  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.
