Trapping Brownian ensemble optimally using Broadcast Stochastic Receding Horizon Control

Collision of suspended entities with surrounding molecules in a fluid environment leads to random movements of these entities, known as Brownian motion. Suppression of this motion in a Brownian ensemble has recently become essential for facilitating emerging applications in biology and in micro and nano scale self-assembled systems. How optimally this suppression can be performed remains an open question of great interest to both the natural science and the control engineering communities. In this paper, we address this question theoretically by introducing a novel "Broadcast Stochastic Receding Horizon Control" strategy for trapping an ensemble of non-interacting Brownian particles. The strategy designs a control input, independent of the number of particles, using measurements from a single particle as the only available feedback information and broadcasts it to all particles in the ensemble. We show the existence of a minimum region in which all particles can be driven and trapped indefinitely using the proposed control action. Under specific conditions, we guarantee the trapping of all particles in this region with probability 1. Finally, we demonstrate the efficacy of our control design in a simulation environment by trapping 100 Brownian particles in one, two and three dimensional homogeneous medium.