We initially study the control of linear systems by state feedback and simultaneously require the feedback gain matrix to satisfy a certain constraint equation. We provide two fundamentally different methods to solve the constrained stabilization problem. The first method parametrizes the class of stabilizing gain matrices and converts the problem to an equivalent nonlinear programming problem. The second method, on the other hand, converts the problem so an equivalent stabilization problem by a static output controller where the static gain is required to take a special structure. The motivation behind introducing this problem and its solution is the application of the results to the design of observer based controllers for the singularly perturbed systems. It is shown that the transient mismatch phenomenon, which occurs in this type of design, may be alleviated if the observer gain of the fast observer is chosen based on the solution of a certain constraint stabilization problem which involves the output matrix of the slow subsystem.