The detection search problem is the identification of search paths for a specified time interval [0,T], so that the expected number of surviving targets at time T is minimized. The problem can be solved in real time only when the two major procedures: (1) estimation of target posterior distribution; and (2) evaluation of optimal controls (search path planning) based on this posterior target distribution can be done on line. The unnormalized target conditional pdf, p{ABOVE ~}(t, {SOME GREEK LETTER}|x^t_0), satisfies a linear partial differential equation, too complex to be solved on line. Hence, a dual stochastic optimal control problem (in fact, a conventional LQG problem) by using a logarithmic transformation. A special case of the dual problem is studied in detail and a recursive formula is obtained for updating the target's conditional density. The search path planning problem is treated in Part II.