### Abstract

We consider the problem of controlling the spatiotemporal probability distribution of a robotic swarm that evolves according to a reflected diffusion process, using the space-and time-dependent drift vector field parameter as the control variable. In contrast to previous work on control of the Fokker-Planck equation, a zero-flux boundary condition is imposed on the partial differential equation that governs the swarm probability distribution, and only bounded vector fields are considered to be admissible as control parameters. Under these constraints, we show that any initial probability distribution can be transported to a target probability distribution under certain assumptions on the regularity of the target distribution. In particular, we show that if the target distribution is (essentially) bounded, has bounded first-order and second-order partial derivatives, and is bounded from below by a strictly positive constant, then this distribution can be reached exactly using a drift vector field that is bounded in space and time. Our proof is constructive and based on classical linear semigroup theoretic concepts.

Original language | English (US) |
---|---|

Title of host publication | 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 2485-2491 |

Number of pages | 7 |

Volume | 2018-January |

ISBN (Electronic) | 9781509028733 |

DOIs | |

State | Published - Jan 18 2018 |

Event | 56th IEEE Annual Conference on Decision and Control, CDC 2017 - Melbourne, Australia Duration: Dec 12 2017 → Dec 15 2017 |

### Other

Other | 56th IEEE Annual Conference on Decision and Control, CDC 2017 |
---|---|

Country | Australia |

City | Melbourne |

Period | 12/12/17 → 12/15/17 |

### Fingerprint

### ASJC Scopus subject areas

- Decision Sciences (miscellaneous)
- Industrial and Manufacturing Engineering
- Control and Optimization

### Cite this

*2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017*(Vol. 2018-January, pp. 2485-2491). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2017.8264014

**Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition.** / Elamvazhuthi, Karthik; Kuiper, Hendrik; Berman, Spring.

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

*2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017.*vol. 2018-January, Institute of Electrical and Electronics Engineers Inc., pp. 2485-2491, 56th IEEE Annual Conference on Decision and Control, CDC 2017, Melbourne, Australia, 12/12/17. https://doi.org/10.1109/CDC.2017.8264014

}

TY - GEN

T1 - Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition

AU - Elamvazhuthi, Karthik

AU - Kuiper, Hendrik

AU - Berman, Spring

PY - 2018/1/18

Y1 - 2018/1/18

N2 - We consider the problem of controlling the spatiotemporal probability distribution of a robotic swarm that evolves according to a reflected diffusion process, using the space-and time-dependent drift vector field parameter as the control variable. In contrast to previous work on control of the Fokker-Planck equation, a zero-flux boundary condition is imposed on the partial differential equation that governs the swarm probability distribution, and only bounded vector fields are considered to be admissible as control parameters. Under these constraints, we show that any initial probability distribution can be transported to a target probability distribution under certain assumptions on the regularity of the target distribution. In particular, we show that if the target distribution is (essentially) bounded, has bounded first-order and second-order partial derivatives, and is bounded from below by a strictly positive constant, then this distribution can be reached exactly using a drift vector field that is bounded in space and time. Our proof is constructive and based on classical linear semigroup theoretic concepts.

AB - We consider the problem of controlling the spatiotemporal probability distribution of a robotic swarm that evolves according to a reflected diffusion process, using the space-and time-dependent drift vector field parameter as the control variable. In contrast to previous work on control of the Fokker-Planck equation, a zero-flux boundary condition is imposed on the partial differential equation that governs the swarm probability distribution, and only bounded vector fields are considered to be admissible as control parameters. Under these constraints, we show that any initial probability distribution can be transported to a target probability distribution under certain assumptions on the regularity of the target distribution. In particular, we show that if the target distribution is (essentially) bounded, has bounded first-order and second-order partial derivatives, and is bounded from below by a strictly positive constant, then this distribution can be reached exactly using a drift vector field that is bounded in space and time. Our proof is constructive and based on classical linear semigroup theoretic concepts.

UR - http://www.scopus.com/inward/record.url?scp=85046123202&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046123202&partnerID=8YFLogxK

U2 - 10.1109/CDC.2017.8264014

DO - 10.1109/CDC.2017.8264014

M3 - Conference contribution

VL - 2018-January

SP - 2485

EP - 2491

BT - 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017

PB - Institute of Electrical and Electronics Engineers Inc.

ER -