# Controllability to Equilibria of the 1-D Fokker-Planck Equation with Zero-Flux Boundary Condition

## TL;DR

This paper demonstrates that under specific regularity and boundary conditions, the probability distribution of a robotic swarm governed by a 1-D Fokker-Planck equation can be exactly controlled from any initial to a desired target distribution using bounded drift fields.

## Contribution

It introduces a control method for the 1-D Fokker-Planck equation with zero-flux boundary conditions, allowing exact distribution steering with bounded controls, which was not previously established.

## Key findings

- Any initial distribution can be transported to a target distribution under certain regularity conditions.
- Exact control is possible with bounded drift vector fields.
- The proof employs classical linear semigroup theory.

## 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.