Title: Controllability to Equilibria of the 1-D Fokker-Planck Equation with Zero-Flux Boundary Condition,
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.