The problem of constructing an "observer" to enable us to implement an approximate optimal control for a distributed parameter system is examined where the state is measured at a few pre-specified points. The observer is formulated as the output of a dynamical system described by a set of ordinary differential equations. Both distributed and boundary control problems are studied and the observer-formulation is set up for both cases. Some reasonable assumptions have been made in order that the approximation introduced by the eigenfunction expansion technique be satisfactory. For the case of the boundary control problem, a simple example is solved to illustrate the method.