### Abstract

This research concerns a noncooperative dynamic game with large number of oscillators. The states are interpreted as the phase angles for a collection of non-homogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. We introduce approximate dynamic programming (ADP) techniques for learning approximating optimal control laws for this model. Two types of parameterizations are considered, each of which is based on analysis of the deterministic PDE model introduced in our prior research. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters. In an online setting, a steepest descent stochastic approximation algorithm is proposed. We provide detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control cost parameter is identified: Above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

Original language | English (US) |
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Title of host publication | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |

Pages | 3125-3132 |

Number of pages | 8 |

DOIs | |

State | Published - Dec 1 2010 |

Event | 2010 49th IEEE Conference on Decision and Control, CDC 2010 - Atlanta, GA, United States Duration: Dec 15 2010 → Dec 17 2010 |

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
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ISSN (Print) | 0191-2216 |

### Other

Other | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |
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Country | United States |

City | Atlanta, GA |

Period | 12/15/10 → 12/17/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*2010 49th IEEE Conference on Decision and Control, CDC 2010*(pp. 3125-3132). [5717142] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2010.5717142

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*2010 49th IEEE Conference on Decision and Control, CDC 2010.*, 5717142, Proceedings of the IEEE Conference on Decision and Control, pp. 3125-3132, 2010 49th IEEE Conference on Decision and Control, CDC 2010, Atlanta, GA, United States, 12/15/10. https://doi.org/10.1109/CDC.2010.5717142

**Learning in mean-field oscillator games.** / Yin, Huibing; Mehta, Prashant G.; Meyn, Sean P.; Shanbhag, Vinayak V.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Learning in mean-field oscillator games

AU - Yin, Huibing

AU - Mehta, Prashant G.

AU - Meyn, Sean P.

AU - Shanbhag, Vinayak V.

PY - 2010/12/1

Y1 - 2010/12/1

N2 - This research concerns a noncooperative dynamic game with large number of oscillators. The states are interpreted as the phase angles for a collection of non-homogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. We introduce approximate dynamic programming (ADP) techniques for learning approximating optimal control laws for this model. Two types of parameterizations are considered, each of which is based on analysis of the deterministic PDE model introduced in our prior research. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters. In an online setting, a steepest descent stochastic approximation algorithm is proposed. We provide detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control cost parameter is identified: Above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

AB - This research concerns a noncooperative dynamic game with large number of oscillators. The states are interpreted as the phase angles for a collection of non-homogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. We introduce approximate dynamic programming (ADP) techniques for learning approximating optimal control laws for this model. Two types of parameterizations are considered, each of which is based on analysis of the deterministic PDE model introduced in our prior research. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters. In an online setting, a steepest descent stochastic approximation algorithm is proposed. We provide detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control cost parameter is identified: Above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=79953147145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953147145&partnerID=8YFLogxK

U2 - 10.1109/CDC.2010.5717142

DO - 10.1109/CDC.2010.5717142

M3 - Conference contribution

SN - 9781424477456

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 3125

EP - 3132

BT - 2010 49th IEEE Conference on Decision and Control, CDC 2010

ER -