# (528b) A Mathematical Theory of Manipulating Suspended Multiple Brownian Particles Simultaneously in a Solution

#### AIChE Annual Meeting

#### 2010

#### 2010 Annual Meeting

#### Computing and Systems Technology Division

#### Control and Optimization of Multiscale Systems

#### Wednesday, November 10, 2010 - 3:35pm to 3:55pm

Manipulation of multiple stochastic single entities simultaneously with a goal of achieving a predefined desired system behavior has long been a topic of study in various science and engineering disciplines. This interest has accelerated in the last decade with a burst of emerging physical and biological applications in micro and nano-scale systems. Designing of smart robots and multi-agent intelligent systems have emanated several similar problems. Most of these emerging problems can be captured into a mother problem of harnessing the motion of multiple Brownian particles suspended in solution simultaneously to achieve a desired system behavior. In this presentation, we attempt to address the problem of manipulating such system behavior theoretically by proposing a novel broadcast stochastic receding horizon control theory.

We begin this presentation with a consideration of simple one dimensional random walks possessed by N particles. We represent the behavior of each particle as a Bernoulli random variable with forward and backward movement on a line. We make assumption of independent and identical behavior of particles. We further assume that there are N parallel paths and each particle is associated with its individual path. Thus the overall system consists of N particles with N parallel paths and is a reasonable representation of N simultaneous 1-D asymmetric random walks. An appropriate objective is to align all the particles on a reference line, perpendicular to all N paths, using a single controller. To achieve this objective, we define a state variable |s_{i, t}| as the absolute perpendicular distance of the i^{th} particle from the reference line. With |s_{i, t}| as a random variable, we define |s_{i, t+Δ}| as |s_{i, t}| ± δ where δ is displacement of a particle in a Δ time step. We associate a probability p_{t} to a particle moving in the direction of the reference target by step δ at time t. We consider the accumulative behavior of the overall system at any time t as sum of |s_{i, t}| over all particles and define as S_{t}. We compute conditional expectation and variance of this random variable S_{t} and show that S_{t} is a super-martingale over a range of p_{t}. This confirms the existence of state transition probability such that S_{t} can approach a limit random variable almost surely (a.s.). With this, we formulate a broadcast finite receding horizon [1] based non-linear constrained control problem and guarantee stability and convergence of the overall system.

Next we extend this idea to 2-D random walks on a plane. In this, we consider n-directional random walk on a 2-D plane and assume that each particle can make a movement of δ in any of the n possible directions in a time period of Δ with associated finite probability. We again assume identical and independent behavior of particles. With a reasonable objective of aligning all the particles in a reference target region, we develop a control problem very similar to the 1-D case and show existence of transition probabilities that drive the overall system behavior to reach the desired target. We further extend this 2-D random walk problem by letting n to infinity. In this situation, each of the particles is free to choose any of the infinite possible directions on a 2-D plane. Now we assume a square relation of displacement to time (δ^{2} = Δ) and let Δ → 0 as a limit. We claim that the independent motion possessed by each of the particles shows the classical Brownian behavior observed by Robert Brown in the motion of pollen particles suspended in a solution but constrained on a 2-D plane. To prove our claim mathematically, we take an approach of weak convergences on probability measures [2] and reconstruct the problem in the space of C[0, 1], a space of continuous, real-valued functions on the unit interval [0, 1] with the uniform metric. On this space, we prove the convergence of the random elements to the Wiener distribution with the assumption of tightness of probability measures. With this, we show a controlled behavior of multiple independent particles possessing 2-D random walks in the limit of infinite direction.

Reference:

[1] G. Kumar and M. V. Kothare. Broadcast Model Predictive Control of Multi-Cellular System. Cast Plenary Session, AICHE Annual Meeting, 2009.

[2] Patrick Billingsley, Convergence of Probability Measures. 2^{nd} Edition, 1999.