in a C 2 domain R c R n+1 and which is reflected from the boundary an . The controller situating outside of the domain, can observe the position of the object only when it hits the boundary a~ , and he can influence its motion by prescribing the angle of the (non-normal) reflexion at each point of the boundary. The controller's objective is to minimize the expected cost (composed of a terminal and an integral part) arising until the first hitting of a target set, where the process is killed. We assume, that the target set is an attainable subset of the boundary, this means that the process terminates almost surely in finite time and that its death is observable for the controller.