures. In: Bociu, L., Désidéri, J.-A., Habbal, A. (eds.) CSMO 2015. IAICT, vol. 494, pp. 200–208. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-55795-3_18 CrossRef

10.

Cavagnari, G., Marigonda, A., Piccoli, B.: Averaged time-optimal control problem in the space of positive Borel measures. ESAIM: Control Optim. Calc. Var. (2017). https://doi.org/10.1051/cocv/2017060

11.

Cavagnari, G., Marigonda, A., Piccoli, B.: Optimal synchronization problem for a multi-agent system. Netw. Heterogen. Media 12(2), 277–295 (2017). https://doi.org/10.3934/nhm.2017012 MathSciNetCrossRefMATH

12.

Cavagnari, G., Marigonda, A., Priuli, F.S.: Attainability property for a probabilistic target in Wasserstein spaces (preprint)

13.

Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9(1), 155–182 (2011)MathSciNetCrossRefMATH

14.

Cristiani, E., Piccoli, B., Tosin, A.: Multiscale Modeling of Pedestrian Dynamics. MS&A Modeling, Simulation and Applications, vol. 12. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06620-2 MATH

15.

Maniglia, S.: Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl. 87(6), 601–626 (2007)MathSciNetCrossRefMATH

16.

Siegfried, G., Harald, L.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000). https://doi.org/10.1007/BFb0103945 MATH

17.

Young, L.C.: Lectures on the calculus of variations and optimal control theory. American Mathematical Society, vol. 304. AMS Chelsea Publishing, Providence (1980)

Title: Superposition Principle for Differential Inclusions
Authors: Giulia Cavagnari
Antonio Marigonda
Benedetto Piccoli
Publisher: Springer International Publishing
Book: Large-Scale Scientific Computing
Print ISBN: 978-3-319-73440-8

Electronic ISBN: 978-3-319-73441-5

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-73441-5_21

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner