Abstract: This paper deals with the problem of designing optimal state observers and optimal state feedback controllers for linear systems based on quadratic performance index. The theory of differential game is used to study the problem by assuming ‘state feedback of observer state’ as the first player and ‘output feedback of state error’ as the second player. The existence conditions for optimal observer and feedback controller are given and the solutions can be obtained by solving two algebraic Riccati equations. It is shown that the optimal state observer does not exist in general. Furthermore, the problem of designing suboptimal state feedback controllers based on state observers is studied and sufficient conditions are given in the forms of LMIs.