This paper presents a scheme for constructing an almost optimal control law for a linear time invariant plant via a bounded input bounded output stable controller. The control law is required to be almost optimal with respect to a quadratic performance index. The controller is also required to stabilize the system. The main result of the paper is a scheme for constructing such an output feedback controller. This linear controller is described by a set of state equations which is of twice the plant dimension and allows for jumps in the state at periodic intervals. It is shown that for the existence of such a controller, it is sufficient that the plant be stabilizable and detectable.