  This paper considers the problem of designing time-dependent, real-time
control policies for controllable nonlinear diffusion processes, with the goal
of obtaining maximally-informative observations about parameters of interest.
More precisely, we maximize the expected Fisher information for the parameter
obtained over the duration of the experiment, conditional on observations made
up to that time. We propose to accomplish this with a two-step strategy: when
the full state vector of the diffusion process is observable continuously, we
formulate this as an optimal control problem and apply numerical techniques
from stochastic optimal control to solve it. When observations are incomplete,
infrequent, or noisy, we propose using standard filtering techniques to first
estimate the state of the system, then apply the optimal control policy using
the posterior expectation of the state. We assess the effectiveness of these
methods in 3 situations: a paradigmatic bistable model from statistical
physics, a model of action potential generation in neurons, and a model of a
simple ecological system.
