Output Feedback Control for Irregular LQ Problem

In this letter, we study the irregular output feedback linear quadratic (LQ) control problem where the state is measured through a linear system with noise. It is interesting to show that the irregular output feedback control with the standard average LQ cost is unsolvable even if the state feedback LQ control is solvable, which is completely different from the classical regular output feedback LQ control. In order to guarantee the solvability of the irregular output feedback control, the LQ cost is modified at the terminal time with the expectation of terminal state in this letter. In the framework of the modified cost function, it is shown that the separation principle holds and the explicitly optimal controller is given in the feedback form of the Kalman filtering. In particular, the feedback gain is calculated by two Riccati equations, independently of the Kalman filtering. The key technique is the analytical solution of the forward and backward differential equations (FBDEs). We also emphasize that the optimal controller at the terminal time is required to be deterministic.