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

Karthik Elamvazhuthi, Hendrik Kuiper, Spring Berman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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 languageEnglish (US)
Title of host publication2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages2485-2491
Number of pages7
Volume2018-January
ISBN (Electronic)9781509028733
DOIs
StatePublished - Jan 18 2018
Event56th IEEE Annual Conference on Decision and Control, CDC 2017 - Melbourne, Australia
Duration: Dec 12 2017Dec 15 2017

Other

Other56th IEEE Annual Conference on Decision and Control, CDC 2017
CountryAustralia
CityMelbourne
Period12/12/1712/15/17

Fingerprint

Fokker Planck equation
Fokker-Planck Equation
Controllability
Probability distributions
Boundary conditions
Fluxes
Probability Distribution
Zero
Vector Field
Target
Reflected Diffusion
Swarm Robotics
Second-order Derivatives
Partial differential equations
Strictly positive
Partial derivative
Robotics
Swarm
Control Parameter
Diffusion Process

ASJC Scopus subject areas

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

Cite this

Elamvazhuthi, K., Kuiper, H., & Berman, S. (2018). Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition. In 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.

2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. Vol. 2018-January Institute of Electrical and Electronics Engineers Inc., 2018. p. 2485-2491.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Elamvazhuthi, K, Kuiper, H & Berman, S 2018, Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition. in 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
Elamvazhuthi K, Kuiper H, Berman S. Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition. In 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. Vol. 2018-January. Institute of Electrical and Electronics Engineers Inc. 2018. p. 2485-2491 https://doi.org/10.1109/CDC.2017.8264014
Elamvazhuthi, Karthik ; Kuiper, Hendrik ; Berman, Spring. / Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition. 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. Vol. 2018-January Institute of Electrical and Electronics Engineers Inc., 2018. pp. 2485-2491
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