The authors present in this study a numerical method which computes the optimal trajectory of a underwater vehicle subject to some mission objectives. The method is applied to a submarine whose goal is to best detect one or several targets, or/and to minimise its own detection range perceived by the other targets. The signal considered is acoustic propagation attenuation. This approach is based on dynamic programming of a finite horizon Markov decision process. A quantisation method is applied to fully discretise the problem and allows a numerically tractable solution. Different scenarios are considered. The authors suppose at first that the position and the velocity of the targets are known and in the second they suppose that they are unknown and estimated by a Kalman type filter in a context of passive tracking.