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

N1 - Funding Information:
This work was supported by National Science Foundation (NSF) Award CMMI-1436960.
Publisher Copyright:
© 2017 IEEE.

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.

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U2 - 10.1109/CDC.2017.8264014

DO - 10.1109/CDC.2017.8264014

M3 - Conference contribution

AN - SCOPUS:85046123202

T3 - 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017

SP - 2485

EP - 2491

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

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 56th IEEE Annual Conference on Decision and Control, CDC 2017

Y2 - 12 December 2017 through 15 December 2017

ER -