The problem of multiple hypothesis testing with observation control is
considered.  First, a setup with a fixed sample size is considered.  For the
binary hypothesis case, the optimal exponent for the maximal error probability
is fully characterized.  The characterization implies that the optimal
exponent
is achievable by a pure stationary (open-loop) control.  Upper and lower
bounds
on the optimal exponent are derived for the multiple hypothesis case.  Second,
a
sequential setup is considered wherein the main interest is in the optimal
tradeoff among expected stopping times under various hypotheses subject to the
constraint of vanishing error probabilities.  Inspired by a classic result of
Chernoff, a sequential test is proposed and its asymptotic optimality is
established.