In this paper we deal with the problem of designing an output feedback controller which guarantees, at the same time, that the closed-loop poles are in specified regions of the complex plane and that the system under control is finite time bounded. This is accomplished by means of a dynamic compensator in the controller-observer form. The design procedure is divided in two steps: first, supposing that the state is available, a state feedback controller which gives the desired closed-loop properties is designed; then a state observer which tries to retain the properties guaranteed by the state feedback controller is synthesized. All the conditions are expressed in terms of Linear Matrix Inequalities and therefore the problem can be solved by efficient numerical optimization algorithms.