In , a Bayesian two-threshold algorithm was obtained for quickest detection of a change in the distribution of a sequence of random variables, subject to constraints of probability of false alarm and observation cost. This algorithm was shown to be asymptotically optimal and to have good trade-off curves. In this paper, the results in  are extended to the more practically relevant minimax setting. Motivated by the structure of the algorithm developed in , a CUSUM based algorithm, called DE-CUSUM is proposed, which can be used for on-off observation control and to detect change as quickly as possible subject to a false alarm constraint. It is shown that the DE-CUSUM algorithm inherits the good qualities of the algorithm in , i.e., it is also asymptotically optimal and has good trade-off curves. Numerical results show that the DE-CUSUM algorithm provides a substantial savings in the observation cost over the naive approach of fractional sampling.