Indirect Optimal Control of Advection-Diffusion Fields through Robotic Swarms

TL;DR

This paper develops an optimal control framework for guiding large underwater robot swarms to indirectly manipulate environmental advection-diffusion fields, using mean-field models and finite element methods.

Contribution

It introduces a novel optimal control formulation for swarm-based environmental field regulation, incorporating stochastic microscopic dynamics and mean-field PDE models.

Findings

Effective control of environmental fields demonstrated in simulations

Optimal vector and scalar actuation fields successfully regulate the field

Method handles complex flow environments like double-gyre flows

Abstract

In this paper, we consider the problem of optimally guiding a large-scale swarm of underwater vehicles that is tasked with the indirect control of an advection-diffusion environmental field. The microscopic vehicle dynamics are governed by a stochastic differential equation with drift. The drift terms model the self-propelled velocity of the vehicle and the velocity field of the currents. In the mean-field setting, the macroscopic vehicle dynamics are governed by a Kolmogorov forward equation in the form of a linear parabolic advection-diffusion equation. The environmental field is governed by an advection-diffusion equation in which the advection term is defined by the fluid velocity field. The vehicles are equipped with on-board actuators that enable the swarm to act as a distributed source in the environmental field, modulated by a scalar control parameter that determines the local source intensity. In this setting, we formulate an optimal control problem to compute the vehicle velocity and actuator intensity fields that drive the environmental field to a desired distribution within a specified amount of time. In other words, we design optimal vector and scalar actuation fields to indirectly control the environmental field through a distributed source, produced by the swarm. After proving an existence result for the solution of the optimal control problem, we discretize and solve the problem using the Finite Element Method (FEM). The FEM discretization naturally provides an operator that represents the bilinear way in which the controls enter into the dynamics of the vehicle swarm and the environmental field. Finally, we show through numerical simulations the effectiveness of our control strategy in regulating the environmental field to zero or to a desired distribution in the presence of a double-gyre flow field.