  In the problem of quickest change detection, a change occurs at some unknown
time in the distribution of a sequence of random vectors that are monitored in
real time, and the goal is to detect this change as quickly as possible subject
to a certain false alarm constraint. In this work we consider this problem in
the presence of parametric uncertainty in the post-change regime and controlled
sensing. That is, the post-change distribution contains an unknown parameter,
and the distribution of each observation, before and after the change, is
affected by a control action. In this context, in addition to a stopping rule
that determines the time at which it is declared that the change has occurred,
one also needs to determine a sequential control policy, which chooses the
control action at each time based on the already collected observations. We
formulate this problem mathematically using Lorden's minimax criterion, and
assuming that there are finitely many possible actions and post-change
parameter values. We then propose a specific procedure for this problem that
employs an adaptive CuSum statistic in which (i) the estimate of the parameter
is based on a fixed number of the more recent observations, and (ii) each
action is selected to maximize the Kullback-Leibler divergence of the next
observation based on the current parameter estimate, apart from a small number
of exploration times. We show that this procedure, which we call the Windowed
Chernoff-CuSum (WCC), is first-order asymptotically optimal under Lorden's
minimax criterion, for every possible possible value of the unknown post-change
parameter, as the mean time to false alarm goes to infinity. We also provide
simulation results to illustrate the performance of the WCC procedure.
