Mean--field control and Riccati equations

Michael  Herty; Lorenzo  Pareschi; Sonja  Steffensen; Michael  Herty; Lorenzo  Pareschi; Sonja  Steffensen

doi:10.3934/nhm.2015.10.699

Networks and Heterogeneous Media

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

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Mean--field control and Riccati equations

1.
RWTH Aachen University, IGPM, Templergraben 55, 52062 Aachen
2.
University of Ferrara, Department of Mathematics and Computer Science, Via Machiavelli 35, 44121 Ferrara

Received: 01 October 2014 Revised: 01 January 2015
Primary: 35Q93, 91A23, 82C40; Secondary: 92D25.

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].
- Riccati equation,
- Boltzmann equation,
- optimal control,
- consensus modeling,
- collective behavior,
- mean-field limit
Citation: Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Mean--field control and Riccati equations[J]. Networks and Heterogeneous Media, 2015, 10(3): 699-715. doi: 10.3934/nhm.2015.10.699

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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].

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