Networks and Heterogeneous Media (NHM)

Mean--field control and Riccati equations

Pages: 699 - 715, Volume 10, Issue 3, September 2015      doi:10.3934/nhm.2015.10.699

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Michael Herty - RWTH Aachen University, IGPM, Templergraben 55, 52062 Aachen, Germany (email)
Lorenzo Pareschi - University of Ferrara, Department of Mathematics and Computer Science, Via Machiavelli 35, 44121 Ferrara, Italy (email)
Sonja Steffensen - RWTH Aachen University, IGPM, Templergraben 55, 52062 Aachen, Germany (email)

Abstract: We present a control approach for large systems of interacting agents based on the Riccati equation. If the agent dynamics enjoys a strong symmetry the arising high dimensional Riccati equation is simplified and the resulting coupled system allows for a formal mean--field limit. The steady--states of the kinetic equation of Boltzmann and Fokker Planck type can be studied analytically. In case of linear dynamics and quadratic objective function the presented approach is optimal and is compared to the model predictive control approach introduced in [2].

Keywords:  Riccati equation, Boltzmann equation, optimal control, consensus modeling, collective behavior, mean-field limit.
Mathematics Subject Classification:  Primary: 35Q93, 91A23, 82C40; Secondary: 92D25.

Received: October 2014;      Revised: January 2015;      Available Online: July 2015.