  This paper describes an optimization framework to design guidance for a
possibly heterogeneous team of multiple mobile sensors to estimate a
spatiotemporal process modeled by a 2D diffusion-advection process. Owing to
the abstract linear system representation of the process, we apply the
Kalman-Bucy filter for estimation, where the sensors provide linear outputs. We
propose an optimization problem that minimizes the sum of the trace of the
covariance operator of the Kalman-Bucy filter and a generic mobility cost of
the mobile sensors, subject to the sensors' motion modeled by linear dynamics.
We establish the existence of a solution to this problem. Moreover, we prove
convergence to the exact optimal solution of the approximate optimal solution.
That is, when evaluating these two solutions using the original cost function,
the difference becomes arbitrarily small as the approximation gets finer. To
compute the approximate solution, we use Pontryagin's minimum principle after
approximating the infinite-dimensional terms originating from the
diffusion-advection process. The approximate solution is applied in simulation
to analyze how a single mobile sensor's performance depends on two important
parameters: sensor noise variance and mobility penalty. We also illustrate the
application of the framework to multiple sensors, in particular the performance
of a heterogeneous team of sensors.
