Lost target search simplified Probabilistic Question

The overall question is the following:
Which actions have to perform the sensing agents to find the lost target?

Without taking into account the time, we can simplify the question to:
Which actions have to perform the sensing agents to maximize the probability of detecting the target?
Or its inverse question:
Which actions have to perform the sensing agents to minimize the probability of non-detecting the target?

From the probabilistic reasoning point of view, this question is formally defined as:

    \[ P(U|\tau,S,Z,\pi) \]

The problem as a joint probability distribution is then stated as:

    \[ P(\tau,S,Z,U|\pi) \]

The question depends on the estimation of the target location P(\tau|Z,U,S), the sensing platform states P(S|U,S=s) and the sensor model P(Z|S,\tau)

1. Assuming no uncertainty within the agents movements that is it exist a deterministic function f_s: S \times U \rightarrow S that makes agent go from state S=s by applying the action U=u to state S=s' unequivocally, we can avoid the using of P(S|U,S=s).

2. Assuming that the sensing platforms while they are searching, the measurements obtained by the sensors are always not detection Z=\overline{D}

3. Assuming that the platforms counts with a prior information of the possible location of the target P(\tau^0|\pi).

4. Assuming that the observations made by the agents are independent (i.e. only depends on the phenomenon and not on the rest of the sensors)

The final question is the following:

    \[ P(\overline{D}|\tau,S,U) = \int_\tau \prod_i P(\overline{D}|\tau,S) P(\tau^0|\pi) \]