We present in this paper a numerical method which computes the trajectory of a vehicle subject to some mission objectives. The method is applied to a submarine whose goal is to best detect one or several targets (we consider signal attenuation due to acoustic propagation) or/and to minimize its own detection range perceived by the other targets. Our approach is based on dynamic programming of a finite horizon Markov decision process. The position and the velocity of the targets are supposed to be known only up to a random estimation error, as a Kalman type filter is used to estimate these quantities from the measurements given by the on board sonar. A quantization method is applied to fully discretize the problem and solve it numerically.