We consider the problem of dynamic output feedback stabilization at an unobservable target point. The challenge lies in according the antagonistic nature of the objective and the properties of the system: the system tends to be less observable as it approaches the target. In the literature, switching techniques rapidly appeared as a suitable approach to deal with this issue. On a case of systems with linear conservative dynamics and nonlinear output, this approach is used in conjunction with an embedding into bilinear systems that admit observers with dissipative error. Combining these two elements, global stabilization by means of a dynamic periodic time-varying output feedback is proved, and numerical simulations are provided.