Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Performance bounds for the mean-field limit of constrained dynamics

Pages: 2023 - 2043, Volume 37, Issue 4, April 2017      doi:10.3934/dcds.2017086

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Michael Herty - Department of Mathematics, IGPM, RWTH Aachen University, Templergraben 55, Aachen, 52062, Germany (email)
Mattia Zanella - University of Ferrara, Department of Mathematics and Computer Science, Via N. Machiavelli 35, 44121, Ferrara, Italy (email)

Abstract: In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

Keywords:  Model predictive control, mean-field limits, consensus models.
Mathematics Subject Classification:  Primary: 35Q93, 49N35, 93B05, 93B52; Secondary: 91A23, 92D25.

Received: August 2016;      Revised: October 2016;      Available Online: December 2016.