  The problem of multi-hypothesis testing with controlled sensing of
observations is considered. The distribution of observations collected under
each control is assumed to follow a single-parameter exponential family
distribution. The goal is to design a policy to find the true hypothesis with
minimum expected delay while ensuring that the probability of error is below a
given constraint. The decision-maker can control the delay by intelligently
choosing the control for observation collection in each time slot. We derive a
policy that satisfies the given constraint on the error probability. We also
show that the policy is asymptotically optimal in the sense that it
asymptotically achieves an information-theoretic lower bound on the expected
delay.
