Solvability condition of output feedback of assuming observation noise and evaluating only the state

In the previous study, output feedback control of performing feedback control directly without using an observer etc. by the output of the system detected by the sensor is proposed. As for the output feedback proposed until now, observation noise is not included although system noise is included in the controlled object. We consider the output feedback to a system including system noise and observation noise. In this paper, as a first step, we consider the cost function of only a state, and show the conditions from which the optimal output feedback law which minimizes this cost function is obtained with a constant feedback gain by it.